id	sid	tid	token	lemma	pos
ejpam-3776	1	1	european	european	PROPN
ejpam-3776	1	2	journal	journal	PROPN
ejpam-3776	1	3	of	of	ADP
ejpam-3776	1	4	pure	pure	ADJ
ejpam-3776	1	5	and	and	CCONJ
ejpam-3776	1	6	applied	apply	VERB
ejpam-3776	1	7	mathematics	mathematic	NOUN
ejpam-3776	1	8	vol	vol	NOUN
ejpam-3776	1	9	.	.	PROPN
ejpam-3776	2	1	13	13	NUM
ejpam-3776	2	2	,	,	PUNCT
ejpam-3776	2	3	no	no	INTJ
ejpam-3776	2	4	.	.	NOUN
ejpam-3776	2	5	4	4	NUM
ejpam-3776	2	6	,	,	PUNCT
ejpam-3776	2	7	2020	2020	NUM
ejpam-3776	2	8	,	,	PUNCT
ejpam-3776	2	9	814	814	NUM
ejpam-3776	2	10	-	-	SYM
ejpam-3776	2	11	829	829	NUM
ejpam-3776	2	12	issn	issn	PROPN
ejpam-3776	2	13	1307	1307	NUM
ejpam-3776	2	14	-	-	SYM
ejpam-3776	2	15	5543	5543	NUM
ejpam-3776	2	16	–	–	PUNCT
ejpam-3776	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3776	2	18	published	publish	VERB
ejpam-3776	2	19	by	by	ADP
ejpam-3776	2	20	new	new	PROPN
ejpam-3776	2	21	york	york	PROPN
ejpam-3776	2	22	business	business	PROPN
ejpam-3776	2	23	global	global	ADJ
ejpam-3776	2	24	new	new	ADJ
ejpam-3776	2	25	refinement	refinement	NOUN
ejpam-3776	2	26	of	of	ADP
ejpam-3776	2	27	niezgoda	niezgoda	PROPN
ejpam-3776	2	28	’s	’s	PART
ejpam-3776	2	29	inequality	inequality	NOUN
ejpam-3776	2	30	with	with	ADP
ejpam-3776	2	31	applications	application	NOUN
ejpam-3776	2	32	to	to	ADP
ejpam-3776	2	33	ky	ky	PROPN
ejpam-3776	2	34	fan	fan	PROPN
ejpam-3776	2	35	inequality	inequality	PROPN
ejpam-3776	2	36	sadia	sadia	PROPN
ejpam-3776	2	37	chanan1,∗	chanan1,∗	PROPN
ejpam-3776	2	38	,	,	PUNCT
ejpam-3776	2	39	asif	asif	PROPN
ejpam-3776	2	40	r.	r.	PROPN
ejpam-3776	2	41	khan1	khan1	PROPN
ejpam-3776	3	1	1	1	NUM
ejpam-3776	3	2	department	department	NOUN
ejpam-3776	3	3	of	of	ADP
ejpam-3776	3	4	mathematics	mathematic	NOUN
ejpam-3776	3	5	,	,	PUNCT
ejpam-3776	3	6	university	university	PROPN
ejpam-3776	3	7	of	of	ADP
ejpam-3776	3	8	karachi	karachi	PROPN
ejpam-3776	3	9	,	,	PUNCT
ejpam-3776	3	10	university	university	NOUN
ejpam-3776	3	11	road	road	NOUN
ejpam-3776	3	12	,	,	PUNCT
ejpam-3776	3	13	karachi-75270	karachi-75270	INTJ
ejpam-3776	3	14	,	,	PUNCT
ejpam-3776	3	15	pakistan	pakistan	PROPN
ejpam-3776	3	16	abstract	abstract	NOUN
ejpam-3776	3	17	.	.	PUNCT
ejpam-3776	4	1	the	the	DET
ejpam-3776	4	2	aim	aim	NOUN
ejpam-3776	4	3	of	of	ADP
ejpam-3776	4	4	this	this	DET
ejpam-3776	4	5	article	article	NOUN
ejpam-3776	4	6	is	be	AUX
ejpam-3776	4	7	to	to	PART
ejpam-3776	4	8	give	give	VERB
ejpam-3776	4	9	the	the	DET
ejpam-3776	4	10	refinement	refinement	NOUN
ejpam-3776	4	11	of	of	ADP
ejpam-3776	4	12	niezgoda	niezgoda	PROPN
ejpam-3776	4	13	’s	’s	PART
ejpam-3776	4	14	inequality	inequality	NOUN
ejpam-3776	4	15	with	with	ADP
ejpam-3776	4	16	its	its	PRON
ejpam-3776	4	17	applications	application	NOUN
ejpam-3776	4	18	to	to	ADP
ejpam-3776	4	19	ky	ky	PROPN
ejpam-3776	4	20	fan	fan	PROPN
ejpam-3776	4	21	inequality	inequality	PROPN
ejpam-3776	4	22	and	and	CCONJ
ejpam-3776	4	23	cyclic	cyclic	ADJ
ejpam-3776	4	24	mixed	mixed	ADJ
ejpam-3776	4	25	symmetric	symmetric	ADJ
ejpam-3776	4	26	means	mean	NOUN
ejpam-3776	4	27	.	.	PUNCT
ejpam-3776	5	1	2020	2020	NUM
ejpam-3776	5	2	mathematics	mathematic	NOUN
ejpam-3776	5	3	subject	subject	NOUN
ejpam-3776	5	4	classifications	classification	NOUN
ejpam-3776	5	5	:	:	PUNCT
ejpam-3776	5	6	26a51	26a51	NUM
ejpam-3776	5	7	,	,	PUNCT
ejpam-3776	5	8	39b62	39b62	NUM
ejpam-3776	5	9	,	,	PUNCT
ejpam-3776	5	10	26d15	26d15	NUM
ejpam-3776	5	11	,	,	PUNCT
ejpam-3776	5	12	26d20	26d20	NUM
ejpam-3776	5	13	,	,	PUNCT
ejpam-3776	5	14	26d99	26d99	NUM
ejpam-3776	5	15	key	key	ADJ
ejpam-3776	5	16	words	word	NOUN
ejpam-3776	5	17	and	and	CCONJ
ejpam-3776	5	18	phrases	phrase	NOUN
ejpam-3776	5	19	:	:	PUNCT
ejpam-3776	5	20	convex	convex	NOUN
ejpam-3776	5	21	functions	function	NOUN
ejpam-3776	5	22	,	,	PUNCT
ejpam-3776	5	23	niezgoda	niezgoda	PROPN
ejpam-3776	5	24	’s	’s	PART
ejpam-3776	5	25	inequality	inequality	NOUN
ejpam-3776	5	26	,	,	PUNCT
ejpam-3776	5	27	ky	ky	PROPN
ejpam-3776	5	28	fan	fan	PROPN
ejpam-3776	5	29	inequality	inequality	PROPN
ejpam-3776	5	30	,	,	PUNCT
ejpam-3776	5	31	cyclic	cyclic	ADJ
ejpam-3776	5	32	mixed	mixed	ADJ
ejpam-3776	5	33	symmetric	symmetric	ADJ
ejpam-3776	5	34	means	mean	NOUN
ejpam-3776	5	35	1	1	NUM
ejpam-3776	5	36	.	.	PUNCT
ejpam-3776	5	37	introduction	introduction	NOUN
ejpam-3776	5	38	and	and	CCONJ
ejpam-3776	5	39	preliminaries	preliminary	NOUN
ejpam-3776	5	40	jensen	jensen	PROPN
ejpam-3776	5	41	’s	’s	PART
ejpam-3776	5	42	inequality	inequality	NOUN
ejpam-3776	5	43	for	for	ADP
ejpam-3776	5	44	convex	convex	NOUN
ejpam-3776	5	45	functions	function	NOUN
ejpam-3776	5	46	is	be	AUX
ejpam-3776	5	47	one	one	NUM
ejpam-3776	5	48	of	of	ADP
ejpam-3776	5	49	the	the	DET
ejpam-3776	5	50	most	most	ADV
ejpam-3776	5	51	celebrated	celebrated	ADJ
ejpam-3776	5	52	inequality	inequality	NOUN
ejpam-3776	5	53	in	in	ADP
ejpam-3776	5	54	mathematics	mathematic	NOUN
ejpam-3776	5	55	and	and	CCONJ
ejpam-3776	5	56	statistics	statistic	NOUN
ejpam-3776	5	57	.	.	PUNCT
ejpam-3776	6	1	due	due	ADP
ejpam-3776	6	2	to	to	ADP
ejpam-3776	6	3	its	its	PRON
ejpam-3776	6	4	high	high	ADJ
ejpam-3776	6	5	importance	importance	NOUN
ejpam-3776	6	6	there	there	PRON
ejpam-3776	6	7	are	be	VERB
ejpam-3776	6	8	given	give	VERB
ejpam-3776	6	9	numerous	numerous	ADJ
ejpam-3776	6	10	variants	variant	NOUN
ejpam-3776	6	11	,	,	PUNCT
ejpam-3776	6	12	generalizations	generalization	NOUN
ejpam-3776	6	13	and	and	CCONJ
ejpam-3776	6	14	refinements	refinement	NOUN
ejpam-3776	6	15	of	of	ADP
ejpam-3776	6	16	jensen	jensen	PROPN
ejpam-3776	6	17	’s	’s	PART
ejpam-3776	6	18	inequalities	inequality	NOUN
ejpam-3776	6	19	(	(	PUNCT
ejpam-3776	6	20	for	for	ADP
ejpam-3776	6	21	reference	reference	NOUN
ejpam-3776	6	22	see	see	VERB
ejpam-3776	6	23	[	[	X
ejpam-3776	6	24	8	8	NUM
ejpam-3776	6	25	,	,	PUNCT
ejpam-3776	6	26	9	9	NUM
ejpam-3776	6	27	,	,	PUNCT
ejpam-3776	6	28	12	12	NUM
ejpam-3776	6	29	,	,	PUNCT
ejpam-3776	6	30	13	13	NUM
ejpam-3776	6	31	,	,	PUNCT
ejpam-3776	6	32	29	29	NUM
ejpam-3776	6	33	]	]	PUNCT
ejpam-3776	6	34	)	)	PUNCT
ejpam-3776	6	35	.	.	PUNCT
ejpam-3776	7	1	we	we	PRON
ejpam-3776	7	2	also	also	ADV
ejpam-3776	7	3	adduce	adduce	VERB
ejpam-3776	7	4	to	to	ADP
ejpam-3776	7	5	[	[	X
ejpam-3776	7	6	25	25	NUM
ejpam-3776	7	7	]	]	PUNCT
ejpam-3776	7	8	and	and	CCONJ
ejpam-3776	7	9	[	[	X
ejpam-3776	7	10	28	28	NUM
ejpam-3776	7	11	]	]	PUNCT
ejpam-3776	7	12	for	for	ADP
ejpam-3776	7	13	detailed	detailed	ADJ
ejpam-3776	7	14	discussion	discussion	NOUN
ejpam-3776	7	15	on	on	ADP
ejpam-3776	7	16	jensen	jensen	PROPN
ejpam-3776	7	17	’s	’s	PART
ejpam-3776	7	18	inequality	inequality	NOUN
ejpam-3776	7	19	and	and	CCONJ
ejpam-3776	7	20	for	for	ADP
ejpam-3776	7	21	some	some	DET
ejpam-3776	7	22	remarks	remark	NOUN
ejpam-3776	7	23	on	on	ADP
ejpam-3776	7	24	literature	literature	NOUN
ejpam-3776	7	25	and	and	CCONJ
ejpam-3776	7	26	history	history	NOUN
ejpam-3776	7	27	of	of	ADP
ejpam-3776	7	28	the	the	DET
ejpam-3776	7	29	topic	topic	NOUN
ejpam-3776	7	30	.	.	PUNCT
ejpam-3776	8	1	a	a	DET
ejpam-3776	8	2	variant	variant	NOUN
ejpam-3776	8	3	of	of	ADP
ejpam-3776	8	4	jensen	jensen	PROPN
ejpam-3776	8	5	’s	’s	PART
ejpam-3776	8	6	inequality	inequality	NOUN
ejpam-3776	8	7	named	name	VERB
ejpam-3776	8	8	as	as	SCONJ
ejpam-3776	8	9	jensen	jensen	PROPN
ejpam-3776	8	10	-	-	PUNCT
ejpam-3776	8	11	mercer	mercer	PROPN
ejpam-3776	8	12	inequality	inequality	NOUN
ejpam-3776	8	13	was	be	AUX
ejpam-3776	8	14	established	establish	VERB
ejpam-3776	8	15	by	by	ADP
ejpam-3776	8	16	mercer	mercer	PROPN
ejpam-3776	8	17	[	[	X
ejpam-3776	8	18	24	24	NUM
ejpam-3776	8	19	]	]	PUNCT
ejpam-3776	8	20	given	give	VERB
ejpam-3776	8	21	as	as	SCONJ
ejpam-3776	8	22	follows	follow	VERB
ejpam-3776	8	23	:	:	PUNCT
ejpam-3776	8	24	theorem	theorem	NOUN
ejpam-3776	8	25	1	1	NUM
ejpam-3776	8	26	.	.	PUNCT
ejpam-3776	9	1	let	let	VERB
ejpam-3776	9	2	x1	x1	NOUN
ejpam-3776	9	3	≤	≤	NUM
ejpam-3776	9	4	x2	x2	ADJ
ejpam-3776	9	5	≤	≤	NOUN
ejpam-3776	9	6	·	·	PUNCT
ejpam-3776	9	7	·	·	PUNCT
ejpam-3776	9	8	·	·	PUNCT
ejpam-3776	10	1	≤	≤	NUM
ejpam-3776	10	2	xn	xn	PUNCT
ejpam-3776	11	1	and	and	CCONJ
ejpam-3776	11	2	let	let	VERB
ejpam-3776	11	3	w1	w1	NOUN
ejpam-3776	11	4	,	,	PUNCT
ejpam-3776	11	5	w2	w2	NOUN
ejpam-3776	11	6	,	,	PUNCT
ejpam-3776	11	7	.	.	PUNCT
ejpam-3776	11	8	.	.	PUNCT
ejpam-3776	12	1	.	.	PUNCT
ejpam-3776	13	1	,	,	PUNCT
ejpam-3776	13	2	wn	wn	PROPN
ejpam-3776	13	3	be	be	AUX
ejpam-3776	13	4	nonnegative	nonnegative	ADJ
ejpam-3776	13	5	real	real	ADJ
ejpam-3776	13	6	numbers	number	NOUN
ejpam-3776	13	7	such	such	ADJ
ejpam-3776	13	8	that	that	SCONJ
ejpam-3776	13	9	n∑	n∑	PROPN
ejpam-3776	13	10	i=1	i=1	PROPN
ejpam-3776	13	11	wi	wi	PROPN
ejpam-3776	13	12	=	=	SYM
ejpam-3776	14	1	1	1	X
ejpam-3776	14	2	.	.	PUNCT
ejpam-3776	15	1	if	if	SCONJ
ejpam-3776	15	2	φ	φ	PROPN
ejpam-3776	15	3	is	be	AUX
ejpam-3776	15	4	a	a	DET
ejpam-3776	15	5	convex	convex	NOUN
ejpam-3776	15	6	function	function	NOUN
ejpam-3776	15	7	defined	define	VERB
ejpam-3776	15	8	on	on	ADP
ejpam-3776	15	9	an	an	DET
ejpam-3776	15	10	interval	interval	NOUN
ejpam-3776	15	11	containing	contain	VERB
ejpam-3776	15	12	all	all	PRON
ejpam-3776	15	13	xi	xi	ADP
ejpam-3776	15	14	’s	’s	NOUN
ejpam-3776	15	15	for	for	ADP
ejpam-3776	15	16	1	1	NUM
ejpam-3776	15	17	≤	≤	NUM
ejpam-3776	15	18	i	i	PRON
ejpam-3776	15	19	≤	≤	PROPN
ejpam-3776	15	20	n.	n.	NOUN
ejpam-3776	16	1	then	then	ADV
ejpam-3776	16	2	φ	φ	PROPN
ejpam-3776	16	3	(	(	PUNCT
ejpam-3776	16	4	x1	x1	PROPN
ejpam-3776	16	5	+	+	NUM
ejpam-3776	16	6	xn	xn	NUM
ejpam-3776	17	1	−	−	NUM
ejpam-3776	17	2	n∑	n∑	NOUN
ejpam-3776	17	3	i=1	i=1	PROPN
ejpam-3776	17	4	wixi	wixi	PROPN
ejpam-3776	17	5	)	)	PUNCT
ejpam-3776	17	6	≤	≤	NUM
ejpam-3776	17	7	φ	φ	PROPN
ejpam-3776	17	8	(	(	PUNCT
ejpam-3776	17	9	x1	x1	PROPN
ejpam-3776	17	10	)	)	PUNCT
ejpam-3776	18	1	+	+	CCONJ
ejpam-3776	18	2	φ	φ	PROPN
ejpam-3776	18	3	(	(	PUNCT
ejpam-3776	18	4	xn)−	xn)−	PROPN
ejpam-3776	18	5	n∑	n∑	PROPN
ejpam-3776	18	6	i=1	i=1	PROPN
ejpam-3776	18	7	wiφ	wiφ	INTJ
ejpam-3776	18	8	(	(	PUNCT
ejpam-3776	18	9	xi	xi	PROPN
ejpam-3776	18	10	)	)	PUNCT
ejpam-3776	18	11	.	.	PUNCT
ejpam-3776	19	1	(	(	PUNCT
ejpam-3776	19	2	1	1	X
ejpam-3776	19	3	)	)	PUNCT
ejpam-3776	19	4	now	now	ADV
ejpam-3776	19	5	,	,	PUNCT
ejpam-3776	19	6	we	we	PRON
ejpam-3776	19	7	recall	recall	VERB
ejpam-3776	19	8	a	a	DET
ejpam-3776	19	9	prerequisite	prerequisite	NOUN
ejpam-3776	19	10	concept	concept	NOUN
ejpam-3776	19	11	of	of	ADP
ejpam-3776	19	12	majorization	majorization	NOUN
ejpam-3776	19	13	from	from	ADP
ejpam-3776	19	14	[	[	X
ejpam-3776	19	15	23	23	NUM
ejpam-3776	19	16	]	]	PUNCT
ejpam-3776	19	17	.	.	PUNCT
ejpam-3776	20	1	let	let	VERB
ejpam-3776	20	2	x	x	PUNCT
ejpam-3776	20	3	=	=	SYM
ejpam-3776	20	4	(	(	PUNCT
ejpam-3776	20	5	x1	x1	PROPN
ejpam-3776	20	6	,	,	PUNCT
ejpam-3776	20	7	.	.	PUNCT
ejpam-3776	20	8	.	.	PUNCT
ejpam-3776	21	1	.	.	PUNCT
ejpam-3776	22	1	,	,	PUNCT
ejpam-3776	22	2	xm	xm	PROPN
ejpam-3776	22	3	)	)	PUNCT
ejpam-3776	22	4	and	and	CCONJ
ejpam-3776	22	5	y	y	PROPN
ejpam-3776	22	6	=	=	SYM
ejpam-3776	22	7	(	(	PUNCT
ejpam-3776	22	8	y1	y1	INTJ
ejpam-3776	22	9	,	,	PUNCT
ejpam-3776	22	10	.	.	PUNCT
ejpam-3776	22	11	.	.	PUNCT
ejpam-3776	22	12	.	.	PUNCT
ejpam-3776	23	1	,	,	PUNCT
ejpam-3776	23	2	ym	ym	PROPN
ejpam-3776	23	3	)	)	PUNCT
ejpam-3776	23	4	denote	denote	VERB
ejpam-3776	23	5	two	two	NUM
ejpam-3776	23	6	m	m	NOUN
ejpam-3776	23	7	-	-	PUNCT
ejpam-3776	23	8	tuples	tuple	NOUN
ejpam-3776	23	9	and	and	CCONJ
ejpam-3776	23	10	x[1	x[1	PROPN
ejpam-3776	23	11	]	]	X
ejpam-3776	23	12	≥	≥	X
ejpam-3776	23	13	·	·	PUNCT
ejpam-3776	23	14	·	·	PUNCT
ejpam-3776	23	15	·	·	PUNCT
ejpam-3776	23	16	≥	≥	X
ejpam-3776	24	1	x[m	x[m	X
ejpam-3776	24	2	]	]	X
ejpam-3776	24	3	,	,	PUNCT
ejpam-3776	24	4	y[1	y[1	PROPN
ejpam-3776	24	5	]	]	X
ejpam-3776	24	6	≥	≥	X
ejpam-3776	24	7	·	·	PUNCT
ejpam-3776	24	8	·	·	PUNCT
ejpam-3776	24	9	·	·	PUNCT
ejpam-3776	24	10	≥	≥	X
ejpam-3776	25	1	y[m	y[m	X
ejpam-3776	25	2	]	]	PUNCT
ejpam-3776	25	3	be	be	VERB
ejpam-3776	25	4	their	their	PRON
ejpam-3776	25	5	ordered	order	VERB
ejpam-3776	25	6	components	component	NOUN
ejpam-3776	25	7	.	.	PUNCT
ejpam-3776	26	1	∗corresponding	∗corresponde	VERB
ejpam-3776	26	2	author	author	NOUN
ejpam-3776	26	3	.	.	PUNCT
ejpam-3776	27	1	doi	doi	NOUN
ejpam-3776	27	2	:	:	PUNCT
ejpam-3776	27	3	https://doi.org/10.29020/nybg.ejpam.v13i4.3776	https://doi.org/10.29020/nybg.ejpam.v13i4.3776	NOUN
ejpam-3776	27	4	email	email	NOUN
ejpam-3776	27	5	addresses	address	VERB
ejpam-3776	27	6	:	:	PUNCT
ejpam-3776	27	7	sadiachanankhan@yahoo.com	sadiachanankhan@yahoo.com	X
ejpam-3776	27	8	(	(	PUNCT
ejpam-3776	27	9	s.	s.	PROPN
ejpam-3776	27	10	chanan	chanan	PROPN
ejpam-3776	27	11	)	)	PUNCT
ejpam-3776	27	12	,	,	PUNCT
ejpam-3776	27	13	asifrk@uok.edu.pk	asifrk@uok.edu.pk	NOUN
ejpam-3776	27	14	(	(	PUNCT
ejpam-3776	27	15	a.	a.	PROPN
ejpam-3776	27	16	r.	r.	PROPN
ejpam-3776	27	17	khan	khan	PROPN
ejpam-3776	27	18	)	)	PUNCT
ejpam-3776	27	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-3776	28	1	814	814	NUM
ejpam-3776	28	2	c	c	NOUN
ejpam-3776	28	3	©	©	NOUN
ejpam-3776	28	4	2020	2020	NUM
ejpam-3776	28	5	ejpam	ejpam	VERB
ejpam-3776	28	6	all	all	DET
ejpam-3776	28	7	rights	right	NOUN
ejpam-3776	28	8	reserved	reserve	VERB
ejpam-3776	28	9	.	.	PUNCT
ejpam-3776	29	1	s.	s.	PROPN
ejpam-3776	29	2	chanan	chanan	PROPN
ejpam-3776	29	3	,	,	PUNCT
ejpam-3776	29	4	a.	a.	PROPN
ejpam-3776	29	5	r.	r.	PROPN
ejpam-3776	29	6	khan	khan	PROPN
ejpam-3776	29	7	/	/	SYM
ejpam-3776	29	8	eur	eur	PROPN
ejpam-3776	29	9	.	.	PUNCT
ejpam-3776	30	1	j.	j.	PROPN
ejpam-3776	30	2	pure	pure	PROPN
ejpam-3776	30	3	appl	appl	PROPN
ejpam-3776	30	4	.	.	PROPN
ejpam-3776	30	5	math	math	PROPN
ejpam-3776	30	6	,	,	PUNCT
ejpam-3776	30	7	13	13	NUM
ejpam-3776	30	8	(	(	PUNCT
ejpam-3776	30	9	4	4	NUM
ejpam-3776	30	10	)	)	PUNCT
ejpam-3776	30	11	(	(	PUNCT
ejpam-3776	30	12	2020	2020	NUM
ejpam-3776	30	13	)	)	PUNCT
ejpam-3776	30	14	,	,	PUNCT
ejpam-3776	30	15	814	814	NUM
ejpam-3776	30	16	-	-	SYM
ejpam-3776	30	17	829	829	NUM
ejpam-3776	30	18	815	815	NUM
ejpam-3776	30	19	definition	definition	NOUN
ejpam-3776	30	20	1	1	NUM
ejpam-3776	30	21	.	.	PUNCT
ejpam-3776	31	1	for	for	ADP
ejpam-3776	31	2	x	x	SYM
ejpam-3776	31	3	,	,	PUNCT
ejpam-3776	31	4	y	y	PROPN
ejpam-3776	31	5	∈	∈	PROPN
ejpam-3776	31	6	rm	rm	PROPN
ejpam-3776	31	7	,	,	PUNCT
ejpam-3776	31	8	x	x	PROPN
ejpam-3776	31	9	≺	≺	NOUN
ejpam-3776	31	10	y	y	NOUN
ejpam-3776	31	11	if	if	SCONJ
ejpam-3776	31	12			PRON
ejpam-3776	31	13	k∑	k∑	VERB
ejpam-3776	31	14	i=1	i=1	PRON
ejpam-3776	31	15	x[i	x[i	NUM
ejpam-3776	31	16	]	]	X
ejpam-3776	31	17	≤	≤	NOUN
ejpam-3776	31	18	k∑	k∑	VERB
ejpam-3776	31	19	i=1	i=1	PROPN
ejpam-3776	32	1	y[i	y[i	X
ejpam-3776	32	2	]	]	PUNCT
ejpam-3776	32	3	,	,	PUNCT
ejpam-3776	32	4	k	k	PROPN
ejpam-3776	32	5	∈	∈	PROPN
ejpam-3776	32	6	{	{	PUNCT
ejpam-3776	32	7	1	1	NUM
ejpam-3776	32	8	,	,	PUNCT
ejpam-3776	32	9	.	.	PUNCT
ejpam-3776	32	10	.	.	PUNCT
ejpam-3776	32	11	.	.	PUNCT
ejpam-3776	33	1	,	,	PUNCT
ejpam-3776	33	2	m−	m−	PROPN
ejpam-3776	33	3	1	1	NUM
ejpam-3776	33	4	}	}	PUNCT
ejpam-3776	33	5	,	,	PUNCT
ejpam-3776	33	6	m∑	m∑	ADV
ejpam-3776	33	7	i=1	i=1	ADP
ejpam-3776	33	8	x[i	x[i	NUM
ejpam-3776	33	9	]	]	X
ejpam-3776	33	10	=	=	PUNCT
ejpam-3776	33	11	m∑	m∑	INTJ
ejpam-3776	33	12	i=1	i=1	PROPN
ejpam-3776	33	13	y[i	y[i	X
ejpam-3776	33	14	]	]	X
ejpam-3776	33	15	when	when	SCONJ
ejpam-3776	33	16	x	x	PROPN
ejpam-3776	33	17	≺	≺	NOUN
ejpam-3776	33	18	y	y	PROPN
ejpam-3776	33	19	,	,	PUNCT
ejpam-3776	33	20	x	x	VERB
ejpam-3776	33	21	is	be	AUX
ejpam-3776	33	22	said	say	VERB
ejpam-3776	33	23	to	to	PART
ejpam-3776	33	24	be	be	AUX
ejpam-3776	33	25	majorized	majorize	VERB
ejpam-3776	33	26	by	by	ADP
ejpam-3776	33	27	y	y	PROPN
ejpam-3776	33	28	or	or	CCONJ
ejpam-3776	33	29	y	y	PROPN
ejpam-3776	33	30	majorizes	majorize	NOUN
ejpam-3776	33	31	x.	x.	NOUN
ejpam-3776	33	32	in	in	ADP
ejpam-3776	33	33	the	the	DET
ejpam-3776	33	34	book	book	NOUN
ejpam-3776	33	35	[	[	X
ejpam-3776	33	36	10	10	NUM
ejpam-3776	33	37	]	]	PUNCT
ejpam-3776	33	38	we	we	PRON
ejpam-3776	33	39	find	find	VERB
ejpam-3776	33	40	a	a	DET
ejpam-3776	33	41	very	very	ADJ
ejpam-3776	33	42	power	power	NOUN
ejpam-3776	33	43	result	result	NOUN
ejpam-3776	33	44	namely	namely	ADV
ejpam-3776	33	45	majorization	majorization	NOUN
ejpam-3776	33	46	theorem	theorem	NOUN
ejpam-3776	33	47	(	(	PUNCT
ejpam-3776	33	48	see	see	VERB
ejpam-3776	33	49	also	also	ADV
ejpam-3776	33	50	[	[	X
ejpam-3776	33	51	23	23	NUM
ejpam-3776	33	52	]	]	SYM
ejpam-3776	33	53	)	)	PUNCT
ejpam-3776	33	54	.	.	PUNCT
ejpam-3776	34	1	theorem	theorem	NOUN
ejpam-3776	34	2	2	2	NUM
ejpam-3776	34	3	.	.	PUNCT
ejpam-3776	35	1	let	let	VERB
ejpam-3776	35	2	x	x	PRON
ejpam-3776	35	3	,	,	PUNCT
ejpam-3776	35	4	y	y	PROPN
ejpam-3776	35	5	∈	∈	PROPN
ejpam-3776	35	6	rn	rn	PROPN
ejpam-3776	35	7	then	then	ADV
ejpam-3776	35	8	following	follow	VERB
ejpam-3776	35	9	inequality	inequality	NOUN
ejpam-3776	35	10	is	be	AUX
ejpam-3776	35	11	true	true	ADJ
ejpam-3776	35	12	for	for	ADP
ejpam-3776	35	13	all	all	DET
ejpam-3776	35	14	continuous	continuous	ADJ
ejpam-3776	35	15	convex	convex	NOUN
ejpam-3776	35	16	functions	function	NOUN
ejpam-3776	35	17	φ	φ	NOUN
ejpam-3776	35	18	:	:	PUNCT
ejpam-3776	35	19	r→	r→	PROPN
ejpam-3776	35	20	r	r	PROPN
ejpam-3776	35	21	,	,	PUNCT
ejpam-3776	35	22	n∑	n∑	PROPN
ejpam-3776	35	23	i=1	i=1	PROPN
ejpam-3776	35	24	φ	φ	PROPN
ejpam-3776	35	25	(	(	PUNCT
ejpam-3776	35	26	xi	xi	PROPN
ejpam-3776	35	27	)	)	PUNCT
ejpam-3776	35	28	≤	≤	NOUN
ejpam-3776	36	1	n∑	n∑	PROPN
ejpam-3776	36	2	i=1	i=1	PROPN
ejpam-3776	36	3	φ	φ	PROPN
ejpam-3776	36	4	(	(	PUNCT
ejpam-3776	36	5	yi	yi	PROPN
ejpam-3776	36	6	)	)	PUNCT
ejpam-3776	36	7	if	if	SCONJ
ejpam-3776	36	8	and	and	CCONJ
ejpam-3776	36	9	only	only	ADV
ejpam-3776	36	10	if	if	SCONJ
ejpam-3776	36	11	x	x	NOUN
ejpam-3776	36	12	≺	≺	VERB
ejpam-3776	36	13	y.	y.	NOUN
ejpam-3776	36	14	we	we	PRON
ejpam-3776	36	15	now	now	ADV
ejpam-3776	36	16	define	define	VERB
ejpam-3776	36	17	an	an	DET
ejpam-3776	36	18	extension	extension	NOUN
ejpam-3776	36	19	of	of	ADP
ejpam-3776	36	20	jensen	jensen	PROPN
ejpam-3776	36	21	-	-	PUNCT
ejpam-3776	36	22	mercer	mercer	PROPN
ejpam-3776	36	23	inequality	inequality	NOUN
ejpam-3776	36	24	which	which	PRON
ejpam-3776	36	25	is	be	AUX
ejpam-3776	36	26	referred	refer	VERB
ejpam-3776	36	27	as	as	ADP
ejpam-3776	36	28	niezgoda	niezgoda	NOUN
ejpam-3776	36	29	’s	’s	PART
ejpam-3776	36	30	inequality	inequality	NOUN
ejpam-3776	36	31	by	by	ADP
ejpam-3776	36	32	niezgoda	niezgoda	NOUN
ejpam-3776	36	33	[	[	X
ejpam-3776	36	34	26	26	NUM
ejpam-3776	36	35	]	]	PUNCT
ejpam-3776	36	36	.	.	PUNCT
ejpam-3776	37	1	for	for	ADP
ejpam-3776	37	2	recent	recent	ADJ
ejpam-3776	37	3	work	work	NOUN
ejpam-3776	37	4	on	on	ADP
ejpam-3776	37	5	niezgoda	niezgoda	NOUN
ejpam-3776	37	6	inequality	inequality	NOUN
ejpam-3776	37	7	we	we	PRON
ejpam-3776	37	8	refer	refer	VERB
ejpam-3776	37	9	the	the	DET
ejpam-3776	37	10	reader	reader	NOUN
ejpam-3776	37	11	[	[	X
ejpam-3776	37	12	1	1	NUM
ejpam-3776	37	13	,	,	PUNCT
ejpam-3776	37	14	15–17	15–17	NUM
ejpam-3776	37	15	,	,	PUNCT
ejpam-3776	37	16	27	27	NUM
ejpam-3776	37	17	]	]	PUNCT
ejpam-3776	37	18	.	.	PUNCT
ejpam-3776	38	1	theorem	theorem	NOUN
ejpam-3776	38	2	3	3	X
ejpam-3776	38	3	.	.	PUNCT
ejpam-3776	38	4	suppose	suppose	VERB
ejpam-3776	38	5	that	that	SCONJ
ejpam-3776	38	6	a	a	PRON
ejpam-3776	38	7	be	be	AUX
ejpam-3776	38	8	an	an	DET
ejpam-3776	38	9	m	m	NOUN
ejpam-3776	38	10	-	-	NOUN
ejpam-3776	38	11	tuple	tuple	NOUN
ejpam-3776	38	12	such	such	ADJ
ejpam-3776	38	13	that	that	SCONJ
ejpam-3776	38	14	ai	ai	VERB
ejpam-3776	38	15	∈	∈	PROPN
ejpam-3776	38	16	j	j	PROPN
ejpam-3776	38	17	and	and	CCONJ
ejpam-3776	38	18	a	a	DET
ejpam-3776	38	19	n	n	NOUN
ejpam-3776	38	20	×m	×m	NOUN
ejpam-3776	38	21	matrix	matrix	NOUN
ejpam-3776	38	22	x	x	PUNCT
ejpam-3776	38	23	=	=	SYM
ejpam-3776	38	24	(	(	PUNCT
ejpam-3776	38	25	xj	xj	PROPN
ejpam-3776	38	26	)	)	PUNCT
ejpam-3776	38	27	=	=	PUNCT
ejpam-3776	38	28	(	(	PUNCT
ejpam-3776	38	29	xij	xij	X
ejpam-3776	38	30	)	)	PUNCT
ejpam-3776	38	31	with	with	ADP
ejpam-3776	38	32	xij	xij	PROPN
ejpam-3776	38	33	∈	∈	PROPN
ejpam-3776	38	34	j	j	PROPN
ejpam-3776	38	35	for	for	ADP
ejpam-3776	38	36	all	all	PRON
ejpam-3776	38	37	i	i	PRON
ejpam-3776	38	38	∈	∈	PROPN
ejpam-3776	38	39	{	{	PUNCT
ejpam-3776	38	40	1	1	NUM
ejpam-3776	38	41	,	,	PUNCT
ejpam-3776	38	42	.	.	PUNCT
ejpam-3776	38	43	.	.	PUNCT
ejpam-3776	39	1	.	.	PUNCT
ejpam-3776	40	1	,	,	PUNCT
ejpam-3776	40	2	n	n	CCONJ
ejpam-3776	40	3	}	}	PUNCT
ejpam-3776	40	4	and	and	CCONJ
ejpam-3776	40	5	j	j	PROPN
ejpam-3776	40	6	∈	∈	PROPN
ejpam-3776	40	7	{	{	PUNCT
ejpam-3776	40	8	1	1	NUM
ejpam-3776	40	9	,	,	PUNCT
ejpam-3776	40	10	.	.	PUNCT
ejpam-3776	40	11	.	.	PUNCT
ejpam-3776	41	1	.	.	PUNCT
ejpam-3776	42	1	,	,	PUNCT
ejpam-3776	42	2	m	m	VERB
ejpam-3776	42	3	}	}	PUNCT
ejpam-3776	42	4	.	.	PUNCT
ejpam-3776	43	1	if	if	SCONJ
ejpam-3776	43	2	a	a	DET
ejpam-3776	43	3	majorizes	majorize	NOUN
ejpam-3776	43	4	each	each	DET
ejpam-3776	43	5	row	row	NOUN
ejpam-3776	43	6	of	of	ADP
ejpam-3776	43	7	x	x	NOUN
ejpam-3776	43	8	,	,	PUNCT
ejpam-3776	43	9	that	that	ADV
ejpam-3776	43	10	is	is	ADV
ejpam-3776	43	11	,	,	PUNCT
ejpam-3776	43	12	xi	xi	PROPN
ejpam-3776	43	13	.	.	PUNCT
ejpam-3776	44	1	=	=	PRON
ejpam-3776	44	2	(	(	PUNCT
ejpam-3776	44	3	xi1	xi1	PROPN
ejpam-3776	44	4	,	,	PUNCT
ejpam-3776	44	5	.	.	PUNCT
ejpam-3776	44	6	.	.	PUNCT
ejpam-3776	45	1	.	.	PUNCT
ejpam-3776	46	1	,	,	PUNCT
ejpam-3776	46	2	xim	xim	PROPN
ejpam-3776	46	3	)	)	PUNCT
ejpam-3776	46	4	≺	≺	NOUN
ejpam-3776	46	5	(	(	PUNCT
ejpam-3776	46	6	a1	a1	NOUN
ejpam-3776	46	7	,	,	PUNCT
ejpam-3776	46	8	.	.	PUNCT
ejpam-3776	46	9	.	.	PUNCT
ejpam-3776	46	10	.	.	PUNCT
ejpam-3776	47	1	,	,	PUNCT
ejpam-3776	47	2	am	be	AUX
ejpam-3776	47	3	)	)	PUNCT
ejpam-3776	47	4	=	=	SYM
ejpam-3776	48	1	a	a	PRON
ejpam-3776	48	2	for	for	ADP
ejpam-3776	48	3	each	each	DET
ejpam-3776	48	4	i	i	PRON
ejpam-3776	48	5	∈	∈	PROPN
ejpam-3776	48	6	{	{	PUNCT
ejpam-3776	48	7	1	1	NUM
ejpam-3776	48	8	,	,	PUNCT
ejpam-3776	48	9	.	.	PUNCT
ejpam-3776	48	10	.	.	PUNCT
ejpam-3776	49	1	.	.	PUNCT
ejpam-3776	50	1	,	,	PUNCT
ejpam-3776	50	2	n	n	CCONJ
ejpam-3776	50	3	}	}	PUNCT
ejpam-3776	50	4	,	,	PUNCT
ejpam-3776	50	5	then	then	ADV
ejpam-3776	50	6	for	for	ADP
ejpam-3776	50	7	a	a	DET
ejpam-3776	50	8	continuous	continuous	ADJ
ejpam-3776	50	9	convex	convex	NOUN
ejpam-3776	50	10	function	function	NOUN
ejpam-3776	50	11	φ	φ	PROPN
ejpam-3776	50	12	on	on	ADP
ejpam-3776	50	13	j	j	PROPN
ejpam-3776	50	14	following	follow	VERB
ejpam-3776	50	15	inequality	inequality	NOUN
ejpam-3776	50	16	holds	hold	VERB
ejpam-3776	50	17	.	.	PUNCT
ejpam-3776	51	1	φ	φ	PROPN
ejpam-3776	51	2			PROPN
ejpam-3776	51	3	m∑	m∑	PROPN
ejpam-3776	51	4	j=1	j=1	PROPN
ejpam-3776	51	5	aj	aj	PROPN
ejpam-3776	51	6	−	−	PROPN
ejpam-3776	51	7	m−1∑	m−1∑	NUM
ejpam-3776	52	1	j=1	j=1	PROPN
ejpam-3776	52	2	n∑	n∑	PROPN
ejpam-3776	52	3	i=1	i=1	PROPN
ejpam-3776	53	1	wixij	wixij	PROPN
ejpam-3776	53	2			PROPN
ejpam-3776	53	3	≤	≤	PROPN
ejpam-3776	53	4	m∑	m∑	CCONJ
ejpam-3776	53	5	j=1	j=1	ADJ
ejpam-3776	53	6	φ(aj)−	φ(aj)−	NOUN
ejpam-3776	54	1	m−1∑	m−1∑	PROPN
ejpam-3776	54	2	j=1	j=1	PROPN
ejpam-3776	54	3	n∑	n∑	PROPN
ejpam-3776	54	4	i=1	i=1	PROPN
ejpam-3776	54	5	wiφ(xij	wiφ(xij	PROPN
ejpam-3776	54	6	)	)	PUNCT
ejpam-3776	54	7	,	,	PUNCT
ejpam-3776	54	8	(	(	PUNCT
ejpam-3776	54	9	2	2	X
ejpam-3776	54	10	)	)	PUNCT
ejpam-3776	54	11	with	with	ADP
ejpam-3776	54	12	wi	wi	PROPN
ejpam-3776	54	13	≥	≥	NUM
ejpam-3776	54	14	0	0	NUM
ejpam-3776	54	15	such	such	ADJ
ejpam-3776	54	16	that	that	SCONJ
ejpam-3776	54	17	∑n	∑n	PROPN
ejpam-3776	54	18	i=1wi	i=1wi	X
ejpam-3776	54	19	=	=	SYM
ejpam-3776	54	20	1	1	X
ejpam-3776	54	21	.	.	PUNCT
ejpam-3776	54	22	especially	especially	ADV
ejpam-3776	54	23	,	,	PUNCT
ejpam-3776	54	24	the	the	DET
ejpam-3776	54	25	inequality	inequality	NOUN
ejpam-3776	54	26	stated	state	VERB
ejpam-3776	54	27	below	below	ADV
ejpam-3776	54	28	is	be	AUX
ejpam-3776	54	29	also	also	ADV
ejpam-3776	54	30	valid	valid	ADJ
ejpam-3776	54	31	for	for	ADP
ejpam-3776	54	32	wi	wi	PROPN
ejpam-3776	54	33	=	=	SYM
ejpam-3776	54	34	1	1	NUM
ejpam-3776	54	35	n	n	NOUN
ejpam-3776	54	36	,	,	PUNCT
ejpam-3776	54	37	i	i	PRON
ejpam-3776	54	38	∈	∈	PROPN
ejpam-3776	54	39	{	{	PUNCT
ejpam-3776	54	40	1	1	NUM
ejpam-3776	54	41	,	,	PUNCT
ejpam-3776	54	42	.	.	PUNCT
ejpam-3776	54	43	.	.	PUNCT
ejpam-3776	55	1	.	.	PUNCT
ejpam-3776	56	1	,	,	PUNCT
ejpam-3776	56	2	n	n	CCONJ
ejpam-3776	56	3	}	}	PUNCT
ejpam-3776	56	4	φ	φ	PROPN
ejpam-3776	56	5			PROPN
ejpam-3776	56	6	m∑	m∑	PROPN
ejpam-3776	56	7	j=1	j=1	PROPN
ejpam-3776	56	8	aj	aj	PROPN
ejpam-3776	57	1	−	−	PROPN
ejpam-3776	57	2	1	1	NUM
ejpam-3776	57	3	n	n	PROPN
ejpam-3776	57	4	m−1∑	m−1∑	NUM
ejpam-3776	58	1	j=1	j=1	PROPN
ejpam-3776	58	2	n∑	n∑	PROPN
ejpam-3776	58	3	i=1	i=1	PROPN
ejpam-3776	59	1	xij	xij	PROPN
ejpam-3776	59	2			PROPN
ejpam-3776	59	3	≤	≤	NUM
ejpam-3776	59	4	m∑	m∑	CCONJ
ejpam-3776	59	5	j=1	j=1	ADJ
ejpam-3776	59	6	φ(aj)−	φ(aj)−	NOUN
ejpam-3776	59	7	1	1	NUM
ejpam-3776	59	8	n	n	PROPN
ejpam-3776	59	9	m−1∑	m−1∑	PROPN
ejpam-3776	60	1	j=1	j=1	NOUN
ejpam-3776	60	2	n∑	n∑	PROPN
ejpam-3776	60	3	i=1	i=1	PROPN
ejpam-3776	60	4	φ(xij	φ(xij	PROPN
ejpam-3776	60	5	)	)	PUNCT
ejpam-3776	60	6	.	.	PUNCT
ejpam-3776	61	1	(	(	PUNCT
ejpam-3776	61	2	3	3	X
ejpam-3776	61	3	)	)	PUNCT
ejpam-3776	61	4	the	the	DET
ejpam-3776	61	5	cyclic	cyclic	ADJ
ejpam-3776	61	6	refinement	refinement	NOUN
ejpam-3776	61	7	of	of	ADP
ejpam-3776	61	8	the	the	DET
ejpam-3776	61	9	jensen	jensen	PROPN
ejpam-3776	61	10	’s	’s	PART
ejpam-3776	61	11	inequality	inequality	NOUN
ejpam-3776	61	12	in	in	ADP
ejpam-3776	61	13	paper	paper	NOUN
ejpam-3776	61	14	[	[	X
ejpam-3776	61	15	3	3	X
ejpam-3776	61	16	]	]	PUNCT
ejpam-3776	61	17	is	be	AUX
ejpam-3776	61	18	given	give	VERB
ejpam-3776	61	19	as	as	SCONJ
ejpam-3776	61	20	follows	follow	VERB
ejpam-3776	61	21	:	:	PUNCT
ejpam-3776	61	22	s.	s.	PROPN
ejpam-3776	61	23	chanan	chanan	PROPN
ejpam-3776	61	24	,	,	PUNCT
ejpam-3776	61	25	a.	a.	PROPN
ejpam-3776	61	26	r.	r.	PROPN
ejpam-3776	61	27	khan	khan	PROPN
ejpam-3776	61	28	/	/	SYM
ejpam-3776	61	29	eur	eur	PROPN
ejpam-3776	61	30	.	.	PUNCT
ejpam-3776	62	1	j.	j.	PROPN
ejpam-3776	62	2	pure	pure	PROPN
ejpam-3776	62	3	appl	appl	PROPN
ejpam-3776	62	4	.	.	PROPN
ejpam-3776	62	5	math	math	PROPN
ejpam-3776	62	6	,	,	PUNCT
ejpam-3776	62	7	13	13	NUM
ejpam-3776	62	8	(	(	PUNCT
ejpam-3776	62	9	4	4	NUM
ejpam-3776	62	10	)	)	PUNCT
ejpam-3776	62	11	(	(	PUNCT
ejpam-3776	62	12	2020	2020	NUM
ejpam-3776	62	13	)	)	PUNCT
ejpam-3776	62	14	,	,	PUNCT
ejpam-3776	62	15	814	814	NUM
ejpam-3776	62	16	-	-	SYM
ejpam-3776	62	17	829	829	NUM
ejpam-3776	62	18	816	816	NUM
ejpam-3776	62	19	theorem	theorem	NOUN
ejpam-3776	62	20	4	4	NUM
ejpam-3776	62	21	.	.	PUNCT
ejpam-3776	63	1	let	let	VERB
ejpam-3776	63	2	φ	φ	NOUN
ejpam-3776	63	3	:	:	PUNCT
ejpam-3776	64	1	i	i	PRON
ejpam-3776	64	2	→	→	PUNCT
ejpam-3776	64	3	r	r	NOUN
ejpam-3776	64	4	be	be	AUX
ejpam-3776	64	5	a	a	DET
ejpam-3776	64	6	convex	convex	NOUN
ejpam-3776	64	7	function	function	NOUN
ejpam-3776	64	8	and	and	CCONJ
ejpam-3776	64	9	i	i	PRON
ejpam-3776	64	10	be	be	VERB
ejpam-3776	64	11	an	an	DET
ejpam-3776	64	12	interval	interval	NOUN
ejpam-3776	64	13	in	in	ADP
ejpam-3776	64	14	r	r	NOUN
ejpam-3776	64	15	,	,	PUNCT
ejpam-3776	64	16	x	x	SYM
ejpam-3776	64	17	=	=	SYM
ejpam-3776	64	18	(	(	PUNCT
ejpam-3776	64	19	x1	x1	PROPN
ejpam-3776	64	20	,	,	PUNCT
ejpam-3776	64	21	.	.	PUNCT
ejpam-3776	64	22	.	.	PUNCT
ejpam-3776	64	23	.	.	PUNCT
ejpam-3776	65	1	,	,	PUNCT
ejpam-3776	65	2	xn	xn	X
ejpam-3776	65	3	)	)	PUNCT
ejpam-3776	65	4	∈	∈	PROPN
ejpam-3776	65	5	in	in	ADP
ejpam-3776	65	6	and	and	CCONJ
ejpam-3776	65	7	λ	λ	X
ejpam-3776	65	8	=	=	SYM
ejpam-3776	65	9	(	(	PUNCT
ejpam-3776	65	10	λ1	λ1	ADJ
ejpam-3776	65	11	,	,	PUNCT
ejpam-3776	65	12	.	.	PUNCT
ejpam-3776	65	13	.	.	PUNCT
ejpam-3776	66	1	.	.	PUNCT
ejpam-3776	67	1	,	,	PUNCT
ejpam-3776	67	2	λn	λn	NOUN
ejpam-3776	67	3	)	)	PUNCT
ejpam-3776	67	4	be	be	AUX
ejpam-3776	67	5	a	a	DET
ejpam-3776	67	6	nonnegative	nonnegative	ADJ
ejpam-3776	67	7	n	n	CCONJ
ejpam-3776	67	8	-	-	PUNCT
ejpam-3776	67	9	tuple	tuple	NOUN
ejpam-3776	67	10	such	such	ADJ
ejpam-3776	67	11	that	that	SCONJ
ejpam-3776	67	12	∑k	∑k	PROPN
ejpam-3776	68	1	i=1	i=1	X
ejpam-3776	68	2	λi	λi	NOUN
ejpam-3776	68	3	=	=	NOUN
ejpam-3776	68	4	1	1	NUM
ejpam-3776	68	5	for	for	ADP
ejpam-3776	68	6	some	some	DET
ejpam-3776	68	7	k	k	NOUN
ejpam-3776	68	8	,	,	PUNCT
ejpam-3776	68	9	2	2	NUM
ejpam-3776	68	10	≤	≤	NUM
ejpam-3776	68	11	k	k	PROPN
ejpam-3776	68	12	≤	≤	PROPN
ejpam-3776	68	13	n.	n.	NOUN
ejpam-3776	68	14	then	then	ADV
ejpam-3776	68	15	φ	φ	PROPN
ejpam-3776	68	16	(	(	PUNCT
ejpam-3776	68	17	1	1	NUM
ejpam-3776	68	18	n	n	NUM
ejpam-3776	68	19	n∑	n∑	NOUN
ejpam-3776	68	20	i=1	i=1	X
ejpam-3776	68	21	xi	xi	X
ejpam-3776	68	22	)	)	PUNCT
ejpam-3776	68	23	≤	≤	NOUN
ejpam-3776	68	24	1	1	NUM
ejpam-3776	69	1	n	n	NUM
ejpam-3776	69	2	n∑	n∑	NOUN
ejpam-3776	69	3	i=1	i=1	PROPN
ejpam-3776	70	1	φ	φ	PROPN
ejpam-3776	70	2	k−1∑	k−1∑	PROPN
ejpam-3776	70	3	j=0	j=0	PROPN
ejpam-3776	70	4	λj+1xi+j	λj+1xi+j	ADP
ejpam-3776	70	5			PROPN
ejpam-3776	70	6	≤	≤	ADV
ejpam-3776	70	7	1	1	NUM
ejpam-3776	70	8	n	n	NUM
ejpam-3776	70	9	n∑	n∑	NOUN
ejpam-3776	70	10	i=1	i=1	PROPN
ejpam-3776	71	1	φ	φ	PROPN
ejpam-3776	71	2	(	(	PUNCT
ejpam-3776	71	3	xi	xi	PROPN
ejpam-3776	71	4	)	)	PUNCT
ejpam-3776	71	5	,	,	PUNCT
ejpam-3776	71	6	(	(	PUNCT
ejpam-3776	71	7	4	4	X
ejpam-3776	71	8	)	)	PUNCT
ejpam-3776	71	9	where	where	SCONJ
ejpam-3776	71	10	i+	i+	NUM
ejpam-3776	71	11	j	j	PROPN
ejpam-3776	71	12	means	mean	VERB
ejpam-3776	71	13	i+	i+	PROPN
ejpam-3776	71	14	j	j	NOUN
ejpam-3776	71	15	−	−	NOUN
ejpam-3776	71	16	n	n	CCONJ
ejpam-3776	71	17	in	in	ADP
ejpam-3776	71	18	case	case	NOUN
ejpam-3776	71	19	of	of	ADP
ejpam-3776	71	20	i+	i+	NUM
ejpam-3776	71	21	j	j	PROPN
ejpam-3776	71	22	>	>	X
ejpam-3776	71	23	n.	n.	PROPN
ejpam-3776	71	24	the	the	DET
ejpam-3776	71	25	cyclic	cyclic	PROPN
ejpam-3776	71	26	refinement	refinement	NOUN
ejpam-3776	71	27	of	of	ADP
ejpam-3776	71	28	jensen	jensen	PROPN
ejpam-3776	71	29	-	-	PUNCT
ejpam-3776	71	30	mercer	mercer	PROPN
ejpam-3776	71	31	inequality	inequality	NOUN
ejpam-3776	71	32	in	in	ADP
ejpam-3776	71	33	paper	paper	NOUN
ejpam-3776	72	1	[	[	X
ejpam-3776	72	2	5	5	NUM
ejpam-3776	72	3	]	]	PUNCT
ejpam-3776	72	4	is	be	AUX
ejpam-3776	72	5	defined	define	VERB
ejpam-3776	72	6	as	as	SCONJ
ejpam-3776	72	7	follows	follow	VERB
ejpam-3776	72	8	:	:	PUNCT
ejpam-3776	72	9	theorem	theorem	NOUN
ejpam-3776	72	10	5	5	NUM
ejpam-3776	72	11	.	.	PUNCT
ejpam-3776	73	1	let	let	VERB
ejpam-3776	73	2	i	i	PRON
ejpam-3776	73	3	⊂	⊂	PRON
ejpam-3776	73	4	r	r	PRON
ejpam-3776	73	5	be	be	VERB
ejpam-3776	73	6	an	an	DET
ejpam-3776	73	7	interval	interval	NOUN
ejpam-3776	73	8	,	,	PUNCT
ejpam-3776	73	9	x	x	SYM
ejpam-3776	73	10	=	=	SYM
ejpam-3776	73	11	(	(	PUNCT
ejpam-3776	73	12	x1	x1	PROPN
ejpam-3776	73	13	,	,	PUNCT
ejpam-3776	73	14	.	.	PUNCT
ejpam-3776	73	15	.	.	PUNCT
ejpam-3776	74	1	.	.	PUNCT
ejpam-3776	75	1	,	,	PUNCT
ejpam-3776	75	2	xn	xn	X
ejpam-3776	75	3	)	)	PUNCT
ejpam-3776	75	4	∈	∈	NOUN
ejpam-3776	75	5	in	in	ADP
ejpam-3776	75	6	such	such	ADJ
ejpam-3776	75	7	that	that	PRON
ejpam-3776	75	8	(	(	PUNCT
ejpam-3776	75	9	c+	c+	X
ejpam-3776	75	10	d−	d−	PROPN
ejpam-3776	75	11	∑k−1	∑k−1	PRON
ejpam-3776	75	12	j=0	j=0	PROPN
ejpam-3776	75	13	λj+1xi+j	λj+1xi+j	ADP
ejpam-3776	75	14	)	)	PUNCT
ejpam-3776	75	15	∈	∈	PROPN
ejpam-3776	75	16	i	i	PRON
ejpam-3776	75	17	and	and	CCONJ
ejpam-3776	75	18	λ	λ	PROPN
ejpam-3776	75	19	=	=	SYM
ejpam-3776	75	20	(	(	PUNCT
ejpam-3776	75	21	λ1	λ1	ADJ
ejpam-3776	75	22	,	,	PUNCT
ejpam-3776	75	23	.	.	PUNCT
ejpam-3776	75	24	.	.	PUNCT
ejpam-3776	76	1	.	.	PUNCT
ejpam-3776	77	1	,	,	PUNCT
ejpam-3776	77	2	λn	λn	NOUN
ejpam-3776	77	3	)	)	PUNCT
ejpam-3776	77	4	be	be	AUX
ejpam-3776	77	5	a	a	DET
ejpam-3776	77	6	positive	positive	ADJ
ejpam-3776	77	7	n	n	CCONJ
ejpam-3776	77	8	-	-	PUNCT
ejpam-3776	77	9	tuple	tuple	NOUN
ejpam-3776	77	10	such	such	ADJ
ejpam-3776	77	11	that	that	SCONJ
ejpam-3776	77	12	∑k	∑k	PROPN
ejpam-3776	78	1	i=1	i=1	X
ejpam-3776	78	2	λi	λi	NOUN
ejpam-3776	78	3	=	=	NOUN
ejpam-3776	78	4	1	1	NUM
ejpam-3776	78	5	for	for	ADP
ejpam-3776	78	6	some	some	DET
ejpam-3776	78	7	k	k	NOUN
ejpam-3776	78	8	,	,	PUNCT
ejpam-3776	78	9	2	2	NUM
ejpam-3776	78	10	≤	≤	NUM
ejpam-3776	78	11	k	k	NOUN
ejpam-3776	78	12	≤	≤	PROPN
ejpam-3776	78	13	n	n	CCONJ
ejpam-3776	78	14	,	,	PUNCT
ejpam-3776	78	15	then	then	ADV
ejpam-3776	78	16	for	for	ADP
ejpam-3776	78	17	convex	convex	PROPN
ejpam-3776	78	18	function	function	NOUN
ejpam-3776	78	19	φ	φ	NOUN
ejpam-3776	78	20	:	:	PUNCT
ejpam-3776	79	1	i	i	PRON
ejpam-3776	79	2	→	→	SYM
ejpam-3776	79	3	r	r	X
ejpam-3776	79	4	,	,	PUNCT
ejpam-3776	79	5	[	[	X
ejpam-3776	79	6	c	c	X
ejpam-3776	79	7	,	,	PUNCT
ejpam-3776	79	8	d	d	X
ejpam-3776	79	9	]	]	X
ejpam-3776	79	10	⊂	⊂	PROPN
ejpam-3776	79	11	i	i	PRON
ejpam-3776	79	12	,	,	PUNCT
ejpam-3776	79	13	following	follow	VERB
ejpam-3776	79	14	inequalities	inequality	NOUN
ejpam-3776	79	15	hold	hold	VERB
ejpam-3776	79	16	:	:	PUNCT
ejpam-3776	79	17	φ	φ	PROPN
ejpam-3776	79	18	(	(	PUNCT
ejpam-3776	79	19	c+	c+	PROPN
ejpam-3776	79	20	d−	d−	PROPN
ejpam-3776	79	21	n∑	n∑	NOUN
ejpam-3776	79	22	i=1	i=1	PROPN
ejpam-3776	79	23	wixi	wixi	ADJ
ejpam-3776	79	24	)	)	PUNCT
ejpam-3776	79	25	≤	≤	NOUN
ejpam-3776	80	1	n∑	n∑	PROPN
ejpam-3776	80	2	i=1	i=1	PROPN
ejpam-3776	80	3	wiφ	wiφ	PROPN
ejpam-3776	80	4	c+	c+	PROPN
ejpam-3776	81	1	d−	d−	PROPN
ejpam-3776	81	2	k−1∑	k−1∑	PROPN
ejpam-3776	81	3	j=0	j=0	PROPN
ejpam-3776	81	4	λj+1xi+j	λj+1xi+j	ADP
ejpam-3776	81	5			PROPN
ejpam-3776	81	6	≤	≤	PROPN
ejpam-3776	81	7	φ	φ	PROPN
ejpam-3776	81	8	(	(	PUNCT
ejpam-3776	81	9	c	c	NOUN
ejpam-3776	81	10	)	)	PUNCT
ejpam-3776	82	1	+	+	CCONJ
ejpam-3776	82	2	φ	φ	PROPN
ejpam-3776	82	3	(	(	PUNCT
ejpam-3776	82	4	d)−	d)−	PROPN
ejpam-3776	82	5	n∑	n∑	PROPN
ejpam-3776	82	6	i=1	i=1	PROPN
ejpam-3776	82	7	wiφ	wiφ	INTJ
ejpam-3776	82	8	(	(	PUNCT
ejpam-3776	82	9	xi	xi	PROPN
ejpam-3776	82	10	)	)	PUNCT
ejpam-3776	82	11	,	,	PUNCT
ejpam-3776	82	12	(	(	PUNCT
ejpam-3776	82	13	5	5	X
ejpam-3776	82	14	)	)	PUNCT
ejpam-3776	82	15	where	where	SCONJ
ejpam-3776	82	16	i+	i+	NUM
ejpam-3776	82	17	j	j	PROPN
ejpam-3776	82	18	means	mean	VERB
ejpam-3776	82	19	i+	i+	PROPN
ejpam-3776	82	20	j	j	NOUN
ejpam-3776	82	21	−	−	NOUN
ejpam-3776	82	22	n	n	CCONJ
ejpam-3776	82	23	in	in	ADP
ejpam-3776	82	24	case	case	NOUN
ejpam-3776	82	25	of	of	ADP
ejpam-3776	82	26	i+	i+	NUM
ejpam-3776	82	27	j	j	PROPN
ejpam-3776	82	28	>	>	X
ejpam-3776	82	29	n.	n.	NOUN
ejpam-3776	82	30	in	in	ADP
ejpam-3776	82	31	this	this	DET
ejpam-3776	82	32	article	article	NOUN
ejpam-3776	82	33	we	we	PRON
ejpam-3776	82	34	are	be	AUX
ejpam-3776	82	35	going	go	VERB
ejpam-3776	82	36	to	to	PART
ejpam-3776	82	37	use	use	VERB
ejpam-3776	82	38	some	some	PRON
ejpam-3776	82	39	of	of	ADP
ejpam-3776	82	40	the	the	DET
ejpam-3776	82	41	following	following	ADJ
ejpam-3776	82	42	assumptions	assumption	NOUN
ejpam-3776	82	43	:	:	PUNCT
ejpam-3776	82	44	•	•	X
ejpam-3776	82	45	(	(	PUNCT
ejpam-3776	82	46	c1	c1	NOUN
ejpam-3776	82	47	):	):	PUNCT
ejpam-3776	82	48	let	let	VERB
ejpam-3776	82	49	φ	φ	PROPN
ejpam-3776	82	50	:	:	PUNCT
ejpam-3776	83	1	j	j	PROPN
ejpam-3776	83	2	→	→	PUNCT
ejpam-3776	83	3	r	r	NOUN
ejpam-3776	83	4	be	be	AUX
ejpam-3776	83	5	a	a	DET
ejpam-3776	83	6	convex	convex	NOUN
ejpam-3776	83	7	function	function	NOUN
ejpam-3776	83	8	.	.	PUNCT
ejpam-3776	84	1	•	•	NUM
ejpam-3776	84	2	(	(	PUNCT
ejpam-3776	84	3	c2	c2	PROPN
ejpam-3776	84	4	):	):	PUNCT
ejpam-3776	84	5	let	let	VERB
ejpam-3776	84	6	a	a	PRON
ejpam-3776	84	7	be	be	AUX
ejpam-3776	84	8	a	a	DET
ejpam-3776	84	9	m	m	NOUN
ejpam-3776	84	10	-	-	NOUN
ejpam-3776	84	11	tuple	tuple	NOUN
ejpam-3776	84	12	such	such	ADJ
ejpam-3776	84	13	that	that	SCONJ
ejpam-3776	84	14	aj	aj	PROPN
ejpam-3776	84	15	∈	∈	PROPN
ejpam-3776	84	16	jn	jn	PROPN
ejpam-3776	84	17	and	and	CCONJ
ejpam-3776	84	18	a	a	DET
ejpam-3776	84	19	n×m	n×m	PROPN
ejpam-3776	84	20	matrix	matrix	NOUN
ejpam-3776	84	21	x	x	PUNCT
ejpam-3776	85	1	=	=	SYM
ejpam-3776	85	2	(	(	PUNCT
ejpam-3776	85	3	xij	xij	X
ejpam-3776	85	4	)	)	PUNCT
ejpam-3776	85	5	∈	∈	PROPN
ejpam-3776	85	6	jn,∀i	jn,∀i	PROPN
ejpam-3776	85	7	∈	∈	PROPN
ejpam-3776	85	8	{	{	PUNCT
ejpam-3776	85	9	1	1	NUM
ejpam-3776	85	10	,	,	PUNCT
ejpam-3776	85	11	.	.	PUNCT
ejpam-3776	85	12	.	.	PUNCT
ejpam-3776	85	13	.	.	PUNCT
ejpam-3776	85	14	,	,	PUNCT
ejpam-3776	85	15	n	n	CCONJ
ejpam-3776	85	16	}	}	PUNCT
ejpam-3776	85	17	and	and	CCONJ
ejpam-3776	85	18	∀j	∀j	PROPN
ejpam-3776	85	19	∈	∈	PROPN
ejpam-3776	85	20	{	{	PUNCT
ejpam-3776	85	21	1	1	NUM
ejpam-3776	85	22	,	,	PUNCT
ejpam-3776	85	23	.	.	PUNCT
ejpam-3776	85	24	.	.	PUNCT
ejpam-3776	85	25	.	.	PUNCT
ejpam-3776	86	1	,	,	PUNCT
ejpam-3776	86	2	m	m	VERB
ejpam-3776	86	3	}	}	PUNCT
ejpam-3776	86	4	such	such	ADJ
ejpam-3776	86	5	that	that	SCONJ
ejpam-3776	86	6	(	(	PUNCT
ejpam-3776	86	7	xi1+k	xi1+k	PROPN
ejpam-3776	86	8	,	,	PUNCT
ejpam-3776	86	9	.	.	PUNCT
ejpam-3776	86	10	.	.	PUNCT
ejpam-3776	87	1	.	.	PUNCT
ejpam-3776	88	1	,	,	PUNCT
ejpam-3776	88	2	xim+k	xim+k	PROPN
ejpam-3776	88	3	)	)	PUNCT
ejpam-3776	88	4	=	=	SYM
ejpam-3776	88	5	(	(	PUNCT
ejpam-3776	88	6	xi1	xi1	PROPN
ejpam-3776	88	7	,	,	PUNCT
ejpam-3776	88	8	.	.	PUNCT
ejpam-3776	88	9	.	.	PUNCT
ejpam-3776	89	1	.	.	PUNCT
ejpam-3776	90	1	,	,	PUNCT
ejpam-3776	90	2	xim),∀i	xim),∀i	PROPN
ejpam-3776	90	3	∈	∈	PROPN
ejpam-3776	90	4	{	{	PUNCT
ejpam-3776	90	5	1	1	NUM
ejpam-3776	90	6	,	,	PUNCT
ejpam-3776	90	7	.	.	PUNCT
ejpam-3776	90	8	.	.	PUNCT
ejpam-3776	90	9	.	.	PUNCT
ejpam-3776	90	10	,	,	PUNCT
ejpam-3776	90	11	n	n	CCONJ
ejpam-3776	90	12	}	}	PUNCT
ejpam-3776	90	13	)	)	PUNCT
ejpam-3776	90	14	and	and	CCONJ
ejpam-3776	90	15	λ	λ	X
ejpam-3776	90	16	:	:	PUNCT
ejpam-3776	90	17	(	(	PUNCT
ejpam-3776	90	18	λ1	λ1	ADJ
ejpam-3776	90	19	,	,	PUNCT
ejpam-3776	90	20	.	.	PUNCT
ejpam-3776	90	21	.	.	PUNCT
ejpam-3776	90	22	.	.	PUNCT
ejpam-3776	91	1	,	,	PUNCT
ejpam-3776	91	2	λn	λn	NOUN
ejpam-3776	91	3	)	)	PUNCT
ejpam-3776	91	4	ba	ba	PROPN
ejpam-3776	91	5	a	a	DET
ejpam-3776	91	6	n	n	NOUN
ejpam-3776	91	7	-	-	PUNCT
ejpam-3776	91	8	tuple	tuple	NOUN
ejpam-3776	91	9	such	such	ADJ
ejpam-3776	91	10	that	that	SCONJ
ejpam-3776	91	11	∑l	∑l	PROPN
ejpam-3776	92	1	k=1	k=1	X
ejpam-3776	92	2	λk	λk	X
ejpam-3776	92	3	=	=	NOUN
ejpam-3776	92	4	1	1	NUM
ejpam-3776	92	5	,	,	PUNCT
ejpam-3776	92	6	l	l	NOUN
ejpam-3776	92	7	∈	∈	PROPN
ejpam-3776	92	8	{	{	PUNCT
ejpam-3776	92	9	2	2	NUM
ejpam-3776	92	10	,	,	PUNCT
ejpam-3776	92	11	.	.	PUNCT
ejpam-3776	92	12	.	.	PUNCT
ejpam-3776	92	13	.	.	PUNCT
ejpam-3776	92	14	,	,	PUNCT
ejpam-3776	92	15	n	n	CCONJ
ejpam-3776	92	16	}	}	PUNCT
ejpam-3776	92	17	.	.	PUNCT
ejpam-3776	93	1	moreover	moreover	ADV
ejpam-3776	93	2	,	,	PUNCT
ejpam-3776	93	3	wi	wi	PROPN
ejpam-3776	93	4	’s	’s	PART
ejpam-3776	93	5	are	be	AUX
ejpam-3776	93	6	non	non	ADJ
ejpam-3776	93	7	-	-	ADJ
ejpam-3776	93	8	negative	negative	ADJ
ejpam-3776	93	9	real	real	ADJ
ejpam-3776	93	10	weights	weight	NOUN
ejpam-3776	93	11	for	for	ADP
ejpam-3776	93	12	1	1	NUM
ejpam-3776	93	13	≤	≤	NUM
ejpam-3776	93	14	i	i	PRON
ejpam-3776	93	15	≤	≤	ADJ
ejpam-3776	94	1	n	n	CCONJ
ejpam-3776	94	2	such	such	ADJ
ejpam-3776	94	3	that	that	SCONJ
ejpam-3776	94	4	∑n	∑n	PROPN
ejpam-3776	94	5	i=1wi	i=1wi	X
ejpam-3776	95	1	=	=	SYM
ejpam-3776	95	2	1	1	NUM
ejpam-3776	95	3	•	•	NOUN
ejpam-3776	95	4	(	(	PUNCT
ejpam-3776	95	5	c3	c3	NOUN
ejpam-3776	95	6	):	):	PUNCT
ejpam-3776	95	7	let	let	VERB
ejpam-3776	95	8	φ	φ	NUM
ejpam-3776	95	9	,	,	PUNCT
ejpam-3776	95	10	ψ	ψ	X
ejpam-3776	95	11	:	:	PUNCT
ejpam-3776	95	12	j	j	PROPN
ejpam-3776	95	13	→	→	PUNCT
ejpam-3776	95	14	r	r	NOUN
ejpam-3776	95	15	be	be	AUX
ejpam-3776	95	16	continuous	continuous	ADJ
ejpam-3776	95	17	and	and	CCONJ
ejpam-3776	95	18	strictly	strictly	ADV
ejpam-3776	95	19	monotone	monotone	ADJ
ejpam-3776	95	20	functions	function	NOUN
ejpam-3776	95	21	.	.	PUNCT
ejpam-3776	96	1	under	under	ADP
ejpam-3776	96	2	the	the	DET
ejpam-3776	96	3	assumptions	assumption	NOUN
ejpam-3776	96	4	stated	state	VERB
ejpam-3776	96	5	above	above	ADP
ejpam-3776	96	6	it	it	PRON
ejpam-3776	96	7	should	should	AUX
ejpam-3776	96	8	be	be	AUX
ejpam-3776	96	9	noted	note	VERB
ejpam-3776	96	10	that	that	NOUN
ejpam-3776	96	11	m∑	m∑	ADP
ejpam-3776	97	1	j=1	j=1	PROPN
ejpam-3776	97	2	aj	aj	PROPN
ejpam-3776	97	3	−	−	PROPN
ejpam-3776	97	4	m−1∑	m−1∑	NUM
ejpam-3776	97	5	j=1	j=1	NOUN
ejpam-3776	97	6	l−1∑	l−1∑	PRON
ejpam-3776	97	7	k=0	k=0	PROPN
ejpam-3776	97	8	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	97	9			PROPN
ejpam-3776	97	10	∈	∈	PROPN
ejpam-3776	97	11	j.	j.	PROPN
ejpam-3776	97	12	the	the	DET
ejpam-3776	97	13	aim	aim	NOUN
ejpam-3776	97	14	of	of	ADP
ejpam-3776	97	15	this	this	DET
ejpam-3776	97	16	paper	paper	NOUN
ejpam-3776	97	17	is	be	AUX
ejpam-3776	97	18	to	to	PART
ejpam-3776	97	19	present	present	VERB
ejpam-3776	97	20	new	new	ADJ
ejpam-3776	97	21	refinement	refinement	NOUN
ejpam-3776	97	22	of	of	ADP
ejpam-3776	97	23	theorem	theorem	NOUN
ejpam-3776	97	24	3	3	NUM
ejpam-3776	97	25	.	.	PUNCT
ejpam-3776	97	26	in	in	ADP
ejpam-3776	97	27	main	main	ADJ
ejpam-3776	97	28	result	result	NOUN
ejpam-3776	97	29	section	section	NOUN
ejpam-3776	97	30	,	,	PUNCT
ejpam-3776	97	31	we	we	PRON
ejpam-3776	97	32	will	will	AUX
ejpam-3776	97	33	give	give	VERB
ejpam-3776	97	34	refinement	refinement	NOUN
ejpam-3776	97	35	for	for	ADP
ejpam-3776	97	36	weighted	weighted	ADJ
ejpam-3776	97	37	version	version	NOUN
ejpam-3776	97	38	of	of	ADP
ejpam-3776	97	39	niezgoda	niezgoda	PROPN
ejpam-3776	97	40	’s	’s	PART
ejpam-3776	97	41	inequality	inequality	NOUN
ejpam-3776	97	42	,	,	PUNCT
ejpam-3776	97	43	then	then	ADV
ejpam-3776	97	44	we	we	PRON
ejpam-3776	97	45	will	will	AUX
ejpam-3776	97	46	define	define	VERB
ejpam-3776	97	47	its	its	PRON
ejpam-3776	97	48	special	special	ADJ
ejpam-3776	97	49	case	case	NOUN
ejpam-3776	97	50	for	for	ADP
ejpam-3776	97	51	equal	equal	ADJ
ejpam-3776	97	52	weights	weight	NOUN
ejpam-3776	97	53	.	.	PUNCT
ejpam-3776	98	1	in	in	ADP
ejpam-3776	98	2	application	application	NOUN
ejpam-3776	98	3	section	section	NOUN
ejpam-3776	98	4	,	,	PUNCT
ejpam-3776	98	5	with	with	ADP
ejpam-3776	98	6	the	the	DET
ejpam-3776	98	7	help	help	NOUN
ejpam-3776	98	8	of	of	ADP
ejpam-3776	98	9	main	main	ADJ
ejpam-3776	98	10	results	result	NOUN
ejpam-3776	98	11	we	we	PRON
ejpam-3776	98	12	will	will	AUX
ejpam-3776	98	13	give	give	VERB
ejpam-3776	98	14	refinements	refinement	NOUN
ejpam-3776	98	15	of	of	ADP
ejpam-3776	98	16	ky	ky	PROPN
ejpam-3776	98	17	fan	fan	PROPN
ejpam-3776	98	18	and	and	CCONJ
ejpam-3776	98	19	arithmetic	arithmetic	ADJ
ejpam-3776	98	20	-	-	PUNCT
ejpam-3776	98	21	geometric	geometric	ADJ
ejpam-3776	98	22	means	mean	NOUN
ejpam-3776	98	23	inequalities	inequality	NOUN
ejpam-3776	98	24	and	and	CCONJ
ejpam-3776	98	25	their	their	PRON
ejpam-3776	98	26	related	related	ADJ
ejpam-3776	98	27	results	result	NOUN
ejpam-3776	98	28	.	.	PUNCT
ejpam-3776	99	1	we	we	PRON
ejpam-3776	99	2	also	also	ADV
ejpam-3776	99	3	define	define	VERB
ejpam-3776	99	4	cyclic	cyclic	ADJ
ejpam-3776	99	5	mixed	mixed	ADJ
ejpam-3776	99	6	symmetric	symmetric	ADJ
ejpam-3776	99	7	means	mean	NOUN
ejpam-3776	99	8	,	,	PUNCT
ejpam-3776	99	9	power	power	NOUN
ejpam-3776	99	10	mean	mean	NOUN
ejpam-3776	99	11	and	and	CCONJ
ejpam-3776	99	12	generalized	generalized	ADJ
ejpam-3776	99	13	quasi	quasi	ADJ
ejpam-3776	99	14	-	-	ADJ
ejpam-3776	99	15	arithmetic	arithmetic	ADJ
ejpam-3776	99	16	means	mean	NOUN
ejpam-3776	99	17	and	and	CCONJ
ejpam-3776	99	18	study	study	VERB
ejpam-3776	99	19	their	their	PRON
ejpam-3776	99	20	properties	property	NOUN
ejpam-3776	99	21	.	.	PUNCT
ejpam-3776	100	1	we	we	PRON
ejpam-3776	100	2	follow	follow	VERB
ejpam-3776	100	3	the	the	DET
ejpam-3776	100	4	techniques	technique	NOUN
ejpam-3776	100	5	given	give	VERB
ejpam-3776	100	6	in	in	ADP
ejpam-3776	100	7	[	[	X
ejpam-3776	100	8	3	3	NUM
ejpam-3776	100	9	]	]	PUNCT
ejpam-3776	100	10	.	.	PUNCT
ejpam-3776	101	1	the	the	DET
ejpam-3776	101	2	final	final	ADJ
ejpam-3776	101	3	section	section	NOUN
ejpam-3776	101	4	gives	give	VERB
ejpam-3776	101	5	suggestions	suggestion	NOUN
ejpam-3776	101	6	for	for	ADP
ejpam-3776	101	7	further	further	ADJ
ejpam-3776	101	8	work	work	NOUN
ejpam-3776	101	9	and	and	CCONJ
ejpam-3776	101	10	future	future	ADJ
ejpam-3776	101	11	ideas	idea	NOUN
ejpam-3776	101	12	.	.	PUNCT
ejpam-3776	102	1	s.	s.	PROPN
ejpam-3776	102	2	chanan	chanan	PROPN
ejpam-3776	102	3	,	,	PUNCT
ejpam-3776	102	4	a.	a.	PROPN
ejpam-3776	102	5	r.	r.	PROPN
ejpam-3776	102	6	khan	khan	PROPN
ejpam-3776	102	7	/	/	SYM
ejpam-3776	102	8	eur	eur	PROPN
ejpam-3776	102	9	.	.	PUNCT
ejpam-3776	103	1	j.	j.	PROPN
ejpam-3776	103	2	pure	pure	PROPN
ejpam-3776	103	3	appl	appl	PROPN
ejpam-3776	103	4	.	.	PROPN
ejpam-3776	103	5	math	math	PROPN
ejpam-3776	103	6	,	,	PUNCT
ejpam-3776	103	7	13	13	NUM
ejpam-3776	103	8	(	(	PUNCT
ejpam-3776	103	9	4	4	NUM
ejpam-3776	103	10	)	)	PUNCT
ejpam-3776	103	11	(	(	PUNCT
ejpam-3776	103	12	2020	2020	NUM
ejpam-3776	103	13	)	)	PUNCT
ejpam-3776	103	14	,	,	PUNCT
ejpam-3776	103	15	814	814	NUM
ejpam-3776	103	16	-	-	SYM
ejpam-3776	103	17	829	829	NUM
ejpam-3776	103	18	817	817	NUM
ejpam-3776	103	19	2	2	NUM
ejpam-3776	103	20	.	.	PUNCT
ejpam-3776	103	21	main	main	ADJ
ejpam-3776	103	22	result	result	NOUN
ejpam-3776	103	23	theorem	theorem	VERB
ejpam-3776	103	24	6	6	NUM
ejpam-3776	103	25	.	.	PUNCT
ejpam-3776	104	1	let	let	VERB
ejpam-3776	104	2	the	the	DET
ejpam-3776	104	3	assumptions	assumption	NOUN
ejpam-3776	104	4	stated	state	VERB
ejpam-3776	104	5	in	in	ADP
ejpam-3776	104	6	theorem	theorem	ADJ
ejpam-3776	104	7	3	3	NUM
ejpam-3776	104	8	be	be	AUX
ejpam-3776	104	9	true	true	ADJ
ejpam-3776	104	10	.	.	PUNCT
ejpam-3776	105	1	in	in	ADP
ejpam-3776	105	2	addition	addition	NOUN
ejpam-3776	105	3	we	we	PRON
ejpam-3776	105	4	suppose	suppose	VERB
ejpam-3776	105	5	that	that	SCONJ
ejpam-3776	105	6	the	the	DET
ejpam-3776	105	7	assumptions	assumption	NOUN
ejpam-3776	105	8	given	give	VERB
ejpam-3776	105	9	in	in	ADP
ejpam-3776	105	10	(	(	PUNCT
ejpam-3776	105	11	c1	c1	PROPN
ejpam-3776	105	12	)	)	PUNCT
ejpam-3776	105	13	and	and	CCONJ
ejpam-3776	105	14	(	(	PUNCT
ejpam-3776	105	15	c2	c2	PROPN
ejpam-3776	105	16	)	)	PUNCT
ejpam-3776	105	17	are	be	AUX
ejpam-3776	105	18	also	also	ADV
ejpam-3776	105	19	valid	valid	ADJ
ejpam-3776	105	20	.	.	PUNCT
ejpam-3776	106	1	then	then	ADV
ejpam-3776	106	2	we	we	PRON
ejpam-3776	106	3	have	have	VERB
ejpam-3776	106	4	φ	φ	VERB
ejpam-3776	106	5			PROPN
ejpam-3776	106	6	m∑	m∑	PROPN
ejpam-3776	106	7	j=1	j=1	PROPN
ejpam-3776	106	8	aj	aj	PROPN
ejpam-3776	106	9	−	−	PROPN
ejpam-3776	106	10	m−1∑	m−1∑	NUM
ejpam-3776	107	1	j=1	j=1	PROPN
ejpam-3776	107	2	n∑	n∑	PROPN
ejpam-3776	107	3	i=1	i=1	PROPN
ejpam-3776	108	1	wixij	wixij	PROPN
ejpam-3776	108	2			PROPN
ejpam-3776	109	1	≤	≤	PROPN
ejpam-3776	110	1	n∑	n∑	PROPN
ejpam-3776	110	2	i=1	i=1	PROPN
ejpam-3776	110	3	wiφ	wiφ	VERB
ejpam-3776	110	4			PROPN
ejpam-3776	110	5	m∑	m∑	PROPN
ejpam-3776	110	6	j=1	j=1	PROPN
ejpam-3776	110	7	aj	aj	PROPN
ejpam-3776	110	8	−	−	PROPN
ejpam-3776	110	9	m−1∑	m−1∑	NUM
ejpam-3776	110	10	j=1	j=1	NOUN
ejpam-3776	110	11	l−1∑	l−1∑	PRON
ejpam-3776	110	12	k=0	k=0	PROPN
ejpam-3776	110	13	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	110	14			PROPN
ejpam-3776	110	15	≤	≤	NUM
ejpam-3776	110	16	m∑	m∑	CCONJ
ejpam-3776	110	17	j=1	j=1	ADJ
ejpam-3776	110	18	φ(aj)−	φ(aj)−	NOUN
ejpam-3776	110	19	m−1∑	m−1∑	PROPN
ejpam-3776	111	1	j=1	j=1	PROPN
ejpam-3776	111	2	n∑	n∑	PROPN
ejpam-3776	111	3	i=1	i=1	PROPN
ejpam-3776	111	4	wiφ	wiφ	INTJ
ejpam-3776	111	5	(	(	PUNCT
ejpam-3776	111	6	xij	xij	X
ejpam-3776	111	7	)	)	PUNCT
ejpam-3776	111	8	.	.	PUNCT
ejpam-3776	112	1	(	(	PUNCT
ejpam-3776	112	2	6	6	X
ejpam-3776	112	3	)	)	PUNCT
ejpam-3776	112	4	proof	proof	NOUN
ejpam-3776	112	5	.	.	PUNCT
ejpam-3776	113	1	to	to	PART
ejpam-3776	113	2	prove	prove	VERB
ejpam-3776	113	3	first	first	ADJ
ejpam-3776	113	4	inequality	inequality	NOUN
ejpam-3776	113	5	of	of	ADP
ejpam-3776	113	6	(	(	PUNCT
ejpam-3776	113	7	6	6	NUM
ejpam-3776	113	8	)	)	PUNCT
ejpam-3776	113	9	,	,	PUNCT
ejpam-3776	113	10	since	since	SCONJ
ejpam-3776	113	11	φ	φ	PROPN
ejpam-3776	113	12	is	be	AUX
ejpam-3776	113	13	a	a	DET
ejpam-3776	113	14	convex	convex	NOUN
ejpam-3776	113	15	function	function	NOUN
ejpam-3776	113	16	and	and	PROPN
ejpam-3776	113	17	m∑	m∑	ADP
ejpam-3776	113	18	j=1	j=1	PROPN
ejpam-3776	113	19	aj	aj	PROPN
ejpam-3776	113	20	−	−	PROPN
ejpam-3776	113	21	m−1∑	m−1∑	NUM
ejpam-3776	113	22	j=1	j=1	NOUN
ejpam-3776	113	23	l−1∑	l−1∑	PRON
ejpam-3776	113	24	k=0	k=0	PROPN
ejpam-3776	113	25	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	113	26			PROPN
ejpam-3776	113	27	∈	∈	PROPN
ejpam-3776	113	28	j	j	PROPN
ejpam-3776	113	29	,	,	PUNCT
ejpam-3776	113	30	therefore	therefore	ADV
ejpam-3776	113	31	by	by	ADP
ejpam-3776	113	32	jensen	jensen	PROPN
ejpam-3776	113	33	’s	’s	PART
ejpam-3776	113	34	inequality	inequality	NOUN
ejpam-3776	113	35	,	,	PUNCT
ejpam-3776	113	36	n∑	n∑	PROPN
ejpam-3776	113	37	i=1	i=1	PROPN
ejpam-3776	113	38	wiφ	wiφ	VERB
ejpam-3776	113	39			PROPN
ejpam-3776	113	40	m∑	m∑	PROPN
ejpam-3776	113	41	j=1	j=1	PROPN
ejpam-3776	113	42	aj	aj	PROPN
ejpam-3776	113	43	−	−	PROPN
ejpam-3776	113	44	m−1∑	m−1∑	NUM
ejpam-3776	113	45	j=1	j=1	NOUN
ejpam-3776	113	46	l−1∑	l−1∑	PRON
ejpam-3776	113	47	k=0	k=0	PROPN
ejpam-3776	113	48	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	113	49			PROPN
ejpam-3776	113	50	≥	≥	NOUN
ejpam-3776	113	51	φ	φ	NUM
ejpam-3776	113	52			PROPN
ejpam-3776	113	53	n∑	n∑	PROPN
ejpam-3776	113	54	i=1	i=1	PROPN
ejpam-3776	113	55	wi	wi	PROPN
ejpam-3776	114	1	m∑	m∑	VERB
ejpam-3776	114	2	j=1	j=1	PROPN
ejpam-3776	114	3	aj	aj	PROPN
ejpam-3776	115	1	−	−	PROPN
ejpam-3776	115	2	n∑	n∑	PROPN
ejpam-3776	115	3	i=1	i=1	PROPN
ejpam-3776	116	1	m−1∑	m−1∑	NUM
ejpam-3776	116	2	j=1	j=1	NOUN
ejpam-3776	116	3	l−1∑	l−1∑	PRON
ejpam-3776	116	4	k=0	k=0	PROPN
ejpam-3776	116	5	wiλk+1xij+k	wiλk+1xij+k	VERB
ejpam-3776	116	6			PROPN
ejpam-3776	116	7	=	=	SYM
ejpam-3776	116	8	φ	φ	PROPN
ejpam-3776	116	9			PROPN
ejpam-3776	116	10	m∑	m∑	PROPN
ejpam-3776	116	11	j=1	j=1	PROPN
ejpam-3776	116	12	aj	aj	PROPN
ejpam-3776	116	13	−	−	PROPN
ejpam-3776	116	14	(	(	PUNCT
ejpam-3776	116	15	l∑	l∑	AUX
ejpam-3776	116	16	k=1	k=1	X
ejpam-3776	116	17	λk	λk	X
ejpam-3776	116	18	)	)	PUNCT
ejpam-3776	117	1	m−1∑	m−1∑	PROPN
ejpam-3776	117	2	j=1	j=1	PROPN
ejpam-3776	117	3	n∑	n∑	PROPN
ejpam-3776	117	4	i=1	i=1	PROPN
ejpam-3776	118	1	wixij	wixij	NOUN
ejpam-3776	118	2			PROPN
ejpam-3776	118	3	=	=	SYM
ejpam-3776	118	4	φ	φ	PROPN
ejpam-3776	118	5			PROPN
ejpam-3776	118	6	m∑	m∑	PROPN
ejpam-3776	118	7	j=1	j=1	PROPN
ejpam-3776	118	8	aj	aj	PROPN
ejpam-3776	118	9	−	−	PROPN
ejpam-3776	118	10	m−1∑	m−1∑	NUM
ejpam-3776	119	1	j=1	j=1	PROPN
ejpam-3776	119	2	n∑	n∑	PROPN
ejpam-3776	119	3	i=1	i=1	PROPN
ejpam-3776	120	1	wixij	wixij	PROPN
ejpam-3776	120	2			PROPN
ejpam-3776	120	3	.	.	PUNCT
ejpam-3776	121	1	on	on	ADP
ejpam-3776	121	2	the	the	DET
ejpam-3776	121	3	other	other	ADJ
ejpam-3776	121	4	hand	hand	NOUN
ejpam-3776	121	5	,	,	PUNCT
ejpam-3776	121	6	to	to	PART
ejpam-3776	121	7	prove	prove	VERB
ejpam-3776	121	8	second	second	ADJ
ejpam-3776	121	9	inequality	inequality	NOUN
ejpam-3776	121	10	of	of	ADP
ejpam-3776	121	11	(	(	PUNCT
ejpam-3776	121	12	6	6	NUM
ejpam-3776	121	13	)	)	PUNCT
ejpam-3776	121	14	,	,	PUNCT
ejpam-3776	121	15	we	we	PRON
ejpam-3776	121	16	consider	consider	VERB
ejpam-3776	121	17	following	follow	VERB
ejpam-3776	121	18	expression	expression	NOUN
ejpam-3776	121	19	φ	φ	NOUN
ejpam-3776	122	1			PROPN
ejpam-3776	122	2	m∑	m∑	PROPN
ejpam-3776	122	3	j=1	j=1	PROPN
ejpam-3776	122	4	aj	aj	PROPN
ejpam-3776	122	5	−	−	PROPN
ejpam-3776	122	6	m−1∑	m−1∑	NUM
ejpam-3776	122	7	j=1	j=1	NOUN
ejpam-3776	122	8	l−1∑	l−1∑	PRON
ejpam-3776	122	9	k=0	k=0	PROPN
ejpam-3776	122	10	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	122	11			PROPN
ejpam-3776	122	12	for	for	ADP
ejpam-3776	122	13	fixed	fix	VERB
ejpam-3776	122	14	i	i	PRON
ejpam-3776	122	15	∈	∈	PROPN
ejpam-3776	122	16	{	{	PUNCT
ejpam-3776	122	17	1	1	NUM
ejpam-3776	122	18	,	,	PUNCT
ejpam-3776	122	19	2	2	NUM
ejpam-3776	122	20	,	,	PUNCT
ejpam-3776	122	21	.	.	PUNCT
ejpam-3776	122	22	.	.	PUNCT
ejpam-3776	123	1	.	.	PUNCT
ejpam-3776	124	1	,	,	PUNCT
ejpam-3776	125	1	n	n	CCONJ
ejpam-3776	125	2	}	}	PUNCT
ejpam-3776	126	1	and	and	CCONJ
ejpam-3776	126	2	proceed	proceed	VERB
ejpam-3776	126	3	as	as	SCONJ
ejpam-3776	126	4	follows	follow	VERB
ejpam-3776	126	5	:	:	PUNCT
ejpam-3776	126	6	φ	φ	PROPN
ejpam-3776	126	7			PROPN
ejpam-3776	126	8	m∑	m∑	PROPN
ejpam-3776	126	9	j=1	j=1	PROPN
ejpam-3776	126	10	aj	aj	PROPN
ejpam-3776	126	11	−	−	PROPN
ejpam-3776	127	1	m−1∑	m−1∑	NUM
ejpam-3776	127	2	j=1	j=1	NOUN
ejpam-3776	127	3	l−1∑	l−1∑	PRON
ejpam-3776	127	4	k=0	k=0	PROPN
ejpam-3776	127	5	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	127	6			PROPN
ejpam-3776	127	7	s.	s.	PROPN
ejpam-3776	127	8	chanan	chanan	PROPN
ejpam-3776	127	9	,	,	PUNCT
ejpam-3776	127	10	a.	a.	PROPN
ejpam-3776	127	11	r.	r.	PROPN
ejpam-3776	127	12	khan	khan	PROPN
ejpam-3776	127	13	/	/	SYM
ejpam-3776	127	14	eur	eur	PROPN
ejpam-3776	127	15	.	.	PUNCT
ejpam-3776	128	1	j.	j.	PROPN
ejpam-3776	128	2	pure	pure	PROPN
ejpam-3776	128	3	appl	appl	PROPN
ejpam-3776	128	4	.	.	PROPN
ejpam-3776	128	5	math	math	PROPN
ejpam-3776	128	6	,	,	PUNCT
ejpam-3776	128	7	13	13	NUM
ejpam-3776	128	8	(	(	PUNCT
ejpam-3776	128	9	4	4	NUM
ejpam-3776	128	10	)	)	PUNCT
ejpam-3776	128	11	(	(	PUNCT
ejpam-3776	128	12	2020	2020	NUM
ejpam-3776	128	13	)	)	PUNCT
ejpam-3776	128	14	,	,	PUNCT
ejpam-3776	128	15	814	814	NUM
ejpam-3776	128	16	-	-	SYM
ejpam-3776	128	17	829	829	NUM
ejpam-3776	128	18	818	818	NUM
ejpam-3776	128	19	=	=	SYM
ejpam-3776	128	20	φ	φ	PROPN
ejpam-3776	128	21			PROPN
ejpam-3776	128	22	m∑	m∑	PROPN
ejpam-3776	128	23	j=1	j=1	PROPN
ejpam-3776	128	24	aj	aj	PROPN
ejpam-3776	129	1	−	−	PROPN
ejpam-3776	129	2	m−1∑	m−1∑	NUM
ejpam-3776	129	3	j=1	j=1	NOUN
ejpam-3776	129	4	l−1∑	l−1∑	ADP
ejpam-3776	129	5	k=0	k=0	PUNCT
ejpam-3776	129	6	λk+1xij	λk+1xij	X
ejpam-3776	129	7			PROPN
ejpam-3776	129	8	=	=	SYM
ejpam-3776	129	9	φ	φ	PROPN
ejpam-3776	129	10			PROPN
ejpam-3776	129	11	m∑	m∑	PROPN
ejpam-3776	129	12	j=1	j=1	PROPN
ejpam-3776	129	13	aj	aj	PROPN
ejpam-3776	129	14	−	−	PROPN
ejpam-3776	129	15	m−1∑	m−1∑	INTJ
ejpam-3776	129	16	j=1	j=1	NOUN
ejpam-3776	129	17	l∑	l∑	PUNCT
ejpam-3776	130	1	k=1	k=1	PUNCT
ejpam-3776	130	2	λkxij	λkxij	PROPN
ejpam-3776	130	3			PROPN
ejpam-3776	130	4	=	=	SYM
ejpam-3776	130	5	φ	φ	PROPN
ejpam-3776	130	6			PROPN
ejpam-3776	130	7	m∑	m∑	PROPN
ejpam-3776	130	8	j=1	j=1	PROPN
ejpam-3776	130	9	aj	aj	PROPN
ejpam-3776	130	10	−	−	PROPN
ejpam-3776	130	11	(	(	PUNCT
ejpam-3776	130	12	l∑	l∑	AUX
ejpam-3776	131	1	k=1	k=1	X
ejpam-3776	131	2	λk	λk	X
ejpam-3776	131	3	)	)	PUNCT
ejpam-3776	132	1	m−1∑	m−1∑	PROPN
ejpam-3776	132	2	j=1	j=1	PROPN
ejpam-3776	132	3	xij	xij	PROPN
ejpam-3776	132	4			PROPN
ejpam-3776	132	5	=	=	SYM
ejpam-3776	132	6	φ	φ	PROPN
ejpam-3776	132	7			PROPN
ejpam-3776	132	8	m∑	m∑	PROPN
ejpam-3776	132	9	j=1	j=1	PROPN
ejpam-3776	132	10	aj	aj	PROPN
ejpam-3776	132	11	−	−	PROPN
ejpam-3776	132	12	m−1∑	m−1∑	NUM
ejpam-3776	132	13	j=1	j=1	NOUN
ejpam-3776	132	14	xij	xij	PROPN
ejpam-3776	132	15			PROPN
ejpam-3776	132	16	using	use	VERB
ejpam-3776	132	17	majorization	majorization	NOUN
ejpam-3776	132	18	property	property	NOUN
ejpam-3776	132	19	we	we	PRON
ejpam-3776	132	20	have	have	VERB
ejpam-3776	132	21	φ	φ	VERB
ejpam-3776	132	22			PROPN
ejpam-3776	132	23	m∑	m∑	PROPN
ejpam-3776	132	24	j=1	j=1	PROPN
ejpam-3776	132	25	aj	aj	PROPN
ejpam-3776	132	26	−	−	PROPN
ejpam-3776	132	27	m−1∑	m−1∑	NUM
ejpam-3776	132	28	j=1	j=1	PROPN
ejpam-3776	132	29	xij	xij	PROPN
ejpam-3776	132	30			PROPN
ejpam-3776	132	31	=	=	SYM
ejpam-3776	132	32	φ	φ	PROPN
ejpam-3776	132	33	(	(	PUNCT
ejpam-3776	132	34	xim	xim	PROPN
ejpam-3776	132	35	)	)	PUNCT
ejpam-3776	132	36	≤	≤	NOUN
ejpam-3776	132	37	m∑	m∑	VERB
ejpam-3776	132	38	j=1	j=1	PROPN
ejpam-3776	132	39	φ	φ	PROPN
ejpam-3776	132	40	(	(	PUNCT
ejpam-3776	132	41	aj)−	aj)−	NOUN
ejpam-3776	132	42	m−1∑	m−1∑	PROPN
ejpam-3776	132	43	j=1	j=1	PROPN
ejpam-3776	132	44	φ	φ	X
ejpam-3776	132	45	(	(	PUNCT
ejpam-3776	132	46	xij	xij	X
ejpam-3776	132	47	)	)	PUNCT
ejpam-3776	132	48	or	or	CCONJ
ejpam-3776	132	49	we	we	PRON
ejpam-3776	132	50	can	can	AUX
ejpam-3776	132	51	write	write	VERB
ejpam-3776	132	52	,	,	PUNCT
ejpam-3776	132	53	φ	φ	PROPN
ejpam-3776	132	54			PROPN
ejpam-3776	132	55	m∑	m∑	PROPN
ejpam-3776	132	56	j=1	j=1	PROPN
ejpam-3776	132	57	aj	aj	PROPN
ejpam-3776	132	58	−	−	PROPN
ejpam-3776	132	59	m−1∑	m−1∑	NUM
ejpam-3776	132	60	j=1	j=1	NOUN
ejpam-3776	132	61	l−1∑	l−1∑	PRON
ejpam-3776	132	62	k=0	k=0	PROPN
ejpam-3776	132	63	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	132	64			PROPN
ejpam-3776	132	65	≤	≤	NUM
ejpam-3776	132	66	m∑	m∑	CCONJ
ejpam-3776	132	67	j=1	j=1	PROPN
ejpam-3776	132	68	φ	φ	PROPN
ejpam-3776	132	69	(	(	PUNCT
ejpam-3776	132	70	aj)−	aj)−	NOUN
ejpam-3776	132	71	m−1∑	m−1∑	PROPN
ejpam-3776	132	72	j=1	j=1	PROPN
ejpam-3776	132	73	φ	φ	PROPN
ejpam-3776	132	74	(	(	PUNCT
ejpam-3776	132	75	xij	xij	X
ejpam-3776	132	76	)	)	PUNCT
ejpam-3776	132	77	.	.	PUNCT
ejpam-3776	133	1	(	(	PUNCT
ejpam-3776	133	2	7	7	X
ejpam-3776	133	3	)	)	PUNCT
ejpam-3776	133	4	now	now	ADV
ejpam-3776	133	5	multiplying	multiply	VERB
ejpam-3776	133	6	inequality	inequality	NOUN
ejpam-3776	133	7	(	(	PUNCT
ejpam-3776	133	8	7	7	NUM
ejpam-3776	133	9	)	)	PUNCT
ejpam-3776	133	10	with	with	ADP
ejpam-3776	133	11	wi	wi	PROPN
ejpam-3776	133	12	and	and	CCONJ
ejpam-3776	133	13	summing	sum	VERB
ejpam-3776	133	14	over	over	ADP
ejpam-3776	133	15	i	i	PRON
ejpam-3776	133	16	from	from	ADP
ejpam-3776	133	17	1	1	NUM
ejpam-3776	133	18	to	to	ADP
ejpam-3776	133	19	n	n	PRON
ejpam-3776	133	20	we	we	PRON
ejpam-3776	133	21	get	get	VERB
ejpam-3776	133	22	our	our	PRON
ejpam-3776	133	23	required	require	VERB
ejpam-3776	133	24	result	result	NOUN
ejpam-3776	133	25	.	.	PUNCT
ejpam-3776	134	1	corollary	corollary	ADJ
ejpam-3776	134	2	1	1	NUM
ejpam-3776	134	3	.	.	PUNCT
ejpam-3776	135	1	under	under	ADP
ejpam-3776	135	2	the	the	DET
ejpam-3776	135	3	assumptions	assumption	NOUN
ejpam-3776	135	4	of	of	ADP
ejpam-3776	135	5	theorem	theorem	NOUN
ejpam-3776	135	6	6	6	NUM
ejpam-3776	135	7	and	and	CCONJ
ejpam-3776	135	8	for	for	ADP
ejpam-3776	135	9	wi	wi	PROPN
ejpam-3776	135	10	=	=	SYM
ejpam-3776	135	11	1	1	NUM
ejpam-3776	135	12	n	n	NOUN
ejpam-3776	135	13	,	,	PUNCT
ejpam-3776	135	14	i	i	PRON
ejpam-3776	135	15	∈	∈	PROPN
ejpam-3776	135	16	{	{	PUNCT
ejpam-3776	135	17	1	1	NUM
ejpam-3776	135	18	,	,	PUNCT
ejpam-3776	135	19	.	.	PUNCT
ejpam-3776	135	20	.	.	PUNCT
ejpam-3776	135	21	.	.	PUNCT
ejpam-3776	135	22	,	,	PUNCT
ejpam-3776	135	23	n	n	CCONJ
ejpam-3776	135	24	}	}	PUNCT
ejpam-3776	135	25	,	,	PUNCT
ejpam-3776	135	26	we	we	PRON
ejpam-3776	135	27	have	have	VERB
ejpam-3776	135	28	φ	φ	VERB
ejpam-3776	135	29			PROPN
ejpam-3776	135	30	m∑	m∑	PROPN
ejpam-3776	135	31	j=1	j=1	PROPN
ejpam-3776	135	32	aj	aj	PROPN
ejpam-3776	136	1	−	−	PROPN
ejpam-3776	136	2	1	1	NUM
ejpam-3776	136	3	n	n	PROPN
ejpam-3776	136	4	m−1∑	m−1∑	NUM
ejpam-3776	137	1	j=1	j=1	PROPN
ejpam-3776	137	2	n∑	n∑	PROPN
ejpam-3776	137	3	i=1	i=1	PROPN
ejpam-3776	138	1	xij	xij	PROPN
ejpam-3776	138	2			PROPN
ejpam-3776	138	3	≤	≤	NUM
ejpam-3776	138	4	1	1	NUM
ejpam-3776	139	1	n	n	NUM
ejpam-3776	139	2	n∑	n∑	NOUN
ejpam-3776	139	3	i=1	i=1	PROPN
ejpam-3776	140	1	φ	φ	PROPN
ejpam-3776	141	1			PROPN
ejpam-3776	141	2	m∑	m∑	PROPN
ejpam-3776	141	3	j=1	j=1	PROPN
ejpam-3776	141	4	aj	aj	PROPN
ejpam-3776	141	5	−	−	PROPN
ejpam-3776	141	6	m−1∑	m−1∑	NUM
ejpam-3776	141	7	j=1	j=1	NOUN
ejpam-3776	141	8	l−1∑	l−1∑	PRON
ejpam-3776	141	9	k=0	k=0	PROPN
ejpam-3776	141	10	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	141	11			PROPN
ejpam-3776	141	12	≤	≤	NUM
ejpam-3776	141	13	m∑	m∑	CCONJ
ejpam-3776	141	14	j=1	j=1	ADJ
ejpam-3776	141	15	φ(aj)−	φ(aj)−	NOUN
ejpam-3776	141	16	1	1	NUM
ejpam-3776	141	17	n	n	PROPN
ejpam-3776	141	18	m−1∑	m−1∑	PROPN
ejpam-3776	142	1	j=1	j=1	PROPN
ejpam-3776	142	2	n∑	n∑	PROPN
ejpam-3776	143	1	i=1	i=1	PROPN
ejpam-3776	143	2	φ	φ	PROPN
ejpam-3776	143	3	(	(	PUNCT
ejpam-3776	143	4	xij	xij	X
ejpam-3776	143	5	)	)	PUNCT
ejpam-3776	143	6	.	.	PUNCT
ejpam-3776	144	1	(	(	PUNCT
ejpam-3776	144	2	8)	8)	NUM
ejpam-3776	144	3	remark	remark	NOUN
ejpam-3776	144	4	1	1	NUM
ejpam-3776	144	5	.	.	PUNCT
ejpam-3776	145	1	if	if	SCONJ
ejpam-3776	145	2	we	we	PRON
ejpam-3776	145	3	set	set	VERB
ejpam-3776	145	4	m	m	PROPN
ejpam-3776	145	5	=	=	SYM
ejpam-3776	145	6	2	2	NUM
ejpam-3776	145	7	,	,	PUNCT
ejpam-3776	145	8	a1	a1	NOUN
ejpam-3776	145	9	=	=	SYM
ejpam-3776	145	10	c	c	X
ejpam-3776	145	11	,	,	PUNCT
ejpam-3776	145	12	a2	a2	PROPN
ejpam-3776	145	13	=	=	SYM
ejpam-3776	146	1	d	d	PROPN
ejpam-3776	146	2	and	and	CCONJ
ejpam-3776	146	3	xi1	xi1	PROPN
ejpam-3776	146	4	=	=	PROPN
ejpam-3776	147	1	xi	xi	PROPN
ejpam-3776	147	2	for	for	ADP
ejpam-3776	147	3	i	i	PROPN
ejpam-3776	147	4	∈	∈	PROPN
ejpam-3776	147	5	{	{	PUNCT
ejpam-3776	147	6	1	1	NUM
ejpam-3776	147	7	,	,	PUNCT
ejpam-3776	147	8	.	.	PUNCT
ejpam-3776	147	9	.	.	PUNCT
ejpam-3776	148	1	.	.	PUNCT
ejpam-3776	149	1	,	,	PUNCT
ejpam-3776	150	1	n	n	CCONJ
ejpam-3776	150	2	}	}	PUNCT
ejpam-3776	150	3	,	,	PUNCT
ejpam-3776	150	4	then	then	ADV
ejpam-3776	150	5	theorem	theorem	VERB
ejpam-3776	150	6	5	5	NUM
ejpam-3776	150	7	will	will	AUX
ejpam-3776	150	8	become	become	VERB
ejpam-3776	150	9	special	special	ADJ
ejpam-3776	150	10	case	case	NOUN
ejpam-3776	150	11	of	of	ADP
ejpam-3776	150	12	theorem	theorem	NOUN
ejpam-3776	150	13	6	6	NUM
ejpam-3776	150	14	.	.	PUNCT
ejpam-3776	150	15	s.	s.	PROPN
ejpam-3776	150	16	chanan	chanan	PROPN
ejpam-3776	150	17	,	,	PUNCT
ejpam-3776	150	18	a.	a.	PROPN
ejpam-3776	150	19	r.	r.	PROPN
ejpam-3776	150	20	khan	khan	PROPN
ejpam-3776	150	21	/	/	SYM
ejpam-3776	150	22	eur	eur	PROPN
ejpam-3776	150	23	.	.	PUNCT
ejpam-3776	151	1	j.	j.	PROPN
ejpam-3776	151	2	pure	pure	PROPN
ejpam-3776	151	3	appl	appl	PROPN
ejpam-3776	151	4	.	.	PROPN
ejpam-3776	151	5	math	math	PROPN
ejpam-3776	151	6	,	,	PUNCT
ejpam-3776	151	7	13	13	NUM
ejpam-3776	151	8	(	(	PUNCT
ejpam-3776	151	9	4	4	NUM
ejpam-3776	151	10	)	)	PUNCT
ejpam-3776	151	11	(	(	PUNCT
ejpam-3776	151	12	2020	2020	NUM
ejpam-3776	151	13	)	)	PUNCT
ejpam-3776	151	14	,	,	PUNCT
ejpam-3776	151	15	814	814	NUM
ejpam-3776	151	16	-	-	SYM
ejpam-3776	151	17	829	829	NUM
ejpam-3776	151	18	819	819	NUM
ejpam-3776	151	19	3	3	NUM
ejpam-3776	151	20	.	.	PUNCT
ejpam-3776	151	21	refinement	refinement	NOUN
ejpam-3776	151	22	of	of	ADP
ejpam-3776	151	23	the	the	DET
ejpam-3776	151	24	ky	ky	PROPN
ejpam-3776	151	25	fan	fan	PROPN
ejpam-3776	151	26	inequality	inequality	PROPN
ejpam-3776	151	27	throughout	throughout	ADP
ejpam-3776	151	28	this	this	DET
ejpam-3776	151	29	section	section	NOUN
ejpam-3776	151	30	,	,	PUNCT
ejpam-3776	151	31	let	let	VERB
ejpam-3776	151	32	the	the	DET
ejpam-3776	151	33	assumptions	assumption	NOUN
ejpam-3776	151	34	stated	state	VERB
ejpam-3776	151	35	in	in	ADP
ejpam-3776	151	36	theorem	theorem	NOUN
ejpam-3776	151	37	6	6	NUM
ejpam-3776	151	38	be	be	AUX
ejpam-3776	151	39	valid	valid	ADJ
ejpam-3776	151	40	with	with	ADP
ejpam-3776	151	41	0	0	NUM
ejpam-3776	151	42	<	<	X
ejpam-3776	151	43	c	c	X
ejpam-3776	151	44	<	<	X
ejpam-3776	151	45	d.	d.	PROPN
ejpam-3776	151	46	we	we	PRON
ejpam-3776	151	47	define	define	VERB
ejpam-3776	151	48	generalized	generalized	ADJ
ejpam-3776	151	49	(	(	PUNCT
ejpam-3776	151	50	or	or	CCONJ
ejpam-3776	151	51	modified	modified	ADJ
ejpam-3776	151	52	)	)	PUNCT
ejpam-3776	151	53	arithmetic	arithmetic	ADJ
ejpam-3776	151	54	,	,	PUNCT
ejpam-3776	151	55	geometric	geometric	ADJ
ejpam-3776	151	56	and	and	CCONJ
ejpam-3776	151	57	harmonic	harmonic	ADJ
ejpam-3776	151	58	mean	mean	NOUN
ejpam-3776	151	59	respectively	respectively	ADV
ejpam-3776	151	60	as	as	ADP
ejpam-3776	151	61	follow	follow	NOUN
ejpam-3776	151	62	(	(	PUNCT
ejpam-3776	151	63	for	for	ADP
ejpam-3776	151	64	general	general	ADJ
ejpam-3776	151	65	discussion	discussion	NOUN
ejpam-3776	151	66	on	on	ADP
ejpam-3776	151	67	mean	mean	ADJ
ejpam-3776	151	68	and	and	CCONJ
ejpam-3776	151	69	related	related	ADJ
ejpam-3776	151	70	inequalities	inequality	NOUN
ejpam-3776	151	71	we	we	PRON
ejpam-3776	151	72	refer	refer	VERB
ejpam-3776	151	73	[	[	X
ejpam-3776	151	74	4	4	NUM
ejpam-3776	151	75	]	]	PUNCT
ejpam-3776	151	76	):	):	PUNCT
ejpam-3776	151	77	ân	ân	NOUN
ejpam-3776	151	78	=	=	PUNCT
ejpam-3776	151	79	m∑	m∑	PROPN
ejpam-3776	151	80	j=1	j=1	PROPN
ejpam-3776	151	81	aj	aj	PROPN
ejpam-3776	151	82	−	−	PROPN
ejpam-3776	151	83	m−1∑	m−1∑	NUM
ejpam-3776	152	1	j=1	j=1	PROPN
ejpam-3776	152	2	n∑	n∑	PROPN
ejpam-3776	152	3	i=1	i=1	PROPN
ejpam-3776	153	1	wixij	wixij	PROPN
ejpam-3776	153	2	,	,	PUNCT
ejpam-3776	153	3	ĝn	ĝn	PROPN
ejpam-3776	153	4	=	=	SYM
ejpam-3776	153	5	m∏	m∏	PROPN
ejpam-3776	153	6	j=1	j=1	PROPN
ejpam-3776	153	7	aj	aj	PROPN
ejpam-3776	153	8	m−1∏	m−1∏	PROPN
ejpam-3776	153	9	j=1	j=1	PROPN
ejpam-3776	153	10	n∏	n∏	PROPN
ejpam-3776	153	11	i=1	i=1	PROPN
ejpam-3776	153	12	(	(	PUNCT
ejpam-3776	153	13	xij	xij	X
ejpam-3776	153	14	)	)	PUNCT
ejpam-3776	153	15	wi	wi	PROPN
ejpam-3776	153	16	,	,	PUNCT
ejpam-3776	153	17	ĥn	ĥn	PROPN
ejpam-3776	153	18	=	=	PUNCT
ejpam-3776	153	19			PROPN
ejpam-3776	153	20	m∑	m∑	ADV
ejpam-3776	153	21	j=1	j=1	PROPN
ejpam-3776	153	22	(	(	PUNCT
ejpam-3776	153	23	aj	aj	PROPN
ejpam-3776	153	24	)	)	PUNCT
ejpam-3776	153	25	−1	−1	NOUN
ejpam-3776	153	26	−	−	PROPN
ejpam-3776	153	27	m−1∑	m−1∑	NUM
ejpam-3776	154	1	j=1	j=1	PROPN
ejpam-3776	154	2	n∑	n∑	PROPN
ejpam-3776	154	3	i=1	i=1	PROPN
ejpam-3776	154	4	wi(xij	wi(xij	PROPN
ejpam-3776	154	5	)	)	PUNCT
ejpam-3776	155	1	−1	−1	NOUN
ejpam-3776	156	1	−1	−1	INTJ
ejpam-3776	156	2	.	.	PUNCT
ejpam-3776	157	1	also	also	ADV
ejpam-3776	157	2	for	for	ADP
ejpam-3776	157	3	xij	xij	PROPN
ejpam-3776	157	4	∈	∈	PROPN
ejpam-3776	157	5	(	(	PUNCT
ejpam-3776	157	6	0	0	NUM
ejpam-3776	157	7	,	,	PUNCT
ejpam-3776	157	8	12	12	NUM
ejpam-3776	157	9	]	]	PUNCT
ejpam-3776	157	10	,	,	PUNCT
ejpam-3776	157	11	we	we	PRON
ejpam-3776	157	12	define	define	VERB
ejpam-3776	157	13	arithmetic	arithmetic	ADJ
ejpam-3776	157	14	,	,	PUNCT
ejpam-3776	157	15	geometric	geometric	ADJ
ejpam-3776	157	16	and	and	CCONJ
ejpam-3776	157	17	harmonic	harmonic	ADJ
ejpam-3776	157	18	means	mean	NOUN
ejpam-3776	157	19	as	as	SCONJ
ejpam-3776	157	20	follows	follow	VERB
ejpam-3776	157	21	:	:	PUNCT
ejpam-3776	157	22	â′n	â′n	NOUN
ejpam-3776	157	23	=	=	PUNCT
ejpam-3776	158	1	m∑	m∑	CCONJ
ejpam-3776	158	2	j=1	j=1	NOUN
ejpam-3776	158	3	(	(	PUNCT
ejpam-3776	158	4	1−	1−	NUM
ejpam-3776	158	5	aj)−	aj)−	NOUN
ejpam-3776	158	6	m−1∑	m−1∑	NUM
ejpam-3776	158	7	j=1	j=1	PROPN
ejpam-3776	158	8	n∑	n∑	PROPN
ejpam-3776	159	1	i=1	i=1	PROPN
ejpam-3776	160	1	wi	wi	PROPN
ejpam-3776	160	2	(	(	PUNCT
ejpam-3776	160	3	1−	1−	NUM
ejpam-3776	160	4	xij	xij	X
ejpam-3776	160	5	)	)	PUNCT
ejpam-3776	160	6	,	,	PUNCT
ejpam-3776	160	7	ĝ′n	ĝ′n	NOUN
ejpam-3776	160	8	=	=	SYM
ejpam-3776	160	9	m∏	m∏	PROPN
ejpam-3776	160	10	j=1	j=1	NOUN
ejpam-3776	160	11	(	(	PUNCT
ejpam-3776	160	12	1−	1−	NUM
ejpam-3776	160	13	aj	aj	PROPN
ejpam-3776	160	14	)	)	PUNCT
ejpam-3776	160	15	m−1∏	m−1∏	PROPN
ejpam-3776	160	16	j=1	j=1	PROPN
ejpam-3776	160	17	n∏	n∏	PROPN
ejpam-3776	160	18	i=1	i=1	PROPN
ejpam-3776	160	19	(	(	PUNCT
ejpam-3776	160	20	(	(	PUNCT
ejpam-3776	160	21	1−	1−	NUM
ejpam-3776	160	22	xij)wi	xij)wi	NOUN
ejpam-3776	160	23	,	,	PUNCT
ejpam-3776	160	24	ĥ	ĥ	PROPN
ejpam-3776	160	25	′n	′n	PROPN
ejpam-3776	160	26	=	=	SYM
ejpam-3776	160	27			PROPN
ejpam-3776	160	28	m∑	m∑	ADV
ejpam-3776	160	29	j=1	j=1	PROPN
ejpam-3776	160	30	(	(	PUNCT
ejpam-3776	160	31	1−	1−	NUM
ejpam-3776	160	32	aj)−1	aj)−1	SYM
ejpam-3776	160	33	−	−	PROPN
ejpam-3776	161	1	m−1∑	m−1∑	NUM
ejpam-3776	161	2	j=1	j=1	PROPN
ejpam-3776	161	3	n∑	n∑	PROPN
ejpam-3776	161	4	i=1	i=1	PROPN
ejpam-3776	161	5	wi	wi	PROPN
ejpam-3776	161	6	(	(	PUNCT
ejpam-3776	161	7	1−	1−	NUM
ejpam-3776	161	8	xij)−1	xij)−1	NOUN
ejpam-3776	161	9	−1	−1	INTJ
ejpam-3776	161	10	.	.	PUNCT
ejpam-3776	162	1	we	we	PRON
ejpam-3776	162	2	also	also	ADV
ejpam-3776	162	3	define	define	VERB
ejpam-3776	162	4	new	new	ADJ
ejpam-3776	162	5	notations	notation	NOUN
ejpam-3776	162	6	â(λ	â(λ	ADP
ejpam-3776	162	7	;	;	PUNCT
ejpam-3776	162	8	x	x	X
ejpam-3776	162	9	)	)	PUNCT
ejpam-3776	162	10	and	and	CCONJ
ejpam-3776	162	11	ĝ(λ	ĝ(λ	ADP
ejpam-3776	162	12	;	;	PUNCT
ejpam-3776	162	13	x	x	X
ejpam-3776	162	14	)	)	PUNCT
ejpam-3776	162	15	as	as	ADP
ejpam-3776	162	16	under	under	ADV
ejpam-3776	162	17	:	:	PUNCT
ejpam-3776	162	18	â(λ	â(λ	NUM
ejpam-3776	162	19	;	;	PUNCT
ejpam-3776	162	20	x	x	X
ejpam-3776	162	21	)	)	PUNCT
ejpam-3776	162	22	=	=	PUNCT
ejpam-3776	163	1	m∑	m∑	PROPN
ejpam-3776	163	2	j=1	j=1	PROPN
ejpam-3776	163	3	aj	aj	PROPN
ejpam-3776	163	4	−	−	PROPN
ejpam-3776	163	5	m−1∑	m−1∑	NUM
ejpam-3776	163	6	j=1	j=1	NOUN
ejpam-3776	163	7	l−1∑	l−1∑	PRON
ejpam-3776	163	8	k=0	k=0	PROPN
ejpam-3776	163	9	λk+1xij+k	λk+1xij+k	PROPN
ejpam-3776	163	10	,	,	PUNCT
ejpam-3776	163	11	ĝ(λ	ĝ(λ	ADP
ejpam-3776	163	12	;	;	PUNCT
ejpam-3776	163	13	x	x	X
ejpam-3776	163	14	)	)	PUNCT
ejpam-3776	163	15	=	=	SYM
ejpam-3776	163	16	m∏	m∏	PROPN
ejpam-3776	163	17	j=1	j=1	PROPN
ejpam-3776	163	18	aj	aj	PROPN
ejpam-3776	163	19	m−1∏	m−1∏	PROPN
ejpam-3776	163	20	j=1	j=1	PROPN
ejpam-3776	163	21	l−1∏	l−1∏	PROPN
ejpam-3776	163	22	k=0	k=0	PROPN
ejpam-3776	163	23	(	(	PUNCT
ejpam-3776	163	24	xij+k	xij+k	PROPN
ejpam-3776	163	25	)	)	PUNCT
ejpam-3776	163	26	λk+1	λk+1	PROPN
ejpam-3776	163	27	.	.	PUNCT
ejpam-3776	164	1	s.	s.	PROPN
ejpam-3776	164	2	chanan	chanan	PROPN
ejpam-3776	164	3	,	,	PUNCT
ejpam-3776	164	4	a.	a.	PROPN
ejpam-3776	164	5	r.	r.	PROPN
ejpam-3776	164	6	khan	khan	PROPN
ejpam-3776	164	7	/	/	SYM
ejpam-3776	164	8	eur	eur	PROPN
ejpam-3776	164	9	.	.	PUNCT
ejpam-3776	165	1	j.	j.	PROPN
ejpam-3776	165	2	pure	pure	PROPN
ejpam-3776	165	3	appl	appl	PROPN
ejpam-3776	165	4	.	.	PROPN
ejpam-3776	165	5	math	math	PROPN
ejpam-3776	165	6	,	,	PUNCT
ejpam-3776	165	7	13	13	NUM
ejpam-3776	165	8	(	(	PUNCT
ejpam-3776	165	9	4	4	NUM
ejpam-3776	165	10	)	)	PUNCT
ejpam-3776	165	11	(	(	PUNCT
ejpam-3776	165	12	2020	2020	NUM
ejpam-3776	165	13	)	)	PUNCT
ejpam-3776	165	14	,	,	PUNCT
ejpam-3776	165	15	814	814	NUM
ejpam-3776	165	16	-	-	SYM
ejpam-3776	165	17	829	829	NUM
ejpam-3776	165	18	820	820	NUM
ejpam-3776	165	19	also	also	ADV
ejpam-3776	165	20	for	for	ADP
ejpam-3776	165	21	xij	xij	PROPN
ejpam-3776	165	22	∈	∈	PROPN
ejpam-3776	165	23	(	(	PUNCT
ejpam-3776	165	24	0	0	NUM
ejpam-3776	165	25	,	,	PUNCT
ejpam-3776	165	26	12	12	NUM
ejpam-3776	165	27	]	]	PUNCT
ejpam-3776	165	28	,	,	PUNCT
ejpam-3776	165	29	we	we	PRON
ejpam-3776	165	30	define	define	VERB
ejpam-3776	165	31	â′(λ	â′(λ	PROPN
ejpam-3776	165	32	;	;	PUNCT
ejpam-3776	165	33	x	x	X
ejpam-3776	165	34	)	)	PUNCT
ejpam-3776	165	35	=	=	PUNCT
ejpam-3776	166	1	m∑	m∑	CCONJ
ejpam-3776	166	2	j=1	j=1	NOUN
ejpam-3776	166	3	(	(	PUNCT
ejpam-3776	166	4	1−	1−	NUM
ejpam-3776	166	5	aj)−	aj)−	NOUN
ejpam-3776	166	6	m−1∑	m−1∑	NUM
ejpam-3776	166	7	j=1	j=1	NOUN
ejpam-3776	166	8	l−1∑	l−1∑	PRON
ejpam-3776	166	9	k=0	k=0	PROPN
ejpam-3776	166	10	λk+1(1−	λk+1(1−	X
ejpam-3776	166	11	xij+k	xij+k	PROPN
ejpam-3776	166	12	)	)	PUNCT
ejpam-3776	166	13	,	,	PUNCT
ejpam-3776	166	14	ĝ′(λ	ĝ′(λ	PROPN
ejpam-3776	166	15	;	;	PUNCT
ejpam-3776	166	16	x	x	X
ejpam-3776	166	17	)	)	PUNCT
ejpam-3776	166	18	=	=	SYM
ejpam-3776	166	19	m∏	m∏	NOUN
ejpam-3776	166	20	j=1	j=1	NOUN
ejpam-3776	166	21	(	(	PUNCT
ejpam-3776	166	22	1−	1−	NUM
ejpam-3776	166	23	aj	aj	PROPN
ejpam-3776	166	24	)	)	PUNCT
ejpam-3776	166	25	m−1∏	m−1∏	PROPN
ejpam-3776	166	26	j=1	j=1	NOUN
ejpam-3776	166	27	l−1∏	l−1∏	PROPN
ejpam-3776	166	28	k=0	k=0	PROPN
ejpam-3776	166	29	(	(	PUNCT
ejpam-3776	166	30	1−	1−	NUM
ejpam-3776	166	31	xij+k)λk+1	xij+k)λk+1	NOUN
ejpam-3776	166	32	.	.	PUNCT
ejpam-3776	167	1	now	now	ADV
ejpam-3776	167	2	,	,	PUNCT
ejpam-3776	167	3	we	we	PRON
ejpam-3776	167	4	present	present	VERB
ejpam-3776	167	5	the	the	DET
ejpam-3776	167	6	refinement	refinement	NOUN
ejpam-3776	167	7	of	of	ADP
ejpam-3776	167	8	the	the	DET
ejpam-3776	167	9	ky	ky	PROPN
ejpam-3776	167	10	fan	fan	PROPN
ejpam-3776	167	11	type	type	PROPN
ejpam-3776	167	12	inequality	inequality	NOUN
ejpam-3776	167	13	.	.	PUNCT
ejpam-3776	168	1	for	for	ADP
ejpam-3776	168	2	ky	ky	PROPN
ejpam-3776	168	3	fan	fan	PROPN
ejpam-3776	168	4	inequality	inequality	PROPN
ejpam-3776	168	5	and	and	CCONJ
ejpam-3776	168	6	related	related	ADJ
ejpam-3776	168	7	results	result	NOUN
ejpam-3776	168	8	,	,	PUNCT
ejpam-3776	168	9	see	see	VERB
ejpam-3776	168	10	[	[	X
ejpam-3776	168	11	2	2	NUM
ejpam-3776	168	12	]	]	PUNCT
ejpam-3776	168	13	,	,	PUNCT
ejpam-3776	168	14	[	[	X
ejpam-3776	168	15	6	6	NUM
ejpam-3776	168	16	]	]	PUNCT
ejpam-3776	168	17	and	and	CCONJ
ejpam-3776	168	18	[	[	X
ejpam-3776	168	19	5	5	NUM
ejpam-3776	168	20	]	]	PUNCT
ejpam-3776	168	21	and	and	CCONJ
ejpam-3776	168	22	references	reference	NOUN
ejpam-3776	168	23	given	give	VERB
ejpam-3776	168	24	therein	therein	ADV
ejpam-3776	168	25	.	.	PUNCT
ejpam-3776	169	1	theorem	theorem	ADJ
ejpam-3776	169	2	7	7	NUM
ejpam-3776	169	3	.	.	PUNCT
ejpam-3776	170	1	let	let	VERB
ejpam-3776	170	2	assumptions	assumption	NOUN
ejpam-3776	170	3	stated	state	VERB
ejpam-3776	170	4	in	in	ADP
ejpam-3776	170	5	theorem	theorem	NOUN
ejpam-3776	170	6	6	6	NUM
ejpam-3776	170	7	be	be	AUX
ejpam-3776	170	8	true	true	ADJ
ejpam-3776	170	9	.	.	PUNCT
ejpam-3776	171	1	then	then	ADV
ejpam-3776	171	2	following	follow	VERB
ejpam-3776	171	3	inequality	inequality	NOUN
ejpam-3776	171	4	holds	hold	VERB
ejpam-3776	171	5	:	:	PUNCT
ejpam-3776	171	6	ân	ân	PROPN
ejpam-3776	171	7	â′n	â′n	NOUN
ejpam-3776	171	8	≤	≤	PROPN
ejpam-3776	171	9	n∏	n∏	PROPN
ejpam-3776	171	10	i=1	i=1	PROPN
ejpam-3776	171	11	(	(	PUNCT
ejpam-3776	171	12	â(λ	â(λ	PROPN
ejpam-3776	171	13	,	,	PUNCT
ejpam-3776	171	14	x	x	NOUN
ejpam-3776	171	15	)	)	PUNCT
ejpam-3776	171	16	â′(λ	â′(λ	PROPN
ejpam-3776	171	17	,	,	PUNCT
ejpam-3776	171	18	x	x	NOUN
ejpam-3776	171	19	)	)	PUNCT
ejpam-3776	171	20	)	)	PUNCT
ejpam-3776	172	1	wi	wi	PROPN
ejpam-3776	172	2	≤	≤	NOUN
ejpam-3776	172	3	ĝn	ĝn	VERB
ejpam-3776	172	4	ĝ′n	ĝ′n	NOUN
ejpam-3776	172	5	.	.	PUNCT
ejpam-3776	173	1	proof	proof	NOUN
ejpam-3776	173	2	.	.	PUNCT
ejpam-3776	174	1	by	by	ADP
ejpam-3776	174	2	applying	apply	VERB
ejpam-3776	174	3	the	the	DET
ejpam-3776	174	4	convex	convex	NOUN
ejpam-3776	174	5	function	function	NOUN
ejpam-3776	174	6	φ(x	φ(x	NOUN
ejpam-3776	174	7	)	)	PUNCT
ejpam-3776	174	8	=	=	SYM
ejpam-3776	174	9	ln	ln	NOUN
ejpam-3776	174	10	(	(	PUNCT
ejpam-3776	174	11	x	x	NOUN
ejpam-3776	174	12	1−x	1−x	NUM
ejpam-3776	174	13	)	)	PUNCT
ejpam-3776	174	14	for	for	ADP
ejpam-3776	174	15	all	all	DET
ejpam-3776	174	16	x	x	SYM
ejpam-3776	174	17	∈	∈	PROPN
ejpam-3776	174	18	(	(	PUNCT
ejpam-3776	174	19	0	0	NUM
ejpam-3776	174	20	,	,	PUNCT
ejpam-3776	174	21	12	12	NUM
ejpam-3776	174	22	]	]	PUNCT
ejpam-3776	174	23	,	,	PUNCT
ejpam-3776	174	24	to	to	ADP
ejpam-3776	174	25	the	the	DET
ejpam-3776	174	26	inequality	inequality	NOUN
ejpam-3776	174	27	(	(	PUNCT
ejpam-3776	174	28	6	6	NUM
ejpam-3776	174	29	)	)	PUNCT
ejpam-3776	174	30	,	,	PUNCT
ejpam-3776	174	31	we	we	PRON
ejpam-3776	174	32	get	get	VERB
ejpam-3776	174	33	,	,	PUNCT
ejpam-3776	174	34	ln	ln	CCONJ
ejpam-3776	174	35	(	(	PUNCT
ejpam-3776	174	36	∑m	∑m	PROPN
ejpam-3776	174	37	j=1	j=1	PROPN
ejpam-3776	174	38	aj	aj	PROPN
ejpam-3776	174	39	−	−	PROPN
ejpam-3776	174	40	∑m−1	∑m−1	NOUN
ejpam-3776	174	41	j=1	j=1	NOUN
ejpam-3776	175	1	∑n	∑n	PROPN
ejpam-3776	175	2	i=1wixij	i=1wixij	PROPN
ejpam-3776	175	3	1−	1−	PROPN
ejpam-3776	175	4	∑m	∑m	PROPN
ejpam-3776	175	5	j=1	j=1	PROPN
ejpam-3776	175	6	aj	aj	PROPN
ejpam-3776	176	1	+	+	SYM
ejpam-3776	176	2	∑m−1	∑m−1	ADJ
ejpam-3776	176	3	j=1	j=1	NOUN
ejpam-3776	176	4	∑n	∑n	PROPN
ejpam-3776	176	5	i=1wixij	i=1wixij	PROPN
ejpam-3776	176	6	)	)	PUNCT
ejpam-3776	176	7	≤	≤	NUM
ejpam-3776	177	1	n∑	n∑	PROPN
ejpam-3776	177	2	i=1	i=1	PROPN
ejpam-3776	178	1	wi	wi	PROPN
ejpam-3776	178	2	ln	ln	PROPN
ejpam-3776	179	1	(	(	PUNCT
ejpam-3776	179	2	∑m	∑m	PROPN
ejpam-3776	179	3	j=1	j=1	PROPN
ejpam-3776	179	4	aj	aj	PROPN
ejpam-3776	179	5	−	−	PROPN
ejpam-3776	179	6	∑m−1	∑m−1	PROPN
ejpam-3776	179	7	j=1	j=1	NOUN
ejpam-3776	179	8	∑l−1	∑l−1	VERB
ejpam-3776	180	1	k=0	k=0	PROPN
ejpam-3776	180	2	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	180	3	1−	1−	NUM
ejpam-3776	180	4	∑m	∑m	PROPN
ejpam-3776	180	5	j=1	j=1	PROPN
ejpam-3776	180	6	aj	aj	PROPN
ejpam-3776	181	1	+	+	SYM
ejpam-3776	181	2	∑m−1	∑m−1	ADJ
ejpam-3776	181	3	j=1	j=1	NOUN
ejpam-3776	181	4	∑l−1	∑l−1	VERB
ejpam-3776	181	5	k=0	k=0	PROPN
ejpam-3776	181	6	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	181	7	)	)	PUNCT
ejpam-3776	181	8	≤	≤	NUM
ejpam-3776	182	1	m∑	m∑	ADP
ejpam-3776	182	2	j=1	j=1	PROPN
ejpam-3776	182	3	ln	ln	ADV
ejpam-3776	182	4	(	(	PUNCT
ejpam-3776	182	5	ai	ai	PROPN
ejpam-3776	182	6	1−	1−	NUM
ejpam-3776	182	7	aj	aj	PROPN
ejpam-3776	182	8	)	)	PUNCT
ejpam-3776	183	1	−	−	PROPN
ejpam-3776	184	1	m−1∑	m−1∑	NUM
ejpam-3776	184	2	j=1	j=1	PROPN
ejpam-3776	184	3	n∑	n∑	PROPN
ejpam-3776	184	4	i=1	i=1	PROPN
ejpam-3776	184	5	wi	wi	PROPN
ejpam-3776	184	6	ln	ln	PROPN
ejpam-3776	184	7	(	(	PUNCT
ejpam-3776	184	8	xij	xij	PROPN
ejpam-3776	184	9	1−	1−	NUM
ejpam-3776	184	10	xij	xij	NOUN
ejpam-3776	184	11	)	)	PUNCT
ejpam-3776	184	12	consequently	consequently	ADV
ejpam-3776	184	13	,	,	PUNCT
ejpam-3776	184	14	ln	ln	X
ejpam-3776	184	15	(	(	PUNCT
ejpam-3776	184	16	ân	ân	NOUN
ejpam-3776	184	17	â′n	â′n	NOUN
ejpam-3776	184	18	)	)	PUNCT
ejpam-3776	184	19	≤	≤	PUNCT
ejpam-3776	185	1	ln	ln	PROPN
ejpam-3776	185	2	n∏	n∏	PROPN
ejpam-3776	185	3	i=1	i=1	PROPN
ejpam-3776	186	1	(	(	PUNCT
ejpam-3776	186	2	∑m	∑m	PROPN
ejpam-3776	186	3	j=1	j=1	PROPN
ejpam-3776	186	4	aj	aj	PROPN
ejpam-3776	186	5	−	−	PROPN
ejpam-3776	186	6	∑m−1	∑m−1	PROPN
ejpam-3776	186	7	j=1	j=1	NOUN
ejpam-3776	186	8	∑l−1	∑l−1	VERB
ejpam-3776	187	1	k=0	k=0	PROPN
ejpam-3776	187	2	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	187	3	1−	1−	NUM
ejpam-3776	187	4	∑m	∑m	PROPN
ejpam-3776	187	5	j=1	j=1	PROPN
ejpam-3776	187	6	aj	aj	PROPN
ejpam-3776	188	1	+	+	SYM
ejpam-3776	188	2	∑m−1	∑m−1	ADJ
ejpam-3776	188	3	j=1	j=1	NOUN
ejpam-3776	188	4	∑l−1	∑l−1	VERB
ejpam-3776	188	5	k=0	k=0	PROPN
ejpam-3776	188	6	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	188	7	)	)	PUNCT
ejpam-3776	188	8	wi	wi	PROPN
ejpam-3776	188	9	≤	≤	PROPN
ejpam-3776	188	10	ln	ln	ADV
ejpam-3776	188	11	(	(	PUNCT
ejpam-3776	188	12	ĝn	ĝn	X
ejpam-3776	188	13	ĝ′n	ĝ′n	NOUN
ejpam-3776	188	14	)	)	PUNCT
ejpam-3776	188	15	finally	finally	ADV
ejpam-3776	188	16	,	,	PUNCT
ejpam-3776	188	17	we	we	PRON
ejpam-3776	188	18	obtain	obtain	VERB
ejpam-3776	188	19	(	(	PUNCT
ejpam-3776	188	20	ân	ân	NOUN
ejpam-3776	188	21	â′n	â′n	NOUN
ejpam-3776	188	22	)	)	PUNCT
ejpam-3776	188	23	≤	≤	PROPN
ejpam-3776	189	1	n∏	n∏	PROPN
ejpam-3776	189	2	i=1	i=1	PROPN
ejpam-3776	190	1	(	(	PUNCT
ejpam-3776	190	2	â(λ	â(λ	PROPN
ejpam-3776	190	3	,	,	PUNCT
ejpam-3776	190	4	x	x	NOUN
ejpam-3776	190	5	)	)	PUNCT
ejpam-3776	190	6	â′(λ	â′(λ	PROPN
ejpam-3776	190	7	,	,	PUNCT
ejpam-3776	190	8	x	x	NOUN
ejpam-3776	190	9	)	)	PUNCT
ejpam-3776	190	10	)	)	PUNCT
ejpam-3776	191	1	wi	wi	PROPN
ejpam-3776	191	2	≤	≤	PROPN
ejpam-3776	191	3	(	(	PUNCT
ejpam-3776	191	4	ĝn	ĝn	NOUN
ejpam-3776	191	5	ĝ′n	ĝ′n	NOUN
ejpam-3776	191	6	)	)	PUNCT
ejpam-3776	191	7	,	,	PUNCT
ejpam-3776	191	8	which	which	PRON
ejpam-3776	191	9	completes	complete	VERB
ejpam-3776	191	10	the	the	DET
ejpam-3776	191	11	proof	proof	NOUN
ejpam-3776	191	12	.	.	PUNCT
ejpam-3776	192	1	s.	s.	PROPN
ejpam-3776	192	2	chanan	chanan	PROPN
ejpam-3776	192	3	,	,	PUNCT
ejpam-3776	192	4	a.	a.	PROPN
ejpam-3776	192	5	r.	r.	PROPN
ejpam-3776	192	6	khan	khan	PROPN
ejpam-3776	192	7	/	/	SYM
ejpam-3776	192	8	eur	eur	PROPN
ejpam-3776	192	9	.	.	PUNCT
ejpam-3776	193	1	j.	j.	PROPN
ejpam-3776	193	2	pure	pure	PROPN
ejpam-3776	193	3	appl	appl	PROPN
ejpam-3776	193	4	.	.	PROPN
ejpam-3776	193	5	math	math	PROPN
ejpam-3776	193	6	,	,	PUNCT
ejpam-3776	193	7	13	13	NUM
ejpam-3776	193	8	(	(	PUNCT
ejpam-3776	193	9	4	4	NUM
ejpam-3776	193	10	)	)	PUNCT
ejpam-3776	193	11	(	(	PUNCT
ejpam-3776	193	12	2020	2020	NUM
ejpam-3776	193	13	)	)	PUNCT
ejpam-3776	193	14	,	,	PUNCT
ejpam-3776	193	15	814	814	NUM
ejpam-3776	193	16	-	-	SYM
ejpam-3776	193	17	829	829	NUM
ejpam-3776	193	18	821	821	NUM
ejpam-3776	193	19	remark	remark	NOUN
ejpam-3776	193	20	2	2	NUM
ejpam-3776	193	21	.	.	PUNCT
ejpam-3776	194	1	for	for	ADP
ejpam-3776	194	2	wi	wi	PROPN
ejpam-3776	194	3	=	=	SYM
ejpam-3776	194	4	1	1	NUM
ejpam-3776	194	5	n	n	NOUN
ejpam-3776	194	6	,	,	PUNCT
ejpam-3776	194	7	we	we	PRON
ejpam-3776	194	8	obtain	obtain	VERB
ejpam-3776	194	9	the	the	DET
ejpam-3776	194	10	special	special	ADJ
ejpam-3776	194	11	case	case	NOUN
ejpam-3776	194	12	of	of	ADP
ejpam-3776	194	13	theorem	theorem	NOUN
ejpam-3776	194	14	7	7	NUM
ejpam-3776	194	15	as	as	SCONJ
ejpam-3776	194	16	follows	follow	VERB
ejpam-3776	194	17	:	:	PUNCT
ejpam-3776	195	1	an	an	DET
ejpam-3776	195	2	a′n	a′n	NOUN
ejpam-3776	195	3	≤	≤	PROPN
ejpam-3776	195	4	n∏	n∏	PROPN
ejpam-3776	195	5	i=1	i=1	PROPN
ejpam-3776	196	1	(	(	PUNCT
ejpam-3776	196	2	a(λ	a(λ	ADV
ejpam-3776	196	3	,	,	PUNCT
ejpam-3776	196	4	x	x	X
ejpam-3776	196	5	)	)	PUNCT
ejpam-3776	197	1	a′(λ	a′(λ	PROPN
ejpam-3776	197	2	,	,	PUNCT
ejpam-3776	197	3	x	x	X
ejpam-3776	197	4	)	)	PUNCT
ejpam-3776	197	5	)	)	PUNCT
ejpam-3776	197	6	1	1	NUM
ejpam-3776	197	7	n	n	NOUN
ejpam-3776	197	8	≤	≤	NOUN
ejpam-3776	198	1	gn	gn	INTJ
ejpam-3776	198	2	g′n	g′n	NOUN
ejpam-3776	198	3	,	,	PUNCT
ejpam-3776	198	4	where	where	SCONJ
ejpam-3776	198	5	an	an	DET
ejpam-3776	198	6	=	=	PUNCT
ejpam-3776	198	7	m∑	m∑	PROPN
ejpam-3776	198	8	j=1	j=1	PROPN
ejpam-3776	198	9	aj	aj	PROPN
ejpam-3776	199	1	−	−	PROPN
ejpam-3776	199	2	1	1	NUM
ejpam-3776	199	3	n	n	PROPN
ejpam-3776	199	4	m−1∑	m−1∑	NUM
ejpam-3776	200	1	j=1	j=1	PROPN
ejpam-3776	200	2	n∑	n∑	PROPN
ejpam-3776	200	3	i=1	i=1	PROPN
ejpam-3776	201	1	xij	xij	PROPN
ejpam-3776	201	2	,	,	PUNCT
ejpam-3776	201	3	gn	gn	PROPN
ejpam-3776	201	4	=	=	PUNCT
ejpam-3776	201	5	∏m	∏m	NOUN
ejpam-3776	202	1	j=1	j=1	NOUN
ejpam-3776	202	2	ajm−1∏	ajm−1∏	NUM
ejpam-3776	202	3	j=1	j=1	PROPN
ejpam-3776	202	4	n∏	n∏	PROPN
ejpam-3776	202	5	i=1	i=1	PROPN
ejpam-3776	203	1	xij	xij	PROPN
ejpam-3776	203	2			PROPN
ejpam-3776	203	3	1	1	NUM
ejpam-3776	203	4	n	n	NOUN
ejpam-3776	203	5	,	,	PUNCT
ejpam-3776	203	6	and	and	CCONJ
ejpam-3776	203	7	a′n	a′n	ADP
ejpam-3776	203	8	=	=	PUNCT
ejpam-3776	203	9	m∑	m∑	CCONJ
ejpam-3776	203	10	j=1	j=1	NOUN
ejpam-3776	203	11	(	(	PUNCT
ejpam-3776	203	12	1−	1−	NUM
ejpam-3776	203	13	aj)−	aj)−	NOUN
ejpam-3776	203	14	1	1	NUM
ejpam-3776	203	15	n	n	PROPN
ejpam-3776	203	16	m−1∑	m−1∑	NUM
ejpam-3776	204	1	j=1	j=1	NOUN
ejpam-3776	204	2	n∑	n∑	PROPN
ejpam-3776	204	3	i=1	i=1	PROPN
ejpam-3776	205	1	(	(	PUNCT
ejpam-3776	205	2	1−	1−	NUM
ejpam-3776	205	3	xij	xij	X
ejpam-3776	205	4	)	)	PUNCT
ejpam-3776	205	5	,	,	PUNCT
ejpam-3776	206	1	g′n	g′n	NOUN
ejpam-3776	206	2	=	=	PUNCT
ejpam-3776	206	3	∏m	∏m	ADJ
ejpam-3776	206	4	j=1(1−	j=1(1−	ADJ
ejpam-3776	206	5	aj)m−1∏	aj)m−1∏	NOUN
ejpam-3776	206	6	j=1	j=1	PROPN
ejpam-3776	206	7	n∏	n∏	PROPN
ejpam-3776	206	8	i=1	i=1	PROPN
ejpam-3776	206	9	(	(	PUNCT
ejpam-3776	206	10	1−	1−	NUM
ejpam-3776	206	11	xij	xij	NOUN
ejpam-3776	206	12	)	)	PUNCT
ejpam-3776	207	1			PROPN
ejpam-3776	207	2	1	1	NUM
ejpam-3776	207	3	n	n	NOUN
ejpam-3776	207	4	.	.	PUNCT
ejpam-3776	208	1	now	now	ADV
ejpam-3776	208	2	,	,	PUNCT
ejpam-3776	208	3	we	we	PRON
ejpam-3776	208	4	present	present	VERB
ejpam-3776	208	5	refinement	refinement	NOUN
ejpam-3776	208	6	of	of	ADP
ejpam-3776	208	7	arithmetic	arithmetic	ADJ
ejpam-3776	208	8	-	-	PUNCT
ejpam-3776	208	9	geometric	geometric	ADJ
ejpam-3776	208	10	mean	mean	NOUN
ejpam-3776	208	11	type	type	NOUN
ejpam-3776	208	12	inequality	inequality	NOUN
ejpam-3776	208	13	as	as	SCONJ
ejpam-3776	208	14	follows	follow	VERB
ejpam-3776	208	15	:	:	PUNCT
ejpam-3776	208	16	corollary	corollary	ADJ
ejpam-3776	208	17	2	2	X
ejpam-3776	208	18	.	.	PUNCT
ejpam-3776	209	1	let	let	VERB
ejpam-3776	209	2	the	the	DET
ejpam-3776	209	3	assumptions	assumption	NOUN
ejpam-3776	209	4	stated	state	VERB
ejpam-3776	209	5	in	in	ADP
ejpam-3776	209	6	theorem	theorem	NOUN
ejpam-3776	209	7	6	6	NUM
ejpam-3776	209	8	be	be	AUX
ejpam-3776	209	9	true	true	ADJ
ejpam-3776	209	10	.	.	PUNCT
ejpam-3776	210	1	then	then	ADV
ejpam-3776	210	2	following	follow	VERB
ejpam-3776	210	3	inequality	inequality	NOUN
ejpam-3776	210	4	holds	hold	VERB
ejpam-3776	210	5	:	:	PUNCT
ejpam-3776	210	6	ân	ân	PROPN
ejpam-3776	210	7	≥	≥	PROPN
ejpam-3776	210	8	n∏	n∏	PROPN
ejpam-3776	210	9	i=1	i=1	PROPN
ejpam-3776	210	10	(	(	PUNCT
ejpam-3776	210	11	â(λ	â(λ	X
ejpam-3776	210	12	,	,	PUNCT
ejpam-3776	210	13	x	x	NOUN
ejpam-3776	210	14	)	)	PUNCT
ejpam-3776	210	15	)	)	PUNCT
ejpam-3776	211	1	wi	wi	PROPN
ejpam-3776	211	2	≥	≥	PROPN
ejpam-3776	211	3	ĝ′n	ĝ′n	NOUN
ejpam-3776	211	4	.	.	PUNCT
ejpam-3776	212	1	proof	proof	NOUN
ejpam-3776	212	2	.	.	PUNCT
ejpam-3776	213	1	by	by	ADP
ejpam-3776	213	2	applying	apply	VERB
ejpam-3776	213	3	the	the	DET
ejpam-3776	213	4	convex	convex	NOUN
ejpam-3776	213	5	function	function	NOUN
ejpam-3776	213	6	φ	φ	PROPN
ejpam-3776	213	7	(	(	PUNCT
ejpam-3776	213	8	x	x	NOUN
ejpam-3776	213	9	)	)	PUNCT
ejpam-3776	213	10	=	=	SYM
ejpam-3776	214	1	−	−	PROPN
ejpam-3776	214	2	ln	ln	INTJ
ejpam-3776	214	3	(	(	PUNCT
ejpam-3776	214	4	x	x	X
ejpam-3776	214	5	)	)	PUNCT
ejpam-3776	214	6	,	,	PUNCT
ejpam-3776	214	7	x	x	PUNCT
ejpam-3776	214	8	∈	∈	PROPN
ejpam-3776	214	9	(	(	PUNCT
ejpam-3776	214	10	0	0	NUM
ejpam-3776	214	11	,	,	PUNCT
ejpam-3776	214	12	12	12	NUM
ejpam-3776	214	13	]	]	PUNCT
ejpam-3776	214	14	to	to	PART
ejpam-3776	214	15	theorem	theorem	VERB
ejpam-3776	214	16	6	6	NUM
ejpam-3776	214	17	we	we	PRON
ejpam-3776	214	18	obtain	obtain	VERB
ejpam-3776	214	19	required	required	ADJ
ejpam-3776	214	20	result	result	NOUN
ejpam-3776	214	21	.	.	PUNCT
ejpam-3776	215	1	now	now	ADV
ejpam-3776	215	2	,	,	PUNCT
ejpam-3776	215	3	we	we	PRON
ejpam-3776	215	4	present	present	VERB
ejpam-3776	215	5	refinement	refinement	NOUN
ejpam-3776	215	6	of	of	ADP
ejpam-3776	215	7	harmonic	harmonic	ADJ
ejpam-3776	215	8	and	and	CCONJ
ejpam-3776	215	9	geometric	geometric	ADJ
ejpam-3776	215	10	means	mean	NOUN
ejpam-3776	215	11	inequality	inequality	NOUN
ejpam-3776	215	12	as	as	SCONJ
ejpam-3776	215	13	follows	follow	VERB
ejpam-3776	215	14	:	:	PUNCT
ejpam-3776	215	15	corollary	corollary	ADJ
ejpam-3776	215	16	3	3	X
ejpam-3776	215	17	.	.	PUNCT
ejpam-3776	216	1	let	let	VERB
ejpam-3776	216	2	the	the	DET
ejpam-3776	216	3	assumptions	assumption	NOUN
ejpam-3776	216	4	stated	state	VERB
ejpam-3776	216	5	in	in	ADP
ejpam-3776	216	6	theorem	theorem	NOUN
ejpam-3776	216	7	6	6	NUM
ejpam-3776	216	8	be	be	AUX
ejpam-3776	216	9	true	true	ADJ
ejpam-3776	216	10	.	.	PUNCT
ejpam-3776	217	1	then	then	ADV
ejpam-3776	217	2	following	follow	VERB
ejpam-3776	217	3	inequalities	inequality	NOUN
ejpam-3776	217	4	hold	hold	VERB
ejpam-3776	217	5	:	:	PUNCT
ejpam-3776	217	6	(	(	PUNCT
ejpam-3776	217	7	ĝ′n	ĝ′n	NOUN
ejpam-3776	217	8	)	)	PUNCT
ejpam-3776	217	9	−1	−1	NOUN
ejpam-3776	217	10	≤	≤	NUM
ejpam-3776	218	1	n∑	n∑	PROPN
ejpam-3776	218	2	i=1	i=1	PROPN
ejpam-3776	219	1	wi	wi	PROPN
ejpam-3776	219	2	(	(	PUNCT
ejpam-3776	219	3	ĝ′(λ	ĝ′(λ	PROPN
ejpam-3776	219	4	,	,	PUNCT
ejpam-3776	219	5	x	x	NOUN
ejpam-3776	219	6	)	)	PUNCT
ejpam-3776	219	7	)	)	PUNCT
ejpam-3776	219	8	−1	−1	NOUN
ejpam-3776	219	9	≤	≤	NOUN
ejpam-3776	219	10	(	(	PUNCT
ejpam-3776	219	11	ĥ	ĥ	PUNCT
ejpam-3776	219	12	′n	′n	NOUN
ejpam-3776	219	13	)	)	PUNCT
ejpam-3776	219	14	−1	−1	NOUN
ejpam-3776	219	15	.	.	PUNCT
ejpam-3776	220	1	s.	s.	PROPN
ejpam-3776	220	2	chanan	chanan	PROPN
ejpam-3776	220	3	,	,	PUNCT
ejpam-3776	220	4	a.	a.	PROPN
ejpam-3776	220	5	r.	r.	PROPN
ejpam-3776	220	6	khan	khan	PROPN
ejpam-3776	220	7	/	/	SYM
ejpam-3776	220	8	eur	eur	PROPN
ejpam-3776	220	9	.	.	PUNCT
ejpam-3776	221	1	j.	j.	PROPN
ejpam-3776	221	2	pure	pure	PROPN
ejpam-3776	221	3	appl	appl	PROPN
ejpam-3776	221	4	.	.	PROPN
ejpam-3776	221	5	math	math	PROPN
ejpam-3776	221	6	,	,	PUNCT
ejpam-3776	221	7	13	13	NUM
ejpam-3776	221	8	(	(	PUNCT
ejpam-3776	221	9	4	4	NUM
ejpam-3776	221	10	)	)	PUNCT
ejpam-3776	221	11	(	(	PUNCT
ejpam-3776	221	12	2020	2020	NUM
ejpam-3776	221	13	)	)	PUNCT
ejpam-3776	221	14	,	,	PUNCT
ejpam-3776	221	15	814	814	NUM
ejpam-3776	221	16	-	-	SYM
ejpam-3776	221	17	829	829	NUM
ejpam-3776	221	18	822	822	NUM
ejpam-3776	221	19	proof	proof	NOUN
ejpam-3776	221	20	.	.	PUNCT
ejpam-3776	222	1	by	by	ADP
ejpam-3776	222	2	applying	apply	VERB
ejpam-3776	222	3	the	the	DET
ejpam-3776	222	4	convex	convex	NOUN
ejpam-3776	222	5	function	function	NOUN
ejpam-3776	222	6	φ	φ	PROPN
ejpam-3776	222	7	(	(	PUNCT
ejpam-3776	222	8	x	x	NOUN
ejpam-3776	222	9	)	)	PUNCT
ejpam-3776	222	10	=	=	SYM
ejpam-3776	222	11	exp	exp	NOUN
ejpam-3776	222	12	(	(	PUNCT
ejpam-3776	222	13	x	x	NOUN
ejpam-3776	222	14	)	)	PUNCT
ejpam-3776	222	15	,	,	PUNCT
ejpam-3776	222	16	x	x	PUNCT
ejpam-3776	222	17	∈	∈	PROPN
ejpam-3776	222	18	(	(	PUNCT
ejpam-3776	222	19	0	0	NUM
ejpam-3776	222	20	,	,	PUNCT
ejpam-3776	222	21	12	12	NUM
ejpam-3776	222	22	]	]	PUNCT
ejpam-3776	222	23	to	to	PART
ejpam-3776	222	24	theorem	theorem	VERB
ejpam-3776	222	25	6	6	NUM
ejpam-3776	222	26	and	and	CCONJ
ejpam-3776	222	27	by	by	ADP
ejpam-3776	222	28	replacing	replace	VERB
ejpam-3776	222	29	aj	aj	PROPN
ejpam-3776	222	30	and	and	CCONJ
ejpam-3776	222	31	xij	xij	PRON
ejpam-3776	222	32	by	by	ADP
ejpam-3776	222	33	ln	ln	PROPN
ejpam-3776	222	34	(	(	PUNCT
ejpam-3776	222	35	1	1	NUM
ejpam-3776	222	36	1−aj	1−aj	NUM
ejpam-3776	222	37	)	)	PUNCT
ejpam-3776	222	38	and	and	CCONJ
ejpam-3776	222	39	ln	ln	ADJ
ejpam-3776	222	40	(	(	PUNCT
ejpam-3776	222	41	1	1	NUM
ejpam-3776	222	42	1−xij	1−xij	NUM
ejpam-3776	222	43	)	)	PUNCT
ejpam-3776	222	44	respectively	respectively	ADV
ejpam-3776	222	45	,	,	PUNCT
ejpam-3776	222	46	we	we	PRON
ejpam-3776	222	47	get	get	VERB
ejpam-3776	222	48	exp	exp	NOUN
ejpam-3776	222	49			PROPN
ejpam-3776	222	50	m∑	m∑	PROPN
ejpam-3776	222	51	j=1	j=1	PROPN
ejpam-3776	222	52	ln	ln	NOUN
ejpam-3776	222	53	(	(	PUNCT
ejpam-3776	222	54	1	1	NUM
ejpam-3776	222	55	1−	1−	NUM
ejpam-3776	222	56	aj	aj	PROPN
ejpam-3776	222	57	)	)	PUNCT
ejpam-3776	223	1	−	−	PROPN
ejpam-3776	224	1	m−1∑	m−1∑	NUM
ejpam-3776	224	2	j=1	j=1	PROPN
ejpam-3776	224	3	n∑	n∑	PROPN
ejpam-3776	224	4	i=1	i=1	PROPN
ejpam-3776	224	5	wi	wi	PROPN
ejpam-3776	224	6	ln	ln	PROPN
ejpam-3776	224	7	(	(	PUNCT
ejpam-3776	224	8	1	1	NUM
ejpam-3776	224	9	1−	1−	NUM
ejpam-3776	224	10	xij	xij	NOUN
ejpam-3776	224	11	)	)	PUNCT
ejpam-3776	225	1			PROPN
ejpam-3776	225	2	≤	≤	NUM
ejpam-3776	225	3	n∑	n∑	PROPN
ejpam-3776	225	4	i=1	i=1	PROPN
ejpam-3776	225	5	wi	wi	PROPN
ejpam-3776	225	6	exp	exp	NOUN
ejpam-3776	226	1			PROPN
ejpam-3776	226	2	m∑	m∑	PROPN
ejpam-3776	226	3	j=1	j=1	PROPN
ejpam-3776	226	4	ln	ln	NOUN
ejpam-3776	226	5	(	(	PUNCT
ejpam-3776	226	6	1	1	NUM
ejpam-3776	226	7	1−	1−	NUM
ejpam-3776	226	8	aj	aj	PROPN
ejpam-3776	226	9	)	)	PUNCT
ejpam-3776	227	1	−	−	PROPN
ejpam-3776	228	1	m−1∑	m−1∑	NUM
ejpam-3776	228	2	j=1	j=1	NOUN
ejpam-3776	228	3	l−1∑	l−1∑	PROPN
ejpam-3776	228	4	k=0	k=0	PROPN
ejpam-3776	228	5	λk+1	λk+1	VERB
ejpam-3776	228	6	ln	ln	NOUN
ejpam-3776	228	7	(	(	PUNCT
ejpam-3776	228	8	1	1	NUM
ejpam-3776	228	9	1−	1−	NUM
ejpam-3776	228	10	xij+k	xij+k	NUM
ejpam-3776	228	11	)	)	PUNCT
ejpam-3776	228	12			PROPN
ejpam-3776	228	13	≤	≤	NUM
ejpam-3776	228	14	m∑	m∑	CCONJ
ejpam-3776	228	15	j=1	j=1	PROPN
ejpam-3776	228	16	exp	exp	PROPN
ejpam-3776	228	17	(	(	PUNCT
ejpam-3776	228	18	ln	ln	NOUN
ejpam-3776	228	19	(	(	PUNCT
ejpam-3776	228	20	1	1	NUM
ejpam-3776	228	21	1−	1−	NUM
ejpam-3776	228	22	aj	aj	PROPN
ejpam-3776	228	23	)	)	PUNCT
ejpam-3776	228	24	)	)	PUNCT
ejpam-3776	229	1	−	−	PROPN
ejpam-3776	230	1	m−1∑	m−1∑	NUM
ejpam-3776	230	2	j=1	j=1	PROPN
ejpam-3776	230	3	n∑	n∑	PROPN
ejpam-3776	230	4	i=0	i=0	PROPN
ejpam-3776	230	5	wi	wi	PROPN
ejpam-3776	230	6	exp	exp	PROPN
ejpam-3776	230	7	(	(	PUNCT
ejpam-3776	230	8	ln	ln	NOUN
ejpam-3776	230	9	(	(	PUNCT
ejpam-3776	230	10	1	1	NUM
ejpam-3776	230	11	1−	1−	NUM
ejpam-3776	230	12	xij	xij	NOUN
ejpam-3776	230	13	)	)	PUNCT
ejpam-3776	230	14	)	)	PUNCT
ejpam-3776	230	15	,	,	PUNCT
ejpam-3776	230	16	consequently	consequently	ADV
ejpam-3776	230	17	,	,	PUNCT
ejpam-3776	230	18	exp	exp	NOUN
ejpam-3776	230	19	(	(	PUNCT
ejpam-3776	230	20	−	−	PROPN
ejpam-3776	230	21	ln	ln	ADJ
ejpam-3776	230	22	(	(	PUNCT
ejpam-3776	230	23	∏m	∏m	X
ejpam-3776	230	24	j=1(1−	j=1(1−	ADJ
ejpam-3776	230	25	aj)∏n	aj)∏n	PUNCT
ejpam-3776	230	26	i=1	i=1	PROPN
ejpam-3776	230	27	∏m−1	∏m−1	PROPN
ejpam-3776	230	28	j=1	j=1	NOUN
ejpam-3776	230	29	(	(	PUNCT
ejpam-3776	230	30	1−	1−	NUM
ejpam-3776	230	31	xij)wi	xij)wi	NOUN
ejpam-3776	230	32	)	)	PUNCT
ejpam-3776	230	33	)	)	PUNCT
ejpam-3776	230	34	≤	≤	NUM
ejpam-3776	231	1	n∑	n∑	PROPN
ejpam-3776	231	2	i=1	i=1	PROPN
ejpam-3776	231	3	wi	wi	PROPN
ejpam-3776	231	4	exp	exp	NOUN
ejpam-3776	231	5	−	−	PROPN
ejpam-3776	231	6	ln	ln	NOUN
ejpam-3776	231	7			X
ejpam-3776	231	8	∏m	∏m	X
ejpam-3776	231	9	j=1(1−	j=1(1−	PROPN
ejpam-3776	231	10	aj)∏m−1	aj)∏m−1	NOUN
ejpam-3776	231	11	j=1	j=1	NOUN
ejpam-3776	231	12	∏l−1	∏l−1	NOUN
ejpam-3776	232	1	k=0	k=0	PROPN
ejpam-3776	232	2	(	(	PUNCT
ejpam-3776	232	3	1	1	NUM
ejpam-3776	232	4	1−xij+k	1−xij+k	NOUN
ejpam-3776	232	5	)	)	PUNCT
ejpam-3776	233	1	λj+1	λj+1	PROPN
ejpam-3776	233	2			NOUN
ejpam-3776	233	3			ADV
ejpam-3776	233	4	≤	≤	NOUN
ejpam-3776	234	1			PROPN
ejpam-3776	234	2	m∑	m∑	ADV
ejpam-3776	234	3	j=1	j=1	NOUN
ejpam-3776	234	4	(	(	PUNCT
ejpam-3776	234	5	1	1	NUM
ejpam-3776	234	6	(	(	PUNCT
ejpam-3776	234	7	1−	1−	NUM
ejpam-3776	234	8	aj	aj	PROPN
ejpam-3776	234	9	)	)	PUNCT
ejpam-3776	234	10	)	)	PUNCT
ejpam-3776	234	11	−	−	PROPN
ejpam-3776	235	1	m−1∑	m−1∑	NUM
ejpam-3776	235	2	j=1	j=1	PROPN
ejpam-3776	235	3	n∑	n∑	PROPN
ejpam-3776	236	1	i=1	i=1	PROPN
ejpam-3776	237	1	wi	wi	PROPN
ejpam-3776	237	2	(	(	PUNCT
ejpam-3776	237	3	1	1	NUM
ejpam-3776	237	4	1−	1−	NUM
ejpam-3776	237	5	xij	xij	NOUN
ejpam-3776	237	6	)	)	PUNCT
ejpam-3776	237	7			PROPN
ejpam-3776	237	8	,	,	PUNCT
ejpam-3776	237	9	which	which	PRON
ejpam-3776	237	10	is	be	AUX
ejpam-3776	237	11	equivalent	equivalent	ADJ
ejpam-3776	237	12	to	to	ADP
ejpam-3776	237	13	(	(	PUNCT
ejpam-3776	237	14	ĝ′n	ĝ′n	NOUN
ejpam-3776	237	15	)	)	PUNCT
ejpam-3776	237	16	−1	−1	NOUN
ejpam-3776	237	17	≤	≤	NUM
ejpam-3776	238	1	n∑	n∑	PROPN
ejpam-3776	238	2	i=1	i=1	PROPN
ejpam-3776	238	3	wi	wi	PROPN
ejpam-3776	238	4	(	(	PUNCT
ejpam-3776	238	5	ĝ′	ĝ′	PROPN
ejpam-3776	238	6	(	(	PUNCT
ejpam-3776	238	7	λ	λ	PROPN
ejpam-3776	238	8	,	,	PUNCT
ejpam-3776	238	9	x	x	NOUN
ejpam-3776	238	10	)	)	PUNCT
ejpam-3776	238	11	)	)	PUNCT
ejpam-3776	238	12	−1	−1	NOUN
ejpam-3776	238	13	≤	≤	NOUN
ejpam-3776	239	1			PROPN
ejpam-3776	239	2	m∑	m∑	ADV
ejpam-3776	239	3	j=1	j=1	NOUN
ejpam-3776	239	4	(	(	PUNCT
ejpam-3776	239	5	1	1	NUM
ejpam-3776	239	6	(	(	PUNCT
ejpam-3776	239	7	1−	1−	NUM
ejpam-3776	239	8	aj	aj	PROPN
ejpam-3776	239	9	)	)	PUNCT
ejpam-3776	239	10	)	)	PUNCT
ejpam-3776	239	11	−	−	PROPN
ejpam-3776	240	1	m−1∑	m−1∑	NUM
ejpam-3776	240	2	j=1	j=1	PROPN
ejpam-3776	240	3	n∑	n∑	PROPN
ejpam-3776	241	1	i=1	i=1	PROPN
ejpam-3776	242	1	wi	wi	PROPN
ejpam-3776	242	2	(	(	PUNCT
ejpam-3776	242	3	1	1	NUM
ejpam-3776	242	4	1−	1−	NUM
ejpam-3776	242	5	xij	xij	NOUN
ejpam-3776	242	6	)	)	PUNCT
ejpam-3776	242	7			PROPN
ejpam-3776	242	8	,	,	PUNCT
ejpam-3776	242	9	finally	finally	ADV
ejpam-3776	242	10	,	,	PUNCT
ejpam-3776	242	11	we	we	PRON
ejpam-3776	242	12	obtain	obtain	VERB
ejpam-3776	242	13	(	(	PUNCT
ejpam-3776	242	14	ĝ′n	ĝ′n	NOUN
ejpam-3776	242	15	)	)	PUNCT
ejpam-3776	242	16	−1	−1	NOUN
ejpam-3776	242	17	≤	≤	NUM
ejpam-3776	243	1	n∑	n∑	PROPN
ejpam-3776	243	2	i=1	i=1	PROPN
ejpam-3776	243	3	wi	wi	PROPN
ejpam-3776	243	4	(	(	PUNCT
ejpam-3776	243	5	ĝ′	ĝ′	PROPN
ejpam-3776	243	6	(	(	PUNCT
ejpam-3776	243	7	λ	λ	PROPN
ejpam-3776	243	8	,	,	PUNCT
ejpam-3776	243	9	x	x	NOUN
ejpam-3776	243	10	)	)	PUNCT
ejpam-3776	243	11	)	)	PUNCT
ejpam-3776	244	1	−1	−1	NOUN
ejpam-3776	244	2	≤	≤	NOUN
ejpam-3776	244	3	(	(	PUNCT
ejpam-3776	244	4	ĥ	ĥ	PUNCT
ejpam-3776	244	5	′n	′n	NOUN
ejpam-3776	244	6	)	)	PUNCT
ejpam-3776	244	7	−1	−1	NOUN
ejpam-3776	244	8	,	,	PUNCT
ejpam-3776	244	9	which	which	PRON
ejpam-3776	244	10	completes	complete	VERB
ejpam-3776	244	11	the	the	DET
ejpam-3776	244	12	proof	proof	NOUN
ejpam-3776	244	13	.	.	PUNCT
ejpam-3776	245	1	we	we	PRON
ejpam-3776	245	2	would	would	AUX
ejpam-3776	245	3	also	also	ADV
ejpam-3776	245	4	establish	establish	VERB
ejpam-3776	245	5	refinements	refinement	NOUN
ejpam-3776	245	6	related	relate	VERB
ejpam-3776	245	7	to	to	ADP
ejpam-3776	245	8	arithmetic	arithmetic	ADJ
ejpam-3776	245	9	-	-	PUNCT
ejpam-3776	245	10	harmonic	harmonic	ADJ
ejpam-3776	245	11	means	mean	NOUN
ejpam-3776	245	12	inequalities	inequality	NOUN
ejpam-3776	245	13	as	as	SCONJ
ejpam-3776	245	14	follows	follow	VERB
ejpam-3776	245	15	:	:	PUNCT
ejpam-3776	245	16	s.	s.	PROPN
ejpam-3776	245	17	chanan	chanan	PROPN
ejpam-3776	245	18	,	,	PUNCT
ejpam-3776	245	19	a.	a.	PROPN
ejpam-3776	245	20	r.	r.	PROPN
ejpam-3776	245	21	khan	khan	PROPN
ejpam-3776	245	22	/	/	SYM
ejpam-3776	245	23	eur	eur	PROPN
ejpam-3776	245	24	.	.	PUNCT
ejpam-3776	246	1	j.	j.	PROPN
ejpam-3776	246	2	pure	pure	PROPN
ejpam-3776	246	3	appl	appl	PROPN
ejpam-3776	246	4	.	.	PROPN
ejpam-3776	246	5	math	math	PROPN
ejpam-3776	246	6	,	,	PUNCT
ejpam-3776	246	7	13	13	NUM
ejpam-3776	246	8	(	(	PUNCT
ejpam-3776	246	9	4	4	NUM
ejpam-3776	246	10	)	)	PUNCT
ejpam-3776	246	11	(	(	PUNCT
ejpam-3776	246	12	2020	2020	NUM
ejpam-3776	246	13	)	)	PUNCT
ejpam-3776	246	14	,	,	PUNCT
ejpam-3776	246	15	814	814	NUM
ejpam-3776	246	16	-	-	SYM
ejpam-3776	246	17	829	829	NUM
ejpam-3776	246	18	823	823	NUM
ejpam-3776	246	19	corollary	corollary	ADJ
ejpam-3776	246	20	4	4	NUM
ejpam-3776	246	21	.	.	PUNCT
ejpam-3776	247	1	let	let	VERB
ejpam-3776	247	2	the	the	DET
ejpam-3776	247	3	assumptions	assumption	NOUN
ejpam-3776	247	4	stated	state	VERB
ejpam-3776	247	5	in	in	ADP
ejpam-3776	247	6	theorem	theorem	NOUN
ejpam-3776	247	7	6	6	NUM
ejpam-3776	247	8	be	be	AUX
ejpam-3776	247	9	true	true	ADJ
ejpam-3776	247	10	.	.	PUNCT
ejpam-3776	248	1	then	then	ADV
ejpam-3776	248	2	following	follow	VERB
ejpam-3776	248	3	inequalities	inequality	NOUN
ejpam-3776	248	4	hold	hold	VERB
ejpam-3776	248	5	:	:	PUNCT
ejpam-3776	248	6	1	1	NUM
ejpam-3776	248	7	ân	ân	NOUN
ejpam-3776	248	8	≤	≤	NUM
ejpam-3776	248	9	n∑	n∑	PROPN
ejpam-3776	248	10	i=1	i=1	PROPN
ejpam-3776	249	1	wi	wi	PROPN
ejpam-3776	249	2	(	(	PUNCT
ejpam-3776	249	3	1	1	NUM
ejpam-3776	249	4	â(λ	â(λ	NOUN
ejpam-3776	249	5	,	,	PUNCT
ejpam-3776	249	6	x	x	NOUN
ejpam-3776	249	7	)	)	PUNCT
ejpam-3776	249	8	)	)	PUNCT
ejpam-3776	249	9	≤	≤	ADV
ejpam-3776	249	10	1	1	NUM
ejpam-3776	249	11	ĥn	ĥn	NOUN
ejpam-3776	249	12	,	,	PUNCT
ejpam-3776	249	13	(	(	PUNCT
ejpam-3776	249	14	9	9	NUM
ejpam-3776	249	15	)	)	SYM
ejpam-3776	249	16	1	1	NUM
ejpam-3776	249	17	â′n	â′n	NOUN
ejpam-3776	249	18	≤	≤	NUM
ejpam-3776	249	19	n∑	n∑	PROPN
ejpam-3776	249	20	i=1	i=1	PROPN
ejpam-3776	249	21	wi	wi	PROPN
ejpam-3776	249	22	(	(	PUNCT
ejpam-3776	249	23	1	1	NUM
ejpam-3776	249	24	â′(λ	â′(λ	PROPN
ejpam-3776	249	25	,	,	PUNCT
ejpam-3776	249	26	x	x	NOUN
ejpam-3776	249	27	)	)	PUNCT
ejpam-3776	249	28	)	)	PUNCT
ejpam-3776	249	29	≤	≤	NOUN
ejpam-3776	249	30	1	1	NUM
ejpam-3776	249	31	ĥ	ĥ	PUNCT
ejpam-3776	249	32	′n	′n	NOUN
ejpam-3776	249	33	.	.	PUNCT
ejpam-3776	250	1	(	(	PUNCT
ejpam-3776	250	2	10	10	NUM
ejpam-3776	250	3	)	)	PUNCT
ejpam-3776	250	4	proof	proof	NOUN
ejpam-3776	250	5	.	.	PUNCT
ejpam-3776	251	1	by	by	ADP
ejpam-3776	251	2	applying	apply	VERB
ejpam-3776	251	3	convex	convex	NOUN
ejpam-3776	251	4	function	function	NOUN
ejpam-3776	251	5	f(x	f(x	PROPN
ejpam-3776	251	6	)	)	PUNCT
ejpam-3776	251	7	=	=	SYM
ejpam-3776	251	8	1	1	NUM
ejpam-3776	251	9	x	x	NOUN
ejpam-3776	251	10	,	,	PUNCT
ejpam-3776	251	11	x	x	SYM
ejpam-3776	251	12	∈	∈	PROPN
ejpam-3776	251	13	(	(	PUNCT
ejpam-3776	251	14	0	0	NUM
ejpam-3776	251	15	,	,	PUNCT
ejpam-3776	251	16	12	12	NUM
ejpam-3776	251	17	]	]	PUNCT
ejpam-3776	251	18	to	to	PART
ejpam-3776	251	19	theorem	theorem	VERB
ejpam-3776	251	20	6	6	NUM
ejpam-3776	251	21	we	we	PRON
ejpam-3776	251	22	get	get	VERB
ejpam-3776	251	23	inequality	inequality	NOUN
ejpam-3776	251	24	(	(	PUNCT
ejpam-3776	251	25	9	9	NUM
ejpam-3776	251	26	)	)	PUNCT
ejpam-3776	251	27	.	.	PUNCT
ejpam-3776	252	1	similarly	similarly	ADV
ejpam-3776	252	2	,	,	PUNCT
ejpam-3776	252	3	by	by	ADP
ejpam-3776	252	4	using	use	VERB
ejpam-3776	252	5	convex	convex	PROPN
ejpam-3776	252	6	function	function	NOUN
ejpam-3776	252	7	f(x	f(x	PROPN
ejpam-3776	252	8	)	)	PUNCT
ejpam-3776	252	9	=	=	PUNCT
ejpam-3776	252	10	1	1	NUM
ejpam-3776	252	11	1−x	1−x	NUM
ejpam-3776	252	12	,	,	PUNCT
ejpam-3776	252	13	x	x	PUNCT
ejpam-3776	252	14	∈	∈	PROPN
ejpam-3776	252	15	(	(	PUNCT
ejpam-3776	252	16	0	0	NUM
ejpam-3776	252	17	,	,	PUNCT
ejpam-3776	252	18	12	12	NUM
ejpam-3776	252	19	]	]	PUNCT
ejpam-3776	252	20	to	to	PART
ejpam-3776	252	21	theorem	theorem	VERB
ejpam-3776	252	22	6	6	NUM
ejpam-3776	252	23	we	we	PRON
ejpam-3776	252	24	get	get	VERB
ejpam-3776	252	25	inequality	inequality	NOUN
ejpam-3776	252	26	(	(	PUNCT
ejpam-3776	252	27	10	10	NUM
ejpam-3776	252	28	)	)	PUNCT
ejpam-3776	252	29	.	.	PUNCT
ejpam-3776	253	1	we	we	PRON
ejpam-3776	253	2	establish	establish	VERB
ejpam-3776	253	3	a	a	DET
ejpam-3776	253	4	refinement	refinement	NOUN
ejpam-3776	253	5	of	of	ADP
ejpam-3776	253	6	the	the	DET
ejpam-3776	253	7	difference	difference	NOUN
ejpam-3776	253	8	of	of	ADP
ejpam-3776	253	9	the	the	DET
ejpam-3776	253	10	arithmetic	arithmetic	ADJ
ejpam-3776	253	11	and	and	CCONJ
ejpam-3776	253	12	harmonic	harmonic	ADJ
ejpam-3776	253	13	means	mean	NOUN
ejpam-3776	253	14	.	.	PUNCT
ejpam-3776	254	1	corollary	corollary	ADJ
ejpam-3776	254	2	5	5	NUM
ejpam-3776	254	3	.	.	PUNCT
ejpam-3776	255	1	let	let	VERB
ejpam-3776	255	2	the	the	DET
ejpam-3776	255	3	assumptions	assumption	NOUN
ejpam-3776	255	4	stated	state	VERB
ejpam-3776	255	5	in	in	ADP
ejpam-3776	255	6	theorem	theorem	NOUN
ejpam-3776	255	7	6	6	NUM
ejpam-3776	255	8	be	be	AUX
ejpam-3776	255	9	true	true	ADJ
ejpam-3776	255	10	.	.	PUNCT
ejpam-3776	256	1	then	then	ADV
ejpam-3776	256	2	following	follow	VERB
ejpam-3776	256	3	inequalities	inequality	NOUN
ejpam-3776	256	4	hold	hold	VERB
ejpam-3776	256	5	:	:	PUNCT
ejpam-3776	256	6	1	1	NUM
ejpam-3776	256	7	ân	ân	NOUN
ejpam-3776	256	8	−	−	NUM
ejpam-3776	256	9	1	1	NUM
ejpam-3776	256	10	â′n	â′n	NOUN
ejpam-3776	256	11	≤	≤	NUM
ejpam-3776	256	12	n∑	n∑	PROPN
ejpam-3776	256	13	i=1	i=1	PROPN
ejpam-3776	256	14	wi	wi	PROPN
ejpam-3776	256	15	(	(	PUNCT
ejpam-3776	256	16	1	1	NUM
ejpam-3776	256	17	â(λ	â(λ	NOUN
ejpam-3776	256	18	,	,	PUNCT
ejpam-3776	256	19	x	x	NOUN
ejpam-3776	256	20	)	)	PUNCT
ejpam-3776	256	21	−	−	PROPN
ejpam-3776	256	22	1	1	NUM
ejpam-3776	257	1	â′(λ	â′(λ	PROPN
ejpam-3776	257	2	,	,	PUNCT
ejpam-3776	257	3	x	x	NOUN
ejpam-3776	257	4	)	)	PUNCT
ejpam-3776	257	5	)	)	PUNCT
ejpam-3776	258	1	≤	≤	ADV
ejpam-3776	258	2	1	1	NUM
ejpam-3776	258	3	ĥn	ĥn	NOUN
ejpam-3776	258	4	−	−	PROPN
ejpam-3776	258	5	1	1	NUM
ejpam-3776	258	6	ĥ	ĥ	X
ejpam-3776	258	7	′n	′n	NOUN
ejpam-3776	258	8	.	.	PUNCT
ejpam-3776	259	1	proof	proof	NOUN
ejpam-3776	259	2	.	.	PUNCT
ejpam-3776	260	1	by	by	ADP
ejpam-3776	260	2	applying	apply	VERB
ejpam-3776	260	3	convex	convex	NOUN
ejpam-3776	260	4	function	function	NOUN
ejpam-3776	260	5	f(x	f(x	PROPN
ejpam-3776	260	6	)	)	PUNCT
ejpam-3776	260	7	=	=	SYM
ejpam-3776	260	8	1	1	NUM
ejpam-3776	260	9	x	x	SYM
ejpam-3776	260	10	−	−	PROPN
ejpam-3776	260	11	1	1	NUM
ejpam-3776	260	12	1−x	1−x	NUM
ejpam-3776	260	13	,	,	PUNCT
ejpam-3776	260	14	x	x	PUNCT
ejpam-3776	260	15	∈	∈	PROPN
ejpam-3776	260	16	(	(	PUNCT
ejpam-3776	260	17	0	0	NUM
ejpam-3776	260	18	,	,	PUNCT
ejpam-3776	260	19	12	12	NUM
ejpam-3776	260	20	]	]	PUNCT
ejpam-3776	260	21	to	to	PART
ejpam-3776	260	22	theorem	theorem	VERB
ejpam-3776	260	23	6	6	NUM
ejpam-3776	260	24	we	we	PRON
ejpam-3776	260	25	obtain	obtain	VERB
ejpam-3776	260	26	required	required	ADJ
ejpam-3776	260	27	result	result	NOUN
ejpam-3776	260	28	.	.	PUNCT
ejpam-3776	261	1	4	4	X
ejpam-3776	261	2	.	.	X
ejpam-3776	261	3	cyclic	cyclic	ADJ
ejpam-3776	261	4	mixed	mix	VERB
ejpam-3776	261	5	symmetric	symmetric	ADJ
ejpam-3776	261	6	means	mean	VERB
ejpam-3776	261	7	the	the	DET
ejpam-3776	261	8	jensen	jensen	PROPN
ejpam-3776	261	9	’s	’s	PART
ejpam-3776	261	10	inequality	inequality	PROPN
ejpam-3776	261	11	and	and	CCONJ
ejpam-3776	261	12	jensen	jensen	PROPN
ejpam-3776	261	13	-	-	PUNCT
ejpam-3776	261	14	mercer	mercer	PROPN
ejpam-3776	261	15	inequality	inequality	NOUN
ejpam-3776	261	16	are	be	AUX
ejpam-3776	261	17	much	much	ADV
ejpam-3776	261	18	fertile	fertile	ADJ
ejpam-3776	261	19	to	to	PART
ejpam-3776	261	20	study	study	VERB
ejpam-3776	261	21	about	about	ADP
ejpam-3776	261	22	mixed	mixed	ADJ
ejpam-3776	261	23	means	mean	NOUN
ejpam-3776	261	24	(	(	PUNCT
ejpam-3776	261	25	see	see	VERB
ejpam-3776	261	26	[	[	X
ejpam-3776	261	27	11	11	NUM
ejpam-3776	261	28	]	]	PUNCT
ejpam-3776	261	29	and	and	CCONJ
ejpam-3776	261	30	[	[	X
ejpam-3776	261	31	21	21	NUM
ejpam-3776	261	32	]	]	PUNCT
ejpam-3776	261	33	)	)	PUNCT
ejpam-3776	261	34	.	.	PUNCT
ejpam-3776	262	1	let	let	VERB
ejpam-3776	262	2	the	the	DET
ejpam-3776	262	3	assumptions	assumption	NOUN
ejpam-3776	262	4	stated	state	VERB
ejpam-3776	262	5	in	in	ADP
ejpam-3776	262	6	theorem	theorem	NOUN
ejpam-3776	262	7	6	6	NUM
ejpam-3776	262	8	be	be	AUX
ejpam-3776	262	9	true	true	ADJ
ejpam-3776	262	10	.	.	PUNCT
ejpam-3776	263	1	then	then	ADV
ejpam-3776	263	2	we	we	PRON
ejpam-3776	263	3	define	define	VERB
ejpam-3776	263	4	power	power	NOUN
ejpam-3776	263	5	mean	mean	NOUN
ejpam-3776	263	6	of	of	ADP
ejpam-3776	263	7	the	the	DET
ejpam-3776	263	8	order	order	NOUN
ejpam-3776	263	9	r	r	NOUN
ejpam-3776	263	10	∈	∈	NOUN
ejpam-3776	263	11	r	r	NOUN
ejpam-3776	263	12	,	,	PUNCT
ejpam-3776	263	13	for	for	ADP
ejpam-3776	263	14	positive	positive	ADJ
ejpam-3776	263	15	n	n	CCONJ
ejpam-3776	263	16	-	-	PUNCT
ejpam-3776	263	17	tuple	tuple	NOUN
ejpam-3776	263	18	x	x	PUNCT
ejpam-3776	263	19	as	as	SCONJ
ejpam-3776	263	20	follows	follow	VERB
ejpam-3776	263	21	:	:	PUNCT
ejpam-3776	264	1	m̂r	m̂r	PROPN
ejpam-3776	264	2	(	(	PUNCT
ejpam-3776	264	3	xij	xij	NOUN
ejpam-3776	264	4	,	,	PUNCT
ejpam-3776	264	5	.	.	PUNCT
ejpam-3776	264	6	.	.	PUNCT
ejpam-3776	264	7	.	.	PUNCT
ejpam-3776	265	1	,	,	PUNCT
ejpam-3776	265	2	xij+l−1;λ1	xij+l−1;λ1	PROPN
ejpam-3776	265	3	,	,	PUNCT
ejpam-3776	265	4	.	.	PUNCT
ejpam-3776	265	5	.	.	PUNCT
ejpam-3776	266	1	.	.	PUNCT
ejpam-3776	267	1	,	,	PUNCT
ejpam-3776	267	2	λl	λl	X
ejpam-3776	267	3	)	)	PUNCT
ejpam-3776	267	4	=	=	SYM
ejpam-3776	267	5			PROPN
ejpam-3776	268	1	∑m	∑m	PROPN
ejpam-3776	268	2	j=1	j=1	PROPN
ejpam-3776	268	3	a	a	DET
ejpam-3776	268	4	r	r	NOUN
ejpam-3776	268	5	j	j	NOUN
ejpam-3776	268	6	−	−	NOUN
ejpam-3776	269	1	m−1∑	m−1∑	NUM
ejpam-3776	269	2	j=1	j=1	NOUN
ejpam-3776	269	3	l−1∑	l−1∑	PRON
ejpam-3776	269	4	k=0	k=0	PROPN
ejpam-3776	269	5	λk+1x	λk+1x	NOUN
ejpam-3776	269	6	r	r	PROPN
ejpam-3776	269	7	ij+k	ij+k	NOUN
ejpam-3776	269	8			PROPN
ejpam-3776	269	9	1	1	NUM
ejpam-3776	269	10	r	r	NOUN
ejpam-3776	269	11	,	,	PUNCT
ejpam-3776	269	12	r	r	NOUN
ejpam-3776	269	13	6=	6=	NUM
ejpam-3776	269	14	0	0	NUM
ejpam-3776	269	15	,	,	PUNCT
ejpam-3776	269	16	m∏	m∏	PROPN
ejpam-3776	269	17	j=1	j=1	PROPN
ejpam-3776	269	18	aj	aj	PROPN
ejpam-3776	269	19	m−1∏	m−1∏	PROPN
ejpam-3776	269	20	j=1	j=1	PROPN
ejpam-3776	269	21	l−1∏	l−1∏	PROPN
ejpam-3776	269	22	k=0	k=0	PROPN
ejpam-3776	269	23	(	(	PUNCT
ejpam-3776	269	24	xij+k	xij+k	PROPN
ejpam-3776	269	25	)	)	PUNCT
ejpam-3776	269	26	λk+1	λk+1	X
ejpam-3776	269	27	,	,	PUNCT
ejpam-3776	269	28	r	r	NOUN
ejpam-3776	269	29	=	=	SYM
ejpam-3776	269	30	0	0	NUM
ejpam-3776	269	31	,	,	PUNCT
ejpam-3776	269	32	s.	s.	PROPN
ejpam-3776	269	33	chanan	chanan	PROPN
ejpam-3776	269	34	,	,	PUNCT
ejpam-3776	269	35	a.	a.	PROPN
ejpam-3776	269	36	r.	r.	PROPN
ejpam-3776	269	37	khan	khan	PROPN
ejpam-3776	269	38	/	/	SYM
ejpam-3776	269	39	eur	eur	PROPN
ejpam-3776	269	40	.	.	PUNCT
ejpam-3776	270	1	j.	j.	PROPN
ejpam-3776	270	2	pure	pure	PROPN
ejpam-3776	270	3	appl	appl	PROPN
ejpam-3776	270	4	.	.	PROPN
ejpam-3776	270	5	math	math	PROPN
ejpam-3776	270	6	,	,	PUNCT
ejpam-3776	270	7	13	13	NUM
ejpam-3776	270	8	(	(	PUNCT
ejpam-3776	270	9	4	4	NUM
ejpam-3776	270	10	)	)	PUNCT
ejpam-3776	270	11	(	(	PUNCT
ejpam-3776	270	12	2020	2020	NUM
ejpam-3776	270	13	)	)	PUNCT
ejpam-3776	270	14	,	,	PUNCT
ejpam-3776	270	15	814	814	NUM
ejpam-3776	270	16	-	-	SYM
ejpam-3776	270	17	829	829	NUM
ejpam-3776	270	18	824	824	NUM
ejpam-3776	270	19	and	and	CCONJ
ejpam-3776	270	20	cyclic	cyclic	ADJ
ejpam-3776	270	21	mixed	mixed	ADJ
ejpam-3776	270	22	symmetric	symmetric	ADJ
ejpam-3776	270	23	means	mean	NOUN
ejpam-3776	270	24	corresponding	correspond	VERB
ejpam-3776	270	25	to	to	ADP
ejpam-3776	270	26	(	(	PUNCT
ejpam-3776	270	27	6	6	NUM
ejpam-3776	270	28	)	)	PUNCT
ejpam-3776	270	29	is	be	AUX
ejpam-3776	270	30	given	give	VERB
ejpam-3776	270	31	as	as	ADP
ejpam-3776	270	32	:	:	PUNCT
ejpam-3776	270	33	m̂r	m̂r	NUM
ejpam-3776	270	34	,	,	PUNCT
ejpam-3776	270	35	s(x	s(x	PROPN
ejpam-3776	270	36	,	,	PUNCT
ejpam-3776	270	37	λ	λ	NOUN
ejpam-3776	270	38	)	)	PUNCT
ejpam-3776	270	39	=	=	SYM
ejpam-3776	270	40			PROPN
ejpam-3776	271	1	(	(	PUNCT
ejpam-3776	271	2	n∑	n∑	NOUN
ejpam-3776	271	3	i=1	i=1	PROPN
ejpam-3776	272	1	wim̂	wim̂	PROPN
ejpam-3776	272	2	s	s	PART
ejpam-3776	272	3	r	r	NOUN
ejpam-3776	272	4	(	(	PUNCT
ejpam-3776	272	5	xij	xij	NOUN
ejpam-3776	272	6	,	,	PUNCT
ejpam-3776	272	7	·	·	PUNCT
ejpam-3776	272	8	·	·	PUNCT
ejpam-3776	272	9	·	·	PUNCT
ejpam-3776	272	10	,	,	PUNCT
ejpam-3776	272	11	xij+l−1	xij+l−1	PROPN
ejpam-3776	272	12	,	,	PUNCT
ejpam-3776	272	13	λ1	λ1	ADJ
ejpam-3776	272	14	,	,	PUNCT
ejpam-3776	272	15	·	·	PUNCT
ejpam-3776	272	16	·	·	PUNCT
ejpam-3776	272	17	·	·	PUNCT
ejpam-3776	272	18	,	,	PUNCT
ejpam-3776	272	19	λl	λl	X
ejpam-3776	272	20	)	)	PUNCT
ejpam-3776	272	21	)	)	PUNCT
ejpam-3776	272	22	1	1	NUM
ejpam-3776	272	23	s	s	NOUN
ejpam-3776	272	24	,	,	PUNCT
ejpam-3776	272	25	s	s	PROPN
ejpam-3776	272	26	6=	6=	NUM
ejpam-3776	272	27	0	0	NUM
ejpam-3776	272	28	,	,	PUNCT
ejpam-3776	272	29	(	(	PUNCT
ejpam-3776	272	30	n∏	n∏	PROPN
ejpam-3776	272	31	i=1	i=1	PROPN
ejpam-3776	272	32	m̂r	m̂r	PROPN
ejpam-3776	272	33	(	(	PUNCT
ejpam-3776	272	34	xij	xij	NOUN
ejpam-3776	272	35	,	,	PUNCT
ejpam-3776	272	36	·	·	PUNCT
ejpam-3776	272	37	·	·	PUNCT
ejpam-3776	272	38	·	·	PUNCT
ejpam-3776	272	39	,	,	PUNCT
ejpam-3776	272	40	xij+l−1	xij+l−1	PROPN
ejpam-3776	272	41	,	,	PUNCT
ejpam-3776	272	42	λ1	λ1	ADJ
ejpam-3776	272	43	,	,	PUNCT
ejpam-3776	272	44	·	·	PUNCT
ejpam-3776	272	45	·	·	PUNCT
ejpam-3776	272	46	·	·	PUNCT
ejpam-3776	272	47	,	,	PUNCT
ejpam-3776	272	48	λl	λl	X
ejpam-3776	272	49	)	)	PUNCT
ejpam-3776	272	50	)	)	PUNCT
ejpam-3776	272	51	wi	wi	PROPN
ejpam-3776	272	52	,	,	PUNCT
ejpam-3776	272	53	s	s	PART
ejpam-3776	272	54	=	=	NOUN
ejpam-3776	272	55	0	0	PROPN
ejpam-3776	272	56	.	.	PUNCT
ejpam-3776	273	1	the	the	DET
ejpam-3776	273	2	standard	standard	ADJ
ejpam-3776	273	3	power	power	NOUN
ejpam-3776	273	4	mean	mean	NOUN
ejpam-3776	273	5	of	of	ADP
ejpam-3776	273	6	order	order	NOUN
ejpam-3776	273	7	r	r	NOUN
ejpam-3776	273	8	∈	∈	NOUN
ejpam-3776	273	9	r	r	NOUN
ejpam-3776	273	10	for	for	ADP
ejpam-3776	273	11	the	the	DET
ejpam-3776	273	12	positive	positive	ADJ
ejpam-3776	273	13	n	n	CCONJ
ejpam-3776	273	14	-	-	PUNCT
ejpam-3776	273	15	tuple	tuple	NOUN
ejpam-3776	273	16	x	x	X
ejpam-3776	273	17	are	be	AUX
ejpam-3776	273	18	defined	define	VERB
ejpam-3776	273	19	as	as	SCONJ
ejpam-3776	273	20	follows	follow	VERB
ejpam-3776	273	21	:	:	PUNCT
ejpam-3776	273	22	m̂r	m̂r	NUM
ejpam-3776	273	23	(	(	PUNCT
ejpam-3776	273	24	x	x	X
ejpam-3776	273	25	)	)	PUNCT
ejpam-3776	273	26	=	=	SYM
ejpam-3776	273	27			PROPN
ejpam-3776	273	28	∑m	∑m	PROPN
ejpam-3776	274	1	j=1	j=1	NOUN
ejpam-3776	274	2	a	a	DET
ejpam-3776	274	3	r	r	NOUN
ejpam-3776	274	4	j	j	NOUN
ejpam-3776	274	5	−	−	PROPN
ejpam-3776	275	1	m−1∑	m−1∑	NUM
ejpam-3776	275	2	j=1	j=1	PROPN
ejpam-3776	275	3	n∑	n∑	PROPN
ejpam-3776	275	4	i=1	i=1	PROPN
ejpam-3776	275	5	wix	wix	PROPN
ejpam-3776	275	6	r	r	NOUN
ejpam-3776	275	7	ij	ij	NUM
ejpam-3776	275	8			PROPN
ejpam-3776	275	9	1	1	NUM
ejpam-3776	275	10	r	r	NOUN
ejpam-3776	275	11	,	,	PUNCT
ejpam-3776	275	12	r	r	NOUN
ejpam-3776	275	13	6=	6=	NUM
ejpam-3776	275	14	0,∏m	0,∏m	NUM
ejpam-3776	275	15	j=1	j=1	PROPN
ejpam-3776	275	16	aj	aj	PROPN
ejpam-3776	275	17	m−1∏	m−1∏	PROPN
ejpam-3776	275	18	j=1	j=1	PROPN
ejpam-3776	275	19	n∏	n∏	PROPN
ejpam-3776	275	20	i=1	i=1	PROPN
ejpam-3776	275	21	(	(	PUNCT
ejpam-3776	275	22	xij	xij	X
ejpam-3776	275	23	)	)	PUNCT
ejpam-3776	275	24	wi	wi	PROPN
ejpam-3776	275	25	,	,	PUNCT
ejpam-3776	275	26	r	r	NOUN
ejpam-3776	275	27	=	=	SYM
ejpam-3776	275	28	0	0	X
ejpam-3776	275	29	.	.	PUNCT
ejpam-3776	275	30	corollary	corollary	ADJ
ejpam-3776	275	31	6	6	NUM
ejpam-3776	275	32	.	.	PUNCT
ejpam-3776	276	1	for	for	ADP
ejpam-3776	276	2	r	r	NOUN
ejpam-3776	276	3	≤	≤	NUM
ejpam-3776	276	4	1	1	NUM
ejpam-3776	276	5	and	and	CCONJ
ejpam-3776	276	6	by	by	ADP
ejpam-3776	276	7	considering	consider	VERB
ejpam-3776	276	8	the	the	DET
ejpam-3776	276	9	assumptions	assumption	NOUN
ejpam-3776	276	10	stated	state	VERB
ejpam-3776	276	11	in	in	ADP
ejpam-3776	276	12	(	(	PUNCT
ejpam-3776	276	13	c1	c1	PROPN
ejpam-3776	276	14	)	)	PUNCT
ejpam-3776	276	15	for	for	ADP
ejpam-3776	276	16	positive	positive	ADJ
ejpam-3776	276	17	m	m	NOUN
ejpam-3776	276	18	-	-	NOUN
ejpam-3776	276	19	tuple	tuple	NOUN
ejpam-3776	276	20	a	a	PRON
ejpam-3776	276	21	and	and	CCONJ
ejpam-3776	276	22	x	x	ADP
ejpam-3776	276	23	,	,	PUNCT
ejpam-3776	276	24	the	the	DET
ejpam-3776	276	25	following	follow	VERB
ejpam-3776	276	26	inequality	inequality	NOUN
ejpam-3776	276	27	hold	hold	NOUN
ejpam-3776	276	28	.	.	PUNCT
ejpam-3776	277	1	m̂r	m̂r	X
ejpam-3776	277	2	(	(	PUNCT
ejpam-3776	277	3	x	x	NOUN
ejpam-3776	277	4	)	)	PUNCT
ejpam-3776	277	5	≤	≤	NUM
ejpam-3776	277	6	m̂r	m̂r	X
ejpam-3776	277	7	(	(	PUNCT
ejpam-3776	277	8	x	x	NOUN
ejpam-3776	277	9	,	,	PUNCT
ejpam-3776	277	10	λ	λ	NOUN
ejpam-3776	277	11	)	)	PUNCT
ejpam-3776	277	12	≤	≤	NOUN
ejpam-3776	278	1	ân	ân	PROPN
ejpam-3776	278	2	.	.	PUNCT
ejpam-3776	279	1	(	(	PUNCT
ejpam-3776	279	2	11	11	NUM
ejpam-3776	279	3	)	)	PUNCT
ejpam-3776	279	4	for	for	ADP
ejpam-3776	279	5	r	r	NOUN
ejpam-3776	279	6	≥	≥	NOUN
ejpam-3776	279	7	0	0	NUM
ejpam-3776	279	8	,	,	PUNCT
ejpam-3776	279	9	the	the	DET
ejpam-3776	279	10	inequality	inequality	NOUN
ejpam-3776	279	11	(	(	PUNCT
ejpam-3776	279	12	11	11	NUM
ejpam-3776	279	13	)	)	PUNCT
ejpam-3776	279	14	is	be	AUX
ejpam-3776	279	15	reversed	reverse	VERB
ejpam-3776	279	16	.	.	PUNCT
ejpam-3776	280	1	proof	proof	NOUN
ejpam-3776	280	2	.	.	PUNCT
ejpam-3776	281	1	for	for	ADP
ejpam-3776	281	2	r	r	NOUN
ejpam-3776	281	3	≤	≤	NUM
ejpam-3776	281	4	1	1	NUM
ejpam-3776	281	5	,	,	PUNCT
ejpam-3776	281	6	r	r	NOUN
ejpam-3776	281	7	6=	6=	PROPN
ejpam-3776	281	8	0	0	NUM
ejpam-3776	281	9	,	,	PUNCT
ejpam-3776	281	10	by	by	ADP
ejpam-3776	281	11	applying	apply	VERB
ejpam-3776	281	12	the	the	DET
ejpam-3776	281	13	convex	convex	NOUN
ejpam-3776	281	14	function	function	NOUN
ejpam-3776	281	15	φ(x	φ(x	NOUN
ejpam-3776	281	16	)	)	PUNCT
ejpam-3776	281	17	=	=	SYM
ejpam-3776	281	18	x	x	SYM
ejpam-3776	281	19	1	1	NUM
ejpam-3776	281	20	r	r	NOUN
ejpam-3776	281	21	to	to	PART
ejpam-3776	281	22	theorem	theorem	VERB
ejpam-3776	281	23	6	6	NUM
ejpam-3776	281	24	and	and	CCONJ
ejpam-3776	281	25	replacing	replace	VERB
ejpam-3776	281	26	aj	aj	PROPN
ejpam-3776	281	27	and	and	CCONJ
ejpam-3776	281	28	xij	xij	PRON
ejpam-3776	281	29	with	with	ADP
ejpam-3776	281	30	arj	arj	PROPN
ejpam-3776	281	31	and	and	CCONJ
ejpam-3776	281	32	xrij	xrij	PROPN
ejpam-3776	281	33	respectively	respectively	ADV
ejpam-3776	281	34	and	and	CCONJ
ejpam-3776	281	35	for	for	ADP
ejpam-3776	281	36	r	r	NOUN
ejpam-3776	281	37	=	=	SYM
ejpam-3776	281	38	0	0	NUM
ejpam-3776	281	39	applying	apply	VERB
ejpam-3776	281	40	convex	convex	NOUN
ejpam-3776	281	41	function	function	NOUN
ejpam-3776	281	42	φ(x	φ(x	NOUN
ejpam-3776	281	43	)	)	PUNCT
ejpam-3776	281	44	=	=	SYM
ejpam-3776	281	45	exp(x	exp(x	PROPN
ejpam-3776	281	46	)	)	PUNCT
ejpam-3776	281	47	to	to	ADP
ejpam-3776	281	48	the	the	DET
ejpam-3776	281	49	theorem	theorem	ADJ
ejpam-3776	281	50	6	6	NUM
ejpam-3776	281	51	,	,	PUNCT
ejpam-3776	281	52	replacing	replace	VERB
ejpam-3776	281	53	ajand	ajand	NOUN
ejpam-3776	281	54	xij	xij	PROPN
ejpam-3776	281	55	with	with	ADP
ejpam-3776	281	56	ln	ln	PROPN
ejpam-3776	281	57	aj	aj	PROPN
ejpam-3776	281	58	and	and	CCONJ
ejpam-3776	281	59	lnxij	lnxij	PROPN
ejpam-3776	281	60	respectively	respectively	ADV
ejpam-3776	281	61	,	,	PUNCT
ejpam-3776	281	62	we	we	PRON
ejpam-3776	281	63	obtain	obtain	VERB
ejpam-3776	281	64	11	11	NUM
ejpam-3776	281	65	.	.	PUNCT
ejpam-3776	282	1	if	if	SCONJ
ejpam-3776	282	2	r	r	NOUN
ejpam-3776	282	3	≥	≥	NUM
ejpam-3776	282	4	1	1	NUM
ejpam-3776	282	5	,	,	PUNCT
ejpam-3776	282	6	then	then	ADV
ejpam-3776	282	7	the	the	DET
ejpam-3776	282	8	function	function	NOUN
ejpam-3776	282	9	φ(x	φ(x	NOUN
ejpam-3776	282	10	)	)	PUNCT
ejpam-3776	282	11	=	=	SYM
ejpam-3776	282	12	x	x	SYM
ejpam-3776	282	13	1	1	NUM
ejpam-3776	282	14	r	r	NOUN
ejpam-3776	282	15	is	be	AUX
ejpam-3776	282	16	concave	concave	VERB
ejpam-3776	282	17	,	,	PUNCT
ejpam-3776	282	18	so	so	SCONJ
ejpam-3776	282	19	the	the	DET
ejpam-3776	282	20	inequalities	inequality	NOUN
ejpam-3776	282	21	in	in	ADP
ejpam-3776	282	22	(	(	PUNCT
ejpam-3776	282	23	11	11	NUM
ejpam-3776	282	24	)	)	PUNCT
ejpam-3776	282	25	is	be	AUX
ejpam-3776	282	26	reversed	reverse	VERB
ejpam-3776	282	27	.	.	PUNCT
ejpam-3776	283	1	now	now	ADV
ejpam-3776	283	2	,	,	PUNCT
ejpam-3776	283	3	we	we	PRON
ejpam-3776	283	4	define	define	VERB
ejpam-3776	283	5	the	the	DET
ejpam-3776	283	6	bounds	bound	NOUN
ejpam-3776	283	7	for	for	ADP
ejpam-3776	283	8	power	power	NOUN
ejpam-3776	283	9	mean	mean	NOUN
ejpam-3776	283	10	and	and	CCONJ
ejpam-3776	283	11	cyclic	cyclic	ADJ
ejpam-3776	283	12	mixed	mixed	ADJ
ejpam-3776	283	13	symmetric	symmetric	ADJ
ejpam-3776	283	14	means	mean	NOUN
ejpam-3776	283	15	as	as	SCONJ
ejpam-3776	283	16	follows	follow	VERB
ejpam-3776	283	17	:	:	PUNCT
ejpam-3776	283	18	corollary	corollary	ADJ
ejpam-3776	283	19	7	7	X
ejpam-3776	283	20	.	.	PUNCT
ejpam-3776	284	1	let	let	VERB
ejpam-3776	284	2	r	r	NOUN
ejpam-3776	284	3	,	,	PUNCT
ejpam-3776	284	4	s	s	PART
ejpam-3776	284	5	∈	∈	NOUN
ejpam-3776	284	6	r	r	NOUN
ejpam-3776	284	7	such	such	ADJ
ejpam-3776	284	8	that	that	DET
ejpam-3776	284	9	r	r	NOUN
ejpam-3776	284	10	≤	≤	NUM
ejpam-3776	284	11	s	s	VERB
ejpam-3776	284	12	and	and	CCONJ
ejpam-3776	284	13	considering	consider	VERB
ejpam-3776	284	14	the	the	DET
ejpam-3776	284	15	assumption	assumption	NOUN
ejpam-3776	284	16	stated	state	VERB
ejpam-3776	284	17	in	in	ADP
ejpam-3776	284	18	(	(	PUNCT
ejpam-3776	284	19	c1	c1	PROPN
ejpam-3776	284	20	)	)	PUNCT
ejpam-3776	284	21	for	for	ADP
ejpam-3776	284	22	positive	positive	ADJ
ejpam-3776	284	23	n	n	CCONJ
ejpam-3776	284	24	-	-	PUNCT
ejpam-3776	284	25	tuple	tuple	NOUN
ejpam-3776	284	26	x	x	NOUN
ejpam-3776	284	27	,	,	PUNCT
ejpam-3776	284	28	following	follow	VERB
ejpam-3776	284	29	inequalities	inequality	NOUN
ejpam-3776	284	30	hold	hold	VERB
ejpam-3776	284	31	.	.	PUNCT
ejpam-3776	285	1	m̂r	m̂r	X
ejpam-3776	285	2	(	(	PUNCT
ejpam-3776	285	3	x	x	NOUN
ejpam-3776	285	4	)	)	PUNCT
ejpam-3776	285	5	≤	≤	NOUN
ejpam-3776	285	6	m̂r	m̂r	NOUN
ejpam-3776	285	7	,	,	PUNCT
ejpam-3776	285	8	s	s	PART
ejpam-3776	285	9	(	(	PUNCT
ejpam-3776	285	10	x	x	NOUN
ejpam-3776	285	11	,	,	PUNCT
ejpam-3776	285	12	λ	λ	NOUN
ejpam-3776	285	13	)	)	PUNCT
ejpam-3776	285	14	≤	≤	NOUN
ejpam-3776	285	15	m̂s	m̂s	X
ejpam-3776	285	16	(	(	PUNCT
ejpam-3776	285	17	x	x	NOUN
ejpam-3776	285	18	)	)	PUNCT
ejpam-3776	285	19	.	.	PUNCT
ejpam-3776	286	1	(	(	PUNCT
ejpam-3776	286	2	12	12	NUM
ejpam-3776	286	3	)	)	PUNCT
ejpam-3776	286	4	proof	proof	NOUN
ejpam-3776	286	5	.	.	PUNCT
ejpam-3776	287	1	let	let	VERB
ejpam-3776	287	2	r	r	NOUN
ejpam-3776	287	3	,	,	PUNCT
ejpam-3776	287	4	s	s	PART
ejpam-3776	287	5	6=	6=	NUM
ejpam-3776	287	6	0	0	NUM
ejpam-3776	287	7	.	.	PUNCT
ejpam-3776	288	1	by	by	ADP
ejpam-3776	288	2	applying	apply	VERB
ejpam-3776	288	3	theorem	theorem	NOUN
ejpam-3776	288	4	6	6	NUM
ejpam-3776	288	5	for	for	ADP
ejpam-3776	288	6	convex	convex	ADJ
ejpam-3776	288	7	function	function	NOUN
ejpam-3776	288	8	φ	φ	PROPN
ejpam-3776	288	9	(	(	PUNCT
ejpam-3776	288	10	x	x	NOUN
ejpam-3776	288	11	)	)	PUNCT
ejpam-3776	288	12	=	=	PUNCT
ejpam-3776	288	13	x	x	SYM
ejpam-3776	288	14	s	s	NOUN
ejpam-3776	288	15	r	r	NOUN
ejpam-3776	288	16	,	,	PUNCT
ejpam-3776	288	17	x	x	X
ejpam-3776	288	18	>	>	X
ejpam-3776	288	19	0	0	PUNCT
ejpam-3776	288	20	and	and	CCONJ
ejpam-3776	288	21	by	by	ADP
ejpam-3776	288	22	replacing	replace	VERB
ejpam-3776	288	23	aj	aj	PROPN
ejpam-3776	288	24	and	and	CCONJ
ejpam-3776	288	25	positive	positive	ADJ
ejpam-3776	288	26	n	n	CCONJ
ejpam-3776	288	27	-	-	PUNCT
ejpam-3776	288	28	tuple	tuple	NOUN
ejpam-3776	288	29	x	x	PUNCT
ejpam-3776	288	30	by	by	ADP
ejpam-3776	288	31	arj	arj	PROPN
ejpam-3776	288	32	and	and	CCONJ
ejpam-3776	288	33	(	(	PUNCT
ejpam-3776	288	34	xr	xr	NOUN
ejpam-3776	288	35	)	)	PUNCT
ejpam-3776	288	36	respectively	respectively	ADV
ejpam-3776	288	37	,	,	PUNCT
ejpam-3776	288	38	and	and	CCONJ
ejpam-3776	288	39	then	then	ADV
ejpam-3776	288	40	raising	raise	VERB
ejpam-3776	288	41	the	the	DET
ejpam-3776	288	42	power	power	NOUN
ejpam-3776	288	43	1	1	NUM
ejpam-3776	288	44	s	s	PART
ejpam-3776	288	45	we	we	PRON
ejpam-3776	288	46	get	get	VERB
ejpam-3776	288	47	,	,	PUNCT
ejpam-3776	288	48	m̂r	m̂r	X
ejpam-3776	288	49	(	(	PUNCT
ejpam-3776	288	50	x	x	NOUN
ejpam-3776	288	51	)	)	PUNCT
ejpam-3776	288	52	≤	≤	NOUN
ejpam-3776	288	53	m̂r	m̂r	NOUN
ejpam-3776	288	54	,	,	PUNCT
ejpam-3776	288	55	s	s	PART
ejpam-3776	288	56	(	(	PUNCT
ejpam-3776	288	57	x	x	NOUN
ejpam-3776	288	58	,	,	PUNCT
ejpam-3776	288	59	λ	λ	NOUN
ejpam-3776	288	60	)	)	PUNCT
ejpam-3776	288	61	≤	≤	NOUN
ejpam-3776	288	62	m̂s	m̂s	X
ejpam-3776	288	63	(	(	PUNCT
ejpam-3776	288	64	x	x	NOUN
ejpam-3776	288	65	)	)	PUNCT
ejpam-3776	288	66	.	.	PUNCT
ejpam-3776	289	1	s.	s.	PROPN
ejpam-3776	289	2	chanan	chanan	PROPN
ejpam-3776	289	3	,	,	PUNCT
ejpam-3776	289	4	a.	a.	PROPN
ejpam-3776	289	5	r.	r.	PROPN
ejpam-3776	289	6	khan	khan	PROPN
ejpam-3776	289	7	/	/	SYM
ejpam-3776	289	8	eur	eur	PROPN
ejpam-3776	289	9	.	.	PUNCT
ejpam-3776	290	1	j.	j.	PROPN
ejpam-3776	290	2	pure	pure	PROPN
ejpam-3776	290	3	appl	appl	PROPN
ejpam-3776	290	4	.	.	PROPN
ejpam-3776	290	5	math	math	PROPN
ejpam-3776	290	6	,	,	PUNCT
ejpam-3776	290	7	13	13	NUM
ejpam-3776	290	8	(	(	PUNCT
ejpam-3776	290	9	4	4	NUM
ejpam-3776	290	10	)	)	PUNCT
ejpam-3776	290	11	(	(	PUNCT
ejpam-3776	290	12	2020	2020	NUM
ejpam-3776	290	13	)	)	PUNCT
ejpam-3776	290	14	,	,	PUNCT
ejpam-3776	290	15	814	814	NUM
ejpam-3776	290	16	-	-	SYM
ejpam-3776	290	17	829	829	NUM
ejpam-3776	290	18	825	825	NUM
ejpam-3776	290	19	for	for	ADP
ejpam-3776	290	20	s	s	NOUN
ejpam-3776	290	21	=	=	SYM
ejpam-3776	290	22	0	0	NUM
ejpam-3776	290	23	or	or	CCONJ
ejpam-3776	290	24	r	r	NOUN
ejpam-3776	290	25	=	=	SYM
ejpam-3776	290	26	0	0	NUM
ejpam-3776	290	27	,	,	PUNCT
ejpam-3776	290	28	we	we	PRON
ejpam-3776	290	29	obtain	obtain	VERB
ejpam-3776	290	30	the	the	DET
ejpam-3776	290	31	required	require	VERB
ejpam-3776	290	32	result	result	NOUN
ejpam-3776	290	33	by	by	ADP
ejpam-3776	290	34	applying	apply	VERB
ejpam-3776	290	35	appropriate	appropriate	ADJ
ejpam-3776	290	36	limits	limit	NOUN
ejpam-3776	290	37	.	.	PUNCT
ejpam-3776	291	1	let	let	VERB
ejpam-3776	291	2	φ	φ	NOUN
ejpam-3776	291	3	:	:	PUNCT
ejpam-3776	292	1	[	[	X
ejpam-3776	292	2	a	a	X
ejpam-3776	292	3	,	,	PUNCT
ejpam-3776	292	4	b	b	NOUN
ejpam-3776	292	5	]	]	X
ejpam-3776	292	6	→	→	PUNCT
ejpam-3776	292	7	r	r	NOUN
ejpam-3776	292	8	be	be	AUX
ejpam-3776	292	9	a	a	DET
ejpam-3776	292	10	continuous	continuous	ADJ
ejpam-3776	292	11	and	and	CCONJ
ejpam-3776	292	12	strictly	strictly	ADV
ejpam-3776	292	13	monotone	monotone	ADJ
ejpam-3776	292	14	function	function	NOUN
ejpam-3776	292	15	then	then	ADV
ejpam-3776	292	16	cyclic	cyclic	ADJ
ejpam-3776	292	17	quasiarithmetic	quasiarithmetic	ADJ
ejpam-3776	292	18	means	mean	NOUN
ejpam-3776	292	19	are	be	AUX
ejpam-3776	292	20	defined	define	VERB
ejpam-3776	292	21	as	as	ADP
ejpam-3776	292	22	m̂φ	m̂φ	X
ejpam-3776	292	23	(	(	PUNCT
ejpam-3776	292	24	x	x	NOUN
ejpam-3776	292	25	)	)	PUNCT
ejpam-3776	292	26	:	:	PUNCT
ejpam-3776	292	27	=	=	SYM
ejpam-3776	292	28	φ−1	φ−1	PROPN
ejpam-3776	292	29			PROPN
ejpam-3776	292	30	m∑	m∑	ADV
ejpam-3776	292	31	j=1	j=1	PROPN
ejpam-3776	292	32	φ	φ	PROPN
ejpam-3776	292	33	(	(	PUNCT
ejpam-3776	292	34	aj)−	aj)−	NOUN
ejpam-3776	292	35	m−1∑	m−1∑	PROPN
ejpam-3776	293	1	j=1	j=1	PROPN
ejpam-3776	293	2	n∑	n∑	PROPN
ejpam-3776	294	1	i=1	i=1	PROPN
ejpam-3776	294	2	wiφ	wiφ	INTJ
ejpam-3776	294	3	(	(	PUNCT
ejpam-3776	294	4	xij	xij	X
ejpam-3776	294	5	)	)	PUNCT
ejpam-3776	294	6			PROPN
ejpam-3776	294	7	(	(	PUNCT
ejpam-3776	294	8	13	13	NUM
ejpam-3776	294	9	)	)	PUNCT
ejpam-3776	294	10	.	.	PUNCT
ejpam-3776	295	1	let	let	VERB
ejpam-3776	295	2	φ	φ	NOUN
ejpam-3776	295	3	,	,	PUNCT
ejpam-3776	295	4	ψ	ψ	X
ejpam-3776	295	5	:	:	PUNCT
ejpam-3776	295	6	[	[	X
ejpam-3776	295	7	a	a	X
ejpam-3776	295	8	,	,	PUNCT
ejpam-3776	295	9	b	b	NOUN
ejpam-3776	295	10	]	]	X
ejpam-3776	295	11	→	→	PUNCT
ejpam-3776	295	12	r	r	NOUN
ejpam-3776	295	13	be	be	AUX
ejpam-3776	295	14	strictly	strictly	ADV
ejpam-3776	295	15	monotonic	monotonic	ADJ
ejpam-3776	295	16	and	and	CCONJ
ejpam-3776	295	17	continuous	continuous	ADJ
ejpam-3776	295	18	functions	function	NOUN
ejpam-3776	295	19	and	and	CCONJ
ejpam-3776	295	20	under	under	ADP
ejpam-3776	295	21	the	the	DET
ejpam-3776	295	22	assumptions	assumption	NOUN
ejpam-3776	295	23	stated	state	VERB
ejpam-3776	295	24	in	in	ADP
ejpam-3776	295	25	(	(	PUNCT
ejpam-3776	295	26	a1	a1	NOUN
ejpam-3776	295	27	)	)	PUNCT
ejpam-3776	295	28	and	and	CCONJ
ejpam-3776	295	29	(	(	PUNCT
ejpam-3776	295	30	a2	a2	PROPN
ejpam-3776	295	31	)	)	PUNCT
ejpam-3776	295	32	,	,	PUNCT
ejpam-3776	295	33	we	we	PRON
ejpam-3776	295	34	define	define	VERB
ejpam-3776	295	35	generalized	generalized	ADJ
ejpam-3776	295	36	means	mean	NOUN
ejpam-3776	295	37	with	with	ADP
ejpam-3776	295	38	respect	respect	NOUN
ejpam-3776	295	39	to	to	ADP
ejpam-3776	295	40	(	(	PUNCT
ejpam-3776	295	41	6	6	NUM
ejpam-3776	295	42	)	)	PUNCT
ejpam-3776	295	43	as	as	SCONJ
ejpam-3776	295	44	follows	follow	VERB
ejpam-3776	295	45	:	:	PUNCT
ejpam-3776	295	46	mφ	mφ	NOUN
ejpam-3776	295	47	,	,	PUNCT
ejpam-3776	295	48	ψ	ψ	X
ejpam-3776	295	49	(	(	PUNCT
ejpam-3776	295	50	x	x	NOUN
ejpam-3776	295	51	,	,	PUNCT
ejpam-3776	295	52	λ	λ	NOUN
ejpam-3776	295	53	)	)	PUNCT
ejpam-3776	295	54	=	=	SYM
ejpam-3776	295	55	φ−1	φ−1	PROPN
ejpam-3776	295	56	φoψ−1	φoψ−1	ADV
ejpam-3776	295	57			PROPN
ejpam-3776	295	58	m∑	m∑	ADV
ejpam-3776	295	59	j=1	j=1	PROPN
ejpam-3776	295	60	aj	aj	PROPN
ejpam-3776	296	1	−	−	PROPN
ejpam-3776	296	2	n∑	n∑	PROPN
ejpam-3776	297	1	i=1	i=1	PROPN
ejpam-3776	298	1	wi	wi	PROPN
ejpam-3776	299	1	(	(	PUNCT
ejpam-3776	299	2	φoψ−1	φoψ−1	NOUN
ejpam-3776	299	3	)	)	PUNCT
ejpam-3776	299	4	m−1∑	m−1∑	VERB
ejpam-3776	299	5	j=1	j=1	NOUN
ejpam-3776	299	6	l−1∑	l−1∑	DET
ejpam-3776	299	7	k=0	k=0	PROPN
ejpam-3776	299	8	λk+1ψ	λk+1ψ	PROPN
ejpam-3776	299	9	(	(	PUNCT
ejpam-3776	299	10	xij+k	xij+k	PROPN
ejpam-3776	299	11	)	)	PUNCT
ejpam-3776	299	12			PROPN
ejpam-3776	299	13	.	.	PUNCT
ejpam-3776	300	1	(	(	PUNCT
ejpam-3776	300	2	14	14	NUM
ejpam-3776	300	3	)	)	PUNCT
ejpam-3776	300	4	now	now	ADV
ejpam-3776	300	5	,	,	PUNCT
ejpam-3776	300	6	we	we	PRON
ejpam-3776	300	7	establish	establish	VERB
ejpam-3776	300	8	the	the	DET
ejpam-3776	300	9	relation	relation	NOUN
ejpam-3776	300	10	among	among	ADP
ejpam-3776	300	11	generalized	generalized	ADJ
ejpam-3776	300	12	means	mean	NOUN
ejpam-3776	300	13	and	and	CCONJ
ejpam-3776	300	14	quasi	quasi	ADJ
ejpam-3776	300	15	-	-	ADJ
ejpam-3776	300	16	arithmetic	arithmetic	ADJ
ejpam-3776	300	17	means	mean	NOUN
ejpam-3776	300	18	as	as	SCONJ
ejpam-3776	300	19	follows	follow	VERB
ejpam-3776	300	20	:	:	PUNCT
ejpam-3776	300	21	corollary	corollary	ADJ
ejpam-3776	300	22	8	8	NUM
ejpam-3776	300	23	.	.	PUNCT
ejpam-3776	301	1	let	let	VERB
ejpam-3776	301	2	assumptions	assumption	NOUN
ejpam-3776	301	3	(	(	PUNCT
ejpam-3776	301	4	c1	c1	NOUN
ejpam-3776	301	5	)	)	PUNCT
ejpam-3776	301	6	and	and	CCONJ
ejpam-3776	301	7	(	(	PUNCT
ejpam-3776	301	8	c3	c3	NOUN
ejpam-3776	301	9	)	)	PUNCT
ejpam-3776	301	10	be	be	AUX
ejpam-3776	301	11	true	true	ADJ
ejpam-3776	301	12	.	.	PUNCT
ejpam-3776	302	1	then	then	ADV
ejpam-3776	302	2	m̂ψ	m̂ψ	NOUN
ejpam-3776	302	3	(	(	PUNCT
ejpam-3776	302	4	x	x	NOUN
ejpam-3776	302	5	)	)	PUNCT
ejpam-3776	302	6	≤	≤	NUM
ejpam-3776	302	7	m̂φ	m̂φ	X
ejpam-3776	302	8	,	,	PUNCT
ejpam-3776	302	9	ψ	ψ	X
ejpam-3776	302	10	(	(	PUNCT
ejpam-3776	302	11	x	x	NOUN
ejpam-3776	302	12	,	,	PUNCT
ejpam-3776	302	13	λ	λ	NOUN
ejpam-3776	302	14	)	)	PUNCT
ejpam-3776	302	15	≤	≤	NUM
ejpam-3776	302	16	m̂φ	m̂φ	NOUN
ejpam-3776	302	17	(	(	PUNCT
ejpam-3776	302	18	x	x	X
ejpam-3776	302	19	)	)	PUNCT
ejpam-3776	302	20	,	,	PUNCT
ejpam-3776	302	21	if	if	SCONJ
ejpam-3776	302	22	either	either	DET
ejpam-3776	302	23	φoψ−1	φoψ−1	PROPN
ejpam-3776	302	24	is	be	AUX
ejpam-3776	302	25	convex	convex	ADJ
ejpam-3776	302	26	and	and	CCONJ
ejpam-3776	302	27	ψ	ψ	NOUN
ejpam-3776	302	28	is	be	AUX
ejpam-3776	302	29	strictly	strictly	ADV
ejpam-3776	302	30	increasing	increase	VERB
ejpam-3776	302	31	or	or	CCONJ
ejpam-3776	302	32	φoψ−1	φoψ−1	NOUN
ejpam-3776	302	33	is	be	AUX
ejpam-3776	302	34	concave	concave	ADJ
ejpam-3776	302	35	and	and	CCONJ
ejpam-3776	302	36	ψ	ψ	NOUN
ejpam-3776	302	37	is	be	AUX
ejpam-3776	302	38	strictly	strictly	ADV
ejpam-3776	302	39	decreasing	decrease	VERB
ejpam-3776	302	40	.	.	PUNCT
ejpam-3776	303	1	proof	proof	NOUN
ejpam-3776	303	2	.	.	PUNCT
ejpam-3776	304	1	by	by	ADP
ejpam-3776	304	2	applying	apply	VERB
ejpam-3776	304	3	theorem	theorem	NOUN
ejpam-3776	304	4	6	6	NUM
ejpam-3776	304	5	to	to	ADP
ejpam-3776	304	6	the	the	DET
ejpam-3776	304	7	convex	convex	NOUN
ejpam-3776	304	8	function	function	VERB
ejpam-3776	304	9	φoψ−1	φoψ−1	PROPN
ejpam-3776	304	10	and	and	CCONJ
ejpam-3776	304	11	replacing	replace	VERB
ejpam-3776	304	12	aj	aj	PROPN
ejpam-3776	304	13	by	by	ADP
ejpam-3776	304	14	ψ	ψ	X
ejpam-3776	304	15	(	(	PUNCT
ejpam-3776	304	16	aj	aj	PROPN
ejpam-3776	304	17	)	)	PUNCT
ejpam-3776	304	18	and	and	CCONJ
ejpam-3776	304	19	n	n	CCONJ
ejpam-3776	304	20	-	-	PUNCT
ejpam-3776	304	21	tuples	tuple	NOUN
ejpam-3776	304	22	x	x	PUNCT
ejpam-3776	304	23	by	by	ADP
ejpam-3776	304	24	ψ	ψ	X
ejpam-3776	304	25	(	(	PUNCT
ejpam-3776	304	26	x	x	X
ejpam-3776	304	27	)	)	PUNCT
ejpam-3776	304	28	,	,	PUNCT
ejpam-3776	304	29	we	we	PRON
ejpam-3776	304	30	get	get	VERB
ejpam-3776	304	31	φoψ−1	φoψ−1	NOUN
ejpam-3776	304	32			PROPN
ejpam-3776	305	1	m∑	m∑	ADV
ejpam-3776	305	2	j=1	j=1	PROPN
ejpam-3776	305	3	ψ(aj)−	ψ(aj)−	NOUN
ejpam-3776	305	4	m−1∑	m−1∑	PROPN
ejpam-3776	305	5	j=1	j=1	PROPN
ejpam-3776	305	6	n∑	n∑	PROPN
ejpam-3776	305	7	i=1	i=1	PROPN
ejpam-3776	305	8	wiψ(xij	wiψ(xij	X
ejpam-3776	305	9	)	)	PUNCT
ejpam-3776	306	1			PROPN
ejpam-3776	306	2	≤	≤	NUM
ejpam-3776	306	3	n∑	n∑	PROPN
ejpam-3776	307	1	i=1	i=1	PROPN
ejpam-3776	308	1	wiφoψ	wiφoψ	PROPN
ejpam-3776	308	2	−1	−1	NOUN
ejpam-3776	309	1			PROPN
ejpam-3776	309	2	m∑	m∑	ADV
ejpam-3776	309	3	j=1	j=1	PROPN
ejpam-3776	309	4	ψ(aj)−	ψ(aj)−	NOUN
ejpam-3776	309	5	l−1∑	l−1∑	PROPN
ejpam-3776	309	6	k=0	k=0	PROPN
ejpam-3776	309	7	m−1∑	m−1∑	PROPN
ejpam-3776	309	8	j=1	j=1	PROPN
ejpam-3776	309	9	λk+1ψ(xij+k	λk+1ψ(xij+k	X
ejpam-3776	309	10	)	)	PUNCT
ejpam-3776	309	11			PROPN
ejpam-3776	309	12	≤	≤	NUM
ejpam-3776	309	13	m∑	m∑	VERB
ejpam-3776	310	1	j=1	j=1	PROPN
ejpam-3776	310	2	φoψ−1	φoψ−1	PROPN
ejpam-3776	310	3	(	(	PUNCT
ejpam-3776	310	4	ψ(aj))−	ψ(aj))−	PROPN
ejpam-3776	310	5	m−1∑	m−1∑	NUM
ejpam-3776	311	1	j=1	j=1	PROPN
ejpam-3776	311	2	n∑	n∑	PROPN
ejpam-3776	311	3	i=1	i=1	PROPN
ejpam-3776	312	1	wiφoψ	wiφoψ	PROPN
ejpam-3776	312	2	−1	−1	NOUN
ejpam-3776	312	3	(	(	PUNCT
ejpam-3776	312	4	ψ(xij	ψ(xij	PROPN
ejpam-3776	312	5	)	)	PUNCT
ejpam-3776	312	6	)	)	PUNCT
ejpam-3776	312	7	,	,	PUNCT
ejpam-3776	312	8	consequently	consequently	ADV
ejpam-3776	312	9	,	,	PUNCT
ejpam-3776	312	10	φoψ−1	φoψ−1	PROPN
ejpam-3776	312	11			PROPN
ejpam-3776	312	12	m∑	m∑	ADV
ejpam-3776	312	13	j=1	j=1	PROPN
ejpam-3776	312	14	ψ(aj)−	ψ(aj)−	NOUN
ejpam-3776	312	15	m−1∑	m−1∑	PROPN
ejpam-3776	312	16	j=1	j=1	PROPN
ejpam-3776	312	17	n∑	n∑	PROPN
ejpam-3776	312	18	i=1	i=1	PROPN
ejpam-3776	312	19	wiψ(xij	wiψ(xij	NOUN
ejpam-3776	312	20	)	)	PUNCT
ejpam-3776	313	1			PROPN
ejpam-3776	313	2	s.	s.	PROPN
ejpam-3776	313	3	chanan	chanan	PROPN
ejpam-3776	313	4	,	,	PUNCT
ejpam-3776	313	5	a.	a.	PROPN
ejpam-3776	313	6	r.	r.	PROPN
ejpam-3776	313	7	khan	khan	PROPN
ejpam-3776	313	8	/	/	SYM
ejpam-3776	313	9	eur	eur	PROPN
ejpam-3776	313	10	.	.	PUNCT
ejpam-3776	314	1	j.	j.	PROPN
ejpam-3776	314	2	pure	pure	PROPN
ejpam-3776	314	3	appl	appl	PROPN
ejpam-3776	314	4	.	.	PROPN
ejpam-3776	314	5	math	math	PROPN
ejpam-3776	314	6	,	,	PUNCT
ejpam-3776	314	7	13	13	NUM
ejpam-3776	314	8	(	(	PUNCT
ejpam-3776	314	9	4	4	NUM
ejpam-3776	314	10	)	)	PUNCT
ejpam-3776	314	11	(	(	PUNCT
ejpam-3776	314	12	2020	2020	NUM
ejpam-3776	314	13	)	)	PUNCT
ejpam-3776	314	14	,	,	PUNCT
ejpam-3776	314	15	814	814	NUM
ejpam-3776	314	16	-	-	SYM
ejpam-3776	314	17	829	829	NUM
ejpam-3776	314	18	826	826	NUM
ejpam-3776	314	19	≤	≤	NUM
ejpam-3776	314	20	n∑	n∑	PROPN
ejpam-3776	314	21	i=1	i=1	PROPN
ejpam-3776	315	1	wiφoψ	wiφoψ	PROPN
ejpam-3776	315	2	−1	−1	NOUN
ejpam-3776	316	1			PROPN
ejpam-3776	316	2	m∑	m∑	ADV
ejpam-3776	316	3	j=1	j=1	PROPN
ejpam-3776	316	4	ψ(aj)−	ψ(aj)−	NOUN
ejpam-3776	316	5	l−1∑	l−1∑	PROPN
ejpam-3776	316	6	k=0	k=0	PROPN
ejpam-3776	316	7	m−1∑	m−1∑	PROPN
ejpam-3776	316	8	j=1	j=1	PROPN
ejpam-3776	316	9	λk+1ψ(xij+k	λk+1ψ(xij+k	X
ejpam-3776	316	10	)	)	PUNCT
ejpam-3776	316	11			PROPN
ejpam-3776	316	12	≤	≤	NUM
ejpam-3776	316	13	m∑	m∑	CCONJ
ejpam-3776	316	14	j=1	j=1	ADJ
ejpam-3776	316	15	φ(aj)−	φ(aj)−	NOUN
ejpam-3776	316	16	m−1∑	m−1∑	PROPN
ejpam-3776	316	17	j=1	j=1	PROPN
ejpam-3776	316	18	n∑	n∑	PROPN
ejpam-3776	316	19	i=1	i=1	PROPN
ejpam-3776	316	20	wiφ(xij	wiφ(xij	PROPN
ejpam-3776	316	21	)	)	PUNCT
ejpam-3776	316	22	,	,	PUNCT
ejpam-3776	316	23	by	by	ADP
ejpam-3776	316	24	applying	apply	VERB
ejpam-3776	316	25	φ−1	φ−1	PROPN
ejpam-3776	316	26	,	,	PUNCT
ejpam-3776	316	27	we	we	PRON
ejpam-3776	316	28	get	get	VERB
ejpam-3776	316	29	φ−1φoψ−1	φ−1φoψ−1	ADP
ejpam-3776	317	1			PROPN
ejpam-3776	317	2	m∑	m∑	ADV
ejpam-3776	317	3	j=1	j=1	PROPN
ejpam-3776	317	4	ψ(aj)−	ψ(aj)−	NOUN
ejpam-3776	317	5	m−1∑	m−1∑	PROPN
ejpam-3776	317	6	j=1	j=1	PROPN
ejpam-3776	317	7	n∑	n∑	PROPN
ejpam-3776	317	8	i=1	i=1	PROPN
ejpam-3776	317	9	wiψ(xij	wiψ(xij	NOUN
ejpam-3776	317	10	)	)	PUNCT
ejpam-3776	318	1			PROPN
ejpam-3776	318	2	≤	≤	ADJ
ejpam-3776	318	3	φ−1	φ−1	PROPN
ejpam-3776	318	4			PROPN
ejpam-3776	318	5	n∑	n∑	PROPN
ejpam-3776	318	6	i=1	i=1	PROPN
ejpam-3776	319	1	wiφoψ	wiφoψ	PROPN
ejpam-3776	319	2	−1	−1	NOUN
ejpam-3776	320	1			PROPN
ejpam-3776	320	2	m∑	m∑	ADV
ejpam-3776	320	3	j=1	j=1	PROPN
ejpam-3776	320	4	ψ(aj)−	ψ(aj)−	NOUN
ejpam-3776	320	5	l−1∑	l−1∑	PROPN
ejpam-3776	320	6	k=0	k=0	PROPN
ejpam-3776	320	7	m−1∑	m−1∑	PROPN
ejpam-3776	320	8	j=1	j=1	PROPN
ejpam-3776	320	9	λk+1ψ(xij+k	λk+1ψ(xij+k	PROPN
ejpam-3776	320	10	)	)	PUNCT
ejpam-3776	320	11			PROPN
ejpam-3776	320	12	≤	≤	PUNCT
ejpam-3776	320	13	φ−1	φ−1	PROPN
ejpam-3776	320	14			PROPN
ejpam-3776	320	15	m∑	m∑	ADV
ejpam-3776	320	16	j=1	j=1	PROPN
ejpam-3776	320	17	φ(aj)−	φ(aj)−	NOUN
ejpam-3776	321	1	m−1∑	m−1∑	PROPN
ejpam-3776	321	2	j=1	j=1	PROPN
ejpam-3776	321	3	n∑	n∑	PROPN
ejpam-3776	322	1	i=1	i=1	PROPN
ejpam-3776	323	1	wiφ(xij	wiφ(xij	PROPN
ejpam-3776	323	2	)	)	PUNCT
ejpam-3776	324	1			PROPN
ejpam-3776	324	2	,	,	PUNCT
ejpam-3776	324	3	and	and	CCONJ
ejpam-3776	324	4	after	after	ADP
ejpam-3776	324	5	some	some	DET
ejpam-3776	324	6	simplification	simplification	NOUN
ejpam-3776	324	7	we	we	PRON
ejpam-3776	324	8	obtained	obtain	VERB
ejpam-3776	324	9	required	required	ADJ
ejpam-3776	324	10	result	result	NOUN
ejpam-3776	324	11	.	.	PUNCT
ejpam-3776	325	1	5	5	X
ejpam-3776	325	2	.	.	X
ejpam-3776	325	3	further	further	ADJ
ejpam-3776	325	4	results	result	NOUN
ejpam-3776	325	5	and	and	CCONJ
ejpam-3776	325	6	future	future	ADJ
ejpam-3776	325	7	work	work	NOUN
ejpam-3776	325	8	under	under	ADP
ejpam-3776	325	9	the	the	DET
ejpam-3776	325	10	assumptions	assumption	NOUN
ejpam-3776	325	11	of	of	ADP
ejpam-3776	325	12	theorem	theorem	NOUN
ejpam-3776	325	13	6	6	NUM
ejpam-3776	325	14	,	,	PUNCT
ejpam-3776	325	15	we	we	PRON
ejpam-3776	325	16	define	define	VERB
ejpam-3776	325	17	two	two	NUM
ejpam-3776	325	18	positive	positive	ADJ
ejpam-3776	325	19	linear	linear	NOUN
ejpam-3776	325	20	functionals	functional	NOUN
ejpam-3776	325	21	as	as	ADP
ejpam-3776	325	22	ϕ1	ϕ1	NOUN
ejpam-3776	325	23	(	(	PUNCT
ejpam-3776	325	24	x	x	NOUN
ejpam-3776	325	25	,	,	PUNCT
ejpam-3776	325	26	λ	λ	PROPN
ejpam-3776	325	27	,	,	PUNCT
ejpam-3776	325	28	φ	φ	NUM
ejpam-3776	325	29	)	)	PUNCT
ejpam-3776	325	30	=	=	PUNCT
ejpam-3776	326	1	m∑	m∑	CCONJ
ejpam-3776	326	2	j=1	j=1	PROPN
ejpam-3776	326	3	φ	φ	PROPN
ejpam-3776	326	4	(	(	PUNCT
ejpam-3776	326	5	aj)−	aj)−	NOUN
ejpam-3776	326	6	m−1∑	m−1∑	PROPN
ejpam-3776	326	7	j=1	j=1	PROPN
ejpam-3776	326	8	n∑	n∑	PROPN
ejpam-3776	327	1	i=1	i=1	PROPN
ejpam-3776	327	2	wiφ	wiφ	INTJ
ejpam-3776	327	3	(	(	PUNCT
ejpam-3776	327	4	xij	xij	X
ejpam-3776	327	5	)	)	PUNCT
ejpam-3776	328	1	−	−	PROPN
ejpam-3776	328	2	n∑	n∑	INTJ
ejpam-3776	328	3	i=1	i=1	PROPN
ejpam-3776	328	4	wiφ	wiφ	VERB
ejpam-3776	328	5			PROPN
ejpam-3776	328	6	m∑	m∑	PROPN
ejpam-3776	328	7	j=1	j=1	PROPN
ejpam-3776	328	8	aj	aj	PROPN
ejpam-3776	328	9	−	−	PROPN
ejpam-3776	328	10	l−1∑	l−1∑	ADJ
ejpam-3776	328	11	k=0	k=0	PUNCT
ejpam-3776	328	12	m−1∑	m−1∑	PRON
ejpam-3776	328	13	j=1	j=1	PROPN
ejpam-3776	328	14	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	328	15			PROPN
ejpam-3776	328	16	ϕ2	ϕ2	ADV
ejpam-3776	328	17	(	(	PUNCT
ejpam-3776	328	18	x	x	NOUN
ejpam-3776	328	19	,	,	PUNCT
ejpam-3776	328	20	λ	λ	PROPN
ejpam-3776	328	21	,	,	PUNCT
ejpam-3776	328	22	φ	φ	NUM
ejpam-3776	328	23	)	)	PUNCT
ejpam-3776	329	1	=	=	SYM
ejpam-3776	329	2	n∑	n∑	PROPN
ejpam-3776	329	3	i=1	i=1	PROPN
ejpam-3776	329	4	wiφ	wiφ	VERB
ejpam-3776	329	5			PROPN
ejpam-3776	329	6	m∑	m∑	PROPN
ejpam-3776	329	7	j=1	j=1	PROPN
ejpam-3776	329	8	aj	aj	PROPN
ejpam-3776	329	9	−	−	PROPN
ejpam-3776	329	10	l−1∑	l−1∑	ADJ
ejpam-3776	329	11	k=0	k=0	PUNCT
ejpam-3776	329	12	m−1∑	m−1∑	PRON
ejpam-3776	329	13	j=1	j=1	PROPN
ejpam-3776	329	14	λk+1xij+k	λk+1xij+k	NOUN
ejpam-3776	329	15			PROPN
ejpam-3776	329	16	−	−	PROPN
ejpam-3776	329	17			PROPN
ejpam-3776	329	18	m∑	m∑	ADV
ejpam-3776	329	19	j=1	j=1	ADJ
ejpam-3776	329	20	φ(aj)−	φ(aj)−	NOUN
ejpam-3776	330	1	m−1∑	m−1∑	PROPN
ejpam-3776	330	2	j=1	j=1	PROPN
ejpam-3776	330	3	n∑	n∑	PROPN
ejpam-3776	330	4	i=1	i=1	PROPN
ejpam-3776	330	5	wiφ	wiφ	INTJ
ejpam-3776	330	6	(	(	PUNCT
ejpam-3776	330	7	xij	xij	X
ejpam-3776	330	8	)	)	PUNCT
ejpam-3776	331	1			PROPN
ejpam-3776	331	2	we	we	PRON
ejpam-3776	331	3	can	can	AUX
ejpam-3776	331	4	state	state	VERB
ejpam-3776	331	5	different	different	ADJ
ejpam-3776	331	6	results	result	NOUN
ejpam-3776	331	7	for	for	ADP
ejpam-3776	331	8	these	these	DET
ejpam-3776	331	9	two	two	NUM
ejpam-3776	331	10	functionals	functional	NOUN
ejpam-3776	331	11	defined	define	VERB
ejpam-3776	331	12	above	above	ADP
ejpam-3776	331	13	which	which	PRON
ejpam-3776	331	14	may	may	AUX
ejpam-3776	331	15	be	be	AUX
ejpam-3776	331	16	listed	list	VERB
ejpam-3776	331	17	as	as	SCONJ
ejpam-3776	331	18	follows	follow	VERB
ejpam-3776	331	19	:	:	PUNCT
ejpam-3776	331	20	(	(	PUNCT
ejpam-3776	331	21	i	i	NOUN
ejpam-3776	331	22	)	)	PUNCT
ejpam-3776	331	23	we	we	PRON
ejpam-3776	331	24	can	can	AUX
ejpam-3776	331	25	state	state	VERB
ejpam-3776	331	26	lagrange	lagrange	NOUN
ejpam-3776	331	27	type	type	NOUN
ejpam-3776	331	28	and	and	CCONJ
ejpam-3776	331	29	cauchy	cauchy	NOUN
ejpam-3776	331	30	type	type	NOUN
ejpam-3776	331	31	mean	mean	NOUN
ejpam-3776	331	32	value	value	NOUN
ejpam-3776	331	33	theorems	theorem	NOUN
ejpam-3776	331	34	and	and	CCONJ
ejpam-3776	331	35	results	result	NOUN
ejpam-3776	331	36	related	relate	VERB
ejpam-3776	331	37	to	to	ADP
ejpam-3776	331	38	n−exponential	n−exponential	ADJ
ejpam-3776	331	39	and	and	CCONJ
ejpam-3776	331	40	logarithmic	logarithmic	ADJ
ejpam-3776	331	41	convexity	convexity	NOUN
ejpam-3776	331	42	by	by	ADP
ejpam-3776	331	43	using	use	VERB
ejpam-3776	331	44	similar	similar	ADJ
ejpam-3776	331	45	techniques	technique	NOUN
ejpam-3776	331	46	as	as	SCONJ
ejpam-3776	331	47	stated	state	VERB
ejpam-3776	331	48	in	in	ADP
ejpam-3776	331	49	[	[	X
ejpam-3776	331	50	3	3	NUM
ejpam-3776	331	51	]	]	PUNCT
ejpam-3776	331	52	and	and	CCONJ
ejpam-3776	331	53	[	[	X
ejpam-3776	331	54	14	14	NUM
ejpam-3776	331	55	]	]	PUNCT
ejpam-3776	331	56	.	.	PUNCT
ejpam-3776	332	1	references	reference	NOUN
ejpam-3776	332	2	827	827	NUM
ejpam-3776	332	3	(	(	PUNCT
ejpam-3776	332	4	ii	ii	NOUN
ejpam-3776	332	5	)	)	PUNCT
ejpam-3776	332	6	we	we	PRON
ejpam-3776	332	7	can	can	AUX
ejpam-3776	332	8	also	also	ADV
ejpam-3776	332	9	state	state	VERB
ejpam-3776	332	10	number	number	NOUN
ejpam-3776	332	11	of	of	ADP
ejpam-3776	332	12	applications	application	NOUN
ejpam-3776	332	13	by	by	ADP
ejpam-3776	332	14	using	use	VERB
ejpam-3776	332	15	method	method	NOUN
ejpam-3776	332	16	of	of	ADP
ejpam-3776	332	17	article	article	NOUN
ejpam-3776	332	18	[	[	X
ejpam-3776	332	19	22	22	NUM
ejpam-3776	332	20	]	]	PUNCT
ejpam-3776	332	21	.	.	PUNCT
ejpam-3776	333	1	(	(	PUNCT
ejpam-3776	333	2	iii	iii	X
ejpam-3776	333	3	)	)	PUNCT
ejpam-3776	333	4	we	we	PRON
ejpam-3776	333	5	can	can	AUX
ejpam-3776	333	6	state	state	VERB
ejpam-3776	333	7	further	further	ADJ
ejpam-3776	333	8	results	result	NOUN
ejpam-3776	333	9	using	use	VERB
ejpam-3776	333	10	technique	technique	NOUN
ejpam-3776	333	11	of	of	ADP
ejpam-3776	333	12	index	index	NOUN
ejpam-3776	333	13	set	set	VERB
ejpam-3776	333	14	function	function	NOUN
ejpam-3776	333	15	with	with	ADP
ejpam-3776	333	16	series	series	NOUN
ejpam-3776	333	17	of	of	ADP
ejpam-3776	333	18	refinements	refinement	NOUN
ejpam-3776	333	19	and	and	CCONJ
ejpam-3776	333	20	plenty	plenty	NOUN
ejpam-3776	333	21	of	of	ADP
ejpam-3776	333	22	applications	application	NOUN
ejpam-3776	333	23	including	include	VERB
ejpam-3776	333	24	rado	rado	PROPN
ejpam-3776	333	25	and	and	CCONJ
ejpam-3776	333	26	popovicu	popovicu	PROPN
ejpam-3776	333	27	series	series	NOUN
ejpam-3776	333	28	of	of	ADP
ejpam-3776	333	29	inequality	inequality	NOUN
ejpam-3776	333	30	by	by	ADP
ejpam-3776	333	31	using	use	VERB
ejpam-3776	333	32	method	method	NOUN
ejpam-3776	333	33	of	of	ADP
ejpam-3776	333	34	[	[	X
ejpam-3776	333	35	18	18	NUM
ejpam-3776	333	36	]	]	PUNCT
ejpam-3776	333	37	and	and	CCONJ
ejpam-3776	333	38	[	[	X
ejpam-3776	333	39	19	19	NUM
ejpam-3776	333	40	]	]	PUNCT
ejpam-3776	333	41	.	.	PUNCT
ejpam-3776	334	1	(	(	PUNCT
ejpam-3776	334	2	iv	iv	X
ejpam-3776	334	3	)	)	PUNCT
ejpam-3776	334	4	we	we	PRON
ejpam-3776	334	5	can	can	AUX
ejpam-3776	334	6	also	also	ADV
ejpam-3776	334	7	prove	prove	VERB
ejpam-3776	334	8	all	all	DET
ejpam-3776	334	9	inequalities	inequality	NOUN
ejpam-3776	334	10	in	in	ADP
ejpam-3776	334	11	reverse	reverse	ADJ
ejpam-3776	334	12	direction	direction	NOUN
ejpam-3776	334	13	by	by	ADP
ejpam-3776	334	14	considering	consider	VERB
ejpam-3776	334	15	concave	concave	NOUN
ejpam-3776	334	16	function	function	NOUN
ejpam-3776	334	17	instead	instead	ADV
ejpam-3776	334	18	of	of	ADP
ejpam-3776	334	19	convex	convex	NOUN
ejpam-3776	334	20	function	function	NOUN
ejpam-3776	334	21	by	by	ADP
ejpam-3776	334	22	using	use	VERB
ejpam-3776	334	23	simple	simple	ADJ
ejpam-3776	334	24	relation	relation	NOUN
ejpam-3776	334	25	:	:	PUNCT
ejpam-3776	334	26	f	f	PROPN
ejpam-3776	334	27	is	be	AUX
ejpam-3776	334	28	concave	concave	VERB
ejpam-3776	334	29	iff	iff	PROPN
ejpam-3776	334	30	and	and	CCONJ
ejpam-3776	334	31	−f	−f	PROPN
ejpam-3776	334	32	is	be	AUX
ejpam-3776	334	33	convex	convex	ADJ
ejpam-3776	334	34	.	.	PUNCT
ejpam-3776	335	1	here	here	ADV
ejpam-3776	335	2	we	we	PRON
ejpam-3776	335	3	state	state	VERB
ejpam-3776	335	4	some	some	DET
ejpam-3776	335	5	future	future	ADJ
ejpam-3776	335	6	ideas	idea	NOUN
ejpam-3776	335	7	for	for	ADP
ejpam-3776	335	8	interested	interested	ADJ
ejpam-3776	335	9	readers	reader	NOUN
ejpam-3776	335	10	:	:	PUNCT
ejpam-3776	335	11	(	(	PUNCT
ejpam-3776	335	12	i	i	NOUN
ejpam-3776	335	13	)	)	PUNCT
ejpam-3776	335	14	one	one	PRON
ejpam-3776	335	15	can	can	AUX
ejpam-3776	335	16	also	also	ADV
ejpam-3776	335	17	work	work	VERB
ejpam-3776	335	18	on	on	ADP
ejpam-3776	335	19	similar	similar	ADJ
ejpam-3776	335	20	results	result	NOUN
ejpam-3776	335	21	as	as	SCONJ
ejpam-3776	335	22	stated	state	VERB
ejpam-3776	335	23	in	in	ADP
ejpam-3776	335	24	this	this	DET
ejpam-3776	335	25	article	article	NOUN
ejpam-3776	335	26	for	for	ADP
ejpam-3776	335	27	generalized	generalized	ADJ
ejpam-3776	335	28	convex	convex	NOUN
ejpam-3776	335	29	functions	function	NOUN
ejpam-3776	335	30	including	include	VERB
ejpam-3776	335	31	functions	function	NOUN
ejpam-3776	335	32	with	with	ADP
ejpam-3776	335	33	nondecreasing	nondecreasing	ADJ
ejpam-3776	335	34	increments	increment	NOUN
ejpam-3776	335	35	and	and	CCONJ
ejpam-3776	335	36	functions	function	NOUN
ejpam-3776	335	37	with	with	ADP
ejpam-3776	335	38	nondecreasing	nondecreasing	ADJ
ejpam-3776	335	39	increments	increment	NOUN
ejpam-3776	335	40	of	of	ADP
ejpam-3776	335	41	convex	convex	ADJ
ejpam-3776	335	42	type	type	NOUN
ejpam-3776	335	43	see	see	NOUN
ejpam-3776	335	44	for	for	ADP
ejpam-3776	335	45	example	example	NOUN
ejpam-3776	335	46	[	[	X
ejpam-3776	335	47	1	1	NUM
ejpam-3776	335	48	]	]	PUNCT
ejpam-3776	335	49	,	,	PUNCT
ejpam-3776	335	50	[	[	X
ejpam-3776	335	51	7	7	X
ejpam-3776	335	52	]	]	PUNCT
ejpam-3776	335	53	and	and	CCONJ
ejpam-3776	335	54	[	[	X
ejpam-3776	335	55	20	20	NUM
ejpam-3776	335	56	]	]	PUNCT
ejpam-3776	335	57	.	.	PUNCT
ejpam-3776	336	1	(	(	PUNCT
ejpam-3776	336	2	ii	ii	NOUN
ejpam-3776	336	3	)	)	PUNCT
ejpam-3776	336	4	one	one	NOUN
ejpam-3776	336	5	can	can	AUX
ejpam-3776	336	6	also	also	ADV
ejpam-3776	336	7	state	state	VERB
ejpam-3776	336	8	similar	similar	ADJ
ejpam-3776	336	9	results	result	NOUN
ejpam-3776	336	10	as	as	SCONJ
ejpam-3776	336	11	stated	state	VERB
ejpam-3776	336	12	in	in	ADP
ejpam-3776	336	13	this	this	DET
ejpam-3776	336	14	article	article	NOUN
ejpam-3776	336	15	for	for	ADP
ejpam-3776	336	16	arbitrary	arbitrary	ADJ
ejpam-3776	336	17	real	real	ADJ
ejpam-3776	336	18	numbers	number	NOUN
ejpam-3776	336	19	(	(	PUNCT
ejpam-3776	336	20	not	not	PART
ejpam-3776	336	21	only	only	ADV
ejpam-3776	336	22	non	non	ADJ
ejpam-3776	336	23	-	-	ADJ
ejpam-3776	336	24	negative	negative	ADJ
ejpam-3776	336	25	real	real	ADJ
ejpam-3776	336	26	numbers	number	NOUN
ejpam-3776	336	27	)	)	PUNCT
ejpam-3776	336	28	for	for	ADP
ejpam-3776	336	29	example	example	NOUN
ejpam-3776	336	30	by	by	ADP
ejpam-3776	336	31	working	work	VERB
ejpam-3776	336	32	with	with	ADP
ejpam-3776	336	33	assumptions	assumption	NOUN
ejpam-3776	336	34	of	of	ADP
ejpam-3776	336	35	jensen	jensen	PROPN
ejpam-3776	336	36	-	-	PUNCT
ejpam-3776	336	37	steffensen	steffensen	PROPN
ejpam-3776	336	38	inequality	inequality	NOUN
ejpam-3776	336	39	.	.	PUNCT
ejpam-3776	337	1	(	(	PUNCT
ejpam-3776	337	2	iii	iii	X
ejpam-3776	337	3	)	)	PUNCT
ejpam-3776	337	4	one	one	NOUN
ejpam-3776	337	5	can	can	AUX
ejpam-3776	337	6	also	also	ADV
ejpam-3776	337	7	try	try	VERB
ejpam-3776	337	8	its	its	PRON
ejpam-3776	337	9	integral	integral	ADJ
ejpam-3776	337	10	version	version	NOUN
ejpam-3776	337	11	as	as	ADV
ejpam-3776	337	12	well	well	ADV
ejpam-3776	337	13	.	.	PUNCT
ejpam-3776	338	1	references	reference	NOUN
ejpam-3776	338	2	[	[	X
ejpam-3776	338	3	1	1	NUM
ejpam-3776	338	4	]	]	PUNCT
ejpam-3776	338	5	m.	m.	PROPN
ejpam-3776	338	6	maqsood	maqsood	PROPN
ejpam-3776	338	7	ali	ali	PROPN
ejpam-3776	338	8	,	,	PUNCT
ejpam-3776	338	9	asif	asif	PROPN
ejpam-3776	338	10	r.	r.	PROPN
ejpam-3776	338	11	kha	kha	PROPN
ejpam-3776	338	12	,	,	PUNCT
ejpam-3776	338	13	inam	inam	PROPN
ejpam-3776	338	14	ullah	ullah	PROPN
ejpam-3776	338	15	khan	khan	PROPN
ejpam-3776	338	16	,	,	PUNCT
ejpam-3776	338	17	and	and	CCONJ
ejpam-3776	338	18	sumayyah	sumayyah	PROPN
ejpam-3776	338	19	saadi	saadi	PROPN
ejpam-3776	338	20	.	.	PUNCT
ejpam-3776	339	1	improvement	improvement	NOUN
ejpam-3776	339	2	of	of	ADP
ejpam-3776	339	3	jensen	jensen	PROPN
ejpam-3776	339	4	and	and	CCONJ
ejpam-3776	339	5	levinson	levinson	PROPN
ejpam-3776	339	6	type	type	NOUN
ejpam-3776	339	7	inequalities	inequality	NOUN
ejpam-3776	339	8	for	for	ADP
ejpam-3776	339	9	functions	function	NOUN
ejpam-3776	339	10	with	with	ADP
ejpam-3776	339	11	nondecreasing	nondecreasing	ADJ
ejpam-3776	339	12	increments	increment	NOUN
ejpam-3776	339	13	.	.	PUNCT
ejpam-3776	340	1	global	global	ADJ
ejpam-3776	340	2	j.	j.	PROPN
ejpam-3776	340	3	pure	pure	PROPN
ejpam-3776	340	4	appl	appl	PROPN
ejpam-3776	340	5	.	.	PUNCT
ejpam-3776	340	6	math	math	PROPN
ejpam-3776	340	7	.	.	PUNCT
ejpam-3776	340	8	,	,	PUNCT
ejpam-3776	340	9	6(15):945–970	6(15):945–970	NUM
ejpam-3776	340	10	,	,	PUNCT
ejpam-3776	340	11	2019	2019	NUM
ejpam-3776	340	12	.	.	PUNCT
ejpam-3776	341	1	[	[	X
ejpam-3776	341	2	2	2	X
ejpam-3776	341	3	]	]	PUNCT
ejpam-3776	341	4	h.	h.	PROPN
ejpam-3776	341	5	alzer	alzer	PROPN
ejpam-3776	341	6	.	.	PUNCT
ejpam-3776	342	1	the	the	DET
ejpam-3776	342	2	inequality	inequality	NOUN
ejpam-3776	342	3	of	of	ADP
ejpam-3776	342	4	ky	ky	PROPN
ejpam-3776	342	5	fan	fan	PROPN
ejpam-3776	342	6	’s	’s	PART
ejpam-3776	342	7	and	and	CCONJ
ejpam-3776	342	8	related	related	ADJ
ejpam-3776	342	9	results	result	NOUN
ejpam-3776	342	10	.	.	PUNCT
ejpam-3776	343	1	acta	acta	PROPN
ejpam-3776	343	2	app	app	PROPN
ejpam-3776	343	3	.	.	PROPN
ejpam-3776	343	4	math	math	PROPN
ejpam-3776	343	5	.	.	PUNCT
ejpam-3776	343	6	,	,	PUNCT
ejpam-3776	344	1	38:305–354	38:305–354	PROPN
ejpam-3776	344	2	,	,	PUNCT
ejpam-3776	344	3	1995	1995	NUM
ejpam-3776	344	4	.	.	PUNCT
ejpam-3776	345	1	[	[	X
ejpam-3776	345	2	3	3	NUM
ejpam-3776	345	3	]	]	X
ejpam-3776	345	4	i.	i.	NOUN
ejpam-3776	345	5	brentic	brentic	PROPN
ejpam-3776	345	6	,	,	PUNCT
ejpam-3776	345	7	k.	k.	PROPN
ejpam-3776	345	8	a.	a.	PROPN
ejpam-3776	345	9	khan	khan	PROPN
ejpam-3776	345	10	,	,	PUNCT
ejpam-3776	345	11	and	and	CCONJ
ejpam-3776	345	12	j.	j.	PROPN
ejpam-3776	345	13	pečarić.	pečarić.	PROPN
ejpam-3776	345	14	refinements	refinement	NOUN
ejpam-3776	345	15	of	of	ADP
ejpam-3776	345	16	jensen	jensen	PROPN
ejpam-3776	345	17	’s	’s	PART
ejpam-3776	345	18	inequality	inequality	NOUN
ejpam-3776	345	19	with	with	ADP
ejpam-3776	345	20	applications	application	NOUN
ejpam-3776	345	21	to	to	ADP
ejpam-3776	345	22	cyclic	cyclic	ADJ
ejpam-3776	345	23	mixed	mixed	ADJ
ejpam-3776	345	24	symmetric	symmetric	ADJ
ejpam-3776	345	25	means	mean	NOUN
ejpam-3776	345	26	and	and	CCONJ
ejpam-3776	345	27	cauchy	cauchy	PROPN
ejpam-3776	345	28	means	mean	NOUN
ejpam-3776	345	29	.	.	PUNCT
ejpam-3776	346	1	j.	j.	PROPN
ejpam-3776	346	2	math	math	PROPN
ejpam-3776	346	3	.	.	PUNCT
ejpam-3776	347	1	inequal	inequal	ADJ
ejpam-3776	347	2	,	,	PUNCT
ejpam-3776	347	3	4(9):1309–1321	4(9):1309–1321	PROPN
ejpam-3776	347	4	,	,	PUNCT
ejpam-3776	347	5	2015	2015	NUM
ejpam-3776	347	6	.	.	PUNCT
ejpam-3776	348	1	[	[	X
ejpam-3776	348	2	4	4	X
ejpam-3776	348	3	]	]	PUNCT
ejpam-3776	348	4	p.	p.	NOUN
ejpam-3776	348	5	s.	s.	PROPN
ejpam-3776	348	6	bullen	bullen	PROPN
ejpam-3776	348	7	,	,	PUNCT
ejpam-3776	348	8	d.	d.	PROPN
ejpam-3776	348	9	s.	s.	PROPN
ejpam-3776	348	10	mitrinović	mitrinović	PROPN
ejpam-3776	348	11	,	,	PUNCT
ejpam-3776	348	12	and	and	CCONJ
ejpam-3776	348	13	p.	p.	NOUN
ejpam-3776	348	14	m.	m.	NOUN
ejpam-3776	349	1	vasić.	vasić.	PROPN
ejpam-3776	349	2	means	mean	VERB
ejpam-3776	349	3	and	and	CCONJ
ejpam-3776	349	4	their	their	PRON
ejpam-3776	349	5	inequalities	inequality	NOUN
ejpam-3776	349	6	.	.	PUNCT
ejpam-3776	350	1	reidel	reidel	PROPN
ejpam-3776	350	2	,	,	PUNCT
ejpam-3776	350	3	dordrech	dordrech	PROPN
ejpam-3776	350	4	,	,	PUNCT
ejpam-3776	350	5	1988	1988	NUM
ejpam-3776	350	6	.	.	PUNCT
ejpam-3776	351	1	[	[	X
ejpam-3776	351	2	5	5	X
ejpam-3776	351	3	]	]	PUNCT
ejpam-3776	351	4	s.	s.	PROPN
ejpam-3776	351	5	chanan	chanan	PROPN
ejpam-3776	351	6	and	and	CCONJ
ejpam-3776	351	7	asif	asif	PROPN
ejpam-3776	351	8	r.	r.	PROPN
ejpam-3776	351	9	khan	khan	PROPN
ejpam-3776	351	10	.	.	PUNCT
ejpam-3776	352	1	on	on	ADP
ejpam-3776	352	2	some	some	DET
ejpam-3776	352	3	refinements	refinement	NOUN
ejpam-3776	352	4	of	of	ADP
ejpam-3776	352	5	jensen	jensen	PROPN
ejpam-3776	352	6	-	-	PUNCT
ejpam-3776	352	7	mercer	mercer	PROPN
ejpam-3776	352	8	inequality	inequality	NOUN
ejpam-3776	352	9	with	with	ADP
ejpam-3776	352	10	applications	application	NOUN
ejpam-3776	352	11	.	.	PUNCT
ejpam-3776	353	1	submitted	submit	VERB
ejpam-3776	353	2	.	.	PUNCT
ejpam-3776	354	1	[	[	X
ejpam-3776	354	2	6	6	NUM
ejpam-3776	354	3	]	]	PUNCT
ejpam-3776	354	4	s.	s.	PROPN
ejpam-3776	354	5	chanan	chanan	PROPN
ejpam-3776	354	6	,	,	PUNCT
ejpam-3776	354	7	asif	asif	PROPN
ejpam-3776	354	8	r.	r.	PROPN
ejpam-3776	354	9	khan	khan	PROPN
ejpam-3776	354	10	,	,	PUNCT
ejpam-3776	354	11	s.	s.	PROPN
ejpam-3776	354	12	ahmed	ahmed	PROPN
ejpam-3776	354	13	,	,	PUNCT
ejpam-3776	354	14	and	and	CCONJ
ejpam-3776	354	15	n.	n.	PROPN
ejpam-3776	354	16	raisat	raisat	PROPN
ejpam-3776	354	17	.	.	PUNCT
ejpam-3776	355	1	generalizations	generalization	NOUN
ejpam-3776	355	2	of	of	ADP
ejpam-3776	355	3	ky	ky	PROPN
ejpam-3776	355	4	fan	fan	PROPN
ejpam-3776	355	5	inequality	inequality	PROPN
ejpam-3776	355	6	and	and	CCONJ
ejpam-3776	355	7	related	related	ADJ
ejpam-3776	355	8	results	result	NOUN
ejpam-3776	355	9	.	.	PUNCT
ejpam-3776	356	1	j.	j.	PROPN
ejpam-3776	356	2	inequal	inequal	PROPN
ejpam-3776	356	3	.	.	PUNCT
ejpam-3776	356	4	and	and	CCONJ
ejpam-3776	356	5	special	special	ADJ
ejpam-3776	356	6	functions	function	NOUN
ejpam-3776	356	7	,	,	PUNCT
ejpam-3776	356	8	10:123–142	10:123–142	NUM
ejpam-3776	356	9	,	,	PUNCT
ejpam-3776	356	10	2019	2019	NUM
ejpam-3776	356	11	.	.	PUNCT
ejpam-3776	357	1	[	[	X
ejpam-3776	357	2	7	7	X
ejpam-3776	357	3	]	]	X
ejpam-3776	357	4	s.	s.	PROPN
ejpam-3776	357	5	chanan	chanan	PROPN
ejpam-3776	357	6	,	,	PUNCT
ejpam-3776	357	7	asif	asif	PROPN
ejpam-3776	357	8	r.	r.	PROPN
ejpam-3776	357	9	khan	khan	PROPN
ejpam-3776	357	10	,	,	PUNCT
ejpam-3776	357	11	and	and	CCONJ
ejpam-3776	357	12	inam	inam	PROPN
ejpam-3776	357	13	ullah	ullah	PROPN
ejpam-3776	357	14	khan	khan	PROPN
ejpam-3776	357	15	.	.	PUNCT
ejpam-3776	358	1	gabler	gabler	PROPN
ejpam-3776	358	2	inequality	inequality	PROPN
ejpam-3776	358	3	for	for	ADP
ejpam-3776	358	4	functions	function	NOUN
ejpam-3776	358	5	with	with	ADP
ejpam-3776	358	6	nondecreasing	nondecreasing	ADJ
ejpam-3776	358	7	increments	increment	NOUN
ejpam-3776	358	8	of	of	ADP
ejpam-3776	358	9	convex	convex	ADJ
ejpam-3776	358	10	type	type	NOUN
ejpam-3776	358	11	.	.	PUNCT
ejpam-3776	359	1	adv	adv	PROPN
ejpam-3776	359	2	.	.	PUNCT
ejpam-3776	359	3	inequal	inequal	PROPN
ejpam-3776	359	4	.	.	PUNCT
ejpam-3776	360	1	appl	appl	PROPN
ejpam-3776	360	2	.	.	PROPN
ejpam-3776	360	3	,	,	PUNCT
ejpam-3776	360	4	10(3	10(3	NUM
ejpam-3776	360	5	)	)	PUNCT
ejpam-3776	360	6	,	,	PUNCT
ejpam-3776	360	7	2020	2020	NUM
ejpam-3776	360	8	.	.	PUNCT
ejpam-3776	361	1	[	[	X
ejpam-3776	361	2	8	8	NUM
ejpam-3776	361	3	]	]	X
ejpam-3776	361	4	m.	m.	NOUN
ejpam-3776	361	5	klariči	klariči	PROPN
ejpam-3776	361	6	´	´	PROPN
ejpam-3776	361	7	c	c	PROPN
ejpam-3776	361	8	bakula	bakula	PROPN
ejpam-3776	361	9	and	and	CCONJ
ejpam-3776	361	10	j.	j.	PROPN
ejpam-3776	361	11	pečarić.	pečarić.	PROPN
ejpam-3776	361	12	on	on	ADP
ejpam-3776	361	13	the	the	DET
ejpam-3776	361	14	jensens	jensens	PROPN
ejpam-3776	361	15	inequality	inequality	NOUN
ejpam-3776	361	16	for	for	ADP
ejpam-3776	361	17	convex	convex	NOUN
ejpam-3776	361	18	functions	function	NOUN
ejpam-3776	361	19	on	on	ADP
ejpam-3776	361	20	the	the	DET
ejpam-3776	361	21	co	co	NOUN
ejpam-3776	361	22	-	-	NOUN
ejpam-3776	361	23	ordinates	ordinate	NOUN
ejpam-3776	361	24	in	in	ADP
ejpam-3776	361	25	a	a	DET
ejpam-3776	361	26	rectangle	rectangle	NOUN
ejpam-3776	361	27	from	from	ADP
ejpam-3776	361	28	the	the	DET
ejpam-3776	361	29	plane	plane	NOUN
ejpam-3776	361	30	.	.	PUNCT
ejpam-3776	362	1	taiwanese	taiwanese	ADJ
ejpam-3776	362	2	j.	j.	PROPN
ejpam-3776	362	3	math	math	PROPN
ejpam-3776	362	4	.	.	PUNCT
ejpam-3776	362	5	,	,	PUNCT
ejpam-3776	362	6	10(5):1271–1292	10(5):1271–1292	NUM
ejpam-3776	362	7	,	,	PUNCT
ejpam-3776	362	8	2006	2006	NUM
ejpam-3776	362	9	.	.	PUNCT
ejpam-3776	362	10	references	reference	NOUN
ejpam-3776	362	11	828	828	NUM
ejpam-3776	362	12	[	[	X
ejpam-3776	362	13	9	9	NUM
ejpam-3776	362	14	]	]	PUNCT
ejpam-3776	362	15	s.	s.	PROPN
ejpam-3776	362	16	s.	s.	PROPN
ejpam-3776	362	17	dragomir	dragomir	PROPN
ejpam-3776	362	18	.	.	PUNCT
ejpam-3776	363	1	a	a	DET
ejpam-3776	363	2	new	new	ADJ
ejpam-3776	363	3	refinement	refinement	NOUN
ejpam-3776	363	4	of	of	ADP
ejpam-3776	363	5	jensen	jensen	PROPN
ejpam-3776	363	6	’s	’s	PART
ejpam-3776	363	7	inequality	inequality	NOUN
ejpam-3776	363	8	in	in	ADP
ejpam-3776	363	9	linear	linear	PROPN
ejpam-3776	363	10	spaces	space	NOUN
ejpam-3776	363	11	with	with	ADP
ejpam-3776	363	12	applications	application	NOUN
ejpam-3776	363	13	.	.	PUNCT
ejpam-3776	364	1	mathematical	mathematical	ADJ
ejpam-3776	364	2	and	and	CCONJ
ejpam-3776	364	3	computer	computer	NOUN
ejpam-3776	364	4	modelling	modelling	NOUN
ejpam-3776	364	5	,	,	PUNCT
ejpam-3776	364	6	52:1497–1505	52:1497–1505	NUM
ejpam-3776	364	7	,	,	PUNCT
ejpam-3776	364	8	2010	2010	NUM
ejpam-3776	364	9	.	.	PUNCT
ejpam-3776	365	1	[	[	X
ejpam-3776	365	2	10	10	NUM
ejpam-3776	365	3	]	]	X
ejpam-3776	365	4	g.	g.	PROPN
ejpam-3776	365	5	h.	h.	PROPN
ejpam-3776	365	6	hardy	hardy	PROPN
ejpam-3776	365	7	,	,	PUNCT
ejpam-3776	365	8	j.	j.	PROPN
ejpam-3776	365	9	e.	e.	PROPN
ejpam-3776	365	10	littlewood	littlewood	PROPN
ejpam-3776	365	11	,	,	PUNCT
ejpam-3776	365	12	and	and	CCONJ
ejpam-3776	365	13	g.	g.	PROPN
ejpam-3776	365	14	pólya	pólya	PROPN
ejpam-3776	365	15	.	.	PUNCT
ejpam-3776	366	1	inequalities	inequality	NOUN
ejpam-3776	366	2	.	.	PUNCT
ejpam-3776	367	1	cambridge	cambridge	PROPN
ejpam-3776	367	2	university	university	PROPN
ejpam-3776	367	3	press	press	PROPN
ejpam-3776	367	4	,	,	PUNCT
ejpam-3776	367	5	cambridge	cambridge	PROPN
ejpam-3776	367	6	,	,	PUNCT
ejpam-3776	367	7	,	,	PUNCT
ejpam-3776	367	8	1978	1978	NUM
ejpam-3776	367	9	.	.	PUNCT
ejpam-3776	368	1	[	[	X
ejpam-3776	368	2	11	11	NUM
ejpam-3776	368	3	]	]	PUNCT
ejpam-3776	368	4	l.	l.	PROPN
ejpam-3776	368	5	horv́ath	horv́ath	PROPN
ejpam-3776	368	6	,	,	PUNCT
ejpam-3776	368	7	k.	k.	PROPN
ejpam-3776	368	8	a.	a.	PROPN
ejpam-3776	368	9	khan	khan	PROPN
ejpam-3776	368	10	,	,	PUNCT
ejpam-3776	368	11	and	and	CCONJ
ejpam-3776	368	12	j.	j.	PROPN
ejpam-3776	368	13	pečarić.	pečarić.	PROPN
ejpam-3776	368	14	further	far	ADV
ejpam-3776	368	15	refinement	refinement	NOUN
ejpam-3776	368	16	of	of	ADP
ejpam-3776	368	17	results	result	NOUN
ejpam-3776	368	18	about	about	ADP
ejpam-3776	368	19	mixed	mixed	ADJ
ejpam-3776	368	20	symmetric	symmetric	ADJ
ejpam-3776	368	21	means	mean	NOUN
ejpam-3776	368	22	and	and	CCONJ
ejpam-3776	368	23	cauchy	cauchy	PROPN
ejpam-3776	368	24	means	mean	NOUN
ejpam-3776	368	25	.	.	PUNCT
ejpam-3776	369	1	advances	advance	NOUN
ejpam-3776	369	2	in	in	ADP
ejpam-3776	369	3	inequalities	inequality	NOUN
ejpam-3776	369	4	and	and	CCONJ
ejpam-3776	369	5	applications	application	NOUN
ejpam-3776	369	6	,	,	PUNCT
ejpam-3776	369	7	1:12–32	1:12–32	NUM
ejpam-3776	369	8	,	,	PUNCT
ejpam-3776	369	9	2012	2012	NUM
ejpam-3776	369	10	.	.	PUNCT
ejpam-3776	370	1	[	[	X
ejpam-3776	370	2	12	12	NUM
ejpam-3776	370	3	]	]	PUNCT
ejpam-3776	370	4	s.	s.	PROPN
ejpam-3776	370	5	hussain	hussain	PROPN
ejpam-3776	370	6	and	and	CCONJ
ejpam-3776	370	7	j.	j.	PROPN
ejpam-3776	370	8	pečarić.	pečarić.	PROPN
ejpam-3776	370	9	an	an	DET
ejpam-3776	370	10	improvement	improvement	NOUN
ejpam-3776	370	11	of	of	ADP
ejpam-3776	370	12	jensens	jensens	PROPN
ejpam-3776	370	13	inequality	inequality	NOUN
ejpam-3776	370	14	with	with	ADP
ejpam-3776	370	15	some	some	DET
ejpam-3776	370	16	applications	application	NOUN
ejpam-3776	370	17	.	.	PUNCT
ejpam-3776	371	1	asian	asian	ADJ
ejpam-3776	371	2	-	-	PUNCT
ejpam-3776	371	3	european	european	PROPN
ejpam-3776	371	4	j.	j.	PROPN
ejpam-3776	371	5	math	math	PROPN
ejpam-3776	371	6	.	.	PUNCT
ejpam-3776	371	7	,	,	PUNCT
ejpam-3776	371	8	2:85–94	2:85–94	NUM
ejpam-3776	371	9	,	,	PUNCT
ejpam-3776	371	10	2009	2009	NUM
ejpam-3776	371	11	.	.	PUNCT
ejpam-3776	372	1	[	[	X
ejpam-3776	372	2	13	13	NUM
ejpam-3776	372	3	]	]	X
ejpam-3776	372	4	asif	asif	PROPN
ejpam-3776	372	5	r.	r.	PROPN
ejpam-3776	372	6	khan	khan	PROPN
ejpam-3776	372	7	,	,	PUNCT
ejpam-3776	372	8	josip	josip	PROPN
ejpam-3776	372	9	pečarić	pečarić	PROPN
ejpam-3776	372	10	,	,	PUNCT
ejpam-3776	372	11	and	and	CCONJ
ejpam-3776	372	12	mirna	mirna	PROPN
ejpam-3776	372	13	rodić	rodić	NOUN
ejpam-3776	373	1	lipanović.	lipanović.	PROPN
ejpam-3776	373	2	nexponential	nexponential	ADJ
ejpam-3776	373	3	convexity	convexity	NOUN
ejpam-3776	373	4	for	for	ADP
ejpam-3776	373	5	jensen	jensen	PROPN
ejpam-3776	373	6	-	-	PUNCT
ejpam-3776	373	7	type	type	NOUN
ejpam-3776	373	8	inequalities	inequality	NOUN
ejpam-3776	373	9	.	.	PUNCT
ejpam-3776	374	1	j.	j.	PROPN
ejpam-3776	374	2	math	math	PROPN
ejpam-3776	374	3	.	.	PUNCT
ejpam-3776	375	1	inequal	inequal	ADJ
ejpam-3776	375	2	.	.	PUNCT
ejpam-3776	375	3	,	,	PUNCT
ejpam-3776	375	4	7:313–335	7:313–335	PROPN
ejpam-3776	375	5	,	,	PUNCT
ejpam-3776	375	6	2013	2013	NUM
ejpam-3776	375	7	.	.	PUNCT
ejpam-3776	376	1	[	[	X
ejpam-3776	376	2	14	14	NUM
ejpam-3776	376	3	]	]	X
ejpam-3776	376	4	asif	asif	PROPN
ejpam-3776	376	5	r.	r.	PROPN
ejpam-3776	376	6	khan	khan	PROPN
ejpam-3776	376	7	,	,	PUNCT
ejpam-3776	376	8	josip	josip	PROPN
ejpam-3776	376	9	pečarić	pečarić	PROPN
ejpam-3776	376	10	,	,	PUNCT
ejpam-3776	376	11	and	and	CCONJ
ejpam-3776	376	12	marjan	marjan	PROPN
ejpam-3776	376	13	praljak	praljak	PROPN
ejpam-3776	376	14	.	.	PUNCT
ejpam-3776	377	1	popoviciu	popoviciu	PROPN
ejpam-3776	377	2	type	type	NOUN
ejpam-3776	377	3	inequalities	inequality	NOUN
ejpam-3776	377	4	for	for	ADP
ejpam-3776	377	5	n	n	CCONJ
ejpam-3776	377	6	-	-	PUNCT
ejpam-3776	377	7	convex	convex	NOUN
ejpam-3776	377	8	functions	function	NOUN
ejpam-3776	377	9	via	via	ADP
ejpam-3776	377	10	extension	extension	NOUN
ejpam-3776	377	11	of	of	ADP
ejpam-3776	377	12	montgomery	montgomery	PROPN
ejpam-3776	377	13	identity	identity	NOUN
ejpam-3776	377	14	.	.	PUNCT
ejpam-3776	378	1	an	an	PRON
ejpam-3776	378	2	.	.	PUNCT
ejpam-3776	379	1	şt	şt	PROPN
ejpam-3776	379	2	.	.	PROPN
ejpam-3776	379	3	univ	univ	PROPN
ejpam-3776	379	4	.	.	PUNCT
ejpam-3776	380	1	ovidius	ovidius	PROPN
ejpam-3776	380	2	constanţa	constanţa	NOUN
ejpam-3776	380	3	,	,	PUNCT
ejpam-3776	380	4	24:161–188	24:161–188	PROPN
ejpam-3776	380	5	,	,	PUNCT
ejpam-3776	380	6	2016	2016	NUM
ejpam-3776	380	7	.	.	PUNCT
ejpam-3776	381	1	[	[	X
ejpam-3776	381	2	15	15	NUM
ejpam-3776	381	3	]	]	X
ejpam-3776	381	4	asif	asif	PROPN
ejpam-3776	381	5	r.	r.	PROPN
ejpam-3776	381	6	khan	khan	PROPN
ejpam-3776	381	7	,	,	PUNCT
ejpam-3776	381	8	josip	josip	PROPN
ejpam-3776	381	9	pečarić	pečarić	PROPN
ejpam-3776	381	10	,	,	PUNCT
ejpam-3776	381	11	and	and	CCONJ
ejpam-3776	381	12	marjan	marjan	PROPN
ejpam-3776	381	13	praljak	praljak	PROPN
ejpam-3776	381	14	.	.	PUNCT
ejpam-3776	382	1	a	a	DET
ejpam-3776	382	2	note	note	NOUN
ejpam-3776	382	3	on	on	ADP
ejpam-3776	382	4	generalized	generalized	ADJ
ejpam-3776	382	5	mercer	mercer	PROPN
ejpam-3776	382	6	’s	’s	PART
ejpam-3776	382	7	inequality	inequality	NOUN
ejpam-3776	382	8	.	.	PUNCT
ejpam-3776	383	1	bull	bull	NOUN
ejpam-3776	383	2	.	.	PUNCT
ejpam-3776	384	1	malays	malays	PROPN
ejpam-3776	384	2	.	.	PUNCT
ejpam-3776	385	1	math	math	NOUN
ejpam-3776	385	2	.	.	PUNCT
ejpam-3776	386	1	sci	sci	PROPN
ejpam-3776	386	2	.	.	PROPN
ejpam-3776	386	3	soc	soc	PROPN
ejpam-3776	386	4	.	.	PUNCT
ejpam-3776	386	5	,	,	PUNCT
ejpam-3776	386	6	2017:1–11	2017:1–11	NUM
ejpam-3776	386	7	,	,	PUNCT
ejpam-3776	386	8	2017	2017	NUM
ejpam-3776	386	9	.	.	PUNCT
ejpam-3776	387	1	[	[	X
ejpam-3776	387	2	16	16	NUM
ejpam-3776	387	3	]	]	X
ejpam-3776	387	4	asif	asif	PROPN
ejpam-3776	387	5	r.	r.	PROPN
ejpam-3776	387	6	khan	khan	PROPN
ejpam-3776	387	7	and	and	CCONJ
ejpam-3776	387	8	inam	inam	PROPN
ejpam-3776	387	9	ullah	ullah	PROPN
ejpam-3776	387	10	khan	khan	PROPN
ejpam-3776	387	11	.	.	PUNCT
ejpam-3776	388	1	some	some	DET
ejpam-3776	388	2	remarks	remark	NOUN
ejpam-3776	388	3	on	on	ADP
ejpam-3776	388	4	niezgoda	niezgoda	PROPN
ejpam-3776	388	5	’s	’s	PART
ejpam-3776	388	6	extension	extension	NOUN
ejpam-3776	388	7	of	of	ADP
ejpam-3776	388	8	jensenmercer	jensenmercer	NOUN
ejpam-3776	388	9	inequality	inequality	NOUN
ejpam-3776	388	10	.	.	PUNCT
ejpam-3776	389	1	adv	adv	PROPN
ejpam-3776	389	2	.	.	PUNCT
ejpam-3776	389	3	inequal	inequal	PROPN
ejpam-3776	389	4	.	.	PUNCT
ejpam-3776	390	1	appl	appl	PROPN
ejpam-3776	390	2	.	.	PROPN
ejpam-3776	390	3	,	,	PUNCT
ejpam-3776	390	4	12:1–11	12:1–11	NUM
ejpam-3776	390	5	,	,	PUNCT
ejpam-3776	390	6	2016	2016	NUM
ejpam-3776	390	7	.	.	PUNCT
ejpam-3776	391	1	[	[	X
ejpam-3776	391	2	17	17	NUM
ejpam-3776	391	3	]	]	X
ejpam-3776	391	4	asif	asif	PROPN
ejpam-3776	391	5	r.	r.	PROPN
ejpam-3776	391	6	khan	khan	PROPN
ejpam-3776	391	7	and	and	CCONJ
ejpam-3776	391	8	inam	inam	PROPN
ejpam-3776	391	9	ullah	ullah	PROPN
ejpam-3776	391	10	khan	khan	PROPN
ejpam-3776	391	11	.	.	PUNCT
ejpam-3776	392	1	an	an	DET
ejpam-3776	392	2	extension	extension	NOUN
ejpam-3776	392	3	of	of	ADP
ejpam-3776	392	4	jensen	jensen	PROPN
ejpam-3776	392	5	-	-	PUNCT
ejpam-3776	392	6	mercer	mercer	PROPN
ejpam-3776	392	7	inequality	inequality	NOUN
ejpam-3776	392	8	for	for	ADP
ejpam-3776	392	9	functions	function	NOUN
ejpam-3776	392	10	with	with	ADP
ejpam-3776	392	11	nondecreasing	nondecreasing	ADJ
ejpam-3776	392	12	increments	increment	NOUN
ejpam-3776	392	13	.	.	PUNCT
ejpam-3776	393	1	j.	j.	PROPN
ejpam-3776	393	2	inequal	inequal	PROPN
ejpam-3776	393	3	.	.	PUNCT
ejpam-3776	394	1	special	special	ADJ
ejpam-3776	394	2	funct	funct	NOUN
ejpam-3776	394	3	,	,	PUNCT
ejpam-3776	394	4	10:1–15	10:1–15	NUM
ejpam-3776	394	5	,	,	PUNCT
ejpam-3776	394	6	2019	2019	NUM
ejpam-3776	394	7	.	.	PUNCT
ejpam-3776	395	1	[	[	X
ejpam-3776	395	2	18	18	NUM
ejpam-3776	395	3	]	]	X
ejpam-3776	395	4	asif	asif	PROPN
ejpam-3776	395	5	r.	r.	PROPN
ejpam-3776	395	6	khan	khan	PROPN
ejpam-3776	395	7	,	,	PUNCT
ejpam-3776	395	8	inam	inam	PROPN
ejpam-3776	395	9	ullah	ullah	PROPN
ejpam-3776	395	10	khan	khan	PROPN
ejpam-3776	395	11	,	,	PUNCT
ejpam-3776	395	12	and	and	CCONJ
ejpam-3776	395	13	shahid	shahid	PROPN
ejpam-3776	395	14	sultan	sultan	PROPN
ejpam-3776	395	15	ali	ali	PROPN
ejpam-3776	395	16	ramji	ramji	PROPN
ejpam-3776	395	17	.	.	PUNCT
ejpam-3776	396	1	generalization	generalization	NOUN
ejpam-3776	396	2	and	and	CCONJ
ejpam-3776	396	3	refinements	refinement	NOUN
ejpam-3776	396	4	of	of	ADP
ejpam-3776	396	5	jensen	jensen	PROPN
ejpam-3776	396	6	-	-	PUNCT
ejpam-3776	396	7	mercer	mercer	PROPN
ejpam-3776	396	8	inequality	inequality	NOUN
ejpam-3776	396	9	with	with	ADP
ejpam-3776	396	10	applications	application	NOUN
ejpam-3776	396	11	.	.	PUNCT
ejpam-3776	397	1	j.	j.	PROPN
ejpam-3776	397	2	math	math	PROPN
ejpam-3776	397	3	.	.	PUNCT
ejpam-3776	398	1	inequal	inequal	ADJ
ejpam-3776	398	2	.	.	PUNCT
ejpam-3776	398	3	,	,	PUNCT
ejpam-3776	398	4	to	to	PART
ejpam-3776	398	5	appear	appear	VERB
ejpam-3776	398	6	.	.	PUNCT
ejpam-3776	399	1	[	[	X
ejpam-3776	399	2	19	19	NUM
ejpam-3776	399	3	]	]	X
ejpam-3776	399	4	asif	asif	PROPN
ejpam-3776	399	5	r.	r.	PROPN
ejpam-3776	399	6	khan	khan	PROPN
ejpam-3776	399	7	and	and	CCONJ
ejpam-3776	399	8	sumayyah	sumayyah	PROPN
ejpam-3776	399	9	saadi	saadi	PROPN
ejpam-3776	399	10	.	.	PUNCT
ejpam-3776	400	1	generalized	generalize	VERB
ejpam-3776	400	2	and	and	CCONJ
ejpam-3776	400	3	refinements	refinement	NOUN
ejpam-3776	400	4	of	of	ADP
ejpam-3776	400	5	generalized	generalized	ADJ
ejpam-3776	400	6	niezgoda	niezgoda	NOUN
ejpam-3776	400	7	-	-	PUNCT
ejpam-3776	400	8	type	type	NOUN
ejpam-3776	400	9	inequality	inequality	NOUN
ejpam-3776	400	10	for	for	ADP
ejpam-3776	400	11	similary	similary	ADJ
ejpam-3776	400	12	separable	separable	ADJ
ejpam-3776	400	13	vectors	vector	NOUN
ejpam-3776	400	14	with	with	ADP
ejpam-3776	400	15	applications	application	NOUN
ejpam-3776	400	16	.	.	PUNCT
ejpam-3776	401	1	filomat	filomat	NOUN
ejpam-3776	401	2	.	.	PUNCT
ejpam-3776	402	1	[	[	X
ejpam-3776	402	2	20	20	NUM
ejpam-3776	402	3	]	]	X
ejpam-3776	402	4	asif	asif	PROPN
ejpam-3776	402	5	r.	r.	PROPN
ejpam-3776	402	6	khan	khan	PROPN
ejpam-3776	402	7	and	and	CCONJ
ejpam-3776	402	8	sumayyah	sumayyah	PROPN
ejpam-3776	402	9	saadi	saadi	PROPN
ejpam-3776	402	10	.	.	PUNCT
ejpam-3776	403	1	generalized	generalize	VERB
ejpam-3776	403	2	jensen	jensen	PROPN
ejpam-3776	403	3	-	-	PUNCT
ejpam-3776	403	4	mercer	mercer	PROPN
ejpam-3776	403	5	inequality	inequality	NOUN
ejpam-3776	403	6	for	for	ADP
ejpam-3776	403	7	functions	function	NOUN
ejpam-3776	403	8	with	with	ADP
ejpam-3776	403	9	nondecreasing	nondecreasing	ADJ
ejpam-3776	403	10	increments	increment	NOUN
ejpam-3776	403	11	.	.	PUNCT
ejpam-3776	404	1	abs	ab	NOUN
ejpam-3776	404	2	.	.	PUNCT
ejpam-3776	405	1	and	and	CCONJ
ejpam-3776	405	2	appl	appl	PROPN
ejpam-3776	405	3	.	.	PROPN
ejpam-3776	406	1	anal	anal	PROPN
ejpam-3776	406	2	.	.	PROPN
ejpam-3776	406	3	,	,	PUNCT
ejpam-3776	406	4	2016:12	2016:12	NUM
ejpam-3776	406	5	,	,	PUNCT
ejpam-3776	406	6	2016	2016	NUM
ejpam-3776	406	7	.	.	PUNCT
ejpam-3776	407	1	[	[	X
ejpam-3776	407	2	21	21	NUM
ejpam-3776	407	3	]	]	PUNCT
ejpam-3776	407	4	k.	k.	PROPN
ejpam-3776	407	5	a.	a.	PROPN
ejpam-3776	407	6	khan	khan	PROPN
ejpam-3776	407	7	,	,	PUNCT
ejpam-3776	407	8	j.	j.	PROPN
ejpam-3776	407	9	pečarić	pečarić	PROPN
ejpam-3776	407	10	,	,	PUNCT
ejpam-3776	407	11	and	and	CCONJ
ejpam-3776	407	12	i.	i.	PROPN
ejpam-3776	407	13	peric	peric	PROPN
ejpam-3776	407	14	.	.	PUNCT
ejpam-3776	408	1	differences	difference	NOUN
ejpam-3776	408	2	of	of	ADP
ejpam-3776	408	3	weighted	weight	VERB
ejpam-3776	408	4	mixed	mixed	ADJ
ejpam-3776	408	5	symmetric	symmetric	ADJ
ejpam-3776	408	6	means	mean	NOUN
ejpam-3776	408	7	and	and	CCONJ
ejpam-3776	408	8	related	related	ADJ
ejpam-3776	408	9	results	result	NOUN
ejpam-3776	408	10	.	.	PUNCT
ejpam-3776	409	1	journal	journal	NOUN
ejpam-3776	409	2	of	of	ADP
ejpam-3776	409	3	inequalities	inequality	NOUN
ejpam-3776	409	4	and	and	CCONJ
ejpam-3776	409	5	applications	application	NOUN
ejpam-3776	409	6	,	,	PUNCT
ejpam-3776	409	7	2010	2010	NUM
ejpam-3776	409	8	,	,	PUNCT
ejpam-3776	409	9	2010	2010	NUM
ejpam-3776	409	10	.	.	PUNCT
ejpam-3776	410	1	[	[	X
ejpam-3776	410	2	22	22	NUM
ejpam-3776	410	3	]	]	PUNCT
ejpam-3776	410	4	m.	m.	NOUN
ejpam-3776	410	5	adil	adil	PROPN
ejpam-3776	410	6	khan	khan	PROPN
ejpam-3776	410	7	and	and	CCONJ
ejpam-3776	410	8	j.	j.	PROPN
ejpam-3776	410	9	peˇ	peˇ	PROPN
ejpam-3776	410	10	carić	carić	PROPN
ejpam-3776	410	11	asif	asif	PROPN
ejpam-3776	410	12	r.	r.	PROPN
ejpam-3776	410	13	khan	khan	PROPN
ejpam-3776	410	14	.	.	PUNCT
ejpam-3776	411	1	on	on	ADP
ejpam-3776	411	2	the	the	DET
ejpam-3776	411	3	refinements	refinement	NOUN
ejpam-3776	411	4	of	of	ADP
ejpam-3776	411	5	jensen	jensen	PROPN
ejpam-3776	411	6	-	-	PUNCT
ejpam-3776	411	7	mercer	mercer	PROPN
ejpam-3776	411	8	’s	’s	PART
ejpam-3776	411	9	inequality	inequality	NOUN
ejpam-3776	411	10	.	.	PUNCT
ejpam-3776	412	1	rev	rev	PROPN
ejpam-3776	412	2	.	.	PROPN
ejpam-3776	412	3	anal	anal	PROPN
ejpam-3776	412	4	.	.	PUNCT
ejpam-3776	413	1	numer	numer	PROPN
ejpam-3776	413	2	.	.	PUNCT
ejpam-3776	414	1	theor	theor	PROPN
ejpam-3776	414	2	.	.	PUNCT
ejpam-3776	415	1	approx	approx	PROPN
ejpam-3776	415	2	.	.	PUNCT
ejpam-3776	415	3	,	,	PUNCT
ejpam-3776	415	4	41:62–81	41:62–81	NUM
ejpam-3776	415	5	,	,	PUNCT
ejpam-3776	415	6	2012	2012	NUM
ejpam-3776	415	7	.	.	PUNCT
ejpam-3776	416	1	[	[	X
ejpam-3776	416	2	23	23	NUM
ejpam-3776	416	3	]	]	PUNCT
ejpam-3776	416	4	a.	a.	PROPN
ejpam-3776	416	5	w.	w.	PROPN
ejpam-3776	416	6	marshall	marshall	PROPN
ejpam-3776	416	7	,	,	PUNCT
ejpam-3776	416	8	i.	i.	PROPN
ejpam-3776	416	9	olkin	olkin	PROPN
ejpam-3776	416	10	,	,	PUNCT
ejpam-3776	416	11	and	and	CCONJ
ejpam-3776	416	12	b.	b.	PROPN
ejpam-3776	416	13	c.	c.	PROPN
ejpam-3776	416	14	arnold	arnold	PROPN
ejpam-3776	416	15	.	.	PUNCT
ejpam-3776	417	1	inequalities	inequality	NOUN
ejpam-3776	417	2	:	:	PUNCT
ejpam-3776	417	3	theory	theory	NOUN
ejpam-3776	417	4	of	of	ADP
ejpam-3776	417	5	majorization	majorization	NOUN
ejpam-3776	417	6	and	and	CCONJ
ejpam-3776	417	7	its	its	PRON
ejpam-3776	417	8	applications	application	NOUN
ejpam-3776	417	9	(	(	PUNCT
ejpam-3776	417	10	second	second	ADJ
ejpam-3776	417	11	edition	edition	NOUN
ejpam-3776	417	12	)	)	PUNCT
ejpam-3776	417	13	.	.	PUNCT
ejpam-3776	418	1	springer	springer	NOUN
ejpam-3776	418	2	series	series	PROPN
ejpam-3776	418	3	in	in	ADP
ejpam-3776	418	4	statistics	statistic	NOUN
ejpam-3776	418	5	,	,	PUNCT
ejpam-3776	418	6	new	new	PROPN
ejpam-3776	418	7	york	york	PROPN
ejpam-3776	418	8	,	,	PUNCT
ejpam-3776	418	9	2011	2011	NUM
ejpam-3776	418	10	.	.	PUNCT
ejpam-3776	419	1	references	reference	NOUN
ejpam-3776	419	2	829	829	NUM
ejpam-3776	419	3	[	[	X
ejpam-3776	419	4	24	24	NUM
ejpam-3776	419	5	]	]	PUNCT
ejpam-3776	419	6	a.	a.	NOUN
ejpam-3776	419	7	mcd	mcd	PROPN
ejpam-3776	419	8	.	.	PUNCT
ejpam-3776	420	1	mercer	mercer	PROPN
ejpam-3776	420	2	.	.	PUNCT
ejpam-3776	421	1	a	a	DET
ejpam-3776	421	2	variant	variant	NOUN
ejpam-3776	421	3	of	of	ADP
ejpam-3776	421	4	jensen	jensen	PROPN
ejpam-3776	421	5	’s	’s	PART
ejpam-3776	421	6	inequality	inequality	NOUN
ejpam-3776	421	7	.	.	PUNCT
ejpam-3776	422	1	j.	j.	PROPN
ejpam-3776	422	2	ineq	ineq	PROPN
ejpam-3776	422	3	.	.	PUNCT
ejpam-3776	423	1	pure	pure	ADJ
ejpam-3776	423	2	and	and	CCONJ
ejpam-3776	423	3	appl	appl	PROPN
ejpam-3776	423	4	.	.	PROPN
ejpam-3776	423	5	math	math	PROPN
ejpam-3776	423	6	.	.	PUNCT
ejpam-3776	424	1	,	,	PUNCT
ejpam-3776	425	1	4	4	NUM
ejpam-3776	425	2	,	,	PUNCT
ejpam-3776	425	3	2013	2013	NUM
ejpam-3776	425	4	.	.	PUNCT
ejpam-3776	426	1	[	[	X
ejpam-3776	426	2	25	25	NUM
ejpam-3776	426	3	]	]	X
ejpam-3776	426	4	d.	d.	PROPN
ejpam-3776	426	5	s.	s.	PROPN
ejpam-3776	426	6	mitrinović	mitrinović	PROPN
ejpam-3776	426	7	,	,	PUNCT
ejpam-3776	426	8	j.	j.	PROPN
ejpam-3776	426	9	e.	e.	PROPN
ejpam-3776	426	10	pečarić	pečarić	PROPN
ejpam-3776	426	11	,	,	PUNCT
ejpam-3776	426	12	and	and	CCONJ
ejpam-3776	426	13	a.	a.	NOUN
ejpam-3776	426	14	m.	m.	NOUN
ejpam-3776	426	15	fink	fink	PROPN
ejpam-3776	426	16	.	.	PUNCT
ejpam-3776	427	1	classical	classical	ADJ
ejpam-3776	427	2	and	and	CCONJ
ejpam-3776	427	3	new	new	ADJ
ejpam-3776	427	4	inequalities	inequality	NOUN
ejpam-3776	427	5	in	in	ADP
ejpam-3776	427	6	analysis	analysis	NOUN
ejpam-3776	427	7	.	.	PUNCT
ejpam-3776	428	1	kluwer	kluwer	NOUN
ejpam-3776	428	2	academic	academic	ADJ
ejpam-3776	428	3	publishers	publisher	NOUN
ejpam-3776	428	4	group	group	NOUN
ejpam-3776	428	5	,	,	PUNCT
ejpam-3776	428	6	dordrecht	dordrecht	PROPN
ejpam-3776	428	7	,	,	PUNCT
ejpam-3776	428	8	dordrecht	dordrecht	PROPN
ejpam-3776	428	9	,	,	PUNCT
ejpam-3776	428	10	1993	1993	NUM
ejpam-3776	428	11	.	.	PUNCT
ejpam-3776	429	1	[	[	X
ejpam-3776	429	2	26	26	NUM
ejpam-3776	429	3	]	]	PUNCT
ejpam-3776	429	4	m.	m.	NOUN
ejpam-3776	429	5	niezgoda	niezgoda	PROPN
ejpam-3776	429	6	.	.	PUNCT
ejpam-3776	430	1	a	a	DET
ejpam-3776	430	2	generalization	generalization	NOUN
ejpam-3776	430	3	of	of	ADP
ejpam-3776	430	4	mercer	mercer	PROPN
ejpam-3776	430	5	’s	’s	PART
ejpam-3776	430	6	result	result	NOUN
ejpam-3776	430	7	on	on	ADP
ejpam-3776	430	8	convex	convex	NOUN
ejpam-3776	430	9	functions	function	NOUN
ejpam-3776	430	10	.	.	PUNCT
ejpam-3776	431	1	nonlinear	nonlinear	ADJ
ejpam-3776	431	2	anal	anal	PROPN
ejpam-3776	431	3	.	.	PUNCT
ejpam-3776	431	4	,	,	PUNCT
ejpam-3776	431	5	71:2771–2779	71:2771–2779	PROPN
ejpam-3776	431	6	,	,	PUNCT
ejpam-3776	431	7	2009	2009	NUM
ejpam-3776	431	8	.	.	PUNCT
ejpam-3776	432	1	[	[	X
ejpam-3776	432	2	27	27	NUM
ejpam-3776	432	3	]	]	PUNCT
ejpam-3776	432	4	m.	m.	NOUN
ejpam-3776	432	5	niezgoda	niezgoda	PROPN
ejpam-3776	432	6	.	.	PUNCT
ejpam-3776	433	1	a	a	DET
ejpam-3776	433	2	generalization	generalization	NOUN
ejpam-3776	433	3	of	of	ADP
ejpam-3776	433	4	mercer	mercer	PROPN
ejpam-3776	433	5	’s	’s	PART
ejpam-3776	433	6	result	result	NOUN
ejpam-3776	433	7	on	on	ADP
ejpam-3776	433	8	convex	convex	NOUN
ejpam-3776	433	9	functions	function	NOUN
ejpam-3776	433	10	,	,	PUNCT
ejpam-3776	433	11	ii	ii	PROPN
ejpam-3776	433	12	.	.	PUNCT
ejpam-3776	433	13	math	math	PROPN
ejpam-3776	433	14	.	.	PUNCT
ejpam-3776	434	1	inequal	inequal	PROPN
ejpam-3776	434	2	.	.	PUNCT
ejpam-3776	435	1	appl	appl	PROPN
ejpam-3776	435	2	.	.	PROPN
ejpam-3776	435	3	,	,	PUNCT
ejpam-3776	435	4	18:1013–1023	18:1013–1023	NUM
ejpam-3776	435	5	,	,	PUNCT
ejpam-3776	435	6	2015	2015	NUM
ejpam-3776	435	7	.	.	PUNCT
ejpam-3776	436	1	[	[	X
ejpam-3776	436	2	28	28	NUM
ejpam-3776	436	3	]	]	X
ejpam-3776	436	4	j.	j.	PROPN
ejpam-3776	436	5	pečarić	pečarić	PROPN
ejpam-3776	436	6	,	,	PUNCT
ejpam-3776	436	7	f.	f.	PROPN
ejpam-3776	436	8	proschan	proschan	PROPN
ejpam-3776	436	9	,	,	PUNCT
ejpam-3776	436	10	and	and	CCONJ
ejpam-3776	436	11	y.	y.	PROPN
ejpam-3776	436	12	l.	l.	PROPN
ejpam-3776	436	13	tong	tong	PROPN
ejpam-3776	436	14	.	.	PUNCT
ejpam-3776	437	1	convex	convex	PROPN
ejpam-3776	437	2	functions	function	NOUN
ejpam-3776	437	3	,	,	PUNCT
ejpam-3776	437	4	partial	partial	ADJ
ejpam-3776	437	5	orderings	ordering	NOUN
ejpam-3776	437	6	and	and	CCONJ
ejpam-3776	437	7	statistical	statistical	ADJ
ejpam-3776	437	8	applications	application	NOUN
ejpam-3776	437	9	.	.	PUNCT
ejpam-3776	438	1	academic	academic	ADJ
ejpam-3776	438	2	press	press	NOUN
ejpam-3776	438	3	,	,	PUNCT
ejpam-3776	438	4	new	new	PROPN
ejpam-3776	438	5	york	york	PROPN
ejpam-3776	438	6	,	,	PUNCT
ejpam-3776	438	7	academic	academic	ADJ
ejpam-3776	438	8	press	press	NOUN
ejpam-3776	438	9	,	,	PUNCT
ejpam-3776	438	10	new	new	PROPN
ejpam-3776	438	11	york	york	PROPN
ejpam-3776	438	12	,	,	PUNCT
ejpam-3776	438	13	1992	1992	NUM
ejpam-3776	438	14	.	.	PUNCT
ejpam-3776	439	1	[	[	X
ejpam-3776	439	2	29	29	NUM
ejpam-3776	439	3	]	]	PUNCT
ejpam-3776	439	4	j.	j.	PROPN
ejpam-3776	439	5	rooin	rooin	PROPN
ejpam-3776	439	6	.	.	PUNCT
ejpam-3776	440	1	some	some	DET
ejpam-3776	440	2	refinements	refinement	NOUN
ejpam-3776	440	3	of	of	ADP
ejpam-3776	440	4	discrete	discrete	ADJ
ejpam-3776	440	5	jensens	jensens	PROPN
ejpam-3776	440	6	inequality	inequality	NOUN
ejpam-3776	440	7	and	and	CCONJ
ejpam-3776	440	8	some	some	PRON
ejpam-3776	440	9	of	of	ADP
ejpam-3776	440	10	its	its	PRON
ejpam-3776	440	11	applications	application	NOUN
ejpam-3776	440	12	.	.	PUNCT
ejpam-3776	441	1	nonlinear	nonlinear	ADJ
ejpam-3776	441	2	functional	functional	ADJ
ejpam-3776	441	3	anal	anal	NOUN
ejpam-3776	441	4	.	.	PUNCT
ejpam-3776	442	1	appl	appl	PROPN
ejpam-3776	442	2	.	.	PROPN
ejpam-3776	442	3	,	,	PUNCT
ejpam-3776	442	4	1:107–118	1:107–118	NUM
ejpam-3776	442	5	,	,	PUNCT
ejpam-3776	442	6	2007	2007	NUM
ejpam-3776	442	7	.	.	PUNCT