id	sid	tid	token	lemma	pos
ejpam-5661	1	1	european	european	PROPN
ejpam-5661	1	2	journal	journal	PROPN
ejpam-5661	1	3	of	of	ADP
ejpam-5661	1	4	pure	pure	ADJ
ejpam-5661	1	5	and	and	CCONJ
ejpam-5661	1	6	applied	applied	ADJ
ejpam-5661	1	7	mathematics	mathematic	NOUN
ejpam-5661	1	8	2025	2025	NUM
ejpam-5661	1	9	,	,	PUNCT
ejpam-5661	1	10	vol	vol	NOUN
ejpam-5661	1	11	.	.	PROPN
ejpam-5661	1	12	18	18	NUM
ejpam-5661	1	13	,	,	PUNCT
ejpam-5661	1	14	issue	issue	NOUN
ejpam-5661	1	15	1	1	NUM
ejpam-5661	1	16	,	,	PUNCT
ejpam-5661	1	17	article	article	NOUN
ejpam-5661	1	18	number	number	NOUN
ejpam-5661	1	19	5661	5661	NUM
ejpam-5661	1	20	issn	issn	VERB
ejpam-5661	1	21	1307	1307	NUM
ejpam-5661	1	22	-	-	SYM
ejpam-5661	1	23	5543	5543	NUM
ejpam-5661	1	24	–	–	PUNCT
ejpam-5661	1	25	ejpam.com	ejpam.com	X
ejpam-5661	1	26	published	publish	VERB
ejpam-5661	1	27	by	by	ADP
ejpam-5661	1	28	new	new	PROPN
ejpam-5661	1	29	york	york	PROPN
ejpam-5661	1	30	business	business	PROPN
ejpam-5661	1	31	global	global	PROPN
ejpam-5661	1	32	on	on	ADP
ejpam-5661	1	33	the	the	DET
ejpam-5661	1	34	extension	extension	NOUN
ejpam-5661	1	35	of	of	ADP
ejpam-5661	1	36	q	q	ADJ
ejpam-5661	1	37	-	-	PUNCT
ejpam-5661	1	38	hermite	hermite	ADJ
ejpam-5661	1	39	-	-	PUNCT
ejpam-5661	1	40	hadamard	hadamard	ADJ
ejpam-5661	1	41	inequalities	inequality	NOUN
ejpam-5661	1	42	for	for	ADP
ejpam-5661	1	43	strong	strong	ADJ
ejpam-5661	1	44	convexity	convexity	NOUN
ejpam-5661	1	45	chanokgan	chanokgan	VERB
ejpam-5661	1	46	sahatsathatsana1,∗	sahatsathatsana1,∗	NOUN
ejpam-5661	1	47	,	,	PUNCT
ejpam-5661	1	48	pongsakorn	pongsakorn	ADJ
ejpam-5661	1	49	yotkaew1	yotkaew1	PROPN
ejpam-5661	1	50	1	1	NUM
ejpam-5661	1	51	department	department	NOUN
ejpam-5661	1	52	of	of	ADP
ejpam-5661	1	53	mathematics	mathematic	NOUN
ejpam-5661	1	54	,	,	PUNCT
ejpam-5661	1	55	faculty	faculty	NOUN
ejpam-5661	1	56	of	of	ADP
ejpam-5661	1	57	science	science	NOUN
ejpam-5661	1	58	,	,	PUNCT
ejpam-5661	1	59	khon	khon	PROPN
ejpam-5661	1	60	kaen	kaen	PROPN
ejpam-5661	1	61	university	university	PROPN
ejpam-5661	1	62	,	,	PUNCT
ejpam-5661	1	63	khon	khon	PROPN
ejpam-5661	1	64	kaen	kaen	PROPN
ejpam-5661	1	65	40002	40002	NUM
ejpam-5661	1	66	,	,	PUNCT
ejpam-5661	1	67	thailand	thailand	PROPN
ejpam-5661	1	68	abstract	abstract	PROPN
ejpam-5661	1	69	.	.	PUNCT
ejpam-5661	2	1	this	this	DET
ejpam-5661	2	2	study	study	NOUN
ejpam-5661	2	3	leverages	leverage	VERB
ejpam-5661	2	4	q	q	NOUN
ejpam-5661	2	5	-	-	PUNCT
ejpam-5661	2	6	calculus	calculus	NOUN
ejpam-5661	2	7	to	to	PART
ejpam-5661	2	8	establish	establish	VERB
ejpam-5661	2	9	q	q	ADJ
ejpam-5661	2	10	-	-	PUNCT
ejpam-5661	2	11	hermite	hermite	ADJ
ejpam-5661	2	12	-	-	PUNCT
ejpam-5661	2	13	hadamard	hadamard	ADJ
ejpam-5661	2	14	type	type	NOUN
ejpam-5661	2	15	inequalities	inequality	NOUN
ejpam-5661	2	16	for	for	ADP
ejpam-5661	2	17	strongly	strongly	ADV
ejpam-5661	2	18	convex	convex	NOUN
ejpam-5661	2	19	functions	function	NOUN
ejpam-5661	2	20	,	,	PUNCT
ejpam-5661	2	21	showcasing	showcase	VERB
ejpam-5661	2	22	possible	possible	ADJ
ejpam-5661	2	23	extensions	extension	NOUN
ejpam-5661	2	24	of	of	ADP
ejpam-5661	2	25	well	well	ADV
ejpam-5661	2	26	-	-	PUNCT
ejpam-5661	2	27	known	know	VERB
ejpam-5661	2	28	results	result	NOUN
ejpam-5661	2	29	in	in	ADP
ejpam-5661	2	30	the	the	DET
ejpam-5661	2	31	field	field	NOUN
ejpam-5661	2	32	.	.	PUNCT
ejpam-5661	3	1	additionally	additionally	ADV
ejpam-5661	3	2	,	,	PUNCT
ejpam-5661	3	3	it	it	PRON
ejpam-5661	3	4	enhances	enhance	VERB
ejpam-5661	3	5	these	these	DET
ejpam-5661	3	6	findings	finding	NOUN
ejpam-5661	3	7	by	by	ADP
ejpam-5661	3	8	exploring	explore	VERB
ejpam-5661	3	9	the	the	DET
ejpam-5661	3	10	strong	strong	ADJ
ejpam-5661	3	11	convexity	convexity	NOUN
ejpam-5661	3	12	of	of	ADP
ejpam-5661	3	13	the	the	DET
ejpam-5661	3	14	function	function	NOUN
ejpam-5661	3	15	φ	φ	NOUN
ejpam-5661	3	16	.	.	PUNCT
ejpam-5661	4	1	furthermore	furthermore	ADV
ejpam-5661	4	2	,	,	PUNCT
ejpam-5661	4	3	the	the	DET
ejpam-5661	4	4	q	q	NOUN
ejpam-5661	4	5	-	-	PUNCT
ejpam-5661	4	6	midpoint	midpoint	NOUN
ejpam-5661	4	7	and	and	CCONJ
ejpam-5661	4	8	q	q	ADJ
ejpam-5661	4	9	-	-	PUNCT
ejpam-5661	4	10	trapezoidal	trapezoidal	ADJ
ejpam-5661	4	11	inequalities	inequality	NOUN
ejpam-5661	4	12	are	be	AUX
ejpam-5661	4	13	unified	unify	VERB
ejpam-5661	4	14	within	within	ADP
ejpam-5661	4	15	a	a	DET
ejpam-5661	4	16	comprehensive	comprehensive	ADJ
ejpam-5661	4	17	framework	framework	NOUN
ejpam-5661	4	18	.	.	PUNCT
ejpam-5661	5	1	ultimately	ultimately	ADV
ejpam-5661	5	2	,	,	PUNCT
ejpam-5661	5	3	the	the	DET
ejpam-5661	5	4	results	result	NOUN
ejpam-5661	5	5	suggest	suggest	VERB
ejpam-5661	5	6	that	that	SCONJ
ejpam-5661	5	7	the	the	DET
ejpam-5661	5	8	newly	newly	ADV
ejpam-5661	5	9	derived	derive	VERB
ejpam-5661	5	10	inequalities	inequality	NOUN
ejpam-5661	5	11	can	can	AUX
ejpam-5661	5	12	be	be	AUX
ejpam-5661	5	13	effectively	effectively	ADV
ejpam-5661	5	14	applied	apply	VERB
ejpam-5661	5	15	in	in	ADP
ejpam-5661	5	16	the	the	DET
ejpam-5661	5	17	context	context	NOUN
ejpam-5661	5	18	of	of	ADP
ejpam-5661	5	19	special	special	ADJ
ejpam-5661	5	20	means	mean	NOUN
ejpam-5661	5	21	.	.	PUNCT
ejpam-5661	6	1	2020	2020	NUM
ejpam-5661	6	2	mathematics	mathematic	NOUN
ejpam-5661	6	3	subject	subject	NOUN
ejpam-5661	6	4	classifications	classification	NOUN
ejpam-5661	6	5	:	:	PUNCT
ejpam-5661	6	6	05a30	05a30	NOUN
ejpam-5661	6	7	,	,	PUNCT
ejpam-5661	6	8	26a51	26a51	NUM
ejpam-5661	6	9	,	,	PUNCT
ejpam-5661	6	10	26d10	26d10	NUM
ejpam-5661	6	11	,	,	PUNCT
ejpam-5661	6	12	26d15	26d15	NUM
ejpam-5661	6	13	,	,	PUNCT
ejpam-5661	6	14	26d25	26d25	NUM
ejpam-5661	6	15	,	,	PUNCT
ejpam-5661	6	16	52a01	52a01	VERB
ejpam-5661	6	17	key	key	ADJ
ejpam-5661	6	18	words	word	NOUN
ejpam-5661	6	19	and	and	CCONJ
ejpam-5661	6	20	phrases	phrase	NOUN
ejpam-5661	6	21	:	:	PUNCT
ejpam-5661	6	22	h	h	ADJ
ejpam-5661	6	23	-	-	PUNCT
ejpam-5661	6	24	h	h	NOUN
ejpam-5661	6	25	type	type	NOUN
ejpam-5661	6	26	inequalities	inequality	NOUN
ejpam-5661	6	27	,	,	PUNCT
ejpam-5661	6	28	strongly	strongly	ADV
ejpam-5661	6	29	convex	convex	NOUN
ejpam-5661	6	30	functions	function	NOUN
ejpam-5661	6	31	,	,	PUNCT
ejpam-5661	6	32	quantum	quantum	NOUN
ejpam-5661	6	33	calculus	calculus	NOUN
ejpam-5661	6	34	1	1	NUM
ejpam-5661	6	35	.	.	PUNCT
ejpam-5661	7	1	introduction	introduction	NOUN
ejpam-5661	7	2	the	the	DET
ejpam-5661	7	3	hermite	hermite	PROPN
ejpam-5661	7	4	-	-	PUNCT
ejpam-5661	7	5	hadamard	hadamard	ADJ
ejpam-5661	7	6	inequality	inequality	NOUN
ejpam-5661	7	7	(	(	PUNCT
ejpam-5661	7	8	h	h	NOUN
ejpam-5661	7	9	-	-	PUNCT
ejpam-5661	7	10	h	h	NOUN
ejpam-5661	7	11	inequality	inequality	NOUN
ejpam-5661	7	12	)	)	PUNCT
ejpam-5661	7	13	was	be	AUX
ejpam-5661	7	14	proposed	propose	VERB
ejpam-5661	7	15	and	and	CCONJ
ejpam-5661	7	16	explored	explore	VERB
ejpam-5661	7	17	by	by	ADP
ejpam-5661	7	18	c.	c.	PROPN
ejpam-5661	7	19	hermite	hermite	PROPN
ejpam-5661	8	1	[	[	X
ejpam-5661	8	2	13	13	NUM
ejpam-5661	8	3	]	]	PUNCT
ejpam-5661	8	4	and	and	CCONJ
ejpam-5661	8	5	j.	j.	PROPN
ejpam-5661	8	6	hadamard	hadamard	PROPN
ejpam-5661	9	1	[	[	X
ejpam-5661	9	2	12	12	NUM
ejpam-5661	9	3	]	]	PUNCT
ejpam-5661	9	4	.	.	PUNCT
ejpam-5661	10	1	this	this	DET
ejpam-5661	10	2	inequality	inequality	NOUN
ejpam-5661	10	3	offers	offer	VERB
ejpam-5661	10	4	an	an	DET
ejpam-5661	10	5	estimate	estimate	NOUN
ejpam-5661	10	6	for	for	ADP
ejpam-5661	10	7	the	the	DET
ejpam-5661	10	8	mean	mean	ADJ
ejpam-5661	10	9	value	value	NOUN
ejpam-5661	10	10	of	of	ADP
ejpam-5661	10	11	a	a	DET
ejpam-5661	10	12	convex	convex	NOUN
ejpam-5661	10	13	function	function	NOUN
ejpam-5661	10	14	and	and	CCONJ
ejpam-5661	10	15	refines	refine	VERB
ejpam-5661	10	16	the	the	DET
ejpam-5661	10	17	jensen	jensen	PROPN
ejpam-5661	10	18	inequality	inequality	NOUN
ejpam-5661	10	19	[	[	X
ejpam-5661	10	20	9	9	NUM
ejpam-5661	10	21	]	]	PUNCT
ejpam-5661	10	22	.	.	PUNCT
ejpam-5661	11	1	in	in	ADP
ejpam-5661	11	2	2018	2018	NUM
ejpam-5661	11	3	,	,	PUNCT
ejpam-5661	11	4	n.	n.	PROPN
ejpam-5661	11	5	alp	alp	PROPN
ejpam-5661	11	6	et	et	PROPN
ejpam-5661	11	7	al	al	PROPN
ejpam-5661	11	8	.	.	PUNCT
ejpam-5661	12	1	[	[	X
ejpam-5661	12	2	3	3	X
ejpam-5661	12	3	]	]	PUNCT
ejpam-5661	12	4	demonstrated	demonstrate	VERB
ejpam-5661	12	5	a	a	DET
ejpam-5661	12	6	version	version	NOUN
ejpam-5661	12	7	of	of	ADP
ejpam-5661	12	8	the	the	DET
ejpam-5661	12	9	q	q	ADJ
ejpam-5661	12	10	-	-	PUNCT
ejpam-5661	12	11	h	h	NOUN
ejpam-5661	12	12	-	-	PUNCT
ejpam-5661	12	13	h	h	NOUN
ejpam-5661	12	14	inequality	inequality	NOUN
ejpam-5661	12	15	for	for	ADP
ejpam-5661	12	16	convex	convex	NOUN
ejpam-5661	12	17	functions	function	NOUN
ejpam-5661	12	18	utilizing	utilize	VERB
ejpam-5661	12	19	left	leave	VERB
ejpam-5661	12	20	qintegrals	qintegral	NOUN
ejpam-5661	12	21	.	.	PUNCT
ejpam-5661	13	1	theorem	theorem	NOUN
ejpam-5661	13	2	1	1	NUM
ejpam-5661	13	3	(	(	PUNCT
ejpam-5661	13	4	[	[	X
ejpam-5661	13	5	3	3	NUM
ejpam-5661	13	6	]	]	PUNCT
ejpam-5661	13	7	)	)	PUNCT
ejpam-5661	13	8	.	.	PUNCT
ejpam-5661	14	1	let	let	VERB
ejpam-5661	14	2	φ	φ	NOUN
ejpam-5661	14	3	:	:	PUNCT
ejpam-5661	15	1	[	[	X
ejpam-5661	15	2	α	α	X
ejpam-5661	15	3	,	,	PUNCT
ejpam-5661	15	4	υ	υ	NOUN
ejpam-5661	15	5	]	]	X
ejpam-5661	15	6	→	→	PUNCT
ejpam-5661	15	7	r	r	NOUN
ejpam-5661	15	8	be	be	AUX
ejpam-5661	15	9	a	a	DET
ejpam-5661	15	10	convex	convex	ADJ
ejpam-5661	15	11	differentiable	differentiable	ADJ
ejpam-5661	15	12	function	function	NOUN
ejpam-5661	15	13	defined	define	VERB
ejpam-5661	15	14	on	on	ADP
ejpam-5661	15	15	[	[	X
ejpam-5661	15	16	α	α	NOUN
ejpam-5661	15	17	,	,	PUNCT
ejpam-5661	15	18	υ	υ	NOUN
ejpam-5661	15	19	]	]	X
ejpam-5661	15	20	with	with	ADP
ejpam-5661	15	21	0	0	NUM
ejpam-5661	15	22	<	<	X
ejpam-5661	15	23	q	q	X
ejpam-5661	15	24	<	<	X
ejpam-5661	15	25	1	1	NUM
ejpam-5661	15	26	.	.	PUNCT
ejpam-5661	16	1	the	the	DET
ejpam-5661	16	2	following	follow	VERB
ejpam-5661	16	3	inequalities	inequality	NOUN
ejpam-5661	16	4	then	then	ADV
ejpam-5661	16	5	hold	hold	VERB
ejpam-5661	16	6	:	:	PUNCT
ejpam-5661	16	7	φ	φ	PROPN
ejpam-5661	16	8	(	(	PUNCT
ejpam-5661	16	9	qα+υ	qα+υ	PROPN
ejpam-5661	17	1	[	[	X
ejpam-5661	17	2	2]q	2]q	NUM
ejpam-5661	17	3	)	)	PUNCT
ejpam-5661	17	4	≤	≤	NOUN
ejpam-5661	17	5	1	1	NUM
ejpam-5661	17	6	υ−	υ−	PROPN
ejpam-5661	17	7	α	α	PRON
ejpam-5661	17	8	∫	∫	PROPN
ejpam-5661	17	9	υ	υ	PROPN
ejpam-5661	17	10	α	α	PROPN
ejpam-5661	17	11	φ(ω	φ(ω	PROPN
ejpam-5661	17	12	)	)	PUNCT
ejpam-5661	17	13	αdqω	αdqω	VERB
ejpam-5661	17	14	≤	≤	NOUN
ejpam-5661	17	15	qφ(α	qφ(α	PUNCT
ejpam-5661	17	16	)	)	PUNCT
ejpam-5661	18	1	+	+	CCONJ
ejpam-5661	18	2	φ(υ	φ(υ	NUM
ejpam-5661	18	3	)	)	PUNCT
ejpam-5661	19	1	[	[	X
ejpam-5661	19	2	2]q	2]q	NUM
ejpam-5661	19	3	,	,	PUNCT
ejpam-5661	19	4	(	(	PUNCT
ejpam-5661	19	5	1	1	X
ejpam-5661	19	6	)	)	PUNCT
ejpam-5661	19	7	where	where	SCONJ
ejpam-5661	19	8	[	[	X
ejpam-5661	19	9	2]q	2]q	NUM
ejpam-5661	19	10	=	=	SYM
ejpam-5661	19	11	1	1	NUM
ejpam-5661	19	12	+	+	CCONJ
ejpam-5661	19	13	q.	q.	NOUN
ejpam-5661	19	14	in	in	ADP
ejpam-5661	19	15	2020	2020	NUM
ejpam-5661	19	16	,	,	PUNCT
ejpam-5661	19	17	s.	s.	PROPN
ejpam-5661	19	18	bermudo	bermudo	PROPN
ejpam-5661	19	19	et	et	PROPN
ejpam-5661	19	20	al	al	PROPN
ejpam-5661	19	21	.	.	PUNCT
ejpam-5661	20	1	[	[	X
ejpam-5661	20	2	5	5	NUM
ejpam-5661	20	3	]	]	PUNCT
ejpam-5661	20	4	established	establish	VERB
ejpam-5661	20	5	the	the	DET
ejpam-5661	20	6	following	follow	VERB
ejpam-5661	20	7	q	q	ADJ
ejpam-5661	20	8	-	-	PUNCT
ejpam-5661	20	9	h	h	NOUN
ejpam-5661	20	10	-	-	PUNCT
ejpam-5661	20	11	h	h	NOUN
ejpam-5661	20	12	inequality	inequality	NOUN
ejpam-5661	20	13	applicable	applicable	ADJ
ejpam-5661	20	14	to	to	ADP
ejpam-5661	20	15	convex	convex	VERB
ejpam-5661	20	16	functions	function	NOUN
ejpam-5661	20	17	.	.	PUNCT
ejpam-5661	21	1	∗corresponding	∗corresponde	VERB
ejpam-5661	21	2	author	author	NOUN
ejpam-5661	21	3	.	.	PUNCT
ejpam-5661	22	1	doi	doi	NOUN
ejpam-5661	22	2	:	:	PUNCT
ejpam-5661	22	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5661	https://doi.org/10.29020/nybg.ejpam.v18i1.5661	PRON
ejpam-5661	22	4	email	email	NOUN
ejpam-5661	22	5	addresses	address	VERB
ejpam-5661	22	6	:	:	PUNCT
ejpam-5661	22	7	chanokgan.na@ksu.ac.th	chanokgan.na@ksu.ac.th	NUM
ejpam-5661	22	8	(	(	PUNCT
ejpam-5661	22	9	c.	c.	PROPN
ejpam-5661	22	10	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	22	11	)	)	PUNCT
ejpam-5661	22	12	,	,	PUNCT
ejpam-5661	22	13	pongyo@kku.ac.th	pongyo@kku.ac.th	PROPN
ejpam-5661	22	14	(	(	PUNCT
ejpam-5661	22	15	p.	p.	PROPN
ejpam-5661	22	16	yotkaew	yotkaew	PROPN
ejpam-5661	22	17	)	)	PUNCT
ejpam-5661	22	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5661	23	1	1	1	NUM
ejpam-5661	23	2	copyright	copyright	NOUN
ejpam-5661	23	3	:	:	PUNCT
ejpam-5661	23	4	©	©	PROPN
ejpam-5661	23	5	2025	2025	NUM
ejpam-5661	23	6	the	the	DET
ejpam-5661	23	7	author(s	author(s	NOUN
ejpam-5661	23	8	)	)	PUNCT
ejpam-5661	23	9	.	.	PUNCT
ejpam-5661	24	1	(	(	PUNCT
ejpam-5661	24	2	cc	cc	NOUN
ejpam-5661	24	3	by	by	ADP
ejpam-5661	24	4	-	-	PUNCT
ejpam-5661	24	5	nc	nc	PROPN
ejpam-5661	24	6	4.0	4.0	NUM
ejpam-5661	24	7	)	)	PUNCT
ejpam-5661	24	8	c.	c.	PROPN
ejpam-5661	24	9	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	24	10	,	,	PUNCT
ejpam-5661	24	11	p.	p.	PROPN
ejpam-5661	24	12	yotkaew	yotkaew	PROPN
ejpam-5661	24	13	/	/	SYM
ejpam-5661	24	14	eur	eur	PROPN
ejpam-5661	24	15	.	.	PUNCT
ejpam-5661	25	1	j.	j.	PROPN
ejpam-5661	25	2	pure	pure	PROPN
ejpam-5661	25	3	appl	appl	PROPN
ejpam-5661	25	4	.	.	PROPN
ejpam-5661	25	5	math	math	PROPN
ejpam-5661	25	6	,	,	PUNCT
ejpam-5661	25	7	18	18	NUM
ejpam-5661	25	8	(	(	PUNCT
ejpam-5661	25	9	1	1	NUM
ejpam-5661	25	10	)	)	PUNCT
ejpam-5661	25	11	(	(	PUNCT
ejpam-5661	25	12	2025	2025	NUM
ejpam-5661	25	13	)	)	PUNCT
ejpam-5661	25	14	,	,	PUNCT
ejpam-5661	25	15	5661	5661	NUM
ejpam-5661	25	16	2	2	NUM
ejpam-5661	25	17	of	of	ADP
ejpam-5661	25	18	24	24	NUM
ejpam-5661	25	19	theorem	theorem	VERB
ejpam-5661	25	20	2	2	NUM
ejpam-5661	25	21	(	(	PUNCT
ejpam-5661	25	22	[	[	X
ejpam-5661	25	23	5	5	NUM
ejpam-5661	25	24	]	]	PUNCT
ejpam-5661	25	25	)	)	PUNCT
ejpam-5661	25	26	.	.	PUNCT
ejpam-5661	26	1	let	let	VERB
ejpam-5661	26	2	φ	φ	NOUN
ejpam-5661	26	3	:	:	PUNCT
ejpam-5661	27	1	[	[	X
ejpam-5661	27	2	α	α	X
ejpam-5661	27	3	,	,	PUNCT
ejpam-5661	27	4	υ	υ	NOUN
ejpam-5661	27	5	]	]	X
ejpam-5661	27	6	→	→	PUNCT
ejpam-5661	27	7	r	r	NOUN
ejpam-5661	27	8	be	be	AUX
ejpam-5661	27	9	a	a	DET
ejpam-5661	27	10	convex	convex	ADJ
ejpam-5661	27	11	differentiable	differentiable	ADJ
ejpam-5661	27	12	function	function	NOUN
ejpam-5661	27	13	defined	define	VERB
ejpam-5661	27	14	on	on	ADP
ejpam-5661	27	15	[	[	X
ejpam-5661	27	16	α	α	NOUN
ejpam-5661	27	17	,	,	PUNCT
ejpam-5661	27	18	υ	υ	NOUN
ejpam-5661	27	19	]	]	X
ejpam-5661	27	20	with	with	ADP
ejpam-5661	27	21	0	0	NUM
ejpam-5661	27	22	<	<	X
ejpam-5661	27	23	q	q	X
ejpam-5661	27	24	<	<	X
ejpam-5661	27	25	1	1	NUM
ejpam-5661	27	26	.	.	PUNCT
ejpam-5661	28	1	the	the	DET
ejpam-5661	28	2	following	follow	VERB
ejpam-5661	28	3	inequalities	inequality	NOUN
ejpam-5661	28	4	are	be	AUX
ejpam-5661	28	5	established	establish	VERB
ejpam-5661	28	6	:	:	PUNCT
ejpam-5661	28	7	φ	φ	PROPN
ejpam-5661	28	8	(	(	PUNCT
ejpam-5661	28	9	α+	α+	PROPN
ejpam-5661	28	10	qυ	qυ	X
ejpam-5661	28	11	[	[	X
ejpam-5661	28	12	2]q	2]q	NUM
ejpam-5661	28	13	)	)	PUNCT
ejpam-5661	28	14	≤	≤	NOUN
ejpam-5661	28	15	1	1	NUM
ejpam-5661	28	16	υ−	υ−	PROPN
ejpam-5661	28	17	α	α	PRON
ejpam-5661	28	18	∫	∫	PROPN
ejpam-5661	28	19	υ	υ	PROPN
ejpam-5661	28	20	α	α	PROPN
ejpam-5661	28	21	φ(ω	φ(ω	PROPN
ejpam-5661	28	22	)	)	PUNCT
ejpam-5661	28	23	υdqω	υdqω	VERB
ejpam-5661	28	24	≤	≤	NOUN
ejpam-5661	28	25	φ(α	φ(α	PROPN
ejpam-5661	28	26	)	)	PUNCT
ejpam-5661	29	1	+	+	CCONJ
ejpam-5661	29	2	qφ(υ	qφ(υ	X
ejpam-5661	29	3	)	)	PUNCT
ejpam-5661	30	1	[	[	X
ejpam-5661	30	2	2]q	2]q	NUM
ejpam-5661	30	3	.	.	PUNCT
ejpam-5661	31	1	(	(	PUNCT
ejpam-5661	31	2	2	2	NUM
ejpam-5661	31	3	)	)	PUNCT
ejpam-5661	31	4	by	by	ADP
ejpam-5661	31	5	combining	combine	VERB
ejpam-5661	31	6	the	the	DET
ejpam-5661	31	7	inequalities	inequality	NOUN
ejpam-5661	31	8	(	(	PUNCT
ejpam-5661	31	9	1	1	NUM
ejpam-5661	31	10	)	)	PUNCT
ejpam-5661	31	11	and	and	CCONJ
ejpam-5661	31	12	(	(	PUNCT
ejpam-5661	31	13	2	2	NUM
ejpam-5661	31	14	)	)	PUNCT
ejpam-5661	31	15	,	,	PUNCT
ejpam-5661	31	16	the	the	DET
ejpam-5661	31	17	result	result	NOUN
ejpam-5661	31	18	is	be	AUX
ejpam-5661	31	19	presented	present	VERB
ejpam-5661	31	20	in	in	ADP
ejpam-5661	31	21	[	[	X
ejpam-5661	31	22	5	5	NUM
ejpam-5661	31	23	]	]	PUNCT
ejpam-5661	31	24	as	as	SCONJ
ejpam-5661	31	25	follows	follow	VERB
ejpam-5661	31	26	:	:	PUNCT
ejpam-5661	31	27	φ	φ	PROPN
ejpam-5661	31	28	(	(	PUNCT
ejpam-5661	31	29	α+υ	α+υ	NUM
ejpam-5661	31	30	2	2	NUM
ejpam-5661	31	31	)	)	PUNCT
ejpam-5661	31	32	≤	≤	NUM
ejpam-5661	31	33	1	1	NUM
ejpam-5661	31	34	2(υ−	2(υ−	NUM
ejpam-5661	31	35	α	α	NOUN
ejpam-5661	31	36	)	)	PUNCT
ejpam-5661	31	37	(	(	PUNCT
ejpam-5661	31	38	∫	∫	PROPN
ejpam-5661	31	39	υ	υ	PROPN
ejpam-5661	31	40	α	α	PROPN
ejpam-5661	31	41	φ(ω	φ(ω	PROPN
ejpam-5661	31	42	)	)	PUNCT
ejpam-5661	31	43	αdqω	αdqω	VERB
ejpam-5661	31	44	+	+	CCONJ
ejpam-5661	31	45	∫	∫	PROPN
ejpam-5661	31	46	υ	υ	PROPN
ejpam-5661	31	47	α	α	PROPN
ejpam-5661	31	48	φ(ω	φ(ω	PROPN
ejpam-5661	31	49	)	)	PUNCT
ejpam-5661	31	50	υdqω	υdqω	PROPN
ejpam-5661	31	51	)	)	PUNCT
ejpam-5661	31	52	≤	≤	NUM
ejpam-5661	31	53	φ(α	φ(α	PROPN
ejpam-5661	31	54	)	)	PUNCT
ejpam-5661	32	1	+	+	CCONJ
ejpam-5661	32	2	φ(υ	φ(υ	PROPN
ejpam-5661	32	3	)	)	PUNCT
ejpam-5661	32	4	2	2	NUM
ejpam-5661	32	5	.	.	PUNCT
ejpam-5661	33	1	(	(	PUNCT
ejpam-5661	33	2	3	3	X
ejpam-5661	33	3	)	)	PUNCT
ejpam-5661	33	4	in	in	ADP
ejpam-5661	33	5	2023	2023	NUM
ejpam-5661	33	6	,	,	PUNCT
ejpam-5661	33	7	m.a	m.a	PROPN
ejpam-5661	33	8	.	.	PROPN
ejpam-5661	33	9	ali	ali	PROPN
ejpam-5661	33	10	et	et	PROPN
ejpam-5661	33	11	al	al	PROPN
ejpam-5661	33	12	.	.	PUNCT
ejpam-5661	34	1	[	[	X
ejpam-5661	34	2	1	1	X
ejpam-5661	34	3	]	]	PUNCT
ejpam-5661	34	4	and	and	CCONJ
ejpam-5661	34	5	t.	t.	PROPN
ejpam-5661	34	6	sitthiwirattham	sitthiwirattham	PROPN
ejpam-5661	34	7	et	et	PROPN
ejpam-5661	34	8	al	al	PROPN
ejpam-5661	34	9	.	.	PUNCT
ejpam-5661	35	1	[	[	X
ejpam-5661	35	2	23	23	NUM
ejpam-5661	35	3	]	]	PUNCT
ejpam-5661	35	4	formulated	formulate	VERB
ejpam-5661	35	5	the	the	DET
ejpam-5661	35	6	following	follow	VERB
ejpam-5661	35	7	q	q	ADJ
ejpam-5661	35	8	-	-	PUNCT
ejpam-5661	35	9	h	h	NOUN
ejpam-5661	35	10	-	-	PUNCT
ejpam-5661	35	11	h	h	NOUN
ejpam-5661	35	12	-	-	PUNCT
ejpam-5661	35	13	type	type	NOUN
ejpam-5661	35	14	inequalities	inequality	NOUN
ejpam-5661	35	15	.	.	PUNCT
ejpam-5661	36	1	theorem	theorem	VERB
ejpam-5661	36	2	3	3	NUM
ejpam-5661	36	3	(	(	PUNCT
ejpam-5661	36	4	[	[	X
ejpam-5661	36	5	1	1	NUM
ejpam-5661	36	6	,	,	PUNCT
ejpam-5661	36	7	23	23	NUM
ejpam-5661	36	8	]	]	PUNCT
ejpam-5661	36	9	)	)	PUNCT
ejpam-5661	36	10	.	.	PUNCT
ejpam-5661	37	1	let	let	VERB
ejpam-5661	37	2	φ	φ	NOUN
ejpam-5661	37	3	:	:	PUNCT
ejpam-5661	38	1	[	[	X
ejpam-5661	38	2	α	α	X
ejpam-5661	38	3	,	,	PUNCT
ejpam-5661	38	4	υ	υ	NOUN
ejpam-5661	38	5	]	]	X
ejpam-5661	38	6	→	→	PUNCT
ejpam-5661	38	7	r	r	NOUN
ejpam-5661	38	8	be	be	AUX
ejpam-5661	38	9	a	a	DET
ejpam-5661	38	10	convex	convex	NOUN
ejpam-5661	38	11	function	function	NOUN
ejpam-5661	38	12	.	.	PUNCT
ejpam-5661	39	1	the	the	DET
ejpam-5661	39	2	following	follow	VERB
ejpam-5661	39	3	inequalities	inequality	NOUN
ejpam-5661	39	4	are	be	AUX
ejpam-5661	39	5	valid	valid	ADJ
ejpam-5661	39	6	:	:	PUNCT
ejpam-5661	39	7	φ	φ	PROPN
ejpam-5661	39	8	(	(	PUNCT
ejpam-5661	39	9	α+υ	α+υ	NUM
ejpam-5661	39	10	2	2	NUM
ejpam-5661	39	11	)	)	PUNCT
ejpam-5661	39	12	≤	≤	NOUN
ejpam-5661	39	13	1	1	NUM
ejpam-5661	39	14	υ−	υ−	PROPN
ejpam-5661	39	15	α	α	PROPN
ejpam-5661	39	16	(	(	PUNCT
ejpam-5661	39	17	∫	∫	PROPN
ejpam-5661	39	18	(	(	PUNCT
ejpam-5661	39	19	α+υ	α+υ	NUM
ejpam-5661	39	20	)	)	PUNCT
ejpam-5661	39	21	2	2	NUM
ejpam-5661	39	22	α	α	NOUN
ejpam-5661	39	23	φ(ω	φ(ω	PROPN
ejpam-5661	39	24	)	)	PUNCT
ejpam-5661	39	25	(	(	PUNCT
ejpam-5661	39	26	α+υ	α+υ	NUM
ejpam-5661	39	27	)	)	PUNCT
ejpam-5661	39	28	2	2	NUM
ejpam-5661	39	29	dqω	dqω	NOUN
ejpam-5661	39	30	+	+	CCONJ
ejpam-5661	40	1	∫	∫	PROPN
ejpam-5661	40	2	υ	υ	X
ejpam-5661	40	3	(	(	PUNCT
ejpam-5661	40	4	α+υ	α+υ	NUM
ejpam-5661	40	5	)	)	PUNCT
ejpam-5661	40	6	2	2	NUM
ejpam-5661	40	7	φ(ω	φ(ω	NOUN
ejpam-5661	40	8	)	)	PUNCT
ejpam-5661	40	9	(	(	PUNCT
ejpam-5661	40	10	α+υ	α+υ	NUM
ejpam-5661	40	11	)	)	PUNCT
ejpam-5661	40	12	2	2	NUM
ejpam-5661	40	13	dqω	dqω	NOUN
ejpam-5661	40	14	)	)	PUNCT
ejpam-5661	40	15	≤	≤	NOUN
ejpam-5661	40	16	φ(α	φ(α	NOUN
ejpam-5661	40	17	)	)	PUNCT
ejpam-5661	41	1	+	+	CCONJ
ejpam-5661	41	2	φ(υ	φ(υ	PROPN
ejpam-5661	41	3	)	)	PUNCT
ejpam-5661	41	4	2	2	NUM
ejpam-5661	41	5	(	(	PUNCT
ejpam-5661	41	6	4	4	NUM
ejpam-5661	41	7	)	)	PUNCT
ejpam-5661	41	8	and	and	CCONJ
ejpam-5661	41	9	φ	φ	NUM
ejpam-5661	41	10	(	(	PUNCT
ejpam-5661	41	11	α+υ	α+υ	NUM
ejpam-5661	41	12	2	2	NUM
ejpam-5661	41	13	)	)	PUNCT
ejpam-5661	41	14	≤	≤	NOUN
ejpam-5661	41	15	1	1	NUM
ejpam-5661	41	16	υ−	υ−	PROPN
ejpam-5661	41	17	α	α	PROPN
ejpam-5661	41	18	(	(	PUNCT
ejpam-5661	41	19	∫	∫	PROPN
ejpam-5661	41	20	(	(	PUNCT
ejpam-5661	41	21	α+υ	α+υ	NUM
ejpam-5661	41	22	)	)	PUNCT
ejpam-5661	41	23	2	2	NUM
ejpam-5661	41	24	α	α	PROPN
ejpam-5661	41	25	φ(ω	φ(ω	PROPN
ejpam-5661	41	26	)	)	PUNCT
ejpam-5661	41	27	αdqω	αdqω	VERB
ejpam-5661	42	1	+	+	CCONJ
ejpam-5661	42	2	∫	∫	PROPN
ejpam-5661	42	3	υ	υ	X
ejpam-5661	42	4	(	(	PUNCT
ejpam-5661	42	5	α+υ	α+υ	NUM
ejpam-5661	42	6	)	)	PUNCT
ejpam-5661	42	7	2	2	NUM
ejpam-5661	42	8	φ(ω	φ(ω	NOUN
ejpam-5661	42	9	)	)	PUNCT
ejpam-5661	42	10	υdqω	υdqω	PROPN
ejpam-5661	42	11	)	)	PUNCT
ejpam-5661	42	12	≤	≤	NUM
ejpam-5661	42	13	φ(α	φ(α	PROPN
ejpam-5661	42	14	)	)	PUNCT
ejpam-5661	43	1	+	+	CCONJ
ejpam-5661	43	2	φ(υ	φ(υ	PROPN
ejpam-5661	43	3	)	)	PUNCT
ejpam-5661	43	4	2	2	NUM
ejpam-5661	43	5	.	.	PUNCT
ejpam-5661	44	1	(	(	PUNCT
ejpam-5661	44	2	5	5	X
ejpam-5661	44	3	)	)	PUNCT
ejpam-5661	44	4	the	the	DET
ejpam-5661	44	5	h	h	NOUN
ejpam-5661	44	6	-	-	PUNCT
ejpam-5661	44	7	h	h	NOUN
ejpam-5661	44	8	inequality	inequality	NOUN
ejpam-5661	44	9	has	have	AUX
ejpam-5661	44	10	garnered	garner	VERB
ejpam-5661	44	11	significant	significant	ADJ
ejpam-5661	44	12	attention	attention	NOUN
ejpam-5661	44	13	in	in	ADP
ejpam-5661	44	14	the	the	DET
ejpam-5661	44	15	literature	literature	NOUN
ejpam-5661	44	16	,	,	PUNCT
ejpam-5661	44	17	particularly	particularly	ADV
ejpam-5661	44	18	concerning	concern	VERB
ejpam-5661	44	19	various	various	ADJ
ejpam-5661	44	20	notions	notion	NOUN
ejpam-5661	44	21	of	of	ADP
ejpam-5661	44	22	convexity	convexity	NOUN
ejpam-5661	44	23	and	and	CCONJ
ejpam-5661	44	24	extensions	extension	NOUN
ejpam-5661	44	25	and	and	CCONJ
ejpam-5661	44	26	refinements	refinement	NOUN
ejpam-5661	44	27	.	.	PUNCT
ejpam-5661	45	1	for	for	ADP
ejpam-5661	45	2	further	further	ADJ
ejpam-5661	45	3	details	detail	NOUN
ejpam-5661	45	4	,	,	PUNCT
ejpam-5661	45	5	interested	interested	ADJ
ejpam-5661	45	6	readers	reader	NOUN
ejpam-5661	45	7	may	may	AUX
ejpam-5661	45	8	refer	refer	VERB
ejpam-5661	45	9	to	to	ADP
ejpam-5661	45	10	[	[	X
ejpam-5661	45	11	6–8	6–8	X
ejpam-5661	45	12	]	]	X
ejpam-5661	45	13	and	and	CCONJ
ejpam-5661	45	14	the	the	DET
ejpam-5661	45	15	references	reference	NOUN
ejpam-5661	45	16	therein	therein	ADV
ejpam-5661	45	17	.	.	PUNCT
ejpam-5661	46	1	in	in	ADP
ejpam-5661	46	2	1966	1966	NUM
ejpam-5661	46	3	,	,	PUNCT
ejpam-5661	46	4	b.t	b.t	PROPN
ejpam-5661	46	5	.	.	PROPN
ejpam-5661	46	6	polyak	polyak	NOUN
ejpam-5661	46	7	[	[	X
ejpam-5661	46	8	19	19	NUM
ejpam-5661	46	9	]	]	PUNCT
ejpam-5661	46	10	introduced	introduce	VERB
ejpam-5661	46	11	the	the	DET
ejpam-5661	46	12	principle	principle	NOUN
ejpam-5661	46	13	of	of	ADP
ejpam-5661	46	14	strongly	strongly	ADV
ejpam-5661	46	15	convex	convex	ADJ
ejpam-5661	46	16	functions	function	NOUN
ejpam-5661	46	17	.	.	PUNCT
ejpam-5661	47	1	this	this	DET
ejpam-5661	47	2	idea	idea	NOUN
ejpam-5661	47	3	is	be	AUX
ejpam-5661	47	4	significant	significant	ADJ
ejpam-5661	47	5	in	in	ADP
ejpam-5661	47	6	mathematical	mathematical	ADJ
ejpam-5661	47	7	programming	programming	NOUN
ejpam-5661	47	8	and	and	CCONJ
ejpam-5661	47	9	the	the	DET
ejpam-5661	47	10	application	application	NOUN
ejpam-5661	47	11	of	of	ADP
ejpam-5661	47	12	mathematical	mathematical	ADJ
ejpam-5661	47	13	models	model	NOUN
ejpam-5661	47	14	.	.	PUNCT
ejpam-5661	48	1	it	it	PRON
ejpam-5661	48	2	has	have	AUX
ejpam-5661	48	3	been	be	AUX
ejpam-5661	48	4	widely	widely	ADV
ejpam-5661	48	5	utilized	utilize	VERB
ejpam-5661	48	6	in	in	ADP
ejpam-5661	48	7	various	various	ADJ
ejpam-5661	48	8	studies	study	NOUN
ejpam-5661	48	9	,	,	PUNCT
ejpam-5661	48	10	such	such	ADJ
ejpam-5661	48	11	as	as	ADP
ejpam-5661	48	12	those	those	PRON
ejpam-5661	48	13	mentioned	mention	VERB
ejpam-5661	48	14	in	in	ADP
ejpam-5661	48	15	[	[	X
ejpam-5661	48	16	4	4	NUM
ejpam-5661	48	17	,	,	PUNCT
ejpam-5661	48	18	17	17	NUM
ejpam-5661	48	19	,	,	PUNCT
ejpam-5661	48	20	18	18	NUM
ejpam-5661	48	21	,	,	PUNCT
ejpam-5661	48	22	21	21	NUM
ejpam-5661	48	23	,	,	PUNCT
ejpam-5661	48	24	22	22	NUM
ejpam-5661	48	25	]	]	PUNCT
ejpam-5661	48	26	.	.	PUNCT
ejpam-5661	49	1	definition	definition	NOUN
ejpam-5661	49	2	1	1	NUM
ejpam-5661	49	3	(	(	PUNCT
ejpam-5661	49	4	[	[	X
ejpam-5661	49	5	19	19	NUM
ejpam-5661	49	6	]	]	NUM
ejpam-5661	49	7	)	)	PUNCT
ejpam-5661	49	8	.	.	PUNCT
ejpam-5661	50	1	let	let	VERB
ejpam-5661	51	1	φ	φ	NOUN
ejpam-5661	51	2	:	:	PUNCT
ejpam-5661	52	1	i	i	PRON
ejpam-5661	52	2	→	→	PUNCT
ejpam-5661	52	3	r	r	NOUN
ejpam-5661	52	4	be	be	AUX
ejpam-5661	52	5	a	a	DET
ejpam-5661	52	6	function	function	NOUN
ejpam-5661	52	7	defined	define	VERB
ejpam-5661	52	8	on	on	ADP
ejpam-5661	52	9	an	an	DET
ejpam-5661	52	10	interval	interval	NOUN
ejpam-5661	52	11	i.	i.	NOUN
ejpam-5661	52	12	we	we	PRON
ejpam-5661	52	13	say	say	VERB
ejpam-5661	52	14	that	that	SCONJ
ejpam-5661	52	15	φ	φ	PROPN
ejpam-5661	52	16	is	be	AUX
ejpam-5661	52	17	a	a	DET
ejpam-5661	52	18	strongly	strongly	ADV
ejpam-5661	52	19	convex	convex	ADJ
ejpam-5661	52	20	function	function	NOUN
ejpam-5661	52	21	with	with	ADP
ejpam-5661	52	22	modulus	modulus	PROPN
ejpam-5661	52	23	φ	φ	PROPN
ejpam-5661	52	24	>	>	X
ejpam-5661	52	25	0	0	PUNCT
ejpam-5661	53	1	if	if	SCONJ
ejpam-5661	53	2	φ(ϖω	φ(ϖω	ADP
ejpam-5661	53	3	+	+	CCONJ
ejpam-5661	53	4	(	(	PUNCT
ejpam-5661	53	5	1−ϖ)ζ	1−ϖ)ζ	NUM
ejpam-5661	53	6	)	)	PUNCT
ejpam-5661	53	7	≤	≤	NOUN
ejpam-5661	53	8	ϖφ(ω	ϖφ(ω	PUNCT
ejpam-5661	53	9	)	)	PUNCT
ejpam-5661	54	1	+	+	CCONJ
ejpam-5661	54	2	(	(	PUNCT
ejpam-5661	54	3	1−ϖ)φ(ζ)−	1−ϖ)φ(ζ)−	PROPN
ejpam-5661	54	4	φϖ(1−ϖ)(ω	φϖ(1−ϖ)(ω	NOUN
ejpam-5661	54	5	−	−	NOUN
ejpam-5661	54	6	ζ)2	ζ)2	NOUN
ejpam-5661	54	7	(	(	PUNCT
ejpam-5661	54	8	6	6	NUM
ejpam-5661	54	9	)	)	PUNCT
ejpam-5661	54	10	for	for	ADP
ejpam-5661	54	11	all	all	DET
ejpam-5661	54	12	ω	ω	NOUN
ejpam-5661	54	13	,	,	PUNCT
ejpam-5661	54	14	ζ	ζ	PROPN
ejpam-5661	54	15	∈	∈	NOUN
ejpam-5661	54	16	i	i	PRON
ejpam-5661	54	17	and	and	CCONJ
ejpam-5661	54	18	ϖ	ϖ	PRON
ejpam-5661	54	19	∈	∈	PROPN
ejpam-5661	55	1	[	[	X
ejpam-5661	55	2	0	0	NUM
ejpam-5661	55	3	,	,	PUNCT
ejpam-5661	55	4	1	1	NUM
ejpam-5661	55	5	]	]	PUNCT
ejpam-5661	55	6	.	.	PUNCT
ejpam-5661	56	1	calculus	calculus	NOUN
ejpam-5661	56	2	is	be	AUX
ejpam-5661	56	3	a	a	DET
ejpam-5661	56	4	vital	vital	ADJ
ejpam-5661	56	5	branch	branch	NOUN
ejpam-5661	56	6	of	of	ADP
ejpam-5661	56	7	mathematics	mathematic	NOUN
ejpam-5661	56	8	focused	focus	VERB
ejpam-5661	56	9	on	on	ADP
ejpam-5661	56	10	the	the	DET
ejpam-5661	56	11	analysis	analysis	NOUN
ejpam-5661	56	12	of	of	ADP
ejpam-5661	56	13	functions	function	NOUN
ejpam-5661	56	14	and	and	CCONJ
ejpam-5661	56	15	their	their	PRON
ejpam-5661	56	16	continuous	continuous	ADJ
ejpam-5661	56	17	changes	change	NOUN
ejpam-5661	56	18	.	.	PUNCT
ejpam-5661	57	1	in	in	ADP
ejpam-5661	57	2	the	the	DET
ejpam-5661	57	3	17th	17th	ADJ
ejpam-5661	57	4	century	century	NOUN
ejpam-5661	57	5	,	,	PUNCT
ejpam-5661	57	6	significant	significant	ADJ
ejpam-5661	57	7	advancements	advancement	NOUN
ejpam-5661	57	8	in	in	ADP
ejpam-5661	57	9	calculus	calculus	NOUN
ejpam-5661	57	10	were	be	AUX
ejpam-5661	57	11	made	make	VERB
ejpam-5661	57	12	by	by	ADP
ejpam-5661	57	13	i.	i.	PROPN
ejpam-5661	57	14	newton	newton	PROPN
ejpam-5661	57	15	and	and	CCONJ
ejpam-5661	57	16	g.w	g.w	PROPN
ejpam-5661	57	17	.	.	PROPN
ejpam-5661	57	18	leibniz	leibniz	PROPN
ejpam-5661	57	19	,	,	PUNCT
ejpam-5661	57	20	laying	lay	VERB
ejpam-5661	57	21	the	the	DET
ejpam-5661	57	22	groundwork	groundwork	NOUN
ejpam-5661	57	23	for	for	ADP
ejpam-5661	57	24	its	its	PRON
ejpam-5661	57	25	modern	modern	ADJ
ejpam-5661	57	26	development	development	NOUN
ejpam-5661	57	27	.	.	PUNCT
ejpam-5661	58	1	subsequently	subsequently	ADV
ejpam-5661	58	2	,	,	PUNCT
ejpam-5661	58	3	l.	l.	PROPN
ejpam-5661	58	4	euler	euler	PROPN
ejpam-5661	58	5	(	(	PUNCT
ejpam-5661	58	6	1707–1783	1707–1783	NOUN
ejpam-5661	58	7	)	)	PUNCT
ejpam-5661	58	8	introduced	introduce	VERB
ejpam-5661	58	9	the	the	DET
ejpam-5661	58	10	concept	concept	NOUN
ejpam-5661	58	11	of	of	ADP
ejpam-5661	58	12	quantum	quantum	NOUN
ejpam-5661	58	13	calculus	calculus	NOUN
ejpam-5661	58	14	(	(	PUNCT
ejpam-5661	58	15	q	q	NOUN
ejpam-5661	58	16	-	-	NOUN
ejpam-5661	58	17	calculus	calculus	NOUN
ejpam-5661	58	18	)	)	PUNCT
ejpam-5661	58	19	,	,	PUNCT
ejpam-5661	58	20	c.	c.	PROPN
ejpam-5661	58	21	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	58	22	,	,	PUNCT
ejpam-5661	58	23	p.	p.	PROPN
ejpam-5661	58	24	yotkaew	yotkaew	PROPN
ejpam-5661	58	25	/	/	SYM
ejpam-5661	58	26	eur	eur	PROPN
ejpam-5661	58	27	.	.	PUNCT
ejpam-5661	59	1	j.	j.	PROPN
ejpam-5661	59	2	pure	pure	PROPN
ejpam-5661	59	3	appl	appl	PROPN
ejpam-5661	59	4	.	.	PROPN
ejpam-5661	59	5	math	math	PROPN
ejpam-5661	59	6	,	,	PUNCT
ejpam-5661	59	7	18	18	NUM
ejpam-5661	59	8	(	(	PUNCT
ejpam-5661	59	9	1	1	NUM
ejpam-5661	59	10	)	)	PUNCT
ejpam-5661	59	11	(	(	PUNCT
ejpam-5661	59	12	2025	2025	NUM
ejpam-5661	59	13	)	)	PUNCT
ejpam-5661	59	14	,	,	PUNCT
ejpam-5661	59	15	5661	5661	NUM
ejpam-5661	59	16	3	3	NUM
ejpam-5661	59	17	of	of	ADP
ejpam-5661	59	18	24	24	NUM
ejpam-5661	59	19	which	which	PRON
ejpam-5661	59	20	operates	operate	VERB
ejpam-5661	59	21	without	without	ADP
ejpam-5661	59	22	the	the	DET
ejpam-5661	59	23	traditional	traditional	ADJ
ejpam-5661	59	24	notion	notion	NOUN
ejpam-5661	59	25	of	of	ADP
ejpam-5661	59	26	limits	limit	NOUN
ejpam-5661	59	27	and	and	CCONJ
ejpam-5661	59	28	forges	forge	VERB
ejpam-5661	59	29	a	a	DET
ejpam-5661	59	30	link	link	NOUN
ejpam-5661	59	31	between	between	ADP
ejpam-5661	59	32	mathematics	mathematic	NOUN
ejpam-5661	59	33	and	and	CCONJ
ejpam-5661	59	34	physics	physics	NOUN
ejpam-5661	59	35	.	.	PUNCT
ejpam-5661	60	1	in	in	ADP
ejpam-5661	60	2	the	the	DET
ejpam-5661	60	3	20th	20th	ADJ
ejpam-5661	60	4	century	century	NOUN
ejpam-5661	60	5	,	,	PUNCT
ejpam-5661	60	6	f.h	f.h	PROPN
ejpam-5661	60	7	.	.	PROPN
ejpam-5661	60	8	jackson	jackson	PROPN
ejpam-5661	61	1	[	[	X
ejpam-5661	61	2	14	14	NUM
ejpam-5661	61	3	,	,	PUNCT
ejpam-5661	61	4	15	15	NUM
ejpam-5661	61	5	]	]	PUNCT
ejpam-5661	61	6	further	far	ADV
ejpam-5661	61	7	expanded	expand	VERB
ejpam-5661	61	8	euler	euler	NOUN
ejpam-5661	61	9	’s	’s	PART
ejpam-5661	61	10	ideas	idea	NOUN
ejpam-5661	61	11	by	by	ADP
ejpam-5661	61	12	formalizing	formalize	VERB
ejpam-5661	61	13	the	the	DET
ejpam-5661	61	14	principles	principle	NOUN
ejpam-5661	61	15	of	of	ADP
ejpam-5661	61	16	q	q	NOUN
ejpam-5661	61	17	-	-	NOUN
ejpam-5661	61	18	calculus	calculus	NOUN
ejpam-5661	61	19	.	.	PUNCT
ejpam-5661	62	1	later	later	ADV
ejpam-5661	62	2	,	,	PUNCT
ejpam-5661	62	3	in	in	ADP
ejpam-5661	62	4	2000	2000	NUM
ejpam-5661	62	5	,	,	PUNCT
ejpam-5661	62	6	v.g	v.g	PROPN
ejpam-5661	62	7	.	.	PROPN
ejpam-5661	62	8	kac	kac	PROPN
ejpam-5661	62	9	and	and	CCONJ
ejpam-5661	62	10	p.	p.	PROPN
ejpam-5661	62	11	cheung	cheung	PROPN
ejpam-5661	63	1	[	[	X
ejpam-5661	63	2	16	16	NUM
ejpam-5661	63	3	]	]	PUNCT
ejpam-5661	63	4	provided	provide	VERB
ejpam-5661	63	5	a	a	DET
ejpam-5661	63	6	comprehensive	comprehensive	ADJ
ejpam-5661	63	7	overview	overview	NOUN
ejpam-5661	63	8	of	of	ADP
ejpam-5661	63	9	the	the	DET
ejpam-5661	63	10	foundational	foundational	ADJ
ejpam-5661	63	11	concepts	concept	NOUN
ejpam-5661	63	12	of	of	ADP
ejpam-5661	63	13	q	q	NOUN
ejpam-5661	63	14	-	-	NOUN
ejpam-5661	63	15	calculus	calculus	NOUN
ejpam-5661	63	16	in	in	ADP
ejpam-5661	63	17	their	their	PRON
ejpam-5661	63	18	publication	publication	NOUN
ejpam-5661	63	19	;	;	PUNCT
ejpam-5661	63	20	additional	additional	ADJ
ejpam-5661	63	21	insights	insight	NOUN
ejpam-5661	63	22	can	can	AUX
ejpam-5661	63	23	be	be	AUX
ejpam-5661	63	24	found	find	VERB
ejpam-5661	63	25	in	in	ADP
ejpam-5661	63	26	[	[	X
ejpam-5661	63	27	10	10	NUM
ejpam-5661	63	28	,	,	PUNCT
ejpam-5661	63	29	11	11	NUM
ejpam-5661	63	30	]	]	PUNCT
ejpam-5661	63	31	.	.	PUNCT
ejpam-5661	64	1	in	in	ADP
ejpam-5661	64	2	[	[	X
ejpam-5661	64	3	24	24	NUM
ejpam-5661	64	4	,	,	PUNCT
ejpam-5661	64	5	25	25	NUM
ejpam-5661	64	6	]	]	PUNCT
ejpam-5661	64	7	,	,	PUNCT
ejpam-5661	64	8	j.	j.	PROPN
ejpam-5661	64	9	tariboon	tariboon	PROPN
ejpam-5661	64	10	and	and	CCONJ
ejpam-5661	64	11	s.k	s.k	PROPN
ejpam-5661	64	12	.	.	PROPN
ejpam-5661	64	13	ntouyas	ntouyas	NOUN
ejpam-5661	64	14	(	(	PUNCT
ejpam-5661	64	15	2013	2013	NUM
ejpam-5661	64	16	,	,	PUNCT
ejpam-5661	64	17	2014	2014	NUM
ejpam-5661	64	18	)	)	PUNCT
ejpam-5661	64	19	introduced	introduce	VERB
ejpam-5661	64	20	the	the	DET
ejpam-5661	64	21	q	q	NOUN
ejpam-5661	64	22	-	-	NOUN
ejpam-5661	64	23	calculus	calculus	NOUN
ejpam-5661	64	24	for	for	ADP
ejpam-5661	64	25	continuous	continuous	ADJ
ejpam-5661	64	26	functions	function	NOUN
ejpam-5661	64	27	defined	define	VERB
ejpam-5661	64	28	on	on	ADP
ejpam-5661	64	29	finite	finite	ADJ
ejpam-5661	64	30	intervals	interval	NOUN
ejpam-5661	64	31	and	and	CCONJ
ejpam-5661	64	32	examined	examine	VERB
ejpam-5661	64	33	some	some	PRON
ejpam-5661	64	34	of	of	ADP
ejpam-5661	64	35	its	its	PRON
ejpam-5661	64	36	properties	property	NOUN
ejpam-5661	64	37	,	,	PUNCT
ejpam-5661	64	38	referred	refer	VERB
ejpam-5661	64	39	to	to	ADP
ejpam-5661	64	40	as	as	ADP
ejpam-5661	64	41	qa	qa	NOUN
ejpam-5661	64	42	-	-	PUNCT
ejpam-5661	64	43	calculus	calculus	NOUN
ejpam-5661	64	44	.	.	PUNCT
ejpam-5661	65	1	inspired	inspire	VERB
ejpam-5661	65	2	by	by	ADP
ejpam-5661	65	3	the	the	DET
ejpam-5661	65	4	earlier	early	ADJ
ejpam-5661	65	5	discussion	discussion	NOUN
ejpam-5661	65	6	of	of	ADP
ejpam-5661	65	7	q	q	NOUN
ejpam-5661	65	8	-	-	PUNCT
ejpam-5661	65	9	calculus	calculus	NOUN
ejpam-5661	65	10	,	,	PUNCT
ejpam-5661	65	11	this	this	DET
ejpam-5661	65	12	study	study	NOUN
ejpam-5661	65	13	develops	develop	VERB
ejpam-5661	65	14	q	q	ADJ
ejpam-5661	65	15	-	-	PUNCT
ejpam-5661	65	16	h	h	NOUN
ejpam-5661	65	17	-	-	PUNCT
ejpam-5661	65	18	h	h	NOUN
ejpam-5661	65	19	inequalities	inequality	NOUN
ejpam-5661	65	20	for	for	ADP
ejpam-5661	65	21	strongly	strongly	ADV
ejpam-5661	65	22	convex	convex	NOUN
ejpam-5661	65	23	functions	function	NOUN
ejpam-5661	65	24	using	use	VERB
ejpam-5661	65	25	the	the	DET
ejpam-5661	65	26	principles	principle	NOUN
ejpam-5661	65	27	of	of	ADP
ejpam-5661	65	28	q	q	NOUN
ejpam-5661	65	29	-	-	NOUN
ejpam-5661	65	30	calculus	calculus	NOUN
ejpam-5661	65	31	.	.	PUNCT
ejpam-5661	66	1	it	it	PRON
ejpam-5661	66	2	refines	refine	VERB
ejpam-5661	66	3	existing	exist	VERB
ejpam-5661	66	4	findings	finding	NOUN
ejpam-5661	66	5	by	by	ADP
ejpam-5661	66	6	utilizing	utilize	VERB
ejpam-5661	66	7	the	the	DET
ejpam-5661	66	8	characterization	characterization	NOUN
ejpam-5661	66	9	of	of	ADP
ejpam-5661	66	10	strong	strong	ADJ
ejpam-5661	66	11	convexity	convexity	NOUN
ejpam-5661	66	12	of	of	ADP
ejpam-5661	66	13	φ	φ	PROPN
ejpam-5661	66	14	.	.	PUNCT
ejpam-5661	67	1	furthermore	furthermore	ADV
ejpam-5661	67	2	,	,	PUNCT
ejpam-5661	67	3	the	the	DET
ejpam-5661	67	4	q	q	NOUN
ejpam-5661	67	5	-	-	PUNCT
ejpam-5661	67	6	midpoint	midpoint	NOUN
ejpam-5661	67	7	and	and	CCONJ
ejpam-5661	67	8	q	q	ADJ
ejpam-5661	67	9	-	-	PUNCT
ejpam-5661	67	10	trapezoidal	trapezoidal	ADJ
ejpam-5661	67	11	inequalities	inequality	NOUN
ejpam-5661	67	12	are	be	AUX
ejpam-5661	67	13	unified	unify	VERB
ejpam-5661	67	14	into	into	ADP
ejpam-5661	67	15	one	one	NUM
ejpam-5661	67	16	.	.	PUNCT
ejpam-5661	68	1	the	the	DET
ejpam-5661	68	2	newly	newly	ADV
ejpam-5661	68	3	established	establish	VERB
ejpam-5661	68	4	inequalities	inequality	NOUN
ejpam-5661	68	5	also	also	ADV
ejpam-5661	68	6	demonstrate	demonstrate	VERB
ejpam-5661	68	7	applications	application	NOUN
ejpam-5661	68	8	to	to	ADP
ejpam-5661	68	9	special	special	ADJ
ejpam-5661	68	10	means	mean	NOUN
ejpam-5661	68	11	.	.	PUNCT
ejpam-5661	69	1	2	2	X
ejpam-5661	69	2	.	.	NUM
ejpam-5661	69	3	preliminaries	preliminary	NOUN
ejpam-5661	69	4	in	in	ADP
ejpam-5661	69	5	this	this	DET
ejpam-5661	69	6	section	section	NOUN
ejpam-5661	69	7	,	,	PUNCT
ejpam-5661	69	8	we	we	PRON
ejpam-5661	69	9	will	will	AUX
ejpam-5661	69	10	review	review	VERB
ejpam-5661	69	11	the	the	DET
ejpam-5661	69	12	definitions	definition	NOUN
ejpam-5661	69	13	and	and	CCONJ
ejpam-5661	69	14	key	key	ADJ
ejpam-5661	69	15	properties	property	NOUN
ejpam-5661	69	16	related	relate	VERB
ejpam-5661	69	17	to	to	ADP
ejpam-5661	69	18	the	the	DET
ejpam-5661	69	19	concept	concept	NOUN
ejpam-5661	69	20	of	of	ADP
ejpam-5661	69	21	q	q	NOUN
ejpam-5661	69	22	-	-	NOUN
ejpam-5661	69	23	calculus	calculus	NOUN
ejpam-5661	69	24	that	that	PRON
ejpam-5661	69	25	are	be	AUX
ejpam-5661	69	26	pertinent	pertinent	ADJ
ejpam-5661	69	27	to	to	ADP
ejpam-5661	69	28	this	this	DET
ejpam-5661	69	29	study	study	NOUN
ejpam-5661	69	30	.	.	PUNCT
ejpam-5661	70	1	for	for	ADP
ejpam-5661	70	2	the	the	DET
ejpam-5661	70	3	purposes	purpose	NOUN
ejpam-5661	70	4	of	of	ADP
ejpam-5661	70	5	this	this	DET
ejpam-5661	70	6	paper	paper	NOUN
ejpam-5661	70	7	,	,	PUNCT
ejpam-5661	70	8	we	we	PRON
ejpam-5661	70	9	will	will	AUX
ejpam-5661	70	10	consider	consider	VERB
ejpam-5661	70	11	α	α	PRON
ejpam-5661	70	12	<	<	X
ejpam-5661	70	13	υ	υ	NOUN
ejpam-5661	70	14	and	and	CCONJ
ejpam-5661	70	15	0	0	NUM
ejpam-5661	70	16	<	<	X
ejpam-5661	70	17	q	q	X
ejpam-5661	70	18	<	<	X
ejpam-5661	70	19	1	1	NUM
ejpam-5661	70	20	as	as	ADP
ejpam-5661	70	21	fixed	fix	VERB
ejpam-5661	70	22	parameters	parameter	NOUN
ejpam-5661	70	23	.	.	PUNCT
ejpam-5661	71	1	[	[	X
ejpam-5661	71	2	η]q	η]q	NOUN
ejpam-5661	71	3	:	:	PUNCT
ejpam-5661	71	4	=	=	SYM
ejpam-5661	71	5	1−	1−	NUM
ejpam-5661	71	6	qη	qη	NOUN
ejpam-5661	71	7	1−	1−	NUM
ejpam-5661	71	8	q	q	NOUN
ejpam-5661	72	1	=	=	NOUN
ejpam-5661	72	2	1	1	NUM
ejpam-5661	72	3	+	+	CCONJ
ejpam-5661	72	4	q	q	NOUN
ejpam-5661	72	5	+	+	NUM
ejpam-5661	72	6	q2	q2	NOUN
ejpam-5661	72	7	+	+	X
ejpam-5661	72	8	·	·	PUNCT
ejpam-5661	72	9	·	·	PUNCT
ejpam-5661	72	10	·	·	PUNCT
ejpam-5661	73	1	+	+	NUM
ejpam-5661	73	2	qη−1	qη−1	PROPN
ejpam-5661	73	3	,	,	PUNCT
ejpam-5661	73	4	η	η	PROPN
ejpam-5661	73	5	∈	∈	PROPN
ejpam-5661	73	6	n.	n.	NOUN
ejpam-5661	73	7	this	this	PRON
ejpam-5661	73	8	represents	represent	VERB
ejpam-5661	73	9	the	the	DET
ejpam-5661	73	10	q	q	NOUN
ejpam-5661	73	11	-	-	PUNCT
ejpam-5661	73	12	analogue	analogue	NOUN
ejpam-5661	73	13	of	of	ADP
ejpam-5661	73	14	η	η	PROPN
ejpam-5661	73	15	;	;	PUNCT
ejpam-5661	73	16	for	for	ADP
ejpam-5661	73	17	further	further	ADJ
ejpam-5661	73	18	information	information	NOUN
ejpam-5661	73	19	,	,	PUNCT
ejpam-5661	73	20	please	please	INTJ
ejpam-5661	73	21	refer	refer	VERB
ejpam-5661	73	22	to	to	ADP
ejpam-5661	73	23	[	[	X
ejpam-5661	73	24	16	16	NUM
ejpam-5661	73	25	]	]	PUNCT
ejpam-5661	73	26	.	.	PUNCT
ejpam-5661	74	1	definition	definition	NOUN
ejpam-5661	74	2	2	2	NUM
ejpam-5661	74	3	(	(	PUNCT
ejpam-5661	74	4	[	[	X
ejpam-5661	74	5	16	16	NUM
ejpam-5661	74	6	,	,	PUNCT
ejpam-5661	74	7	24	24	NUM
ejpam-5661	74	8	]	]	PUNCT
ejpam-5661	74	9	)	)	PUNCT
ejpam-5661	74	10	.	.	PUNCT
ejpam-5661	75	1	let	let	VERB
ejpam-5661	75	2	φ	φ	NOUN
ejpam-5661	75	3	:	:	PUNCT
ejpam-5661	76	1	[	[	X
ejpam-5661	76	2	α	α	X
ejpam-5661	76	3	,	,	PUNCT
ejpam-5661	76	4	υ	υ	NOUN
ejpam-5661	76	5	]	]	X
ejpam-5661	76	6	→	→	PUNCT
ejpam-5661	76	7	r	r	NOUN
ejpam-5661	76	8	be	be	AUX
ejpam-5661	76	9	a	a	DET
ejpam-5661	76	10	continuous	continuous	ADJ
ejpam-5661	76	11	function	function	NOUN
ejpam-5661	76	12	.	.	PUNCT
ejpam-5661	77	1	the	the	DET
ejpam-5661	77	2	qα	qα	PROPN
ejpam-5661	77	3	-	-	PUNCT
ejpam-5661	77	4	derivative	derivative	NOUN
ejpam-5661	77	5	of	of	ADP
ejpam-5661	77	6	φ	φ	PROPN
ejpam-5661	77	7	at	at	ADP
ejpam-5661	77	8	ω	ω	PROPN
ejpam-5661	77	9	∈	∈	PROPN
ejpam-5661	77	10	[	[	X
ejpam-5661	77	11	α	α	NOUN
ejpam-5661	77	12	,	,	PUNCT
ejpam-5661	77	13	υ	υ	NOUN
ejpam-5661	77	14	]	]	X
ejpam-5661	77	15	is	be	AUX
ejpam-5661	77	16	,	,	PUNCT
ejpam-5661	77	17	consequently	consequently	ADV
ejpam-5661	77	18	,	,	PUNCT
ejpam-5661	77	19	defined	define	VERB
ejpam-5661	77	20	by	by	ADP
ejpam-5661	77	21	the	the	DET
ejpam-5661	77	22	following	follow	VERB
ejpam-5661	77	23	expression	expression	NOUN
ejpam-5661	77	24	αdqφ(ω	αdqφ(ω	NOUN
ejpam-5661	77	25	)	)	PUNCT
ejpam-5661	78	1	=	=	SYM
ejpam-5661	78	2	φ(ω)−	φ(ω)−	NOUN
ejpam-5661	78	3	φ(qω	φ(qω	PROPN
ejpam-5661	78	4	+	+	CCONJ
ejpam-5661	78	5	(	(	PUNCT
ejpam-5661	78	6	1−	1−	NUM
ejpam-5661	78	7	q)α	q)α	X
ejpam-5661	78	8	)	)	PUNCT
ejpam-5661	78	9	(	(	PUNCT
ejpam-5661	78	10	1−	1−	NUM
ejpam-5661	78	11	q)(ω	q)(ω	VERB
ejpam-5661	78	12	−	−	PROPN
ejpam-5661	78	13	α	α	NOUN
ejpam-5661	78	14	)	)	PUNCT
ejpam-5661	78	15	,	,	PUNCT
ejpam-5661	79	1	ω	ω	NUM
ejpam-5661	79	2	̸=	̸=	PROPN
ejpam-5661	79	3	α	α	NOUN
ejpam-5661	79	4	.	.	PUNCT
ejpam-5661	80	1	(	(	PUNCT
ejpam-5661	80	2	7	7	X
ejpam-5661	80	3	)	)	PUNCT
ejpam-5661	80	4	if	if	SCONJ
ejpam-5661	80	5	ω	ω	PROPN
ejpam-5661	80	6	=	=	SYM
ejpam-5661	80	7	α	α	X
ejpam-5661	80	8	,	,	PUNCT
ejpam-5661	80	9	we	we	PRON
ejpam-5661	80	10	define	define	VERB
ejpam-5661	80	11	αdqφ(α	αdqφ(α	NOUN
ejpam-5661	80	12	)	)	PUNCT
ejpam-5661	80	13	=	=	SYM
ejpam-5661	80	14	lim	lim	PROPN
ejpam-5661	80	15	ω→a	ω→a	NUM
ejpam-5661	80	16	αdqφ(ω	αdqφ(ω	PROPN
ejpam-5661	80	17	)	)	PUNCT
ejpam-5661	80	18	,	,	PUNCT
ejpam-5661	80	19	whether	whether	SCONJ
ejpam-5661	80	20	it	it	PRON
ejpam-5661	80	21	exists	exist	VERB
ejpam-5661	80	22	and	and	CCONJ
ejpam-5661	80	23	is	be	AUX
ejpam-5661	80	24	finite	finite	ADJ
ejpam-5661	80	25	.	.	PUNCT
ejpam-5661	81	1	if	if	SCONJ
ejpam-5661	81	2	we	we	PRON
ejpam-5661	81	3	take	take	VERB
ejpam-5661	81	4	α	α	NOUN
ejpam-5661	81	5	=	=	NOUN
ejpam-5661	81	6	0	0	NUM
ejpam-5661	81	7	in	in	ADP
ejpam-5661	81	8	(	(	PUNCT
ejpam-5661	81	9	7	7	NUM
ejpam-5661	81	10	)	)	PUNCT
ejpam-5661	81	11	,	,	PUNCT
ejpam-5661	81	12	then	then	ADV
ejpam-5661	81	13	we	we	PRON
ejpam-5661	81	14	have	have	VERB
ejpam-5661	81	15	0dqφ(ω	0dqφ(ω	NUM
ejpam-5661	81	16	)	)	PUNCT
ejpam-5661	81	17	=	=	SYM
ejpam-5661	81	18	dqφ(ω	dqφ(ω	PROPN
ejpam-5661	81	19	)	)	PUNCT
ejpam-5661	81	20	,	,	PUNCT
ejpam-5661	81	21	which	which	PRON
ejpam-5661	81	22	can	can	AUX
ejpam-5661	81	23	be	be	AUX
ejpam-5661	81	24	simplified	simplify	VERB
ejpam-5661	81	25	to	to	ADP
ejpam-5661	81	26	dqφ(ω	dqφ(ω	PROPN
ejpam-5661	81	27	)	)	PUNCT
ejpam-5661	82	1	=	=	SYM
ejpam-5661	82	2	φ(ω)−	φ(ω)−	PROPN
ejpam-5661	82	3	φ(qω	φ(qω	PROPN
ejpam-5661	82	4	)	)	PUNCT
ejpam-5661	82	5	(	(	PUNCT
ejpam-5661	82	6	1−	1−	NUM
ejpam-5661	82	7	q)ω	q)ω	PUNCT
ejpam-5661	82	8	,	,	PUNCT
ejpam-5661	82	9	ω	ω	NUM
ejpam-5661	82	10	̸=	̸=	PROPN
ejpam-5661	82	11	0	0	NUM
ejpam-5661	82	12	.	.	PUNCT
ejpam-5661	83	1	this	this	PRON
ejpam-5661	83	2	represents	represent	VERB
ejpam-5661	83	3	the	the	DET
ejpam-5661	83	4	q	q	PROPN
ejpam-5661	83	5	-	-	PUNCT
ejpam-5661	83	6	jackson	jackson	PROPN
ejpam-5661	83	7	derivative	derivative	NOUN
ejpam-5661	83	8	.	.	PUNCT
ejpam-5661	84	1	definition	definition	NOUN
ejpam-5661	84	2	3	3	NUM
ejpam-5661	84	3	(	(	PUNCT
ejpam-5661	84	4	[	[	X
ejpam-5661	84	5	5	5	NUM
ejpam-5661	84	6	]	]	PUNCT
ejpam-5661	84	7	)	)	PUNCT
ejpam-5661	84	8	.	.	PUNCT
ejpam-5661	85	1	let	let	VERB
ejpam-5661	85	2	φ	φ	NOUN
ejpam-5661	85	3	:	:	PUNCT
ejpam-5661	86	1	[	[	X
ejpam-5661	86	2	α	α	X
ejpam-5661	86	3	,	,	PUNCT
ejpam-5661	86	4	υ	υ	NOUN
ejpam-5661	86	5	]	]	X
ejpam-5661	86	6	→	→	PUNCT
ejpam-5661	86	7	r	r	NOUN
ejpam-5661	86	8	be	be	AUX
ejpam-5661	86	9	a	a	DET
ejpam-5661	86	10	continuous	continuous	ADJ
ejpam-5661	86	11	function	function	NOUN
ejpam-5661	86	12	.	.	PUNCT
ejpam-5661	87	1	then	then	ADV
ejpam-5661	87	2	,	,	PUNCT
ejpam-5661	87	3	the	the	DET
ejpam-5661	87	4	qυ	qυ	NOUN
ejpam-5661	87	5	-	-	NOUN
ejpam-5661	87	6	derivative	derivative	NOUN
ejpam-5661	87	7	of	of	ADP
ejpam-5661	87	8	φ	φ	PROPN
ejpam-5661	87	9	at	at	ADP
ejpam-5661	87	10	ω	ω	PROPN
ejpam-5661	87	11	∈	∈	PROPN
ejpam-5661	87	12	[	[	X
ejpam-5661	87	13	α	α	NOUN
ejpam-5661	87	14	,	,	PUNCT
ejpam-5661	87	15	b	b	AUX
ejpam-5661	87	16	]	]	PUNCT
ejpam-5661	87	17	is	be	AUX
ejpam-5661	87	18	defined	define	VERB
ejpam-5661	87	19	by	by	ADP
ejpam-5661	87	20	υdqφ(ω	υdqφ(ω	PROPN
ejpam-5661	87	21	)	)	PUNCT
ejpam-5661	87	22	=	=	SYM
ejpam-5661	87	23	φ(qω	φ(qω	PROPN
ejpam-5661	87	24	+	+	SYM
ejpam-5661	87	25	(	(	PUNCT
ejpam-5661	87	26	1−	1−	NUM
ejpam-5661	87	27	q)υ)−	q)υ)−	NOUN
ejpam-5661	87	28	φ(ω	φ(ω	NOUN
ejpam-5661	87	29	)	)	PUNCT
ejpam-5661	87	30	(	(	PUNCT
ejpam-5661	87	31	1−	1−	NUM
ejpam-5661	87	32	q)(υ−	q)(υ−	PROPN
ejpam-5661	87	33	ω	ω	NUM
ejpam-5661	87	34	)	)	PUNCT
ejpam-5661	87	35	,	,	PUNCT
ejpam-5661	88	1	ω	ω	NUM
ejpam-5661	88	2	̸=	̸=	PROPN
ejpam-5661	88	3	υ	υ	NOUN
ejpam-5661	88	4	.	.	PUNCT
ejpam-5661	89	1	(	(	PUNCT
ejpam-5661	89	2	8)	8)	NUM
ejpam-5661	89	3	c.	c.	PROPN
ejpam-5661	89	4	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	89	5	,	,	PUNCT
ejpam-5661	89	6	p.	p.	PROPN
ejpam-5661	89	7	yotkaew	yotkaew	PROPN
ejpam-5661	89	8	/	/	SYM
ejpam-5661	89	9	eur	eur	PROPN
ejpam-5661	89	10	.	.	PUNCT
ejpam-5661	90	1	j.	j.	PROPN
ejpam-5661	90	2	pure	pure	PROPN
ejpam-5661	90	3	appl	appl	PROPN
ejpam-5661	90	4	.	.	PROPN
ejpam-5661	90	5	math	math	PROPN
ejpam-5661	90	6	,	,	PUNCT
ejpam-5661	90	7	18	18	NUM
ejpam-5661	90	8	(	(	PUNCT
ejpam-5661	90	9	1	1	NUM
ejpam-5661	90	10	)	)	PUNCT
ejpam-5661	90	11	(	(	PUNCT
ejpam-5661	90	12	2025	2025	NUM
ejpam-5661	90	13	)	)	PUNCT
ejpam-5661	90	14	,	,	PUNCT
ejpam-5661	90	15	5661	5661	NUM
ejpam-5661	90	16	4	4	NUM
ejpam-5661	90	17	of	of	ADP
ejpam-5661	90	18	24	24	NUM
ejpam-5661	90	19	if	if	SCONJ
ejpam-5661	90	20	ω	ω	NUM
ejpam-5661	90	21	=	=	SYM
ejpam-5661	90	22	υ	υ	NOUN
ejpam-5661	90	23	,	,	PUNCT
ejpam-5661	90	24	we	we	PRON
ejpam-5661	90	25	define	define	VERB
ejpam-5661	90	26	υdqφ(b	υdqφ(b	ADP
ejpam-5661	90	27	)	)	PUNCT
ejpam-5661	90	28	=	=	SYM
ejpam-5661	90	29	lim	lim	PROPN
ejpam-5661	90	30	ω→υ	ω→υ	NUM
ejpam-5661	90	31	υdqφ(ω	υdqφ(ω	NOUN
ejpam-5661	90	32	)	)	PUNCT
ejpam-5661	90	33	,	,	PUNCT
ejpam-5661	90	34	whether	whether	SCONJ
ejpam-5661	90	35	it	it	PRON
ejpam-5661	90	36	exists	exist	VERB
ejpam-5661	90	37	and	and	CCONJ
ejpam-5661	90	38	is	be	AUX
ejpam-5661	90	39	finite	finite	ADJ
ejpam-5661	90	40	.	.	PUNCT
ejpam-5661	91	1	definition	definition	NOUN
ejpam-5661	91	2	4	4	NUM
ejpam-5661	91	3	(	(	PUNCT
ejpam-5661	91	4	[	[	X
ejpam-5661	91	5	20	20	NUM
ejpam-5661	91	6	]	]	NUM
ejpam-5661	91	7	)	)	PUNCT
ejpam-5661	91	8	.	.	PUNCT
ejpam-5661	92	1	let	let	VERB
ejpam-5661	92	2	φ	φ	NOUN
ejpam-5661	92	3	:	:	PUNCT
ejpam-5661	93	1	[	[	X
ejpam-5661	93	2	α	α	X
ejpam-5661	93	3	,	,	PUNCT
ejpam-5661	93	4	υ	υ	NOUN
ejpam-5661	93	5	]	]	X
ejpam-5661	93	6	→	→	PUNCT
ejpam-5661	93	7	r	r	NOUN
ejpam-5661	93	8	be	be	AUX
ejpam-5661	93	9	a	a	DET
ejpam-5661	93	10	continuous	continuous	ADJ
ejpam-5661	93	11	function	function	NOUN
ejpam-5661	93	12	and	and	CCONJ
ejpam-5661	93	13	ω	ω	NUM
ejpam-5661	93	14	∈	∈	PROPN
ejpam-5661	93	15	[	[	X
ejpam-5661	93	16	α	α	NOUN
ejpam-5661	93	17	,	,	PUNCT
ejpam-5661	93	18	υ	υ	NOUN
ejpam-5661	93	19	]	]	X
ejpam-5661	93	20	.	.	PUNCT
ejpam-5661	94	1	the	the	DET
ejpam-5661	94	2	qα	qα	PROPN
ejpam-5661	94	3	-	-	PUNCT
ejpam-5661	94	4	integral	integral	ADJ
ejpam-5661	94	5	of	of	ADP
ejpam-5661	94	6	φ	φ	PROPN
ejpam-5661	94	7	on	on	ADP
ejpam-5661	94	8	[	[	X
ejpam-5661	94	9	α	α	X
ejpam-5661	94	10	,	,	PUNCT
ejpam-5661	94	11	ω	ω	PROPN
ejpam-5661	94	12	]	]	X
ejpam-5661	94	13	is	be	AUX
ejpam-5661	94	14	defined	define	VERB
ejpam-5661	94	15	by∫	by∫	PROPN
ejpam-5661	94	16	ω	ω	NUM
ejpam-5661	94	17	α	α	PROPN
ejpam-5661	94	18	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	94	19	)	)	PUNCT
ejpam-5661	94	20	αdqϖ	αdqϖ	NOUN
ejpam-5661	95	1	=	=	SYM
ejpam-5661	95	2	(	(	PUNCT
ejpam-5661	95	3	1−	1−	NUM
ejpam-5661	95	4	q)(ω	q)(ω	VERB
ejpam-5661	95	5	−	−	PROPN
ejpam-5661	95	6	α	α	NOUN
ejpam-5661	95	7	)	)	PUNCT
ejpam-5661	95	8	∞∑	∞∑	PROPN
ejpam-5661	95	9	n=0	n=0	NUM
ejpam-5661	95	10	qnφ	qnφ	NOUN
ejpam-5661	95	11	(	(	PUNCT
ejpam-5661	95	12	qnω	qnω	NOUN
ejpam-5661	95	13	+	+	CCONJ
ejpam-5661	95	14	(	(	PUNCT
ejpam-5661	95	15	1−	1−	NUM
ejpam-5661	95	16	qn)α	qn)α	ADV
ejpam-5661	95	17	)	)	PUNCT
ejpam-5661	95	18	.	.	PUNCT
ejpam-5661	96	1	(	(	PUNCT
ejpam-5661	96	2	9	9	X
ejpam-5661	96	3	)	)	PUNCT
ejpam-5661	96	4	note	note	NOUN
ejpam-5661	96	5	that	that	SCONJ
ejpam-5661	96	6	∫	∫	PROPN
ejpam-5661	96	7	ω	ω	NUM
ejpam-5661	96	8	α	α	PROPN
ejpam-5661	96	9	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	96	10	)	)	PUNCT
ejpam-5661	96	11	αdqϖ	αdqϖ	NOUN
ejpam-5661	96	12	=	=	SYM
ejpam-5661	96	13	(	(	PUNCT
ejpam-5661	96	14	ω	ω	NOUN
ejpam-5661	96	15	−	−	PROPN
ejpam-5661	96	16	α	α	NOUN
ejpam-5661	96	17	)	)	PUNCT
ejpam-5661	96	18	∫	∫	PROPN
ejpam-5661	96	19	1	1	NUM
ejpam-5661	96	20	0	0	NUM
ejpam-5661	96	21	φ	φ	PROPN
ejpam-5661	96	22	(	(	PUNCT
ejpam-5661	96	23	(	(	PUNCT
ejpam-5661	96	24	1−ϖ)α+ϖω	1−ϖ)α+ϖω	ADJ
ejpam-5661	96	25	)	)	PUNCT
ejpam-5661	96	26	αdqϖ	αdqϖ	NOUN
ejpam-5661	96	27	,	,	PUNCT
ejpam-5661	96	28	and	and	CCONJ
ejpam-5661	96	29	if	if	SCONJ
ejpam-5661	96	30	α	α	PRON
ejpam-5661	96	31	=	=	SYM
ejpam-5661	96	32	0	0	NUM
ejpam-5661	96	33	,	,	PUNCT
ejpam-5661	96	34	then	then	ADV
ejpam-5661	96	35	(	(	PUNCT
ejpam-5661	96	36	9	9	X
ejpam-5661	96	37	)	)	PUNCT
ejpam-5661	96	38	reduces	reduce	VERB
ejpam-5661	96	39	to∫	to∫	NOUN
ejpam-5661	96	40	ω	ω	NUM
ejpam-5661	96	41	0	0	NUM
ejpam-5661	96	42	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	96	43	)	)	PUNCT
ejpam-5661	97	1	0dqϖ	0dqϖ	NUM
ejpam-5661	97	2	=	=	SYM
ejpam-5661	98	1	∫	∫	PROPN
ejpam-5661	98	2	ω	ω	NOUN
ejpam-5661	98	3	0	0	PUNCT
ejpam-5661	98	4	φ(ϖ)dqϖ	φ(ϖ)dqϖ	NOUN
ejpam-5661	98	5	=	=	SYM
ejpam-5661	98	6	(	(	PUNCT
ejpam-5661	98	7	1−	1−	NUM
ejpam-5661	98	8	q)ω	q)ω	PUNCT
ejpam-5661	98	9	∞∑	∞∑	PRON
ejpam-5661	98	10	n=0	n=0	NUM
ejpam-5661	98	11	qnφ(qnω	qnφ(qnω	NOUN
ejpam-5661	98	12	)	)	PUNCT
ejpam-5661	98	13	.	.	PUNCT
ejpam-5661	99	1	this	this	PRON
ejpam-5661	99	2	represents	represent	VERB
ejpam-5661	99	3	the	the	DET
ejpam-5661	99	4	qα	qα	PROPN
ejpam-5661	99	5	-	-	PUNCT
ejpam-5661	99	6	integral	integral	ADJ
ejpam-5661	99	7	,	,	PUNCT
ejpam-5661	99	8	for	for	ADP
ejpam-5661	99	9	further	further	ADJ
ejpam-5661	99	10	details	detail	NOUN
ejpam-5661	99	11	,	,	PUNCT
ejpam-5661	99	12	refer	refer	VERB
ejpam-5661	99	13	to	to	ADP
ejpam-5661	99	14	[	[	X
ejpam-5661	99	15	16	16	NUM
ejpam-5661	99	16	,	,	PUNCT
ejpam-5661	99	17	24	24	NUM
ejpam-5661	99	18	]	]	PUNCT
ejpam-5661	99	19	.	.	PUNCT
ejpam-5661	100	1	definition	definition	NOUN
ejpam-5661	100	2	5	5	NUM
ejpam-5661	100	3	(	(	PUNCT
ejpam-5661	100	4	[	[	X
ejpam-5661	100	5	5	5	NUM
ejpam-5661	100	6	]	]	PUNCT
ejpam-5661	100	7	)	)	PUNCT
ejpam-5661	100	8	.	.	PUNCT
ejpam-5661	101	1	let	let	VERB
ejpam-5661	101	2	φ	φ	NOUN
ejpam-5661	101	3	:	:	PUNCT
ejpam-5661	102	1	[	[	X
ejpam-5661	102	2	α	α	X
ejpam-5661	102	3	,	,	PUNCT
ejpam-5661	102	4	υ	υ	NOUN
ejpam-5661	102	5	]	]	X
ejpam-5661	102	6	→	→	PUNCT
ejpam-5661	102	7	r	r	NOUN
ejpam-5661	102	8	be	be	AUX
ejpam-5661	102	9	a	a	DET
ejpam-5661	102	10	continuous	continuous	ADJ
ejpam-5661	102	11	function	function	NOUN
ejpam-5661	102	12	and	and	CCONJ
ejpam-5661	102	13	ω	ω	NUM
ejpam-5661	102	14	∈	∈	PROPN
ejpam-5661	102	15	[	[	X
ejpam-5661	102	16	α	α	NOUN
ejpam-5661	102	17	,	,	PUNCT
ejpam-5661	102	18	υ	υ	NOUN
ejpam-5661	102	19	]	]	X
ejpam-5661	102	20	.	.	PUNCT
ejpam-5661	103	1	the	the	DET
ejpam-5661	103	2	qυ	qυ	NOUN
ejpam-5661	103	3	-	-	ADJ
ejpam-5661	103	4	integral	integral	ADJ
ejpam-5661	103	5	of	of	ADP
ejpam-5661	103	6	φ	φ	PROPN
ejpam-5661	103	7	on	on	ADP
ejpam-5661	103	8	[	[	X
ejpam-5661	103	9	ω	ω	NOUN
ejpam-5661	103	10	,	,	PUNCT
ejpam-5661	103	11	υ	υ	NOUN
ejpam-5661	103	12	]	]	PUNCT
ejpam-5661	103	13	is	be	AUX
ejpam-5661	103	14	defined	define	VERB
ejpam-5661	103	15	by∫	by∫	PROPN
ejpam-5661	103	16	υ	υ	PROPN
ejpam-5661	103	17	ω	ω	PROPN
ejpam-5661	103	18	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	103	19	)	)	PUNCT
ejpam-5661	104	1	υdqϖ	υdqϖ	NOUN
ejpam-5661	104	2	=	=	SYM
ejpam-5661	104	3	(	(	PUNCT
ejpam-5661	104	4	1−	1−	NUM
ejpam-5661	104	5	q)(υ−	q)(υ−	PROPN
ejpam-5661	104	6	ω	ω	NOUN
ejpam-5661	104	7	)	)	PUNCT
ejpam-5661	105	1	∞∑	∞∑	PROPN
ejpam-5661	105	2	n=0	n=0	NUM
ejpam-5661	105	3	qnφ	qnφ	NOUN
ejpam-5661	105	4	(	(	PUNCT
ejpam-5661	105	5	qnω	qnω	NOUN
ejpam-5661	105	6	+	+	CCONJ
ejpam-5661	105	7	(	(	PUNCT
ejpam-5661	105	8	1−	1−	NUM
ejpam-5661	105	9	qn)υ	qn)υ	ADJ
ejpam-5661	105	10	)	)	PUNCT
ejpam-5661	105	11	.	.	PUNCT
ejpam-5661	106	1	(	(	PUNCT
ejpam-5661	106	2	10	10	NUM
ejpam-5661	106	3	)	)	PUNCT
ejpam-5661	106	4	note	note	NOUN
ejpam-5661	106	5	that	that	SCONJ
ejpam-5661	106	6	∫	∫	PROPN
ejpam-5661	106	7	υ	υ	PROPN
ejpam-5661	106	8	ω	ω	PROPN
ejpam-5661	106	9	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	106	10	)	)	PUNCT
ejpam-5661	106	11	υdqϖ	υdqϖ	NOUN
ejpam-5661	106	12	=	=	SYM
ejpam-5661	106	13	(	(	PUNCT
ejpam-5661	106	14	υ−	υ−	PROPN
ejpam-5661	106	15	ω	ω	NOUN
ejpam-5661	106	16	)	)	PUNCT
ejpam-5661	106	17	∫	∫	PROPN
ejpam-5661	106	18	1	1	NUM
ejpam-5661	106	19	0	0	NUM
ejpam-5661	106	20	φ	φ	PROPN
ejpam-5661	106	21	(	(	PUNCT
ejpam-5661	106	22	ϖυ+	ϖυ+	X
ejpam-5661	106	23	(	(	PUNCT
ejpam-5661	106	24	1−ϖ)ω	1−ϖ)ω	NUM
ejpam-5661	106	25	)	)	PUNCT
ejpam-5661	106	26	1dqϖ	1dqϖ	NUM
ejpam-5661	106	27	,	,	PUNCT
ejpam-5661	106	28	and	and	CCONJ
ejpam-5661	106	29	if	if	SCONJ
ejpam-5661	106	30	ω	ω	PROPN
ejpam-5661	106	31	=	=	SYM
ejpam-5661	106	32	0	0	NUM
ejpam-5661	106	33	,	,	PUNCT
ejpam-5661	106	34	then	then	ADV
ejpam-5661	106	35	(	(	PUNCT
ejpam-5661	106	36	10	10	NUM
ejpam-5661	106	37	)	)	PUNCT
ejpam-5661	106	38	reduces	reduce	VERB
ejpam-5661	106	39	to∫	to∫	INTJ
ejpam-5661	106	40	υ	υ	NOUN
ejpam-5661	106	41	0	0	SYM
ejpam-5661	106	42	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	106	43	)	)	PUNCT
ejpam-5661	106	44	υdqϖ	υdqϖ	NOUN
ejpam-5661	106	45	=	=	SYM
ejpam-5661	106	46	(	(	PUNCT
ejpam-5661	106	47	1−	1−	NUM
ejpam-5661	106	48	q)υ	q)υ	X
ejpam-5661	106	49	∞∑	∞∑	NUM
ejpam-5661	106	50	n=0	n=0	PRON
ejpam-5661	106	51	qnφ((1−	qnφ((1−	NOUN
ejpam-5661	106	52	qn)υ	qn)υ	PROPN
ejpam-5661	106	53	)	)	PUNCT
ejpam-5661	106	54	.	.	PUNCT
ejpam-5661	107	1	this	this	PRON
ejpam-5661	107	2	represents	represent	VERB
ejpam-5661	107	3	the	the	DET
ejpam-5661	107	4	qυ	qυ	ADJ
ejpam-5661	107	5	-	-	ADJ
ejpam-5661	107	6	integral	integral	ADJ
ejpam-5661	107	7	.	.	PUNCT
ejpam-5661	108	1	3	3	X
ejpam-5661	108	2	.	.	X
ejpam-5661	108	3	refinements	refinement	NOUN
ejpam-5661	108	4	of	of	ADP
ejpam-5661	108	5	q	q	ADJ
ejpam-5661	108	6	-	-	PUNCT
ejpam-5661	108	7	h	h	NOUN
ejpam-5661	108	8	-	-	PUNCT
ejpam-5661	108	9	h	h	NOUN
ejpam-5661	108	10	-	-	PUNCT
ejpam-5661	108	11	type	type	NOUN
ejpam-5661	108	12	inequalities	inequality	NOUN
ejpam-5661	108	13	we	we	PRON
ejpam-5661	108	14	begin	begin	VERB
ejpam-5661	108	15	by	by	ADP
ejpam-5661	108	16	refining	refining	NOUN
ejpam-5661	108	17	theorem	theorem	NOUN
ejpam-5661	108	18	3.1	3.1	NUM
ejpam-5661	108	19	in	in	ADP
ejpam-5661	108	20	n.	n.	PROPN
ejpam-5661	108	21	alp	alp	PROPN
ejpam-5661	108	22	et	et	PROPN
ejpam-5661	108	23	al	al	PROPN
ejpam-5661	108	24	.	.	PUNCT
ejpam-5661	109	1	[	[	X
ejpam-5661	109	2	2	2	X
ejpam-5661	109	3	]	]	PUNCT
ejpam-5661	109	4	using	use	VERB
ejpam-5661	109	5	the	the	DET
ejpam-5661	109	6	characterization	characterization	NOUN
ejpam-5661	109	7	of	of	ADP
ejpam-5661	109	8	strong	strong	ADJ
ejpam-5661	109	9	convexity	convexity	NOUN
ejpam-5661	109	10	.	.	PUNCT
ejpam-5661	110	1	c.	c.	PROPN
ejpam-5661	110	2	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	110	3	,	,	PUNCT
ejpam-5661	110	4	p.	p.	PROPN
ejpam-5661	110	5	yotkaew	yotkaew	PROPN
ejpam-5661	110	6	/	/	SYM
ejpam-5661	110	7	eur	eur	PROPN
ejpam-5661	110	8	.	.	PUNCT
ejpam-5661	111	1	j.	j.	PROPN
ejpam-5661	111	2	pure	pure	PROPN
ejpam-5661	111	3	appl	appl	PROPN
ejpam-5661	111	4	.	.	PROPN
ejpam-5661	111	5	math	math	PROPN
ejpam-5661	111	6	,	,	PUNCT
ejpam-5661	111	7	18	18	NUM
ejpam-5661	111	8	(	(	PUNCT
ejpam-5661	111	9	1	1	NUM
ejpam-5661	111	10	)	)	PUNCT
ejpam-5661	111	11	(	(	PUNCT
ejpam-5661	111	12	2025	2025	NUM
ejpam-5661	111	13	)	)	PUNCT
ejpam-5661	111	14	,	,	PUNCT
ejpam-5661	111	15	5661	5661	NUM
ejpam-5661	111	16	5	5	NUM
ejpam-5661	111	17	of	of	ADP
ejpam-5661	111	18	24	24	NUM
ejpam-5661	111	19	theorem	theorem	NOUN
ejpam-5661	111	20	4	4	NUM
ejpam-5661	111	21	.	.	PUNCT
ejpam-5661	112	1	let	let	VERB
ejpam-5661	112	2	φ	φ	NOUN
ejpam-5661	112	3	:	:	PUNCT
ejpam-5661	113	1	[	[	X
ejpam-5661	113	2	α	α	X
ejpam-5661	113	3	,	,	PUNCT
ejpam-5661	113	4	υ	υ	NOUN
ejpam-5661	113	5	]	]	X
ejpam-5661	113	6	→	→	PUNCT
ejpam-5661	113	7	r	r	NOUN
ejpam-5661	113	8	be	be	AUX
ejpam-5661	113	9	a	a	DET
ejpam-5661	113	10	strongly	strongly	ADV
ejpam-5661	113	11	convex	convex	ADJ
ejpam-5661	113	12	function	function	NOUN
ejpam-5661	113	13	for	for	ADP
ejpam-5661	113	14	φ	φ	PROPN
ejpam-5661	113	15	>	>	X
ejpam-5661	113	16	0	0	PROPN
ejpam-5661	113	17	.	.	PUNCT
ejpam-5661	114	1	then	then	ADV
ejpam-5661	114	2	the	the	DET
ejpam-5661	114	3	following	follow	VERB
ejpam-5661	114	4	inequalities	inequality	NOUN
ejpam-5661	114	5	are	be	AUX
ejpam-5661	114	6	established	establish	VERB
ejpam-5661	114	7	:	:	PUNCT
ejpam-5661	114	8	φ	φ	PROPN
ejpam-5661	114	9	(	(	PUNCT
ejpam-5661	114	10	α+υ	α+υ	NUM
ejpam-5661	114	11	2	2	NUM
ejpam-5661	114	12	)	)	PUNCT
ejpam-5661	114	13	≤	≤	NOUN
ejpam-5661	114	14	φ	φ	PROPN
ejpam-5661	114	15	(	(	PUNCT
ejpam-5661	114	16	α+υ	α+υ	NUM
ejpam-5661	114	17	2	2	NUM
ejpam-5661	114	18	)	)	PUNCT
ejpam-5661	115	1	+	+	CCONJ
ejpam-5661	115	2	φ(υ−	φ(υ−	NUM
ejpam-5661	115	3	α)2	α)2	NOUN
ejpam-5661	115	4	4	4	NUM
ejpam-5661	115	5	(	(	PUNCT
ejpam-5661	115	6	4θ2	4θ2	NUM
ejpam-5661	116	1	[	[	X
ejpam-5661	116	2	3]q	3]q	NUM
ejpam-5661	116	3	−	−	NUM
ejpam-5661	116	4	4θ	4θ	NOUN
ejpam-5661	116	5	[	[	X
ejpam-5661	116	6	2]q	2]q	NUM
ejpam-5661	116	7	+	+	CCONJ
ejpam-5661	116	8	1	1	NUM
ejpam-5661	116	9	)	)	PUNCT
ejpam-5661	116	10	≤	≤	NUM
ejpam-5661	116	11	1	1	NUM
ejpam-5661	116	12	2θ(υ−	2θ(υ−	NUM
ejpam-5661	116	13	α	α	NOUN
ejpam-5661	116	14	)	)	PUNCT
ejpam-5661	116	15	(	(	PUNCT
ejpam-5661	116	16	∫	∫	PROPN
ejpam-5661	116	17	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	116	18	α	α	PROPN
ejpam-5661	116	19	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	116	20	+	+	CCONJ
ejpam-5661	116	21	∫	∫	PROPN
ejpam-5661	116	22	b	b	X
ejpam-5661	116	23	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	116	24	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	116	25	)	)	PUNCT
ejpam-5661	116	26	υdqϖ	υdqϖ	NOUN
ejpam-5661	116	27	)	)	PUNCT
ejpam-5661	116	28	≤	≤	NUM
ejpam-5661	116	29	φ(α	φ(α	NOUN
ejpam-5661	116	30	)	)	PUNCT
ejpam-5661	117	1	+	+	CCONJ
ejpam-5661	117	2	φ(υ	φ(υ	PROPN
ejpam-5661	117	3	)	)	PUNCT
ejpam-5661	117	4	2	2	NUM
ejpam-5661	117	5	−	−	NOUN
ejpam-5661	117	6	φ(υ−	φ(υ−	NOUN
ejpam-5661	117	7	α)2	α)2	NOUN
ejpam-5661	117	8	(	(	PUNCT
ejpam-5661	117	9	θ	θ	PROPN
ejpam-5661	118	1	[	[	X
ejpam-5661	118	2	2]q	2]q	NUM
ejpam-5661	118	3	−	−	NOUN
ejpam-5661	118	4	θ2	θ2	PROPN
ejpam-5661	118	5	[	[	X
ejpam-5661	118	6	3]q	3]q	NUM
ejpam-5661	118	7	)	)	PUNCT
ejpam-5661	118	8	≤	≤	NOUN
ejpam-5661	118	9	φ(α	φ(α	NOUN
ejpam-5661	118	10	)	)	PUNCT
ejpam-5661	119	1	+	+	CCONJ
ejpam-5661	119	2	φ(υ	φ(υ	PROPN
ejpam-5661	119	3	)	)	PUNCT
ejpam-5661	119	4	2	2	NUM
ejpam-5661	119	5	(	(	PUNCT
ejpam-5661	119	6	11	11	NUM
ejpam-5661	119	7	)	)	PUNCT
ejpam-5661	119	8	for	for	ADP
ejpam-5661	119	9	all	all	DET
ejpam-5661	119	10	θ	θ	PROPN
ejpam-5661	119	11	∈	∈	PROPN
ejpam-5661	119	12	(	(	PUNCT
ejpam-5661	119	13	0	0	NUM
ejpam-5661	119	14	,	,	PUNCT
ejpam-5661	119	15	1	1	NUM
ejpam-5661	119	16	]	]	PUNCT
ejpam-5661	119	17	.	.	PUNCT
ejpam-5661	120	1	proof	proof	NOUN
ejpam-5661	120	2	.	.	PUNCT
ejpam-5661	121	1	it	it	PRON
ejpam-5661	121	2	follows	follow	VERB
ejpam-5661	121	3	from	from	ADP
ejpam-5661	121	4	strong	strong	ADJ
ejpam-5661	121	5	convexity	convexity	NOUN
ejpam-5661	121	6	of	of	ADP
ejpam-5661	121	7	φ	φ	PROPN
ejpam-5661	121	8	on	on	ADP
ejpam-5661	121	9	[	[	X
ejpam-5661	121	10	α	α	NOUN
ejpam-5661	121	11	,	,	PUNCT
ejpam-5661	121	12	υ	υ	NOUN
ejpam-5661	121	13	]	]	X
ejpam-5661	121	14	that	that	SCONJ
ejpam-5661	121	15	φ(ϖs+	φ(ϖs+	PROPN
ejpam-5661	121	16	(	(	PUNCT
ejpam-5661	121	17	1−ϖ)s1	1−ϖ)s1	NOUN
ejpam-5661	121	18	)	)	PUNCT
ejpam-5661	121	19	≤	≤	NOUN
ejpam-5661	121	20	ϖφ(s	ϖφ(s	NUM
ejpam-5661	121	21	)	)	PUNCT
ejpam-5661	122	1	+	+	CCONJ
ejpam-5661	122	2	(	(	PUNCT
ejpam-5661	122	3	1−ϖ)φ(s1)−	1−ϖ)φ(s1)−	ADJ
ejpam-5661	122	4	φϖ(1−ϖ)(s−	φϖ(1−ϖ)(s−	ADJ
ejpam-5661	122	5	s1	s1	NOUN
ejpam-5661	122	6	)	)	PUNCT
ejpam-5661	122	7	2	2	NUM
ejpam-5661	122	8	,	,	PUNCT
ejpam-5661	122	9	(	(	PUNCT
ejpam-5661	122	10	12	12	NUM
ejpam-5661	122	11	)	)	PUNCT
ejpam-5661	122	12	for	for	ADP
ejpam-5661	122	13	all	all	DET
ejpam-5661	122	14	s	s	NOUN
ejpam-5661	122	15	,	,	PUNCT
ejpam-5661	122	16	s1	s1	PROPN
ejpam-5661	122	17	∈	∈	PROPN
ejpam-5661	122	18	[	[	X
ejpam-5661	122	19	α	α	NOUN
ejpam-5661	122	20	,	,	PUNCT
ejpam-5661	122	21	υ	υ	NOUN
ejpam-5661	122	22	]	]	X
ejpam-5661	122	23	and	and	CCONJ
ejpam-5661	122	24	ϖ	ϖ	PRON
ejpam-5661	122	25	∈	∈	PROPN
ejpam-5661	123	1	[	[	X
ejpam-5661	123	2	0	0	NUM
ejpam-5661	123	3	,	,	PUNCT
ejpam-5661	123	4	1	1	NUM
ejpam-5661	123	5	]	]	PUNCT
ejpam-5661	123	6	.	.	PUNCT
ejpam-5661	124	1	substituting	substitute	VERB
ejpam-5661	124	2	ϖ	ϖ	X
ejpam-5661	124	3	=	=	SYM
ejpam-5661	124	4	1/2	1/2	NUM
ejpam-5661	124	5	into	into	ADP
ejpam-5661	124	6	the	the	DET
ejpam-5661	124	7	inequality	inequality	NOUN
ejpam-5661	124	8	(	(	PUNCT
ejpam-5661	124	9	12	12	NUM
ejpam-5661	124	10	)	)	PUNCT
ejpam-5661	124	11	,	,	PUNCT
ejpam-5661	124	12	then	then	ADV
ejpam-5661	124	13	φ	φ	PROPN
ejpam-5661	124	14	(	(	PUNCT
ejpam-5661	124	15	s+	s+	ADV
ejpam-5661	124	16	s1	s1	PROPN
ejpam-5661	124	17	2	2	NUM
ejpam-5661	124	18	)	)	PUNCT
ejpam-5661	124	19	≤	≤	NOUN
ejpam-5661	124	20	φ(s	φ(s	NOUN
ejpam-5661	124	21	)	)	PUNCT
ejpam-5661	124	22	+	+	CCONJ
ejpam-5661	124	23	φ(s1	φ(s1	NOUN
ejpam-5661	124	24	)	)	PUNCT
ejpam-5661	124	25	2	2	NUM
ejpam-5661	124	26	−	−	PROPN
ejpam-5661	124	27	φ(s−	φ(s−	ADJ
ejpam-5661	124	28	s1	s1	NOUN
ejpam-5661	124	29	)	)	PUNCT
ejpam-5661	124	30	2	2	NUM
ejpam-5661	124	31	4	4	NUM
ejpam-5661	124	32	.	.	PUNCT
ejpam-5661	125	1	(	(	PUNCT
ejpam-5661	125	2	13	13	NUM
ejpam-5661	125	3	)	)	PUNCT
ejpam-5661	125	4	let	let	VERB
ejpam-5661	125	5	θ	θ	PROPN
ejpam-5661	125	6	∈	∈	PROPN
ejpam-5661	125	7	(	(	PUNCT
ejpam-5661	125	8	0	0	NUM
ejpam-5661	125	9	,	,	PUNCT
ejpam-5661	125	10	1	1	NUM
ejpam-5661	125	11	]	]	PUNCT
ejpam-5661	125	12	and	and	CCONJ
ejpam-5661	125	13	putting	put	VERB
ejpam-5661	125	14	s	s	PART
ejpam-5661	125	15	=	=	NOUN
ejpam-5661	125	16	θϖυ	θϖυ	NOUN
ejpam-5661	125	17	+	+	CCONJ
ejpam-5661	125	18	(	(	PUNCT
ejpam-5661	125	19	1	1	NUM
ejpam-5661	125	20	−	−	PROPN
ejpam-5661	125	21	θϖ)α	θϖ)α	PROPN
ejpam-5661	125	22	and	and	CCONJ
ejpam-5661	125	23	s1	s1	PROPN
ejpam-5661	125	24	=	=	PUNCT
ejpam-5661	125	25	θϖα	θϖα	NOUN
ejpam-5661	125	26	+	+	CCONJ
ejpam-5661	125	27	(	(	PUNCT
ejpam-5661	125	28	1	1	NUM
ejpam-5661	125	29	−	−	PROPN
ejpam-5661	125	30	θϖ)υ	θϖ)υ	PROPN
ejpam-5661	125	31	in	in	ADP
ejpam-5661	125	32	the	the	DET
ejpam-5661	125	33	inequality	inequality	NOUN
ejpam-5661	125	34	(	(	PUNCT
ejpam-5661	125	35	13	13	NUM
ejpam-5661	125	36	)	)	PUNCT
ejpam-5661	125	37	,	,	PUNCT
ejpam-5661	125	38	we	we	PRON
ejpam-5661	125	39	have	have	VERB
ejpam-5661	125	40	φ	φ	PROPN
ejpam-5661	125	41	(	(	PUNCT
ejpam-5661	125	42	α+υ	α+υ	NUM
ejpam-5661	125	43	2	2	NUM
ejpam-5661	125	44	)	)	PUNCT
ejpam-5661	125	45	≤	≤	NOUN
ejpam-5661	125	46	φ(θϖυ+	φ(θϖυ+	X
ejpam-5661	125	47	(	(	PUNCT
ejpam-5661	125	48	1−θϖ)α	1−θϖ)α	NOUN
ejpam-5661	125	49	)	)	PUNCT
ejpam-5661	125	50	+	+	NUM
ejpam-5661	126	1	φ(θϖα+	φ(θϖα+	X
ejpam-5661	126	2	(	(	PUNCT
ejpam-5661	126	3	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	126	4	)	)	PUNCT
ejpam-5661	126	5	2	2	NUM
ejpam-5661	126	6	−	−	NOUN
ejpam-5661	126	7	φ(υ−	φ(υ−	NOUN
ejpam-5661	126	8	α)2	α)2	NOUN
ejpam-5661	126	9	(	(	PUNCT
ejpam-5661	126	10	2θϖ	2θϖ	ADJ
ejpam-5661	126	11	−	−	PROPN
ejpam-5661	126	12	1)2	1)2	NUM
ejpam-5661	126	13	4	4	NUM
ejpam-5661	126	14	.	.	PUNCT
ejpam-5661	127	1	(	(	PUNCT
ejpam-5661	127	2	14	14	NUM
ejpam-5661	127	3	)	)	PUNCT
ejpam-5661	127	4	it	it	PRON
ejpam-5661	127	5	follows	follow	VERB
ejpam-5661	127	6	from	from	ADP
ejpam-5661	127	7	the	the	DET
ejpam-5661	127	8	definitions	definition	NOUN
ejpam-5661	127	9	4	4	NUM
ejpam-5661	127	10	and	and	CCONJ
ejpam-5661	127	11	5	5	NUM
ejpam-5661	127	12	,	,	PUNCT
ejpam-5661	127	13	and	and	CCONJ
ejpam-5661	127	14	by	by	ADP
ejpam-5661	127	15	applying	apply	VERB
ejpam-5661	127	16	q	q	NOUN
ejpam-5661	127	17	-	-	PUNCT
ejpam-5661	127	18	integration	integration	NOUN
ejpam-5661	127	19	to	to	ADP
ejpam-5661	127	20	both	both	DET
ejpam-5661	127	21	sides	side	NOUN
ejpam-5661	127	22	of	of	ADP
ejpam-5661	127	23	the	the	DET
ejpam-5661	127	24	inequality	inequality	NOUN
ejpam-5661	127	25	(	(	PUNCT
ejpam-5661	127	26	14	14	NUM
ejpam-5661	127	27	)	)	PUNCT
ejpam-5661	127	28	,	,	PUNCT
ejpam-5661	127	29	we	we	PRON
ejpam-5661	127	30	find	find	VERB
ejpam-5661	127	31	that	that	SCONJ
ejpam-5661	127	32	φ	φ	PROPN
ejpam-5661	127	33	(	(	PUNCT
ejpam-5661	127	34	α+υ	α+υ	NUM
ejpam-5661	127	35	2	2	NUM
ejpam-5661	127	36	)	)	PUNCT
ejpam-5661	127	37	≤	≤	NUM
ejpam-5661	127	38	∫	∫	PROPN
ejpam-5661	127	39	1	1	NUM
ejpam-5661	127	40	0	0	NUM
ejpam-5661	127	41	(	(	PUNCT
ejpam-5661	127	42	φ(θϖυ+	φ(θϖυ+	X
ejpam-5661	127	43	(	(	PUNCT
ejpam-5661	127	44	1−θϖ)α	1−θϖ)α	NUM
ejpam-5661	127	45	)	)	PUNCT
ejpam-5661	127	46	+	+	NUM
ejpam-5661	127	47	φ(θϖα+	φ(θϖα+	X
ejpam-5661	127	48	(	(	PUNCT
ejpam-5661	127	49	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	127	50	)	)	PUNCT
ejpam-5661	127	51	2	2	NUM
ejpam-5661	127	52	−	−	NOUN
ejpam-5661	127	53	φ(υ−	φ(υ−	NOUN
ejpam-5661	127	54	α)2	α)2	NOUN
ejpam-5661	127	55	(	(	PUNCT
ejpam-5661	127	56	2θϖ	2θϖ	ADJ
ejpam-5661	127	57	−	−	PROPN
ejpam-5661	127	58	1)2	1)2	NUM
ejpam-5661	127	59	4	4	NUM
ejpam-5661	127	60	)	)	PUNCT
ejpam-5661	127	61	dqϖ	dqϖ	NOUN
ejpam-5661	127	62	=	=	PUNCT
ejpam-5661	127	63	∫	∫	PROPN
ejpam-5661	127	64	1	1	NUM
ejpam-5661	127	65	0	0	NUM
ejpam-5661	127	66	φ(θϖυ+	φ(θϖυ+	X
ejpam-5661	127	67	(	(	PUNCT
ejpam-5661	127	68	1−θϖ)α	1−θϖ)α	NUM
ejpam-5661	127	69	)	)	PUNCT
ejpam-5661	127	70	+	+	NUM
ejpam-5661	127	71	φ(θϖα+	φ(θϖα+	X
ejpam-5661	127	72	(	(	PUNCT
ejpam-5661	127	73	1−θϖ)υ	1−θϖ)υ	NOUN
ejpam-5661	127	74	)	)	PUNCT
ejpam-5661	127	75	2	2	NUM
ejpam-5661	127	76	dqϖ	dqϖ	NOUN
ejpam-5661	127	77	−	−	PROPN
ejpam-5661	127	78	∫	∫	PROPN
ejpam-5661	127	79	1	1	NUM
ejpam-5661	127	80	0	0	NUM
ejpam-5661	127	81	φ(υ−	φ(υ−	NOUN
ejpam-5661	127	82	α)2	α)2	NOUN
ejpam-5661	127	83	(	(	PUNCT
ejpam-5661	127	84	4θ2ϖ2	4θ2ϖ2	NOUN
ejpam-5661	127	85	−	−	PROPN
ejpam-5661	127	86	4θϖ	4θϖ	PROPN
ejpam-5661	127	87	+	+	CCONJ
ejpam-5661	127	88	1	1	X
ejpam-5661	127	89	)	)	SYM
ejpam-5661	127	90	4	4	NUM
ejpam-5661	127	91	dqϖ	dqϖ	NOUN
ejpam-5661	127	92	=	=	SYM
ejpam-5661	127	93	(	(	PUNCT
ejpam-5661	127	94	1−	1−	NUM
ejpam-5661	127	95	q	q	NOUN
ejpam-5661	127	96	)	)	PUNCT
ejpam-5661	127	97	∞∑	∞∑	PROPN
ejpam-5661	127	98	n=0	n=0	PUNCT
ejpam-5661	127	99	qn	qn	NOUN
ejpam-5661	127	100	φ(θqnυ+	φ(θqnυ+	PROPN
ejpam-5661	127	101	(	(	PUNCT
ejpam-5661	127	102	1−θqn)α	1−θqn)α	NUM
ejpam-5661	127	103	)	)	PUNCT
ejpam-5661	127	104	+	+	CCONJ
ejpam-5661	127	105	φ(θqnα+	φ(θqnα+	NUM
ejpam-5661	127	106	(	(	PUNCT
ejpam-5661	127	107	1−θqn)b	1−θqn)b	NUM
ejpam-5661	127	108	)	)	PUNCT
ejpam-5661	127	109	2	2	NUM
ejpam-5661	127	110	c.	c.	PROPN
ejpam-5661	127	111	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	127	112	,	,	PUNCT
ejpam-5661	127	113	p.	p.	PROPN
ejpam-5661	127	114	yotkaew	yotkaew	PROPN
ejpam-5661	127	115	/	/	SYM
ejpam-5661	127	116	eur	eur	PROPN
ejpam-5661	127	117	.	.	PUNCT
ejpam-5661	128	1	j.	j.	PROPN
ejpam-5661	128	2	pure	pure	PROPN
ejpam-5661	128	3	appl	appl	PROPN
ejpam-5661	128	4	.	.	PROPN
ejpam-5661	128	5	math	math	PROPN
ejpam-5661	128	6	,	,	PUNCT
ejpam-5661	128	7	18	18	NUM
ejpam-5661	128	8	(	(	PUNCT
ejpam-5661	128	9	1	1	NUM
ejpam-5661	128	10	)	)	PUNCT
ejpam-5661	128	11	(	(	PUNCT
ejpam-5661	128	12	2025	2025	NUM
ejpam-5661	128	13	)	)	PUNCT
ejpam-5661	128	14	,	,	PUNCT
ejpam-5661	128	15	5661	5661	NUM
ejpam-5661	128	16	6	6	NUM
ejpam-5661	128	17	of	of	ADP
ejpam-5661	128	18	24	24	NUM
ejpam-5661	128	19	−	−	NOUN
ejpam-5661	128	20	φ(υ−	φ(υ−	NOUN
ejpam-5661	128	21	α)2	α)2	NOUN
ejpam-5661	128	22	4	4	NUM
ejpam-5661	128	23	(	(	PUNCT
ejpam-5661	128	24	4θ2	4θ2	NUM
ejpam-5661	129	1	[	[	X
ejpam-5661	129	2	3]q	3]q	NUM
ejpam-5661	129	3	−	−	NUM
ejpam-5661	129	4	4θ	4θ	NOUN
ejpam-5661	129	5	[	[	X
ejpam-5661	129	6	2]q	2]q	NUM
ejpam-5661	129	7	+	+	CCONJ
ejpam-5661	129	8	1	1	NUM
ejpam-5661	129	9	)	)	PUNCT
ejpam-5661	129	10	=	=	SYM
ejpam-5661	129	11	θ(υ−	θ(υ−	PROPN
ejpam-5661	129	12	α)(1−	α)(1−	PROPN
ejpam-5661	129	13	q	q	PROPN
ejpam-5661	129	14	)	)	PUNCT
ejpam-5661	129	15	2θ(υ−	2θ(υ−	NUM
ejpam-5661	129	16	α	α	NOUN
ejpam-5661	129	17	)	)	PUNCT
ejpam-5661	129	18	(	(	PUNCT
ejpam-5661	129	19	∞∑	∞∑	PROPN
ejpam-5661	129	20	n=0	n=0	NUM
ejpam-5661	129	21	qnφ	qnφ	NOUN
ejpam-5661	129	22	(	(	PUNCT
ejpam-5661	129	23	(	(	PUNCT
ejpam-5661	129	24	θυ	θυ	PROPN
ejpam-5661	129	25	+	+	CCONJ
ejpam-5661	129	26	(	(	PUNCT
ejpam-5661	129	27	1−θ)α)qn	1−θ)α)qn	NUM
ejpam-5661	129	28	+	+	CCONJ
ejpam-5661	129	29	(	(	PUNCT
ejpam-5661	129	30	1−	1−	NUM
ejpam-5661	129	31	qn)α	qn)α	ADV
ejpam-5661	129	32	)	)	PUNCT
ejpam-5661	130	1	+	+	CCONJ
ejpam-5661	130	2	∞∑	∞∑	NUM
ejpam-5661	130	3	n=0	n=0	NUM
ejpam-5661	130	4	qnφ	qnφ	NOUN
ejpam-5661	130	5	(	(	PUNCT
ejpam-5661	130	6	(	(	PUNCT
ejpam-5661	130	7	θα+	θα+	NOUN
ejpam-5661	130	8	(	(	PUNCT
ejpam-5661	130	9	1−θ)υ)qn	1−θ)υ)qn	NUM
ejpam-5661	130	10	+	+	CCONJ
ejpam-5661	130	11	(	(	PUNCT
ejpam-5661	130	12	1−	1−	NUM
ejpam-5661	130	13	qn)υ	qn)υ	ADJ
ejpam-5661	130	14	)	)	PUNCT
ejpam-5661	130	15	)	)	PUNCT
ejpam-5661	130	16	−	−	PUNCT
ejpam-5661	131	1	φ(υ−	φ(υ−	X
ejpam-5661	131	2	α)2	α)2	NOUN
ejpam-5661	131	3	4	4	NUM
ejpam-5661	131	4	(	(	PUNCT
ejpam-5661	131	5	4θ2	4θ2	NUM
ejpam-5661	132	1	[	[	X
ejpam-5661	132	2	3]q	3]q	NUM
ejpam-5661	132	3	−	−	NUM
ejpam-5661	132	4	4θ	4θ	NOUN
ejpam-5661	132	5	[	[	X
ejpam-5661	132	6	2]q	2]q	NUM
ejpam-5661	132	7	+	+	CCONJ
ejpam-5661	132	8	1	1	NUM
ejpam-5661	132	9	)	)	PUNCT
ejpam-5661	132	10	=	=	SYM
ejpam-5661	132	11	1	1	NUM
ejpam-5661	132	12	2θ(υ−	2θ(υ−	NUM
ejpam-5661	132	13	α	α	NOUN
ejpam-5661	132	14	)	)	PUNCT
ejpam-5661	132	15	(	(	PUNCT
ejpam-5661	132	16	∫	∫	PROPN
ejpam-5661	132	17	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	132	18	α	α	PROPN
ejpam-5661	132	19	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	132	20	+	+	CCONJ
ejpam-5661	132	21	∫	∫	PROPN
ejpam-5661	132	22	υ	υ	PROPN
ejpam-5661	132	23	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	132	24	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	132	25	)	)	PUNCT
ejpam-5661	132	26	υdqϖ	υdqϖ	NOUN
ejpam-5661	132	27	)	)	PUNCT
ejpam-5661	132	28	−	−	PUNCT
ejpam-5661	133	1	φ(υ−	φ(υ−	X
ejpam-5661	133	2	α)2	α)2	NOUN
ejpam-5661	133	3	4	4	NUM
ejpam-5661	133	4	(	(	PUNCT
ejpam-5661	133	5	4θ2	4θ2	NUM
ejpam-5661	134	1	[	[	X
ejpam-5661	134	2	3]q	3]q	NUM
ejpam-5661	134	3	−	−	NUM
ejpam-5661	134	4	4θ	4θ	NOUN
ejpam-5661	134	5	[	[	X
ejpam-5661	134	6	2]q	2]q	NUM
ejpam-5661	134	7	+	+	CCONJ
ejpam-5661	134	8	1	1	NUM
ejpam-5661	134	9	)	)	PUNCT
ejpam-5661	134	10	.	.	PUNCT
ejpam-5661	135	1	(	(	PUNCT
ejpam-5661	135	2	15	15	X
ejpam-5661	135	3	)	)	PUNCT
ejpam-5661	135	4	note	note	NOUN
ejpam-5661	135	5	that	that	SCONJ
ejpam-5661	135	6	4θ2/[3]q	4θ2/[3]q	NUM
ejpam-5661	135	7	−	−	NOUN
ejpam-5661	135	8	4θ/[2]q	4θ/[2]q	NOUN
ejpam-5661	136	1	+	+	CCONJ
ejpam-5661	136	2	1	1	NUM
ejpam-5661	136	3	≥	≥	NOUN
ejpam-5661	136	4	0	0	NUM
ejpam-5661	136	5	.	.	PUNCT
ejpam-5661	137	1	this	this	PRON
ejpam-5661	137	2	implies	imply	VERB
ejpam-5661	137	3	that	that	SCONJ
ejpam-5661	137	4	φ	φ	PROPN
ejpam-5661	137	5	(	(	PUNCT
ejpam-5661	137	6	α+υ	α+υ	NUM
ejpam-5661	137	7	2	2	NUM
ejpam-5661	137	8	)	)	PUNCT
ejpam-5661	137	9	≤	≤	NOUN
ejpam-5661	137	10	φ	φ	PROPN
ejpam-5661	137	11	(	(	PUNCT
ejpam-5661	137	12	α+υ	α+υ	NUM
ejpam-5661	137	13	2	2	NUM
ejpam-5661	137	14	)	)	PUNCT
ejpam-5661	138	1	+	+	CCONJ
ejpam-5661	138	2	φ(υ−	φ(υ−	NUM
ejpam-5661	138	3	α)2	α)2	NOUN
ejpam-5661	138	4	4	4	NUM
ejpam-5661	138	5	(	(	PUNCT
ejpam-5661	138	6	4θ2	4θ2	NUM
ejpam-5661	139	1	[	[	X
ejpam-5661	139	2	3]q	3]q	NUM
ejpam-5661	139	3	−	−	NUM
ejpam-5661	139	4	4θ	4θ	NOUN
ejpam-5661	139	5	[	[	X
ejpam-5661	139	6	2]q	2]q	NUM
ejpam-5661	139	7	+	+	CCONJ
ejpam-5661	139	8	1	1	NUM
ejpam-5661	139	9	)	)	PUNCT
ejpam-5661	139	10	≤	≤	NUM
ejpam-5661	139	11	1	1	NUM
ejpam-5661	139	12	2θ(υ−	2θ(υ−	NUM
ejpam-5661	139	13	α	α	NOUN
ejpam-5661	139	14	)	)	PUNCT
ejpam-5661	139	15	(	(	PUNCT
ejpam-5661	139	16	∫	∫	PROPN
ejpam-5661	139	17	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	139	18	α	α	PROPN
ejpam-5661	139	19	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	139	20	+	+	CCONJ
ejpam-5661	139	21	∫	∫	PROPN
ejpam-5661	139	22	υ	υ	PROPN
ejpam-5661	139	23	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	139	24	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	139	25	)	)	PUNCT
ejpam-5661	139	26	υdqϖ	υdqϖ	NOUN
ejpam-5661	139	27	)	)	PUNCT
ejpam-5661	139	28	.	.	PUNCT
ejpam-5661	140	1	this	this	PRON
ejpam-5661	140	2	gives	give	VERB
ejpam-5661	140	3	the	the	DET
ejpam-5661	140	4	first	first	ADJ
ejpam-5661	140	5	and	and	CCONJ
ejpam-5661	140	6	second	second	ADJ
ejpam-5661	140	7	inequalities	inequality	NOUN
ejpam-5661	140	8	of	of	ADP
ejpam-5661	140	9	(	(	PUNCT
ejpam-5661	140	10	11	11	NUM
ejpam-5661	140	11	)	)	PUNCT
ejpam-5661	140	12	.	.	PUNCT
ejpam-5661	141	1	from	from	ADP
ejpam-5661	141	2	the	the	DET
ejpam-5661	141	3	inequality	inequality	NOUN
ejpam-5661	141	4	(	(	PUNCT
ejpam-5661	141	5	14	14	NUM
ejpam-5661	141	6	)	)	PUNCT
ejpam-5661	141	7	and	and	CCONJ
ejpam-5661	141	8	by	by	ADP
ejpam-5661	141	9	strong	strong	ADJ
ejpam-5661	141	10	convexity	convexity	NOUN
ejpam-5661	141	11	of	of	ADP
ejpam-5661	141	12	φ	φ	PROPN
ejpam-5661	141	13	,	,	PUNCT
ejpam-5661	141	14	we	we	PRON
ejpam-5661	141	15	obtain	obtain	VERB
ejpam-5661	141	16	φ(θϖυ+	φ(θϖυ+	X
ejpam-5661	141	17	(	(	PUNCT
ejpam-5661	141	18	1−θϖ)α	1−θϖ)α	NOUN
ejpam-5661	141	19	)	)	PUNCT
ejpam-5661	141	20	+	+	NUM
ejpam-5661	142	1	φ(θϖα+	φ(θϖα+	X
ejpam-5661	142	2	(	(	PUNCT
ejpam-5661	142	3	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	142	4	)	)	PUNCT
ejpam-5661	142	5	2	2	NUM
ejpam-5661	142	6	−	−	NOUN
ejpam-5661	142	7	φ(υ−	φ(υ−	NOUN
ejpam-5661	142	8	α)2	α)2	NOUN
ejpam-5661	142	9	(	(	PUNCT
ejpam-5661	142	10	2θϖ	2θϖ	ADJ
ejpam-5661	142	11	−	−	PROPN
ejpam-5661	142	12	1)2	1)2	NUM
ejpam-5661	142	13	4	4	NUM
ejpam-5661	142	14	≤	≤	NUM
ejpam-5661	142	15	θϖφ(α	θϖφ(α	PROPN
ejpam-5661	142	16	)	)	PUNCT
ejpam-5661	143	1	+	+	CCONJ
ejpam-5661	143	2	(	(	PUNCT
ejpam-5661	143	3	1−θϖ)φ(υ)−	1−θϖ)φ(υ)−	NUM
ejpam-5661	143	4	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	143	5	α)2	α)2	PROPN
ejpam-5661	143	6	2	2	NUM
ejpam-5661	143	7	+	+	CCONJ
ejpam-5661	143	8	θϖφ(υ	θϖφ(υ	NUM
ejpam-5661	143	9	)	)	PUNCT
ejpam-5661	144	1	+	+	CCONJ
ejpam-5661	144	2	(	(	PUNCT
ejpam-5661	144	3	1−θϖ)φ(α)−	1−θϖ)φ(α)−	NUM
ejpam-5661	144	4	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	144	5	α)2	α)2	PROPN
ejpam-5661	144	6	2	2	NUM
ejpam-5661	144	7	−	−	NOUN
ejpam-5661	144	8	φ(υ−	φ(υ−	NOUN
ejpam-5661	144	9	α)2	α)2	NOUN
ejpam-5661	144	10	(	(	PUNCT
ejpam-5661	144	11	2θϖ	2θϖ	ADJ
ejpam-5661	144	12	−	−	NOUN
ejpam-5661	144	13	1)2	1)2	NUM
ejpam-5661	144	14	4	4	NUM
ejpam-5661	144	15	=	=	SYM
ejpam-5661	144	16	φ(α	φ(α	PROPN
ejpam-5661	144	17	)	)	PUNCT
ejpam-5661	145	1	+	+	CCONJ
ejpam-5661	145	2	φ(υ	φ(υ	PROPN
ejpam-5661	145	3	)	)	PUNCT
ejpam-5661	145	4	2	2	NUM
ejpam-5661	145	5	−	−	NOUN
ejpam-5661	145	6	φ(υ−	φ(υ−	NOUN
ejpam-5661	145	7	α)2	α)2	NOUN
ejpam-5661	145	8	4	4	NUM
ejpam-5661	145	9	.	.	PUNCT
ejpam-5661	146	1	applying	apply	VERB
ejpam-5661	146	2	q	q	NOUN
ejpam-5661	146	3	-	-	PUNCT
ejpam-5661	146	4	integration	integration	NOUN
ejpam-5661	146	5	to	to	ADP
ejpam-5661	146	6	both	both	DET
ejpam-5661	146	7	sides	side	NOUN
ejpam-5661	146	8	of	of	ADP
ejpam-5661	146	9	the	the	DET
ejpam-5661	146	10	above	above	ADJ
ejpam-5661	146	11	inequality	inequality	NOUN
ejpam-5661	146	12	yields∫	yields∫	VERB
ejpam-5661	146	13	1	1	NUM
ejpam-5661	146	14	0	0	NUM
ejpam-5661	146	15	(	(	PUNCT
ejpam-5661	146	16	φ(θϖυ+	φ(θϖυ+	X
ejpam-5661	146	17	(	(	PUNCT
ejpam-5661	146	18	1−θϖ)α	1−θϖ)α	NUM
ejpam-5661	146	19	)	)	PUNCT
ejpam-5661	146	20	+	+	NUM
ejpam-5661	147	1	φ(θϖα+	φ(θϖα+	X
ejpam-5661	147	2	(	(	PUNCT
ejpam-5661	147	3	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	147	4	)	)	PUNCT
ejpam-5661	147	5	2	2	NUM
ejpam-5661	147	6	−	−	NOUN
ejpam-5661	147	7	φ(υ−	φ(υ−	NOUN
ejpam-5661	147	8	α)2	α)2	NOUN
ejpam-5661	147	9	(	(	PUNCT
ejpam-5661	147	10	2θϖ	2θϖ	ADJ
ejpam-5661	147	11	−	−	PROPN
ejpam-5661	147	12	1)2	1)2	NUM
ejpam-5661	147	13	4	4	NUM
ejpam-5661	147	14	)	)	PUNCT
ejpam-5661	147	15	dqϖ	dqϖ	VERB
ejpam-5661	147	16	≤	≤	NUM
ejpam-5661	147	17	∫	∫	PROPN
ejpam-5661	147	18	1	1	NUM
ejpam-5661	147	19	0	0	NUM
ejpam-5661	147	20	(	(	PUNCT
ejpam-5661	147	21	φ(α	φ(α	PROPN
ejpam-5661	147	22	)	)	PUNCT
ejpam-5661	147	23	+	+	CCONJ
ejpam-5661	148	1	φ(υ	φ(υ	PROPN
ejpam-5661	148	2	)	)	PUNCT
ejpam-5661	148	3	2	2	NUM
ejpam-5661	148	4	−	−	NOUN
ejpam-5661	148	5	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	148	6	α)2	α)2	NOUN
ejpam-5661	148	7	−	−	NOUN
ejpam-5661	148	8	φ(υ−	φ(υ−	NOUN
ejpam-5661	148	9	α)2	α)2	NOUN
ejpam-5661	148	10	(	(	PUNCT
ejpam-5661	148	11	2θϖ	2θϖ	ADJ
ejpam-5661	148	12	−	−	PROPN
ejpam-5661	148	13	1)2	1)2	NUM
ejpam-5661	148	14	4	4	NUM
ejpam-5661	148	15	)	)	PUNCT
ejpam-5661	148	16	dqϖ	dqϖ	NOUN
ejpam-5661	148	17	=	=	SYM
ejpam-5661	148	18	φ(α	φ(α	PROPN
ejpam-5661	148	19	)	)	PUNCT
ejpam-5661	148	20	+	+	CCONJ
ejpam-5661	148	21	φ(υ	φ(υ	PROPN
ejpam-5661	148	22	)	)	PUNCT
ejpam-5661	148	23	2	2	NUM
ejpam-5661	148	24	−	−	NOUN
ejpam-5661	148	25	φ(υ−	φ(υ−	NOUN
ejpam-5661	148	26	α)2	α)2	NOUN
ejpam-5661	148	27	4	4	NUM
ejpam-5661	148	28	.	.	PUNCT
ejpam-5661	149	1	(	(	PUNCT
ejpam-5661	149	2	16	16	NUM
ejpam-5661	149	3	)	)	PUNCT
ejpam-5661	149	4	c.	c.	PROPN
ejpam-5661	149	5	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	149	6	,	,	PUNCT
ejpam-5661	149	7	p.	p.	PROPN
ejpam-5661	149	8	yotkaew	yotkaew	PROPN
ejpam-5661	149	9	/	/	SYM
ejpam-5661	149	10	eur	eur	PROPN
ejpam-5661	149	11	.	.	PUNCT
ejpam-5661	150	1	j.	j.	PROPN
ejpam-5661	150	2	pure	pure	PROPN
ejpam-5661	150	3	appl	appl	PROPN
ejpam-5661	150	4	.	.	PROPN
ejpam-5661	150	5	math	math	PROPN
ejpam-5661	150	6	,	,	PUNCT
ejpam-5661	150	7	18	18	NUM
ejpam-5661	150	8	(	(	PUNCT
ejpam-5661	150	9	1	1	NUM
ejpam-5661	150	10	)	)	PUNCT
ejpam-5661	150	11	(	(	PUNCT
ejpam-5661	150	12	2025	2025	NUM
ejpam-5661	150	13	)	)	PUNCT
ejpam-5661	150	14	,	,	PUNCT
ejpam-5661	150	15	5661	5661	NUM
ejpam-5661	150	16	7	7	NUM
ejpam-5661	150	17	of	of	ADP
ejpam-5661	150	18	24	24	NUM
ejpam-5661	150	19	from	from	ADP
ejpam-5661	150	20	the	the	DET
ejpam-5661	150	21	inequalities	inequality	NOUN
ejpam-5661	150	22	(	(	PUNCT
ejpam-5661	150	23	15	15	NUM
ejpam-5661	150	24	)	)	PUNCT
ejpam-5661	150	25	and	and	CCONJ
ejpam-5661	150	26	(	(	PUNCT
ejpam-5661	150	27	16	16	NUM
ejpam-5661	150	28	)	)	PUNCT
ejpam-5661	151	1	,	,	PUNCT
ejpam-5661	151	2	we	we	PRON
ejpam-5661	151	3	obtain∫	obtain∫	VERB
ejpam-5661	151	4	1	1	NUM
ejpam-5661	151	5	0	0	NUM
ejpam-5661	151	6	(	(	PUNCT
ejpam-5661	151	7	φ(θϖυ+	φ(θϖυ+	X
ejpam-5661	151	8	(	(	PUNCT
ejpam-5661	151	9	1−θϖ)α	1−θϖ)α	NUM
ejpam-5661	151	10	)	)	PUNCT
ejpam-5661	151	11	+	+	NUM
ejpam-5661	152	1	φ(θϖα+	φ(θϖα+	X
ejpam-5661	152	2	(	(	PUNCT
ejpam-5661	152	3	1−θϖ)b	1−θϖ)b	NUM
ejpam-5661	152	4	)	)	PUNCT
ejpam-5661	152	5	2	2	NUM
ejpam-5661	152	6	−	−	NOUN
ejpam-5661	152	7	φ(υ−	φ(υ−	NOUN
ejpam-5661	152	8	α)2	α)2	NOUN
ejpam-5661	152	9	(	(	PUNCT
ejpam-5661	152	10	2θϖ	2θϖ	ADJ
ejpam-5661	152	11	−	−	PROPN
ejpam-5661	152	12	1)2	1)2	NUM
ejpam-5661	152	13	4	4	NUM
ejpam-5661	152	14	)	)	PUNCT
ejpam-5661	152	15	dqϖ	dqϖ	NOUN
ejpam-5661	152	16	=	=	SYM
ejpam-5661	152	17	1	1	NUM
ejpam-5661	152	18	2θ(υ−	2θ(υ−	NUM
ejpam-5661	152	19	α	α	NOUN
ejpam-5661	152	20	)	)	PUNCT
ejpam-5661	152	21	(	(	PUNCT
ejpam-5661	152	22	∫	∫	PROPN
ejpam-5661	152	23	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	152	24	a	a	DET
ejpam-5661	152	25	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	152	26	+	+	NUM
ejpam-5661	152	27	∫	∫	PROPN
ejpam-5661	152	28	υ	υ	PROPN
ejpam-5661	152	29	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	152	30	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	152	31	)	)	PUNCT
ejpam-5661	152	32	υdqϖ	υdqϖ	NOUN
ejpam-5661	152	33	)	)	PUNCT
ejpam-5661	152	34	−	−	PUNCT
ejpam-5661	152	35	φ(υ−	φ(υ−	X
ejpam-5661	152	36	α)2	α)2	NOUN
ejpam-5661	152	37	4	4	NUM
ejpam-5661	152	38	(	(	PUNCT
ejpam-5661	152	39	4θ2	4θ2	NUM
ejpam-5661	152	40	[	[	X
ejpam-5661	152	41	3]q	3]q	NUM
ejpam-5661	152	42	−	−	NUM
ejpam-5661	152	43	4θ	4θ	NOUN
ejpam-5661	152	44	[	[	X
ejpam-5661	152	45	2]q	2]q	NUM
ejpam-5661	152	46	+	+	CCONJ
ejpam-5661	152	47	1	1	NUM
ejpam-5661	152	48	)	)	PUNCT
ejpam-5661	152	49	≤	≤	NOUN
ejpam-5661	152	50	φ(α	φ(α	NOUN
ejpam-5661	152	51	)	)	PUNCT
ejpam-5661	153	1	+	+	CCONJ
ejpam-5661	153	2	φ(υ	φ(υ	PROPN
ejpam-5661	153	3	)	)	PUNCT
ejpam-5661	153	4	2	2	NUM
ejpam-5661	153	5	−	−	NOUN
ejpam-5661	153	6	φ(υ−	φ(υ−	NOUN
ejpam-5661	153	7	α)2	α)2	NOUN
ejpam-5661	153	8	4	4	NUM
ejpam-5661	153	9	.	.	PUNCT
ejpam-5661	154	1	note	note	VERB
ejpam-5661	154	2	that	that	SCONJ
ejpam-5661	154	3	θ/[2]q	θ/[2]q	PROPN
ejpam-5661	154	4	−θ2/[3]q	−θ2/[3]q	PROPN
ejpam-5661	154	5	≥	≥	NUM
ejpam-5661	154	6	0	0	NUM
ejpam-5661	154	7	.	.	PUNCT
ejpam-5661	155	1	this	this	PRON
ejpam-5661	155	2	implies	imply	VERB
ejpam-5661	155	3	that	that	SCONJ
ejpam-5661	155	4	1	1	NUM
ejpam-5661	155	5	2θ(υ−	2θ(υ−	NUM
ejpam-5661	155	6	α	α	NOUN
ejpam-5661	155	7	)	)	PUNCT
ejpam-5661	155	8	(	(	PUNCT
ejpam-5661	155	9	∫	∫	PROPN
ejpam-5661	155	10	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	155	11	α	α	PROPN
ejpam-5661	155	12	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	155	13	+	+	CCONJ
ejpam-5661	155	14	∫	∫	PROPN
ejpam-5661	155	15	υ	υ	PROPN
ejpam-5661	155	16	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	155	17	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	155	18	)	)	PUNCT
ejpam-5661	155	19	υdqϖ	υdqϖ	NOUN
ejpam-5661	155	20	)	)	PUNCT
ejpam-5661	155	21	≤	≤	NUM
ejpam-5661	155	22	φ(α	φ(α	NOUN
ejpam-5661	155	23	)	)	PUNCT
ejpam-5661	156	1	+	+	CCONJ
ejpam-5661	156	2	φ(υ	φ(υ	PROPN
ejpam-5661	156	3	)	)	PUNCT
ejpam-5661	156	4	2	2	NUM
ejpam-5661	156	5	−	−	NOUN
ejpam-5661	156	6	φ(υ−	φ(υ−	NOUN
ejpam-5661	156	7	α)2	α)2	NOUN
ejpam-5661	156	8	(	(	PUNCT
ejpam-5661	156	9	θ	θ	PROPN
ejpam-5661	157	1	[	[	X
ejpam-5661	157	2	2]q	2]q	NUM
ejpam-5661	157	3	−	−	NOUN
ejpam-5661	157	4	θ2	θ2	PROPN
ejpam-5661	157	5	[	[	X
ejpam-5661	157	6	3]q	3]q	NUM
ejpam-5661	157	7	)	)	PUNCT
ejpam-5661	157	8	≤	≤	NOUN
ejpam-5661	157	9	φ(α	φ(α	NOUN
ejpam-5661	157	10	)	)	PUNCT
ejpam-5661	158	1	+	+	CCONJ
ejpam-5661	158	2	φ(υ	φ(υ	PROPN
ejpam-5661	158	3	)	)	PUNCT
ejpam-5661	158	4	2	2	NUM
ejpam-5661	158	5	.	.	PUNCT
ejpam-5661	159	1	this	this	DET
ejpam-5661	159	2	finalizes	finalize	VERB
ejpam-5661	159	3	the	the	DET
ejpam-5661	159	4	proof	proof	NOUN
ejpam-5661	159	5	.	.	PUNCT
ejpam-5661	160	1	remark	remark	NOUN
ejpam-5661	160	2	1	1	NUM
ejpam-5661	160	3	.	.	PUNCT
ejpam-5661	161	1	setting	set	VERB
ejpam-5661	161	2	θ	θ	PROPN
ejpam-5661	161	3	=	=	SYM
ejpam-5661	161	4	1	1	NUM
ejpam-5661	161	5	,	,	PUNCT
ejpam-5661	161	6	in	in	ADP
ejpam-5661	161	7	theorem	theorem	NOUN
ejpam-5661	161	8	4	4	NUM
ejpam-5661	161	9	,	,	PUNCT
ejpam-5661	161	10	then	then	ADV
ejpam-5661	161	11	we	we	PRON
ejpam-5661	161	12	obtain	obtain	VERB
ejpam-5661	161	13	φ	φ	PROPN
ejpam-5661	161	14	(	(	PUNCT
ejpam-5661	161	15	α+υ	α+υ	NUM
ejpam-5661	161	16	2	2	NUM
ejpam-5661	161	17	)	)	PUNCT
ejpam-5661	161	18	≤	≤	NOUN
ejpam-5661	161	19	φ	φ	PROPN
ejpam-5661	161	20	(	(	PUNCT
ejpam-5661	161	21	α+υ	α+υ	NUM
ejpam-5661	161	22	2	2	NUM
ejpam-5661	161	23	)	)	PUNCT
ejpam-5661	162	1	+	+	CCONJ
ejpam-5661	162	2	φ(υ−	φ(υ−	NUM
ejpam-5661	162	3	α)2	α)2	NOUN
ejpam-5661	162	4	4	4	NUM
ejpam-5661	162	5	(	(	PUNCT
ejpam-5661	162	6	q3	q3	NOUN
ejpam-5661	162	7	−	−	PROPN
ejpam-5661	162	8	2q2	2q2	NUM
ejpam-5661	163	1	+	+	CCONJ
ejpam-5661	163	2	2q	2q	NUM
ejpam-5661	163	3	−	−	NOUN
ejpam-5661	163	4	1	1	NUM
ejpam-5661	164	1	[	[	X
ejpam-5661	164	2	2]q[3]q	2]q[3]q	NUM
ejpam-5661	164	3	)	)	PUNCT
ejpam-5661	164	4	≤	≤	NUM
ejpam-5661	164	5	1	1	NUM
ejpam-5661	164	6	2(υ−	2(υ−	NUM
ejpam-5661	164	7	α	α	NOUN
ejpam-5661	164	8	)	)	PUNCT
ejpam-5661	164	9	(	(	PUNCT
ejpam-5661	164	10	∫	∫	PROPN
ejpam-5661	164	11	υ	υ	PROPN
ejpam-5661	164	12	α	α	PROPN
ejpam-5661	164	13	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	ADJ
ejpam-5661	165	1	+	+	CCONJ
ejpam-5661	165	2	∫	∫	PROPN
ejpam-5661	165	3	υ	υ	PROPN
ejpam-5661	165	4	α	α	PROPN
ejpam-5661	165	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	165	6	)	)	PUNCT
ejpam-5661	165	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	165	8	)	)	PUNCT
ejpam-5661	165	9	≤	≤	NUM
ejpam-5661	166	1	φ(α	φ(α	NOUN
ejpam-5661	166	2	)	)	PUNCT
ejpam-5661	167	1	+	+	CCONJ
ejpam-5661	167	2	φ(υ	φ(υ	PROPN
ejpam-5661	167	3	)	)	PUNCT
ejpam-5661	167	4	2	2	NUM
ejpam-5661	167	5	−	−	NOUN
ejpam-5661	167	6	φq2(υ−	φq2(υ−	NOUN
ejpam-5661	167	7	α)2	α)2	NOUN
ejpam-5661	167	8	[	[	X
ejpam-5661	167	9	2]q[3]q	2]q[3]q	NUM
ejpam-5661	167	10	≤	≤	NOUN
ejpam-5661	167	11	φ(α	φ(α	NOUN
ejpam-5661	167	12	)	)	PUNCT
ejpam-5661	168	1	+	+	CCONJ
ejpam-5661	168	2	φ(υ	φ(υ	PROPN
ejpam-5661	168	3	)	)	PUNCT
ejpam-5661	168	4	2	2	NUM
ejpam-5661	168	5	,	,	PUNCT
ejpam-5661	168	6	which	which	PRON
ejpam-5661	168	7	appeared	appear	VERB
ejpam-5661	168	8	in	in	ADP
ejpam-5661	168	9	[	[	X
ejpam-5661	168	10	21	21	NUM
ejpam-5661	168	11	]	]	PUNCT
ejpam-5661	168	12	.	.	PUNCT
ejpam-5661	169	1	remark	remark	PROPN
ejpam-5661	169	2	2	2	NUM
ejpam-5661	169	3	.	.	PUNCT
ejpam-5661	169	4	setting	set	VERB
ejpam-5661	169	5	θ	θ	PROPN
ejpam-5661	169	6	=	=	SYM
ejpam-5661	169	7	1/[2]q	1/[2]q	NUM
ejpam-5661	169	8	,	,	PUNCT
ejpam-5661	169	9	in	in	ADP
ejpam-5661	169	10	theorem	theorem	NOUN
ejpam-5661	169	11	4	4	NUM
ejpam-5661	169	12	,	,	PUNCT
ejpam-5661	169	13	leads	lead	VERB
ejpam-5661	169	14	us	we	PRON
ejpam-5661	169	15	to	to	ADP
ejpam-5661	169	16	the	the	DET
ejpam-5661	169	17	new	new	ADJ
ejpam-5661	169	18	q	q	ADJ
ejpam-5661	169	19	-	-	PUNCT
ejpam-5661	169	20	h	h	NOUN
ejpam-5661	169	21	-	-	PUNCT
ejpam-5661	169	22	h	h	NOUN
ejpam-5661	169	23	inequalities	inequality	NOUN
ejpam-5661	169	24	:	:	PUNCT
ejpam-5661	169	25	φ	φ	PROPN
ejpam-5661	169	26	(	(	PUNCT
ejpam-5661	169	27	α+υ	α+υ	NUM
ejpam-5661	169	28	2	2	NUM
ejpam-5661	169	29	)	)	PUNCT
ejpam-5661	169	30	≤	≤	NOUN
ejpam-5661	169	31	φ	φ	PROPN
ejpam-5661	169	32	(	(	PUNCT
ejpam-5661	169	33	α+υ	α+υ	NUM
ejpam-5661	169	34	2	2	NUM
ejpam-5661	169	35	)	)	PUNCT
ejpam-5661	170	1	+	+	PUNCT
ejpam-5661	170	2	φ(υ−	φ(υ−	NOUN
ejpam-5661	170	3	α)2	α)2	NOUN
ejpam-5661	170	4	4([2]q)2[3]q	4([2]q)2[3]q	NUM
ejpam-5661	170	5	(	(	PUNCT
ejpam-5661	170	6	4−	4−	NUM
ejpam-5661	171	1	4[3]q	4[3]q	NUM
ejpam-5661	171	2	+	+	CCONJ
ejpam-5661	171	3	(	(	PUNCT
ejpam-5661	171	4	[	[	X
ejpam-5661	171	5	2]q	2]q	NUM
ejpam-5661	171	6	)	)	PUNCT
ejpam-5661	171	7	2[3]q	2[3]q	NUM
ejpam-5661	171	8	)	)	PUNCT
ejpam-5661	171	9	≤	≤	NOUN
ejpam-5661	172	1	[	[	PUNCT
ejpam-5661	172	2	2]q	2]q	NUM
ejpam-5661	172	3	2(υ−	2(υ−	NUM
ejpam-5661	172	4	α	α	NOUN
ejpam-5661	172	5	)	)	PUNCT
ejpam-5661	172	6	(	(	PUNCT
ejpam-5661	172	7	∫	∫	PROPN
ejpam-5661	172	8	(	(	PUNCT
ejpam-5661	172	9	qα+υ	qα+υ	NOUN
ejpam-5661	172	10	)	)	PUNCT
ejpam-5661	173	1	[	[	X
ejpam-5661	173	2	2]q	2]q	NUM
ejpam-5661	173	3	α	α	PRON
ejpam-5661	173	4	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	173	5	+	+	CCONJ
ejpam-5661	173	6	∫	∫	PROPN
ejpam-5661	173	7	υ	υ	PROPN
ejpam-5661	173	8	(	(	PUNCT
ejpam-5661	173	9	α+qυ	α+qυ	PROPN
ejpam-5661	173	10	)	)	PUNCT
ejpam-5661	174	1	[	[	X
ejpam-5661	174	2	2]q	2]q	NUM
ejpam-5661	174	3	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	174	4	)	)	PUNCT
ejpam-5661	174	5	υdqϖ	υdqϖ	NOUN
ejpam-5661	174	6	)	)	PUNCT
ejpam-5661	174	7	c.	c.	PROPN
ejpam-5661	174	8	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	174	9	,	,	PUNCT
ejpam-5661	174	10	p.	p.	PROPN
ejpam-5661	174	11	yotkaew	yotkaew	PROPN
ejpam-5661	174	12	/	/	SYM
ejpam-5661	174	13	eur	eur	PROPN
ejpam-5661	174	14	.	.	PUNCT
ejpam-5661	175	1	j.	j.	PROPN
ejpam-5661	175	2	pure	pure	PROPN
ejpam-5661	175	3	appl	appl	PROPN
ejpam-5661	175	4	.	.	PROPN
ejpam-5661	175	5	math	math	PROPN
ejpam-5661	175	6	,	,	PUNCT
ejpam-5661	175	7	18	18	NUM
ejpam-5661	175	8	(	(	PUNCT
ejpam-5661	175	9	1	1	NUM
ejpam-5661	175	10	)	)	PUNCT
ejpam-5661	175	11	(	(	PUNCT
ejpam-5661	175	12	2025	2025	NUM
ejpam-5661	175	13	)	)	PUNCT
ejpam-5661	175	14	,	,	PUNCT
ejpam-5661	175	15	5661	5661	NUM
ejpam-5661	175	16	8	8	NUM
ejpam-5661	175	17	of	of	ADP
ejpam-5661	175	18	24	24	NUM
ejpam-5661	175	19	≤	≤	NOUN
ejpam-5661	175	20	φ(α	φ(α	PROPN
ejpam-5661	175	21	)	)	PUNCT
ejpam-5661	176	1	+	+	CCONJ
ejpam-5661	176	2	φ(υ	φ(υ	PROPN
ejpam-5661	176	3	)	)	PUNCT
ejpam-5661	176	4	2	2	NUM
ejpam-5661	176	5	−	−	NOUN
ejpam-5661	176	6	φ(υ−	φ(υ−	NOUN
ejpam-5661	176	7	α)2	α)2	NOUN
ejpam-5661	176	8	(	(	PUNCT
ejpam-5661	176	9	[	[	X
ejpam-5661	176	10	2]q)2[3]q	2]q)2[3]q	NUM
ejpam-5661	176	11	(	(	PUNCT
ejpam-5661	176	12	[	[	X
ejpam-5661	176	13	3]q	3]q	NUM
ejpam-5661	176	14	−	−	NOUN
ejpam-5661	176	15	1	1	NUM
ejpam-5661	176	16	)	)	PUNCT
ejpam-5661	176	17	≤	≤	NOUN
ejpam-5661	176	18	φ(α	φ(α	NOUN
ejpam-5661	176	19	)	)	PUNCT
ejpam-5661	177	1	+	+	CCONJ
ejpam-5661	177	2	φ(υ	φ(υ	PROPN
ejpam-5661	177	3	)	)	PUNCT
ejpam-5661	177	4	2	2	NUM
ejpam-5661	177	5	.	.	PUNCT
ejpam-5661	178	1	next	next	ADV
ejpam-5661	178	2	,	,	PUNCT
ejpam-5661	178	3	we	we	PRON
ejpam-5661	178	4	refine	refine	VERB
ejpam-5661	178	5	theorem	theorem	ADJ
ejpam-5661	178	6	3.5	3.5	NUM
ejpam-5661	178	7	from	from	ADP
ejpam-5661	178	8	n.	n.	PROPN
ejpam-5661	178	9	alp	alp	PROPN
ejpam-5661	178	10	et	et	PROPN
ejpam-5661	178	11	al	al	PROPN
ejpam-5661	178	12	.	.	PUNCT
ejpam-5661	179	1	[	[	X
ejpam-5661	179	2	2	2	X
ejpam-5661	179	3	]	]	PUNCT
ejpam-5661	179	4	by	by	ADP
ejpam-5661	179	5	leveraging	leverage	VERB
ejpam-5661	179	6	the	the	DET
ejpam-5661	179	7	properties	property	NOUN
ejpam-5661	179	8	associated	associate	VERB
ejpam-5661	179	9	with	with	ADP
ejpam-5661	179	10	strong	strong	ADJ
ejpam-5661	179	11	convexity	convexity	NOUN
ejpam-5661	179	12	theorem	theorem	NOUN
ejpam-5661	179	13	5	5	NUM
ejpam-5661	179	14	.	.	PUNCT
ejpam-5661	180	1	let	let	VERB
ejpam-5661	180	2	φ	φ	NOUN
ejpam-5661	180	3	:	:	PUNCT
ejpam-5661	181	1	[	[	X
ejpam-5661	181	2	α	α	X
ejpam-5661	181	3	,	,	PUNCT
ejpam-5661	181	4	υ	υ	NOUN
ejpam-5661	181	5	]	]	X
ejpam-5661	181	6	→	→	PUNCT
ejpam-5661	181	7	r	r	NOUN
ejpam-5661	181	8	be	be	AUX
ejpam-5661	181	9	a	a	DET
ejpam-5661	181	10	strongly	strongly	ADV
ejpam-5661	181	11	convex	convex	ADJ
ejpam-5661	181	12	function	function	NOUN
ejpam-5661	181	13	for	for	ADP
ejpam-5661	181	14	φ	φ	PROPN
ejpam-5661	181	15	>	>	X
ejpam-5661	181	16	0	0	PROPN
ejpam-5661	181	17	and	and	CCONJ
ejpam-5661	181	18	θ	θ	PROPN
ejpam-5661	181	19	∈	∈	PROPN
ejpam-5661	181	20	(	(	PUNCT
ejpam-5661	181	21	0	0	NUM
ejpam-5661	181	22	,	,	PUNCT
ejpam-5661	181	23	1	1	NUM
ejpam-5661	181	24	]	]	PUNCT
ejpam-5661	181	25	.	.	PUNCT
ejpam-5661	182	1	then	then	ADV
ejpam-5661	182	2	the	the	DET
ejpam-5661	182	3	following	follow	VERB
ejpam-5661	182	4	inequalities	inequality	NOUN
ejpam-5661	182	5	are	be	AUX
ejpam-5661	182	6	established	establish	VERB
ejpam-5661	182	7	:	:	PUNCT
ejpam-5661	182	8	φ	φ	PROPN
ejpam-5661	182	9	(	(	PUNCT
ejpam-5661	182	10	α+υ	α+υ	NUM
ejpam-5661	182	11	2	2	NUM
ejpam-5661	182	12	)	)	PUNCT
ejpam-5661	182	13	≤	≤	NOUN
ejpam-5661	182	14	φ	φ	PROPN
ejpam-5661	182	15	(	(	PUNCT
ejpam-5661	182	16	α+υ	α+υ	NUM
ejpam-5661	182	17	2	2	NUM
ejpam-5661	182	18	)	)	PUNCT
ejpam-5661	182	19	+	+	CCONJ
ejpam-5661	182	20	φ	φ	PROPN
ejpam-5661	182	21	4	4	NUM
ejpam-5661	182	22	(	(	PUNCT
ejpam-5661	182	23	(	(	PUNCT
ejpam-5661	182	24	1−	1−	NUM
ejpam-5661	182	25	q)(υ−	q)(υ−	PROPN
ejpam-5661	182	26	α	α	NOUN
ejpam-5661	182	27	)	)	PUNCT
ejpam-5661	183	1	[	[	X
ejpam-5661	183	2	2]q	2]q	NUM
ejpam-5661	183	3	)	)	PUNCT
ejpam-5661	183	4	2	2	NUM
ejpam-5661	183	5	≤	≤	NUM
ejpam-5661	183	6	1	1	NUM
ejpam-5661	183	7	2	2	NUM
ejpam-5661	183	8	(	(	PUNCT
ejpam-5661	183	9	φ	φ	PROPN
ejpam-5661	183	10	(	(	PUNCT
ejpam-5661	183	11	α+	α+	PROPN
ejpam-5661	183	12	qυ	qυ	X
ejpam-5661	184	1	[	[	X
ejpam-5661	184	2	2]q	2]q	NUM
ejpam-5661	184	3	)	)	PUNCT
ejpam-5661	185	1	+	+	NOUN
ejpam-5661	185	2	φ	φ	PROPN
ejpam-5661	185	3	(	(	PUNCT
ejpam-5661	185	4	qα+υ	qα+υ	PROPN
ejpam-5661	185	5	[	[	X
ejpam-5661	185	6	2]q	2]q	NUM
ejpam-5661	185	7	)	)	PUNCT
ejpam-5661	185	8	)	)	PUNCT
ejpam-5661	185	9	≤	≤	ADV
ejpam-5661	185	10	1	1	NUM
ejpam-5661	185	11	+	+	NUM
ejpam-5661	185	12	q2	q2	NOUN
ejpam-5661	185	13	2qθ[2]q(υ−	2qθ[2]q(υ−	NUM
ejpam-5661	185	14	α	α	NOUN
ejpam-5661	185	15	)	)	PUNCT
ejpam-5661	185	16	(	(	PUNCT
ejpam-5661	185	17	∫	∫	PROPN
ejpam-5661	185	18	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	185	19	α	α	PROPN
ejpam-5661	185	20	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	185	21	+	+	CCONJ
ejpam-5661	185	22	∫	∫	PROPN
ejpam-5661	185	23	υ	υ	PROPN
ejpam-5661	185	24	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	185	25	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	185	26	)	)	PUNCT
ejpam-5661	185	27	υdqϖ	υdqϖ	NOUN
ejpam-5661	185	28	)	)	PUNCT
ejpam-5661	186	1	−	−	PROPN
ejpam-5661	187	1	1−	1−	NUM
ejpam-5661	187	2	q	q	NOUN
ejpam-5661	187	3	2q[2]q	2q[2]q	NOUN
ejpam-5661	187	4	(	(	PUNCT
ejpam-5661	187	5	φ(θα+	φ(θα+	X
ejpam-5661	187	6	(	(	PUNCT
ejpam-5661	187	7	1−θ)υ	1−θ)υ	NUM
ejpam-5661	187	8	)	)	PUNCT
ejpam-5661	187	9	+	+	CCONJ
ejpam-5661	187	10	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	187	11	(	(	PUNCT
ejpam-5661	187	12	1−θ)α	1−θ)α	NUM
ejpam-5661	187	13	)	)	PUNCT
ejpam-5661	187	14	)	)	PUNCT
ejpam-5661	187	15	≤	≤	NUM
ejpam-5661	187	16	φ(α	φ(α	ADJ
ejpam-5661	187	17	)	)	PUNCT
ejpam-5661	188	1	+	+	CCONJ
ejpam-5661	189	1	φ(υ	φ(υ	PROPN
ejpam-5661	189	2	)	)	PUNCT
ejpam-5661	189	3	2	2	NUM
ejpam-5661	189	4	−	−	NOUN
ejpam-5661	189	5	φqθ(υ−	φqθ(υ−	VERB
ejpam-5661	189	6	a)2	a)2	PROPN
ejpam-5661	190	1	[	[	X
ejpam-5661	190	2	2]q	2]q	NUM
ejpam-5661	190	3	(	(	PUNCT
ejpam-5661	190	4	2	2	NUM
ejpam-5661	190	5	[	[	PUNCT
ejpam-5661	190	6	2]q	2]q	NUM
ejpam-5661	190	7	−	−	NOUN
ejpam-5661	190	8	qθ	qθ	NOUN
ejpam-5661	191	1	[	[	X
ejpam-5661	191	2	3]q	3]q	NUM
ejpam-5661	191	3	−	−	NOUN
ejpam-5661	191	4	θ	θ	PROPN
ejpam-5661	191	5	[	[	X
ejpam-5661	191	6	3]q	3]q	NUM
ejpam-5661	191	7	)	)	PUNCT
ejpam-5661	191	8	≤	≤	NOUN
ejpam-5661	191	9	φ(α	φ(α	NOUN
ejpam-5661	191	10	)	)	PUNCT
ejpam-5661	192	1	+	+	CCONJ
ejpam-5661	192	2	φ(υ	φ(υ	PROPN
ejpam-5661	192	3	)	)	PUNCT
ejpam-5661	192	4	2	2	NUM
ejpam-5661	192	5	.	.	PUNCT
ejpam-5661	193	1	(	(	PUNCT
ejpam-5661	193	2	17	17	NUM
ejpam-5661	193	3	)	)	PUNCT
ejpam-5661	193	4	proof	proof	NOUN
ejpam-5661	193	5	.	.	PUNCT
ejpam-5661	194	1	it	it	PRON
ejpam-5661	194	2	follows	follow	VERB
ejpam-5661	194	3	from	from	ADP
ejpam-5661	194	4	strong	strong	ADJ
ejpam-5661	194	5	convexity	convexity	NOUN
ejpam-5661	194	6	of	of	ADP
ejpam-5661	194	7	φ	φ	PROPN
ejpam-5661	194	8	on	on	ADP
ejpam-5661	194	9	[	[	X
ejpam-5661	194	10	α	α	NOUN
ejpam-5661	194	11	,	,	PUNCT
ejpam-5661	194	12	υ	υ	NOUN
ejpam-5661	194	13	]	]	X
ejpam-5661	194	14	that	that	SCONJ
ejpam-5661	194	15	φ	φ	PROPN
ejpam-5661	194	16	(	(	PUNCT
ejpam-5661	194	17	α+υ	α+υ	NUM
ejpam-5661	194	18	2	2	NUM
ejpam-5661	194	19	)	)	PUNCT
ejpam-5661	194	20	=	=	SYM
ejpam-5661	194	21	φ	φ	PROPN
ejpam-5661	194	22	(	(	PUNCT
ejpam-5661	194	23	α+	α+	PRON
ejpam-5661	194	24	qυ+	qυ+	NOUN
ejpam-5661	194	25	qα+υ	qα+υ	VERB
ejpam-5661	194	26	2[2]q	2[2]q	NUM
ejpam-5661	194	27	)	)	PUNCT
ejpam-5661	194	28	≤	≤	NUM
ejpam-5661	194	29	1	1	NUM
ejpam-5661	194	30	2	2	NUM
ejpam-5661	194	31	(	(	PUNCT
ejpam-5661	194	32	φ	φ	PROPN
ejpam-5661	194	33	(	(	PUNCT
ejpam-5661	194	34	α+	α+	PROPN
ejpam-5661	194	35	qυ	qυ	X
ejpam-5661	194	36	[	[	X
ejpam-5661	194	37	2]q	2]q	NUM
ejpam-5661	194	38	)	)	PUNCT
ejpam-5661	195	1	+	+	NOUN
ejpam-5661	195	2	φ	φ	PROPN
ejpam-5661	195	3	(	(	PUNCT
ejpam-5661	195	4	qα+υ	qα+υ	PROPN
ejpam-5661	195	5	[	[	X
ejpam-5661	195	6	2]q	2]q	NUM
ejpam-5661	195	7	)	)	PUNCT
ejpam-5661	195	8	)	)	PUNCT
ejpam-5661	196	1	−	−	PROPN
ejpam-5661	197	1	φ	φ	NUM
ejpam-5661	197	2	4	4	NUM
ejpam-5661	197	3	(	(	PUNCT
ejpam-5661	197	4	(	(	PUNCT
ejpam-5661	197	5	1−	1−	NUM
ejpam-5661	197	6	q)(υ−	q)(υ−	PROPN
ejpam-5661	197	7	α	α	NOUN
ejpam-5661	197	8	)	)	PUNCT
ejpam-5661	198	1	[	[	X
ejpam-5661	198	2	2]q	2]q	NUM
ejpam-5661	198	3	)	)	PUNCT
ejpam-5661	198	4	2	2	NUM
ejpam-5661	198	5	,	,	PUNCT
ejpam-5661	198	6	which	which	PRON
ejpam-5661	198	7	implies	imply	VERB
ejpam-5661	198	8	that	that	SCONJ
ejpam-5661	198	9	φ	φ	PROPN
ejpam-5661	198	10	(	(	PUNCT
ejpam-5661	198	11	α+υ	α+υ	NUM
ejpam-5661	198	12	2	2	NUM
ejpam-5661	198	13	)	)	PUNCT
ejpam-5661	198	14	≤	≤	NOUN
ejpam-5661	198	15	φ	φ	PROPN
ejpam-5661	198	16	(	(	PUNCT
ejpam-5661	198	17	α+υ	α+υ	NUM
ejpam-5661	198	18	2	2	NUM
ejpam-5661	198	19	)	)	PUNCT
ejpam-5661	198	20	+	+	CCONJ
ejpam-5661	198	21	φ	φ	PROPN
ejpam-5661	198	22	4	4	NUM
ejpam-5661	198	23	(	(	PUNCT
ejpam-5661	198	24	(	(	PUNCT
ejpam-5661	198	25	1−	1−	NUM
ejpam-5661	198	26	q)(υ−	q)(υ−	PROPN
ejpam-5661	198	27	α	α	NOUN
ejpam-5661	198	28	)	)	PUNCT
ejpam-5661	199	1	[	[	X
ejpam-5661	199	2	2]q	2]q	NUM
ejpam-5661	199	3	)	)	PUNCT
ejpam-5661	199	4	2	2	NUM
ejpam-5661	199	5	≤	≤	NUM
ejpam-5661	199	6	1	1	NUM
ejpam-5661	199	7	2	2	NUM
ejpam-5661	199	8	(	(	PUNCT
ejpam-5661	199	9	φ	φ	PROPN
ejpam-5661	199	10	(	(	PUNCT
ejpam-5661	199	11	α+	α+	PROPN
ejpam-5661	199	12	qυ	qυ	X
ejpam-5661	200	1	[	[	X
ejpam-5661	200	2	2]q	2]q	NUM
ejpam-5661	200	3	)	)	PUNCT
ejpam-5661	201	1	+	+	NOUN
ejpam-5661	201	2	φ	φ	PROPN
ejpam-5661	201	3	(	(	PUNCT
ejpam-5661	201	4	qα+υ	qα+υ	PROPN
ejpam-5661	202	1	[	[	X
ejpam-5661	202	2	2]q	2]q	NUM
ejpam-5661	202	3	)	)	PUNCT
ejpam-5661	202	4	)	)	PUNCT
ejpam-5661	202	5	.	.	PUNCT
ejpam-5661	203	1	this	this	PRON
ejpam-5661	203	2	proves	prove	VERB
ejpam-5661	203	3	the	the	DET
ejpam-5661	203	4	first	first	ADJ
ejpam-5661	203	5	and	and	CCONJ
ejpam-5661	203	6	second	second	ADJ
ejpam-5661	203	7	inequalities	inequality	NOUN
ejpam-5661	203	8	of	of	ADP
ejpam-5661	203	9	(	(	PUNCT
ejpam-5661	203	10	17	17	NUM
ejpam-5661	203	11	)	)	PUNCT
ejpam-5661	203	12	.	.	PUNCT
ejpam-5661	204	1	by	by	ADP
ejpam-5661	204	2	strong	strong	ADJ
ejpam-5661	204	3	convexity	convexity	NOUN
ejpam-5661	204	4	of	of	ADP
ejpam-5661	204	5	φ	φ	PROPN
ejpam-5661	204	6	,	,	PUNCT
ejpam-5661	204	7	and	and	CCONJ
ejpam-5661	204	8	ϖ	ϖ	PRON
ejpam-5661	204	9	∈	∈	PROPN
ejpam-5661	205	1	[	[	X
ejpam-5661	205	2	0	0	NUM
ejpam-5661	205	3	,	,	PUNCT
ejpam-5661	205	4	1	1	NUM
ejpam-5661	205	5	]	]	PUNCT
ejpam-5661	205	6	and	and	CCONJ
ejpam-5661	205	7	θ	θ	PROPN
ejpam-5661	205	8	∈	∈	PROPN
ejpam-5661	205	9	(	(	PUNCT
ejpam-5661	205	10	0	0	NUM
ejpam-5661	205	11	,	,	PUNCT
ejpam-5661	205	12	1	1	NUM
ejpam-5661	205	13	]	]	PUNCT
ejpam-5661	205	14	,	,	PUNCT
ejpam-5661	205	15	we	we	PRON
ejpam-5661	205	16	can	can	AUX
ejpam-5661	205	17	write	write	VERB
ejpam-5661	205	18	φ	φ	PROPN
ejpam-5661	205	19	(	(	PUNCT
ejpam-5661	205	20	α+	α+	PROPN
ejpam-5661	205	21	qυ	qυ	X
ejpam-5661	206	1	[	[	X
ejpam-5661	206	2	2]q	2]q	NUM
ejpam-5661	206	3	)	)	PUNCT
ejpam-5661	207	1	=	=	SYM
ejpam-5661	207	2	φ	φ	PROPN
ejpam-5661	207	3	(	(	PUNCT
ejpam-5661	207	4	(	(	PUNCT
ejpam-5661	207	5	qθϖυ+	qθϖυ+	X
ejpam-5661	207	6	(	(	PUNCT
ejpam-5661	207	7	1−	1−	NUM
ejpam-5661	207	8	qθϖ)α	qθϖ)α	PROPN
ejpam-5661	207	9	)	)	PUNCT
ejpam-5661	208	1	+	+	NUM
ejpam-5661	208	2	q(θϖα+	q(θϖα+	NOUN
ejpam-5661	208	3	(	(	PUNCT
ejpam-5661	208	4	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	208	5	)	)	PUNCT
ejpam-5661	208	6	[	[	X
ejpam-5661	208	7	2]q	2]q	NUM
ejpam-5661	208	8	)	)	PUNCT
ejpam-5661	208	9	c.	c.	PROPN
ejpam-5661	208	10	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	208	11	,	,	PUNCT
ejpam-5661	208	12	p.	p.	PROPN
ejpam-5661	208	13	yotkaew	yotkaew	PROPN
ejpam-5661	208	14	/	/	SYM
ejpam-5661	208	15	eur	eur	PROPN
ejpam-5661	208	16	.	.	PUNCT
ejpam-5661	209	1	j.	j.	PROPN
ejpam-5661	209	2	pure	pure	PROPN
ejpam-5661	209	3	appl	appl	PROPN
ejpam-5661	209	4	.	.	PROPN
ejpam-5661	209	5	math	math	PROPN
ejpam-5661	209	6	,	,	PUNCT
ejpam-5661	209	7	18	18	NUM
ejpam-5661	209	8	(	(	PUNCT
ejpam-5661	209	9	1	1	NUM
ejpam-5661	209	10	)	)	PUNCT
ejpam-5661	209	11	(	(	PUNCT
ejpam-5661	209	12	2025	2025	NUM
ejpam-5661	209	13	)	)	PUNCT
ejpam-5661	209	14	,	,	PUNCT
ejpam-5661	209	15	5661	5661	NUM
ejpam-5661	209	16	9	9	NUM
ejpam-5661	209	17	of	of	ADP
ejpam-5661	209	18	24	24	NUM
ejpam-5661	209	19	=	=	SYM
ejpam-5661	209	20	φ	φ	X
ejpam-5661	209	21	(	(	PUNCT
ejpam-5661	209	22	(	(	PUNCT
ejpam-5661	209	23	qθϖυ+	qθϖυ+	X
ejpam-5661	209	24	(	(	PUNCT
ejpam-5661	209	25	1−	1−	NUM
ejpam-5661	209	26	qθϖ)α	qθϖ)α	NOUN
ejpam-5661	209	27	)	)	PUNCT
ejpam-5661	210	1	[	[	X
ejpam-5661	210	2	2]q	2]q	NUM
ejpam-5661	210	3	+	+	NUM
ejpam-5661	210	4	q(θϖα+	q(θϖα+	NOUN
ejpam-5661	210	5	(	(	PUNCT
ejpam-5661	210	6	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	210	7	)	)	PUNCT
ejpam-5661	210	8	[	[	X
ejpam-5661	210	9	2]q	2]q	NUM
ejpam-5661	210	10	)	)	PUNCT
ejpam-5661	210	11	≤	≤	NOUN
ejpam-5661	210	12	φ(qθϖυ+	φ(qθϖυ+	X
ejpam-5661	210	13	(	(	PUNCT
ejpam-5661	210	14	1−	1−	NUM
ejpam-5661	210	15	qθϖ)α	qθϖ)α	NOUN
ejpam-5661	210	16	)	)	PUNCT
ejpam-5661	211	1	[	[	X
ejpam-5661	211	2	2]q	2]q	NUM
ejpam-5661	211	3	+	+	CCONJ
ejpam-5661	211	4	qφ(θϖα+	qφ(θϖα+	PROPN
ejpam-5661	211	5	(	(	PUNCT
ejpam-5661	211	6	1−θϖ)b	1−θϖ)b	NUM
ejpam-5661	211	7	)	)	PUNCT
ejpam-5661	212	1	[	[	X
ejpam-5661	212	2	2]q	2]q	NUM
ejpam-5661	212	3	≤	≤	NUM
ejpam-5661	212	4	qθϖφ(υ	qθϖφ(υ	NOUN
ejpam-5661	212	5	)	)	PUNCT
ejpam-5661	212	6	+	+	CCONJ
ejpam-5661	212	7	(	(	PUNCT
ejpam-5661	212	8	1−	1−	NUM
ejpam-5661	212	9	qθϖ)φ(α)−	qθϖ)φ(α)−	VERB
ejpam-5661	212	10	φqθϖ(1−	φqθϖ(1−	ADJ
ejpam-5661	212	11	qθϖ)(υ−	qθϖ)(υ−	NOUN
ejpam-5661	212	12	a)2	a)2	NOUN
ejpam-5661	213	1	[	[	X
ejpam-5661	213	2	2]q	2]q	NUM
ejpam-5661	213	3	+	+	NOUN
ejpam-5661	213	4	q	q	ADJ
ejpam-5661	213	5	(	(	PUNCT
ejpam-5661	213	6	θϖφ(α	θϖφ(α	PROPN
ejpam-5661	213	7	)	)	PUNCT
ejpam-5661	213	8	+	+	CCONJ
ejpam-5661	213	9	(	(	PUNCT
ejpam-5661	213	10	1−θϖ)φ(υ)−	1−θϖ)φ(υ)−	NUM
ejpam-5661	213	11	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	213	12	α)2	α)2	NOUN
ejpam-5661	213	13	)	)	PUNCT
ejpam-5661	214	1	[	[	X
ejpam-5661	214	2	2]q	2]q	NUM
ejpam-5661	214	3	=	=	SYM
ejpam-5661	214	4	φ(α	φ(α	ADJ
ejpam-5661	214	5	)	)	PUNCT
ejpam-5661	215	1	+	+	CCONJ
ejpam-5661	215	2	qφ(υ	qφ(υ	X
ejpam-5661	215	3	)	)	PUNCT
ejpam-5661	216	1	[	[	X
ejpam-5661	216	2	2]q	2]q	NUM
ejpam-5661	216	3	−	−	ADP
ejpam-5661	216	4	φqθ(υ−	φqθ(υ−	NOUN
ejpam-5661	216	5	α)2	α)2	VERB
ejpam-5661	216	6	[	[	X
ejpam-5661	216	7	2]q	2]q	NUM
ejpam-5661	216	8	(	(	PUNCT
ejpam-5661	216	9	2ϖ	2ϖ	NOUN
ejpam-5661	216	10	−	−	NOUN
ejpam-5661	216	11	qθϖ2	qθϖ2	PROPN
ejpam-5661	216	12	−θϖ2	−θϖ2	PROPN
ejpam-5661	216	13	)	)	PUNCT
ejpam-5661	216	14	,	,	PUNCT
ejpam-5661	216	15	which	which	PRON
ejpam-5661	216	16	implies	imply	VERB
ejpam-5661	216	17	that	that	SCONJ
ejpam-5661	217	1	φ	φ	PROPN
ejpam-5661	217	2	(	(	PUNCT
ejpam-5661	217	3	α+	α+	PROPN
ejpam-5661	217	4	qυ	qυ	X
ejpam-5661	217	5	[	[	X
ejpam-5661	217	6	2]q	2]q	NUM
ejpam-5661	217	7	)	)	PUNCT
ejpam-5661	217	8	≤	≤	NOUN
ejpam-5661	217	9	φ(qθϖυ+	φ(qθϖυ+	X
ejpam-5661	217	10	(	(	PUNCT
ejpam-5661	217	11	1−	1−	NUM
ejpam-5661	217	12	qθϖ)α	qθϖ)α	NOUN
ejpam-5661	217	13	)	)	PUNCT
ejpam-5661	218	1	[	[	X
ejpam-5661	218	2	2]q	2]q	NUM
ejpam-5661	218	3	+	+	CCONJ
ejpam-5661	218	4	qφ(θϖα+	qφ(θϖα+	PROPN
ejpam-5661	218	5	(	(	PUNCT
ejpam-5661	218	6	1−θϖ)υ	1−θϖ)υ	NOUN
ejpam-5661	218	7	)	)	PUNCT
ejpam-5661	219	1	[	[	X
ejpam-5661	219	2	2]q	2]q	NUM
ejpam-5661	219	3	≤	≤	NOUN
ejpam-5661	219	4	φ(α	φ(α	ADJ
ejpam-5661	219	5	)	)	PUNCT
ejpam-5661	220	1	+	+	CCONJ
ejpam-5661	220	2	qφ(υ	qφ(υ	X
ejpam-5661	220	3	)	)	PUNCT
ejpam-5661	221	1	[	[	X
ejpam-5661	221	2	2]q	2]q	NUM
ejpam-5661	221	3	−	−	ADP
ejpam-5661	221	4	φqθ(υ−	φqθ(υ−	NOUN
ejpam-5661	221	5	α)2	α)2	VERB
ejpam-5661	221	6	[	[	X
ejpam-5661	221	7	2]q	2]q	NUM
ejpam-5661	221	8	(	(	PUNCT
ejpam-5661	221	9	2ϖ	2ϖ	NOUN
ejpam-5661	221	10	−	−	NOUN
ejpam-5661	221	11	qθϖ2	qθϖ2	PROPN
ejpam-5661	221	12	−θϖ2	−θϖ2	PROPN
ejpam-5661	221	13	)	)	PUNCT
ejpam-5661	221	14	.	.	PUNCT
ejpam-5661	222	1	by	by	ADP
ejpam-5661	222	2	performing	perform	VERB
ejpam-5661	222	3	q	q	NOUN
ejpam-5661	222	4	-	-	PUNCT
ejpam-5661	222	5	integration	integration	NOUN
ejpam-5661	222	6	on	on	ADP
ejpam-5661	222	7	both	both	DET
ejpam-5661	222	8	sides	side	NOUN
ejpam-5661	222	9	of	of	ADP
ejpam-5661	222	10	the	the	DET
ejpam-5661	222	11	inequality	inequality	NOUN
ejpam-5661	222	12	above	above	ADV
ejpam-5661	222	13	,	,	PUNCT
ejpam-5661	222	14	we	we	PRON
ejpam-5661	222	15	derive	derive	VERB
ejpam-5661	222	16	φ	φ	PROPN
ejpam-5661	222	17	(	(	PUNCT
ejpam-5661	222	18	α+	α+	PROPN
ejpam-5661	222	19	qυ	qυ	X
ejpam-5661	223	1	[	[	X
ejpam-5661	223	2	2]q	2]q	NUM
ejpam-5661	223	3	)	)	PUNCT
ejpam-5661	223	4	≤	≤	NUM
ejpam-5661	223	5	∫	∫	PROPN
ejpam-5661	224	1	1	1	NUM
ejpam-5661	224	2	0	0	NUM
ejpam-5661	225	1	(	(	PUNCT
ejpam-5661	225	2	φ(qθϖυ+	φ(qθϖυ+	PROPN
ejpam-5661	225	3	(	(	PUNCT
ejpam-5661	225	4	1−	1−	NUM
ejpam-5661	225	5	qθϖ)α	qθϖ)α	NOUN
ejpam-5661	225	6	)	)	PUNCT
ejpam-5661	226	1	[	[	X
ejpam-5661	226	2	2]q	2]q	NUM
ejpam-5661	226	3	+	+	CCONJ
ejpam-5661	226	4	qφ(θϖα+	qφ(θϖα+	PROPN
ejpam-5661	226	5	(	(	PUNCT
ejpam-5661	226	6	1−θϖ)b	1−θϖ)b	NUM
ejpam-5661	226	7	)	)	PUNCT
ejpam-5661	227	1	[	[	X
ejpam-5661	227	2	2]q	2]q	NUM
ejpam-5661	227	3	)	)	PUNCT
ejpam-5661	227	4	dqϖ	dqϖ	NOUN
ejpam-5661	227	5	(	(	PUNCT
ejpam-5661	227	6	18	18	NUM
ejpam-5661	227	7	)	)	PUNCT
ejpam-5661	227	8	≤	≤	NUM
ejpam-5661	227	9	∫	∫	PROPN
ejpam-5661	227	10	1	1	NUM
ejpam-5661	227	11	0	0	NUM
ejpam-5661	227	12	(	(	PUNCT
ejpam-5661	227	13	φ(α	φ(α	PROPN
ejpam-5661	227	14	)	)	PUNCT
ejpam-5661	227	15	+	+	CCONJ
ejpam-5661	227	16	qφ(υ	qφ(υ	X
ejpam-5661	227	17	)	)	PUNCT
ejpam-5661	228	1	[	[	X
ejpam-5661	228	2	2]q	2]q	NUM
ejpam-5661	228	3	−	−	ADP
ejpam-5661	228	4	φqθ(υ−	φqθ(υ−	NOUN
ejpam-5661	228	5	α)2	α)2	VERB
ejpam-5661	228	6	[	[	X
ejpam-5661	228	7	2]q	2]q	NUM
ejpam-5661	228	8	(	(	PUNCT
ejpam-5661	228	9	2ϖ	2ϖ	NOUN
ejpam-5661	228	10	−	−	NOUN
ejpam-5661	228	11	qθϖ2	qθϖ2	PROPN
ejpam-5661	228	12	−θϖ2	−θϖ2	PROPN
ejpam-5661	228	13	)	)	PUNCT
ejpam-5661	228	14	)	)	PUNCT
ejpam-5661	228	15	dqϖ	dqϖ	NOUN
ejpam-5661	229	1	=	=	SYM
ejpam-5661	229	2	φ(α	φ(α	PROPN
ejpam-5661	229	3	)	)	PUNCT
ejpam-5661	230	1	+	+	CCONJ
ejpam-5661	230	2	qφ(υ	qφ(υ	X
ejpam-5661	230	3	)	)	PUNCT
ejpam-5661	231	1	[	[	X
ejpam-5661	231	2	2]q	2]q	NUM
ejpam-5661	231	3	−	−	ADP
ejpam-5661	231	4	φqθ(υ−	φqθ(υ−	NOUN
ejpam-5661	231	5	α)2	α)2	VERB
ejpam-5661	231	6	[	[	X
ejpam-5661	231	7	2]q	2]q	NUM
ejpam-5661	231	8	(	(	PUNCT
ejpam-5661	231	9	2	2	NUM
ejpam-5661	231	10	[	[	PUNCT
ejpam-5661	231	11	2]q	2]q	NUM
ejpam-5661	231	12	−	−	NOUN
ejpam-5661	231	13	qθ	qθ	NOUN
ejpam-5661	231	14	[	[	X
ejpam-5661	231	15	3]q	3]q	NUM
ejpam-5661	231	16	−	−	NOUN
ejpam-5661	231	17	θ	θ	PROPN
ejpam-5661	232	1	[	[	X
ejpam-5661	232	2	3]q	3]q	NUM
ejpam-5661	232	3	)	)	PUNCT
ejpam-5661	232	4	.	.	PUNCT
ejpam-5661	233	1	the	the	DET
ejpam-5661	233	2	inequality	inequality	NOUN
ejpam-5661	233	3	of	of	ADP
ejpam-5661	233	4	(	(	PUNCT
ejpam-5661	233	5	18	18	NUM
ejpam-5661	233	6	)	)	PUNCT
ejpam-5661	233	7	can	can	AUX
ejpam-5661	233	8	be	be	AUX
ejpam-5661	233	9	computed	compute	VERB
ejpam-5661	233	10	as	as	ADP
ejpam-5661	233	11	φ	φ	PROPN
ejpam-5661	233	12	(	(	PUNCT
ejpam-5661	233	13	α+	α+	PROPN
ejpam-5661	233	14	qυ	qυ	X
ejpam-5661	234	1	[	[	X
ejpam-5661	234	2	2]q	2]q	NUM
ejpam-5661	234	3	)	)	PUNCT
ejpam-5661	234	4	≤	≤	NUM
ejpam-5661	234	5	∫	∫	PROPN
ejpam-5661	235	1	1	1	NUM
ejpam-5661	235	2	0	0	NUM
ejpam-5661	236	1	(	(	PUNCT
ejpam-5661	236	2	φ(qθϖυ+	φ(qθϖυ+	PROPN
ejpam-5661	236	3	(	(	PUNCT
ejpam-5661	236	4	1−	1−	NUM
ejpam-5661	236	5	qθϖ)α	qθϖ)α	NOUN
ejpam-5661	236	6	)	)	PUNCT
ejpam-5661	237	1	[	[	X
ejpam-5661	237	2	2]q	2]q	NUM
ejpam-5661	237	3	+	+	CCONJ
ejpam-5661	237	4	qφ(θϖα+	qφ(θϖα+	PROPN
ejpam-5661	237	5	(	(	PUNCT
ejpam-5661	237	6	1−θϖ)υ	1−θϖ)υ	NOUN
ejpam-5661	237	7	)	)	PUNCT
ejpam-5661	237	8	[	[	X
ejpam-5661	237	9	2]q	2]q	NUM
ejpam-5661	237	10	)	)	PUNCT
ejpam-5661	237	11	dqϖ	dqϖ	NOUN
ejpam-5661	237	12	=	=	SYM
ejpam-5661	237	13	(	(	PUNCT
ejpam-5661	237	14	1−	1−	NUM
ejpam-5661	237	15	q	q	NOUN
ejpam-5661	237	16	)	)	PUNCT
ejpam-5661	237	17	∞∑	∞∑	PROPN
ejpam-5661	237	18	n=0	n=0	NUM
ejpam-5661	237	19	qn	qn	NOUN
ejpam-5661	237	20	φ(θqn+1υ+	φ(θqn+1υ+	PROPN
ejpam-5661	237	21	(	(	PUNCT
ejpam-5661	237	22	1−	1−	NUM
ejpam-5661	237	23	qn+1θ)α	qn+1θ)α	NUM
ejpam-5661	237	24	)	)	PUNCT
ejpam-5661	238	1	+	+	NUM
ejpam-5661	238	2	qφ(qnθα+	qφ(qnθα+	NOUN
ejpam-5661	238	3	(	(	PUNCT
ejpam-5661	238	4	1−θqn)υ	1−θqn)υ	NUM
ejpam-5661	238	5	)	)	PUNCT
ejpam-5661	239	1	[	[	X
ejpam-5661	239	2	2]q	2]q	NUM
ejpam-5661	239	3	=	=	SYM
ejpam-5661	239	4	1	1	NUM
ejpam-5661	239	5	θ[2]q(υ−	θ[2]q(υ−	PROPN
ejpam-5661	239	6	α	α	NOUN
ejpam-5661	239	7	)	)	PUNCT
ejpam-5661	239	8	(	(	PUNCT
ejpam-5661	239	9	1	1	NUM
ejpam-5661	239	10	q	q	NOUN
ejpam-5661	239	11	∫	∫	PROPN
ejpam-5661	239	12	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	240	1	α	α	PROPN
ejpam-5661	240	2	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	240	3	+	+	X
ejpam-5661	240	4	q	q	PROPN
ejpam-5661	240	5	∫	∫	PROPN
ejpam-5661	240	6	υ	υ	PROPN
ejpam-5661	240	7	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	240	8	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	240	9	)	)	PUNCT
ejpam-5661	240	10	υdqϖ	υdqϖ	NOUN
ejpam-5661	240	11	)	)	PUNCT
ejpam-5661	241	1	−	−	PROPN
ejpam-5661	241	2	(	(	PUNCT
ejpam-5661	241	3	1−	1−	NUM
ejpam-5661	241	4	q	q	NOUN
ejpam-5661	241	5	)	)	PUNCT
ejpam-5661	241	6	q[2]q	q[2]q	PROPN
ejpam-5661	241	7	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	241	8	(	(	PUNCT
ejpam-5661	241	9	1−θ)α	1−θ)α	NUM
ejpam-5661	241	10	)	)	PUNCT
ejpam-5661	241	11	.	.	PUNCT
ejpam-5661	242	1	so	so	ADV
ejpam-5661	242	2	,	,	PUNCT
ejpam-5661	242	3	we	we	PRON
ejpam-5661	242	4	can	can	AUX
ejpam-5661	242	5	write	write	VERB
ejpam-5661	242	6	φ	φ	PROPN
ejpam-5661	242	7	(	(	PUNCT
ejpam-5661	242	8	α+	α+	PROPN
ejpam-5661	242	9	qυ	qυ	X
ejpam-5661	243	1	[	[	X
ejpam-5661	243	2	2]q	2]q	NUM
ejpam-5661	243	3	)	)	PUNCT
ejpam-5661	243	4	≤	≤	NOUN
ejpam-5661	243	5	1	1	NUM
ejpam-5661	243	6	θ[2]q(υ−	θ[2]q(υ−	PROPN
ejpam-5661	243	7	α	α	NOUN
ejpam-5661	243	8	)	)	PUNCT
ejpam-5661	243	9	(	(	PUNCT
ejpam-5661	243	10	1	1	NUM
ejpam-5661	243	11	q	q	NOUN
ejpam-5661	243	12	∫	∫	PROPN
ejpam-5661	243	13	θυ+(1−θ)a	θυ+(1−θ)a	VERB
ejpam-5661	243	14	α	α	PROPN
ejpam-5661	243	15	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	ADJ
ejpam-5661	243	16	c.	c.	PROPN
ejpam-5661	243	17	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	243	18	,	,	PUNCT
ejpam-5661	243	19	p.	p.	PROPN
ejpam-5661	243	20	yotkaew	yotkaew	PROPN
ejpam-5661	243	21	/	/	SYM
ejpam-5661	243	22	eur	eur	PROPN
ejpam-5661	243	23	.	.	PUNCT
ejpam-5661	244	1	j.	j.	PROPN
ejpam-5661	244	2	pure	pure	PROPN
ejpam-5661	244	3	appl	appl	PROPN
ejpam-5661	244	4	.	.	PROPN
ejpam-5661	244	5	math	math	PROPN
ejpam-5661	244	6	,	,	PUNCT
ejpam-5661	244	7	18	18	NUM
ejpam-5661	244	8	(	(	PUNCT
ejpam-5661	244	9	1	1	NUM
ejpam-5661	244	10	)	)	PUNCT
ejpam-5661	244	11	(	(	PUNCT
ejpam-5661	244	12	2025	2025	NUM
ejpam-5661	244	13	)	)	PUNCT
ejpam-5661	244	14	,	,	PUNCT
ejpam-5661	244	15	5661	5661	NUM
ejpam-5661	244	16	10	10	NUM
ejpam-5661	244	17	of	of	ADP
ejpam-5661	244	18	24	24	NUM
ejpam-5661	244	19	+	+	PROPN
ejpam-5661	244	20	q	q	NOUN
ejpam-5661	244	21	∫	∫	PROPN
ejpam-5661	244	22	υ	υ	PROPN
ejpam-5661	244	23	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	244	24	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	244	25	)	)	PUNCT
ejpam-5661	244	26	υdqϖ	υdqϖ	NOUN
ejpam-5661	244	27	)	)	PUNCT
ejpam-5661	245	1	−	−	PROPN
ejpam-5661	245	2	(	(	PUNCT
ejpam-5661	245	3	1−	1−	NUM
ejpam-5661	245	4	q	q	NOUN
ejpam-5661	245	5	)	)	PUNCT
ejpam-5661	245	6	q[2]q	q[2]q	PRON
ejpam-5661	245	7	f(θυ	f(θυ	PROPN
ejpam-5661	245	8	+	+	CCONJ
ejpam-5661	245	9	(	(	PUNCT
ejpam-5661	245	10	1−θ)α	1−θ)α	NUM
ejpam-5661	245	11	)	)	PUNCT
ejpam-5661	245	12	≤	≤	NOUN
ejpam-5661	245	13	φ(α	φ(α	NOUN
ejpam-5661	245	14	)	)	PUNCT
ejpam-5661	245	15	+	+	CCONJ
ejpam-5661	245	16	qφ(υ	qφ(υ	X
ejpam-5661	245	17	)	)	PUNCT
ejpam-5661	246	1	[	[	X
ejpam-5661	246	2	2]q	2]q	NUM
ejpam-5661	246	3	−	−	ADP
ejpam-5661	246	4	φqθ(υ−	φqθ(υ−	NOUN
ejpam-5661	246	5	α)2	α)2	VERB
ejpam-5661	246	6	[	[	X
ejpam-5661	246	7	2]q	2]q	NUM
ejpam-5661	246	8	(	(	PUNCT
ejpam-5661	246	9	2	2	NUM
ejpam-5661	246	10	[	[	PUNCT
ejpam-5661	246	11	2]q	2]q	NUM
ejpam-5661	246	12	−	−	NOUN
ejpam-5661	246	13	qθ	qθ	NOUN
ejpam-5661	246	14	[	[	X
ejpam-5661	246	15	3]q	3]q	NUM
ejpam-5661	246	16	−	−	NOUN
ejpam-5661	246	17	θ	θ	PROPN
ejpam-5661	247	1	[	[	X
ejpam-5661	247	2	3]q	3]q	NUM
ejpam-5661	247	3	)	)	PUNCT
ejpam-5661	247	4	.	.	PUNCT
ejpam-5661	248	1	(	(	PUNCT
ejpam-5661	248	2	19	19	NUM
ejpam-5661	248	3	)	)	PUNCT
ejpam-5661	248	4	alternatively	alternatively	ADV
ejpam-5661	248	5	,	,	PUNCT
ejpam-5661	248	6	by	by	ADP
ejpam-5661	248	7	employing	employ	VERB
ejpam-5661	248	8	a	a	DET
ejpam-5661	248	9	comparable	comparable	ADJ
ejpam-5661	248	10	technique	technique	NOUN
ejpam-5661	248	11	,	,	PUNCT
ejpam-5661	248	12	we	we	PRON
ejpam-5661	248	13	can	can	AUX
ejpam-5661	248	14	formulate	formulate	VERB
ejpam-5661	248	15	φ	φ	PROPN
ejpam-5661	248	16	(	(	PUNCT
ejpam-5661	248	17	qα+υ	qα+υ	PROPN
ejpam-5661	249	1	[	[	X
ejpam-5661	249	2	2]q	2]q	NUM
ejpam-5661	249	3	)	)	PUNCT
ejpam-5661	249	4	≤	≤	NOUN
ejpam-5661	249	5	(	(	PUNCT
ejpam-5661	249	6	qφ(θϖυ+	qφ(θϖυ+	PROPN
ejpam-5661	249	7	(	(	PUNCT
ejpam-5661	249	8	1−θϖ)α	1−θϖ)α	NUM
ejpam-5661	249	9	)	)	PUNCT
ejpam-5661	250	1	+	+	NUM
ejpam-5661	250	2	φ(qθϖα+	φ(qθϖα+	NUM
ejpam-5661	250	3	(	(	PUNCT
ejpam-5661	250	4	1−	1−	NUM
ejpam-5661	250	5	qθϖ)υ	qθϖ)υ	NOUN
ejpam-5661	250	6	)	)	PUNCT
ejpam-5661	251	1	[	[	X
ejpam-5661	251	2	2]q	2]q	NUM
ejpam-5661	251	3	)	)	PUNCT
ejpam-5661	251	4	≤	≤	NOUN
ejpam-5661	251	5	qφ(α	qφ(α	NOUN
ejpam-5661	251	6	)	)	PUNCT
ejpam-5661	252	1	+	+	CCONJ
ejpam-5661	252	2	φ(υ	φ(υ	NUM
ejpam-5661	252	3	)	)	PUNCT
ejpam-5661	253	1	[	[	X
ejpam-5661	253	2	2]q	2]q	NUM
ejpam-5661	253	3	−	−	ADP
ejpam-5661	253	4	φqθ(υ−	φqθ(υ−	NOUN
ejpam-5661	253	5	α)2	α)2	VERB
ejpam-5661	253	6	[	[	X
ejpam-5661	253	7	2]q	2]q	NUM
ejpam-5661	253	8	(	(	PUNCT
ejpam-5661	253	9	2ϖ	2ϖ	NOUN
ejpam-5661	253	10	−	−	NOUN
ejpam-5661	253	11	qθϖ2	qθϖ2	PROPN
ejpam-5661	253	12	−θϖ2	−θϖ2	PROPN
ejpam-5661	253	13	)	)	PUNCT
ejpam-5661	253	14	.	.	PUNCT
ejpam-5661	254	1	the	the	DET
ejpam-5661	254	2	process	process	NOUN
ejpam-5661	254	3	of	of	ADP
ejpam-5661	254	4	q	q	NOUN
ejpam-5661	254	5	-	-	PUNCT
ejpam-5661	254	6	integrating	integrate	VERB
ejpam-5661	254	7	both	both	DET
ejpam-5661	254	8	sides	side	NOUN
ejpam-5661	254	9	of	of	ADP
ejpam-5661	254	10	the	the	DET
ejpam-5661	254	11	inequality	inequality	NOUN
ejpam-5661	254	12	above	above	ADP
ejpam-5661	254	13	yields	yields	PROPN
ejpam-5661	254	14	φ	φ	PROPN
ejpam-5661	254	15	(	(	PUNCT
ejpam-5661	254	16	qα+υ	qα+υ	PROPN
ejpam-5661	255	1	[	[	X
ejpam-5661	255	2	2]q	2]q	NUM
ejpam-5661	255	3	)	)	PUNCT
ejpam-5661	255	4	≤	≤	NOUN
ejpam-5661	255	5	1	1	NUM
ejpam-5661	255	6	θ[2]q(υ−	θ[2]q(υ−	PROPN
ejpam-5661	255	7	α	α	NOUN
ejpam-5661	255	8	)	)	PUNCT
ejpam-5661	255	9	(	(	PUNCT
ejpam-5661	255	10	q	q	PUNCT
ejpam-5661	255	11	∫	∫	INTJ
ejpam-5661	255	12	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	255	13	α	α	PRON
ejpam-5661	255	14	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	255	15	+	+	X
ejpam-5661	255	16	1	1	NUM
ejpam-5661	255	17	q	q	NOUN
ejpam-5661	255	18	∫	∫	PROPN
ejpam-5661	255	19	υ	υ	PROPN
ejpam-5661	255	20	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	255	21	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	255	22	)	)	PUNCT
ejpam-5661	255	23	υdqϖ	υdqϖ	NOUN
ejpam-5661	255	24	)	)	PUNCT
ejpam-5661	256	1	−	−	PROPN
ejpam-5661	256	2	(	(	PUNCT
ejpam-5661	256	3	1−	1−	NUM
ejpam-5661	256	4	q	q	NOUN
ejpam-5661	256	5	)	)	PUNCT
ejpam-5661	256	6	q[2]q	q[2]q	NOUN
ejpam-5661	256	7	φ(θα+	φ(θα+	NUM
ejpam-5661	256	8	(	(	PUNCT
ejpam-5661	256	9	1−θ)υ	1−θ)υ	NUM
ejpam-5661	256	10	)	)	PUNCT
ejpam-5661	256	11	≤	≤	NOUN
ejpam-5661	256	12	qφ(α	qφ(α	NOUN
ejpam-5661	256	13	)	)	PUNCT
ejpam-5661	257	1	+	+	CCONJ
ejpam-5661	257	2	φ(υ	φ(υ	NUM
ejpam-5661	257	3	)	)	PUNCT
ejpam-5661	258	1	[	[	X
ejpam-5661	258	2	2]q	2]q	NUM
ejpam-5661	258	3	−	−	ADP
ejpam-5661	258	4	φqθ(υ−	φqθ(υ−	NOUN
ejpam-5661	258	5	α)2	α)2	VERB
ejpam-5661	258	6	[	[	X
ejpam-5661	258	7	2]q	2]q	NUM
ejpam-5661	258	8	(	(	PUNCT
ejpam-5661	258	9	2	2	NUM
ejpam-5661	258	10	[	[	PUNCT
ejpam-5661	258	11	2]q	2]q	NUM
ejpam-5661	258	12	−	−	NOUN
ejpam-5661	258	13	qθ	qθ	NOUN
ejpam-5661	258	14	[	[	X
ejpam-5661	258	15	3]q	3]q	NUM
ejpam-5661	258	16	−	−	NOUN
ejpam-5661	258	17	θ	θ	PROPN
ejpam-5661	259	1	[	[	X
ejpam-5661	259	2	3]q	3]q	NUM
ejpam-5661	259	3	)	)	PUNCT
ejpam-5661	259	4	.	.	PUNCT
ejpam-5661	260	1	(	(	PUNCT
ejpam-5661	260	2	20	20	X
ejpam-5661	260	3	)	)	PUNCT
ejpam-5661	260	4	note	note	VERB
ejpam-5661	260	5	that	that	SCONJ
ejpam-5661	260	6	2/[2]q	2/[2]q	NUM
ejpam-5661	260	7	−	−	NOUN
ejpam-5661	260	8	qθ/[3]q	qθ/[3]q	PROPN
ejpam-5661	260	9	−	−	PROPN
ejpam-5661	260	10	θ/[3]q	θ/[3]q	VERB
ejpam-5661	260	11	≥	≥	NOUN
ejpam-5661	260	12	0	0	NUM
ejpam-5661	260	13	.	.	PUNCT
ejpam-5661	261	1	by	by	ADP
ejpam-5661	261	2	combinating	combinate	VERB
ejpam-5661	261	3	the	the	DET
ejpam-5661	261	4	inequalities	inequality	NOUN
ejpam-5661	261	5	(	(	PUNCT
ejpam-5661	261	6	19	19	NUM
ejpam-5661	261	7	)	)	PUNCT
ejpam-5661	261	8	and	and	CCONJ
ejpam-5661	261	9	(	(	PUNCT
ejpam-5661	261	10	20	20	NUM
ejpam-5661	261	11	)	)	PUNCT
ejpam-5661	261	12	,	,	PUNCT
ejpam-5661	261	13	we	we	PRON
ejpam-5661	261	14	obtain	obtain	VERB
ejpam-5661	261	15	φ	φ	PROPN
ejpam-5661	261	16	(	(	PUNCT
ejpam-5661	261	17	α+	α+	PROPN
ejpam-5661	261	18	qυ	qυ	X
ejpam-5661	262	1	[	[	X
ejpam-5661	262	2	2]q	2]q	NUM
ejpam-5661	262	3	)	)	PUNCT
ejpam-5661	263	1	+	+	NOUN
ejpam-5661	263	2	φ	φ	PROPN
ejpam-5661	263	3	(	(	PUNCT
ejpam-5661	263	4	qα+υ	qα+υ	PROPN
ejpam-5661	263	5	[	[	X
ejpam-5661	263	6	2]q	2]q	NUM
ejpam-5661	263	7	)	)	PUNCT
ejpam-5661	263	8	≤	≤	NOUN
ejpam-5661	263	9	1	1	NUM
ejpam-5661	263	10	θ[2]q(υ−	θ[2]q(υ−	PROPN
ejpam-5661	263	11	α	α	NOUN
ejpam-5661	263	12	)	)	PUNCT
ejpam-5661	263	13	(	(	PUNCT
ejpam-5661	263	14	(	(	PUNCT
ejpam-5661	263	15	1	1	NUM
ejpam-5661	263	16	q	q	NOUN
ejpam-5661	263	17	+	+	NUM
ejpam-5661	263	18	q	q	X
ejpam-5661	263	19	)	)	PUNCT
ejpam-5661	263	20	∫	∫	NOUN
ejpam-5661	263	21	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	263	22	α	α	PROPN
ejpam-5661	263	23	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	263	24	+	+	CCONJ
ejpam-5661	263	25	(	(	PUNCT
ejpam-5661	263	26	1	1	NUM
ejpam-5661	263	27	q	q	NOUN
ejpam-5661	263	28	+	+	NUM
ejpam-5661	263	29	q	q	X
ejpam-5661	263	30	)	)	PUNCT
ejpam-5661	263	31	∫	∫	PROPN
ejpam-5661	263	32	υ	υ	PROPN
ejpam-5661	263	33	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	263	34	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	263	35	)	)	PUNCT
ejpam-5661	263	36	υdqϖ	υdqϖ	NOUN
ejpam-5661	263	37	)	)	PUNCT
ejpam-5661	264	1	−	−	PROPN
ejpam-5661	264	2	(	(	PUNCT
ejpam-5661	264	3	1−	1−	NUM
ejpam-5661	264	4	q	q	NOUN
ejpam-5661	264	5	)	)	PUNCT
ejpam-5661	264	6	q[2]q	q[2]q	NOUN
ejpam-5661	264	7	(	(	PUNCT
ejpam-5661	264	8	φ(θυ	φ(θυ	X
ejpam-5661	264	9	+	+	CCONJ
ejpam-5661	264	10	(	(	PUNCT
ejpam-5661	264	11	1−θ)α	1−θ)α	NUM
ejpam-5661	264	12	)	)	PUNCT
ejpam-5661	265	1	+	+	CCONJ
ejpam-5661	265	2	φ(θα+	φ(θα+	NUM
ejpam-5661	265	3	(	(	PUNCT
ejpam-5661	265	4	1−θ)υ	1−θ)υ	NUM
ejpam-5661	265	5	)	)	PUNCT
ejpam-5661	265	6	)	)	PUNCT
ejpam-5661	265	7	≤	≤	NUM
ejpam-5661	265	8	φ(α	φ(α	ADJ
ejpam-5661	265	9	)	)	PUNCT
ejpam-5661	266	1	+	+	CCONJ
ejpam-5661	266	2	φ(υ)−	φ(υ)−	NOUN
ejpam-5661	266	3	2φqθ(υ−	2φqθ(υ−	NUM
ejpam-5661	266	4	α)2	α)2	NOUN
ejpam-5661	266	5	[	[	X
ejpam-5661	266	6	2]q	2]q	NUM
ejpam-5661	266	7	(	(	PUNCT
ejpam-5661	266	8	2	2	NUM
ejpam-5661	266	9	[	[	PUNCT
ejpam-5661	266	10	2]q	2]q	NUM
ejpam-5661	266	11	−	−	NOUN
ejpam-5661	266	12	qθ	qθ	NOUN
ejpam-5661	266	13	[	[	X
ejpam-5661	266	14	3]q	3]q	NUM
ejpam-5661	266	15	−	−	NOUN
ejpam-5661	266	16	θ	θ	PROPN
ejpam-5661	267	1	[	[	X
ejpam-5661	267	2	3]q	3]q	NUM
ejpam-5661	267	3	)	)	PUNCT
ejpam-5661	267	4	≤	≤	NOUN
ejpam-5661	267	5	φ(α	φ(α	NOUN
ejpam-5661	267	6	)	)	PUNCT
ejpam-5661	268	1	+	+	CCONJ
ejpam-5661	268	2	φ(υ	φ(υ	PROPN
ejpam-5661	268	3	)	)	PUNCT
ejpam-5661	268	4	.	.	PUNCT
ejpam-5661	269	1	multiplying	multiply	VERB
ejpam-5661	269	2	the	the	DET
ejpam-5661	269	3	inequality	inequality	NOUN
ejpam-5661	269	4	above	above	ADP
ejpam-5661	269	5	by	by	ADP
ejpam-5661	269	6	1/2	1/2	NUM
ejpam-5661	269	7	results	result	NOUN
ejpam-5661	269	8	in	in	ADP
ejpam-5661	269	9	the	the	DET
ejpam-5661	269	10	third	third	ADJ
ejpam-5661	269	11	,	,	PUNCT
ejpam-5661	269	12	fourth	fourth	ADJ
ejpam-5661	269	13	,	,	PUNCT
ejpam-5661	269	14	and	and	CCONJ
ejpam-5661	269	15	fifth	fifth	ADJ
ejpam-5661	269	16	inequalities	inequality	NOUN
ejpam-5661	269	17	of	of	ADP
ejpam-5661	269	18	(	(	PUNCT
ejpam-5661	269	19	17	17	NUM
ejpam-5661	269	20	)	)	PUNCT
ejpam-5661	269	21	.	.	PUNCT
ejpam-5661	270	1	the	the	DET
ejpam-5661	270	2	following	follow	VERB
ejpam-5661	270	3	result	result	NOUN
ejpam-5661	270	4	gives	give	VERB
ejpam-5661	270	5	the	the	DET
ejpam-5661	270	6	refinements	refinement	NOUN
ejpam-5661	270	7	of	of	ADP
ejpam-5661	270	8	inequalities	inequality	NOUN
ejpam-5661	270	9	(	(	PUNCT
ejpam-5661	270	10	5	5	NUM
ejpam-5661	270	11	)	)	PUNCT
ejpam-5661	270	12	.	.	PUNCT
ejpam-5661	271	1	corollary	corollary	ADJ
ejpam-5661	271	2	1	1	NUM
ejpam-5661	271	3	.	.	PUNCT
ejpam-5661	271	4	setting	set	VERB
ejpam-5661	271	5	θ	θ	NOUN
ejpam-5661	271	6	=	=	SYM
ejpam-5661	271	7	1/2	1/2	NUM
ejpam-5661	271	8	in	in	ADP
ejpam-5661	271	9	theorem	theorem	NOUN
ejpam-5661	271	10	5	5	NUM
ejpam-5661	271	11	,	,	PUNCT
ejpam-5661	271	12	then	then	ADV
ejpam-5661	271	13	we	we	PRON
ejpam-5661	271	14	obtain	obtain	VERB
ejpam-5661	271	15	φ	φ	PROPN
ejpam-5661	271	16	(	(	PUNCT
ejpam-5661	271	17	α+υ	α+υ	NUM
ejpam-5661	271	18	2	2	NUM
ejpam-5661	271	19	)	)	PUNCT
ejpam-5661	271	20	≤	≤	NOUN
ejpam-5661	271	21	φ1(q	φ1(q	NUM
ejpam-5661	271	22	)	)	PUNCT
ejpam-5661	271	23	≤	≤	NUM
ejpam-5661	271	24	φ2(q	φ2(q	NOUN
ejpam-5661	271	25	)	)	PUNCT
ejpam-5661	271	26	c.	c.	PROPN
ejpam-5661	271	27	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	271	28	,	,	PUNCT
ejpam-5661	271	29	p.	p.	PROPN
ejpam-5661	271	30	yotkaew	yotkaew	PROPN
ejpam-5661	271	31	/	/	SYM
ejpam-5661	271	32	eur	eur	PROPN
ejpam-5661	271	33	.	.	PUNCT
ejpam-5661	272	1	j.	j.	PROPN
ejpam-5661	272	2	pure	pure	PROPN
ejpam-5661	272	3	appl	appl	PROPN
ejpam-5661	272	4	.	.	PROPN
ejpam-5661	272	5	math	math	PROPN
ejpam-5661	272	6	,	,	PUNCT
ejpam-5661	272	7	18	18	NUM
ejpam-5661	272	8	(	(	PUNCT
ejpam-5661	272	9	1	1	NUM
ejpam-5661	272	10	)	)	PUNCT
ejpam-5661	272	11	(	(	PUNCT
ejpam-5661	272	12	2025	2025	NUM
ejpam-5661	272	13	)	)	PUNCT
ejpam-5661	272	14	,	,	PUNCT
ejpam-5661	272	15	5661	5661	NUM
ejpam-5661	272	16	11	11	NUM
ejpam-5661	272	17	of	of	ADP
ejpam-5661	272	18	24	24	NUM
ejpam-5661	272	19	≤	≤	NUM
ejpam-5661	272	20	1	1	NUM
ejpam-5661	272	21	2(υ−	2(υ−	NUM
ejpam-5661	272	22	α	α	NOUN
ejpam-5661	272	23	)	)	PUNCT
ejpam-5661	272	24	(	(	PUNCT
ejpam-5661	272	25	∫	∫	PROPN
ejpam-5661	272	26	α+υ	α+υ	NUM
ejpam-5661	272	27	2	2	NUM
ejpam-5661	272	28	α	α	NOUN
ejpam-5661	272	29	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	272	30	+	+	CCONJ
ejpam-5661	272	31	∫	∫	PROPN
ejpam-5661	272	32	υ	υ	X
ejpam-5661	272	33	α+υ	α+υ	NUM
ejpam-5661	272	34	2	2	NUM
ejpam-5661	272	35	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	272	36	)	)	PUNCT
ejpam-5661	272	37	υdqϖ	υdqϖ	NOUN
ejpam-5661	272	38	)	)	PUNCT
ejpam-5661	272	39	≤	≤	PUNCT
ejpam-5661	273	1	φ3(q	φ3(q	PROPN
ejpam-5661	273	2	)	)	PUNCT
ejpam-5661	273	3	≤	≤	NOUN
ejpam-5661	273	4	φ(α	φ(α	NOUN
ejpam-5661	273	5	)	)	PUNCT
ejpam-5661	274	1	+	+	CCONJ
ejpam-5661	274	2	φ(υ	φ(υ	PROPN
ejpam-5661	274	3	)	)	PUNCT
ejpam-5661	274	4	2	2	NUM
ejpam-5661	274	5	,	,	PUNCT
ejpam-5661	274	6	(	(	PUNCT
ejpam-5661	274	7	21	21	NUM
ejpam-5661	274	8	)	)	PUNCT
ejpam-5661	274	9	where	where	SCONJ
ejpam-5661	274	10	φ1(q	φ1(q	X
ejpam-5661	274	11	)	)	PUNCT
ejpam-5661	274	12	=	=	SYM
ejpam-5661	274	13	q[2]q	q[2]q	NOUN
ejpam-5661	274	14	(	(	PUNCT
ejpam-5661	274	15	1	1	NUM
ejpam-5661	274	16	+	+	NUM
ejpam-5661	274	17	q2	q2	NOUN
ejpam-5661	274	18	)	)	PUNCT
ejpam-5661	274	19	(	(	PUNCT
ejpam-5661	274	20	φ	φ	X
ejpam-5661	274	21	(	(	PUNCT
ejpam-5661	274	22	α+υ	α+υ	NUM
ejpam-5661	274	23	2	2	NUM
ejpam-5661	274	24	)	)	PUNCT
ejpam-5661	274	25	+	+	CCONJ
ejpam-5661	274	26	φ	φ	PROPN
ejpam-5661	274	27	4	4	NUM
ejpam-5661	274	28	(	(	PUNCT
ejpam-5661	274	29	(	(	PUNCT
ejpam-5661	274	30	1−	1−	NUM
ejpam-5661	274	31	q)(υ−	q)(υ−	PROPN
ejpam-5661	274	32	α	α	NOUN
ejpam-5661	274	33	)	)	PUNCT
ejpam-5661	275	1	[	[	X
ejpam-5661	275	2	2]q	2]q	NUM
ejpam-5661	275	3	)	)	PUNCT
ejpam-5661	275	4	2	2	NUM
ejpam-5661	275	5	)	)	PUNCT
ejpam-5661	275	6	+	+	CCONJ
ejpam-5661	275	7	(	(	PUNCT
ejpam-5661	275	8	1−	1−	NUM
ejpam-5661	275	9	q	q	NOUN
ejpam-5661	275	10	)	)	PUNCT
ejpam-5661	275	11	(	(	PUNCT
ejpam-5661	275	12	1	1	NUM
ejpam-5661	275	13	+	+	NUM
ejpam-5661	275	14	q2	q2	NOUN
ejpam-5661	275	15	)	)	PUNCT
ejpam-5661	275	16	φ	φ	PROPN
ejpam-5661	275	17	(	(	PUNCT
ejpam-5661	275	18	α+υ	α+υ	NUM
ejpam-5661	275	19	2	2	NUM
ejpam-5661	275	20	)	)	PUNCT
ejpam-5661	275	21	,	,	PUNCT
ejpam-5661	275	22	φ2(q	φ2(q	X
ejpam-5661	275	23	)	)	PUNCT
ejpam-5661	275	24	=	=	PUNCT
ejpam-5661	275	25	q[2]q	q[2]q	PRON
ejpam-5661	275	26	2(1	2(1	NUM
ejpam-5661	275	27	+	+	NUM
ejpam-5661	275	28	q2	q2	NOUN
ejpam-5661	275	29	)	)	PUNCT
ejpam-5661	275	30	(	(	PUNCT
ejpam-5661	275	31	φ	φ	X
ejpam-5661	275	32	(	(	PUNCT
ejpam-5661	275	33	α+	α+	PROPN
ejpam-5661	275	34	qυ	qυ	X
ejpam-5661	276	1	[	[	X
ejpam-5661	276	2	2]q	2]q	NUM
ejpam-5661	276	3	)	)	PUNCT
ejpam-5661	277	1	+	+	NOUN
ejpam-5661	277	2	φ	φ	PROPN
ejpam-5661	277	3	(	(	PUNCT
ejpam-5661	277	4	qα+υ	qα+υ	PROPN
ejpam-5661	278	1	[	[	X
ejpam-5661	278	2	2]q	2]q	NUM
ejpam-5661	278	3	)	)	PUNCT
ejpam-5661	278	4	)	)	PUNCT
ejpam-5661	279	1	+	+	CCONJ
ejpam-5661	279	2	(	(	PUNCT
ejpam-5661	279	3	1−	1−	NUM
ejpam-5661	279	4	q	q	NOUN
ejpam-5661	279	5	)	)	PUNCT
ejpam-5661	279	6	(	(	PUNCT
ejpam-5661	279	7	1	1	NUM
ejpam-5661	279	8	+	+	NUM
ejpam-5661	279	9	q2	q2	NOUN
ejpam-5661	279	10	)	)	PUNCT
ejpam-5661	279	11	φ	φ	PROPN
ejpam-5661	279	12	(	(	PUNCT
ejpam-5661	279	13	α+υ	α+υ	NUM
ejpam-5661	279	14	2	2	NUM
ejpam-5661	279	15	)	)	PUNCT
ejpam-5661	279	16	and	and	CCONJ
ejpam-5661	279	17	φ3(q	φ3(q	PROPN
ejpam-5661	279	18	)	)	PUNCT
ejpam-5661	280	1	=	=	SYM
ejpam-5661	280	2	q[2]q	q[2]q	NOUN
ejpam-5661	280	3	(	(	PUNCT
ejpam-5661	280	4	1	1	NUM
ejpam-5661	280	5	+	+	NUM
ejpam-5661	280	6	q2	q2	NOUN
ejpam-5661	280	7	)	)	PUNCT
ejpam-5661	280	8	(	(	PUNCT
ejpam-5661	280	9	φ(α	φ(α	PROPN
ejpam-5661	280	10	)	)	PUNCT
ejpam-5661	280	11	+	+	CCONJ
ejpam-5661	280	12	φ(υ	φ(υ	PROPN
ejpam-5661	280	13	)	)	PUNCT
ejpam-5661	280	14	2	2	NUM
ejpam-5661	280	15	−	−	PROPN
ejpam-5661	280	16	φq(υ−	φq(υ−	NOUN
ejpam-5661	280	17	α)2	α)2	VERB
ejpam-5661	280	18	4([2]q)2[3]q	4([2]q)2[3]q	NUM
ejpam-5661	280	19	(	(	PUNCT
ejpam-5661	280	20	3	3	NUM
ejpam-5661	280	21	+	+	NUM
ejpam-5661	280	22	2q	2q	NOUN
ejpam-5661	280	23	+	+	CCONJ
ejpam-5661	280	24	3q2	3q2	NUM
ejpam-5661	280	25	)	)	PUNCT
ejpam-5661	280	26	)	)	PUNCT
ejpam-5661	281	1	+	+	CCONJ
ejpam-5661	281	2	(	(	PUNCT
ejpam-5661	281	3	1−	1−	NUM
ejpam-5661	281	4	q	q	NOUN
ejpam-5661	281	5	)	)	PUNCT
ejpam-5661	281	6	(	(	PUNCT
ejpam-5661	281	7	1	1	NUM
ejpam-5661	281	8	+	+	NUM
ejpam-5661	281	9	q2	q2	NOUN
ejpam-5661	281	10	)	)	PUNCT
ejpam-5661	281	11	φ	φ	PROPN
ejpam-5661	281	12	(	(	PUNCT
ejpam-5661	281	13	α+υ	α+υ	NUM
ejpam-5661	281	14	2	2	NUM
ejpam-5661	281	15	)	)	PUNCT
ejpam-5661	281	16	.	.	PUNCT
ejpam-5661	282	1	proof	proof	NOUN
ejpam-5661	282	2	.	.	PUNCT
ejpam-5661	283	1	setting	set	VERB
ejpam-5661	283	2	θ	θ	NOUN
ejpam-5661	283	3	=	=	SYM
ejpam-5661	283	4	1/2	1/2	NUM
ejpam-5661	283	5	in	in	ADP
ejpam-5661	283	6	theorem	theorem	NOUN
ejpam-5661	283	7	5	5	NUM
ejpam-5661	283	8	,	,	PUNCT
ejpam-5661	283	9	gives	give	VERB
ejpam-5661	283	10	us	we	PRON
ejpam-5661	283	11	φ	φ	PROPN
ejpam-5661	283	12	(	(	PUNCT
ejpam-5661	283	13	α+υ	α+υ	NUM
ejpam-5661	283	14	2	2	NUM
ejpam-5661	283	15	)	)	PUNCT
ejpam-5661	283	16	≤	≤	NOUN
ejpam-5661	283	17	φ	φ	PROPN
ejpam-5661	283	18	(	(	PUNCT
ejpam-5661	283	19	α+υ	α+υ	NUM
ejpam-5661	283	20	2	2	NUM
ejpam-5661	283	21	)	)	PUNCT
ejpam-5661	283	22	+	+	CCONJ
ejpam-5661	283	23	φ	φ	PROPN
ejpam-5661	283	24	4	4	NUM
ejpam-5661	283	25	(	(	PUNCT
ejpam-5661	283	26	(	(	PUNCT
ejpam-5661	283	27	1−	1−	NUM
ejpam-5661	283	28	q)(υ−	q)(υ−	PROPN
ejpam-5661	283	29	α	α	NOUN
ejpam-5661	283	30	)	)	PUNCT
ejpam-5661	284	1	[	[	X
ejpam-5661	284	2	2]q	2]q	NUM
ejpam-5661	284	3	)	)	PUNCT
ejpam-5661	284	4	2	2	NUM
ejpam-5661	284	5	≤	≤	NUM
ejpam-5661	284	6	1	1	NUM
ejpam-5661	284	7	2	2	NUM
ejpam-5661	284	8	(	(	PUNCT
ejpam-5661	284	9	φ	φ	PROPN
ejpam-5661	284	10	(	(	PUNCT
ejpam-5661	284	11	α+	α+	PROPN
ejpam-5661	284	12	qυ	qυ	X
ejpam-5661	285	1	[	[	X
ejpam-5661	285	2	2]q	2]q	NUM
ejpam-5661	285	3	)	)	PUNCT
ejpam-5661	286	1	+	+	NOUN
ejpam-5661	286	2	φ	φ	PROPN
ejpam-5661	286	3	(	(	PUNCT
ejpam-5661	286	4	qα+υ	qα+υ	PROPN
ejpam-5661	286	5	[	[	X
ejpam-5661	286	6	2]q	2]q	NUM
ejpam-5661	286	7	)	)	PUNCT
ejpam-5661	286	8	)	)	PUNCT
ejpam-5661	286	9	≤	≤	NUM
ejpam-5661	286	10	1	1	NUM
ejpam-5661	286	11	+	+	NUM
ejpam-5661	286	12	q2	q2	PROPN
ejpam-5661	286	13	q[2]q(υ−	q[2]q(υ−	PROPN
ejpam-5661	286	14	α	α	NOUN
ejpam-5661	286	15	)	)	PUNCT
ejpam-5661	286	16	(	(	PUNCT
ejpam-5661	286	17	∫	∫	PROPN
ejpam-5661	286	18	α+υ	α+υ	NUM
ejpam-5661	286	19	2	2	NUM
ejpam-5661	286	20	α	α	NOUN
ejpam-5661	286	21	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	286	22	+	+	CCONJ
ejpam-5661	286	23	∫	∫	PROPN
ejpam-5661	286	24	υ	υ	X
ejpam-5661	286	25	α+υ	α+υ	NUM
ejpam-5661	286	26	2	2	NUM
ejpam-5661	286	27	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	286	28	)	)	PUNCT
ejpam-5661	286	29	υdqϖ	υdqϖ	NOUN
ejpam-5661	286	30	)	)	PUNCT
ejpam-5661	287	1	−	−	PROPN
ejpam-5661	288	1	1−	1−	NUM
ejpam-5661	288	2	q	q	PROPN
ejpam-5661	289	1	q[2]q	q[2]q	PROPN
ejpam-5661	289	2	φ	φ	X
ejpam-5661	289	3	(	(	PUNCT
ejpam-5661	289	4	α+υ	α+υ	NUM
ejpam-5661	289	5	2	2	NUM
ejpam-5661	289	6	)	)	PUNCT
ejpam-5661	289	7	≤	≤	NOUN
ejpam-5661	289	8	φ(α	φ(α	NOUN
ejpam-5661	289	9	)	)	PUNCT
ejpam-5661	290	1	+	+	CCONJ
ejpam-5661	291	1	φ(υ	φ(υ	PROPN
ejpam-5661	291	2	)	)	PUNCT
ejpam-5661	291	3	2	2	NUM
ejpam-5661	291	4	−	−	PROPN
ejpam-5661	291	5	φq(υ−	φq(υ−	NOUN
ejpam-5661	291	6	a)2	a)2	PROPN
ejpam-5661	291	7	4([2]q)2[3]q	4([2]q)2[3]q	NUM
ejpam-5661	291	8	(	(	PUNCT
ejpam-5661	291	9	3	3	NUM
ejpam-5661	291	10	+	+	NUM
ejpam-5661	291	11	2q	2q	NOUN
ejpam-5661	291	12	+	+	CCONJ
ejpam-5661	291	13	3q2	3q2	NUM
ejpam-5661	291	14	)	)	PUNCT
ejpam-5661	292	1	≤	≤	NUM
ejpam-5661	292	2	φ(α	φ(α	ADJ
ejpam-5661	292	3	)	)	PUNCT
ejpam-5661	292	4	+	+	CCONJ
ejpam-5661	292	5	φ(υ	φ(υ	PROPN
ejpam-5661	292	6	)	)	PUNCT
ejpam-5661	292	7	2	2	NUM
ejpam-5661	292	8	.	.	PUNCT
ejpam-5661	293	1	this	this	PRON
ejpam-5661	293	2	implies	imply	VERB
ejpam-5661	293	3	that	that	SCONJ
ejpam-5661	293	4	(	(	PUNCT
ejpam-5661	293	5	1	1	NUM
ejpam-5661	293	6	+	+	NUM
ejpam-5661	293	7	q2	q2	X
ejpam-5661	293	8	q[2]q	q[2]q	NOUN
ejpam-5661	293	9	)	)	PUNCT
ejpam-5661	293	10	φ	φ	PROPN
ejpam-5661	293	11	(	(	PUNCT
ejpam-5661	293	12	α+υ	α+υ	NUM
ejpam-5661	293	13	2	2	NUM
ejpam-5661	293	14	)	)	PUNCT
ejpam-5661	294	1	=	=	SYM
ejpam-5661	294	2	φ	φ	PROPN
ejpam-5661	294	3	(	(	PUNCT
ejpam-5661	294	4	α+υ	α+υ	NUM
ejpam-5661	294	5	2	2	NUM
ejpam-5661	294	6	)	)	PUNCT
ejpam-5661	294	7	+	+	CCONJ
ejpam-5661	294	8	(	(	PUNCT
ejpam-5661	294	9	1−	1−	NUM
ejpam-5661	294	10	q	q	NOUN
ejpam-5661	294	11	)	)	PUNCT
ejpam-5661	294	12	q[2]q	q[2]q	NOUN
ejpam-5661	294	13	φ	φ	PROPN
ejpam-5661	294	14	(	(	PUNCT
ejpam-5661	294	15	α+υ	α+υ	NUM
ejpam-5661	294	16	2	2	NUM
ejpam-5661	294	17	)	)	PUNCT
ejpam-5661	294	18	≤	≤	NOUN
ejpam-5661	294	19	φ	φ	PROPN
ejpam-5661	294	20	(	(	PUNCT
ejpam-5661	294	21	α+υ	α+υ	NUM
ejpam-5661	294	22	2	2	NUM
ejpam-5661	294	23	)	)	PUNCT
ejpam-5661	294	24	+	+	CCONJ
ejpam-5661	294	25	(	(	PUNCT
ejpam-5661	294	26	1−	1−	NUM
ejpam-5661	294	27	q	q	NOUN
ejpam-5661	294	28	)	)	PUNCT
ejpam-5661	294	29	q[2]q	q[2]q	NOUN
ejpam-5661	294	30	φ	φ	X
ejpam-5661	294	31	(	(	PUNCT
ejpam-5661	294	32	a+υ	a+υ	X
ejpam-5661	294	33	2	2	NUM
ejpam-5661	294	34	)	)	PUNCT
ejpam-5661	294	35	+	+	CCONJ
ejpam-5661	294	36	φ	φ	PROPN
ejpam-5661	294	37	4	4	NUM
ejpam-5661	294	38	(	(	PUNCT
ejpam-5661	294	39	(	(	PUNCT
ejpam-5661	294	40	1−	1−	NUM
ejpam-5661	294	41	q)(υ−	q)(υ−	PROPN
ejpam-5661	294	42	α	α	NOUN
ejpam-5661	294	43	)	)	PUNCT
ejpam-5661	295	1	[	[	X
ejpam-5661	295	2	2]q	2]q	NUM
ejpam-5661	295	3	)	)	PUNCT
ejpam-5661	295	4	2	2	NUM
ejpam-5661	295	5	≤	≤	NUM
ejpam-5661	295	6	1	1	NUM
ejpam-5661	295	7	2	2	NUM
ejpam-5661	295	8	(	(	PUNCT
ejpam-5661	295	9	φ	φ	PROPN
ejpam-5661	295	10	(	(	PUNCT
ejpam-5661	295	11	α+	α+	PROPN
ejpam-5661	295	12	qυ	qυ	X
ejpam-5661	296	1	[	[	X
ejpam-5661	296	2	2]q	2]q	NUM
ejpam-5661	296	3	)	)	PUNCT
ejpam-5661	297	1	+	+	NOUN
ejpam-5661	297	2	φ	φ	PROPN
ejpam-5661	297	3	(	(	PUNCT
ejpam-5661	297	4	qα+υ	qα+υ	PROPN
ejpam-5661	298	1	[	[	X
ejpam-5661	298	2	2]q	2]q	NUM
ejpam-5661	298	3	)	)	PUNCT
ejpam-5661	298	4	)	)	PUNCT
ejpam-5661	299	1	+	+	CCONJ
ejpam-5661	299	2	(	(	PUNCT
ejpam-5661	299	3	1−	1−	NUM
ejpam-5661	299	4	q	q	NOUN
ejpam-5661	299	5	)	)	PUNCT
ejpam-5661	299	6	q[2]q	q[2]q	NOUN
ejpam-5661	299	7	φ	φ	PROPN
ejpam-5661	299	8	(	(	PUNCT
ejpam-5661	299	9	α+υ	α+υ	NUM
ejpam-5661	299	10	2	2	X
ejpam-5661	299	11	)	)	PUNCT
ejpam-5661	299	12	c.	c.	PROPN
ejpam-5661	299	13	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	299	14	,	,	PUNCT
ejpam-5661	299	15	p.	p.	PROPN
ejpam-5661	299	16	yotkaew	yotkaew	PROPN
ejpam-5661	299	17	/	/	SYM
ejpam-5661	299	18	eur	eur	PROPN
ejpam-5661	299	19	.	.	PUNCT
ejpam-5661	300	1	j.	j.	PROPN
ejpam-5661	300	2	pure	pure	PROPN
ejpam-5661	300	3	appl	appl	PROPN
ejpam-5661	300	4	.	.	PROPN
ejpam-5661	300	5	math	math	PROPN
ejpam-5661	300	6	,	,	PUNCT
ejpam-5661	300	7	18	18	NUM
ejpam-5661	300	8	(	(	PUNCT
ejpam-5661	300	9	1	1	NUM
ejpam-5661	300	10	)	)	PUNCT
ejpam-5661	300	11	(	(	PUNCT
ejpam-5661	300	12	2025	2025	NUM
ejpam-5661	300	13	)	)	PUNCT
ejpam-5661	300	14	,	,	PUNCT
ejpam-5661	300	15	5661	5661	NUM
ejpam-5661	300	16	12	12	NUM
ejpam-5661	300	17	of	of	ADP
ejpam-5661	300	18	24	24	NUM
ejpam-5661	300	19	≤	≤	NUM
ejpam-5661	300	20	1	1	NUM
ejpam-5661	300	21	+	+	NUM
ejpam-5661	300	22	q2	q2	PROPN
ejpam-5661	300	23	q[2]q(υ−	q[2]q(υ−	PROPN
ejpam-5661	300	24	α	α	NOUN
ejpam-5661	300	25	)	)	PUNCT
ejpam-5661	300	26	(	(	PUNCT
ejpam-5661	300	27	∫	∫	PROPN
ejpam-5661	300	28	α+υ	α+υ	NUM
ejpam-5661	300	29	2	2	NUM
ejpam-5661	300	30	α	α	NOUN
ejpam-5661	300	31	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	300	32	+	+	CCONJ
ejpam-5661	300	33	∫	∫	PROPN
ejpam-5661	300	34	υ	υ	X
ejpam-5661	300	35	α+υ	α+υ	NUM
ejpam-5661	300	36	2	2	NUM
ejpam-5661	300	37	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	300	38	)	)	PUNCT
ejpam-5661	300	39	υdqϖ	υdqϖ	NOUN
ejpam-5661	300	40	)	)	PUNCT
ejpam-5661	300	41	−	−	PROPN
ejpam-5661	301	1	(	(	PUNCT
ejpam-5661	301	2	1−	1−	NUM
ejpam-5661	301	3	q	q	NOUN
ejpam-5661	301	4	)	)	PUNCT
ejpam-5661	301	5	q[2]q	q[2]q	NOUN
ejpam-5661	301	6	φ	φ	PROPN
ejpam-5661	301	7	(	(	PUNCT
ejpam-5661	301	8	α+υ	α+υ	NUM
ejpam-5661	301	9	2	2	NUM
ejpam-5661	301	10	)	)	PUNCT
ejpam-5661	301	11	+	+	CCONJ
ejpam-5661	301	12	(	(	PUNCT
ejpam-5661	301	13	1−	1−	NUM
ejpam-5661	301	14	q	q	NOUN
ejpam-5661	301	15	)	)	PUNCT
ejpam-5661	301	16	q[2]q	q[2]q	NOUN
ejpam-5661	301	17	φ	φ	PROPN
ejpam-5661	301	18	(	(	PUNCT
ejpam-5661	301	19	α+υ	α+υ	NUM
ejpam-5661	301	20	2	2	NUM
ejpam-5661	301	21	)	)	PUNCT
ejpam-5661	301	22	≤	≤	NOUN
ejpam-5661	301	23	φ(α	φ(α	NOUN
ejpam-5661	301	24	)	)	PUNCT
ejpam-5661	301	25	+	+	CCONJ
ejpam-5661	301	26	φ(υ	φ(υ	PROPN
ejpam-5661	301	27	)	)	PUNCT
ejpam-5661	301	28	2	2	NUM
ejpam-5661	301	29	−	−	PROPN
ejpam-5661	301	30	φq(υ−	φq(υ−	NOUN
ejpam-5661	301	31	α)2	α)2	VERB
ejpam-5661	301	32	4([2]q)2[3]q	4([2]q)2[3]q	NUM
ejpam-5661	301	33	(	(	PUNCT
ejpam-5661	301	34	3	3	NUM
ejpam-5661	301	35	+	+	NUM
ejpam-5661	301	36	2q	2q	NOUN
ejpam-5661	301	37	+	+	CCONJ
ejpam-5661	301	38	3q2	3q2	NUM
ejpam-5661	301	39	)	)	PUNCT
ejpam-5661	302	1	+	+	CCONJ
ejpam-5661	302	2	(	(	PUNCT
ejpam-5661	302	3	1−	1−	NUM
ejpam-5661	302	4	q	q	NOUN
ejpam-5661	302	5	)	)	PUNCT
ejpam-5661	302	6	q[2]q	q[2]q	NOUN
ejpam-5661	302	7	φ	φ	PROPN
ejpam-5661	302	8	(	(	PUNCT
ejpam-5661	302	9	α+	α+	PROPN
ejpam-5661	302	10	b	b	PROPN
ejpam-5661	302	11	2	2	X
ejpam-5661	302	12	)	)	PUNCT
ejpam-5661	302	13	≤	≤	NOUN
ejpam-5661	302	14	φ(α	φ(α	NOUN
ejpam-5661	302	15	)	)	PUNCT
ejpam-5661	302	16	+	+	CCONJ
ejpam-5661	302	17	φ(υ	φ(υ	PROPN
ejpam-5661	302	18	)	)	PUNCT
ejpam-5661	302	19	2	2	NUM
ejpam-5661	302	20	+	+	SYM
ejpam-5661	302	21	1−	1−	NUM
ejpam-5661	302	22	q	q	NOUN
ejpam-5661	302	23	q[2]q	q[2]q	PROPN
ejpam-5661	302	24	(	(	PUNCT
ejpam-5661	302	25	φ(α	φ(α	PROPN
ejpam-5661	302	26	)	)	PUNCT
ejpam-5661	302	27	+	+	CCONJ
ejpam-5661	302	28	φ(υ	φ(υ	PROPN
ejpam-5661	302	29	)	)	PUNCT
ejpam-5661	302	30	2	2	NUM
ejpam-5661	302	31	)	)	PUNCT
ejpam-5661	302	32	=	=	SYM
ejpam-5661	302	33	1	1	NUM
ejpam-5661	302	34	+	+	NUM
ejpam-5661	302	35	q2	q2	X
ejpam-5661	302	36	q[2]q	q[2]q	NOUN
ejpam-5661	302	37	(	(	PUNCT
ejpam-5661	302	38	φ(α	φ(α	PROPN
ejpam-5661	302	39	)	)	PUNCT
ejpam-5661	302	40	+	+	ADJ
ejpam-5661	302	41	φ(b	φ(b	NOUN
ejpam-5661	302	42	)	)	PUNCT
ejpam-5661	302	43	2	2	NUM
ejpam-5661	302	44	)	)	PUNCT
ejpam-5661	302	45	.	.	PUNCT
ejpam-5661	303	1	(	(	PUNCT
ejpam-5661	303	2	22	22	NUM
ejpam-5661	303	3	)	)	PUNCT
ejpam-5661	303	4	multiply	multiply	ADP
ejpam-5661	303	5	the	the	DET
ejpam-5661	303	6	inequality	inequality	NOUN
ejpam-5661	303	7	(	(	PUNCT
ejpam-5661	303	8	22	22	NUM
ejpam-5661	303	9	)	)	PUNCT
ejpam-5661	303	10	by	by	ADP
ejpam-5661	303	11	q[2]q/(1	q[2]q/(1	PROPN
ejpam-5661	303	12	+	+	NUM
ejpam-5661	303	13	q2	q2	NOUN
ejpam-5661	303	14	)	)	PUNCT
ejpam-5661	304	1	leads	lead	VERB
ejpam-5661	304	2	us	we	PRON
ejpam-5661	304	3	to	to	ADP
ejpam-5661	304	4	the	the	DET
ejpam-5661	304	5	desired	desire	VERB
ejpam-5661	304	6	result	result	NOUN
ejpam-5661	304	7	.	.	PUNCT
ejpam-5661	305	1	the	the	DET
ejpam-5661	305	2	result	result	NOUN
ejpam-5661	305	3	presented	present	VERB
ejpam-5661	305	4	below	below	ADV
ejpam-5661	305	5	provides	provide	VERB
ejpam-5661	305	6	refinements	refinement	NOUN
ejpam-5661	305	7	for	for	ADP
ejpam-5661	305	8	the	the	DET
ejpam-5661	305	9	inequalities	inequality	NOUN
ejpam-5661	305	10	(	(	PUNCT
ejpam-5661	305	11	3	3	NUM
ejpam-5661	305	12	)	)	PUNCT
ejpam-5661	305	13	.	.	PUNCT
ejpam-5661	306	1	corollary	corollary	ADJ
ejpam-5661	306	2	2	2	NUM
ejpam-5661	306	3	.	.	PUNCT
ejpam-5661	306	4	setting	set	VERB
ejpam-5661	306	5	θ	θ	NOUN
ejpam-5661	306	6	=	=	SYM
ejpam-5661	306	7	1	1	NUM
ejpam-5661	306	8	in	in	ADP
ejpam-5661	306	9	theorem	theorem	NOUN
ejpam-5661	306	10	5	5	NUM
ejpam-5661	306	11	,	,	PUNCT
ejpam-5661	306	12	then	then	ADV
ejpam-5661	306	13	we	we	PRON
ejpam-5661	306	14	obtain	obtain	VERB
ejpam-5661	306	15	φ	φ	PROPN
ejpam-5661	306	16	(	(	PUNCT
ejpam-5661	306	17	α+υ	α+υ	NUM
ejpam-5661	306	18	2	2	NUM
ejpam-5661	306	19	)	)	PUNCT
ejpam-5661	306	20	≤	≤	NOUN
ejpam-5661	306	21	φ4(q	φ4(q	NOUN
ejpam-5661	306	22	)	)	PUNCT
ejpam-5661	306	23	≤	≤	NUM
ejpam-5661	306	24	φ5(q	φ5(q	NOUN
ejpam-5661	306	25	)	)	PUNCT
ejpam-5661	306	26	≤	≤	NUM
ejpam-5661	306	27	1	1	NUM
ejpam-5661	306	28	2(υ−	2(υ−	NUM
ejpam-5661	306	29	α	α	NOUN
ejpam-5661	306	30	)	)	PUNCT
ejpam-5661	306	31	(	(	PUNCT
ejpam-5661	306	32	∫	∫	PROPN
ejpam-5661	306	33	υ	υ	PROPN
ejpam-5661	306	34	α	α	PROPN
ejpam-5661	306	35	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	ADJ
ejpam-5661	307	1	+	+	CCONJ
ejpam-5661	307	2	∫	∫	PROPN
ejpam-5661	307	3	υ	υ	PROPN
ejpam-5661	307	4	α	α	PROPN
ejpam-5661	307	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	307	6	)	)	PUNCT
ejpam-5661	307	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	307	8	)	)	PUNCT
ejpam-5661	307	9	≤	≤	NOUN
ejpam-5661	307	10	φ6(q	φ6(q	NOUN
ejpam-5661	307	11	)	)	PUNCT
ejpam-5661	307	12	≤	≤	NOUN
ejpam-5661	308	1	φ(α	φ(α	NOUN
ejpam-5661	308	2	)	)	PUNCT
ejpam-5661	309	1	+	+	CCONJ
ejpam-5661	309	2	φ(υ	φ(υ	PROPN
ejpam-5661	309	3	)	)	PUNCT
ejpam-5661	309	4	2	2	NUM
ejpam-5661	309	5	,	,	PUNCT
ejpam-5661	309	6	(	(	PUNCT
ejpam-5661	309	7	23	23	NUM
ejpam-5661	309	8	)	)	PUNCT
ejpam-5661	309	9	where	where	SCONJ
ejpam-5661	309	10	φ4(q	φ4(q	X
ejpam-5661	309	11	)	)	PUNCT
ejpam-5661	309	12	=	=	SYM
ejpam-5661	309	13	q[2]q	q[2]q	NOUN
ejpam-5661	309	14	(	(	PUNCT
ejpam-5661	309	15	1	1	NUM
ejpam-5661	309	16	+	+	NUM
ejpam-5661	309	17	q2	q2	NOUN
ejpam-5661	309	18	)	)	PUNCT
ejpam-5661	310	1	(	(	PUNCT
ejpam-5661	310	2	φ	φ	X
ejpam-5661	310	3	(	(	PUNCT
ejpam-5661	310	4	α+υ	α+υ	NUM
ejpam-5661	310	5	2	2	NUM
ejpam-5661	310	6	)	)	PUNCT
ejpam-5661	310	7	+	+	CCONJ
ejpam-5661	310	8	φ	φ	PROPN
ejpam-5661	310	9	4	4	NUM
ejpam-5661	310	10	(	(	PUNCT
ejpam-5661	310	11	(	(	PUNCT
ejpam-5661	310	12	1−	1−	NUM
ejpam-5661	310	13	q)(υ−	q)(υ−	PROPN
ejpam-5661	310	14	α	α	NOUN
ejpam-5661	310	15	)	)	PUNCT
ejpam-5661	311	1	[	[	X
ejpam-5661	311	2	2]q	2]q	NUM
ejpam-5661	311	3	)	)	PUNCT
ejpam-5661	311	4	2	2	NUM
ejpam-5661	311	5	)	)	PUNCT
ejpam-5661	311	6	+	+	CCONJ
ejpam-5661	311	7	(	(	PUNCT
ejpam-5661	311	8	1−	1−	NUM
ejpam-5661	311	9	q	q	NOUN
ejpam-5661	311	10	)	)	PUNCT
ejpam-5661	311	11	(	(	PUNCT
ejpam-5661	311	12	1	1	NUM
ejpam-5661	311	13	+	+	NUM
ejpam-5661	311	14	q2	q2	NOUN
ejpam-5661	311	15	)	)	PUNCT
ejpam-5661	311	16	φ	φ	PROPN
ejpam-5661	311	17	(	(	PUNCT
ejpam-5661	311	18	α+υ	α+υ	NUM
ejpam-5661	311	19	2	2	NUM
ejpam-5661	311	20	)	)	PUNCT
ejpam-5661	311	21	,	,	PUNCT
ejpam-5661	311	22	φ5(q	φ5(q	PROPN
ejpam-5661	311	23	)	)	PUNCT
ejpam-5661	311	24	=	=	PUNCT
ejpam-5661	311	25	q[2]q	q[2]q	PRON
ejpam-5661	311	26	2(1	2(1	NUM
ejpam-5661	311	27	+	+	NUM
ejpam-5661	311	28	q2	q2	NOUN
ejpam-5661	311	29	)	)	PUNCT
ejpam-5661	311	30	(	(	PUNCT
ejpam-5661	311	31	φ	φ	X
ejpam-5661	311	32	(	(	PUNCT
ejpam-5661	311	33	α+	α+	PROPN
ejpam-5661	311	34	qυ	qυ	X
ejpam-5661	312	1	[	[	X
ejpam-5661	312	2	2]q	2]q	NUM
ejpam-5661	312	3	)	)	PUNCT
ejpam-5661	313	1	+	+	NOUN
ejpam-5661	313	2	φ	φ	PROPN
ejpam-5661	313	3	(	(	PUNCT
ejpam-5661	313	4	qα+υ	qα+υ	PROPN
ejpam-5661	314	1	[	[	X
ejpam-5661	314	2	2]q	2]q	NUM
ejpam-5661	314	3	)	)	PUNCT
ejpam-5661	314	4	)	)	PUNCT
ejpam-5661	315	1	+	+	CCONJ
ejpam-5661	315	2	(	(	PUNCT
ejpam-5661	315	3	1−	1−	NUM
ejpam-5661	315	4	q	q	NOUN
ejpam-5661	315	5	)	)	PUNCT
ejpam-5661	315	6	(	(	PUNCT
ejpam-5661	315	7	1	1	NUM
ejpam-5661	315	8	+	+	NUM
ejpam-5661	315	9	q2	q2	NOUN
ejpam-5661	315	10	)	)	PUNCT
ejpam-5661	315	11	φ	φ	PROPN
ejpam-5661	315	12	(	(	PUNCT
ejpam-5661	315	13	α+υ	α+υ	NUM
ejpam-5661	315	14	2	2	NUM
ejpam-5661	315	15	)	)	PUNCT
ejpam-5661	315	16	and	and	CCONJ
ejpam-5661	315	17	φ6(q	φ6(q	NUM
ejpam-5661	315	18	)	)	PUNCT
ejpam-5661	315	19	=	=	SYM
ejpam-5661	315	20	q[2]q	q[2]q	NOUN
ejpam-5661	315	21	(	(	PUNCT
ejpam-5661	315	22	1	1	NUM
ejpam-5661	315	23	+	+	NUM
ejpam-5661	315	24	q2	q2	NOUN
ejpam-5661	315	25	)	)	PUNCT
ejpam-5661	315	26	(	(	PUNCT
ejpam-5661	315	27	φ(α	φ(α	PROPN
ejpam-5661	315	28	)	)	PUNCT
ejpam-5661	315	29	+	+	CCONJ
ejpam-5661	315	30	φ(υ	φ(υ	PROPN
ejpam-5661	315	31	)	)	PUNCT
ejpam-5661	315	32	2	2	NUM
ejpam-5661	315	33	)	)	PUNCT
ejpam-5661	315	34	−	−	PROPN
ejpam-5661	315	35	φq2(υ−	φq2(υ−	NOUN
ejpam-5661	315	36	α)2	α)2	NOUN
ejpam-5661	316	1	[	[	X
ejpam-5661	316	2	2]q[3]q	2]q[3]q	NUM
ejpam-5661	316	3	+	+	CCONJ
ejpam-5661	316	4	(	(	PUNCT
ejpam-5661	316	5	1−	1−	NUM
ejpam-5661	316	6	q	q	NOUN
ejpam-5661	316	7	)	)	PUNCT
ejpam-5661	316	8	(	(	PUNCT
ejpam-5661	316	9	1	1	NUM
ejpam-5661	316	10	+	+	NUM
ejpam-5661	316	11	q2	q2	NOUN
ejpam-5661	316	12	)	)	PUNCT
ejpam-5661	316	13	(	(	PUNCT
ejpam-5661	316	14	φ(α	φ(α	PROPN
ejpam-5661	316	15	)	)	PUNCT
ejpam-5661	316	16	+	+	CCONJ
ejpam-5661	316	17	φ(υ	φ(υ	PROPN
ejpam-5661	316	18	)	)	PUNCT
ejpam-5661	316	19	2	2	NUM
ejpam-5661	316	20	)	)	PUNCT
ejpam-5661	316	21	.	.	PUNCT
ejpam-5661	317	1	proof	proof	NOUN
ejpam-5661	317	2	.	.	PUNCT
ejpam-5661	318	1	setting	set	VERB
ejpam-5661	318	2	θ	θ	NOUN
ejpam-5661	318	3	=	=	SYM
ejpam-5661	318	4	1	1	NUM
ejpam-5661	318	5	in	in	ADP
ejpam-5661	318	6	theorem	theorem	NOUN
ejpam-5661	318	7	5	5	NUM
ejpam-5661	318	8	gives	give	VERB
ejpam-5661	318	9	us	we	PRON
ejpam-5661	318	10	φ	φ	PROPN
ejpam-5661	318	11	(	(	PUNCT
ejpam-5661	318	12	α+υ	α+υ	NUM
ejpam-5661	318	13	2	2	NUM
ejpam-5661	318	14	)	)	PUNCT
ejpam-5661	318	15	≤	≤	NOUN
ejpam-5661	318	16	φ	φ	PROPN
ejpam-5661	318	17	(	(	PUNCT
ejpam-5661	318	18	α+υ	α+υ	NUM
ejpam-5661	318	19	2	2	NUM
ejpam-5661	318	20	)	)	PUNCT
ejpam-5661	318	21	+	+	CCONJ
ejpam-5661	318	22	φ	φ	PROPN
ejpam-5661	318	23	4	4	NUM
ejpam-5661	318	24	(	(	PUNCT
ejpam-5661	318	25	(	(	PUNCT
ejpam-5661	318	26	1−	1−	NUM
ejpam-5661	318	27	q)(υ−	q)(υ−	PROPN
ejpam-5661	318	28	α	α	NOUN
ejpam-5661	318	29	)	)	PUNCT
ejpam-5661	319	1	[	[	X
ejpam-5661	319	2	2]q	2]q	NUM
ejpam-5661	319	3	)	)	PUNCT
ejpam-5661	319	4	2	2	NUM
ejpam-5661	319	5	≤	≤	NUM
ejpam-5661	319	6	1	1	NUM
ejpam-5661	319	7	2	2	NUM
ejpam-5661	319	8	(	(	PUNCT
ejpam-5661	319	9	φ	φ	PROPN
ejpam-5661	319	10	(	(	PUNCT
ejpam-5661	319	11	α+	α+	PROPN
ejpam-5661	319	12	qυ	qυ	X
ejpam-5661	320	1	[	[	X
ejpam-5661	320	2	2]q	2]q	NUM
ejpam-5661	320	3	)	)	PUNCT
ejpam-5661	321	1	+	+	NOUN
ejpam-5661	321	2	φ	φ	PROPN
ejpam-5661	321	3	(	(	PUNCT
ejpam-5661	321	4	qα+υ	qα+υ	PROPN
ejpam-5661	322	1	[	[	X
ejpam-5661	322	2	2]q	2]q	NUM
ejpam-5661	322	3	)	)	PUNCT
ejpam-5661	322	4	)	)	PUNCT
ejpam-5661	322	5	c.	c.	PROPN
ejpam-5661	322	6	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	322	7	,	,	PUNCT
ejpam-5661	322	8	p.	p.	PROPN
ejpam-5661	322	9	yotkaew	yotkaew	PROPN
ejpam-5661	322	10	/	/	SYM
ejpam-5661	322	11	eur	eur	PROPN
ejpam-5661	322	12	.	.	PUNCT
ejpam-5661	323	1	j.	j.	PROPN
ejpam-5661	323	2	pure	pure	PROPN
ejpam-5661	323	3	appl	appl	PROPN
ejpam-5661	323	4	.	.	PROPN
ejpam-5661	323	5	math	math	PROPN
ejpam-5661	323	6	,	,	PUNCT
ejpam-5661	323	7	18	18	NUM
ejpam-5661	323	8	(	(	PUNCT
ejpam-5661	323	9	1	1	NUM
ejpam-5661	323	10	)	)	PUNCT
ejpam-5661	323	11	(	(	PUNCT
ejpam-5661	323	12	2025	2025	NUM
ejpam-5661	323	13	)	)	PUNCT
ejpam-5661	323	14	,	,	PUNCT
ejpam-5661	323	15	5661	5661	NUM
ejpam-5661	323	16	13	13	NUM
ejpam-5661	323	17	of	of	ADP
ejpam-5661	323	18	24	24	NUM
ejpam-5661	323	19	≤	≤	NUM
ejpam-5661	323	20	1	1	NUM
ejpam-5661	323	21	+	+	NUM
ejpam-5661	323	22	q2	q2	NOUN
ejpam-5661	323	23	2q[2]q(υ−	2q[2]q(υ−	NUM
ejpam-5661	323	24	α	α	NOUN
ejpam-5661	323	25	)	)	PUNCT
ejpam-5661	323	26	(	(	PUNCT
ejpam-5661	323	27	∫	∫	PROPN
ejpam-5661	323	28	υ	υ	PROPN
ejpam-5661	323	29	α	α	PROPN
ejpam-5661	323	30	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	ADJ
ejpam-5661	324	1	+	+	CCONJ
ejpam-5661	324	2	∫	∫	PROPN
ejpam-5661	324	3	υ	υ	PROPN
ejpam-5661	324	4	α	α	PROPN
ejpam-5661	324	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	324	6	)	)	PUNCT
ejpam-5661	324	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	324	8	)	)	PUNCT
ejpam-5661	325	1	−	−	PROPN
ejpam-5661	326	1	1−	1−	NUM
ejpam-5661	326	2	q	q	NOUN
ejpam-5661	326	3	2q[2]q	2q[2]q	NOUN
ejpam-5661	326	4	(	(	PUNCT
ejpam-5661	326	5	φ(α	φ(α	PROPN
ejpam-5661	326	6	)	)	PUNCT
ejpam-5661	326	7	+	+	CCONJ
ejpam-5661	326	8	φ(υ	φ(υ	PROPN
ejpam-5661	326	9	)	)	PUNCT
ejpam-5661	326	10	)	)	PUNCT
ejpam-5661	326	11	≤	≤	NUM
ejpam-5661	326	12	φ(α	φ(α	NOUN
ejpam-5661	326	13	)	)	PUNCT
ejpam-5661	326	14	+	+	CCONJ
ejpam-5661	326	15	φ(υ	φ(υ	PROPN
ejpam-5661	326	16	)	)	PUNCT
ejpam-5661	326	17	2	2	NUM
ejpam-5661	326	18	−	−	PROPN
ejpam-5661	326	19	φq(1	φq(1	PROPN
ejpam-5661	326	20	+	+	NUM
ejpam-5661	326	21	q2)(υ−	q2)(υ−	NOUN
ejpam-5661	326	22	α)2	α)2	NOUN
ejpam-5661	326	23	(	(	PUNCT
ejpam-5661	326	24	[	[	X
ejpam-5661	326	25	2]q)2[3]q	2]q)2[3]q	NUM
ejpam-5661	326	26	≤	≤	X
ejpam-5661	326	27	φ(α	φ(α	NOUN
ejpam-5661	326	28	)	)	PUNCT
ejpam-5661	326	29	+	+	CCONJ
ejpam-5661	326	30	φ(υ	φ(υ	PROPN
ejpam-5661	326	31	)	)	PUNCT
ejpam-5661	326	32	2	2	NUM
ejpam-5661	326	33	.	.	PUNCT
ejpam-5661	326	34	which	which	PRON
ejpam-5661	326	35	gives	give	VERB
ejpam-5661	326	36	(	(	PUNCT
ejpam-5661	326	37	1	1	NUM
ejpam-5661	326	38	+	+	NUM
ejpam-5661	326	39	q2	q2	X
ejpam-5661	326	40	q[2]q	q[2]q	NOUN
ejpam-5661	326	41	)	)	PUNCT
ejpam-5661	326	42	φ	φ	PROPN
ejpam-5661	326	43	(	(	PUNCT
ejpam-5661	326	44	α+υ	α+υ	NUM
ejpam-5661	326	45	2	2	NUM
ejpam-5661	326	46	)	)	PUNCT
ejpam-5661	326	47	=	=	SYM
ejpam-5661	326	48	φ	φ	PROPN
ejpam-5661	326	49	(	(	PUNCT
ejpam-5661	326	50	α+υ	α+υ	NUM
ejpam-5661	326	51	2	2	NUM
ejpam-5661	326	52	)	)	PUNCT
ejpam-5661	326	53	+	+	CCONJ
ejpam-5661	326	54	(	(	PUNCT
ejpam-5661	326	55	1−	1−	NUM
ejpam-5661	326	56	q	q	NOUN
ejpam-5661	326	57	)	)	PUNCT
ejpam-5661	326	58	q[2]q	q[2]q	NOUN
ejpam-5661	326	59	φ	φ	PROPN
ejpam-5661	326	60	(	(	PUNCT
ejpam-5661	326	61	α+υ	α+υ	NUM
ejpam-5661	326	62	2	2	NUM
ejpam-5661	326	63	)	)	PUNCT
ejpam-5661	326	64	≤	≤	NOUN
ejpam-5661	326	65	φ	φ	PROPN
ejpam-5661	326	66	(	(	PUNCT
ejpam-5661	326	67	α+υ	α+υ	NUM
ejpam-5661	326	68	2	2	NUM
ejpam-5661	326	69	)	)	PUNCT
ejpam-5661	326	70	+	+	CCONJ
ejpam-5661	326	71	φ	φ	PROPN
ejpam-5661	326	72	4	4	NUM
ejpam-5661	326	73	(	(	PUNCT
ejpam-5661	326	74	(	(	PUNCT
ejpam-5661	326	75	1−	1−	NUM
ejpam-5661	326	76	q)(υ−	q)(υ−	PROPN
ejpam-5661	326	77	α	α	NOUN
ejpam-5661	326	78	)	)	PUNCT
ejpam-5661	327	1	[	[	X
ejpam-5661	327	2	2]q	2]q	NUM
ejpam-5661	327	3	)	)	PUNCT
ejpam-5661	327	4	2	2	NUM
ejpam-5661	327	5	+	+	CCONJ
ejpam-5661	327	6	(	(	PUNCT
ejpam-5661	327	7	1−	1−	NUM
ejpam-5661	327	8	q	q	NOUN
ejpam-5661	327	9	)	)	PUNCT
ejpam-5661	327	10	q[2]q	q[2]q	NOUN
ejpam-5661	327	11	φ	φ	PROPN
ejpam-5661	327	12	(	(	PUNCT
ejpam-5661	327	13	α+υ	α+υ	NUM
ejpam-5661	327	14	2	2	NUM
ejpam-5661	327	15	)	)	PUNCT
ejpam-5661	327	16	≤	≤	NOUN
ejpam-5661	327	17	1	1	NUM
ejpam-5661	327	18	2	2	NUM
ejpam-5661	327	19	(	(	PUNCT
ejpam-5661	327	20	φ	φ	PROPN
ejpam-5661	327	21	(	(	PUNCT
ejpam-5661	327	22	α+	α+	PROPN
ejpam-5661	327	23	qυ	qυ	X
ejpam-5661	328	1	[	[	X
ejpam-5661	328	2	2]q	2]q	NUM
ejpam-5661	328	3	)	)	PUNCT
ejpam-5661	329	1	+	+	NOUN
ejpam-5661	329	2	φ	φ	PROPN
ejpam-5661	329	3	(	(	PUNCT
ejpam-5661	329	4	qα+υ	qα+υ	PROPN
ejpam-5661	330	1	[	[	X
ejpam-5661	330	2	2]q	2]q	NUM
ejpam-5661	330	3	)	)	PUNCT
ejpam-5661	330	4	)	)	PUNCT
ejpam-5661	331	1	+	+	CCONJ
ejpam-5661	331	2	(	(	PUNCT
ejpam-5661	331	3	1−	1−	NUM
ejpam-5661	331	4	q	q	NOUN
ejpam-5661	331	5	)	)	PUNCT
ejpam-5661	331	6	q[2]q	q[2]q	NOUN
ejpam-5661	331	7	φ	φ	PROPN
ejpam-5661	331	8	(	(	PUNCT
ejpam-5661	331	9	α+υ	α+υ	NUM
ejpam-5661	331	10	2	2	NUM
ejpam-5661	331	11	)	)	PUNCT
ejpam-5661	331	12	≤	≤	NOUN
ejpam-5661	331	13	1	1	NUM
ejpam-5661	331	14	+	+	NUM
ejpam-5661	331	15	q2	q2	NOUN
ejpam-5661	331	16	2q[2]q(υ−	2q[2]q(υ−	NUM
ejpam-5661	331	17	α	α	NOUN
ejpam-5661	331	18	)	)	PUNCT
ejpam-5661	331	19	(	(	PUNCT
ejpam-5661	331	20	∫	∫	PROPN
ejpam-5661	331	21	υ	υ	PROPN
ejpam-5661	331	22	α	α	PROPN
ejpam-5661	331	23	f(ϖ)αdqϖ	f(ϖ)αdqϖ	PROPN
ejpam-5661	332	1	+	+	CCONJ
ejpam-5661	332	2	∫	∫	PROPN
ejpam-5661	332	3	υ	υ	PROPN
ejpam-5661	332	4	α	α	PROPN
ejpam-5661	332	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	332	6	)	)	PUNCT
ejpam-5661	332	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	332	8	)	)	PUNCT
ejpam-5661	333	1	−	−	PROPN
ejpam-5661	333	2	(	(	PUNCT
ejpam-5661	333	3	1−	1−	NUM
ejpam-5661	333	4	q	q	NOUN
ejpam-5661	333	5	)	)	PUNCT
ejpam-5661	333	6	q[2]q	q[2]q	NOUN
ejpam-5661	333	7	(	(	PUNCT
ejpam-5661	333	8	φ(α	φ(α	PROPN
ejpam-5661	333	9	)	)	PUNCT
ejpam-5661	333	10	+	+	CCONJ
ejpam-5661	333	11	φ(υ	φ(υ	PROPN
ejpam-5661	333	12	)	)	PUNCT
ejpam-5661	333	13	2	2	NUM
ejpam-5661	333	14	)	)	PUNCT
ejpam-5661	334	1	+	+	CCONJ
ejpam-5661	334	2	(	(	PUNCT
ejpam-5661	334	3	1−	1−	NUM
ejpam-5661	334	4	q	q	NOUN
ejpam-5661	334	5	)	)	PUNCT
ejpam-5661	334	6	q[2]q	q[2]q	NOUN
ejpam-5661	334	7	(	(	PUNCT
ejpam-5661	334	8	φ(α	φ(α	PROPN
ejpam-5661	334	9	)	)	PUNCT
ejpam-5661	335	1	+	+	CCONJ
ejpam-5661	335	2	φ(υ	φ(υ	PROPN
ejpam-5661	335	3	)	)	PUNCT
ejpam-5661	335	4	2	2	NUM
ejpam-5661	335	5	)	)	PUNCT
ejpam-5661	335	6	≤	≤	NOUN
ejpam-5661	335	7	φ(α	φ(α	NOUN
ejpam-5661	335	8	)	)	PUNCT
ejpam-5661	336	1	+	+	CCONJ
ejpam-5661	336	2	φ(υ	φ(υ	PROPN
ejpam-5661	336	3	)	)	PUNCT
ejpam-5661	336	4	2	2	NUM
ejpam-5661	336	5	−	−	PROPN
ejpam-5661	336	6	φq(1	φq(1	PROPN
ejpam-5661	336	7	+	+	NUM
ejpam-5661	336	8	q2)(υ−	q2)(υ−	NOUN
ejpam-5661	336	9	α)2	α)2	NOUN
ejpam-5661	336	10	(	(	PUNCT
ejpam-5661	337	1	[	[	X
ejpam-5661	337	2	2]q)2[3]q	2]q)2[3]q	NUM
ejpam-5661	337	3	+	+	CCONJ
ejpam-5661	337	4	(	(	PUNCT
ejpam-5661	337	5	1−	1−	NUM
ejpam-5661	337	6	q	q	NOUN
ejpam-5661	337	7	)	)	PUNCT
ejpam-5661	337	8	q[2]q	q[2]q	NOUN
ejpam-5661	337	9	(	(	PUNCT
ejpam-5661	337	10	φ(α	φ(α	PROPN
ejpam-5661	337	11	)	)	PUNCT
ejpam-5661	337	12	+	+	CCONJ
ejpam-5661	337	13	φ(υ	φ(υ	PROPN
ejpam-5661	337	14	)	)	PUNCT
ejpam-5661	337	15	2	2	NUM
ejpam-5661	337	16	)	)	PUNCT
ejpam-5661	337	17	≤	≤	NOUN
ejpam-5661	337	18	φ(α	φ(α	NOUN
ejpam-5661	337	19	)	)	PUNCT
ejpam-5661	338	1	+	+	CCONJ
ejpam-5661	338	2	φ(υ	φ(υ	PROPN
ejpam-5661	338	3	)	)	PUNCT
ejpam-5661	338	4	2	2	NUM
ejpam-5661	339	1	+	+	CCONJ
ejpam-5661	339	2	(	(	PUNCT
ejpam-5661	339	3	1−	1−	NUM
ejpam-5661	339	4	q	q	NOUN
ejpam-5661	339	5	)	)	PUNCT
ejpam-5661	339	6	q[2]q	q[2]q	NOUN
ejpam-5661	339	7	(	(	PUNCT
ejpam-5661	339	8	φ(α	φ(α	PROPN
ejpam-5661	339	9	)	)	PUNCT
ejpam-5661	339	10	+	+	CCONJ
ejpam-5661	339	11	φ(υ	φ(υ	PROPN
ejpam-5661	339	12	)	)	PUNCT
ejpam-5661	339	13	2	2	NUM
ejpam-5661	339	14	)	)	PUNCT
ejpam-5661	339	15	=	=	PUNCT
ejpam-5661	340	1	(	(	PUNCT
ejpam-5661	340	2	1	1	NUM
ejpam-5661	340	3	+	+	NUM
ejpam-5661	340	4	q2	q2	X
ejpam-5661	340	5	q[2]q	q[2]q	NOUN
ejpam-5661	340	6	)	)	PUNCT
ejpam-5661	340	7	(	(	PUNCT
ejpam-5661	340	8	φ(α	φ(α	PROPN
ejpam-5661	340	9	)	)	PUNCT
ejpam-5661	340	10	+	+	CCONJ
ejpam-5661	340	11	φ(υ	φ(υ	PROPN
ejpam-5661	340	12	)	)	PUNCT
ejpam-5661	340	13	2	2	NUM
ejpam-5661	340	14	)	)	PUNCT
ejpam-5661	340	15	.	.	PUNCT
ejpam-5661	341	1	(	(	PUNCT
ejpam-5661	341	2	24	24	NUM
ejpam-5661	341	3	)	)	PUNCT
ejpam-5661	341	4	multiplying	multiply	VERB
ejpam-5661	341	5	the	the	DET
ejpam-5661	341	6	inequality	inequality	NOUN
ejpam-5661	341	7	(	(	PUNCT
ejpam-5661	341	8	24	24	NUM
ejpam-5661	341	9	)	)	PUNCT
ejpam-5661	341	10	by	by	ADP
ejpam-5661	341	11	q[2]q/(1	q[2]q/(1	PROPN
ejpam-5661	341	12	+	+	NUM
ejpam-5661	341	13	q2	q2	NOUN
ejpam-5661	341	14	)	)	PUNCT
ejpam-5661	341	15	yields	yield	VERB
ejpam-5661	341	16	the	the	DET
ejpam-5661	341	17	desired	desire	VERB
ejpam-5661	341	18	result	result	NOUN
ejpam-5661	341	19	.	.	PUNCT
ejpam-5661	342	1	4	4	X
ejpam-5661	342	2	.	.	X
ejpam-5661	342	3	parameterized	parameterized	ADJ
ejpam-5661	342	4	q	q	ADJ
ejpam-5661	342	5	-	-	ADJ
ejpam-5661	342	6	integral	integral	ADJ
ejpam-5661	342	7	inequalities	inequality	NOUN
ejpam-5661	342	8	we	we	PRON
ejpam-5661	342	9	prove	prove	VERB
ejpam-5661	342	10	the	the	DET
ejpam-5661	342	11	h	h	NOUN
ejpam-5661	342	12	-	-	PUNCT
ejpam-5661	342	13	h	h	NOUN
ejpam-5661	342	14	inequalities	inequality	NOUN
ejpam-5661	342	15	(	(	PUNCT
ejpam-5661	342	16	11	11	NUM
ejpam-5661	342	17	)	)	PUNCT
ejpam-5661	342	18	of	of	ADP
ejpam-5661	342	19	theorem	theorem	NOUN
ejpam-5661	342	20	4	4	NUM
ejpam-5661	342	21	by	by	ADP
ejpam-5661	342	22	utilizing	utilize	VERB
ejpam-5661	342	23	the	the	DET
ejpam-5661	342	24	q	q	NOUN
ejpam-5661	342	25	-	-	PUNCT
ejpam-5661	342	26	differentiability	differentiability	NOUN
ejpam-5661	342	27	of	of	ADP
ejpam-5661	342	28	the	the	DET
ejpam-5661	342	29	function	function	NOUN
ejpam-5661	342	30	.	.	PUNCT
ejpam-5661	343	1	lemma	lemma	PROPN
ejpam-5661	343	2	1	1	NUM
ejpam-5661	343	3	(	(	PUNCT
ejpam-5661	343	4	[	[	X
ejpam-5661	343	5	2	2	NUM
ejpam-5661	343	6	]	]	PUNCT
ejpam-5661	343	7	)	)	PUNCT
ejpam-5661	343	8	.	.	PUNCT
ejpam-5661	344	1	let	let	VERB
ejpam-5661	344	2	φ	φ	NOUN
ejpam-5661	344	3	:	:	PUNCT
ejpam-5661	345	1	[	[	X
ejpam-5661	345	2	α	α	X
ejpam-5661	345	3	,	,	PUNCT
ejpam-5661	345	4	υ	υ	NOUN
ejpam-5661	345	5	]	]	X
ejpam-5661	345	6	→	→	PUNCT
ejpam-5661	345	7	r	r	NOUN
ejpam-5661	345	8	be	be	AUX
ejpam-5661	345	9	a	a	DET
ejpam-5661	345	10	q	q	ADJ
ejpam-5661	345	11	-	-	PUNCT
ejpam-5661	345	12	differentiable	differentiable	ADJ
ejpam-5661	345	13	function	function	NOUN
ejpam-5661	345	14	.	.	PUNCT
ejpam-5661	346	1	if	if	SCONJ
ejpam-5661	346	2	αdqφ	αdqφ	ADJ
ejpam-5661	346	3	and	and	CCONJ
ejpam-5661	346	4	υdqφ	υdqφ	NOUN
ejpam-5661	346	5	are	be	AUX
ejpam-5661	346	6	two	two	NUM
ejpam-5661	346	7	continuous	continuous	ADJ
ejpam-5661	346	8	and	and	CCONJ
ejpam-5661	346	9	integrable	integrable	ADJ
ejpam-5661	346	10	functions	function	NOUN
ejpam-5661	346	11	on	on	ADP
ejpam-5661	346	12	[	[	X
ejpam-5661	346	13	α	α	NOUN
ejpam-5661	346	14	,	,	PUNCT
ejpam-5661	346	15	υ	υ	NOUN
ejpam-5661	346	16	]	]	X
ejpam-5661	346	17	,	,	PUNCT
ejpam-5661	346	18	then	then	ADV
ejpam-5661	346	19	we	we	PRON
ejpam-5661	346	20	have	have	VERB
ejpam-5661	346	21	:	:	PUNCT
ejpam-5661	346	22	θ(υ−	θ(υ−	PROPN
ejpam-5661	346	23	α	α	NOUN
ejpam-5661	346	24	)	)	PUNCT
ejpam-5661	346	25	2	2	NUM
ejpam-5661	346	26	∫	∫	NOUN
ejpam-5661	346	27	1	1	NUM
ejpam-5661	346	28	0	0	NUM
ejpam-5661	346	29	qϖ	qϖ	NOUN
ejpam-5661	346	30	(	(	PUNCT
ejpam-5661	346	31	υdqφ(θϖα+	υdqφ(θϖα+	X
ejpam-5661	346	32	(	(	PUNCT
ejpam-5661	346	33	1−θϖ)υ)−	1−θϖ)υ)−	NUM
ejpam-5661	346	34	αdqφ(θϖυ+	αdqφ(θϖυ+	NOUN
ejpam-5661	346	35	(	(	PUNCT
ejpam-5661	346	36	1−θϖ)α	1−θϖ)α	NUM
ejpam-5661	346	37	)	)	PUNCT
ejpam-5661	346	38	)	)	PUNCT
ejpam-5661	346	39	dqϖ	dqϖ	VERB
ejpam-5661	346	40	c.	c.	PROPN
ejpam-5661	346	41	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	346	42	,	,	PUNCT
ejpam-5661	346	43	p.	p.	PROPN
ejpam-5661	346	44	yotkaew	yotkaew	PROPN
ejpam-5661	346	45	/	/	SYM
ejpam-5661	346	46	eur	eur	PROPN
ejpam-5661	346	47	.	.	PUNCT
ejpam-5661	347	1	j.	j.	PROPN
ejpam-5661	347	2	pure	pure	PROPN
ejpam-5661	347	3	appl	appl	PROPN
ejpam-5661	347	4	.	.	PROPN
ejpam-5661	347	5	math	math	PROPN
ejpam-5661	347	6	,	,	PUNCT
ejpam-5661	347	7	18	18	NUM
ejpam-5661	347	8	(	(	PUNCT
ejpam-5661	347	9	1	1	NUM
ejpam-5661	347	10	)	)	PUNCT
ejpam-5661	347	11	(	(	PUNCT
ejpam-5661	347	12	2025	2025	NUM
ejpam-5661	347	13	)	)	PUNCT
ejpam-5661	347	14	,	,	PUNCT
ejpam-5661	347	15	5661	5661	NUM
ejpam-5661	347	16	14	14	NUM
ejpam-5661	347	17	of	of	ADP
ejpam-5661	347	18	24	24	NUM
ejpam-5661	347	19	=	=	SYM
ejpam-5661	347	20	1	1	NUM
ejpam-5661	347	21	2θ(υ−	2θ(υ−	NUM
ejpam-5661	347	22	α	α	NOUN
ejpam-5661	347	23	)	)	PUNCT
ejpam-5661	347	24	(	(	PUNCT
ejpam-5661	347	25	∫	∫	PROPN
ejpam-5661	347	26	υ	υ	PROPN
ejpam-5661	347	27	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	347	28	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	347	29	)	)	PUNCT
ejpam-5661	347	30	υdqϖ	υdqϖ	NOUN
ejpam-5661	348	1	+	+	CCONJ
ejpam-5661	349	1	∫	∫	PROPN
ejpam-5661	349	2	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	349	3	α	α	PROPN
ejpam-5661	349	4	φ(ϖ)αdqϖ	φ(ϖ)αdqϖ	NOUN
ejpam-5661	349	5	)	)	PUNCT
ejpam-5661	350	1	−	−	PROPN
ejpam-5661	350	2	φ(θα+	φ(θα+	NUM
ejpam-5661	350	3	(	(	PUNCT
ejpam-5661	350	4	1−θ)υ	1−θ)υ	NUM
ejpam-5661	350	5	)	)	PUNCT
ejpam-5661	350	6	+	+	CCONJ
ejpam-5661	350	7	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	350	8	(	(	PUNCT
ejpam-5661	350	9	1−θ)α	1−θ)α	NUM
ejpam-5661	350	10	)	)	PUNCT
ejpam-5661	350	11	2	2	NUM
ejpam-5661	350	12	.	.	PUNCT
ejpam-5661	351	1	(	(	PUNCT
ejpam-5661	351	2	25	25	NUM
ejpam-5661	351	3	)	)	PUNCT
ejpam-5661	351	4	proof	proof	NOUN
ejpam-5661	351	5	.	.	PUNCT
ejpam-5661	352	1	by	by	ADP
ejpam-5661	352	2	definition	definition	NOUN
ejpam-5661	352	3	3	3	NUM
ejpam-5661	352	4	,	,	PUNCT
ejpam-5661	352	5	we	we	PRON
ejpam-5661	352	6	have	have	VERB
ejpam-5661	352	7	i1	i1	NOUN
ejpam-5661	352	8	:	:	PUNCT
ejpam-5661	353	1	=	=	SYM
ejpam-5661	353	2	∫	∫	PROPN
ejpam-5661	353	3	1	1	NUM
ejpam-5661	353	4	0	0	NUM
ejpam-5661	353	5	qϖ	qϖ	NOUN
ejpam-5661	353	6	(	(	PUNCT
ejpam-5661	353	7	υdqφ(θϖα+	υdqφ(θϖα+	X
ejpam-5661	353	8	(	(	PUNCT
ejpam-5661	353	9	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	353	10	)	)	PUNCT
ejpam-5661	353	11	)	)	PUNCT
ejpam-5661	353	12	dqϖ	dqϖ	NOUN
ejpam-5661	354	1	=	=	PUNCT
ejpam-5661	354	2	∫	∫	PROPN
ejpam-5661	354	3	1	1	NUM
ejpam-5661	354	4	0	0	NUM
ejpam-5661	354	5	qϖ	qϖ	NOUN
ejpam-5661	354	6	φ(qθϖα+	φ(qθϖα+	PROPN
ejpam-5661	354	7	(	(	PUNCT
ejpam-5661	354	8	1−	1−	NUM
ejpam-5661	354	9	qθϖ)υ)−	qθϖ)υ)−	NUM
ejpam-5661	354	10	φ(θϖα+	φ(θϖα+	PRON
ejpam-5661	354	11	(	(	PUNCT
ejpam-5661	354	12	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	354	13	)	)	PUNCT
ejpam-5661	354	14	(	(	PUNCT
ejpam-5661	354	15	1−	1−	NUM
ejpam-5661	354	16	q)θϖ(υ−	q)θϖ(υ−	NOUN
ejpam-5661	354	17	α	α	X
ejpam-5661	354	18	)	)	PUNCT
ejpam-5661	354	19	dqϖ	dqϖ	NOUN
ejpam-5661	354	20	=	=	NOUN
ejpam-5661	355	1	q	q	X
ejpam-5661	355	2	(	(	PUNCT
ejpam-5661	355	3	υ−	υ−	PROPN
ejpam-5661	355	4	α)θ	α)θ	PROPN
ejpam-5661	355	5	∞∑	∞∑	PROPN
ejpam-5661	355	6	n=0	n=0	NUM
ejpam-5661	355	7	qnφ(θqn+1α+	qnφ(θqn+1α+	X
ejpam-5661	355	8	(	(	PUNCT
ejpam-5661	355	9	1−θqn+1)υ	1−θqn+1)υ	NUM
ejpam-5661	355	10	)	)	PUNCT
ejpam-5661	355	11	−	−	PROPN
ejpam-5661	356	1	q	q	NOUN
ejpam-5661	356	2	(	(	PUNCT
ejpam-5661	356	3	υ−	υ−	PROPN
ejpam-5661	356	4	α)θ	α)θ	PROPN
ejpam-5661	356	5	∞∑	∞∑	PROPN
ejpam-5661	356	6	n=0	n=0	SYM
ejpam-5661	356	7	qnφ(θqnα+	qnφ(θqnα+	PROPN
ejpam-5661	356	8	(	(	PUNCT
ejpam-5661	356	9	1−θqn)υ	1−θqn)υ	NUM
ejpam-5661	356	10	)	)	PUNCT
ejpam-5661	356	11	=	=	SYM
ejpam-5661	356	12	1	1	NUM
ejpam-5661	356	13	(	(	PUNCT
ejpam-5661	356	14	υ−	υ−	PROPN
ejpam-5661	356	15	α)θ	α)θ	PROPN
ejpam-5661	357	1	∞∑	∞∑	PROPN
ejpam-5661	357	2	n=0	n=0	NUM
ejpam-5661	357	3	qnφ(θqnα+	qnφ(θqnα+	PROPN
ejpam-5661	357	4	(	(	PUNCT
ejpam-5661	357	5	1−θqn)υ)−	1−θqn)υ)−	NUM
ejpam-5661	357	6	1	1	NUM
ejpam-5661	357	7	(	(	PUNCT
ejpam-5661	357	8	υ−	υ−	PROPN
ejpam-5661	357	9	α)θ	α)θ	NUM
ejpam-5661	357	10	φ(θα+	φ(θα+	NUM
ejpam-5661	357	11	(	(	PUNCT
ejpam-5661	357	12	1−θ)υ	1−θ)υ	NUM
ejpam-5661	357	13	)	)	PUNCT
ejpam-5661	357	14	−	−	PROPN
ejpam-5661	358	1	q	q	NOUN
ejpam-5661	358	2	(	(	PUNCT
ejpam-5661	358	3	υ−	υ−	PROPN
ejpam-5661	358	4	α)θ	α)θ	PROPN
ejpam-5661	358	5	∞∑	∞∑	PROPN
ejpam-5661	358	6	n=0	n=0	SYM
ejpam-5661	358	7	qnφ(θqnα+	qnφ(θqnα+	PROPN
ejpam-5661	358	8	(	(	PUNCT
ejpam-5661	358	9	1−θqn)υ	1−θqn)υ	NUM
ejpam-5661	358	10	)	)	PUNCT
ejpam-5661	358	11	=	=	SYM
ejpam-5661	359	1	1−	1−	NUM
ejpam-5661	359	2	q	q	NOUN
ejpam-5661	359	3	(	(	PUNCT
ejpam-5661	359	4	υ−	υ−	PROPN
ejpam-5661	359	5	α)θ	α)θ	PROPN
ejpam-5661	360	1	∞∑	∞∑	PROPN
ejpam-5661	360	2	n=0	n=0	NUM
ejpam-5661	360	3	qnφ(θqnα+	qnφ(θqnα+	PROPN
ejpam-5661	360	4	(	(	PUNCT
ejpam-5661	360	5	1−θqn)υ)−	1−θqn)υ)−	NUM
ejpam-5661	360	6	1	1	NUM
ejpam-5661	360	7	(	(	PUNCT
ejpam-5661	360	8	υ−	υ−	PROPN
ejpam-5661	360	9	α)θ	α)θ	NUM
ejpam-5661	360	10	φ(θα+	φ(θα+	NUM
ejpam-5661	360	11	(	(	PUNCT
ejpam-5661	360	12	1−θ)b	1−θ)b	NUM
ejpam-5661	360	13	)	)	PUNCT
ejpam-5661	360	14	=	=	SYM
ejpam-5661	360	15	1	1	NUM
ejpam-5661	360	16	(	(	PUNCT
ejpam-5661	360	17	υ−	υ−	PROPN
ejpam-5661	360	18	α)2θ2	α)2θ2	PRON
ejpam-5661	360	19	∫	∫	NOUN
ejpam-5661	360	20	υ	υ	PROPN
ejpam-5661	360	21	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	360	22	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	360	23	)	)	PUNCT
ejpam-5661	360	24	υdqϖ	υdqϖ	NOUN
ejpam-5661	360	25	−	−	PROPN
ejpam-5661	360	26	1	1	NUM
ejpam-5661	360	27	(	(	PUNCT
ejpam-5661	360	28	υ−	υ−	PROPN
ejpam-5661	360	29	α)θ	α)θ	NUM
ejpam-5661	360	30	φ(θα+	φ(θα+	NUM
ejpam-5661	360	31	(	(	PUNCT
ejpam-5661	360	32	1−θ)υ	1−θ)υ	NUM
ejpam-5661	360	33	)	)	PUNCT
ejpam-5661	360	34	.	.	PUNCT
ejpam-5661	361	1	similarly	similarly	ADV
ejpam-5661	361	2	,	,	PUNCT
ejpam-5661	361	3	by	by	ADP
ejpam-5661	361	4	definition	definition	NOUN
ejpam-5661	361	5	2	2	NUM
ejpam-5661	361	6	,	,	PUNCT
ejpam-5661	361	7	we	we	PRON
ejpam-5661	361	8	have	have	VERB
ejpam-5661	361	9	i2	i2	NOUN
ejpam-5661	361	10	:	:	PUNCT
ejpam-5661	362	1	=	=	SYM
ejpam-5661	362	2	∫	∫	PROPN
ejpam-5661	362	3	1	1	NUM
ejpam-5661	362	4	0	0	NUM
ejpam-5661	362	5	qϖ	qϖ	NOUN
ejpam-5661	362	6	(	(	PUNCT
ejpam-5661	362	7	αdqφ(θϖυ+	αdqφ(θϖυ+	PROPN
ejpam-5661	362	8	(	(	PUNCT
ejpam-5661	362	9	1−θϖ)α	1−θϖ)α	NUM
ejpam-5661	362	10	)	)	PUNCT
ejpam-5661	362	11	)	)	PUNCT
ejpam-5661	362	12	dqϖ	dqϖ	NOUN
ejpam-5661	363	1	=	=	SYM
ejpam-5661	363	2	1	1	NUM
ejpam-5661	363	3	(	(	PUNCT
ejpam-5661	363	4	υ−	υ−	PROPN
ejpam-5661	363	5	α)θ	α)θ	PROPN
ejpam-5661	363	6	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	363	7	(	(	PUNCT
ejpam-5661	363	8	1−θ)α)−	1−θ)α)−	NUM
ejpam-5661	363	9	1	1	NUM
ejpam-5661	363	10	(	(	PUNCT
ejpam-5661	363	11	υ−	υ−	PROPN
ejpam-5661	363	12	α)2θ2	α)2θ2	NOUN
ejpam-5661	363	13	∫	∫	NOUN
ejpam-5661	363	14	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	363	15	α	α	PROPN
ejpam-5661	363	16	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	363	17	)	)	PUNCT
ejpam-5661	363	18	αdqϖ.	αdqϖ.	NOUN
ejpam-5661	364	1	then	then	ADV
ejpam-5661	364	2	it	it	PRON
ejpam-5661	364	3	follows	follow	VERB
ejpam-5661	364	4	that	that	SCONJ
ejpam-5661	364	5	θ(υ−	θ(υ−	PROPN
ejpam-5661	364	6	α	α	NOUN
ejpam-5661	364	7	)	)	PUNCT
ejpam-5661	364	8	2	2	NUM
ejpam-5661	364	9	(	(	PUNCT
ejpam-5661	364	10	i1	i1	PROPN
ejpam-5661	364	11	−	−	PROPN
ejpam-5661	364	12	i2	i2	PROPN
ejpam-5661	364	13	)	)	PUNCT
ejpam-5661	364	14	=	=	SYM
ejpam-5661	364	15	θ(υ−	θ(υ−	PROPN
ejpam-5661	364	16	α	α	NOUN
ejpam-5661	364	17	)	)	PUNCT
ejpam-5661	364	18	2	2	NUM
ejpam-5661	364	19	(	(	PUNCT
ejpam-5661	364	20	1	1	NUM
ejpam-5661	364	21	(	(	PUNCT
ejpam-5661	364	22	υ−	υ−	PROPN
ejpam-5661	364	23	α)2θ2	α)2θ2	NOUN
ejpam-5661	364	24	∫	∫	NOUN
ejpam-5661	364	25	υ	υ	PROPN
ejpam-5661	364	26	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	364	27	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	364	28	)	)	PUNCT
ejpam-5661	364	29	υdqϖ	υdqϖ	NOUN
ejpam-5661	364	30	−	−	PROPN
ejpam-5661	364	31	1	1	NUM
ejpam-5661	364	32	(	(	PUNCT
ejpam-5661	364	33	υ−	υ−	PROPN
ejpam-5661	364	34	α)θ	α)θ	NUM
ejpam-5661	364	35	φ(θα+	φ(θα+	NUM
ejpam-5661	364	36	(	(	PUNCT
ejpam-5661	364	37	1−θ)υ	1−θ)υ	NUM
ejpam-5661	364	38	)	)	PUNCT
ejpam-5661	364	39	−	−	PROPN
ejpam-5661	364	40	1	1	NUM
ejpam-5661	364	41	(	(	PUNCT
ejpam-5661	364	42	υ−	υ−	PROPN
ejpam-5661	364	43	α)θ	α)θ	PROPN
ejpam-5661	364	44	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	364	45	(	(	PUNCT
ejpam-5661	364	46	1−θ)α	1−θ)α	NUM
ejpam-5661	364	47	)	)	PUNCT
ejpam-5661	365	1	+	+	CCONJ
ejpam-5661	365	2	1	1	NUM
ejpam-5661	365	3	(	(	PUNCT
ejpam-5661	365	4	υ−	υ−	PROPN
ejpam-5661	365	5	α)2θ2	α)2θ2	NOUN
ejpam-5661	365	6	∫	∫	NOUN
ejpam-5661	365	7	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	365	8	α	α	PROPN
ejpam-5661	365	9	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	365	10	)	)	PUNCT
ejpam-5661	365	11	αdqϖ	αdqϖ	ADJ
ejpam-5661	365	12	)	)	PUNCT
ejpam-5661	365	13	c.	c.	PROPN
ejpam-5661	365	14	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	365	15	,	,	PUNCT
ejpam-5661	365	16	p.	p.	PROPN
ejpam-5661	365	17	yotkaew	yotkaew	PROPN
ejpam-5661	365	18	/	/	SYM
ejpam-5661	365	19	eur	eur	PROPN
ejpam-5661	365	20	.	.	PUNCT
ejpam-5661	366	1	j.	j.	PROPN
ejpam-5661	366	2	pure	pure	PROPN
ejpam-5661	366	3	appl	appl	PROPN
ejpam-5661	366	4	.	.	PROPN
ejpam-5661	366	5	math	math	PROPN
ejpam-5661	366	6	,	,	PUNCT
ejpam-5661	366	7	18	18	NUM
ejpam-5661	366	8	(	(	PUNCT
ejpam-5661	366	9	1	1	NUM
ejpam-5661	366	10	)	)	PUNCT
ejpam-5661	366	11	(	(	PUNCT
ejpam-5661	366	12	2025	2025	NUM
ejpam-5661	366	13	)	)	PUNCT
ejpam-5661	366	14	,	,	PUNCT
ejpam-5661	366	15	5661	5661	NUM
ejpam-5661	366	16	15	15	NUM
ejpam-5661	366	17	of	of	ADP
ejpam-5661	366	18	24	24	NUM
ejpam-5661	366	19	=	=	SYM
ejpam-5661	366	20	θ(υ−	θ(υ−	PROPN
ejpam-5661	366	21	α	α	NOUN
ejpam-5661	366	22	)	)	PUNCT
ejpam-5661	366	23	2	2	NUM
ejpam-5661	366	24	(	(	PUNCT
ejpam-5661	366	25	∫	∫	PROPN
ejpam-5661	366	26	υ	υ	PROPN
ejpam-5661	366	27	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	366	28	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	366	29	)	)	PUNCT
ejpam-5661	366	30	υdqϖ	υdqϖ	NOUN
ejpam-5661	367	1	+	+	CCONJ
ejpam-5661	368	1	∫	∫	PROPN
ejpam-5661	368	2	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	368	3	α	α	PROPN
ejpam-5661	368	4	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	368	5	)	)	PUNCT
ejpam-5661	368	6	αdqϖ	αdqϖ	NOUN
ejpam-5661	368	7	)	)	PUNCT
ejpam-5661	369	1	−	−	PROPN
ejpam-5661	369	2	φ(θα+	φ(θα+	NUM
ejpam-5661	369	3	(	(	PUNCT
ejpam-5661	369	4	1−θ)υ	1−θ)υ	NUM
ejpam-5661	369	5	)	)	PUNCT
ejpam-5661	369	6	+	+	CCONJ
ejpam-5661	369	7	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	369	8	(	(	PUNCT
ejpam-5661	369	9	1−θ)α	1−θ)α	NUM
ejpam-5661	369	10	)	)	PUNCT
ejpam-5661	369	11	2	2	NUM
ejpam-5661	369	12	.	.	PUNCT
ejpam-5661	370	1	this	this	DET
ejpam-5661	370	2	finalizes	finalize	VERB
ejpam-5661	370	3	the	the	DET
ejpam-5661	370	4	proof	proof	NOUN
ejpam-5661	370	5	.	.	PUNCT
ejpam-5661	371	1	theorem	theorem	ADJ
ejpam-5661	371	2	6	6	NUM
ejpam-5661	371	3	.	.	PUNCT
ejpam-5661	372	1	let	let	VERB
ejpam-5661	372	2	φ	φ	NOUN
ejpam-5661	372	3	:	:	PUNCT
ejpam-5661	373	1	[	[	X
ejpam-5661	373	2	α	α	X
ejpam-5661	373	3	,	,	PUNCT
ejpam-5661	373	4	υ	υ	NOUN
ejpam-5661	373	5	]	]	X
ejpam-5661	373	6	→	→	PUNCT
ejpam-5661	373	7	r	r	NOUN
ejpam-5661	373	8	be	be	AUX
ejpam-5661	373	9	a	a	DET
ejpam-5661	373	10	q	q	ADJ
ejpam-5661	373	11	-	-	PUNCT
ejpam-5661	373	12	differentiable	differentiable	ADJ
ejpam-5661	373	13	function	function	NOUN
ejpam-5661	373	14	.	.	PUNCT
ejpam-5661	374	1	if	if	SCONJ
ejpam-5661	374	2	|αdqφ|	|αdqφ|	PRON
ejpam-5661	374	3	and	and	CCONJ
ejpam-5661	374	4	|	|	ADV
ejpam-5661	374	5	υdqφ|	υdqφ|	NOUN
ejpam-5661	374	6	are	be	AUX
ejpam-5661	374	7	strongly	strongly	ADV
ejpam-5661	374	8	convex	convex	ADJ
ejpam-5661	374	9	functions	function	NOUN
ejpam-5661	374	10	on	on	ADP
ejpam-5661	374	11	[	[	X
ejpam-5661	374	12	α	α	NOUN
ejpam-5661	374	13	,	,	PUNCT
ejpam-5661	374	14	υ	υ	NOUN
ejpam-5661	374	15	]	]	X
ejpam-5661	374	16	for	for	ADP
ejpam-5661	374	17	φ	φ	PROPN
ejpam-5661	374	18	>	>	X
ejpam-5661	374	19	0	0	PROPN
ejpam-5661	374	20	.	.	PUNCT
ejpam-5661	375	1	then	then	ADV
ejpam-5661	375	2	the	the	DET
ejpam-5661	375	3	following	follow	VERB
ejpam-5661	375	4	inequalities	inequality	NOUN
ejpam-5661	375	5	are	be	AUX
ejpam-5661	375	6	established:∣∣∣∣∣θ(υ−	established:∣∣∣∣∣θ(υ−	PROPN
ejpam-5661	375	7	α	α	NOUN
ejpam-5661	375	8	)	)	PUNCT
ejpam-5661	375	9	2	2	NUM
ejpam-5661	375	10	(	(	PUNCT
ejpam-5661	375	11	∫	∫	PROPN
ejpam-5661	375	12	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	375	13	α	α	PROPN
ejpam-5661	375	14	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	375	15	)	)	PUNCT
ejpam-5661	375	16	αdqϖ	αdqϖ	PROPN
ejpam-5661	376	1	+	+	CCONJ
ejpam-5661	376	2	∫	∫	PROPN
ejpam-5661	376	3	υ	υ	PROPN
ejpam-5661	376	4	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	376	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	376	6	)	)	PUNCT
ejpam-5661	376	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	376	8	)	)	PUNCT
ejpam-5661	377	1	−φ(θα+	−φ(θα+	PROPN
ejpam-5661	377	2	(	(	PUNCT
ejpam-5661	377	3	1−θ)υ	1−θ)υ	NUM
ejpam-5661	377	4	)	)	PUNCT
ejpam-5661	378	1	+	+	CCONJ
ejpam-5661	378	2	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	378	3	(	(	PUNCT
ejpam-5661	378	4	1−θ)α	1−θ)α	NUM
ejpam-5661	378	5	)	)	PUNCT
ejpam-5661	378	6	2	2	NUM
ejpam-5661	378	7	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5661	378	8	≤	≤	PUNCT
ejpam-5661	378	9	θq(υ−	θq(υ−	PROPN
ejpam-5661	378	10	α	α	NOUN
ejpam-5661	378	11	)	)	PUNCT
ejpam-5661	378	12	2[2]q[3]q	2[2]q[3]q	NUM
ejpam-5661	379	1	(	(	PUNCT
ejpam-5661	379	2	[	[	X
ejpam-5661	379	3	2]qθ	2]qθ	X
ejpam-5661	379	4	(	(	PUNCT
ejpam-5661	379	5	|	|	ADV
ejpam-5661	379	6	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	379	7	|	|	ADV
ejpam-5661	379	8	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	379	9	)	)	PUNCT
ejpam-5661	380	1	+	+	CCONJ
ejpam-5661	380	2	(	(	PUNCT
ejpam-5661	380	3	[	[	X
ejpam-5661	380	4	3]q	3]q	NUM
ejpam-5661	380	5	−	−	NOUN
ejpam-5661	380	6	[	[	X
ejpam-5661	380	7	2]qθ	2]qθ	NUM
ejpam-5661	380	8	)	)	PUNCT
ejpam-5661	380	9	(	(	PUNCT
ejpam-5661	380	10	|	|	ADV
ejpam-5661	380	11	υdqφ(υ)|	υdqφ(υ)|	VERB
ejpam-5661	380	12	+	+	PROPN
ejpam-5661	380	13	|	|	ADV
ejpam-5661	380	14	αdqφ(α)|	αdqφ(α)|	NOUN
ejpam-5661	380	15	)	)	PUNCT
ejpam-5661	380	16	)	)	PUNCT
ejpam-5661	380	17	−θ2qφ(υ−	−θ2qφ(υ−	VERB
ejpam-5661	380	18	α)3	α)3	PROPN
ejpam-5661	380	19	(	(	PUNCT
ejpam-5661	381	1	[	[	X
ejpam-5661	381	2	4]q	4]q	X
ejpam-5661	381	3	−	−	PUNCT
ejpam-5661	382	1	[	[	X
ejpam-5661	382	2	3]qθ	3]qθ	NUM
ejpam-5661	382	3	[	[	X
ejpam-5661	382	4	3]q[4]q	3]q[4]q	NUM
ejpam-5661	382	5	)	)	PUNCT
ejpam-5661	382	6	≤	≤	PROPN
ejpam-5661	382	7	θq(υ−	θq(υ−	PROPN
ejpam-5661	382	8	α	α	X
ejpam-5661	382	9	)	)	PUNCT
ejpam-5661	382	10	2[2]q[3]q	2[2]q[3]q	NUM
ejpam-5661	382	11	(	(	PUNCT
ejpam-5661	382	12	[	[	X
ejpam-5661	382	13	2]qθ	2]qθ	X
ejpam-5661	382	14	(	(	PUNCT
ejpam-5661	382	15	|	|	ADV
ejpam-5661	382	16	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	382	17	|	|	ADV
ejpam-5661	382	18	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	382	19	)	)	PUNCT
ejpam-5661	383	1	+	+	CCONJ
ejpam-5661	383	2	(	(	PUNCT
ejpam-5661	383	3	[	[	X
ejpam-5661	383	4	3]q	3]q	NUM
ejpam-5661	383	5	−	−	NOUN
ejpam-5661	383	6	[	[	X
ejpam-5661	383	7	2]qθ	2]qθ	NUM
ejpam-5661	383	8	)	)	PUNCT
ejpam-5661	383	9	(	(	PUNCT
ejpam-5661	383	10	|	|	ADV
ejpam-5661	383	11	υdqφ(υ)|	υdqφ(υ)|	VERB
ejpam-5661	383	12	+	+	PROPN
ejpam-5661	383	13	|	|	ADV
ejpam-5661	383	14	αdqφ(α)|	αdqφ(α)|	VERB
ejpam-5661	383	15	)	)	PUNCT
ejpam-5661	383	16	)	)	PUNCT
ejpam-5661	383	17	.	.	PUNCT
ejpam-5661	384	1	(	(	PUNCT
ejpam-5661	384	2	26	26	NUM
ejpam-5661	384	3	)	)	PUNCT
ejpam-5661	384	4	proof	proof	NOUN
ejpam-5661	384	5	.	.	PUNCT
ejpam-5661	385	1	it	it	PRON
ejpam-5661	385	2	follows	follow	VERB
ejpam-5661	385	3	from	from	ADP
ejpam-5661	385	4	lemma	lemma	PROPN
ejpam-5661	385	5	1	1	NUM
ejpam-5661	385	6	and	and	CCONJ
ejpam-5661	385	7	|αdqφ|	|αdqφ|	PRON
ejpam-5661	385	8	and	and	CCONJ
ejpam-5661	385	9	|υdqφ|	|υdqφ|	NOUN
ejpam-5661	385	10	are	be	AUX
ejpam-5661	385	11	strongly	strongly	ADV
ejpam-5661	385	12	convex	convex	ADJ
ejpam-5661	385	13	functions	function	NOUN
ejpam-5661	385	14	that	that	PRON
ejpam-5661	385	15	∣∣∣∣∣θ(υ−	∣∣∣∣∣θ(υ−	NUM
ejpam-5661	385	16	α	α	NUM
ejpam-5661	385	17	)	)	PUNCT
ejpam-5661	385	18	2	2	NUM
ejpam-5661	385	19	(	(	PUNCT
ejpam-5661	385	20	∫	∫	PROPN
ejpam-5661	385	21	υ	υ	PROPN
ejpam-5661	385	22	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	385	23	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	385	24	)	)	PUNCT
ejpam-5661	385	25	υdqϖ	υdqϖ	NOUN
ejpam-5661	386	1	+	+	CCONJ
ejpam-5661	386	2	∫	∫	PROPN
ejpam-5661	386	3	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	386	4	α	α	PROPN
ejpam-5661	386	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	386	6	)	)	PUNCT
ejpam-5661	386	7	αdqϖ	αdqϖ	NOUN
ejpam-5661	386	8	)	)	PUNCT
ejpam-5661	387	1	−φ(θα+	−φ(θα+	PROPN
ejpam-5661	387	2	(	(	PUNCT
ejpam-5661	387	3	1−θ)υ	1−θ)υ	NUM
ejpam-5661	387	4	)	)	PUNCT
ejpam-5661	388	1	+	+	CCONJ
ejpam-5661	388	2	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	388	3	(	(	PUNCT
ejpam-5661	388	4	1−θ)α	1−θ)α	NUM
ejpam-5661	388	5	)	)	PUNCT
ejpam-5661	388	6	2	2	NUM
ejpam-5661	388	7	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5661	388	8	≤	≤	NUM
ejpam-5661	388	9	θ(υ−	θ(υ−	PROPN
ejpam-5661	388	10	α	α	NOUN
ejpam-5661	388	11	)	)	PUNCT
ejpam-5661	388	12	2	2	NUM
ejpam-5661	388	13	∫	∫	NOUN
ejpam-5661	388	14	1	1	NUM
ejpam-5661	388	15	0	0	NUM
ejpam-5661	388	16	qϖ	qϖ	NOUN
ejpam-5661	388	17	∣∣	∣∣	NUM
ejpam-5661	388	18	υdqφ(θϖα+	υdqφ(θϖα+	PROPN
ejpam-5661	388	19	(	(	PUNCT
ejpam-5661	388	20	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	388	21	)	)	PUNCT
ejpam-5661	388	22	∣∣	∣∣	NUM
ejpam-5661	388	23	dqϖ	dqϖ	VERB
ejpam-5661	388	24	+	+	CCONJ
ejpam-5661	388	25	θ(υ−	θ(υ−	PROPN
ejpam-5661	388	26	α	α	NOUN
ejpam-5661	388	27	)	)	PUNCT
ejpam-5661	388	28	2	2	NUM
ejpam-5661	388	29	∫	∫	NOUN
ejpam-5661	388	30	1	1	NUM
ejpam-5661	388	31	0	0	NUM
ejpam-5661	388	32	qϖ	qϖ	NOUN
ejpam-5661	388	33	|	|	ADV
ejpam-5661	388	34	αdqφ(θϖυ+	αdqφ(θϖυ+	NOUN
ejpam-5661	388	35	(	(	PUNCT
ejpam-5661	388	36	1−θϖ)α)|	1−θϖ)α)|	NUM
ejpam-5661	388	37	dqϖ	dqϖ	NOUN
ejpam-5661	388	38	≤	≤	NUM
ejpam-5661	388	39	θ(υ−	θ(υ−	PROPN
ejpam-5661	388	40	α	α	NOUN
ejpam-5661	388	41	)	)	PUNCT
ejpam-5661	388	42	2	2	NUM
ejpam-5661	388	43	∫	∫	NOUN
ejpam-5661	388	44	1	1	NUM
ejpam-5661	388	45	0	0	NUM
ejpam-5661	388	46	qϖ	qϖ	NOUN
ejpam-5661	388	47	(	(	PUNCT
ejpam-5661	388	48	θϖ|	θϖ|	NOUN
ejpam-5661	388	49	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	388	50	(	(	PUNCT
ejpam-5661	388	51	1−θϖ)|	1−θϖ)|	X
ejpam-5661	388	52	υdqf(υ)|	υdqf(υ)|	ADJ
ejpam-5661	388	53	−	−	PROPN
ejpam-5661	388	54	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	388	55	α)2	α)2	NOUN
ejpam-5661	388	56	)	)	PUNCT
ejpam-5661	388	57	dqϖ	dqϖ	VERB
ejpam-5661	389	1	+	+	CCONJ
ejpam-5661	389	2	θ(υ−	θ(υ−	PROPN
ejpam-5661	389	3	α	α	NOUN
ejpam-5661	389	4	)	)	PUNCT
ejpam-5661	389	5	2	2	NUM
ejpam-5661	389	6	∫	∫	NOUN
ejpam-5661	389	7	1	1	NUM
ejpam-5661	389	8	0	0	NUM
ejpam-5661	389	9	qϖ	qϖ	NOUN
ejpam-5661	389	10	(	(	PUNCT
ejpam-5661	389	11	θϖ|	θϖ|	NOUN
ejpam-5661	389	12	αdqφ(υ)|	αdqφ(υ)|	PROPN
ejpam-5661	389	13	c.	c.	PROPN
ejpam-5661	389	14	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	389	15	,	,	PUNCT
ejpam-5661	389	16	p.	p.	PROPN
ejpam-5661	389	17	yotkaew	yotkaew	PROPN
ejpam-5661	389	18	/	/	SYM
ejpam-5661	389	19	eur	eur	PROPN
ejpam-5661	389	20	.	.	PUNCT
ejpam-5661	390	1	j.	j.	PROPN
ejpam-5661	390	2	pure	pure	PROPN
ejpam-5661	390	3	appl	appl	PROPN
ejpam-5661	390	4	.	.	PROPN
ejpam-5661	390	5	math	math	PROPN
ejpam-5661	390	6	,	,	PUNCT
ejpam-5661	390	7	18	18	NUM
ejpam-5661	390	8	(	(	PUNCT
ejpam-5661	390	9	1	1	NUM
ejpam-5661	390	10	)	)	PUNCT
ejpam-5661	390	11	(	(	PUNCT
ejpam-5661	390	12	2025	2025	NUM
ejpam-5661	390	13	)	)	PUNCT
ejpam-5661	390	14	,	,	PUNCT
ejpam-5661	390	15	5661	5661	NUM
ejpam-5661	390	16	16	16	NUM
ejpam-5661	390	17	of	of	ADP
ejpam-5661	390	18	24	24	NUM
ejpam-5661	390	19	+	+	CCONJ
ejpam-5661	390	20	(	(	PUNCT
ejpam-5661	390	21	1−θϖ)|	1−θϖ)|	NUM
ejpam-5661	390	22	αdqφ(α)|	αdqφ(α)|	VERB
ejpam-5661	390	23	−	−	PROPN
ejpam-5661	390	24	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	390	25	α)2	α)2	NOUN
ejpam-5661	390	26	)	)	PUNCT
ejpam-5661	390	27	dqϖ	dqϖ	NOUN
ejpam-5661	390	28	=	=	PUNCT
ejpam-5661	390	29	θq(υ−	θq(υ−	PROPN
ejpam-5661	390	30	α	α	X
ejpam-5661	390	31	)	)	PUNCT
ejpam-5661	390	32	2[2]q[3]q	2[2]q[3]q	NUM
ejpam-5661	391	1	(	(	PUNCT
ejpam-5661	391	2	[	[	X
ejpam-5661	391	3	2]qθ	2]qθ	X
ejpam-5661	391	4	(	(	PUNCT
ejpam-5661	391	5	|	|	ADV
ejpam-5661	391	6	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	391	7	|	|	ADV
ejpam-5661	391	8	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	391	9	)	)	PUNCT
ejpam-5661	392	1	+	+	CCONJ
ejpam-5661	392	2	(	(	PUNCT
ejpam-5661	392	3	[	[	X
ejpam-5661	392	4	3]q	3]q	NUM
ejpam-5661	392	5	−	−	NOUN
ejpam-5661	392	6	[	[	X
ejpam-5661	392	7	2]qθ	2]qθ	NUM
ejpam-5661	392	8	)	)	PUNCT
ejpam-5661	392	9	(	(	PUNCT
ejpam-5661	392	10	|	|	ADV
ejpam-5661	392	11	υdqφ(υ)|	υdqφ(υ)|	VERB
ejpam-5661	392	12	+	+	PROPN
ejpam-5661	392	13	|	|	ADV
ejpam-5661	392	14	αdqφ(α)|	αdqφ(α)|	NOUN
ejpam-5661	392	15	)	)	PUNCT
ejpam-5661	392	16	)	)	PUNCT
ejpam-5661	392	17	−θ2qφ(υ−	−θ2qφ(υ−	VERB
ejpam-5661	392	18	α)3	α)3	PROPN
ejpam-5661	392	19	(	(	PUNCT
ejpam-5661	393	1	[	[	X
ejpam-5661	393	2	4]q	4]q	X
ejpam-5661	393	3	−	−	PUNCT
ejpam-5661	394	1	[	[	X
ejpam-5661	394	2	3]qθ	3]qθ	NUM
ejpam-5661	394	3	[	[	X
ejpam-5661	394	4	3]q[4]q	3]q[4]q	NUM
ejpam-5661	394	5	)	)	PUNCT
ejpam-5661	394	6	≤	≤	PROPN
ejpam-5661	394	7	θq(υ−	θq(υ−	PROPN
ejpam-5661	394	8	α	α	X
ejpam-5661	394	9	)	)	PUNCT
ejpam-5661	394	10	2[2]q[3]q	2[2]q[3]q	NUM
ejpam-5661	394	11	(	(	PUNCT
ejpam-5661	394	12	[	[	X
ejpam-5661	394	13	2]qθ	2]qθ	X
ejpam-5661	394	14	(	(	PUNCT
ejpam-5661	394	15	|	|	ADV
ejpam-5661	394	16	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	394	17	|	|	ADV
ejpam-5661	394	18	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	394	19	)	)	PUNCT
ejpam-5661	395	1	+	+	CCONJ
ejpam-5661	395	2	(	(	PUNCT
ejpam-5661	395	3	[	[	X
ejpam-5661	395	4	3]q	3]q	NUM
ejpam-5661	395	5	−	−	NOUN
ejpam-5661	395	6	[	[	X
ejpam-5661	395	7	2]qθ	2]qθ	NUM
ejpam-5661	395	8	)	)	PUNCT
ejpam-5661	395	9	(	(	PUNCT
ejpam-5661	395	10	|	|	ADV
ejpam-5661	395	11	υdqφ(υ)|	υdqφ(υ)|	VERB
ejpam-5661	395	12	+	+	PROPN
ejpam-5661	395	13	|	|	ADV
ejpam-5661	395	14	αdqφ(α)|	αdqφ(α)|	VERB
ejpam-5661	395	15	)	)	PUNCT
ejpam-5661	395	16	)	)	PUNCT
ejpam-5661	395	17	.	.	PUNCT
ejpam-5661	396	1	this	this	DET
ejpam-5661	396	2	finalizes	finalize	VERB
ejpam-5661	396	3	the	the	DET
ejpam-5661	396	4	proof	proof	NOUN
ejpam-5661	396	5	.	.	PUNCT
ejpam-5661	397	1	remark	remark	VERB
ejpam-5661	397	2	3	3	NUM
ejpam-5661	397	3	.	.	PUNCT
ejpam-5661	398	1	when	when	SCONJ
ejpam-5661	398	2	θ	θ	X
ejpam-5661	398	3	=	=	SYM
ejpam-5661	398	4	1	1	NUM
ejpam-5661	398	5	in	in	ADP
ejpam-5661	398	6	theorem	theorem	NOUN
ejpam-5661	398	7	6	6	NUM
ejpam-5661	398	8	,	,	PUNCT
ejpam-5661	398	9	we	we	PRON
ejpam-5661	398	10	derive	derive	VERB
ejpam-5661	398	11	trapezoid	trapezoid	ADJ
ejpam-5661	398	12	-	-	PUNCT
ejpam-5661	398	13	type	type	NOUN
ejpam-5661	398	14	inequalities:∣∣∣∣∣(υ−	inequalities:∣∣∣∣∣(υ−	NOUN
ejpam-5661	398	15	α	α	NOUN
ejpam-5661	398	16	)	)	PUNCT
ejpam-5661	398	17	2	2	NUM
ejpam-5661	398	18	(	(	PUNCT
ejpam-5661	398	19	∫	∫	PROPN
ejpam-5661	398	20	υ	υ	PROPN
ejpam-5661	398	21	α	α	PROPN
ejpam-5661	398	22	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	398	23	)	)	PUNCT
ejpam-5661	398	24	αdqϖ	αdqϖ	PROPN
ejpam-5661	399	1	+	+	CCONJ
ejpam-5661	399	2	∫	∫	PROPN
ejpam-5661	399	3	υ	υ	PROPN
ejpam-5661	399	4	α	α	PROPN
ejpam-5661	399	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	399	6	)	)	PUNCT
ejpam-5661	399	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	399	8	)	)	PUNCT
ejpam-5661	400	1	−	−	PROPN
ejpam-5661	400	2	φ(α	φ(α	PROPN
ejpam-5661	400	3	)	)	PUNCT
ejpam-5661	401	1	+	+	CCONJ
ejpam-5661	402	1	φ(υ	φ(υ	PROPN
ejpam-5661	402	2	)	)	PUNCT
ejpam-5661	402	3	2	2	NUM
ejpam-5661	402	4	∣∣∣∣∣	∣∣∣∣∣	SYM
ejpam-5661	402	5	≤	≤	PROPN
ejpam-5661	402	6	q(υ−	q(υ−	PROPN
ejpam-5661	402	7	α	α	NOUN
ejpam-5661	402	8	)	)	PUNCT
ejpam-5661	402	9	2[2]q[3]q	2[2]q[3]q	NUM
ejpam-5661	403	1	(	(	PUNCT
ejpam-5661	403	2	[	[	X
ejpam-5661	403	3	2]q	2]q	X
ejpam-5661	403	4	(	(	PUNCT
ejpam-5661	403	5	|	|	ADV
ejpam-5661	403	6	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	403	7	|	|	ADV
ejpam-5661	403	8	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	403	9	)	)	PUNCT
ejpam-5661	404	1	+	+	NUM
ejpam-5661	404	2	q2	q2	NOUN
ejpam-5661	404	3	(	(	PUNCT
ejpam-5661	404	4	|	|	ADV
ejpam-5661	404	5	υdqφ(υ)|+	υdqφ(υ)|+	ADV
ejpam-5661	404	6	|	|	ADV
ejpam-5661	404	7	αdqφ(α)|	αdqφ(α)|	VERB
ejpam-5661	404	8	)	)	PUNCT
ejpam-5661	404	9	)	)	PUNCT
ejpam-5661	405	1	−	−	PROPN
ejpam-5661	405	2	q4φ(υ−	q4φ(υ−	PROPN
ejpam-5661	405	3	α)3	α)3	X
ejpam-5661	405	4	[	[	X
ejpam-5661	405	5	3]q[4]q	3]q[4]q	NUM
ejpam-5661	405	6	≤	≤	NUM
ejpam-5661	405	7	q(υ−	q(υ−	PROPN
ejpam-5661	405	8	α	α	NOUN
ejpam-5661	405	9	)	)	PUNCT
ejpam-5661	405	10	2[2]q[3]q	2[2]q[3]q	NUM
ejpam-5661	406	1	(	(	PUNCT
ejpam-5661	406	2	[	[	X
ejpam-5661	406	3	2]q	2]q	X
ejpam-5661	406	4	(	(	PUNCT
ejpam-5661	406	5	|	|	ADV
ejpam-5661	406	6	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	406	7	|	|	ADV
ejpam-5661	406	8	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	406	9	)	)	PUNCT
ejpam-5661	407	1	+	+	NUM
ejpam-5661	407	2	q2	q2	NOUN
ejpam-5661	407	3	(	(	PUNCT
ejpam-5661	407	4	|	|	ADV
ejpam-5661	407	5	υdqφ(υ)|+	υdqφ(υ)|+	ADV
ejpam-5661	407	6	|	|	ADV
ejpam-5661	407	7	αdqφ(α)|	αdqφ(α)|	VERB
ejpam-5661	407	8	)	)	PUNCT
ejpam-5661	407	9	)	)	PUNCT
ejpam-5661	407	10	.	.	PUNCT
ejpam-5661	408	1	remark	remark	VERB
ejpam-5661	408	2	4	4	NUM
ejpam-5661	408	3	.	.	PUNCT
ejpam-5661	409	1	when	when	SCONJ
ejpam-5661	409	2	θ	θ	PROPN
ejpam-5661	409	3	=	=	SYM
ejpam-5661	409	4	1/2	1/2	NUM
ejpam-5661	409	5	in	in	ADP
ejpam-5661	409	6	theorem	theorem	NOUN
ejpam-5661	409	7	6	6	NUM
ejpam-5661	409	8	,	,	PUNCT
ejpam-5661	409	9	we	we	PRON
ejpam-5661	409	10	derive	derive	VERB
ejpam-5661	409	11	midpoint	midpoint	NOUN
ejpam-5661	409	12	-	-	PUNCT
ejpam-5661	409	13	type	type	NOUN
ejpam-5661	409	14	inequalities:∣∣∣∣∣(υ−	inequalities:∣∣∣∣∣(υ−	NOUN
ejpam-5661	409	15	α	α	NOUN
ejpam-5661	409	16	)	)	PUNCT
ejpam-5661	409	17	2	2	NUM
ejpam-5661	409	18	(	(	PUNCT
ejpam-5661	409	19	∫	∫	PROPN
ejpam-5661	409	20	(	(	PUNCT
ejpam-5661	409	21	α+υ	α+υ	NUM
ejpam-5661	409	22	)	)	PUNCT
ejpam-5661	409	23	2	2	NUM
ejpam-5661	409	24	α	α	PRON
ejpam-5661	409	25	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	409	26	)	)	PUNCT
ejpam-5661	409	27	αdqϖ	αdqϖ	PROPN
ejpam-5661	410	1	+	+	CCONJ
ejpam-5661	410	2	∫	∫	PROPN
ejpam-5661	410	3	υ	υ	X
ejpam-5661	410	4	(	(	PUNCT
ejpam-5661	410	5	α+υ	α+υ	NUM
ejpam-5661	410	6	)	)	PUNCT
ejpam-5661	410	7	2	2	NUM
ejpam-5661	410	8	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	410	9	)	)	PUNCT
ejpam-5661	410	10	υdqϖ	υdqϖ	NOUN
ejpam-5661	410	11	)	)	PUNCT
ejpam-5661	411	1	−	−	PROPN
ejpam-5661	411	2	φ	φ	PROPN
ejpam-5661	411	3	(	(	PUNCT
ejpam-5661	411	4	α+υ	α+υ	NUM
ejpam-5661	411	5	2	2	NUM
ejpam-5661	411	6	)	)	PUNCT
ejpam-5661	411	7	∣∣∣∣∣	∣∣∣∣∣	ADP
ejpam-5661	411	8	≤	≤	PROPN
ejpam-5661	411	9	q(υ−	q(υ−	PROPN
ejpam-5661	411	10	α	α	NOUN
ejpam-5661	411	11	)	)	PUNCT
ejpam-5661	411	12	8[2]q[3]q	8[2]q[3]q	NUM
ejpam-5661	411	13	(	(	PUNCT
ejpam-5661	411	14	[	[	X
ejpam-5661	411	15	2]q	2]q	NUM
ejpam-5661	411	16	(	(	PUNCT
ejpam-5661	411	17	|	|	ADV
ejpam-5661	411	18	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	411	19	|	|	ADV
ejpam-5661	411	20	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	411	21	)	)	PUNCT
ejpam-5661	412	1	+	+	CCONJ
ejpam-5661	412	2	(	(	PUNCT
ejpam-5661	412	3	[	[	X
ejpam-5661	412	4	3]q	3]q	NUM
ejpam-5661	412	5	+	+	SYM
ejpam-5661	412	6	q2	q2	NOUN
ejpam-5661	412	7	)	)	PUNCT
ejpam-5661	412	8	(	(	PUNCT
ejpam-5661	412	9	|	|	ADV
ejpam-5661	412	10	υdqφ(υ)|	υdqφ(υ)|	VERB
ejpam-5661	412	11	+	+	PROPN
ejpam-5661	412	12	|	|	ADV
ejpam-5661	412	13	αdqφ(α)|	αdqφ(α)|	NOUN
ejpam-5661	412	14	)	)	PUNCT
ejpam-5661	412	15	)	)	PUNCT
ejpam-5661	413	1	−	−	PROPN
ejpam-5661	413	2	q4φ(υ−	q4φ(υ−	PROPN
ejpam-5661	413	3	α)3	α)3	NOUN
ejpam-5661	413	4	(	(	PUNCT
ejpam-5661	413	5	q3	q3	PROPN
ejpam-5661	413	6	+	+	PROPN
ejpam-5661	414	1	[	[	X
ejpam-5661	414	2	4]q	4]q	X
ejpam-5661	414	3	[	[	X
ejpam-5661	414	4	3]q[4]q	3]q[4]q	NUM
ejpam-5661	414	5	)	)	PUNCT
ejpam-5661	414	6	≤	≤	NUM
ejpam-5661	414	7	q(υ−	q(υ−	PROPN
ejpam-5661	414	8	α	α	NOUN
ejpam-5661	414	9	)	)	PUNCT
ejpam-5661	414	10	8[2]q[3]q	8[2]q[3]q	NUM
ejpam-5661	414	11	(	(	PUNCT
ejpam-5661	414	12	[	[	X
ejpam-5661	414	13	2]q	2]q	NUM
ejpam-5661	414	14	(	(	PUNCT
ejpam-5661	414	15	|	|	ADV
ejpam-5661	414	16	υdqφ(α)|+	υdqφ(α)|+	NOUN
ejpam-5661	414	17	|	|	ADV
ejpam-5661	414	18	αdqφ(υ)|	αdqφ(υ)|	NOUN
ejpam-5661	414	19	)	)	PUNCT
ejpam-5661	415	1	+	+	CCONJ
ejpam-5661	415	2	(	(	PUNCT
ejpam-5661	415	3	[	[	X
ejpam-5661	415	4	3]q	3]q	NUM
ejpam-5661	415	5	+	+	SYM
ejpam-5661	415	6	q2	q2	NOUN
ejpam-5661	415	7	)	)	PUNCT
ejpam-5661	415	8	(	(	PUNCT
ejpam-5661	415	9	|	|	ADV
ejpam-5661	415	10	υdqφ(υ)|	υdqφ(υ)|	VERB
ejpam-5661	415	11	+	+	PROPN
ejpam-5661	415	12	|	|	ADV
ejpam-5661	415	13	αdqφ(α)|	αdqφ(α)|	VERB
ejpam-5661	415	14	)	)	PUNCT
ejpam-5661	415	15	)	)	PUNCT
ejpam-5661	415	16	.	.	PUNCT
ejpam-5661	416	1	c.	c.	PROPN
ejpam-5661	416	2	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	416	3	,	,	PUNCT
ejpam-5661	416	4	p.	p.	PROPN
ejpam-5661	416	5	yotkaew	yotkaew	PROPN
ejpam-5661	416	6	/	/	SYM
ejpam-5661	416	7	eur	eur	PROPN
ejpam-5661	416	8	.	.	PUNCT
ejpam-5661	417	1	j.	j.	PROPN
ejpam-5661	417	2	pure	pure	PROPN
ejpam-5661	417	3	appl	appl	PROPN
ejpam-5661	417	4	.	.	PROPN
ejpam-5661	417	5	math	math	PROPN
ejpam-5661	417	6	,	,	PUNCT
ejpam-5661	417	7	18	18	NUM
ejpam-5661	417	8	(	(	PUNCT
ejpam-5661	417	9	1	1	NUM
ejpam-5661	417	10	)	)	PUNCT
ejpam-5661	417	11	(	(	PUNCT
ejpam-5661	417	12	2025	2025	NUM
ejpam-5661	417	13	)	)	PUNCT
ejpam-5661	417	14	,	,	PUNCT
ejpam-5661	417	15	5661	5661	NUM
ejpam-5661	417	16	17	17	NUM
ejpam-5661	417	17	of	of	ADP
ejpam-5661	417	18	24	24	NUM
ejpam-5661	417	19	theorem	theorem	NOUN
ejpam-5661	417	20	7	7	NUM
ejpam-5661	417	21	.	.	PUNCT
ejpam-5661	418	1	let	let	VERB
ejpam-5661	418	2	φ	φ	NOUN
ejpam-5661	418	3	:	:	PUNCT
ejpam-5661	419	1	[	[	X
ejpam-5661	419	2	α	α	X
ejpam-5661	419	3	,	,	PUNCT
ejpam-5661	419	4	υ	υ	NOUN
ejpam-5661	419	5	]	]	X
ejpam-5661	419	6	→	→	PUNCT
ejpam-5661	419	7	r	r	NOUN
ejpam-5661	419	8	be	be	AUX
ejpam-5661	419	9	a	a	DET
ejpam-5661	419	10	q	q	ADJ
ejpam-5661	419	11	-	-	PUNCT
ejpam-5661	419	12	differentiable	differentiable	ADJ
ejpam-5661	419	13	function	function	NOUN
ejpam-5661	419	14	.	.	PUNCT
ejpam-5661	420	1	if	if	SCONJ
ejpam-5661	420	2	|αdqφ|c	|αdqφ|c	NUM
ejpam-5661	420	3	and	and	CCONJ
ejpam-5661	420	4	|	|	ADV
ejpam-5661	420	5	υdqφ|c	υdqφ|c	ADJ
ejpam-5661	420	6	,	,	PUNCT
ejpam-5661	420	7	c	c	AUX
ejpam-5661	420	8	>	>	X
ejpam-5661	420	9	1	1	NUM
ejpam-5661	420	10	are	be	AUX
ejpam-5661	420	11	strongly	strongly	ADV
ejpam-5661	420	12	convex	convex	ADJ
ejpam-5661	420	13	functions	function	NOUN
ejpam-5661	420	14	on	on	ADP
ejpam-5661	420	15	[	[	X
ejpam-5661	420	16	α	α	NOUN
ejpam-5661	420	17	,	,	PUNCT
ejpam-5661	420	18	υ	υ	NOUN
ejpam-5661	420	19	]	]	X
ejpam-5661	420	20	for	for	ADP
ejpam-5661	420	21	φ	φ	PROPN
ejpam-5661	420	22	>	>	X
ejpam-5661	420	23	0	0	PROPN
ejpam-5661	420	24	,	,	PUNCT
ejpam-5661	420	25	then	then	ADV
ejpam-5661	420	26	the	the	DET
ejpam-5661	420	27	following	follow	VERB
ejpam-5661	420	28	inequalities	inequality	NOUN
ejpam-5661	420	29	are	be	AUX
ejpam-5661	420	30	established:∣∣∣∣∣θ(υ−	established:∣∣∣∣∣θ(υ−	PROPN
ejpam-5661	420	31	α	α	NOUN
ejpam-5661	420	32	)	)	PUNCT
ejpam-5661	420	33	2	2	NUM
ejpam-5661	420	34	(	(	PUNCT
ejpam-5661	420	35	∫	∫	PROPN
ejpam-5661	420	36	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	420	37	α	α	PROPN
ejpam-5661	420	38	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	420	39	)	)	PUNCT
ejpam-5661	420	40	αdqϖ	αdqϖ	PROPN
ejpam-5661	421	1	+	+	CCONJ
ejpam-5661	421	2	∫	∫	PROPN
ejpam-5661	421	3	υ	υ	PROPN
ejpam-5661	421	4	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	421	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	421	6	)	)	PUNCT
ejpam-5661	421	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	421	8	)	)	PUNCT
ejpam-5661	422	1	−φ(θα+	−φ(θα+	PROPN
ejpam-5661	422	2	(	(	PUNCT
ejpam-5661	422	3	1−θ)υ	1−θ)υ	NUM
ejpam-5661	422	4	)	)	PUNCT
ejpam-5661	423	1	+	+	CCONJ
ejpam-5661	423	2	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	423	3	(	(	PUNCT
ejpam-5661	423	4	1−θ)α	1−θ)α	NUM
ejpam-5661	423	5	)	)	PUNCT
ejpam-5661	423	6	2	2	NUM
ejpam-5661	423	7	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5661	423	8	≤	≤	NOUN
ejpam-5661	423	9	qθ(υ−	qθ(υ−	NUM
ejpam-5661	423	10	α	α	NOUN
ejpam-5661	423	11	)	)	PUNCT
ejpam-5661	423	12	2	2	NUM
ejpam-5661	423	13	(	(	PUNCT
ejpam-5661	423	14	1	1	NUM
ejpam-5661	423	15	[	[	X
ejpam-5661	423	16	d+	d+	PUNCT
ejpam-5661	423	17	1]q	1]q	NUM
ejpam-5661	423	18	)	)	PUNCT
ejpam-5661	423	19	1	1	NUM
ejpam-5661	423	20	d	d	NOUN
ejpam-5661	423	21	(	(	PUNCT
ejpam-5661	423	22	(	(	PUNCT
ejpam-5661	423	23	θ|	θ|	PROPN
ejpam-5661	423	24	αdqφ(υ)|c	αdqφ(υ)|c	X
ejpam-5661	424	1	[	[	X
ejpam-5661	424	2	2]q	2]q	NUM
ejpam-5661	424	3	+	+	CCONJ
ejpam-5661	424	4	(	(	PUNCT
ejpam-5661	424	5	1−	1−	NUM
ejpam-5661	424	6	θ	θ	PROPN
ejpam-5661	425	1	[	[	X
ejpam-5661	425	2	2]q	2]q	NUM
ejpam-5661	425	3	)	)	PUNCT
ejpam-5661	426	1	|	|	ADV
ejpam-5661	426	2	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	426	3	−	−	NOUN
ejpam-5661	427	1	φθ	φθ	INTJ
ejpam-5661	427	2	(	(	PUNCT
ejpam-5661	427	3	1	1	NUM
ejpam-5661	427	4	[	[	X
ejpam-5661	427	5	2]q	2]q	NUM
ejpam-5661	427	6	−	−	ADP
ejpam-5661	427	7	θ	θ	PROPN
ejpam-5661	428	1	[	[	X
ejpam-5661	428	2	3]q	3]q	NUM
ejpam-5661	428	3	)	)	PUNCT
ejpam-5661	428	4	(	(	PUNCT
ejpam-5661	428	5	υ−	υ−	PROPN
ejpam-5661	428	6	α)2	α)2	NOUN
ejpam-5661	428	7	)	)	PUNCT
ejpam-5661	428	8	1	1	NUM
ejpam-5661	428	9	c	c	NOUN
ejpam-5661	428	10	+	+	CCONJ
ejpam-5661	428	11	(	(	PUNCT
ejpam-5661	428	12	θ|	θ|	PROPN
ejpam-5661	428	13	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	429	1	[	[	X
ejpam-5661	429	2	2]q	2]q	NUM
ejpam-5661	429	3	+	+	CCONJ
ejpam-5661	429	4	(	(	PUNCT
ejpam-5661	429	5	1−	1−	NUM
ejpam-5661	429	6	θ	θ	PROPN
ejpam-5661	430	1	[	[	X
ejpam-5661	430	2	2]q	2]q	NUM
ejpam-5661	430	3	)	)	PUNCT
ejpam-5661	431	1	|	|	ADV
ejpam-5661	431	2	υdqφ(υ)|c	υdqφ(υ)|c	X
ejpam-5661	431	3	−	−	PROPN
ejpam-5661	431	4	φθ	φθ	INTJ
ejpam-5661	431	5	(	(	PUNCT
ejpam-5661	431	6	1	1	NUM
ejpam-5661	432	1	[	[	X
ejpam-5661	432	2	2]q	2]q	NUM
ejpam-5661	432	3	−	−	ADP
ejpam-5661	432	4	θ	θ	PROPN
ejpam-5661	432	5	[	[	X
ejpam-5661	432	6	3]q	3]q	NUM
ejpam-5661	432	7	)	)	PUNCT
ejpam-5661	432	8	(	(	PUNCT
ejpam-5661	432	9	υ−	υ−	PROPN
ejpam-5661	432	10	α)2	α)2	NOUN
ejpam-5661	432	11	)	)	PUNCT
ejpam-5661	432	12	1	1	NUM
ejpam-5661	432	13	c	c	NOUN
ejpam-5661	432	14	)	)	PUNCT
ejpam-5661	432	15	≤	≤	NOUN
ejpam-5661	432	16	qθ(υ−	qθ(υ−	NUM
ejpam-5661	432	17	α	α	NOUN
ejpam-5661	432	18	)	)	PUNCT
ejpam-5661	432	19	2	2	NUM
ejpam-5661	432	20	(	(	PUNCT
ejpam-5661	432	21	1	1	NUM
ejpam-5661	432	22	[	[	X
ejpam-5661	432	23	d+	d+	PUNCT
ejpam-5661	432	24	1]q	1]q	NUM
ejpam-5661	432	25	)	)	PUNCT
ejpam-5661	432	26	1	1	NUM
ejpam-5661	432	27	d	d	NOUN
ejpam-5661	432	28	(	(	PUNCT
ejpam-5661	432	29	(	(	PUNCT
ejpam-5661	432	30	θ|	θ|	PROPN
ejpam-5661	432	31	αdqf(υ)|c	αdqf(υ)|c	VERB
ejpam-5661	432	32	[	[	X
ejpam-5661	432	33	2]q	2]q	NUM
ejpam-5661	432	34	+	+	CCONJ
ejpam-5661	432	35	(	(	PUNCT
ejpam-5661	432	36	[	[	X
ejpam-5661	432	37	2]q	2]q	NUM
ejpam-5661	432	38	−θ	−θ	ADJ
ejpam-5661	432	39	[	[	X
ejpam-5661	432	40	2]q	2]q	NUM
ejpam-5661	432	41	)	)	PUNCT
ejpam-5661	432	42	|	|	ADV
ejpam-5661	432	43	αdqf(α)|c	αdqf(α)|c	NUM
ejpam-5661	432	44	)	)	PUNCT
ejpam-5661	432	45	1	1	NUM
ejpam-5661	432	46	c	c	NOUN
ejpam-5661	432	47	+	+	CCONJ
ejpam-5661	432	48	(	(	PUNCT
ejpam-5661	432	49	θ|	θ|	PROPN
ejpam-5661	432	50	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	433	1	[	[	X
ejpam-5661	433	2	2]q	2]q	NUM
ejpam-5661	433	3	+	+	CCONJ
ejpam-5661	433	4	(	(	PUNCT
ejpam-5661	433	5	[	[	X
ejpam-5661	433	6	2]q	2]q	NUM
ejpam-5661	433	7	−θ	−θ	ADJ
ejpam-5661	433	8	[	[	X
ejpam-5661	433	9	2]q	2]q	NUM
ejpam-5661	433	10	)	)	PUNCT
ejpam-5661	433	11	|	|	ADV
ejpam-5661	433	12	υdqφ(υ)|c	υdqφ(υ)|c	PUNCT
ejpam-5661	433	13	)	)	PUNCT
ejpam-5661	433	14	1	1	NUM
ejpam-5661	433	15	c	c	NOUN
ejpam-5661	433	16	)	)	PUNCT
ejpam-5661	433	17	,	,	PUNCT
ejpam-5661	433	18	(	(	PUNCT
ejpam-5661	433	19	27	27	NUM
ejpam-5661	433	20	)	)	PUNCT
ejpam-5661	433	21	where	where	SCONJ
ejpam-5661	433	22	1	1	NUM
ejpam-5661	433	23	/	/	SYM
ejpam-5661	433	24	c+	c+	VERB
ejpam-5661	433	25	1	1	NUM
ejpam-5661	433	26	/	/	SYM
ejpam-5661	433	27	d	d	NOUN
ejpam-5661	433	28	=	=	SYM
ejpam-5661	433	29	1	1	X
ejpam-5661	433	30	.	.	PUNCT
ejpam-5661	433	31	proof	proof	NOUN
ejpam-5661	433	32	.	.	PUNCT
ejpam-5661	434	1	it	it	PRON
ejpam-5661	434	2	follows	follow	VERB
ejpam-5661	434	3	from	from	ADP
ejpam-5661	434	4	lemma	lemma	PROPN
ejpam-5661	434	5	1	1	NUM
ejpam-5661	434	6	,	,	PUNCT
ejpam-5661	434	7	hölder	hölder	PROPN
ejpam-5661	434	8	’s	’s	PART
ejpam-5661	434	9	inequality	inequality	NOUN
ejpam-5661	434	10	,	,	PUNCT
ejpam-5661	434	11	and	and	CCONJ
ejpam-5661	434	12	|αdqφ|c	|αdqφ|c	NUM
ejpam-5661	434	13	and	and	CCONJ
ejpam-5661	434	14	|υdqφ|c	|υdqφ|c	NUM
ejpam-5661	434	15	are	be	AUX
ejpam-5661	434	16	strongly	strongly	ADV
ejpam-5661	434	17	convex	convex	NOUN
ejpam-5661	434	18	functions	function	NOUN
ejpam-5661	434	19	that∣∣∣∣∣θ(υ−	that∣∣∣∣∣θ(υ−	X
ejpam-5661	434	20	α	α	NOUN
ejpam-5661	434	21	)	)	PUNCT
ejpam-5661	434	22	2	2	NUM
ejpam-5661	434	23	(	(	PUNCT
ejpam-5661	434	24	∫	∫	PROPN
ejpam-5661	434	25	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	434	26	α	α	PROPN
ejpam-5661	434	27	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	434	28	)	)	PUNCT
ejpam-5661	434	29	αdqϖ	αdqϖ	PROPN
ejpam-5661	435	1	+	+	CCONJ
ejpam-5661	435	2	∫	∫	PROPN
ejpam-5661	435	3	υ	υ	PROPN
ejpam-5661	435	4	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	435	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	435	6	)	)	PUNCT
ejpam-5661	435	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	435	8	)	)	PUNCT
ejpam-5661	436	1	−φ(θα+	−φ(θα+	PROPN
ejpam-5661	436	2	(	(	PUNCT
ejpam-5661	436	3	1−θ)υ	1−θ)υ	NUM
ejpam-5661	436	4	)	)	PUNCT
ejpam-5661	437	1	+	+	CCONJ
ejpam-5661	437	2	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	437	3	(	(	PUNCT
ejpam-5661	437	4	1−θ)α	1−θ)α	NUM
ejpam-5661	437	5	)	)	PUNCT
ejpam-5661	437	6	2	2	NUM
ejpam-5661	437	7	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5661	437	8	≤	≤	NUM
ejpam-5661	437	9	θ(υ−	θ(υ−	PROPN
ejpam-5661	437	10	α	α	NOUN
ejpam-5661	437	11	)	)	PUNCT
ejpam-5661	437	12	2	2	NUM
ejpam-5661	437	13	(	(	PUNCT
ejpam-5661	437	14	∫	∫	PROPN
ejpam-5661	437	15	1	1	NUM
ejpam-5661	437	16	0	0	NUM
ejpam-5661	437	17	(	(	PUNCT
ejpam-5661	437	18	qϖ)ddqϖ	qϖ)ddqϖ	NOUN
ejpam-5661	437	19	)	)	PUNCT
ejpam-5661	437	20	1	1	NUM
ejpam-5661	437	21	d	d	NOUN
ejpam-5661	437	22	(	(	PUNCT
ejpam-5661	437	23	∫	∫	PROPN
ejpam-5661	437	24	1	1	NUM
ejpam-5661	437	25	0	0	NUM
ejpam-5661	437	26	|	|	ADV
ejpam-5661	437	27	αdqφ(θϖb+	αdqφ(θϖb+	NOUN
ejpam-5661	437	28	(	(	PUNCT
ejpam-5661	437	29	1−θϖ)α)|c	1−θϖ)α)|c	NUM
ejpam-5661	437	30	dqϖ	dqϖ	NOUN
ejpam-5661	437	31	)	)	PUNCT
ejpam-5661	437	32	1	1	NUM
ejpam-5661	437	33	c	c	NOUN
ejpam-5661	437	34	+	+	CCONJ
ejpam-5661	437	35	θ(υ−	θ(υ−	PROPN
ejpam-5661	437	36	α	α	NOUN
ejpam-5661	437	37	)	)	PUNCT
ejpam-5661	437	38	2	2	NUM
ejpam-5661	437	39	(	(	PUNCT
ejpam-5661	437	40	∫	∫	PROPN
ejpam-5661	437	41	1	1	NUM
ejpam-5661	437	42	0	0	NUM
ejpam-5661	437	43	(	(	PUNCT
ejpam-5661	437	44	qϖ)ddqϖ	qϖ)ddqϖ	NOUN
ejpam-5661	437	45	)	)	PUNCT
ejpam-5661	437	46	1	1	NUM
ejpam-5661	437	47	d	d	NOUN
ejpam-5661	437	48	(	(	PUNCT
ejpam-5661	437	49	∫	∫	PROPN
ejpam-5661	437	50	1	1	NUM
ejpam-5661	437	51	0	0	NUM
ejpam-5661	437	52	∣∣	∣∣	NUM
ejpam-5661	437	53	υdqφ(θϖα+	υdqφ(θϖα+	X
ejpam-5661	437	54	(	(	PUNCT
ejpam-5661	437	55	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	437	56	)	)	PUNCT
ejpam-5661	437	57	∣∣c	∣∣c	PROPN
ejpam-5661	437	58	dqϖ	dqϖ	NOUN
ejpam-5661	437	59	)	)	PUNCT
ejpam-5661	437	60	1	1	NUM
ejpam-5661	437	61	c	c	PROPN
ejpam-5661	437	62	≤	≤	NUM
ejpam-5661	437	63	θ(υ−	θ(υ−	PROPN
ejpam-5661	437	64	α	α	NOUN
ejpam-5661	437	65	)	)	PUNCT
ejpam-5661	437	66	2	2	NUM
ejpam-5661	437	67	(	(	PUNCT
ejpam-5661	437	68	∫	∫	PROPN
ejpam-5661	437	69	1	1	NUM
ejpam-5661	437	70	0	0	NUM
ejpam-5661	437	71	(	(	PUNCT
ejpam-5661	437	72	qϖ)ddqϖ	qϖ)ddqϖ	NOUN
ejpam-5661	437	73	)	)	PUNCT
ejpam-5661	437	74	1	1	NUM
ejpam-5661	437	75	d	d	NOUN
ejpam-5661	437	76	(	(	PUNCT
ejpam-5661	437	77	∫	∫	PROPN
ejpam-5661	437	78	1	1	NUM
ejpam-5661	437	79	0	0	NUM
ejpam-5661	437	80	(	(	PUNCT
ejpam-5661	437	81	θϖ|	θϖ|	NOUN
ejpam-5661	437	82	αdqφ(υ)|c	αdqφ(υ)|c	PROPN
ejpam-5661	437	83	+	+	CCONJ
ejpam-5661	437	84	(	(	PUNCT
ejpam-5661	437	85	1−θϖ)|	1−θϖ)|	ADJ
ejpam-5661	437	86	αdqφ(α)|c	αdqφ(α)|c	PROPN
ejpam-5661	437	87	c.	c.	PROPN
ejpam-5661	437	88	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	437	89	,	,	PUNCT
ejpam-5661	437	90	p.	p.	PROPN
ejpam-5661	437	91	yotkaew	yotkaew	PROPN
ejpam-5661	437	92	/	/	SYM
ejpam-5661	437	93	eur	eur	PROPN
ejpam-5661	437	94	.	.	PUNCT
ejpam-5661	438	1	j.	j.	PROPN
ejpam-5661	438	2	pure	pure	PROPN
ejpam-5661	438	3	appl	appl	PROPN
ejpam-5661	438	4	.	.	PROPN
ejpam-5661	438	5	math	math	PROPN
ejpam-5661	438	6	,	,	PUNCT
ejpam-5661	438	7	18	18	NUM
ejpam-5661	438	8	(	(	PUNCT
ejpam-5661	438	9	1	1	NUM
ejpam-5661	438	10	)	)	PUNCT
ejpam-5661	438	11	(	(	PUNCT
ejpam-5661	438	12	2025	2025	NUM
ejpam-5661	438	13	)	)	PUNCT
ejpam-5661	438	14	,	,	PUNCT
ejpam-5661	438	15	5661	5661	NUM
ejpam-5661	438	16	18	18	NUM
ejpam-5661	438	17	of	of	ADP
ejpam-5661	438	18	24	24	NUM
ejpam-5661	438	19	−	−	NOUN
ejpam-5661	438	20	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	438	21	α)2)dqϖ	α)2)dqϖ	NUM
ejpam-5661	438	22	)	)	PUNCT
ejpam-5661	438	23	1	1	NUM
ejpam-5661	438	24	c	c	NOUN
ejpam-5661	438	25	+	+	CCONJ
ejpam-5661	438	26	θ(υ−	θ(υ−	PROPN
ejpam-5661	438	27	α	α	NOUN
ejpam-5661	438	28	)	)	PUNCT
ejpam-5661	438	29	2	2	NUM
ejpam-5661	438	30	(	(	PUNCT
ejpam-5661	438	31	∫	∫	PROPN
ejpam-5661	438	32	1	1	NUM
ejpam-5661	438	33	0	0	NUM
ejpam-5661	438	34	(	(	PUNCT
ejpam-5661	438	35	qϖ)ddqϖ	qϖ)ddqϖ	NOUN
ejpam-5661	438	36	)	)	PUNCT
ejpam-5661	438	37	1	1	NUM
ejpam-5661	438	38	d	d	NOUN
ejpam-5661	438	39	(	(	PUNCT
ejpam-5661	438	40	∫	∫	PROPN
ejpam-5661	438	41	1	1	NUM
ejpam-5661	438	42	0	0	NUM
ejpam-5661	438	43	(	(	PUNCT
ejpam-5661	438	44	θϖ	θϖ	PRON
ejpam-5661	438	45	|	|	ADV
ejpam-5661	438	46	bdqφ(α)|c	bdqφ(α)|c	PROPN
ejpam-5661	438	47	+	+	CCONJ
ejpam-5661	438	48	(	(	PUNCT
ejpam-5661	438	49	1−θϖ)|	1−θϖ)|	NUM
ejpam-5661	438	50	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	438	51	−	−	PROPN
ejpam-5661	438	52	φθϖ(1−θϖ)(b−	φθϖ(1−θϖ)(b−	NOUN
ejpam-5661	438	53	α)2)dqϖ	α)2)dqϖ	NUM
ejpam-5661	438	54	)	)	PUNCT
ejpam-5661	438	55	1	1	NUM
ejpam-5661	438	56	c	c	NOUN
ejpam-5661	438	57	≤	≤	X
ejpam-5661	438	58	qθ(υ−	qθ(υ−	NUM
ejpam-5661	438	59	α	α	NOUN
ejpam-5661	438	60	)	)	PUNCT
ejpam-5661	438	61	2	2	NUM
ejpam-5661	438	62	(	(	PUNCT
ejpam-5661	438	63	1	1	NUM
ejpam-5661	438	64	[	[	X
ejpam-5661	438	65	d+	d+	PUNCT
ejpam-5661	438	66	1]q	1]q	NUM
ejpam-5661	438	67	)	)	PUNCT
ejpam-5661	438	68	1	1	NUM
ejpam-5661	438	69	d	d	NOUN
ejpam-5661	438	70	(	(	PUNCT
ejpam-5661	438	71	(	(	PUNCT
ejpam-5661	438	72	θ|	θ|	PROPN
ejpam-5661	438	73	αdqφ(υ)|c	αdqφ(υ)|c	X
ejpam-5661	439	1	[	[	X
ejpam-5661	439	2	2]q	2]q	NUM
ejpam-5661	439	3	+	+	CCONJ
ejpam-5661	439	4	(	(	PUNCT
ejpam-5661	439	5	1−	1−	NUM
ejpam-5661	439	6	θ	θ	PROPN
ejpam-5661	440	1	[	[	X
ejpam-5661	440	2	2]q	2]q	NUM
ejpam-5661	440	3	)	)	PUNCT
ejpam-5661	440	4	|	|	ADV
ejpam-5661	440	5	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	440	6	)	)	PUNCT
ejpam-5661	440	7	1	1	NUM
ejpam-5661	440	8	c	c	NOUN
ejpam-5661	440	9	+	+	CCONJ
ejpam-5661	440	10	(	(	PUNCT
ejpam-5661	440	11	θ|	θ|	PROPN
ejpam-5661	440	12	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	441	1	[	[	X
ejpam-5661	441	2	2]q	2]q	NUM
ejpam-5661	441	3	+	+	CCONJ
ejpam-5661	441	4	(	(	PUNCT
ejpam-5661	441	5	1−	1−	NUM
ejpam-5661	441	6	θ	θ	PROPN
ejpam-5661	442	1	[	[	X
ejpam-5661	442	2	2]q	2]q	NUM
ejpam-5661	442	3	)	)	PUNCT
ejpam-5661	442	4	|	|	ADV
ejpam-5661	442	5	υdqφ(υ)|c	υdqφ(υ)|c	PUNCT
ejpam-5661	442	6	)	)	PUNCT
ejpam-5661	442	7	1	1	NUM
ejpam-5661	442	8	c	c	NOUN
ejpam-5661	442	9	)	)	PUNCT
ejpam-5661	442	10	≤	≤	NOUN
ejpam-5661	442	11	qθ(υ−	qθ(υ−	NUM
ejpam-5661	442	12	α	α	NOUN
ejpam-5661	442	13	)	)	PUNCT
ejpam-5661	442	14	2	2	NUM
ejpam-5661	442	15	(	(	PUNCT
ejpam-5661	442	16	1	1	NUM
ejpam-5661	442	17	[	[	X
ejpam-5661	442	18	d+	d+	PUNCT
ejpam-5661	442	19	1]q	1]q	NUM
ejpam-5661	442	20	)	)	PUNCT
ejpam-5661	442	21	1	1	NUM
ejpam-5661	442	22	d	d	NOUN
ejpam-5661	442	23	(	(	PUNCT
ejpam-5661	442	24	(	(	PUNCT
ejpam-5661	442	25	θ|	θ|	PROPN
ejpam-5661	442	26	αdqφ(υ)|c	αdqφ(υ)|c	X
ejpam-5661	443	1	[	[	X
ejpam-5661	443	2	2]q	2]q	NUM
ejpam-5661	443	3	+	+	CCONJ
ejpam-5661	443	4	(	(	PUNCT
ejpam-5661	443	5	[	[	X
ejpam-5661	443	6	2]q	2]q	NUM
ejpam-5661	443	7	−θ	−θ	ADJ
ejpam-5661	443	8	[	[	X
ejpam-5661	443	9	2]q	2]q	NUM
ejpam-5661	443	10	)	)	PUNCT
ejpam-5661	443	11	|	|	ADV
ejpam-5661	443	12	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	443	13	)	)	PUNCT
ejpam-5661	443	14	1	1	NUM
ejpam-5661	443	15	c	c	NOUN
ejpam-5661	443	16	+	+	CCONJ
ejpam-5661	443	17	(	(	PUNCT
ejpam-5661	443	18	θ|	θ|	PROPN
ejpam-5661	443	19	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	443	20	[	[	X
ejpam-5661	443	21	2]q	2]q	NUM
ejpam-5661	443	22	+	+	CCONJ
ejpam-5661	443	23	(	(	PUNCT
ejpam-5661	443	24	[	[	X
ejpam-5661	443	25	2]q	2]q	NUM
ejpam-5661	443	26	−θ	−θ	ADJ
ejpam-5661	443	27	[	[	X
ejpam-5661	443	28	2]q	2]q	NUM
ejpam-5661	443	29	)	)	PUNCT
ejpam-5661	443	30	|	|	ADV
ejpam-5661	443	31	υdqφ(υ)|c	υdqφ(υ)|c	PUNCT
ejpam-5661	443	32	)	)	PUNCT
ejpam-5661	443	33	1	1	NUM
ejpam-5661	443	34	c	c	NOUN
ejpam-5661	443	35	)	)	PUNCT
ejpam-5661	443	36	.	.	PUNCT
ejpam-5661	444	1	this	this	PRON
ejpam-5661	444	2	concludes	conclude	VERB
ejpam-5661	444	3	the	the	DET
ejpam-5661	444	4	proof	proof	NOUN
ejpam-5661	444	5	.	.	PUNCT
ejpam-5661	445	1	remark	remark	NOUN
ejpam-5661	445	2	5	5	NUM
ejpam-5661	445	3	.	.	PUNCT
ejpam-5661	446	1	when	when	SCONJ
ejpam-5661	446	2	θ	θ	X
ejpam-5661	446	3	=	=	SYM
ejpam-5661	446	4	1	1	NUM
ejpam-5661	446	5	in	in	ADP
ejpam-5661	446	6	theorem	theorem	NOUN
ejpam-5661	446	7	7	7	NUM
ejpam-5661	446	8	,	,	PUNCT
ejpam-5661	446	9	we	we	PRON
ejpam-5661	446	10	derive	derive	VERB
ejpam-5661	446	11	trapezoid	trapezoid	ADJ
ejpam-5661	446	12	-	-	PUNCT
ejpam-5661	446	13	type	type	NOUN
ejpam-5661	446	14	inequalities:∣∣∣∣∣(υ−	inequalities:∣∣∣∣∣(υ−	NOUN
ejpam-5661	446	15	α	α	NOUN
ejpam-5661	446	16	)	)	PUNCT
ejpam-5661	446	17	2	2	NUM
ejpam-5661	446	18	(	(	PUNCT
ejpam-5661	446	19	∫	∫	PROPN
ejpam-5661	446	20	υ	υ	PROPN
ejpam-5661	446	21	α	α	PROPN
ejpam-5661	446	22	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	446	23	)	)	PUNCT
ejpam-5661	446	24	αdqϖ	αdqϖ	PROPN
ejpam-5661	447	1	+	+	CCONJ
ejpam-5661	447	2	∫	∫	PROPN
ejpam-5661	447	3	υ	υ	PROPN
ejpam-5661	447	4	α	α	PROPN
ejpam-5661	447	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	447	6	)	)	PUNCT
ejpam-5661	447	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	447	8	)	)	PUNCT
ejpam-5661	448	1	−	−	PROPN
ejpam-5661	448	2	φ(α	φ(α	PROPN
ejpam-5661	448	3	)	)	PUNCT
ejpam-5661	449	1	+	+	CCONJ
ejpam-5661	450	1	φ(υ	φ(υ	PROPN
ejpam-5661	450	2	)	)	PUNCT
ejpam-5661	450	3	2	2	NUM
ejpam-5661	450	4	∣∣∣∣∣	∣∣∣∣∣	SYM
ejpam-5661	450	5	≤	≤	PROPN
ejpam-5661	450	6	q(υ−	q(υ−	PROPN
ejpam-5661	450	7	α	α	NOUN
ejpam-5661	450	8	)	)	PUNCT
ejpam-5661	450	9	2	2	NUM
ejpam-5661	450	10	(	(	PUNCT
ejpam-5661	450	11	1	1	NUM
ejpam-5661	450	12	[	[	X
ejpam-5661	450	13	d+	d+	PUNCT
ejpam-5661	450	14	1]q	1]q	NUM
ejpam-5661	450	15	)	)	PUNCT
ejpam-5661	450	16	1	1	NUM
ejpam-5661	450	17	d	d	NOUN
ejpam-5661	450	18	(	(	PUNCT
ejpam-5661	450	19	(	(	PUNCT
ejpam-5661	450	20	|	|	ADV
ejpam-5661	450	21	αdqφ(υ)|c	αdqφ(υ)|c	X
ejpam-5661	451	1	[	[	X
ejpam-5661	451	2	2]q	2]q	NUM
ejpam-5661	451	3	+	+	CCONJ
ejpam-5661	451	4	(	(	PUNCT
ejpam-5661	451	5	1−	1−	NUM
ejpam-5661	451	6	1	1	NUM
ejpam-5661	451	7	[	[	X
ejpam-5661	451	8	2]q	2]q	NUM
ejpam-5661	451	9	)	)	PUNCT
ejpam-5661	452	1	|	|	ADV
ejpam-5661	452	2	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	452	3	−	−	NOUN
ejpam-5661	453	1	φ	φ	PROPN
ejpam-5661	453	2	(	(	PUNCT
ejpam-5661	453	3	1	1	NUM
ejpam-5661	453	4	[	[	X
ejpam-5661	453	5	2]q	2]q	NUM
ejpam-5661	453	6	−	−	ADP
ejpam-5661	453	7	1	1	NUM
ejpam-5661	453	8	[	[	X
ejpam-5661	453	9	3]q	3]q	NUM
ejpam-5661	453	10	)	)	PUNCT
ejpam-5661	453	11	(	(	PUNCT
ejpam-5661	453	12	υ−	υ−	PROPN
ejpam-5661	453	13	α)2	α)2	NOUN
ejpam-5661	453	14	)	)	PUNCT
ejpam-5661	453	15	1	1	NUM
ejpam-5661	453	16	c	c	NOUN
ejpam-5661	453	17	+	+	CCONJ
ejpam-5661	453	18	(	(	PUNCT
ejpam-5661	453	19	|	|	ADV
ejpam-5661	453	20	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	453	21	[	[	X
ejpam-5661	453	22	2]q	2]q	NUM
ejpam-5661	453	23	+	+	CCONJ
ejpam-5661	453	24	(	(	PUNCT
ejpam-5661	453	25	1−	1−	NUM
ejpam-5661	453	26	1	1	NUM
ejpam-5661	453	27	[	[	X
ejpam-5661	453	28	2]q	2]q	NUM
ejpam-5661	453	29	)	)	PUNCT
ejpam-5661	453	30	|	|	ADV
ejpam-5661	453	31	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	453	32	−	−	PROPN
ejpam-5661	453	33	φ	φ	PROPN
ejpam-5661	453	34	(	(	PUNCT
ejpam-5661	453	35	1	1	NUM
ejpam-5661	453	36	[	[	X
ejpam-5661	453	37	2]q	2]q	NUM
ejpam-5661	453	38	−	−	ADP
ejpam-5661	453	39	1	1	NUM
ejpam-5661	453	40	[	[	X
ejpam-5661	453	41	3]q	3]q	NUM
ejpam-5661	453	42	)	)	PUNCT
ejpam-5661	453	43	(	(	PUNCT
ejpam-5661	453	44	υ−	υ−	PROPN
ejpam-5661	453	45	α)2	α)2	NOUN
ejpam-5661	453	46	)	)	PUNCT
ejpam-5661	453	47	1	1	NUM
ejpam-5661	453	48	c	c	NOUN
ejpam-5661	453	49	)	)	PUNCT
ejpam-5661	453	50	≤	≤	PROPN
ejpam-5661	453	51	q(υ−	q(υ−	PROPN
ejpam-5661	453	52	α	α	NOUN
ejpam-5661	453	53	)	)	PUNCT
ejpam-5661	453	54	2	2	NUM
ejpam-5661	453	55	(	(	PUNCT
ejpam-5661	453	56	1	1	NUM
ejpam-5661	453	57	[	[	X
ejpam-5661	453	58	d+	d+	PUNCT
ejpam-5661	453	59	1]q	1]q	NUM
ejpam-5661	453	60	)	)	PUNCT
ejpam-5661	453	61	1	1	NUM
ejpam-5661	453	62	d	d	NOUN
ejpam-5661	453	63	(	(	PUNCT
ejpam-5661	453	64	(	(	PUNCT
ejpam-5661	453	65	|	|	ADV
ejpam-5661	453	66	αdqφ(υ)|c	αdqφ(υ)|c	PROPN
ejpam-5661	453	67	+	+	CCONJ
ejpam-5661	453	68	q|	q|	NOUN
ejpam-5661	453	69	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	454	1	[	[	X
ejpam-5661	454	2	2]q	2]q	NUM
ejpam-5661	454	3	)	)	PUNCT
ejpam-5661	454	4	1	1	NUM
ejpam-5661	454	5	c	c	NOUN
ejpam-5661	454	6	+	+	CCONJ
ejpam-5661	454	7	(	(	PUNCT
ejpam-5661	454	8	|	|	ADV
ejpam-5661	454	9	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	454	10	+	+	CCONJ
ejpam-5661	454	11	q|	q|	NOUN
ejpam-5661	454	12	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	454	13	[	[	X
ejpam-5661	454	14	2]2	2]2	NUM
ejpam-5661	454	15	)	)	PUNCT
ejpam-5661	454	16	1	1	NUM
ejpam-5661	454	17	c	c	NOUN
ejpam-5661	454	18	)	)	PUNCT
ejpam-5661	454	19	.	.	PUNCT
ejpam-5661	455	1	remark	remark	VERB
ejpam-5661	455	2	6	6	NUM
ejpam-5661	455	3	.	.	PUNCT
ejpam-5661	456	1	when	when	SCONJ
ejpam-5661	456	2	θ	θ	PROPN
ejpam-5661	456	3	=	=	SYM
ejpam-5661	456	4	1/2	1/2	NUM
ejpam-5661	456	5	in	in	ADP
ejpam-5661	456	6	theorem	theorem	NOUN
ejpam-5661	456	7	7	7	NUM
ejpam-5661	456	8	,	,	PUNCT
ejpam-5661	456	9	we	we	PRON
ejpam-5661	456	10	derive	derive	VERB
ejpam-5661	456	11	midpoint	midpoint	NOUN
ejpam-5661	456	12	-	-	PUNCT
ejpam-5661	456	13	type	type	NOUN
ejpam-5661	456	14	inequalities:∣∣∣∣∣(υ−	inequalities:∣∣∣∣∣(υ−	NOUN
ejpam-5661	456	15	α	α	NOUN
ejpam-5661	456	16	)	)	PUNCT
ejpam-5661	456	17	2	2	NUM
ejpam-5661	456	18	(	(	PUNCT
ejpam-5661	456	19	∫	∫	PROPN
ejpam-5661	456	20	(	(	PUNCT
ejpam-5661	456	21	α+υ	α+υ	NUM
ejpam-5661	456	22	)	)	PUNCT
ejpam-5661	456	23	2	2	NUM
ejpam-5661	456	24	α	α	PRON
ejpam-5661	456	25	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	456	26	)	)	PUNCT
ejpam-5661	456	27	αdqϖ	αdqϖ	PROPN
ejpam-5661	457	1	+	+	CCONJ
ejpam-5661	457	2	∫	∫	PROPN
ejpam-5661	457	3	υ	υ	X
ejpam-5661	457	4	(	(	PUNCT
ejpam-5661	457	5	α+υ	α+υ	NUM
ejpam-5661	457	6	)	)	PUNCT
ejpam-5661	457	7	2	2	NUM
ejpam-5661	457	8	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	457	9	)	)	PUNCT
ejpam-5661	457	10	υdqϖ	υdqϖ	NOUN
ejpam-5661	457	11	)	)	PUNCT
ejpam-5661	458	1	−	−	PROPN
ejpam-5661	458	2	φ	φ	PROPN
ejpam-5661	458	3	(	(	PUNCT
ejpam-5661	458	4	α+υ	α+υ	NUM
ejpam-5661	458	5	2	2	NUM
ejpam-5661	458	6	)	)	PUNCT
ejpam-5661	458	7	∣∣∣∣∣	∣∣∣∣∣	PROPN
ejpam-5661	458	8	c.	c.	PROPN
ejpam-5661	458	9	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	458	10	,	,	PUNCT
ejpam-5661	458	11	p.	p.	PROPN
ejpam-5661	458	12	yotkaew	yotkaew	PROPN
ejpam-5661	458	13	/	/	SYM
ejpam-5661	458	14	eur	eur	PROPN
ejpam-5661	458	15	.	.	PUNCT
ejpam-5661	459	1	j.	j.	PROPN
ejpam-5661	459	2	pure	pure	PROPN
ejpam-5661	459	3	appl	appl	PROPN
ejpam-5661	459	4	.	.	PROPN
ejpam-5661	459	5	math	math	PROPN
ejpam-5661	459	6	,	,	PUNCT
ejpam-5661	459	7	18	18	NUM
ejpam-5661	459	8	(	(	PUNCT
ejpam-5661	459	9	1	1	NUM
ejpam-5661	459	10	)	)	PUNCT
ejpam-5661	459	11	(	(	PUNCT
ejpam-5661	459	12	2025	2025	NUM
ejpam-5661	459	13	)	)	PUNCT
ejpam-5661	459	14	,	,	PUNCT
ejpam-5661	459	15	5661	5661	NUM
ejpam-5661	459	16	19	19	NUM
ejpam-5661	459	17	of	of	ADP
ejpam-5661	459	18	24	24	NUM
ejpam-5661	459	19	≤	≤	NUM
ejpam-5661	459	20	q(υ−	q(υ−	PROPN
ejpam-5661	459	21	α	α	NOUN
ejpam-5661	459	22	)	)	PUNCT
ejpam-5661	459	23	2	2	NUM
ejpam-5661	459	24	(	(	PUNCT
ejpam-5661	459	25	1	1	NUM
ejpam-5661	459	26	[	[	X
ejpam-5661	459	27	d+	d+	PUNCT
ejpam-5661	459	28	1]q	1]q	NUM
ejpam-5661	459	29	)	)	PUNCT
ejpam-5661	459	30	1	1	NUM
ejpam-5661	459	31	d	d	NOUN
ejpam-5661	459	32	(	(	PUNCT
ejpam-5661	459	33	(	(	PUNCT
ejpam-5661	459	34	|	|	ADV
ejpam-5661	459	35	αdqφ(υ)|c	αdqφ(υ)|c	NOUN
ejpam-5661	460	1	2[2]q	2[2]q	NUM
ejpam-5661	460	2	+	+	CCONJ
ejpam-5661	460	3	(	(	PUNCT
ejpam-5661	460	4	1−	1−	NUM
ejpam-5661	460	5	1	1	NUM
ejpam-5661	460	6	2[2]q	2[2]q	NUM
ejpam-5661	460	7	)	)	PUNCT
ejpam-5661	460	8	|	|	ADV
ejpam-5661	460	9	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	460	10	−	−	NOUN
ejpam-5661	460	11	φ	φ	NUM
ejpam-5661	460	12	2	2	NUM
ejpam-5661	460	13	(	(	PUNCT
ejpam-5661	460	14	1	1	NUM
ejpam-5661	460	15	[	[	X
ejpam-5661	460	16	2]q	2]q	NUM
ejpam-5661	460	17	−	−	ADP
ejpam-5661	460	18	1	1	NUM
ejpam-5661	460	19	2[3]q	2[3]q	NUM
ejpam-5661	460	20	)	)	PUNCT
ejpam-5661	460	21	(	(	PUNCT
ejpam-5661	460	22	υ−	υ−	PROPN
ejpam-5661	460	23	α)2	α)2	NOUN
ejpam-5661	460	24	)	)	PUNCT
ejpam-5661	460	25	1	1	NUM
ejpam-5661	460	26	c	c	NOUN
ejpam-5661	460	27	+	+	CCONJ
ejpam-5661	460	28	(	(	PUNCT
ejpam-5661	460	29	|	|	ADV
ejpam-5661	460	30	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	460	31	2[2]q	2[2]q	PROPN
ejpam-5661	460	32	+	+	CCONJ
ejpam-5661	460	33	(	(	PUNCT
ejpam-5661	460	34	1−	1−	NUM
ejpam-5661	460	35	1	1	NUM
ejpam-5661	460	36	2[2]q	2[2]q	NUM
ejpam-5661	460	37	)	)	PUNCT
ejpam-5661	460	38	|	|	ADV
ejpam-5661	460	39	υdqφ(υ)|c	υdqφ(υ)|c	ADP
ejpam-5661	460	40	−	−	PROPN
ejpam-5661	460	41	φ	φ	NUM
ejpam-5661	460	42	2	2	NUM
ejpam-5661	460	43	(	(	PUNCT
ejpam-5661	460	44	1	1	NUM
ejpam-5661	460	45	[	[	X
ejpam-5661	460	46	2]q	2]q	NUM
ejpam-5661	460	47	−	−	ADP
ejpam-5661	460	48	1	1	NUM
ejpam-5661	460	49	2[3]q	2[3]q	NUM
ejpam-5661	460	50	)	)	PUNCT
ejpam-5661	460	51	(	(	PUNCT
ejpam-5661	460	52	υ−	υ−	PROPN
ejpam-5661	460	53	α)2	α)2	NOUN
ejpam-5661	460	54	)	)	PUNCT
ejpam-5661	460	55	1	1	NUM
ejpam-5661	460	56	c	c	NOUN
ejpam-5661	460	57	)	)	PUNCT
ejpam-5661	460	58	≤	≤	PROPN
ejpam-5661	460	59	q(υ−	q(υ−	PROPN
ejpam-5661	460	60	α	α	NOUN
ejpam-5661	460	61	)	)	PUNCT
ejpam-5661	460	62	4	4	NUM
ejpam-5661	460	63	(	(	PUNCT
ejpam-5661	460	64	1	1	NUM
ejpam-5661	460	65	[	[	X
ejpam-5661	460	66	d+	d+	PUNCT
ejpam-5661	460	67	1]q	1]q	NUM
ejpam-5661	460	68	)	)	PUNCT
ejpam-5661	460	69	1	1	NUM
ejpam-5661	460	70	d	d	NOUN
ejpam-5661	460	71	(	(	PUNCT
ejpam-5661	460	72	(	(	PUNCT
ejpam-5661	460	73	|	|	ADV
ejpam-5661	460	74	αdqφ(υ)|c	αdqφ(υ)|c	PROPN
ejpam-5661	460	75	+	+	CCONJ
ejpam-5661	460	76	(	(	PUNCT
ejpam-5661	461	1	[	[	X
ejpam-5661	461	2	2]q	2]q	NUM
ejpam-5661	461	3	+	+	NUM
ejpam-5661	461	4	q)|	q)|	NOUN
ejpam-5661	461	5	αdqφ(α)|c	αdqφ(α)|c	PUNCT
ejpam-5661	462	1	[	[	X
ejpam-5661	462	2	2]q	2]q	NUM
ejpam-5661	462	3	)	)	PUNCT
ejpam-5661	462	4	1	1	NUM
ejpam-5661	462	5	c	c	NOUN
ejpam-5661	462	6	+	+	CCONJ
ejpam-5661	462	7	(	(	PUNCT
ejpam-5661	462	8	|	|	ADV
ejpam-5661	462	9	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	462	10	+	+	CCONJ
ejpam-5661	462	11	(	(	PUNCT
ejpam-5661	462	12	[	[	X
ejpam-5661	462	13	2]q	2]q	NUM
ejpam-5661	462	14	+	+	CCONJ
ejpam-5661	462	15	q)|	q)|	PRON
ejpam-5661	462	16	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	462	17	[	[	X
ejpam-5661	462	18	2]2	2]2	NUM
ejpam-5661	462	19	)	)	PUNCT
ejpam-5661	462	20	1	1	NUM
ejpam-5661	462	21	c	c	NOUN
ejpam-5661	462	22	)	)	PUNCT
ejpam-5661	462	23	.	.	PUNCT
ejpam-5661	463	1	theorem	theorem	ADJ
ejpam-5661	463	2	8	8	NUM
ejpam-5661	463	3	.	.	PUNCT
ejpam-5661	464	1	let	let	VERB
ejpam-5661	464	2	φ	φ	NOUN
ejpam-5661	464	3	:	:	PUNCT
ejpam-5661	465	1	[	[	X
ejpam-5661	465	2	α	α	X
ejpam-5661	465	3	,	,	PUNCT
ejpam-5661	465	4	υ	υ	NOUN
ejpam-5661	465	5	]	]	X
ejpam-5661	465	6	→	→	PUNCT
ejpam-5661	465	7	r	r	NOUN
ejpam-5661	465	8	be	be	AUX
ejpam-5661	465	9	a	a	DET
ejpam-5661	465	10	q	q	ADJ
ejpam-5661	465	11	-	-	PUNCT
ejpam-5661	465	12	differentiable	differentiable	ADJ
ejpam-5661	465	13	function	function	NOUN
ejpam-5661	465	14	.	.	PUNCT
ejpam-5661	466	1	if	if	SCONJ
ejpam-5661	466	2	|αdqφ|c	|αdqφ|c	NUM
ejpam-5661	466	3	and	and	CCONJ
ejpam-5661	466	4	|	|	ADV
ejpam-5661	466	5	υdqφ|c	υdqφ|c	ADJ
ejpam-5661	466	6	,	,	PUNCT
ejpam-5661	466	7	c	c	PROPN
ejpam-5661	466	8	≥	≥	NUM
ejpam-5661	466	9	1	1	NUM
ejpam-5661	466	10	are	be	AUX
ejpam-5661	466	11	strongly	strongly	ADV
ejpam-5661	466	12	convex	convex	ADJ
ejpam-5661	466	13	functions	function	NOUN
ejpam-5661	466	14	on	on	ADP
ejpam-5661	466	15	[	[	X
ejpam-5661	466	16	α	α	NOUN
ejpam-5661	466	17	,	,	PUNCT
ejpam-5661	466	18	υ	υ	NOUN
ejpam-5661	466	19	]	]	X
ejpam-5661	466	20	for	for	ADP
ejpam-5661	466	21	φ	φ	PROPN
ejpam-5661	466	22	>	>	X
ejpam-5661	466	23	0	0	PROPN
ejpam-5661	466	24	,	,	PUNCT
ejpam-5661	466	25	then	then	ADV
ejpam-5661	466	26	the	the	DET
ejpam-5661	466	27	following	follow	VERB
ejpam-5661	466	28	inequalities	inequality	NOUN
ejpam-5661	466	29	are	be	AUX
ejpam-5661	466	30	established:∣∣∣∣∣θ(υ−	established:∣∣∣∣∣θ(υ−	PROPN
ejpam-5661	466	31	α	α	NOUN
ejpam-5661	466	32	)	)	PUNCT
ejpam-5661	466	33	2	2	NUM
ejpam-5661	466	34	(	(	PUNCT
ejpam-5661	466	35	∫	∫	PROPN
ejpam-5661	466	36	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	466	37	α	α	PROPN
ejpam-5661	466	38	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	466	39	)	)	PUNCT
ejpam-5661	466	40	αdqϖ	αdqϖ	PROPN
ejpam-5661	467	1	+	+	CCONJ
ejpam-5661	467	2	∫	∫	PROPN
ejpam-5661	467	3	υ	υ	PROPN
ejpam-5661	467	4	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	467	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	467	6	)	)	PUNCT
ejpam-5661	467	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	467	8	)	)	PUNCT
ejpam-5661	468	1	−φ(θα+	−φ(θα+	PROPN
ejpam-5661	468	2	(	(	PUNCT
ejpam-5661	468	3	1−θ)υ	1−θ)υ	NUM
ejpam-5661	468	4	)	)	PUNCT
ejpam-5661	469	1	+	+	CCONJ
ejpam-5661	469	2	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	469	3	(	(	PUNCT
ejpam-5661	469	4	1−θ)α	1−θ)α	NUM
ejpam-5661	469	5	)	)	PUNCT
ejpam-5661	469	6	2	2	NUM
ejpam-5661	469	7	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5661	469	8	≤	≤	NOUN
ejpam-5661	469	9	qθ(υ−	qθ(υ−	NUM
ejpam-5661	469	10	α	α	NOUN
ejpam-5661	469	11	)	)	PUNCT
ejpam-5661	469	12	2	2	NUM
ejpam-5661	469	13	(	(	PUNCT
ejpam-5661	469	14	1	1	NUM
ejpam-5661	469	15	[	[	X
ejpam-5661	469	16	2]q	2]q	NUM
ejpam-5661	469	17	)	)	PUNCT
ejpam-5661	469	18	1−	1−	NUM
ejpam-5661	469	19	1	1	NUM
ejpam-5661	469	20	c	c	NOUN
ejpam-5661	469	21	(	(	PUNCT
ejpam-5661	469	22	(	(	PUNCT
ejpam-5661	469	23	θ|	θ|	PROPN
ejpam-5661	469	24	αdqf(υ)|c	αdqf(υ)|c	VERB
ejpam-5661	469	25	[	[	X
ejpam-5661	469	26	3]q	3]q	NUM
ejpam-5661	469	27	+	+	CCONJ
ejpam-5661	469	28	(	(	PUNCT
ejpam-5661	469	29	1	1	NUM
ejpam-5661	470	1	[	[	X
ejpam-5661	470	2	2]q	2]q	NUM
ejpam-5661	470	3	−	−	ADP
ejpam-5661	470	4	θ	θ	PROPN
ejpam-5661	470	5	[	[	X
ejpam-5661	470	6	3]q	3]q	NUM
ejpam-5661	470	7	)	)	PUNCT
ejpam-5661	470	8	|	|	ADV
ejpam-5661	470	9	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	470	10	−	−	NOUN
ejpam-5661	471	1	φθ	φθ	INTJ
ejpam-5661	471	2	(	(	PUNCT
ejpam-5661	471	3	1	1	NUM
ejpam-5661	471	4	[	[	X
ejpam-5661	471	5	3]q	3]q	NUM
ejpam-5661	471	6	−	−	NUM
ejpam-5661	471	7	θ	θ	PROPN
ejpam-5661	472	1	[	[	X
ejpam-5661	472	2	4]q	4]q	X
ejpam-5661	472	3	)	)	PUNCT
ejpam-5661	472	4	(	(	PUNCT
ejpam-5661	472	5	υ−	υ−	PROPN
ejpam-5661	472	6	α)2	α)2	NOUN
ejpam-5661	472	7	)	)	PUNCT
ejpam-5661	472	8	1	1	NUM
ejpam-5661	472	9	c	c	NOUN
ejpam-5661	472	10	+	+	CCONJ
ejpam-5661	472	11	(	(	PUNCT
ejpam-5661	472	12	θ|	θ|	PROPN
ejpam-5661	472	13	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	473	1	[	[	X
ejpam-5661	474	1	3]q	3]q	NUM
ejpam-5661	474	2	+	+	CCONJ
ejpam-5661	474	3	(	(	PUNCT
ejpam-5661	474	4	1	1	NUM
ejpam-5661	474	5	[	[	X
ejpam-5661	474	6	2]q	2]q	NUM
ejpam-5661	474	7	−	−	ADP
ejpam-5661	474	8	θ	θ	PROPN
ejpam-5661	475	1	[	[	X
ejpam-5661	475	2	3]q	3]q	NUM
ejpam-5661	475	3	)	)	PUNCT
ejpam-5661	476	1	|	|	ADV
ejpam-5661	476	2	υdqφ(υ)|c	υdqφ(υ)|c	X
ejpam-5661	477	1	−	−	PROPN
ejpam-5661	477	2	φθ	φθ	INTJ
ejpam-5661	477	3	(	(	PUNCT
ejpam-5661	477	4	1	1	NUM
ejpam-5661	477	5	[	[	X
ejpam-5661	477	6	3]q	3]q	NUM
ejpam-5661	477	7	−	−	NUM
ejpam-5661	477	8	θ	θ	PROPN
ejpam-5661	478	1	[	[	X
ejpam-5661	478	2	4]q	4]q	X
ejpam-5661	478	3	)	)	PUNCT
ejpam-5661	478	4	(	(	PUNCT
ejpam-5661	478	5	υ−	υ−	PROPN
ejpam-5661	478	6	α)2	α)2	NOUN
ejpam-5661	478	7	)	)	PUNCT
ejpam-5661	478	8	1	1	NUM
ejpam-5661	478	9	c	c	NOUN
ejpam-5661	478	10	)	)	PUNCT
ejpam-5661	478	11	≤	≤	NOUN
ejpam-5661	479	1	qθ(υ−	qθ(υ−	NUM
ejpam-5661	479	2	α	α	NOUN
ejpam-5661	479	3	)	)	PUNCT
ejpam-5661	479	4	2[2]q	2[2]q	PROPN
ejpam-5661	479	5	(	(	PUNCT
ejpam-5661	479	6	(	(	PUNCT
ejpam-5661	480	1	[	[	X
ejpam-5661	480	2	2]qθ|	2]qθ|	NUM
ejpam-5661	480	3	αdqφ(υ)|c	αdqφ(υ)|c	VERB
ejpam-5661	480	4	+	+	CCONJ
ejpam-5661	480	5	(	(	PUNCT
ejpam-5661	480	6	[	[	X
ejpam-5661	480	7	3]q	3]q	NUM
ejpam-5661	480	8	−θ[2]q)|	−θ[2]q)|	NOUN
ejpam-5661	480	9	αdqφ(α)|c	αdqφ(α)|c	PUNCT
ejpam-5661	480	10	[	[	X
ejpam-5661	480	11	3]q	3]q	NUM
ejpam-5661	480	12	)	)	PUNCT
ejpam-5661	480	13	1	1	NUM
ejpam-5661	480	14	c	c	NOUN
ejpam-5661	480	15	+	+	CCONJ
ejpam-5661	480	16	(	(	PUNCT
ejpam-5661	480	17	[	[	X
ejpam-5661	480	18	2]qθ|	2]qθ|	NUM
ejpam-5661	480	19	υdqφ(α)|c	υdqφ(α)|c	NOUN
ejpam-5661	480	20	+	+	CCONJ
ejpam-5661	480	21	(	(	PUNCT
ejpam-5661	480	22	[	[	X
ejpam-5661	480	23	3]q	3]q	NUM
ejpam-5661	480	24	−θ[2]q)|	−θ[2]q)|	ADP
ejpam-5661	480	25	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	480	26	[	[	X
ejpam-5661	480	27	3]q	3]q	NUM
ejpam-5661	480	28	)	)	PUNCT
ejpam-5661	480	29	1	1	NUM
ejpam-5661	480	30	c	c	NOUN
ejpam-5661	480	31	)	)	PUNCT
ejpam-5661	480	32	.	.	PUNCT
ejpam-5661	481	1	(	(	PUNCT
ejpam-5661	481	2	28	28	NUM
ejpam-5661	481	3	)	)	PUNCT
ejpam-5661	481	4	proof	proof	NOUN
ejpam-5661	481	5	.	.	PUNCT
ejpam-5661	482	1	it	it	PRON
ejpam-5661	482	2	can	can	AUX
ejpam-5661	482	3	be	be	AUX
ejpam-5661	482	4	deduced	deduce	VERB
ejpam-5661	482	5	from	from	ADP
ejpam-5661	482	6	lemma	lemma	PROPN
ejpam-5661	482	7	1	1	NUM
ejpam-5661	482	8	,	,	PUNCT
ejpam-5661	482	9	the	the	DET
ejpam-5661	482	10	power	power	NOUN
ejpam-5661	482	11	mean	mean	VERB
ejpam-5661	482	12	inequality	inequality	NOUN
ejpam-5661	482	13	and	and	CCONJ
ejpam-5661	482	14	and	and	CCONJ
ejpam-5661	482	15	the	the	DET
ejpam-5661	482	16	strong	strong	ADJ
ejpam-5661	482	17	convexity	convexity	NOUN
ejpam-5661	482	18	of	of	ADP
ejpam-5661	482	19	|αdqφ|c	|αdqφ|c	NUM
ejpam-5661	482	20	and	and	CCONJ
ejpam-5661	482	21	|υdqφ|c	|υdqφ|c	NUM
ejpam-5661	482	22	that∣∣∣∣∣θ(υ−	that∣∣∣∣∣θ(υ−	NOUN
ejpam-5661	482	23	α	α	NOUN
ejpam-5661	482	24	)	)	PUNCT
ejpam-5661	482	25	2	2	NUM
ejpam-5661	482	26	(	(	PUNCT
ejpam-5661	482	27	∫	∫	PROPN
ejpam-5661	482	28	θυ+(1−θ)α	θυ+(1−θ)α	PROPN
ejpam-5661	482	29	α	α	PROPN
ejpam-5661	482	30	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	482	31	)	)	PUNCT
ejpam-5661	482	32	αdqϖ	αdqϖ	PROPN
ejpam-5661	483	1	+	+	CCONJ
ejpam-5661	483	2	∫	∫	PROPN
ejpam-5661	483	3	υ	υ	PROPN
ejpam-5661	483	4	θα+(1−θ)υ	θα+(1−θ)υ	PROPN
ejpam-5661	483	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	483	6	)	)	PUNCT
ejpam-5661	483	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	483	8	)	)	PUNCT
ejpam-5661	483	9	c.	c.	PROPN
ejpam-5661	483	10	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	483	11	,	,	PUNCT
ejpam-5661	483	12	p.	p.	PROPN
ejpam-5661	483	13	yotkaew	yotkaew	PROPN
ejpam-5661	483	14	/	/	SYM
ejpam-5661	483	15	eur	eur	PROPN
ejpam-5661	483	16	.	.	PUNCT
ejpam-5661	484	1	j.	j.	PROPN
ejpam-5661	484	2	pure	pure	PROPN
ejpam-5661	484	3	appl	appl	PROPN
ejpam-5661	484	4	.	.	PROPN
ejpam-5661	484	5	math	math	PROPN
ejpam-5661	484	6	,	,	PUNCT
ejpam-5661	484	7	18	18	NUM
ejpam-5661	484	8	(	(	PUNCT
ejpam-5661	484	9	1	1	NUM
ejpam-5661	484	10	)	)	PUNCT
ejpam-5661	484	11	(	(	PUNCT
ejpam-5661	484	12	2025	2025	NUM
ejpam-5661	484	13	)	)	PUNCT
ejpam-5661	484	14	,	,	PUNCT
ejpam-5661	484	15	5661	5661	NUM
ejpam-5661	484	16	20	20	NUM
ejpam-5661	484	17	of	of	ADP
ejpam-5661	484	18	24	24	NUM
ejpam-5661	484	19	−φ(θα+	−φ(θα+	PROPN
ejpam-5661	484	20	(	(	PUNCT
ejpam-5661	484	21	1−θ)υ	1−θ)υ	NUM
ejpam-5661	484	22	)	)	PUNCT
ejpam-5661	485	1	+	+	CCONJ
ejpam-5661	485	2	φ(θυ+	φ(θυ+	ADJ
ejpam-5661	485	3	(	(	PUNCT
ejpam-5661	485	4	1−θ)α	1−θ)α	NUM
ejpam-5661	485	5	)	)	PUNCT
ejpam-5661	485	6	2	2	NUM
ejpam-5661	485	7	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5661	485	8	≤	≤	NUM
ejpam-5661	485	9	θ(υ−	θ(υ−	PROPN
ejpam-5661	485	10	α	α	NOUN
ejpam-5661	485	11	)	)	PUNCT
ejpam-5661	485	12	2	2	NUM
ejpam-5661	485	13	(	(	PUNCT
ejpam-5661	485	14	∫	∫	PROPN
ejpam-5661	485	15	1	1	NUM
ejpam-5661	485	16	0	0	NUM
ejpam-5661	485	17	qϖ	qϖ	NOUN
ejpam-5661	485	18	dqϖ	dqϖ	NOUN
ejpam-5661	485	19	)	)	PUNCT
ejpam-5661	485	20	1−	1−	NUM
ejpam-5661	485	21	1	1	NUM
ejpam-5661	485	22	c	c	NOUN
ejpam-5661	485	23	(	(	PUNCT
ejpam-5661	485	24	∫	∫	PROPN
ejpam-5661	485	25	1	1	NUM
ejpam-5661	485	26	0	0	NUM
ejpam-5661	485	27	qϖ	qϖ	NOUN
ejpam-5661	485	28	|	|	ADV
ejpam-5661	485	29	αdqφ(θϖυ+	αdqφ(θϖυ+	NOUN
ejpam-5661	485	30	(	(	PUNCT
ejpam-5661	485	31	1−θϖ)α)|c	1−θϖ)α)|c	NUM
ejpam-5661	485	32	dqϖ	dqϖ	NOUN
ejpam-5661	485	33	)	)	PUNCT
ejpam-5661	485	34	1	1	NUM
ejpam-5661	485	35	c	c	NOUN
ejpam-5661	485	36	+	+	CCONJ
ejpam-5661	485	37	θ(υ−	θ(υ−	PROPN
ejpam-5661	485	38	α	α	NOUN
ejpam-5661	485	39	)	)	PUNCT
ejpam-5661	485	40	2	2	NUM
ejpam-5661	485	41	(	(	PUNCT
ejpam-5661	485	42	∫	∫	PROPN
ejpam-5661	485	43	1	1	NUM
ejpam-5661	485	44	0	0	NUM
ejpam-5661	485	45	qϖ	qϖ	NOUN
ejpam-5661	485	46	dqϖ	dqϖ	NOUN
ejpam-5661	485	47	)	)	PUNCT
ejpam-5661	485	48	1−	1−	NUM
ejpam-5661	485	49	1	1	NUM
ejpam-5661	485	50	c	c	NOUN
ejpam-5661	485	51	(	(	PUNCT
ejpam-5661	485	52	∫	∫	PROPN
ejpam-5661	485	53	1	1	NUM
ejpam-5661	485	54	0	0	NUM
ejpam-5661	485	55	qϖ	qϖ	NOUN
ejpam-5661	485	56	∣∣	∣∣	NUM
ejpam-5661	485	57	υdqφ(θϖα+	υdqφ(θϖα+	PROPN
ejpam-5661	485	58	(	(	PUNCT
ejpam-5661	485	59	1−θϖ)υ	1−θϖ)υ	NUM
ejpam-5661	485	60	)	)	PUNCT
ejpam-5661	485	61	∣∣c	∣∣c	PROPN
ejpam-5661	485	62	dqϖ	dqϖ	NOUN
ejpam-5661	485	63	)	)	PUNCT
ejpam-5661	485	64	1	1	NUM
ejpam-5661	485	65	c	c	PROPN
ejpam-5661	485	66	≤	≤	NUM
ejpam-5661	485	67	θ(υ−	θ(υ−	PROPN
ejpam-5661	485	68	α	α	NOUN
ejpam-5661	485	69	)	)	PUNCT
ejpam-5661	485	70	2	2	NUM
ejpam-5661	485	71	(	(	PUNCT
ejpam-5661	485	72	∫	∫	PROPN
ejpam-5661	485	73	1	1	NUM
ejpam-5661	485	74	0	0	NUM
ejpam-5661	485	75	qϖ	qϖ	NOUN
ejpam-5661	485	76	dqϖ	dqϖ	NOUN
ejpam-5661	485	77	)	)	PUNCT
ejpam-5661	485	78	1−	1−	NUM
ejpam-5661	485	79	1	1	NUM
ejpam-5661	485	80	c	c	NOUN
ejpam-5661	485	81	(	(	PUNCT
ejpam-5661	485	82	∫	∫	PROPN
ejpam-5661	485	83	1	1	NUM
ejpam-5661	485	84	0	0	NUM
ejpam-5661	485	85	qϖ(θϖ|	qϖ(θϖ|	NOUN
ejpam-5661	485	86	αdqφ(υ)|c	αdqφ(υ)|c	PROPN
ejpam-5661	485	87	+	+	CCONJ
ejpam-5661	485	88	(	(	PUNCT
ejpam-5661	485	89	1−θϖ)|	1−θϖ)|	ADJ
ejpam-5661	485	90	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	485	91	−	−	NOUN
ejpam-5661	485	92	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	485	93	α)2)dqϖ	α)2)dqϖ	NUM
ejpam-5661	485	94	)	)	PUNCT
ejpam-5661	485	95	1	1	NUM
ejpam-5661	485	96	c	c	NOUN
ejpam-5661	485	97	+	+	CCONJ
ejpam-5661	485	98	θ(υ−	θ(υ−	PROPN
ejpam-5661	485	99	α	α	NOUN
ejpam-5661	485	100	)	)	PUNCT
ejpam-5661	485	101	2	2	NUM
ejpam-5661	485	102	(	(	PUNCT
ejpam-5661	485	103	∫	∫	PROPN
ejpam-5661	485	104	1	1	NUM
ejpam-5661	485	105	0	0	NUM
ejpam-5661	485	106	qϖ	qϖ	NOUN
ejpam-5661	485	107	dqϖ	dqϖ	NOUN
ejpam-5661	485	108	)	)	PUNCT
ejpam-5661	485	109	1−	1−	NUM
ejpam-5661	485	110	1	1	NUM
ejpam-5661	485	111	c	c	NOUN
ejpam-5661	485	112	(	(	PUNCT
ejpam-5661	485	113	∫	∫	PROPN
ejpam-5661	485	114	1	1	NUM
ejpam-5661	485	115	0	0	NUM
ejpam-5661	485	116	qϖ	qϖ	NOUN
ejpam-5661	485	117	(	(	PUNCT
ejpam-5661	485	118	θϖ|	θϖ|	NOUN
ejpam-5661	485	119	υdqφ(α)|c	υdqφ(α)|c	NOUN
ejpam-5661	485	120	+	+	CCONJ
ejpam-5661	485	121	(	(	PUNCT
ejpam-5661	485	122	1−θϖ)|	1−θϖ)|	X
ejpam-5661	485	123	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	485	124	−	−	NOUN
ejpam-5661	485	125	φθϖ(1−θϖ)(υ−	φθϖ(1−θϖ)(υ−	NOUN
ejpam-5661	485	126	α)2)dqϖ	α)2)dqϖ	NUM
ejpam-5661	485	127	)	)	PUNCT
ejpam-5661	485	128	1	1	NUM
ejpam-5661	485	129	c	c	NOUN
ejpam-5661	485	130	=	=	SYM
ejpam-5661	485	131	θ(υ−	θ(υ−	PROPN
ejpam-5661	485	132	α	α	NOUN
ejpam-5661	485	133	)	)	PUNCT
ejpam-5661	485	134	2	2	NUM
ejpam-5661	485	135	(	(	PUNCT
ejpam-5661	485	136	q	q	X
ejpam-5661	486	1	[	[	X
ejpam-5661	486	2	2]q	2]q	NUM
ejpam-5661	486	3	)	)	PUNCT
ejpam-5661	486	4	1−	1−	NUM
ejpam-5661	486	5	1	1	NUM
ejpam-5661	486	6	c	c	X
ejpam-5661	486	7	(	(	PUNCT
ejpam-5661	486	8	qθ|	qθ|	PROPN
ejpam-5661	486	9	αdqφ(υ)|c	αdqφ(υ)|c	PROPN
ejpam-5661	487	1	[	[	X
ejpam-5661	487	2	3]q	3]q	NUM
ejpam-5661	487	3	+	+	CCONJ
ejpam-5661	487	4	(	(	PUNCT
ejpam-5661	487	5	q	q	X
ejpam-5661	488	1	[	[	X
ejpam-5661	488	2	2]q	2]q	NUM
ejpam-5661	488	3	−	−	NOUN
ejpam-5661	488	4	qθ	qθ	NOUN
ejpam-5661	488	5	[	[	X
ejpam-5661	488	6	3]q	3]q	NUM
ejpam-5661	488	7	)	)	PUNCT
ejpam-5661	488	8	|	|	ADV
ejpam-5661	488	9	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	488	10	−	−	PROPN
ejpam-5661	489	1	φqθ	φqθ	PROPN
ejpam-5661	489	2	(	(	PUNCT
ejpam-5661	489	3	1	1	NUM
ejpam-5661	490	1	[	[	X
ejpam-5661	490	2	3]q	3]q	NUM
ejpam-5661	490	3	−	−	NUM
ejpam-5661	490	4	θ	θ	PROPN
ejpam-5661	491	1	[	[	X
ejpam-5661	491	2	4]q	4]q	X
ejpam-5661	491	3	)	)	PUNCT
ejpam-5661	491	4	(	(	PUNCT
ejpam-5661	491	5	υ−	υ−	PROPN
ejpam-5661	491	6	α)2	α)2	NOUN
ejpam-5661	491	7	)	)	PUNCT
ejpam-5661	491	8	1	1	NUM
ejpam-5661	491	9	c	c	NOUN
ejpam-5661	491	10	+	+	CCONJ
ejpam-5661	491	11	θ(υ−	θ(υ−	PROPN
ejpam-5661	491	12	α	α	NOUN
ejpam-5661	491	13	)	)	PUNCT
ejpam-5661	491	14	2	2	NUM
ejpam-5661	491	15	(	(	PUNCT
ejpam-5661	491	16	q	q	X
ejpam-5661	492	1	[	[	X
ejpam-5661	492	2	2]q	2]q	NUM
ejpam-5661	492	3	)	)	PUNCT
ejpam-5661	492	4	1−	1−	NUM
ejpam-5661	492	5	1	1	NUM
ejpam-5661	492	6	c	c	X
ejpam-5661	492	7	(	(	PUNCT
ejpam-5661	492	8	qθ|	qθ|	PROPN
ejpam-5661	492	9	υdqφ(α)|c	υdqφ(α)|c	PROPN
ejpam-5661	492	10	[	[	X
ejpam-5661	492	11	3]q	3]q	NUM
ejpam-5661	492	12	+	+	CCONJ
ejpam-5661	492	13	(	(	PUNCT
ejpam-5661	492	14	q	q	X
ejpam-5661	493	1	[	[	X
ejpam-5661	493	2	2]q	2]q	NUM
ejpam-5661	493	3	qθ	qθ	PROPN
ejpam-5661	493	4	[	[	X
ejpam-5661	493	5	3]q	3]q	NUM
ejpam-5661	493	6	)	)	PUNCT
ejpam-5661	493	7	|	|	ADV
ejpam-5661	493	8	υdqφ(υ)|c	υdqφ(υ)|c	PUNCT
ejpam-5661	493	9	−	−	PROPN
ejpam-5661	493	10	φqθ	φqθ	PROPN
ejpam-5661	493	11	(	(	PUNCT
ejpam-5661	493	12	1	1	NUM
ejpam-5661	493	13	[	[	X
ejpam-5661	493	14	3]q	3]q	NUM
ejpam-5661	493	15	−	−	NUM
ejpam-5661	493	16	θ	θ	PROPN
ejpam-5661	494	1	[	[	X
ejpam-5661	494	2	4]q	4]q	X
ejpam-5661	494	3	)	)	PUNCT
ejpam-5661	494	4	(	(	PUNCT
ejpam-5661	494	5	υ−	υ−	PROPN
ejpam-5661	494	6	α)2	α)2	NOUN
ejpam-5661	494	7	)	)	PUNCT
ejpam-5661	494	8	1	1	NUM
ejpam-5661	494	9	c	c	NOUN
ejpam-5661	494	10	≤	≤	X
ejpam-5661	494	11	qθ(υ−	qθ(υ−	NUM
ejpam-5661	494	12	α	α	NOUN
ejpam-5661	494	13	)	)	PUNCT
ejpam-5661	494	14	2[2]q	2[2]q	PROPN
ejpam-5661	494	15	(	(	PUNCT
ejpam-5661	494	16	(	(	PUNCT
ejpam-5661	495	1	[	[	X
ejpam-5661	495	2	2]qθ|	2]qθ|	NUM
ejpam-5661	495	3	αdqφ(υ)|c	αdqφ(υ)|c	VERB
ejpam-5661	495	4	+	+	CCONJ
ejpam-5661	495	5	(	(	PUNCT
ejpam-5661	495	6	[	[	X
ejpam-5661	495	7	3]q	3]q	NUM
ejpam-5661	495	8	−θ[2]q)|	−θ[2]q)|	NOUN
ejpam-5661	495	9	αdqφ(α)|c	αdqφ(α)|c	PUNCT
ejpam-5661	495	10	[	[	X
ejpam-5661	495	11	3]q	3]q	NUM
ejpam-5661	495	12	)	)	PUNCT
ejpam-5661	495	13	1	1	NUM
ejpam-5661	495	14	c	c	NOUN
ejpam-5661	495	15	+	+	CCONJ
ejpam-5661	495	16	(	(	PUNCT
ejpam-5661	495	17	[	[	X
ejpam-5661	495	18	2]qθ|	2]qθ|	NUM
ejpam-5661	495	19	υdqφ(α)|c	υdqφ(α)|c	NOUN
ejpam-5661	495	20	+	+	CCONJ
ejpam-5661	495	21	(	(	PUNCT
ejpam-5661	495	22	[	[	X
ejpam-5661	495	23	3]q	3]q	NUM
ejpam-5661	495	24	−θ[2]q)|	−θ[2]q)|	ADP
ejpam-5661	495	25	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	495	26	[	[	X
ejpam-5661	495	27	3]q	3]q	NUM
ejpam-5661	495	28	)	)	PUNCT
ejpam-5661	495	29	1	1	NUM
ejpam-5661	495	30	c	c	NOUN
ejpam-5661	495	31	)	)	PUNCT
ejpam-5661	495	32	.	.	PUNCT
ejpam-5661	496	1	this	this	PRON
ejpam-5661	496	2	concludes	conclude	VERB
ejpam-5661	496	3	the	the	DET
ejpam-5661	496	4	proof	proof	NOUN
ejpam-5661	496	5	.	.	PUNCT
ejpam-5661	497	1	remark	remark	VERB
ejpam-5661	497	2	7	7	NUM
ejpam-5661	497	3	.	.	PUNCT
ejpam-5661	498	1	when	when	SCONJ
ejpam-5661	498	2	θ	θ	PROPN
ejpam-5661	498	3	=	=	SYM
ejpam-5661	498	4	1	1	NUM
ejpam-5661	498	5	in	in	ADP
ejpam-5661	498	6	theorem	theorem	NOUN
ejpam-5661	498	7	8	8	NUM
ejpam-5661	498	8	,	,	PUNCT
ejpam-5661	498	9	we	we	PRON
ejpam-5661	498	10	derive	derive	VERB
ejpam-5661	498	11	trapezoid	trapezoid	ADJ
ejpam-5661	498	12	-	-	PUNCT
ejpam-5661	498	13	type	type	NOUN
ejpam-5661	498	14	inequalities:∣∣∣∣∣(υ−	inequalities:∣∣∣∣∣(υ−	NOUN
ejpam-5661	498	15	α	α	NOUN
ejpam-5661	498	16	)	)	PUNCT
ejpam-5661	498	17	2	2	NUM
ejpam-5661	498	18	(	(	PUNCT
ejpam-5661	498	19	∫	∫	PROPN
ejpam-5661	498	20	υ	υ	PROPN
ejpam-5661	498	21	α	α	PROPN
ejpam-5661	498	22	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	498	23	)	)	PUNCT
ejpam-5661	498	24	αdqϖ	αdqϖ	PROPN
ejpam-5661	499	1	+	+	CCONJ
ejpam-5661	499	2	∫	∫	PROPN
ejpam-5661	499	3	υ	υ	PROPN
ejpam-5661	499	4	α	α	PROPN
ejpam-5661	499	5	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	499	6	)	)	PUNCT
ejpam-5661	499	7	υdqϖ	υdqϖ	NOUN
ejpam-5661	499	8	)	)	PUNCT
ejpam-5661	500	1	−	−	PROPN
ejpam-5661	500	2	φ(α	φ(α	PROPN
ejpam-5661	500	3	)	)	PUNCT
ejpam-5661	501	1	+	+	CCONJ
ejpam-5661	502	1	φ(υ	φ(υ	PROPN
ejpam-5661	502	2	)	)	PUNCT
ejpam-5661	502	3	2	2	NUM
ejpam-5661	502	4	∣∣∣∣∣	∣∣∣∣∣	SYM
ejpam-5661	502	5	≤	≤	PROPN
ejpam-5661	502	6	q(υ−	q(υ−	PROPN
ejpam-5661	502	7	α	α	NOUN
ejpam-5661	502	8	)	)	PUNCT
ejpam-5661	502	9	2	2	NUM
ejpam-5661	502	10	(	(	PUNCT
ejpam-5661	502	11	1	1	NUM
ejpam-5661	503	1	[	[	X
ejpam-5661	503	2	2]q	2]q	NUM
ejpam-5661	503	3	)	)	PUNCT
ejpam-5661	503	4	1−	1−	NUM
ejpam-5661	503	5	1	1	NUM
ejpam-5661	503	6	c	c	NOUN
ejpam-5661	503	7	(	(	PUNCT
ejpam-5661	503	8	(	(	PUNCT
ejpam-5661	503	9	|	|	ADV
ejpam-5661	503	10	αdqφ(υ)|c	αdqφ(υ)|c	X
ejpam-5661	504	1	[	[	X
ejpam-5661	504	2	3]q	3]q	NUM
ejpam-5661	504	3	+	+	CCONJ
ejpam-5661	504	4	(	(	PUNCT
ejpam-5661	504	5	1	1	NUM
ejpam-5661	504	6	[	[	X
ejpam-5661	504	7	2]q	2]q	NUM
ejpam-5661	504	8	−	−	ADP
ejpam-5661	504	9	1	1	NUM
ejpam-5661	504	10	[	[	X
ejpam-5661	504	11	3]q	3]q	NUM
ejpam-5661	504	12	)	)	PUNCT
ejpam-5661	504	13	|	|	ADV
ejpam-5661	504	14	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	504	15	c.	c.	PROPN
ejpam-5661	504	16	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	504	17	,	,	PUNCT
ejpam-5661	504	18	p.	p.	PROPN
ejpam-5661	504	19	yotkaew	yotkaew	PROPN
ejpam-5661	504	20	/	/	SYM
ejpam-5661	504	21	eur	eur	PROPN
ejpam-5661	504	22	.	.	PUNCT
ejpam-5661	505	1	j.	j.	PROPN
ejpam-5661	505	2	pure	pure	PROPN
ejpam-5661	505	3	appl	appl	PROPN
ejpam-5661	505	4	.	.	PROPN
ejpam-5661	505	5	math	math	PROPN
ejpam-5661	505	6	,	,	PUNCT
ejpam-5661	505	7	18	18	NUM
ejpam-5661	505	8	(	(	PUNCT
ejpam-5661	505	9	1	1	NUM
ejpam-5661	505	10	)	)	PUNCT
ejpam-5661	505	11	(	(	PUNCT
ejpam-5661	505	12	2025	2025	NUM
ejpam-5661	505	13	)	)	PUNCT
ejpam-5661	505	14	,	,	PUNCT
ejpam-5661	505	15	5661	5661	NUM
ejpam-5661	505	16	21	21	NUM
ejpam-5661	505	17	of	of	ADP
ejpam-5661	505	18	24	24	NUM
ejpam-5661	505	19	−	−	PROPN
ejpam-5661	505	20	φ	φ	PROPN
ejpam-5661	505	21	(	(	PUNCT
ejpam-5661	505	22	1	1	NUM
ejpam-5661	506	1	[	[	X
ejpam-5661	507	1	3]q	3]q	NUM
ejpam-5661	507	2	−	−	NOUN
ejpam-5661	507	3	1	1	NUM
ejpam-5661	507	4	[	[	X
ejpam-5661	507	5	4]q	4]q	NUM
ejpam-5661	507	6	)	)	PUNCT
ejpam-5661	507	7	(	(	PUNCT
ejpam-5661	507	8	υ−	υ−	PROPN
ejpam-5661	507	9	α)2	α)2	NOUN
ejpam-5661	507	10	)	)	PUNCT
ejpam-5661	507	11	1	1	NUM
ejpam-5661	507	12	c	c	NOUN
ejpam-5661	507	13	+	+	CCONJ
ejpam-5661	507	14	(	(	PUNCT
ejpam-5661	507	15	|	|	ADV
ejpam-5661	507	16	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	507	17	[	[	X
ejpam-5661	507	18	3]q	3]q	NUM
ejpam-5661	507	19	+	+	CCONJ
ejpam-5661	507	20	(	(	PUNCT
ejpam-5661	507	21	1	1	NUM
ejpam-5661	508	1	[	[	X
ejpam-5661	508	2	2]q	2]q	NUM
ejpam-5661	508	3	−	−	ADP
ejpam-5661	508	4	1	1	NUM
ejpam-5661	508	5	[	[	X
ejpam-5661	508	6	3]q	3]q	NUM
ejpam-5661	508	7	)	)	PUNCT
ejpam-5661	509	1	|	|	ADV
ejpam-5661	509	2	υdqφ(υ)|c	υdqφ(υ)|c	SYM
ejpam-5661	509	3	−	−	PROPN
ejpam-5661	509	4	φ	φ	PROPN
ejpam-5661	509	5	(	(	PUNCT
ejpam-5661	509	6	1	1	NUM
ejpam-5661	509	7	[	[	X
ejpam-5661	509	8	3]q	3]q	NUM
ejpam-5661	509	9	−	−	NOUN
ejpam-5661	509	10	1	1	NUM
ejpam-5661	509	11	[	[	X
ejpam-5661	509	12	4]q	4]q	NUM
ejpam-5661	509	13	)	)	PUNCT
ejpam-5661	509	14	(	(	PUNCT
ejpam-5661	509	15	υ−	υ−	PROPN
ejpam-5661	509	16	α)2	α)2	NOUN
ejpam-5661	509	17	)	)	PUNCT
ejpam-5661	509	18	1	1	NUM
ejpam-5661	509	19	c	c	NOUN
ejpam-5661	509	20	)	)	PUNCT
ejpam-5661	509	21	≤	≤	PROPN
ejpam-5661	509	22	q(υ−	q(υ−	PROPN
ejpam-5661	509	23	α	α	NOUN
ejpam-5661	509	24	)	)	PUNCT
ejpam-5661	509	25	2[2]q	2[2]q	PROPN
ejpam-5661	509	26	(	(	PUNCT
ejpam-5661	509	27	(	(	PUNCT
ejpam-5661	509	28	[	[	X
ejpam-5661	509	29	2]qθ|	2]qθ|	NUM
ejpam-5661	509	30	αdqφ(υ)|c	αdqφ(υ)|c	VERB
ejpam-5661	509	31	+	+	X
ejpam-5661	509	32	q2|	q2|	NOUN
ejpam-5661	509	33	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	510	1	[	[	X
ejpam-5661	510	2	3]q	3]q	NUM
ejpam-5661	510	3	)	)	PUNCT
ejpam-5661	510	4	1	1	NUM
ejpam-5661	510	5	c	c	NOUN
ejpam-5661	510	6	+	+	PUNCT
ejpam-5661	510	7	(	(	PUNCT
ejpam-5661	510	8	[	[	X
ejpam-5661	510	9	2]q|	2]q|	NUM
ejpam-5661	510	10	υdqφ(α)|c	υdqφ(α)|c	NOUN
ejpam-5661	510	11	+	+	CCONJ
ejpam-5661	510	12	q2|	q2|	NOUN
ejpam-5661	510	13	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	510	14	[	[	X
ejpam-5661	510	15	3]q	3]q	NUM
ejpam-5661	510	16	)	)	PUNCT
ejpam-5661	510	17	1	1	NUM
ejpam-5661	510	18	c	c	NOUN
ejpam-5661	510	19	)	)	PUNCT
ejpam-5661	510	20	.	.	PUNCT
ejpam-5661	511	1	remark	remark	VERB
ejpam-5661	511	2	8	8	NUM
ejpam-5661	511	3	.	.	PUNCT
ejpam-5661	512	1	when	when	SCONJ
ejpam-5661	512	2	θ	θ	PROPN
ejpam-5661	512	3	=	=	SYM
ejpam-5661	512	4	1/2	1/2	NUM
ejpam-5661	512	5	in	in	ADP
ejpam-5661	512	6	theorem	theorem	NOUN
ejpam-5661	512	7	8	8	NUM
ejpam-5661	512	8	,	,	PUNCT
ejpam-5661	512	9	we	we	PRON
ejpam-5661	512	10	derive	derive	VERB
ejpam-5661	512	11	midpoint	midpoint	NOUN
ejpam-5661	512	12	-	-	PUNCT
ejpam-5661	512	13	type	type	NOUN
ejpam-5661	512	14	inequalities:∣∣∣∣∣(υ−	inequalities:∣∣∣∣∣(υ−	NOUN
ejpam-5661	512	15	α	α	NOUN
ejpam-5661	512	16	)	)	PUNCT
ejpam-5661	512	17	2	2	NUM
ejpam-5661	512	18	(	(	PUNCT
ejpam-5661	512	19	∫	∫	PROPN
ejpam-5661	512	20	(	(	PUNCT
ejpam-5661	512	21	α+υ	α+υ	NUM
ejpam-5661	512	22	)	)	PUNCT
ejpam-5661	512	23	2	2	NUM
ejpam-5661	512	24	α	α	PRON
ejpam-5661	512	25	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	512	26	)	)	PUNCT
ejpam-5661	512	27	αdqϖ	αdqϖ	PROPN
ejpam-5661	513	1	+	+	CCONJ
ejpam-5661	513	2	∫	∫	PROPN
ejpam-5661	513	3	υ	υ	X
ejpam-5661	513	4	(	(	PUNCT
ejpam-5661	513	5	α+υ	α+υ	NUM
ejpam-5661	513	6	)	)	PUNCT
ejpam-5661	513	7	2	2	NUM
ejpam-5661	513	8	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	513	9	)	)	PUNCT
ejpam-5661	513	10	υdqϖ	υdqϖ	NOUN
ejpam-5661	513	11	)	)	PUNCT
ejpam-5661	514	1	−	−	PROPN
ejpam-5661	514	2	φ	φ	PROPN
ejpam-5661	514	3	(	(	PUNCT
ejpam-5661	514	4	α+υ	α+υ	NUM
ejpam-5661	514	5	2	2	NUM
ejpam-5661	514	6	)	)	PUNCT
ejpam-5661	514	7	∣∣∣∣∣	∣∣∣∣∣	ADP
ejpam-5661	514	8	≤	≤	PROPN
ejpam-5661	514	9	q(υ−	q(υ−	PROPN
ejpam-5661	514	10	α	α	NOUN
ejpam-5661	514	11	)	)	PUNCT
ejpam-5661	514	12	2	2	NUM
ejpam-5661	514	13	(	(	PUNCT
ejpam-5661	514	14	1	1	NUM
ejpam-5661	514	15	[	[	X
ejpam-5661	514	16	2]q	2]q	NUM
ejpam-5661	514	17	)	)	PUNCT
ejpam-5661	514	18	1−	1−	NUM
ejpam-5661	514	19	1	1	NUM
ejpam-5661	514	20	c	c	NOUN
ejpam-5661	514	21	(	(	PUNCT
ejpam-5661	514	22	(	(	PUNCT
ejpam-5661	514	23	|	|	ADV
ejpam-5661	514	24	αdqφ(b)|c	αdqφ(b)|c	ADP
ejpam-5661	514	25	2[3]q	2[3]q	NUM
ejpam-5661	514	26	+	+	CCONJ
ejpam-5661	514	27	(	(	PUNCT
ejpam-5661	514	28	1	1	NUM
ejpam-5661	514	29	[	[	X
ejpam-5661	514	30	2]q	2]q	NUM
ejpam-5661	514	31	−	−	ADP
ejpam-5661	514	32	1	1	NUM
ejpam-5661	514	33	2[3]q	2[3]q	NUM
ejpam-5661	514	34	)	)	PUNCT
ejpam-5661	514	35	|	|	ADV
ejpam-5661	514	36	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	514	37	−	−	NOUN
ejpam-5661	515	1	φ	φ	NUM
ejpam-5661	515	2	2	2	NUM
ejpam-5661	515	3	(	(	PUNCT
ejpam-5661	515	4	1	1	NUM
ejpam-5661	515	5	[	[	X
ejpam-5661	515	6	3]q	3]q	NUM
ejpam-5661	515	7	−	−	NOUN
ejpam-5661	515	8	1	1	NUM
ejpam-5661	515	9	2[4]q	2[4]q	NUM
ejpam-5661	515	10	)	)	PUNCT
ejpam-5661	515	11	(	(	PUNCT
ejpam-5661	515	12	υ−	υ−	PROPN
ejpam-5661	515	13	α)2	α)2	NOUN
ejpam-5661	515	14	)	)	PUNCT
ejpam-5661	515	15	1	1	NUM
ejpam-5661	515	16	c	c	NOUN
ejpam-5661	515	17	+	+	CCONJ
ejpam-5661	515	18	(	(	PUNCT
ejpam-5661	515	19	|	|	ADV
ejpam-5661	515	20	υdqφ(α)|c	υdqφ(α)|c	VERB
ejpam-5661	515	21	2[3]q	2[3]q	NUM
ejpam-5661	515	22	+	+	CCONJ
ejpam-5661	515	23	(	(	PUNCT
ejpam-5661	515	24	1	1	NUM
ejpam-5661	515	25	[	[	X
ejpam-5661	515	26	2]q	2]q	NUM
ejpam-5661	515	27	−	−	ADP
ejpam-5661	515	28	1	1	NUM
ejpam-5661	515	29	2[3]q	2[3]q	NUM
ejpam-5661	515	30	)	)	PUNCT
ejpam-5661	515	31	|	|	ADV
ejpam-5661	515	32	υdqφ(υ)|c	υdqφ(υ)|c	ADP
ejpam-5661	515	33	−	−	PROPN
ejpam-5661	515	34	φ	φ	NUM
ejpam-5661	515	35	2	2	NUM
ejpam-5661	515	36	(	(	PUNCT
ejpam-5661	515	37	1	1	NUM
ejpam-5661	515	38	[	[	X
ejpam-5661	515	39	3]q	3]q	NUM
ejpam-5661	515	40	−	−	NOUN
ejpam-5661	515	41	1	1	NUM
ejpam-5661	515	42	2[4]q	2[4]q	NUM
ejpam-5661	515	43	)	)	PUNCT
ejpam-5661	515	44	(	(	PUNCT
ejpam-5661	515	45	υ−	υ−	PROPN
ejpam-5661	515	46	α)2	α)2	NOUN
ejpam-5661	515	47	)	)	PUNCT
ejpam-5661	515	48	1	1	NUM
ejpam-5661	515	49	c	c	NOUN
ejpam-5661	515	50	)	)	PUNCT
ejpam-5661	515	51	≤	≤	PROPN
ejpam-5661	516	1	q(υ−	q(υ−	PROPN
ejpam-5661	516	2	α	α	NOUN
ejpam-5661	516	3	)	)	PUNCT
ejpam-5661	516	4	2[2]q	2[2]q	PROPN
ejpam-5661	516	5	(	(	PUNCT
ejpam-5661	516	6	(	(	PUNCT
ejpam-5661	516	7	[	[	X
ejpam-5661	516	8	2]q|	2]q|	NUM
ejpam-5661	516	9	αdqφ(υ)|c	αdqφ(υ)|c	NOUN
ejpam-5661	517	1	+	+	CCONJ
ejpam-5661	517	2	(	(	PUNCT
ejpam-5661	517	3	[	[	X
ejpam-5661	517	4	3]q	3]q	NUM
ejpam-5661	517	5	+	+	CCONJ
ejpam-5661	517	6	q2)|	q2)|	NOUN
ejpam-5661	517	7	αdqφ(α)|c	αdqφ(α)|c	NUM
ejpam-5661	517	8	2[3]q	2[3]q	NUM
ejpam-5661	517	9	)	)	PUNCT
ejpam-5661	517	10	1	1	NUM
ejpam-5661	517	11	c	c	NOUN
ejpam-5661	517	12	+	+	PUNCT
ejpam-5661	517	13	(	(	PUNCT
ejpam-5661	517	14	(	(	PUNCT
ejpam-5661	517	15	[	[	X
ejpam-5661	517	16	2]q|	2]q|	NUM
ejpam-5661	517	17	υdqφ(α)|c	υdqφ(α)|c	NOUN
ejpam-5661	517	18	+	+	CCONJ
ejpam-5661	517	19	(	(	PUNCT
ejpam-5661	517	20	[	[	X
ejpam-5661	517	21	3]q	3]q	NUM
ejpam-5661	517	22	+	+	NUM
ejpam-5661	517	23	q2)|	q2)|	NOUN
ejpam-5661	517	24	υdqφ(υ)|c	υdqφ(υ)|c	NOUN
ejpam-5661	517	25	2[3]q	2[3]q	NUM
ejpam-5661	517	26	)	)	PUNCT
ejpam-5661	517	27	1	1	NUM
ejpam-5661	517	28	c	c	NOUN
ejpam-5661	517	29	.	.	PUNCT
ejpam-5661	518	1	5	5	X
ejpam-5661	518	2	.	.	X
ejpam-5661	518	3	applications	application	NOUN
ejpam-5661	518	4	we	we	PRON
ejpam-5661	518	5	show	show	VERB
ejpam-5661	518	6	how	how	SCONJ
ejpam-5661	518	7	special	special	ADJ
ejpam-5661	518	8	means	mean	NOUN
ejpam-5661	518	9	can	can	AUX
ejpam-5661	518	10	be	be	AUX
ejpam-5661	518	11	used	use	VERB
ejpam-5661	518	12	to	to	PART
ejpam-5661	518	13	prove	prove	VERB
ejpam-5661	518	14	the	the	DET
ejpam-5661	518	15	results	result	NOUN
ejpam-5661	518	16	in	in	ADP
ejpam-5661	518	17	theorems	theorem	NOUN
ejpam-5661	518	18	6	6	NUM
ejpam-5661	518	19	,	,	PUNCT
ejpam-5661	518	20	7	7	NUM
ejpam-5661	518	21	,	,	PUNCT
ejpam-5661	518	22	and	and	CCONJ
ejpam-5661	518	23	8	8	NUM
ejpam-5661	518	24	.	.	PUNCT
ejpam-5661	518	25	to	to	PART
ejpam-5661	518	26	establish	establish	VERB
ejpam-5661	518	27	arbitrary	arbitrary	ADJ
ejpam-5661	518	28	positive	positive	ADJ
ejpam-5661	518	29	numbers	number	NOUN
ejpam-5661	518	30	w1	w1	NOUN
ejpam-5661	518	31	and	and	CCONJ
ejpam-5661	518	32	w2	w2	PROPN
ejpam-5661	518	33	(	(	PUNCT
ejpam-5661	518	34	w1	w1	NOUN
ejpam-5661	518	35	̸=	̸=	PROPN
ejpam-5661	518	36	w2	w2	NOUN
ejpam-5661	518	37	)	)	PUNCT
ejpam-5661	518	38	,	,	PUNCT
ejpam-5661	518	39	we	we	PRON
ejpam-5661	518	40	define	define	VERB
ejpam-5661	518	41	the	the	DET
ejpam-5661	518	42	means	mean	NOUN
ejpam-5661	518	43	as	as	SCONJ
ejpam-5661	518	44	follows	follow	VERB
ejpam-5661	518	45	:	:	PUNCT
ejpam-5661	518	46	(	(	PUNCT
ejpam-5661	518	47	i	i	NOUN
ejpam-5661	518	48	)	)	PUNCT
ejpam-5661	518	49	the	the	DET
ejpam-5661	518	50	arithmetic	arithmetic	ADJ
ejpam-5661	518	51	mean	mean	VERB
ejpam-5661	519	1	a	a	DET
ejpam-5661	519	2	=	=	PUNCT
ejpam-5661	519	3	a(w1,w2	a(w1,w2	ADJ
ejpam-5661	519	4	)	)	PUNCT
ejpam-5661	520	1	=	=	SYM
ejpam-5661	520	2	w1	w1	NOUN
ejpam-5661	520	3	+	+	NOUN
ejpam-5661	520	4	w2	w2	NOUN
ejpam-5661	520	5	2	2	NUM
ejpam-5661	520	6	.	.	PUNCT
ejpam-5661	521	1	c.	c.	PROPN
ejpam-5661	521	2	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	521	3	,	,	PUNCT
ejpam-5661	521	4	p.	p.	PROPN
ejpam-5661	521	5	yotkaew	yotkaew	PROPN
ejpam-5661	521	6	/	/	SYM
ejpam-5661	521	7	eur	eur	PROPN
ejpam-5661	521	8	.	.	PUNCT
ejpam-5661	522	1	j.	j.	PROPN
ejpam-5661	522	2	pure	pure	PROPN
ejpam-5661	522	3	appl	appl	PROPN
ejpam-5661	522	4	.	.	PROPN
ejpam-5661	522	5	math	math	PROPN
ejpam-5661	522	6	,	,	PUNCT
ejpam-5661	522	7	18	18	NUM
ejpam-5661	522	8	(	(	PUNCT
ejpam-5661	522	9	1	1	NUM
ejpam-5661	522	10	)	)	PUNCT
ejpam-5661	522	11	(	(	PUNCT
ejpam-5661	522	12	2025	2025	NUM
ejpam-5661	522	13	)	)	PUNCT
ejpam-5661	522	14	,	,	PUNCT
ejpam-5661	522	15	5661	5661	NUM
ejpam-5661	522	16	22	22	NUM
ejpam-5661	522	17	of	of	ADP
ejpam-5661	522	18	24	24	NUM
ejpam-5661	522	19	(	(	PUNCT
ejpam-5661	522	20	ii	ii	NOUN
ejpam-5661	522	21	)	)	PUNCT
ejpam-5661	522	22	the	the	DET
ejpam-5661	522	23	logarithmic	logarithmic	ADJ
ejpam-5661	522	24	mean	mean	NOUN
ejpam-5661	522	25	lp	lp	NOUN
ejpam-5661	522	26	=	=	SYM
ejpam-5661	522	27	lp	lp	NOUN
ejpam-5661	522	28	(	(	PUNCT
ejpam-5661	522	29	w1,w2	w1,w2	PROPN
ejpam-5661	522	30	)	)	PUNCT
ejpam-5661	522	31	=	=	SYM
ejpam-5661	522	32	wp+1	wp+1	NOUN
ejpam-5661	522	33	2	2	NUM
ejpam-5661	522	34	−wp+1	−wp+1	PROPN
ejpam-5661	522	35	1	1	NUM
ejpam-5661	522	36	(	(	PUNCT
ejpam-5661	522	37	p	p	NOUN
ejpam-5661	522	38	+	+	PROPN
ejpam-5661	522	39	1)(w1	1)(w1	NUM
ejpam-5661	522	40	−w2	−w2	PROPN
ejpam-5661	522	41	)	)	PUNCT
ejpam-5661	522	42	.	.	PUNCT
ejpam-5661	523	1	proposition	proposition	NOUN
ejpam-5661	523	2	1	1	NUM
ejpam-5661	523	3	.	.	PUNCT
ejpam-5661	523	4	given	give	VERB
ejpam-5661	523	5	that	that	PRON
ejpam-5661	523	6	0	0	PUNCT
ejpam-5661	523	7	<	<	X
ejpam-5661	523	8	α	α	X
ejpam-5661	523	9	<	<	X
ejpam-5661	523	10	υ	υ	PROPN
ejpam-5661	523	11	,	,	PUNCT
ejpam-5661	523	12	the	the	DET
ejpam-5661	523	13	following	follow	VERB
ejpam-5661	523	14	inequalities	inequality	NOUN
ejpam-5661	523	15	are	be	AUX
ejpam-5661	523	16	valid:∣∣∣∣	valid:∣∣∣∣	ADJ
ejpam-5661	523	17	1	1	NUM
ejpam-5661	523	18	c+	c+	NOUN
ejpam-5661	523	19	1	1	NUM
ejpam-5661	523	20	(	(	PUNCT
ejpam-5661	523	21	θ2(υ−	θ2(υ−	PROPN
ejpam-5661	523	22	α)2a(k1	α)2a(k1	PROPN
ejpam-5661	523	23	,	,	PUNCT
ejpam-5661	523	24	k2)−a(w	k2)−a(w	VERB
ejpam-5661	523	25	c+1	c+1	NUM
ejpam-5661	523	26	1	1	NUM
ejpam-5661	523	27	,	,	PUNCT
ejpam-5661	523	28	w	w	PROPN
ejpam-5661	523	29	c+1	c+1	SYM
ejpam-5661	523	30	2	2	NUM
ejpam-5661	523	31	)	)	PUNCT
ejpam-5661	523	32	)	)	PUNCT
ejpam-5661	524	1	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5661	524	2	≤	≤	NOUN
ejpam-5661	524	3	qθ(υ−	qθ(υ−	NUM
ejpam-5661	524	4	α	α	NUM
ejpam-5661	524	5	)	)	PUNCT
ejpam-5661	524	6	2[2]q[3]q	2[2]q[3]q	NUM
ejpam-5661	525	1	(	(	PUNCT
ejpam-5661	525	2	[	[	X
ejpam-5661	525	3	2]qθ(lc(qα+	2]qθ(lc(qα+	NOUN
ejpam-5661	525	4	(	(	PUNCT
ejpam-5661	525	5	1−	1−	NUM
ejpam-5661	525	6	q)υ	q)υ	NOUN
ejpam-5661	525	7	,	,	PUNCT
ejpam-5661	525	8	α	α	X
ejpam-5661	525	9	)	)	PUNCT
ejpam-5661	526	1	+	+	CCONJ
ejpam-5661	526	2	lc(qυ+	lc(qυ+	PROPN
ejpam-5661	526	3	(	(	PUNCT
ejpam-5661	526	4	1−	1−	NUM
ejpam-5661	526	5	q)α	q)α	X
ejpam-5661	526	6	,	,	PUNCT
ejpam-5661	526	7	υ	υ	NOUN
ejpam-5661	526	8	)	)	PUNCT
ejpam-5661	526	9	)	)	PUNCT
ejpam-5661	527	1	+	+	CCONJ
ejpam-5661	527	2	(	(	PUNCT
ejpam-5661	527	3	[	[	X
ejpam-5661	527	4	3]q	3]q	NUM
ejpam-5661	527	5	−	−	NOUN
ejpam-5661	528	1	[	[	X
ejpam-5661	528	2	2]qθ)(αc	2]qθ)(αc	NUM
ejpam-5661	528	3	+	+	ADJ
ejpam-5661	528	4	υc	υc	NUM
ejpam-5661	528	5	)	)	PUNCT
ejpam-5661	528	6	)	)	PUNCT
ejpam-5661	529	1	−	−	PROPN
ejpam-5661	529	2	θ2qφ(υ−	θ2qφ(υ−	NOUN
ejpam-5661	529	3	α)3([4]q	α)3([4]q	X
ejpam-5661	530	1	−	−	NOUN
ejpam-5661	531	1	[	[	X
ejpam-5661	531	2	3]qθ	3]qθ	NUM
ejpam-5661	531	3	)	)	PUNCT
ejpam-5661	532	1	[	[	X
ejpam-5661	532	2	3]q[4]q	3]q[4]q	NUM
ejpam-5661	532	3	,	,	PUNCT
ejpam-5661	532	4	where	where	SCONJ
ejpam-5661	532	5	k1	k1	NOUN
ejpam-5661	532	6	=	=	SYM
ejpam-5661	532	7	(	(	PUNCT
ejpam-5661	532	8	1−	1−	NUM
ejpam-5661	532	9	q	q	NOUN
ejpam-5661	532	10	)	)	PUNCT
ejpam-5661	532	11	∞∑	∞∑	PROPN
ejpam-5661	532	12	n=0	n=0	ADJ
ejpam-5661	532	13	qn	qn	NOUN
ejpam-5661	532	14	(	(	PUNCT
ejpam-5661	532	15	qnθυ+	qnθυ+	CCONJ
ejpam-5661	532	16	(	(	PUNCT
ejpam-5661	532	17	1−	1−	NUM
ejpam-5661	532	18	qnθ)α)c+1	qnθ)α)c+1	NOUN
ejpam-5661	532	19	k2	k2	NOUN
ejpam-5661	532	20	=	=	SYM
ejpam-5661	532	21	(	(	PUNCT
ejpam-5661	532	22	1−	1−	NUM
ejpam-5661	532	23	q	q	NOUN
ejpam-5661	532	24	)	)	PUNCT
ejpam-5661	532	25	∞∑	∞∑	PROPN
ejpam-5661	532	26	n=0	n=0	ADJ
ejpam-5661	532	27	qn	qn	NOUN
ejpam-5661	532	28	(	(	PUNCT
ejpam-5661	532	29	qnθα+	qnθα+	NOUN
ejpam-5661	532	30	(	(	PUNCT
ejpam-5661	532	31	1−	1−	NUM
ejpam-5661	532	32	qnθ)υ)c+1	qnθ)υ)c+1	NOUN
ejpam-5661	532	33	and	and	CCONJ
ejpam-5661	532	34	w1	w1	NOUN
ejpam-5661	532	35	=	=	SYM
ejpam-5661	532	36	θα+	θα+	NOUN
ejpam-5661	532	37	(	(	PUNCT
ejpam-5661	532	38	1−θ)υ	1−θ)υ	NUM
ejpam-5661	532	39	,	,	PUNCT
ejpam-5661	532	40	w2	w2	NOUN
ejpam-5661	532	41	=	=	SYM
ejpam-5661	532	42	θυ+	θυ+	ADJ
ejpam-5661	532	43	(	(	PUNCT
ejpam-5661	532	44	1−θ)α	1−θ)α	NUM
ejpam-5661	532	45	.	.	PUNCT
ejpam-5661	533	1	proof	proof	NOUN
ejpam-5661	533	2	.	.	PUNCT
ejpam-5661	534	1	by	by	ADP
ejpam-5661	534	2	substituting	substitute	VERB
ejpam-5661	534	3	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	534	4	)	)	PUNCT
ejpam-5661	534	5	=	=	PUNCT
ejpam-5661	534	6	ϖc+1/(c	ϖc+1/(c	PROPN
ejpam-5661	534	7	+	+	PROPN
ejpam-5661	534	8	1	1	NUM
ejpam-5661	534	9	)	)	PUNCT
ejpam-5661	534	10	,	,	PUNCT
ejpam-5661	534	11	where	where	SCONJ
ejpam-5661	534	12	ϖ	ϖ	X
ejpam-5661	534	13	>	>	X
ejpam-5661	534	14	0	0	NUM
ejpam-5661	534	15	in	in	ADP
ejpam-5661	534	16	the	the	DET
ejpam-5661	534	17	inequalities	inequality	NOUN
ejpam-5661	534	18	(	(	PUNCT
ejpam-5661	534	19	26	26	NUM
ejpam-5661	534	20	)	)	PUNCT
ejpam-5661	534	21	of	of	ADP
ejpam-5661	534	22	theorem	theorem	NOUN
ejpam-5661	534	23	6	6	NUM
ejpam-5661	534	24	,	,	PUNCT
ejpam-5661	534	25	then	then	ADV
ejpam-5661	534	26	we	we	PRON
ejpam-5661	534	27	obtain	obtain	VERB
ejpam-5661	534	28	the	the	DET
ejpam-5661	534	29	result	result	NOUN
ejpam-5661	534	30	.	.	PUNCT
ejpam-5661	535	1	proposition	proposition	NOUN
ejpam-5661	535	2	2	2	NUM
ejpam-5661	535	3	.	.	PUNCT
ejpam-5661	535	4	given	give	VERB
ejpam-5661	535	5	that	that	PRON
ejpam-5661	535	6	0	0	NUM
ejpam-5661	535	7	<	<	X
ejpam-5661	535	8	α	α	X
ejpam-5661	535	9	<	<	X
ejpam-5661	535	10	υ	υ	PROPN
ejpam-5661	535	11	,	,	PUNCT
ejpam-5661	535	12	the	the	DET
ejpam-5661	535	13	following	follow	VERB
ejpam-5661	535	14	inequalities	inequality	NOUN
ejpam-5661	535	15	are	be	AUX
ejpam-5661	535	16	valid:∣∣∣∣	valid:∣∣∣∣	ADJ
ejpam-5661	535	17	1	1	NUM
ejpam-5661	535	18	c+	c+	NOUN
ejpam-5661	535	19	1	1	NUM
ejpam-5661	535	20	(	(	PUNCT
ejpam-5661	535	21	θ2(υ−	θ2(υ−	PROPN
ejpam-5661	535	22	α)2a(k1	α)2a(k1	PROPN
ejpam-5661	535	23	,	,	PUNCT
ejpam-5661	535	24	k2)−a(w	k2)−a(w	VERB
ejpam-5661	535	25	c+1	c+1	NUM
ejpam-5661	535	26	1	1	NUM
ejpam-5661	535	27	,	,	PUNCT
ejpam-5661	535	28	w	w	PROPN
ejpam-5661	535	29	c+1	c+1	SYM
ejpam-5661	535	30	2	2	NUM
ejpam-5661	535	31	)	)	PUNCT
ejpam-5661	535	32	)	)	PUNCT
ejpam-5661	536	1	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5661	536	2	≤	≤	NOUN
ejpam-5661	536	3	qθ(υ−	qθ(υ−	NUM
ejpam-5661	536	4	α	α	NOUN
ejpam-5661	536	5	)	)	PUNCT
ejpam-5661	536	6	2	2	NUM
ejpam-5661	536	7	(	(	PUNCT
ejpam-5661	536	8	1	1	NUM
ejpam-5661	536	9	[	[	X
ejpam-5661	536	10	d+	d+	PUNCT
ejpam-5661	536	11	1]q	1]q	NUM
ejpam-5661	536	12	)	)	PUNCT
ejpam-5661	536	13	1	1	NUM
ejpam-5661	536	14	d	d	NOUN
ejpam-5661	536	15	(	(	PUNCT
ejpam-5661	536	16	(	(	PUNCT
ejpam-5661	536	17	θ|lc(qυ+	θ|lc(qυ+	NUM
ejpam-5661	536	18	(	(	PUNCT
ejpam-5661	536	19	1−	1−	NUM
ejpam-5661	536	20	q)α	q)α	NOUN
ejpam-5661	536	21	,	,	PUNCT
ejpam-5661	536	22	υ)|c	υ)|c	PROPN
ejpam-5661	537	1	[	[	X
ejpam-5661	537	2	2]q	2]q	NUM
ejpam-5661	537	3	+	+	CCONJ
ejpam-5661	537	4	(	(	PUNCT
ejpam-5661	537	5	1−	1−	NUM
ejpam-5661	537	6	θ	θ	PROPN
ejpam-5661	538	1	[	[	X
ejpam-5661	538	2	2]q	2]q	NUM
ejpam-5661	538	3	)	)	PUNCT
ejpam-5661	538	4	αc	αc	ADP
ejpam-5661	538	5	−	−	PROPN
ejpam-5661	538	6	φθ	φθ	INTJ
ejpam-5661	538	7	(	(	PUNCT
ejpam-5661	538	8	1	1	NUM
ejpam-5661	538	9	[	[	X
ejpam-5661	538	10	2]q	2]q	NUM
ejpam-5661	538	11	−	−	ADP
ejpam-5661	538	12	θ	θ	PROPN
ejpam-5661	539	1	[	[	X
ejpam-5661	539	2	3]q	3]q	NUM
ejpam-5661	539	3	)	)	PUNCT
ejpam-5661	539	4	(	(	PUNCT
ejpam-5661	539	5	υ−	υ−	PROPN
ejpam-5661	539	6	α)2	α)2	NOUN
ejpam-5661	539	7	)	)	PUNCT
ejpam-5661	539	8	1	1	NUM
ejpam-5661	539	9	c	c	NOUN
ejpam-5661	539	10	+	+	CCONJ
ejpam-5661	539	11	(	(	PUNCT
ejpam-5661	539	12	θ|lc(qα+	θ|lc(qα+	NOUN
ejpam-5661	539	13	(	(	PUNCT
ejpam-5661	539	14	1−	1−	NUM
ejpam-5661	539	15	q)υ	q)υ	NOUN
ejpam-5661	539	16	,	,	PUNCT
ejpam-5661	539	17	α)|c	α)|c	VERB
ejpam-5661	539	18	[	[	X
ejpam-5661	539	19	2]q	2]q	NUM
ejpam-5661	539	20	+	+	CCONJ
ejpam-5661	539	21	(	(	PUNCT
ejpam-5661	539	22	1−	1−	NUM
ejpam-5661	539	23	θ	θ	PROPN
ejpam-5661	540	1	[	[	X
ejpam-5661	540	2	2]q	2]q	NUM
ejpam-5661	540	3	)	)	PUNCT
ejpam-5661	540	4	υc	υc	ADP
ejpam-5661	540	5	−	−	PROPN
ejpam-5661	540	6	φθ	φθ	INTJ
ejpam-5661	540	7	(	(	PUNCT
ejpam-5661	540	8	1	1	NUM
ejpam-5661	541	1	[	[	X
ejpam-5661	541	2	2]q	2]q	NUM
ejpam-5661	541	3	−	−	ADP
ejpam-5661	541	4	θ	θ	PROPN
ejpam-5661	541	5	[	[	X
ejpam-5661	541	6	3]q	3]q	NUM
ejpam-5661	541	7	)	)	PUNCT
ejpam-5661	541	8	(	(	PUNCT
ejpam-5661	541	9	υ−	υ−	PROPN
ejpam-5661	541	10	α)2	α)2	NOUN
ejpam-5661	541	11	)	)	PUNCT
ejpam-5661	541	12	1	1	NUM
ejpam-5661	541	13	c	c	NOUN
ejpam-5661	541	14	)	)	PUNCT
ejpam-5661	541	15	.	.	PUNCT
ejpam-5661	542	1	proof	proof	NOUN
ejpam-5661	542	2	.	.	PUNCT
ejpam-5661	543	1	by	by	ADP
ejpam-5661	543	2	substituting	substitute	VERB
ejpam-5661	543	3	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	543	4	)	)	PUNCT
ejpam-5661	543	5	=	=	PUNCT
ejpam-5661	543	6	ϖc+1/(c	ϖc+1/(c	PROPN
ejpam-5661	543	7	+	+	PROPN
ejpam-5661	543	8	1	1	NUM
ejpam-5661	543	9	)	)	PUNCT
ejpam-5661	543	10	,	,	PUNCT
ejpam-5661	543	11	where	where	SCONJ
ejpam-5661	543	12	ϖ	ϖ	X
ejpam-5661	543	13	>	>	X
ejpam-5661	543	14	0	0	NUM
ejpam-5661	543	15	in	in	ADP
ejpam-5661	543	16	the	the	DET
ejpam-5661	543	17	inequalities	inequality	NOUN
ejpam-5661	543	18	(	(	PUNCT
ejpam-5661	543	19	27	27	NUM
ejpam-5661	543	20	)	)	PUNCT
ejpam-5661	543	21	of	of	ADP
ejpam-5661	543	22	theorem	theorem	NOUN
ejpam-5661	543	23	7	7	NUM
ejpam-5661	543	24	,	,	PUNCT
ejpam-5661	543	25	we	we	PRON
ejpam-5661	543	26	obtain	obtain	VERB
ejpam-5661	543	27	the	the	DET
ejpam-5661	543	28	result	result	NOUN
ejpam-5661	543	29	.	.	PUNCT
ejpam-5661	544	1	c.	c.	PROPN
ejpam-5661	544	2	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	544	3	,	,	PUNCT
ejpam-5661	544	4	p.	p.	PROPN
ejpam-5661	544	5	yotkaew	yotkaew	PROPN
ejpam-5661	544	6	/	/	SYM
ejpam-5661	544	7	eur	eur	PROPN
ejpam-5661	544	8	.	.	PUNCT
ejpam-5661	545	1	j.	j.	PROPN
ejpam-5661	545	2	pure	pure	PROPN
ejpam-5661	545	3	appl	appl	PROPN
ejpam-5661	545	4	.	.	PROPN
ejpam-5661	545	5	math	math	PROPN
ejpam-5661	545	6	,	,	PUNCT
ejpam-5661	545	7	18	18	NUM
ejpam-5661	545	8	(	(	PUNCT
ejpam-5661	545	9	1	1	NUM
ejpam-5661	545	10	)	)	PUNCT
ejpam-5661	545	11	(	(	PUNCT
ejpam-5661	545	12	2025	2025	NUM
ejpam-5661	545	13	)	)	PUNCT
ejpam-5661	545	14	,	,	PUNCT
ejpam-5661	545	15	5661	5661	NUM
ejpam-5661	545	16	23	23	NUM
ejpam-5661	545	17	of	of	ADP
ejpam-5661	545	18	24	24	NUM
ejpam-5661	545	19	proposition	proposition	NOUN
ejpam-5661	545	20	3	3	NUM
ejpam-5661	545	21	.	.	PUNCT
ejpam-5661	546	1	given	give	VERB
ejpam-5661	546	2	that	that	PRON
ejpam-5661	546	3	0	0	NUM
ejpam-5661	546	4	<	<	X
ejpam-5661	546	5	α	α	X
ejpam-5661	546	6	<	<	X
ejpam-5661	546	7	υ	υ	PROPN
ejpam-5661	546	8	,	,	PUNCT
ejpam-5661	546	9	the	the	DET
ejpam-5661	546	10	following	follow	VERB
ejpam-5661	546	11	inequalities	inequality	NOUN
ejpam-5661	546	12	are	be	AUX
ejpam-5661	546	13	valid:∣∣∣∣	valid:∣∣∣∣	ADJ
ejpam-5661	546	14	1	1	NUM
ejpam-5661	546	15	c+	c+	NOUN
ejpam-5661	546	16	1	1	NUM
ejpam-5661	546	17	(	(	PUNCT
ejpam-5661	546	18	θ2(υ−	θ2(υ−	PROPN
ejpam-5661	546	19	α)2a(k1	α)2a(k1	PROPN
ejpam-5661	546	20	,	,	PUNCT
ejpam-5661	546	21	k2)−a(w	k2)−a(w	VERB
ejpam-5661	546	22	c+1	c+1	NUM
ejpam-5661	546	23	1	1	NUM
ejpam-5661	546	24	,	,	PUNCT
ejpam-5661	546	25	w	w	PROPN
ejpam-5661	546	26	c+1	c+1	SYM
ejpam-5661	546	27	2	2	NUM
ejpam-5661	546	28	)	)	PUNCT
ejpam-5661	546	29	)	)	PUNCT
ejpam-5661	547	1	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5661	547	2	≤	≤	NOUN
ejpam-5661	547	3	qθ(υ−	qθ(υ−	NUM
ejpam-5661	547	4	α	α	NOUN
ejpam-5661	547	5	)	)	PUNCT
ejpam-5661	547	6	2	2	NUM
ejpam-5661	547	7	(	(	PUNCT
ejpam-5661	547	8	1	1	NUM
ejpam-5661	547	9	[	[	X
ejpam-5661	547	10	2]q	2]q	NUM
ejpam-5661	547	11	)	)	PUNCT
ejpam-5661	547	12	1−	1−	NUM
ejpam-5661	547	13	1	1	NUM
ejpam-5661	547	14	c	c	NOUN
ejpam-5661	547	15	(	(	PUNCT
ejpam-5661	547	16	(	(	PUNCT
ejpam-5661	547	17	θ|lc(qυ+	θ|lc(qυ+	NUM
ejpam-5661	547	18	(	(	PUNCT
ejpam-5661	547	19	1−	1−	NUM
ejpam-5661	547	20	q)α	q)α	NOUN
ejpam-5661	547	21	,	,	PUNCT
ejpam-5661	547	22	υ)|c	υ)|c	PROPN
ejpam-5661	548	1	[	[	X
ejpam-5661	548	2	3]q	3]q	NUM
ejpam-5661	548	3	+	+	CCONJ
ejpam-5661	548	4	(	(	PUNCT
ejpam-5661	548	5	1	1	NUM
ejpam-5661	548	6	[	[	X
ejpam-5661	548	7	2]q	2]q	NUM
ejpam-5661	548	8	−	−	ADP
ejpam-5661	548	9	θ	θ	PROPN
ejpam-5661	548	10	[	[	X
ejpam-5661	548	11	3]q	3]q	NUM
ejpam-5661	548	12	)	)	PUNCT
ejpam-5661	548	13	αc	αc	ADP
ejpam-5661	549	1	−	−	PROPN
ejpam-5661	549	2	φθ	φθ	INTJ
ejpam-5661	549	3	(	(	PUNCT
ejpam-5661	549	4	1	1	NUM
ejpam-5661	549	5	[	[	X
ejpam-5661	549	6	3]q	3]q	NUM
ejpam-5661	549	7	−	−	NUM
ejpam-5661	549	8	θ	θ	PROPN
ejpam-5661	550	1	[	[	X
ejpam-5661	550	2	4]q	4]q	X
ejpam-5661	550	3	)	)	PUNCT
ejpam-5661	550	4	(	(	PUNCT
ejpam-5661	550	5	b−	b−	NOUN
ejpam-5661	550	6	α)2	α)2	VERB
ejpam-5661	550	7	)	)	PUNCT
ejpam-5661	551	1	1	1	NUM
ejpam-5661	551	2	c	c	NOUN
ejpam-5661	551	3	+	+	CCONJ
ejpam-5661	551	4	(	(	PUNCT
ejpam-5661	551	5	θ|lc(qα+	θ|lc(qα+	NOUN
ejpam-5661	551	6	(	(	PUNCT
ejpam-5661	551	7	1−	1−	NUM
ejpam-5661	551	8	q)υ	q)υ	NOUN
ejpam-5661	551	9	,	,	PUNCT
ejpam-5661	551	10	α)|c	α)|c	VERB
ejpam-5661	551	11	[	[	X
ejpam-5661	551	12	2]q	2]q	NUM
ejpam-5661	551	13	+	+	CCONJ
ejpam-5661	551	14	(	(	PUNCT
ejpam-5661	551	15	1	1	NUM
ejpam-5661	551	16	[	[	X
ejpam-5661	551	17	2]q	2]q	NUM
ejpam-5661	551	18	−	−	ADP
ejpam-5661	551	19	θ	θ	PROPN
ejpam-5661	552	1	[	[	X
ejpam-5661	552	2	3]q	3]q	NUM
ejpam-5661	552	3	)	)	PUNCT
ejpam-5661	552	4	υc	υc	ADP
ejpam-5661	552	5	−	−	PROPN
ejpam-5661	552	6	φθ	φθ	INTJ
ejpam-5661	552	7	(	(	PUNCT
ejpam-5661	552	8	1	1	NUM
ejpam-5661	552	9	[	[	X
ejpam-5661	552	10	3]q	3]q	NUM
ejpam-5661	552	11	−	−	NUM
ejpam-5661	552	12	θ	θ	PROPN
ejpam-5661	553	1	[	[	X
ejpam-5661	553	2	4]q	4]q	X
ejpam-5661	553	3	)	)	PUNCT
ejpam-5661	553	4	(	(	PUNCT
ejpam-5661	553	5	υ−	υ−	PROPN
ejpam-5661	553	6	α)2	α)2	NOUN
ejpam-5661	553	7	)	)	PUNCT
ejpam-5661	553	8	1	1	NUM
ejpam-5661	553	9	c	c	NOUN
ejpam-5661	553	10	)	)	PUNCT
ejpam-5661	553	11	.	.	PUNCT
ejpam-5661	554	1	proof	proof	NOUN
ejpam-5661	554	2	.	.	PUNCT
ejpam-5661	555	1	by	by	ADP
ejpam-5661	555	2	substituting	substitute	VERB
ejpam-5661	555	3	φ(ϖ	φ(ϖ	NOUN
ejpam-5661	555	4	)	)	PUNCT
ejpam-5661	555	5	=	=	PUNCT
ejpam-5661	555	6	ϖc+1/(c	ϖc+1/(c	PROPN
ejpam-5661	555	7	+	+	PROPN
ejpam-5661	555	8	1	1	NUM
ejpam-5661	555	9	)	)	PUNCT
ejpam-5661	555	10	,	,	PUNCT
ejpam-5661	555	11	where	where	SCONJ
ejpam-5661	555	12	ϖ	ϖ	X
ejpam-5661	555	13	>	>	X
ejpam-5661	555	14	0	0	NUM
ejpam-5661	555	15	in	in	ADP
ejpam-5661	555	16	the	the	DET
ejpam-5661	555	17	inequalities	inequality	NOUN
ejpam-5661	555	18	(	(	PUNCT
ejpam-5661	555	19	28	28	NUM
ejpam-5661	555	20	)	)	PUNCT
ejpam-5661	555	21	of	of	ADP
ejpam-5661	555	22	theorem	theorem	NOUN
ejpam-5661	555	23	8	8	NUM
ejpam-5661	555	24	,	,	PUNCT
ejpam-5661	555	25	we	we	PRON
ejpam-5661	555	26	obtain	obtain	VERB
ejpam-5661	555	27	the	the	DET
ejpam-5661	555	28	result	result	NOUN
ejpam-5661	555	29	.	.	PUNCT
ejpam-5661	556	1	6	6	X
ejpam-5661	556	2	.	.	X
ejpam-5661	556	3	conclusion	conclusion	NOUN
ejpam-5661	556	4	this	this	DET
ejpam-5661	556	5	study	study	NOUN
ejpam-5661	556	6	focused	focus	VERB
ejpam-5661	556	7	on	on	ADP
ejpam-5661	556	8	using	use	VERB
ejpam-5661	556	9	the	the	DET
ejpam-5661	556	10	principles	principle	NOUN
ejpam-5661	556	11	of	of	ADP
ejpam-5661	556	12	q	q	NOUN
ejpam-5661	556	13	-	-	NOUN
ejpam-5661	556	14	calculus	calculus	NOUN
ejpam-5661	556	15	to	to	PART
ejpam-5661	556	16	prove	prove	VERB
ejpam-5661	556	17	varieties	variety	NOUN
ejpam-5661	556	18	of	of	ADP
ejpam-5661	556	19	q	q	ADJ
ejpam-5661	556	20	-	-	PUNCT
ejpam-5661	556	21	h	h	NOUN
ejpam-5661	556	22	-	-	PUNCT
ejpam-5661	556	23	h	h	NOUN
ejpam-5661	556	24	type	type	NOUN
ejpam-5661	556	25	inequalities	inequality	NOUN
ejpam-5661	556	26	for	for	ADP
ejpam-5661	556	27	strongly	strongly	ADV
ejpam-5661	556	28	convex	convex	ADJ
ejpam-5661	556	29	functions	function	NOUN
ejpam-5661	556	30	.	.	PUNCT
ejpam-5661	557	1	our	our	PRON
ejpam-5661	557	2	findings	finding	NOUN
ejpam-5661	557	3	indicate	indicate	VERB
ejpam-5661	557	4	the	the	DET
ejpam-5661	557	5	potential	potential	NOUN
ejpam-5661	557	6	for	for	ADP
ejpam-5661	557	7	generalizing	generalize	VERB
ejpam-5661	557	8	existing	exist	VERB
ejpam-5661	557	9	comparable	comparable	ADJ
ejpam-5661	557	10	results	result	NOUN
ejpam-5661	557	11	in	in	ADP
ejpam-5661	557	12	the	the	DET
ejpam-5661	557	13	literature	literature	NOUN
ejpam-5661	557	14	.	.	PUNCT
ejpam-5661	558	1	in	in	ADP
ejpam-5661	558	2	addition	addition	NOUN
ejpam-5661	558	3	,	,	PUNCT
ejpam-5661	558	4	we	we	PRON
ejpam-5661	558	5	also	also	ADV
ejpam-5661	558	6	refine	refine	VERB
ejpam-5661	558	7	the	the	DET
ejpam-5661	558	8	results	result	NOUN
ejpam-5661	558	9	in	in	ADP
ejpam-5661	558	10	the	the	DET
ejpam-5661	558	11	literature	literature	NOUN
ejpam-5661	558	12	of	of	ADP
ejpam-5661	558	13	n.	n.	PROPN
ejpam-5661	558	14	alp	alp	PROPN
ejpam-5661	558	15	et	et	PROPN
ejpam-5661	558	16	al	al	PROPN
ejpam-5661	558	17	.	.	PUNCT
ejpam-5661	559	1	[	[	X
ejpam-5661	559	2	2	2	NUM
ejpam-5661	559	3	]	]	PUNCT
ejpam-5661	559	4	by	by	ADP
ejpam-5661	559	5	using	use	VERB
ejpam-5661	559	6	the	the	DET
ejpam-5661	559	7	characterization	characterization	NOUN
ejpam-5661	559	8	of	of	ADP
ejpam-5661	559	9	strong	strong	ADJ
ejpam-5661	559	10	convexity	convexity	NOUN
ejpam-5661	559	11	of	of	ADP
ejpam-5661	559	12	φ	φ	PROPN
ejpam-5661	559	13	.	.	PUNCT
ejpam-5661	560	1	the	the	DET
ejpam-5661	560	2	methodologies	methodology	NOUN
ejpam-5661	560	3	from	from	ADP
ejpam-5661	560	4	this	this	DET
ejpam-5661	560	5	study	study	NOUN
ejpam-5661	560	6	can	can	AUX
ejpam-5661	560	7	be	be	AUX
ejpam-5661	560	8	widely	widely	ADV
ejpam-5661	560	9	utilized	utilize	VERB
ejpam-5661	560	10	to	to	PART
ejpam-5661	560	11	establish	establish	VERB
ejpam-5661	560	12	q	q	ADJ
ejpam-5661	560	13	-	-	NOUN
ejpam-5661	560	14	calculus	calculus	ADJ
ejpam-5661	560	15	and	and	CCONJ
ejpam-5661	560	16	similar	similar	ADJ
ejpam-5661	560	17	inequalities	inequality	NOUN
ejpam-5661	560	18	for	for	ADP
ejpam-5661	560	19	various	various	ADJ
ejpam-5661	560	20	types	type	NOUN
ejpam-5661	560	21	of	of	ADP
ejpam-5661	560	22	convexities	convexity	NOUN
ejpam-5661	560	23	.	.	PUNCT
ejpam-5661	561	1	acknowledgements	acknowledgement	NOUN
ejpam-5661	561	2	we	we	PRON
ejpam-5661	561	3	express	express	VERB
ejpam-5661	561	4	our	our	PRON
ejpam-5661	561	5	sincere	sincere	ADJ
ejpam-5661	561	6	gratitude	gratitude	NOUN
ejpam-5661	561	7	to	to	ADP
ejpam-5661	561	8	the	the	DET
ejpam-5661	561	9	anonymous	anonymous	ADJ
ejpam-5661	561	10	reviewers	reviewer	NOUN
ejpam-5661	561	11	for	for	ADP
ejpam-5661	561	12	their	their	PRON
ejpam-5661	561	13	invaluable	invaluable	ADJ
ejpam-5661	561	14	feedback	feedback	NOUN
ejpam-5661	561	15	and	and	CCONJ
ejpam-5661	561	16	constructive	constructive	ADJ
ejpam-5661	561	17	suggestions	suggestion	NOUN
ejpam-5661	561	18	,	,	PUNCT
ejpam-5661	561	19	which	which	PRON
ejpam-5661	561	20	have	have	AUX
ejpam-5661	561	21	significantly	significantly	ADV
ejpam-5661	561	22	contributed	contribute	VERB
ejpam-5661	561	23	to	to	ADP
ejpam-5661	561	24	improving	improve	VERB
ejpam-5661	561	25	the	the	DET
ejpam-5661	561	26	quality	quality	NOUN
ejpam-5661	561	27	of	of	ADP
ejpam-5661	561	28	this	this	DET
ejpam-5661	561	29	manuscript	manuscript	NOUN
ejpam-5661	561	30	.	.	PUNCT
ejpam-5661	562	1	we	we	PRON
ejpam-5661	562	2	deeply	deeply	ADV
ejpam-5661	562	3	appreciate	appreciate	VERB
ejpam-5661	562	4	khon	khon	PROPN
ejpam-5661	562	5	kaen	kaen	PROPN
ejpam-5661	562	6	university	university	PROPN
ejpam-5661	562	7	(	(	PUNCT
ejpam-5661	562	8	kku	kku	PROPN
ejpam-5661	562	9	)	)	PUNCT
ejpam-5661	562	10	for	for	ADP
ejpam-5661	562	11	its	its	PRON
ejpam-5661	562	12	continuous	continuous	ADJ
ejpam-5661	562	13	encouragement	encouragement	NOUN
ejpam-5661	562	14	and	and	CCONJ
ejpam-5661	562	15	the	the	DET
ejpam-5661	562	16	resources	resource	NOUN
ejpam-5661	562	17	provided	provide	VERB
ejpam-5661	562	18	throughout	throughout	ADP
ejpam-5661	562	19	the	the	DET
ejpam-5661	562	20	research	research	NOUN
ejpam-5661	562	21	process	process	NOUN
ejpam-5661	562	22	.	.	PUNCT
ejpam-5661	563	1	references	reference	NOUN
ejpam-5661	563	2	[	[	X
ejpam-5661	563	3	1	1	NUM
ejpam-5661	563	4	]	]	X
ejpam-5661	563	5	m.a	m.a	PROPN
ejpam-5661	563	6	.	.	PROPN
ejpam-5661	563	7	ali	ali	PROPN
ejpam-5661	563	8	,	,	PUNCT
ejpam-5661	563	9	h.	h.	PROPN
ejpam-5661	563	10	budak	budak	PROPN
ejpam-5661	563	11	,	,	PUNCT
ejpam-5661	563	12	m.	m.	NOUN
ejpam-5661	563	13	fečkan	fečkan	PROPN
ejpam-5661	563	14	,	,	PUNCT
ejpam-5661	563	15	and	and	CCONJ
ejpam-5661	563	16	s.	s.	PROPN
ejpam-5661	563	17	khan	khan	PROPN
ejpam-5661	563	18	.	.	PUNCT
ejpam-5661	564	1	a	a	DET
ejpam-5661	564	2	new	new	ADJ
ejpam-5661	564	3	version	version	NOUN
ejpam-5661	564	4	of	of	ADP
ejpam-5661	564	5	q	q	ADJ
ejpam-5661	564	6	-	-	PUNCT
ejpam-5661	564	7	hermite	hermite	ADJ
ejpam-5661	564	8	–	–	PUNCT
ejpam-5661	564	9	hadamard	hadamard	NOUN
ejpam-5661	564	10	’s	’s	PART
ejpam-5661	564	11	midpoint	midpoint	NOUN
ejpam-5661	564	12	and	and	CCONJ
ejpam-5661	564	13	trapezoid	trapezoid	ADJ
ejpam-5661	564	14	type	type	NOUN
ejpam-5661	564	15	inequalities	inequality	NOUN
ejpam-5661	564	16	for	for	ADP
ejpam-5661	564	17	convex	convex	NOUN
ejpam-5661	564	18	functions	function	NOUN
ejpam-5661	564	19	.	.	PUNCT
ejpam-5661	565	1	math	math	NOUN
ejpam-5661	565	2	.	.	PUNCT
ejpam-5661	566	1	slovaca	slovaca	PROPN
ejpam-5661	566	2	,	,	PUNCT
ejpam-5661	566	3	73(2):369–386	73(2):369–386	PROPN
ejpam-5661	566	4	,	,	PUNCT
ejpam-5661	566	5	2023	2023	NUM
ejpam-5661	566	6	.	.	PUNCT
ejpam-5661	567	1	[	[	X
ejpam-5661	567	2	2	2	NUM
ejpam-5661	567	3	]	]	X
ejpam-5661	567	4	n.	n.	PROPN
ejpam-5661	567	5	alp	alp	PROPN
ejpam-5661	567	6	,	,	PUNCT
ejpam-5661	567	7	h.	h.	PROPN
ejpam-5661	567	8	budak	budak	PROPN
ejpam-5661	567	9	,	,	PUNCT
ejpam-5661	567	10	m.z	m.z	PROPN
ejpam-5661	567	11	.	.	PROPN
ejpam-5661	567	12	sarikaya	sarikaya	PROPN
ejpam-5661	567	13	,	,	PUNCT
ejpam-5661	567	14	and	and	CCONJ
ejpam-5661	567	15	m.a	m.a	PROPN
ejpam-5661	567	16	.	.	PROPN
ejpam-5661	567	17	ali	ali	PROPN
ejpam-5661	567	18	.	.	PROPN
ejpam-5661	568	1	on	on	ADP
ejpam-5661	568	2	new	new	ADJ
ejpam-5661	568	3	refinements	refinement	NOUN
ejpam-5661	568	4	and	and	CCONJ
ejpam-5661	568	5	generalizations	generalization	NOUN
ejpam-5661	568	6	of	of	ADP
ejpam-5661	568	7	q	q	ADJ
ejpam-5661	568	8	-	-	PUNCT
ejpam-5661	568	9	hermite	hermite	ADJ
ejpam-5661	568	10	-	-	PUNCT
ejpam-5661	568	11	hadamard	hadamard	ADJ
ejpam-5661	568	12	inequalities	inequality	NOUN
ejpam-5661	568	13	for	for	ADP
ejpam-5661	568	14	convex	convex	NOUN
ejpam-5661	568	15	functions	function	NOUN
ejpam-5661	568	16	.	.	PUNCT
ejpam-5661	569	1	rocky	rocky	ADJ
ejpam-5661	569	2	mountain	mountain	PROPN
ejpam-5661	569	3	j.	j.	PROPN
ejpam-5661	569	4	math	math	PROPN
ejpam-5661	569	5	.	.	PUNCT
ejpam-5661	569	6	,	,	PUNCT
ejpam-5661	570	1	54(2):361–374	54(2):361–374	PROPN
ejpam-5661	570	2	,	,	PUNCT
ejpam-5661	570	3	2024	2024	NUM
ejpam-5661	570	4	.	.	PUNCT
ejpam-5661	571	1	[	[	X
ejpam-5661	571	2	3	3	X
ejpam-5661	571	3	]	]	X
ejpam-5661	571	4	n.	n.	PROPN
ejpam-5661	571	5	alp	alp	PROPN
ejpam-5661	571	6	,	,	PUNCT
ejpam-5661	571	7	m.z	m.z	PROPN
ejpam-5661	571	8	.	.	PROPN
ejpam-5661	571	9	sarikaya	sarikaya	PROPN
ejpam-5661	571	10	,	,	PUNCT
ejpam-5661	571	11	and	and	CCONJ
ejpam-5661	571	12	m.	m.	PROPN
ejpam-5661	571	13	kunt	kunt	PROPN
ejpam-5661	571	14	i̇mdat	i̇mdat	ADP
ejpam-5661	571	15	i̇şcan	i̇şcan	PROPN
ejpam-5661	571	16	.	.	PUNCT
ejpam-5661	572	1	q	q	ADJ
ejpam-5661	572	2	-	-	PUNCT
ejpam-5661	572	3	hermite	hermite	ADJ
ejpam-5661	572	4	hadamard	hadamard	ADJ
ejpam-5661	572	5	inequalities	inequality	NOUN
ejpam-5661	572	6	and	and	CCONJ
ejpam-5661	572	7	quantum	quantum	NOUN
ejpam-5661	572	8	estimates	estimate	NOUN
ejpam-5661	572	9	for	for	ADP
ejpam-5661	572	10	midpoint	midpoint	NOUN
ejpam-5661	572	11	type	type	NOUN
ejpam-5661	572	12	inequalities	inequality	NOUN
ejpam-5661	572	13	via	via	ADP
ejpam-5661	572	14	convex	convex	NOUN
ejpam-5661	572	15	and	and	CCONJ
ejpam-5661	572	16	quasi	quasi	ADJ
ejpam-5661	572	17	-	-	ADJ
ejpam-5661	572	18	convex	convex	ADJ
ejpam-5661	572	19	functions	function	NOUN
ejpam-5661	572	20	.	.	PUNCT
ejpam-5661	573	1	j.	j.	PROPN
ejpam-5661	573	2	king	king	PROPN
ejpam-5661	573	3	saud	saud	PROPN
ejpam-5661	573	4	univ	univ	PROPN
ejpam-5661	573	5	.	.	PUNCT
ejpam-5661	574	1	sci	sci	PROPN
ejpam-5661	574	2	.	.	PROPN
ejpam-5661	574	3	,	,	PUNCT
ejpam-5661	574	4	30(2):193–203	30(2):193–203	NUM
ejpam-5661	574	5	,	,	PUNCT
ejpam-5661	574	6	2018	2018	NUM
ejpam-5661	574	7	.	.	PUNCT
ejpam-5661	575	1	c.	c.	PROPN
ejpam-5661	575	2	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	575	3	,	,	PUNCT
ejpam-5661	575	4	p.	p.	PROPN
ejpam-5661	575	5	yotkaew	yotkaew	PROPN
ejpam-5661	575	6	/	/	SYM
ejpam-5661	575	7	eur	eur	PROPN
ejpam-5661	575	8	.	.	PUNCT
ejpam-5661	576	1	j.	j.	PROPN
ejpam-5661	576	2	pure	pure	PROPN
ejpam-5661	576	3	appl	appl	PROPN
ejpam-5661	576	4	.	.	PROPN
ejpam-5661	576	5	math	math	PROPN
ejpam-5661	576	6	,	,	PUNCT
ejpam-5661	576	7	18	18	NUM
ejpam-5661	576	8	(	(	PUNCT
ejpam-5661	576	9	1	1	NUM
ejpam-5661	576	10	)	)	PUNCT
ejpam-5661	576	11	(	(	PUNCT
ejpam-5661	576	12	2025	2025	NUM
ejpam-5661	576	13	)	)	PUNCT
ejpam-5661	576	14	,	,	PUNCT
ejpam-5661	576	15	5661	5661	NUM
ejpam-5661	576	16	24	24	NUM
ejpam-5661	576	17	of	of	ADP
ejpam-5661	576	18	24	24	NUM
ejpam-5661	576	19	[	[	SYM
ejpam-5661	576	20	4	4	NUM
ejpam-5661	576	21	]	]	PUNCT
ejpam-5661	576	22	h.	h.	PROPN
ejpam-5661	576	23	angulo	angulo	PROPN
ejpam-5661	576	24	,	,	PUNCT
ejpam-5661	576	25	j.	j.	PROPN
ejpam-5661	576	26	gimenez	gimenez	PROPN
ejpam-5661	576	27	,	,	PUNCT
ejpam-5661	576	28	a.m.	a.m.	PROPN
ejpam-5661	576	29	moros	moros	PROPN
ejpam-5661	576	30	,	,	PUNCT
ejpam-5661	576	31	and	and	CCONJ
ejpam-5661	576	32	k.	k.	PROPN
ejpam-5661	576	33	nikodem	nikodem	PROPN
ejpam-5661	576	34	.	.	PUNCT
ejpam-5661	577	1	on	on	ADP
ejpam-5661	577	2	strongly	strongly	ADV
ejpam-5661	577	3	h	h	ADJ
ejpam-5661	577	4	-	-	PUNCT
ejpam-5661	577	5	convex	convex	NOUN
ejpam-5661	577	6	functions	function	NOUN
ejpam-5661	577	7	.	.	PUNCT
ejpam-5661	578	1	ann	ann	PROPN
ejpam-5661	578	2	.	.	PUNCT
ejpam-5661	578	3	funct	funct	PROPN
ejpam-5661	578	4	.	.	PUNCT
ejpam-5661	579	1	anal	anal	PROPN
ejpam-5661	579	2	.	.	PROPN
ejpam-5661	579	3	,	,	PUNCT
ejpam-5661	579	4	2(2):85–91	2(2):85–91	NUM
ejpam-5661	579	5	,	,	PUNCT
ejpam-5661	579	6	2011	2011	NUM
ejpam-5661	579	7	.	.	PUNCT
ejpam-5661	580	1	[	[	X
ejpam-5661	580	2	5	5	X
ejpam-5661	580	3	]	]	PUNCT
ejpam-5661	580	4	s.	s.	PROPN
ejpam-5661	580	5	bermudo	bermudo	PROPN
ejpam-5661	580	6	,	,	PUNCT
ejpam-5661	580	7	p.	p.	NOUN
ejpam-5661	580	8	kórus	kórus	PROPN
ejpam-5661	580	9	,	,	PUNCT
ejpam-5661	580	10	and	and	CCONJ
ejpam-5661	580	11	j.e	j.e	PROPN
ejpam-5661	580	12	.	.	PROPN
ejpam-5661	580	13	nápoles	nápoles	PROPN
ejpam-5661	580	14	valdés	valdés	PROPN
ejpam-5661	580	15	.	.	PUNCT
ejpam-5661	581	1	on	on	ADP
ejpam-5661	581	2	q	q	ADJ
ejpam-5661	581	3	-	-	PUNCT
ejpam-5661	581	4	hermite	hermite	ADJ
ejpam-5661	581	5	–	–	PUNCT
ejpam-5661	581	6	hadamard	hadamard	ADJ
ejpam-5661	581	7	inequalities	inequality	NOUN
ejpam-5661	581	8	for	for	ADP
ejpam-5661	581	9	general	general	ADJ
ejpam-5661	581	10	convex	convex	NOUN
ejpam-5661	581	11	functions	function	NOUN
ejpam-5661	581	12	.	.	PUNCT
ejpam-5661	582	1	acta	acta	PROPN
ejpam-5661	582	2	math	math	PROPN
ejpam-5661	582	3	.	.	PUNCT
ejpam-5661	583	1	hungar	hungar	PROPN
ejpam-5661	583	2	.	.	PUNCT
ejpam-5661	584	1	,	,	PUNCT
ejpam-5661	584	2	162:364–374	162:364–374	NUM
ejpam-5661	584	3	,	,	PUNCT
ejpam-5661	584	4	2020	2020	NUM
ejpam-5661	584	5	.	.	PUNCT
ejpam-5661	585	1	[	[	X
ejpam-5661	585	2	6	6	NUM
ejpam-5661	585	3	]	]	X
ejpam-5661	585	4	s.i	s.i	PROPN
ejpam-5661	585	5	.	.	PROPN
ejpam-5661	585	6	butt	butt	PROPN
ejpam-5661	585	7	,	,	PUNCT
ejpam-5661	585	8	m.n	m.n	PROPN
ejpam-5661	585	9	.	.	PROPN
ejpam-5661	585	10	aftab	aftab	PROPN
ejpam-5661	585	11	,	,	PUNCT
ejpam-5661	585	12	h.a	h.a	PROPN
ejpam-5661	585	13	.	.	PROPN
ejpam-5661	585	14	nabwey	nabwey	PROPN
ejpam-5661	585	15	,	,	PUNCT
ejpam-5661	585	16	and	and	CCONJ
ejpam-5661	585	17	s.	s.	PROPN
ejpam-5661	585	18	etemad	etemad	PROPN
ejpam-5661	585	19	.	.	PUNCT
ejpam-5661	586	1	some	some	DET
ejpam-5661	586	2	hermite	hermite	PROPN
ejpam-5661	586	3	-	-	PUNCT
ejpam-5661	586	4	hadamard	hadamard	ADJ
ejpam-5661	586	5	and	and	CCONJ
ejpam-5661	586	6	midpoint	midpoint	NOUN
ejpam-5661	586	7	type	type	NOUN
ejpam-5661	586	8	inequalities	inequality	NOUN
ejpam-5661	586	9	in	in	ADP
ejpam-5661	586	10	symmetric	symmetric	ADJ
ejpam-5661	586	11	quantum	quantum	NOUN
ejpam-5661	586	12	calculus	calculus	NOUN
ejpam-5661	586	13	.	.	PUNCT
ejpam-5661	587	1	aims	aim	VERB
ejpam-5661	587	2	math	math	NOUN
ejpam-5661	587	3	.	.	PUNCT
ejpam-5661	587	4	,	,	PUNCT
ejpam-5661	587	5	9(3):5523	9(3):5523	NUM
ejpam-5661	587	6	–	–	PUNCT
ejpam-5661	587	7	5549	5549	NUM
ejpam-5661	587	8	,	,	PUNCT
ejpam-5661	587	9	2024	2024	NUM
ejpam-5661	587	10	.	.	PUNCT
ejpam-5661	588	1	[	[	X
ejpam-5661	588	2	7	7	X
ejpam-5661	588	3	]	]	X
ejpam-5661	588	4	j.l	j.l	PROPN
ejpam-5661	588	5	.	.	PROPN
ejpam-5661	588	6	cardoso	cardoso	PROPN
ejpam-5661	588	7	and	and	CCONJ
ejpam-5661	588	8	e.m	e.m	PROPN
ejpam-5661	588	9	.	.	PROPN
ejpam-5661	588	10	shehata	shehata	PROPN
ejpam-5661	588	11	.	.	PUNCT
ejpam-5661	589	1	hermite	hermite	PROPN
ejpam-5661	589	2	-	-	PUNCT
ejpam-5661	589	3	hadamard	hadamard	ADJ
ejpam-5661	589	4	inequalities	inequality	NOUN
ejpam-5661	589	5	for	for	ADP
ejpam-5661	589	6	quantum	quantum	ADJ
ejpam-5661	589	7	integrals	integral	NOUN
ejpam-5661	589	8	:	:	PUNCT
ejpam-5661	589	9	a	a	DET
ejpam-5661	589	10	unified	unified	ADJ
ejpam-5661	589	11	approach	approach	NOUN
ejpam-5661	589	12	.	.	PUNCT
ejpam-5661	590	1	appl	appl	PROPN
ejpam-5661	590	2	.	.	PROPN
ejpam-5661	590	3	math	math	PROPN
ejpam-5661	590	4	.	.	PUNCT
ejpam-5661	591	1	comput	comput	NOUN
ejpam-5661	591	2	.	.	PUNCT
ejpam-5661	591	3	,	,	PUNCT
ejpam-5661	591	4	463:128345	463:128345	NUM
ejpam-5661	591	5	,	,	PUNCT
ejpam-5661	591	6	2024	2024	NUM
ejpam-5661	591	7	.	.	PUNCT
ejpam-5661	592	1	[	[	X
ejpam-5661	592	2	8	8	NUM
ejpam-5661	592	3	]	]	PUNCT
ejpam-5661	592	4	l.	l.	PROPN
ejpam-5661	592	5	ciurdariu	ciurdariu	PROPN
ejpam-5661	592	6	and	and	CCONJ
ejpam-5661	592	7	e.	e.	PROPN
ejpam-5661	592	8	grecu	grecu	PROPN
ejpam-5661	592	9	.	.	PUNCT
ejpam-5661	593	1	several	several	ADJ
ejpam-5661	593	2	quantum	quantum	ADJ
ejpam-5661	593	3	hermite	hermite	NOUN
ejpam-5661	593	4	–	–	PUNCT
ejpam-5661	593	5	hadamard	hadamard	NOUN
ejpam-5661	593	6	-	-	PUNCT
ejpam-5661	593	7	type	type	NOUN
ejpam-5661	593	8	integral	integral	ADJ
ejpam-5661	593	9	inequalities	inequality	NOUN
ejpam-5661	593	10	for	for	ADP
ejpam-5661	593	11	convex	convex	NOUN
ejpam-5661	593	12	functions	function	NOUN
ejpam-5661	593	13	.	.	PUNCT
ejpam-5661	594	1	fractal	fractal	ADJ
ejpam-5661	594	2	fract	fract	PROPN
ejpam-5661	594	3	.	.	PUNCT
ejpam-5661	594	4	,	,	PUNCT
ejpam-5661	594	5	7(6):463	7(6):463	NUM
ejpam-5661	594	6	,	,	PUNCT
ejpam-5661	594	7	2023	2023	NUM
ejpam-5661	594	8	.	.	PUNCT
ejpam-5661	595	1	[	[	X
ejpam-5661	595	2	9	9	NUM
ejpam-5661	595	3	]	]	X
ejpam-5661	595	4	s.s	s.s	PROPN
ejpam-5661	595	5	.	.	PROPN
ejpam-5661	595	6	dragomir	dragomir	PROPN
ejpam-5661	595	7	and	and	CCONJ
ejpam-5661	595	8	c.	c.	PROPN
ejpam-5661	595	9	pearce	pearce	PROPN
ejpam-5661	595	10	.	.	PUNCT
ejpam-5661	596	1	selected	select	VERB
ejpam-5661	596	2	topics	topic	NOUN
ejpam-5661	596	3	on	on	ADP
ejpam-5661	596	4	hermite	hermite	ADJ
ejpam-5661	596	5	-	-	PUNCT
ejpam-5661	596	6	hadamard	hadamard	ADJ
ejpam-5661	596	7	inequalities	inequality	NOUN
ejpam-5661	596	8	and	and	CCONJ
ejpam-5661	596	9	applications	application	NOUN
ejpam-5661	596	10	.	.	PUNCT
ejpam-5661	597	1	science	science	NOUN
ejpam-5661	597	2	direct	direct	ADJ
ejpam-5661	597	3	working	working	NOUN
ejpam-5661	597	4	paper	paper	NOUN
ejpam-5661	597	5	,	,	PUNCT
ejpam-5661	597	6	(	(	PUNCT
ejpam-5661	597	7	s1574	s1574	NOUN
ejpam-5661	597	8	-	-	PUNCT
ejpam-5661	597	9	0358	0358	NUM
ejpam-5661	597	10	)	)	PUNCT
ejpam-5661	597	11	,	,	PUNCT
ejpam-5661	597	12	2003	2003	NUM
ejpam-5661	597	13	.	.	PUNCT
ejpam-5661	598	1	[	[	X
ejpam-5661	598	2	10	10	NUM
ejpam-5661	598	3	]	]	X
ejpam-5661	598	4	t.	t.	PROPN
ejpam-5661	598	5	ernst	ernst	PROPN
ejpam-5661	598	6	.	.	PUNCT
ejpam-5661	599	1	a	a	DET
ejpam-5661	599	2	method	method	NOUN
ejpam-5661	599	3	for	for	ADP
ejpam-5661	599	4	q	q	NOUN
ejpam-5661	599	5	-	-	NOUN
ejpam-5661	599	6	calculus	calculus	NOUN
ejpam-5661	599	7	.	.	PUNCT
ejpam-5661	600	1	j.	j.	PROPN
ejpam-5661	600	2	nonlinear	nonlinear	PROPN
ejpam-5661	600	3	math	math	PROPN
ejpam-5661	600	4	.	.	PUNCT
ejpam-5661	601	1	phys	phy	NOUN
ejpam-5661	601	2	.	.	PUNCT
ejpam-5661	601	3	,	,	PUNCT
ejpam-5661	601	4	10:487–525	10:487–525	PROPN
ejpam-5661	601	5	,	,	PUNCT
ejpam-5661	601	6	2003	2003	NUM
ejpam-5661	601	7	.	.	PUNCT
ejpam-5661	602	1	[	[	X
ejpam-5661	602	2	11	11	NUM
ejpam-5661	602	3	]	]	PUNCT
ejpam-5661	602	4	t.	t.	PROPN
ejpam-5661	602	5	ernst	ernst	PROPN
ejpam-5661	602	6	.	.	PUNCT
ejpam-5661	603	1	a	a	DET
ejpam-5661	603	2	comprehensive	comprehensive	ADJ
ejpam-5661	603	3	treatment	treatment	NOUN
ejpam-5661	603	4	of	of	ADP
ejpam-5661	603	5	q	q	NOUN
ejpam-5661	603	6	-	-	PUNCT
ejpam-5661	603	7	calculus	calculus	NOUN
ejpam-5661	603	8	.	.	PUNCT
ejpam-5661	604	1	springer	springer	NOUN
ejpam-5661	604	2	,	,	PUNCT
ejpam-5661	604	3	basel	basel	PROPN
ejpam-5661	604	4	,	,	PUNCT
ejpam-5661	604	5	2012	2012	NUM
ejpam-5661	604	6	.	.	PUNCT
ejpam-5661	605	1	[	[	X
ejpam-5661	605	2	12	12	NUM
ejpam-5661	605	3	]	]	PUNCT
ejpam-5661	605	4	j.	j.	PROPN
ejpam-5661	605	5	hadamard	hadamard	PROPN
ejpam-5661	605	6	.	.	PUNCT
ejpam-5661	606	1	étude	étude	VERB
ejpam-5661	606	2	sur	sur	PROPN
ejpam-5661	606	3	les	les	PROPN
ejpam-5661	606	4	propriétés	propriétés	PROPN
ejpam-5661	606	5	des	des	PROPN
ejpam-5661	606	6	fonctions	fonctions	PROPN
ejpam-5661	606	7	entières	entière	NOUN
ejpam-5661	606	8	et	et	PROPN
ejpam-5661	606	9	en	en	X
ejpam-5661	606	10	particulier	particulier	NOUN
ejpam-5661	606	11	d’une	d’une	CCONJ
ejpam-5661	606	12	fonction	fonction	PROPN
ejpam-5661	606	13	considérée	considérée	PROPN
ejpam-5661	606	14	par	par	PROPN
ejpam-5661	606	15	riemann	riemann	PROPN
ejpam-5661	606	16	.	.	PUNCT
ejpam-5661	607	1	j.	j.	PROPN
ejpam-5661	607	2	math	math	PROPN
ejpam-5661	607	3	.	.	PUNCT
ejpam-5661	608	1	pures	pure	NOUN
ejpam-5661	608	2	appl	appl	PROPN
ejpam-5661	608	3	.	.	PROPN
ejpam-5661	608	4	,	,	PUNCT
ejpam-5661	608	5	9:171–215	9:171–215	NUM
ejpam-5661	608	6	,	,	PUNCT
ejpam-5661	608	7	1893	1893	NUM
ejpam-5661	608	8	.	.	PUNCT
ejpam-5661	609	1	[	[	X
ejpam-5661	609	2	13	13	NUM
ejpam-5661	609	3	]	]	X
ejpam-5661	609	4	c.	c.	PROPN
ejpam-5661	609	5	hermite	hermite	PROPN
ejpam-5661	609	6	.	.	PUNCT
ejpam-5661	610	1	sur	sur	PROPN
ejpam-5661	610	2	deux	deux	PROPN
ejpam-5661	610	3	limites	limites	PROPN
ejpam-5661	610	4	d’une	d’une	PROPN
ejpam-5661	610	5	intégrale	intégrale	NOUN
ejpam-5661	610	6	définie	définie	PROPN
ejpam-5661	610	7	.	.	PROPN
ejpam-5661	610	8	mathesis	mathesis	PROPN
ejpam-5661	610	9	,	,	PUNCT
ejpam-5661	610	10	3(1):1–82	3(1):1–82	NUM
ejpam-5661	610	11	,	,	PUNCT
ejpam-5661	610	12	1883	1883	NUM
ejpam-5661	610	13	.	.	PUNCT
ejpam-5661	611	1	[	[	X
ejpam-5661	611	2	14	14	NUM
ejpam-5661	611	3	]	]	X
ejpam-5661	611	4	f.h	f.h	PROPN
ejpam-5661	611	5	.	.	PROPN
ejpam-5661	611	6	jackson	jackson	PROPN
ejpam-5661	611	7	.	.	PUNCT
ejpam-5661	612	1	on	on	ADP
ejpam-5661	612	2	a	a	DET
ejpam-5661	612	3	q	q	ADJ
ejpam-5661	612	4	-	-	PUNCT
ejpam-5661	612	5	definite	definite	ADJ
ejpam-5661	612	6	integrals	integral	NOUN
ejpam-5661	612	7	.	.	PUNCT
ejpam-5661	613	1	q.	q.	PROPN
ejpam-5661	613	2	j.	j.	PROPN
ejpam-5661	613	3	pure	pure	PROPN
ejpam-5661	613	4	appl	appl	PROPN
ejpam-5661	613	5	.	.	PUNCT
ejpam-5661	613	6	math	math	PROPN
ejpam-5661	613	7	.	.	PUNCT
ejpam-5661	613	8	,	,	PUNCT
ejpam-5661	613	9	4:193–203	4:193–203	PROPN
ejpam-5661	613	10	,	,	PUNCT
ejpam-5661	613	11	1910	1910	NUM
ejpam-5661	613	12	.	.	PUNCT
ejpam-5661	614	1	[	[	X
ejpam-5661	614	2	15	15	NUM
ejpam-5661	614	3	]	]	X
ejpam-5661	614	4	f.h	f.h	PROPN
ejpam-5661	614	5	.	.	PROPN
ejpam-5661	614	6	jackson	jackson	PROPN
ejpam-5661	614	7	.	.	PUNCT
ejpam-5661	615	1	q	q	ADJ
ejpam-5661	615	2	-	-	PUNCT
ejpam-5661	615	3	difference	difference	NOUN
ejpam-5661	615	4	equations	equation	NOUN
ejpam-5661	615	5	.	.	PUNCT
ejpam-5661	616	1	am	be	AUX
ejpam-5661	616	2	.	.	PUNCT
ejpam-5661	617	1	j.	j.	PROPN
ejpam-5661	617	2	math	math	PROPN
ejpam-5661	617	3	.	.	PUNCT
ejpam-5661	617	4	,	,	PUNCT
ejpam-5661	617	5	32:305–314	32:305–314	PROPN
ejpam-5661	617	6	,	,	PUNCT
ejpam-5661	617	7	1910	1910	NUM
ejpam-5661	617	8	.	.	PUNCT
ejpam-5661	618	1	[	[	X
ejpam-5661	618	2	16	16	NUM
ejpam-5661	618	3	]	]	X
ejpam-5661	618	4	v.g	v.g	PROPN
ejpam-5661	618	5	.	.	PROPN
ejpam-5661	618	6	kac	kac	PROPN
ejpam-5661	618	7	and	and	CCONJ
ejpam-5661	618	8	p.	p.	PROPN
ejpam-5661	618	9	cheung	cheung	PROPN
ejpam-5661	618	10	.	.	PUNCT
ejpam-5661	619	1	quantum	quantum	PROPN
ejpam-5661	619	2	calculus	calculus	NOUN
ejpam-5661	619	3	.	.	PUNCT
ejpam-5661	620	1	springer	springer	NOUN
ejpam-5661	620	2	-	-	PUNCT
ejpam-5661	620	3	verlag	verlag	PROPN
ejpam-5661	620	4	,	,	PUNCT
ejpam-5661	620	5	new	new	PROPN
ejpam-5661	620	6	york	york	PROPN
ejpam-5661	620	7	,	,	PUNCT
ejpam-5661	620	8	2000	2000	NUM
ejpam-5661	620	9	.	.	PUNCT
ejpam-5661	621	1	[	[	X
ejpam-5661	621	2	17	17	NUM
ejpam-5661	621	3	]	]	X
ejpam-5661	621	4	n.	n.	NOUN
ejpam-5661	621	5	merentes	merente	NOUN
ejpam-5661	621	6	and	and	CCONJ
ejpam-5661	621	7	k.	k.	PROPN
ejpam-5661	621	8	nikodem	nikodem	PROPN
ejpam-5661	621	9	.	.	PUNCT
ejpam-5661	622	1	remarks	remark	NOUN
ejpam-5661	622	2	on	on	ADP
ejpam-5661	622	3	strongly	strongly	ADV
ejpam-5661	622	4	convex	convex	ADJ
ejpam-5661	622	5	functions	function	NOUN
ejpam-5661	622	6	.	.	PUNCT
ejpam-5661	623	1	aequat	aequat	PROPN
ejpam-5661	623	2	.	.	PUNCT
ejpam-5661	624	1	math	math	NOUN
ejpam-5661	624	2	.	.	PUNCT
ejpam-5661	625	1	,	,	PUNCT
ejpam-5661	625	2	80:193–199	80:193–199	NUM
ejpam-5661	625	3	,	,	PUNCT
ejpam-5661	625	4	2010	2010	NUM
ejpam-5661	625	5	.	.	PUNCT
ejpam-5661	626	1	[	[	X
ejpam-5661	626	2	18	18	NUM
ejpam-5661	626	3	]	]	PUNCT
ejpam-5661	626	4	k.	k.	PROPN
ejpam-5661	626	5	nikodem	nikodem	PROPN
ejpam-5661	626	6	and	and	CCONJ
ejpam-5661	626	7	zs	zs	PROPN
ejpam-5661	626	8	.	.	PUNCT
ejpam-5661	626	9	pales	pale	NOUN
ejpam-5661	626	10	.	.	PUNCT
ejpam-5661	627	1	characterizations	characterization	NOUN
ejpam-5661	627	2	of	of	ADP
ejpam-5661	627	3	inner	inner	ADJ
ejpam-5661	627	4	product	product	NOUN
ejpam-5661	627	5	spaces	space	NOUN
ejpam-5661	627	6	by	by	ADP
ejpam-5661	627	7	strongly	strongly	ADV
ejpam-5661	627	8	convex	convex	ADJ
ejpam-5661	627	9	functions	function	NOUN
ejpam-5661	627	10	.	.	PUNCT
ejpam-5661	628	1	banach	banach	NOUN
ejpam-5661	628	2	j.	j.	PROPN
ejpam-5661	628	3	math	math	PROPN
ejpam-5661	628	4	.	.	PUNCT
ejpam-5661	629	1	anal	anal	PROPN
ejpam-5661	629	2	.	.	PROPN
ejpam-5661	629	3	,	,	PUNCT
ejpam-5661	629	4	5(1):83–87	5(1):83–87	NUM
ejpam-5661	629	5	,	,	PUNCT
ejpam-5661	629	6	2011	2011	NUM
ejpam-5661	629	7	.	.	PUNCT
ejpam-5661	630	1	[	[	X
ejpam-5661	630	2	19	19	NUM
ejpam-5661	630	3	]	]	SYM
ejpam-5661	630	4	b.t	b.t	PROPN
ejpam-5661	630	5	.	.	PROPN
ejpam-5661	630	6	polyak	polyak	PROPN
ejpam-5661	630	7	.	.	PUNCT
ejpam-5661	631	1	existence	existence	NOUN
ejpam-5661	631	2	theorems	theorem	NOUN
ejpam-5661	631	3	and	and	CCONJ
ejpam-5661	631	4	convergence	convergence	NOUN
ejpam-5661	631	5	of	of	ADP
ejpam-5661	631	6	minimizing	minimize	VERB
ejpam-5661	631	7	sequences	sequence	NOUN
ejpam-5661	631	8	in	in	ADP
ejpam-5661	631	9	extremum	extremum	ADJ
ejpam-5661	631	10	problems	problem	NOUN
ejpam-5661	631	11	with	with	ADP
ejpam-5661	631	12	restrictions	restriction	NOUN
ejpam-5661	631	13	.	.	PUNCT
ejpam-5661	632	1	soviet	soviet	ADJ
ejpam-5661	632	2	math	math	NOUN
ejpam-5661	632	3	.	.	PUNCT
ejpam-5661	633	1	dokl	dokl	NOUN
ejpam-5661	633	2	.	.	PUNCT
ejpam-5661	633	3	,	,	PUNCT
ejpam-5661	633	4	7:72–75	7:72–75	NUM
ejpam-5661	633	5	,	,	PUNCT
ejpam-5661	633	6	1966	1966	NUM
ejpam-5661	633	7	.	.	PUNCT
ejpam-5661	634	1	[	[	X
ejpam-5661	634	2	20	20	NUM
ejpam-5661	634	3	]	]	PUNCT
ejpam-5661	634	4	p.m.	p.m.	NOUN
ejpam-5661	635	1	rajkovic	rajkovic	PROPN
ejpam-5661	635	2	,	,	PUNCT
ejpam-5661	635	3	m.s	m.s	PROPN
ejpam-5661	635	4	.	.	PROPN
ejpam-5661	635	5	stankovic	stankovic	PROPN
ejpam-5661	635	6	,	,	PUNCT
ejpam-5661	635	7	and	and	CCONJ
ejpam-5661	635	8	s.d	s.d	PROPN
ejpam-5661	635	9	.	.	PROPN
ejpam-5661	635	10	marinkovic	marinkovic	PROPN
ejpam-5661	635	11	.	.	PUNCT
ejpam-5661	636	1	the	the	DET
ejpam-5661	636	2	zeros	zero	NOUN
ejpam-5661	636	3	of	of	ADP
ejpam-5661	636	4	polynomials	polynomial	NOUN
ejpam-5661	636	5	orthogonal	orthogonal	ADJ
ejpam-5661	636	6	with	with	ADP
ejpam-5661	636	7	respect	respect	NOUN
ejpam-5661	636	8	to	to	ADP
ejpam-5661	636	9	q	q	NOUN
ejpam-5661	636	10	-	-	ADJ
ejpam-5661	636	11	integral	integral	ADJ
ejpam-5661	636	12	on	on	ADP
ejpam-5661	636	13	several	several	ADJ
ejpam-5661	636	14	intervals	interval	NOUN
ejpam-5661	636	15	in	in	ADP
ejpam-5661	636	16	the	the	DET
ejpam-5661	636	17	complex	complex	ADJ
ejpam-5661	636	18	plane	plane	NOUN
ejpam-5661	636	19	.	.	PUNCT
ejpam-5661	637	1	geom	geom	PROPN
ejpam-5661	637	2	.	.	PUNCT
ejpam-5661	638	1	integrability	integrability	PROPN
ejpam-5661	638	2	&	&	CCONJ
ejpam-5661	638	3	quantization	quantization	NOUN
ejpam-5661	638	4	,	,	PUNCT
ejpam-5661	638	5	pages	page	NOUN
ejpam-5661	638	6	178–188	178–188	NUM
ejpam-5661	638	7	,	,	PUNCT
ejpam-5661	638	8	2004	2004	NUM
ejpam-5661	638	9	.	.	PUNCT
ejpam-5661	639	1	[	[	X
ejpam-5661	639	2	21	21	NUM
ejpam-5661	639	3	]	]	X
ejpam-5661	639	4	c.	c.	PROPN
ejpam-5661	639	5	sahatsathatsana	sahatsathatsana	PROPN
ejpam-5661	639	6	and	and	CCONJ
ejpam-5661	639	7	k.	k.	X
ejpam-5661	639	8	nonlaopon	nonlaopon	PROPN
ejpam-5661	639	9	.	.	PUNCT
ejpam-5661	640	1	on	on	ADP
ejpam-5661	640	2	hermite	hermite	PROPN
ejpam-5661	640	3	-	-	PUNCT
ejpam-5661	640	4	hadamard	hadamard	ADJ
ejpam-5661	640	5	and	and	CCONJ
ejpam-5661	640	6	ostrowski	ostrowski	ADJ
ejpam-5661	640	7	type	type	NOUN
ejpam-5661	640	8	inequalities	inequality	NOUN
ejpam-5661	640	9	for	for	ADP
ejpam-5661	640	10	strongly	strongly	ADV
ejpam-5661	640	11	convex	convex	ADJ
ejpam-5661	640	12	functions	function	NOUN
ejpam-5661	640	13	via	via	ADP
ejpam-5661	640	14	quantum	quantum	ADJ
ejpam-5661	640	15	calculus	calculus	NOUN
ejpam-5661	640	16	with	with	ADP
ejpam-5661	640	17	applications	application	NOUN
ejpam-5661	640	18	.	.	PUNCT
ejpam-5661	641	1	j.	j.	PROPN
ejpam-5661	641	2	math	math	PROPN
ejpam-5661	641	3	.	.	PUNCT
ejpam-5661	642	1	comput	comput	NOUN
ejpam-5661	642	2	.	.	PUNCT
ejpam-5661	643	1	sci	sci	PROPN
ejpam-5661	643	2	.	.	PROPN
ejpam-5661	643	3	,	,	PUNCT
ejpam-5661	643	4	36(1):35–51	36(1):35–51	NUM
ejpam-5661	643	5	,	,	PUNCT
ejpam-5661	643	6	2025	2025	NUM
ejpam-5661	643	7	.	.	PUNCT
ejpam-5661	644	1	[	[	X
ejpam-5661	644	2	22	22	NUM
ejpam-5661	644	3	]	]	X
ejpam-5661	644	4	m.z	m.z	PROPN
ejpam-5661	644	5	.	.	PROPN
ejpam-5661	644	6	sarikaya	sarikaya	PROPN
ejpam-5661	644	7	.	.	PUNCT
ejpam-5661	645	1	on	on	ADP
ejpam-5661	645	2	hermite	hermite	PROPN
ejpam-5661	645	3	-	-	PUNCT
ejpam-5661	645	4	hadamard	hadamard	ADJ
ejpam-5661	645	5	type	type	NOUN
ejpam-5661	645	6	inequalities	inequality	NOUN
ejpam-5661	645	7	for	for	ADP
ejpam-5661	645	8	strongly	strongly	ADV
ejpam-5661	645	9	φ	φ	VERB
ejpam-5661	645	10	-	-	ADJ
ejpam-5661	645	11	convex	convex	ADJ
ejpam-5661	645	12	functions	function	NOUN
ejpam-5661	645	13	.	.	PUNCT
ejpam-5661	646	1	southeast	southeast	ADJ
ejpam-5661	646	2	asian	asian	ADJ
ejpam-5661	646	3	bull	bull	PROPN
ejpam-5661	646	4	.	.	PUNCT
ejpam-5661	647	1	math	math	NOUN
ejpam-5661	647	2	.	.	PUNCT
ejpam-5661	647	3	,	,	PUNCT
ejpam-5661	648	1	39(1):123–132	39(1):123–132	PROPN
ejpam-5661	648	2	,	,	PUNCT
ejpam-5661	648	3	2015	2015	NUM
ejpam-5661	648	4	.	.	PUNCT
ejpam-5661	649	1	[	[	X
ejpam-5661	649	2	23	23	NUM
ejpam-5661	649	3	]	]	PUNCT
ejpam-5661	649	4	t.	t.	NOUN
ejpam-5661	649	5	sitthiwirattham	sitthiwirattham	PROPN
ejpam-5661	649	6	,	,	PUNCT
ejpam-5661	649	7	m.a	m.a	PROPN
ejpam-5661	649	8	.	.	PROPN
ejpam-5661	649	9	ali	ali	PROPN
ejpam-5661	649	10	,	,	PUNCT
ejpam-5661	649	11	a.	a.	PROPN
ejpam-5661	649	12	ali	ali	PROPN
ejpam-5661	649	13	,	,	PUNCT
ejpam-5661	649	14	and	and	CCONJ
ejpam-5661	649	15	h.	h.	PROPN
ejpam-5661	649	16	budak	budak	PROPN
ejpam-5661	649	17	.	.	PUNCT
ejpam-5661	650	1	a	a	DET
ejpam-5661	650	2	new	new	ADJ
ejpam-5661	650	3	q	q	ADJ
ejpam-5661	650	4	-	-	PUNCT
ejpam-5661	650	5	hermite	hermite	ADJ
ejpam-5661	650	6	-	-	PUNCT
ejpam-5661	650	7	hadamard	hadamard	NOUN
ejpam-5661	650	8	’s	’s	PART
ejpam-5661	650	9	inequality	inequality	NOUN
ejpam-5661	650	10	and	and	CCONJ
ejpam-5661	650	11	estimates	estimate	NOUN
ejpam-5661	650	12	for	for	ADP
ejpam-5661	650	13	midpoint	midpoint	NOUN
ejpam-5661	650	14	type	type	NOUN
ejpam-5661	650	15	inequalities	inequality	NOUN
ejpam-5661	650	16	for	for	ADP
ejpam-5661	650	17	convex	convex	NOUN
ejpam-5661	650	18	functions	function	NOUN
ejpam-5661	650	19	.	.	PUNCT
ejpam-5661	651	1	miskolc	miskolc	ADJ
ejpam-5661	651	2	math	math	NOUN
ejpam-5661	651	3	.	.	PUNCT
ejpam-5661	652	1	notes	note	NOUN
ejpam-5661	652	2	,	,	PUNCT
ejpam-5661	652	3	24(3):1555–1567	24(3):1555–1567	NUM
ejpam-5661	652	4	,	,	PUNCT
ejpam-5661	652	5	2023	2023	NUM
ejpam-5661	652	6	.	.	PUNCT
ejpam-5661	653	1	[	[	X
ejpam-5661	653	2	24	24	NUM
ejpam-5661	653	3	]	]	PUNCT
ejpam-5661	653	4	j.	j.	PROPN
ejpam-5661	653	5	tariboon	tariboon	PROPN
ejpam-5661	653	6	and	and	CCONJ
ejpam-5661	653	7	s.k	s.k	PROPN
ejpam-5661	653	8	.	.	PROPN
ejpam-5661	653	9	ntouyas	ntouyas	PROPN
ejpam-5661	653	10	.	.	PUNCT
ejpam-5661	654	1	quantum	quantum	ADJ
ejpam-5661	654	2	calculus	calculus	NOUN
ejpam-5661	654	3	on	on	ADP
ejpam-5661	654	4	finite	finite	ADJ
ejpam-5661	654	5	intervals	interval	NOUN
ejpam-5661	654	6	and	and	CCONJ
ejpam-5661	654	7	applications	application	NOUN
ejpam-5661	654	8	to	to	ADP
ejpam-5661	654	9	impulsive	impulsive	ADJ
ejpam-5661	654	10	difference	difference	NOUN
ejpam-5661	654	11	equations	equation	NOUN
ejpam-5661	654	12	.	.	PUNCT
ejpam-5661	655	1	adv	adv	PROPN
ejpam-5661	655	2	.	.	PROPN
ejpam-5661	655	3	differ	differ	VERB
ejpam-5661	655	4	.	.	PUNCT
ejpam-5661	656	1	equ	equ	PROPN
ejpam-5661	656	2	.	.	PROPN
ejpam-5661	656	3	,	,	PUNCT
ejpam-5661	656	4	282	282	NUM
ejpam-5661	656	5	,	,	PUNCT
ejpam-5661	656	6	2013	2013	NUM
ejpam-5661	656	7	.	.	PUNCT
ejpam-5661	657	1	[	[	X
ejpam-5661	657	2	25	25	NUM
ejpam-5661	657	3	]	]	PUNCT
ejpam-5661	657	4	j.	j.	PROPN
ejpam-5661	657	5	tariboon	tariboon	PROPN
ejpam-5661	657	6	and	and	CCONJ
ejpam-5661	657	7	s.k	s.k	PROPN
ejpam-5661	657	8	.	.	PROPN
ejpam-5661	657	9	ntouyas	ntouyas	PROPN
ejpam-5661	657	10	.	.	PUNCT
ejpam-5661	658	1	quantum	quantum	ADJ
ejpam-5661	658	2	integral	integral	ADJ
ejpam-5661	658	3	inequalities	inequality	NOUN
ejpam-5661	658	4	on	on	ADP
ejpam-5661	658	5	finite	finite	ADJ
ejpam-5661	658	6	intervals	interval	NOUN
ejpam-5661	658	7	.	.	PUNCT
ejpam-5661	659	1	j.	j.	PROPN
ejpam-5661	659	2	inequal	inequal	PROPN
ejpam-5661	659	3	.	.	PUNCT
ejpam-5661	660	1	appl	appl	PROPN
ejpam-5661	660	2	.	.	PROPN
ejpam-5661	660	3	,	,	PUNCT
ejpam-5661	660	4	121	121	NUM
ejpam-5661	660	5	,	,	PUNCT
ejpam-5661	660	6	2014	2014	NUM
ejpam-5661	660	7	.	.	PUNCT