id	sid	tid	token	lemma	pos
ma-317	1	1	2025	2025	NUM
ma-317	1	2	ada	ada	PROPN
ma-317	1	3	academica	academica	PROPN
ma-317	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-317	1	5	.	.	PUNCT
ma-317	2	1	j.	j.	PROPN
ma-317	2	2	math	math	PROPN
ma-317	2	3	.	.	PUNCT
ma-317	3	1	anal	anal	ADJ
ma-317	3	2	.	.	PUNCT
ma-317	4	1	5	5	NUM
ma-317	4	2	(	(	PUNCT
ma-317	4	3	2025	2025	NUM
ma-317	4	4	)	)	PUNCT
ma-317	4	5	13doi	13doi	NOUN
ma-317	4	6	:	:	PUNCT
ma-317	4	7	10.28924	10.28924	NUM
ma-317	4	8	/	/	SYM
ma-317	4	9	ada	ada	PROPN
ma-317	4	10	/	/	SYM
ma-317	4	11	ma.5.13	ma.5.13	PROPN
ma-317	4	12	some	some	DET
ma-317	4	13	analytical	analytical	ADJ
ma-317	4	14	properties	property	NOUN
ma-317	4	15	of	of	ADP
ma-317	4	16	the	the	DET
ma-317	4	17	remainder	remainder	NOUN
ma-317	4	18	of	of	ADP
ma-317	4	19	binet	binet	NOUN
ma-317	4	20	-	-	PUNCT
ma-317	4	21	like	like	ADJ
ma-317	4	22	expression	expression	NOUN
ma-317	4	23	for	for	ADP
ma-317	4	24	the	the	DET
ma-317	4	25	barnes	barnes	PROPN
ma-317	4	26	g	g	NOUN
ma-317	4	27	-	-	PUNCT
ma-317	4	28	function	function	NOUN
ma-317	4	29	emmanuel	emmanuel	PROPN
ma-317	4	30	ansong	ansong	PROPN
ma-317	4	31	adjei	adjei	PROPN
ma-317	4	32	,	,	PUNCT
ma-317	4	33	kwara	kwara	PROPN
ma-317	4	34	nantomah∗	nantomah∗	PROPN
ma-317	4	35	,	,	PUNCT
ma-317	4	36	morgan	morgan	PROPN
ma-317	4	37	yindobil	yindobil	PROPN
ma-317	4	38	zubil	zubil	PROPN
ma-317	4	39	department	department	PROPN
ma-317	4	40	of	of	ADP
ma-317	4	41	mathematics	mathematic	NOUN
ma-317	4	42	,	,	PUNCT
ma-317	4	43	school	school	NOUN
ma-317	4	44	of	of	ADP
ma-317	4	45	mathematical	mathematical	ADJ
ma-317	4	46	sciences	sciences	PROPN
ma-317	4	47	,	,	PUNCT
ma-317	4	48	c.	c.	PROPN
ma-317	4	49	k.	k.	PROPN
ma-317	4	50	tedam	tedam	PROPN
ma-317	4	51	university	university	PROPN
ma-317	4	52	of	of	ADP
ma-317	4	53	technology	technology	NOUN
ma-317	4	54	and	and	CCONJ
ma-317	4	55	applied	apply	VERB
ma-317	4	56	sciences	science	NOUN
ma-317	4	57	,	,	PUNCT
ma-317	4	58	p.	p.	PROPN
ma-317	4	59	o.	o.	PROPN
ma-317	4	60	box	box	PROPN
ma-317	4	61	24	24	NUM
ma-317	4	62	,	,	PUNCT
ma-317	4	63	navrongo	navrongo	NOUN
ma-317	4	64	,	,	PUNCT
ma-317	4	65	upper	upper	ADJ
ma-317	4	66	-	-	PUNCT
ma-317	4	67	east	east	NOUN
ma-317	4	68	region	region	NOUN
ma-317	4	69	,	,	PUNCT
ma-317	4	70	ghana	ghana	PROPN
ma-317	4	71	adjeid2@gmail.com	adjeid2@gmail.com	PROPN
ma-317	4	72	,	,	PUNCT
ma-317	4	73	knantomah@cktutas.edu.gh	knantomah@cktutas.edu.gh	PROPN
ma-317	4	74	,	,	PUNCT
ma-317	4	75	morganzubil@gmail.com	morganzubil@gmail.com	X
ma-317	5	1	∗correspondence	∗correspondence	NOUN
ma-317	5	2	:	:	PUNCT
ma-317	5	3	knantomah@cktutas.edu.gh	knantomah@cktutas.edu.gh	NOUN
ma-317	5	4	abstract	abstract	NOUN
ma-317	5	5	.	.	PUNCT
ma-317	6	1	in	in	ADP
ma-317	6	2	this	this	DET
ma-317	6	3	paper	paper	NOUN
ma-317	6	4	,	,	PUNCT
ma-317	6	5	we	we	PRON
ma-317	6	6	prove	prove	VERB
ma-317	6	7	some	some	DET
ma-317	6	8	properties	property	NOUN
ma-317	6	9	such	such	ADJ
ma-317	6	10	as	as	ADP
ma-317	6	11	monotonicity	monotonicity	NOUN
ma-317	6	12	,	,	PUNCT
ma-317	6	13	complete	complete	ADJ
ma-317	6	14	monotonicity	monotonicity	NOUN
ma-317	6	15	,	,	PUNCT
ma-317	6	16	log	log	NOUN
ma-317	6	17	-	-	PUNCT
ma-317	6	18	arithmic	arithmic	ADJ
ma-317	6	19	convexity	convexity	NOUN
ma-317	6	20	,	,	PUNCT
ma-317	6	21	inequalities	inequality	NOUN
ma-317	6	22	,	,	PUNCT
ma-317	6	23	subadditivity	subadditivity	NOUN
ma-317	6	24	and	and	CCONJ
ma-317	6	25	starshapedness	starshapedness	NOUN
ma-317	6	26	,	,	PUNCT
ma-317	6	27	involving	involve	VERB
ma-317	6	28	the	the	DET
ma-317	6	29	binet	binet	NOUN
ma-317	6	30	-	-	PUNCT
ma-317	6	31	like	like	ADJ
ma-317	6	32	remainderof	remainderof	NOUN
ma-317	6	33	the	the	DET
ma-317	6	34	barnes	barnes	PROPN
ma-317	6	35	g	g	NOUN
ma-317	6	36	-	-	PUNCT
ma-317	6	37	function	function	NOUN
ma-317	6	38	.	.	PUNCT
ma-317	7	1	the	the	DET
ma-317	7	2	methods	method	NOUN
ma-317	7	3	of	of	ADP
ma-317	7	4	proofs	proof	NOUN
ma-317	7	5	are	be	AUX
ma-317	7	6	analytical	analytical	ADJ
ma-317	7	7	in	in	ADP
ma-317	7	8	nature	nature	NOUN
ma-317	7	9	.	.	PUNCT
ma-317	8	1	1	1	X
ma-317	8	2	.	.	X
ma-317	8	3	introduction	introduction	NOUN
ma-317	8	4	special	special	ADJ
ma-317	8	5	functions	function	NOUN
ma-317	8	6	are	be	AUX
ma-317	8	7	usually	usually	ADV
ma-317	8	8	encountered	encounter	VERB
ma-317	8	9	in	in	ADP
ma-317	8	10	almost	almost	ADV
ma-317	8	11	every	every	PRON
ma-317	8	12	scientific	scientific	ADJ
ma-317	8	13	discipline	discipline	NOUN
ma-317	8	14	.	.	PUNCT
ma-317	9	1	particularly	particularly	ADV
ma-317	9	2	,	,	PUNCT
ma-317	9	3	theyplay	theyplay	VERB
ma-317	9	4	important	important	ADJ
ma-317	9	5	roles	role	NOUN
ma-317	9	6	in	in	ADP
ma-317	9	7	areas	area	NOUN
ma-317	9	8	such	such	ADJ
ma-317	9	9	as	as	ADP
ma-317	9	10	mathematics	mathematic	NOUN
ma-317	9	11	,	,	PUNCT
ma-317	9	12	physics	physics	NOUN
ma-317	9	13	and	and	CCONJ
ma-317	9	14	engineering	engineering	NOUN
ma-317	9	15	.	.	PUNCT
ma-317	10	1	the	the	DET
ma-317	10	2	gamma	gamma	PROPN
ma-317	10	3	function	function	NOUN
ma-317	10	4	,	,	PUNCT
ma-317	10	5	which	which	PRON
ma-317	10	6	is	be	AUX
ma-317	10	7	an	an	DET
ma-317	10	8	extension	extension	NOUN
ma-317	10	9	of	of	ADP
ma-317	10	10	the	the	DET
ma-317	10	11	factorial	factorial	ADJ
ma-317	10	12	function	function	NOUN
ma-317	10	13	,	,	PUNCT
ma-317	10	14	is	be	AUX
ma-317	10	15	arguably	arguably	ADV
ma-317	10	16	the	the	DET
ma-317	10	17	most	most	ADV
ma-317	10	18	important	important	ADJ
ma-317	10	19	special	special	ADJ
ma-317	10	20	function	function	NOUN
ma-317	10	21	.	.	PUNCT
ma-317	11	1	thisis	thisi	NOUN
ma-317	11	2	largely	largely	ADV
ma-317	11	3	due	due	ADP
ma-317	11	4	to	to	ADP
ma-317	11	5	its	its	PRON
ma-317	11	6	vast	vast	ADJ
ma-317	11	7	areas	area	NOUN
ma-317	11	8	of	of	ADP
ma-317	11	9	applications	application	NOUN
ma-317	11	10	as	as	ADV
ma-317	11	11	well	well	ADV
ma-317	11	12	as	as	ADP
ma-317	11	13	its	its	PRON
ma-317	11	14	connection	connection	NOUN
ma-317	11	15	with	with	ADP
ma-317	11	16	other	other	ADJ
ma-317	11	17	special	special	ADJ
ma-317	11	18	functions.it	functions.it	NOUN
ma-317	11	19	is	be	AUX
ma-317	11	20	usually	usually	ADV
ma-317	11	21	defined	define	VERB
ma-317	11	22	as	as	ADP
ma-317	11	23	γ(z	γ(z	PROPN
ma-317	11	24	)	)	PUNCT
ma-317	11	25	=	=	SYM
ma-317	12	1	∫	∫	PROPN
ma-317	12	2	∞	∞	PROPN
ma-317	12	3	0	0	NUM
ma-317	13	1	tz−1e−tdt	tz−1e−tdt	ADJ
ma-317	13	2	(	(	PUNCT
ma-317	13	3	1	1	NUM
ma-317	13	4	)	)	PUNCT
ma-317	13	5	for	for	ADP
ma-317	13	6	z	z	PROPN
ma-317	13	7	>	>	X
ma-317	13	8	0	0	X
ma-317	13	9	.	.	PUNCT
ma-317	14	1	the	the	DET
ma-317	14	2	binet	binet	NOUN
ma-317	14	3	’s	’s	PART
ma-317	14	4	formula	formula	NOUN
ma-317	14	5	for	for	ADP
ma-317	14	6	logarithm	logarithm	NOUN
ma-317	14	7	of	of	ADP
ma-317	14	8	the	the	DET
ma-317	14	9	gamma	gamma	NOUN
ma-317	14	10	function	function	NOUN
ma-317	14	11	is	be	AUX
ma-317	14	12	given	give	VERB
ma-317	14	13	as	as	ADP
ma-317	14	14	ln	ln	ADJ
ma-317	14	15	γ(z	γ(z	PROPN
ma-317	14	16	)	)	PUNCT
ma-317	14	17	=	=	PUNCT
ma-317	15	1	(	(	PUNCT
ma-317	15	2	z	z	NOUN
ma-317	15	3	−	−	NOUN
ma-317	15	4	1	1	NUM
ma-317	15	5	2	2	NUM
ma-317	15	6	)	)	PUNCT
ma-317	15	7	ln	ln	NOUN
ma-317	15	8	z	z	NOUN
ma-317	15	9	−	−	PROPN
ma-317	15	10	z	z	NOUN
ma-317	16	1	+	+	CCONJ
ma-317	17	1	ln	ln	ADJ
ma-317	17	2	√	√	PROPN
ma-317	17	3	2π	2π	NOUN
ma-317	17	4	+	+	CCONJ
ma-317	17	5	θ(z	θ(z	NOUN
ma-317	17	6	)	)	PUNCT
ma-317	17	7	(	(	PUNCT
ma-317	17	8	2	2	X
ma-317	17	9	)	)	PUNCT
ma-317	17	10	for	for	ADP
ma-317	17	11	z	z	PROPN
ma-317	17	12	>	>	X
ma-317	17	13	0	0	NUM
ma-317	17	14	,	,	PUNCT
ma-317	17	15	where	where	SCONJ
ma-317	17	16	θ(z	θ(z	NOUN
ma-317	17	17	)	)	PUNCT
ma-317	17	18	=	=	SYM
ma-317	18	1	∫	∫	PROPN
ma-317	18	2	∞	∞	NUM
ma-317	18	3	0	0	NUM
ma-317	19	1	(	(	PUNCT
ma-317	19	2	1	1	NUM
ma-317	19	3	et	et	NOUN
ma-317	19	4	−	−	NOUN
ma-317	19	5	1	1	NUM
ma-317	19	6	−	−	PROPN
ma-317	19	7	1	1	NUM
ma-317	19	8	t	t	NOUN
ma-317	19	9	+	+	CCONJ
ma-317	19	10	1	1	NUM
ma-317	19	11	2	2	NUM
ma-317	19	12	)	)	PUNCT
ma-317	19	13	e−zt	e−zt	PROPN
ma-317	19	14	t	t	PROPN
ma-317	19	15	dt	dt	X
ma-317	19	16	(	(	PUNCT
ma-317	19	17	3	3	X
ma-317	19	18	)	)	PUNCT
ma-317	19	19	is	be	AUX
ma-317	19	20	known	know	VERB
ma-317	19	21	as	as	ADP
ma-317	19	22	the	the	DET
ma-317	19	23	remainder	remainder	NOUN
ma-317	19	24	of	of	ADP
ma-317	19	25	binet	binet	NOUN
ma-317	19	26	’s	’s	PART
ma-317	19	27	formula	formula	NOUN
ma-317	19	28	.	.	PUNCT
ma-317	20	1	due	due	ADP
ma-317	20	2	to	to	ADP
ma-317	20	3	the	the	DET
ma-317	20	4	important	important	ADJ
ma-317	20	5	properties	property	NOUN
ma-317	20	6	exhibited	exhibit	VERB
ma-317	20	7	by	by	ADP
ma-317	20	8	thefunction	thefunction	NOUN
ma-317	20	9	θ(z	θ(z	NOUN
ma-317	20	10	)	)	PUNCT
ma-317	20	11	,	,	PUNCT
ma-317	20	12	it	it	PRON
ma-317	20	13	has	have	AUX
ma-317	20	14	been	be	AUX
ma-317	20	15	investigated	investigate	VERB
ma-317	20	16	in	in	ADP
ma-317	20	17	multiple	multiple	ADJ
ma-317	20	18	ways	way	NOUN
ma-317	20	19	.	.	PUNCT
ma-317	21	1	in	in	ADP
ma-317	21	2	[	[	X
ma-317	21	3	5	5	NUM
ma-317	21	4	]	]	PUNCT
ma-317	21	5	,	,	PUNCT
ma-317	21	6	the	the	DET
ma-317	21	7	authors	author	NOUN
ma-317	21	8	proved	prove	VERB
ma-317	21	9	among	among	ADP
ma-317	21	10	otherthings	otherthing	NOUN
ma-317	21	11	that	that	SCONJ
ma-317	21	12	,	,	PUNCT
ma-317	21	13	for	for	ADP
ma-317	21	14	p	p	PROPN
ma-317	21	15	∈	∈	PROPN
ma-317	21	16	(	(	PUNCT
ma-317	21	17	0	0	NUM
ma-317	21	18	,	,	PUNCT
ma-317	21	19	1	1	NUM
ma-317	21	20	]	]	PUNCT
ma-317	21	21	,	,	PUNCT
ma-317	21	22	the	the	DET
ma-317	21	23	function	function	NOUN
ma-317	21	24	fp(z	fp(z	PUNCT
ma-317	21	25	)	)	PUNCT
ma-317	21	26	=	=	PRON
ma-317	21	27	θ(pz)−	θ(pz)−	X
ma-317	21	28	pθ(z	pθ(z	NOUN
ma-317	21	29	)	)	PUNCT
ma-317	21	30	(	(	PUNCT
ma-317	21	31	4	4	X
ma-317	21	32	)	)	PUNCT
ma-317	21	33	received	receive	VERB
ma-317	21	34	:	:	PUNCT
ma-317	21	35	20	20	NUM
ma-317	21	36	dec	dec	PROPN
ma-317	21	37	2024.2010	2024.2010	NUM
ma-317	21	38	mathematics	mathematic	NOUN
ma-317	21	39	subject	subject	ADJ
ma-317	21	40	classification	classification	NOUN
ma-317	21	41	.	.	PUNCT
ma-317	22	1	33b15	33b15	NUM
ma-317	22	2	,	,	PUNCT
ma-317	22	3	26a48	26a48	NUM
ma-317	22	4	,	,	PUNCT
ma-317	22	5	26a51	26a51	NUM
ma-317	22	6	.	.	PUNCT
ma-317	23	1	key	key	ADJ
ma-317	23	2	words	word	NOUN
ma-317	23	3	and	and	CCONJ
ma-317	23	4	phrases	phrase	NOUN
ma-317	23	5	.	.	PUNCT
ma-317	24	1	multiple	multiple	ADJ
ma-317	24	2	gamma	gamma	NOUN
ma-317	24	3	function	function	NOUN
ma-317	24	4	;	;	PUNCT
ma-317	24	5	barnes	barnes	PROPN
ma-317	24	6	g	g	NOUN
ma-317	24	7	-	-	PUNCT
ma-317	24	8	function	function	NOUN
ma-317	24	9	;	;	PUNCT
ma-317	24	10	binet	binet	NOUN
ma-317	24	11	remainder	remainder	NOUN
ma-317	24	12	;	;	PUNCT
ma-317	24	13	binet	binet	NOUN
ma-317	24	14	-	-	PUNCT
ma-317	24	15	like	like	ADJ
ma-317	24	16	remainder	remainder	NOUN
ma-317	24	17	;	;	PUNCT
ma-317	24	18	log	log	NOUN
ma-317	24	19	-	-	PUNCT
ma-317	24	20	convex	convex	NOUN
ma-317	24	21	function	function	NOUN
ma-317	24	22	;	;	PUNCT
ma-317	24	23	completely	completely	ADV
ma-317	24	24	monotonic	monotonic	ADJ
ma-317	24	25	function	function	NOUN
ma-317	24	26	;	;	PUNCT
ma-317	24	27	subadditive	subadditive	ADJ
ma-317	24	28	function	function	NOUN
ma-317	24	29	;	;	PUNCT
ma-317	24	30	starshaped	starshape	VERB
ma-317	24	31	function.1	function.1	PROPN
ma-317	24	32	https://adac.ee	https://adac.ee	PROPN
ma-317	24	33	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	PROPN
ma-317	24	34	eur	eur	PROPN
ma-317	24	35	.	.	PUNCT
ma-317	25	1	j.	j.	PROPN
ma-317	25	2	math	math	PROPN
ma-317	25	3	.	.	PUNCT
ma-317	26	1	anal	anal	PROPN
ma-317	26	2	.	.	PUNCT
ma-317	27	1	10.28924	10.28924	NUM
ma-317	27	2	/	/	SYM
ma-317	27	3	ada	ada	PROPN
ma-317	27	4	/	/	SYM
ma-317	27	5	ma.5.13	ma.5.13	PROPN
ma-317	27	6	2is	2is	NOUN
ma-317	27	7	completely	completely	ADV
ma-317	27	8	monotonic	monotonic	ADJ
ma-317	27	9	on	on	ADP
ma-317	27	10	(	(	PUNCT
ma-317	27	11	0,∞	0,∞	NOUN
ma-317	27	12	)	)	PUNCT
ma-317	27	13	.	.	PUNCT
ma-317	28	1	in	in	ADP
ma-317	28	2	[	[	X
ma-317	28	3	7	7	NUM
ma-317	28	4	]	]	PUNCT
ma-317	28	5	,	,	PUNCT
ma-317	28	6	the	the	DET
ma-317	28	7	authors	author	NOUN
ma-317	28	8	investigated	investigate	VERB
ma-317	28	9	the	the	DET
ma-317	28	10	complete	complete	ADJ
ma-317	28	11	monotonicity	monotonicity	NOUN
ma-317	28	12	ofthe	ofthe	NOUN
ma-317	28	13	function	function	NOUN
ma-317	28	14	fp	fp	PROPN
ma-317	28	15	,	,	PUNCT
ma-317	28	16	q	q	NOUN
ma-317	28	17	,	,	PUNCT
ma-317	28	18	r	r	NOUN
ma-317	28	19	(	(	PUNCT
ma-317	28	20	z	z	NOUN
ma-317	28	21	)	)	PUNCT
ma-317	28	22	=	=	SYM
ma-317	29	1	r	r	NOUN
ma-317	29	2	[	[	X
ma-317	29	3	θ(pz)−	θ(pz)−	NUM
ma-317	29	4	qθ(z	qθ(z	NOUN
ma-317	29	5	)	)	PUNCT
ma-317	29	6	]	]	PUNCT
ma-317	29	7	(	(	PUNCT
ma-317	29	8	5	5	X
ma-317	29	9	)	)	PUNCT
ma-317	29	10	where	where	SCONJ
ma-317	29	11	p	p	NOUN
ma-317	29	12	>	>	X
ma-317	29	13	0	0	PROPN
ma-317	29	14	,	,	PUNCT
ma-317	29	15	q	q	PUNCT
ma-317	29	16	is	be	AUX
ma-317	29	17	a	a	DET
ma-317	29	18	real	real	ADJ
ma-317	29	19	number	number	NOUN
ma-317	29	20	and	and	CCONJ
ma-317	29	21	r	r	NOUN
ma-317	29	22	6=	6=	PROPN
ma-317	29	23	0	0	NUM
ma-317	29	24	.	.	PUNCT
ma-317	30	1	they	they	PRON
ma-317	30	2	further	far	ADV
ma-317	30	3	established	establish	VERB
ma-317	30	4	that	that	SCONJ
ma-317	30	5	θ(z	θ(z	NOUN
ma-317	30	6	)	)	PUNCT
ma-317	30	7	is	be	AUX
ma-317	30	8	subadditiveon	subadditiveon	NOUN
ma-317	30	9	(	(	PUNCT
ma-317	30	10	0,∞	0,∞	NOUN
ma-317	30	11	)	)	PUNCT
ma-317	30	12	and	and	CCONJ
ma-317	30	13	−θ(z	−θ(z	X
ma-317	30	14	)	)	PUNCT
ma-317	30	15	is	be	AUX
ma-317	30	16	starshaped	starshape	VERB
ma-317	30	17	on	on	ADP
ma-317	30	18	(	(	PUNCT
ma-317	30	19	0,∞	0,∞	NUM
ma-317	30	20	)	)	PUNCT
ma-317	30	21	.	.	PUNCT
ma-317	31	1	in	in	ADP
ma-317	31	2	[	[	X
ma-317	31	3	11	11	NUM
ma-317	31	4	]	]	PUNCT
ma-317	31	5	,	,	PUNCT
ma-317	31	6	the	the	DET
ma-317	31	7	authors	author	NOUN
ma-317	31	8	considered	consider	VERB
ma-317	31	9	a	a	DET
ma-317	31	10	generalization	generalization	NOUN
ma-317	31	11	ofthe	ofthe	NOUN
ma-317	31	12	function	function	NOUN
ma-317	31	13	θ(z	θ(z	NOUN
ma-317	31	14	)	)	PUNCT
ma-317	31	15	which	which	PRON
ma-317	31	16	is	be	AUX
ma-317	31	17	denoted	denote	VERB
ma-317	31	18	by	by	ADP
ma-317	31	19	θα(z	θα(z	NOUN
ma-317	31	20	)	)	PUNCT
ma-317	31	21	.	.	PUNCT
ma-317	32	1	they	they	PRON
ma-317	32	2	investigated	investigate	VERB
ma-317	32	3	the	the	DET
ma-317	32	4	complete	complete	ADJ
ma-317	32	5	monotonicity	monotonicity	NOUN
ma-317	32	6	of	of	ADP
ma-317	32	7	thefunction	thefunction	NOUN
ma-317	32	8	fp	fp	NOUN
ma-317	32	9	,	,	PUNCT
ma-317	32	10	q	q	NOUN
ma-317	32	11	,	,	PUNCT
ma-317	32	12	α(z	α(z	NOUN
ma-317	32	13	)	)	PUNCT
ma-317	32	14	=	=	PUNCT
ma-317	33	1	θα(pz)−	θα(pz)−	NUM
ma-317	33	2	qθα(z	qθα(z	PROPN
ma-317	33	3	)	)	PUNCT
ma-317	33	4	(	(	PUNCT
ma-317	33	5	6	6	NUM
ma-317	33	6	)	)	PUNCT
ma-317	33	7	where	where	SCONJ
ma-317	33	8	p	p	NOUN
ma-317	33	9	>	>	X
ma-317	33	10	0	0	PROPN
ma-317	33	11	,	,	PUNCT
ma-317	33	12	q	q	PUNCT
ma-317	33	13	is	be	AUX
ma-317	33	14	a	a	DET
ma-317	33	15	real	real	ADJ
ma-317	33	16	number	number	NOUN
ma-317	33	17	and	and	CCONJ
ma-317	33	18	α	α	NOUN
ma-317	33	19	>	>	X
ma-317	33	20	0	0	NUM
ma-317	33	21	.	.	PUNCT
ma-317	34	1	among	among	ADP
ma-317	34	2	other	other	ADJ
ma-317	34	3	things	thing	NOUN
ma-317	34	4	,	,	PUNCT
ma-317	34	5	they	they	PRON
ma-317	34	6	further	far	ADV
ma-317	34	7	established	establish	VERB
ma-317	34	8	that	that	SCONJ
ma-317	34	9	thefunction	thefunction	NOUN
ma-317	34	10	θα(z	θα(z	PRON
ma-317	34	11	)	)	PUNCT
ma-317	34	12	is	be	AUX
ma-317	34	13	subadditive	subadditive	ADJ
ma-317	34	14	.	.	PUNCT
ma-317	35	1	in	in	ADP
ma-317	35	2	[	[	X
ma-317	35	3	3	3	NUM
ma-317	35	4	]	]	PUNCT
ma-317	35	5	,	,	PUNCT
ma-317	35	6	the	the	DET
ma-317	35	7	authors	author	NOUN
ma-317	35	8	considered	consider	VERB
ma-317	35	9	an	an	DET
ma-317	35	10	inequality	inequality	NOUN
ma-317	35	11	for	for	ADP
ma-317	35	12	the	the	DET
ma-317	35	13	r	r	NOUN
ma-317	35	14	-	-	PUNCT
ma-317	35	15	th	th	VERB
ma-317	35	16	derivative	derivative	NOUN
ma-317	35	17	of	of	ADP
ma-317	35	18	θ(z	θ(z	NOUN
ma-317	35	19	)	)	PUNCT
ma-317	35	20	.	.	PUNCT
ma-317	36	1	subsequently	subsequently	ADV
ma-317	36	2	,	,	PUNCT
ma-317	36	3	they	they	PRON
ma-317	36	4	obtain	obtain	VERB
ma-317	36	5	the	the	DET
ma-317	36	6	turan	turan	NOUN
ma-317	36	7	-	-	PUNCT
ma-317	36	8	type	type	NOUN
ma-317	36	9	inequality	inequality	NOUN
ma-317	36	10	(	(	PUNCT
ma-317	36	11	θ(r+1)(z	θ(r+1)(z	NUM
ma-317	36	12	)	)	PUNCT
ma-317	36	13	)	)	PUNCT
ma-317	36	14	2	2	NUM
ma-317	36	15	≤	≤	NUM
ma-317	36	16	θ(r)(z)θ(r+2)(z	θ(r)(z)θ(r+2)(z	PUNCT
ma-317	36	17	)	)	PUNCT
ma-317	36	18	(	(	PUNCT
ma-317	36	19	7	7	X
ma-317	36	20	)	)	PUNCT
ma-317	36	21	where	where	SCONJ
ma-317	36	22	r	r	NOUN
ma-317	36	23	∈	∈	PROPN
ma-317	36	24	n.	n.	NOUN
ma-317	36	25	a	a	DET
ma-317	36	26	generalization	generalization	NOUN
ma-317	36	27	of	of	ADP
ma-317	36	28	(	(	PUNCT
ma-317	36	29	7	7	X
ma-317	36	30	)	)	PUNCT
ma-317	36	31	can	can	AUX
ma-317	36	32	be	be	AUX
ma-317	36	33	found	find	VERB
ma-317	36	34	in	in	ADP
ma-317	36	35	[	[	PUNCT
ma-317	36	36	2].the	2].the	DET
ma-317	36	37	multiple	multiple	ADJ
ma-317	36	38	gamma	gamma	NOUN
ma-317	36	39	function	function	NOUN
ma-317	36	40	,	,	PUNCT
ma-317	36	41	which	which	PRON
ma-317	36	42	is	be	AUX
ma-317	36	43	a	a	DET
ma-317	36	44	generalization	generalization	NOUN
ma-317	36	45	of	of	ADP
ma-317	36	46	the	the	DET
ma-317	36	47	ordinary	ordinary	ADJ
ma-317	36	48	gamma	gamma	NOUN
ma-317	36	49	function	function	NOUN
ma-317	36	50	,	,	PUNCT
ma-317	36	51	wasdefined	wasdefine	VERB
ma-317	36	52	by	by	ADP
ma-317	36	53	barnes	barne	NOUN
ma-317	36	54	as	as	ADP
ma-317	36	55	[	[	X
ma-317	36	56	1	1	NUM
ma-317	36	57	]	]	PUNCT
ma-317	36	58	γr+1(z	γr+1(z	X
ma-317	37	1	+	+	CCONJ
ma-317	37	2	1	1	X
ma-317	37	3	)	)	PUNCT
ma-317	37	4	=	=	VERB
ma-317	37	5	γr+1(z	γr+1(z	NOUN
ma-317	38	1	+	+	CCONJ
ma-317	38	2	1	1	X
ma-317	38	3	)	)	PUNCT
ma-317	38	4	γr	γr	PROPN
ma-317	38	5	(	(	PUNCT
ma-317	38	6	z	z	NOUN
ma-317	38	7	)	)	PUNCT
ma-317	38	8	,	,	PUNCT
ma-317	38	9	z	z	NOUN
ma-317	38	10	∈	∈	PROPN
ma-317	38	11	c	c	X
ma-317	38	12	,	,	PUNCT
ma-317	38	13	r	r	NOUN
ma-317	38	14	∈	∈	PROPN
ma-317	38	15	n	n	CCONJ
ma-317	38	16	,	,	PUNCT
ma-317	38	17	γ1(z	γ1(z	NUM
ma-317	38	18	)	)	PUNCT
ma-317	38	19	=	=	SYM
ma-317	38	20	γ(z	γ(z	PROPN
ma-317	38	21	)	)	PUNCT
ma-317	38	22	,	,	PUNCT
ma-317	38	23	γr	γr	X
ma-317	38	24	(	(	PUNCT
ma-317	38	25	1	1	NUM
ma-317	38	26	)	)	PUNCT
ma-317	38	27	=	=	SYM
ma-317	38	28	1	1	X
ma-317	38	29	.	.	PUNCT
ma-317	39	1	the	the	DET
ma-317	39	2	particular	particular	ADJ
ma-317	39	3	case	case	NOUN
ma-317	39	4	g(z	g(z	PROPN
ma-317	39	5	)	)	PUNCT
ma-317	39	6	=	=	SYM
ma-317	39	7	1	1	NUM
ma-317	39	8	γ2(z	γ2(z	NOUN
ma-317	39	9	)	)	PUNCT
ma-317	39	10	is	be	AUX
ma-317	39	11	referred	refer	VERB
ma-317	39	12	to	to	ADP
ma-317	39	13	as	as	ADP
ma-317	39	14	the	the	DET
ma-317	39	15	barnes	barnes	PROPN
ma-317	39	16	g	g	NOUN
ma-317	39	17	-	-	PUNCT
ma-317	39	18	function	function	NOUN
ma-317	39	19	or	or	CCONJ
ma-317	39	20	the	the	DET
ma-317	39	21	double	double	ADJ
ma-317	39	22	gammafunction	gammafunction	NOUN
ma-317	39	23	.	.	PUNCT
ma-317	40	1	it	it	PRON
ma-317	40	2	satisfies	satisfy	VERB
ma-317	40	3	the	the	DET
ma-317	40	4	following	follow	VERB
ma-317	40	5	basic	basic	ADJ
ma-317	40	6	properties	property	NOUN
ma-317	40	7	[	[	X
ma-317	40	8	1	1	NUM
ma-317	40	9	]	]	PUNCT
ma-317	40	10	.	.	PUNCT
ma-317	41	1	g(z	g(z	ADJ
ma-317	41	2	+	+	CCONJ
ma-317	41	3	1	1	X
ma-317	41	4	)	)	PUNCT
ma-317	41	5	=	=	NOUN
ma-317	41	6	g(z)γ(z	g(z)γ(z	NOUN
ma-317	41	7	)	)	PUNCT
ma-317	41	8	,	,	PUNCT
ma-317	41	9	z	z	NOUN
ma-317	41	10	∈	∈	PROPN
ma-317	41	11	c	c	X
ma-317	41	12	,	,	PUNCT
ma-317	41	13	g(1	g(1	NOUN
ma-317	41	14	)	)	PUNCT
ma-317	41	15	=	=	SYM
ma-317	41	16	1	1	NUM
ma-317	41	17	,	,	PUNCT
ma-317	41	18	(	(	PUNCT
ma-317	41	19	lng(z))′′′	lng(z))′′′	NOUN
ma-317	41	20	≥	≥	NUM
ma-317	41	21	0	0	NUM
ma-317	41	22	,	,	PUNCT
ma-317	41	23	z	z	NOUN
ma-317	41	24	>	>	X
ma-317	41	25	0	0	X
ma-317	41	26	.	.	PUNCT
ma-317	42	1	for	for	ADP
ma-317	42	2	further	further	ADJ
ma-317	42	3	properties	property	NOUN
ma-317	42	4	of	of	ADP
ma-317	42	5	the	the	DET
ma-317	42	6	function	function	NOUN
ma-317	42	7	g(z	g(z	PROPN
ma-317	42	8	)	)	PUNCT
ma-317	42	9	,	,	PUNCT
ma-317	42	10	one	one	PRON
ma-317	42	11	may	may	AUX
ma-317	42	12	refer	refer	VERB
ma-317	42	13	to	to	ADP
ma-317	42	14	the	the	DET
ma-317	42	15	papers	paper	NOUN
ma-317	42	16	[	[	X
ma-317	42	17	10	10	NUM
ma-317	42	18	]	]	PUNCT
ma-317	42	19	and	and	CCONJ
ma-317	42	20	[	[	X
ma-317	42	21	12	12	NUM
ma-317	42	22	]	]	PUNCT
ma-317	42	23	and	and	CCONJ
ma-317	42	24	thereferences	thereference	NOUN
ma-317	42	25	in	in	ADP
ma-317	42	26	there.a	there.a	PRON
ma-317	42	27	binet	binet	NOUN
ma-317	42	28	-	-	PUNCT
ma-317	42	29	like	like	ADJ
ma-317	42	30	expression	expression	NOUN
ma-317	42	31	for	for	ADP
ma-317	42	32	logarithm	logarithm	NOUN
ma-317	42	33	of	of	ADP
ma-317	42	34	the	the	DET
ma-317	42	35	double	double	ADJ
ma-317	42	36	gamma	gamma	NOUN
ma-317	42	37	function	function	NOUN
ma-317	42	38	is	be	AUX
ma-317	42	39	given	give	VERB
ma-317	42	40	by	by	ADP
ma-317	42	41	choi	choi	NOUN
ma-317	42	42	[	[	X
ma-317	42	43	6	6	NUM
ma-317	42	44	]	]	PUNCT
ma-317	42	45	as	as	ADP
ma-317	42	46	ln	ln	ADJ
ma-317	42	47	γ2(z	γ2(z	NOUN
ma-317	42	48	)	)	PUNCT
ma-317	42	49	=	=	SYM
ma-317	42	50	lna−	lna−	PROPN
ma-317	42	51	z2	z2	PROPN
ma-317	42	52	4	4	NUM
ma-317	42	53	+	+	CCONJ
ma-317	42	54	(	(	PUNCT
ma-317	42	55	z2	z2	NUM
ma-317	42	56	2	2	NUM
ma-317	42	57	−	−	PROPN
ma-317	42	58	z	z	NOUN
ma-317	42	59	2	2	NUM
ma-317	42	60	+	+	CCONJ
ma-317	42	61	1	1	NUM
ma-317	42	62	12	12	NUM
ma-317	42	63	)	)	PUNCT
ma-317	43	1	ln	ln	NOUN
ma-317	43	2	z	z	NOUN
ma-317	44	1	+	+	CCONJ
ma-317	45	1	(	(	PUNCT
ma-317	45	2	1−	1−	NUM
ma-317	45	3	z	z	NOUN
ma-317	45	4	)	)	PUNCT
ma-317	45	5	ln	ln	ADJ
ma-317	45	6	γ(z	γ(z	PROPN
ma-317	45	7	)	)	PUNCT
ma-317	46	1	+	+	NUM
ma-317	46	2	θ(z	θ(z	NOUN
ma-317	46	3	)	)	PUNCT
ma-317	46	4	(	(	PUNCT
ma-317	46	5	8)	8)	NUM
ma-317	46	6	where	where	SCONJ
ma-317	46	7	a	a	DET
ma-317	46	8	=	=	SYM
ma-317	46	9	1.282427	1.282427	NUM
ma-317	46	10	...	...	PUNCT
ma-317	46	11	is	be	AUX
ma-317	46	12	the	the	DET
ma-317	46	13	glaisher	glaisher	PROPN
ma-317	46	14	-	-	PUNCT
ma-317	46	15	kinkelin	kinkelin	PROPN
ma-317	46	16	constant	constant	ADJ
ma-317	46	17	and	and	CCONJ
ma-317	46	18	θ(z	θ(z	NOUN
ma-317	46	19	)	)	PUNCT
ma-317	46	20	=	=	SYM
ma-317	47	1	∫	∫	PROPN
ma-317	47	2	∞	∞	NUM
ma-317	47	3	0	0	NUM
ma-317	48	1	(	(	PUNCT
ma-317	48	2	1	1	NUM
ma-317	48	3	t	t	NOUN
ma-317	48	4	−	−	NUM
ma-317	48	5	1	1	NUM
ma-317	48	6	et	et	NOUN
ma-317	48	7	−	−	NOUN
ma-317	48	8	1	1	NUM
ma-317	48	9	−	−	NOUN
ma-317	48	10	1	1	NUM
ma-317	48	11	2	2	NUM
ma-317	48	12	+	+	NUM
ma-317	48	13	t	t	PROPN
ma-317	48	14	12	12	NUM
ma-317	48	15	)	)	PUNCT
ma-317	48	16	e−zt	e−zt	PROPN
ma-317	48	17	t2	t2	PROPN
ma-317	48	18	dt	dt	PROPN
ma-317	48	19	.	.	PUNCT
ma-317	49	1	(	(	PUNCT
ma-317	49	2	9	9	X
ma-317	49	3	)	)	PUNCT
ma-317	49	4	is	be	AUX
ma-317	49	5	what	what	PRON
ma-317	49	6	is	be	AUX
ma-317	49	7	referred	refer	VERB
ma-317	49	8	to	to	ADP
ma-317	49	9	as	as	ADP
ma-317	49	10	the	the	DET
ma-317	49	11	binet	binet	NOUN
ma-317	49	12	-	-	PUNCT
ma-317	49	13	like	like	ADJ
ma-317	49	14	remainder	remainder	NOUN
ma-317	49	15	.	.	PUNCT
ma-317	50	1	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	NUM
ma-317	50	2	eur	eur	PROPN
ma-317	50	3	.	.	PUNCT
ma-317	51	1	j.	j.	PROPN
ma-317	51	2	math	math	PROPN
ma-317	51	3	.	.	PUNCT
ma-317	52	1	anal	anal	PROPN
ma-317	52	2	.	.	PUNCT
ma-317	53	1	10.28924	10.28924	NUM
ma-317	53	2	/	/	SYM
ma-317	53	3	ada	ada	PROPN
ma-317	53	4	/	/	SYM
ma-317	53	5	ma.5.13	ma.5.13	PROPN
ma-317	53	6	3it	3it	NOUN
ma-317	53	7	is	be	AUX
ma-317	53	8	interesting	interesting	ADJ
ma-317	53	9	to	to	PART
ma-317	53	10	observe	observe	VERB
ma-317	53	11	that	that	SCONJ
ma-317	53	12	the	the	DET
ma-317	53	13	binet	binet	NOUN
ma-317	53	14	-	-	PUNCT
ma-317	53	15	like	like	ADJ
ma-317	53	16	remainder	remainder	NOUN
ma-317	53	17	,	,	PUNCT
ma-317	53	18	θ(z	θ(z	NOUN
ma-317	53	19	)	)	PUNCT
ma-317	53	20	has	have	VERB
ma-317	53	21	some	some	DET
ma-317	53	22	resemblance	resemblance	NOUN
ma-317	53	23	withthe	withthe	ADJ
ma-317	53	24	binet	binet	NOUN
ma-317	53	25	remainder	remainder	NOUN
ma-317	53	26	,	,	PUNCT
ma-317	53	27	θ(z	θ(z	NOUN
ma-317	53	28	)	)	PUNCT
ma-317	53	29	.	.	PUNCT
ma-317	54	1	the	the	DET
ma-317	54	2	natural	natural	ADJ
ma-317	54	3	question	question	NOUN
ma-317	54	4	that	that	PRON
ma-317	54	5	arise	arise	NOUN
ma-317	54	6	is	be	AUX
ma-317	54	7	:	:	PUNCT
ma-317	54	8	does	do	AUX
ma-317	54	9	the	the	DET
ma-317	54	10	function	function	NOUN
ma-317	54	11	θ(z	θ(z	NOUN
ma-317	54	12	)	)	PUNCT
ma-317	54	13	satisfythe	satisfythe	ADJ
ma-317	54	14	properties	property	NOUN
ma-317	54	15	satisfied	satisfy	VERB
ma-317	54	16	by	by	ADP
ma-317	54	17	the	the	DET
ma-317	54	18	function	function	NOUN
ma-317	54	19	θ(z	θ(z	NOUN
ma-317	54	20	)	)	PUNCT
ma-317	54	21	?	?	PUNCT
ma-317	55	1	motivated	motivate	VERB
ma-317	55	2	by	by	ADP
ma-317	55	3	the	the	DET
ma-317	55	4	papers	paper	NOUN
ma-317	55	5	[	[	X
ma-317	55	6	3	3	NUM
ma-317	55	7	,	,	PUNCT
ma-317	55	8	5	5	NUM
ma-317	55	9	,	,	PUNCT
ma-317	55	10	7	7	NUM
ma-317	55	11	,	,	PUNCT
ma-317	55	12	11	11	NUM
ma-317	55	13	]	]	PUNCT
ma-317	55	14	,	,	PUNCT
ma-317	55	15	the	the	DET
ma-317	55	16	objectiveof	objectiveof	ADJ
ma-317	55	17	this	this	DET
ma-317	55	18	paper	paper	NOUN
ma-317	55	19	is	be	AUX
ma-317	55	20	to	to	PART
ma-317	55	21	answer	answer	VERB
ma-317	55	22	this	this	DET
ma-317	55	23	question	question	NOUN
ma-317	55	24	.	.	PUNCT
ma-317	56	1	we	we	PRON
ma-317	56	2	establish	establish	VERB
ma-317	56	3	some	some	DET
ma-317	56	4	properties	property	NOUN
ma-317	56	5	of	of	ADP
ma-317	56	6	the	the	DET
ma-317	56	7	function	function	NOUN
ma-317	56	8	θ(z)such	θ(z)such	ADJ
ma-317	56	9	as	as	ADP
ma-317	56	10	monotonicity	monotonicity	NOUN
ma-317	56	11	,	,	PUNCT
ma-317	56	12	complete	complete	ADJ
ma-317	56	13	monotonicity	monotonicity	NOUN
ma-317	56	14	,	,	PUNCT
ma-317	56	15	logarithmic	logarithmic	ADJ
ma-317	56	16	convexity	convexity	NOUN
ma-317	56	17	,	,	PUNCT
ma-317	56	18	inequalities	inequality	NOUN
ma-317	56	19	,	,	PUNCT
ma-317	56	20	subadditivity	subadditivity	NOUN
ma-317	56	21	andstarshapedness	andstarshapedness	ADV
ma-317	56	22	,	,	PUNCT
ma-317	56	23	among	among	ADP
ma-317	56	24	others	other	NOUN
ma-317	56	25	.	.	PUNCT
ma-317	57	1	we	we	PRON
ma-317	57	2	present	present	VERB
ma-317	57	3	our	our	PRON
ma-317	57	4	findings	finding	NOUN
ma-317	57	5	in	in	ADP
ma-317	57	6	the	the	DET
ma-317	57	7	next	next	ADJ
ma-317	57	8	section	section	NOUN
ma-317	57	9	.	.	PUNCT
ma-317	58	1	before	before	ADP
ma-317	58	2	that	that	PRON
ma-317	58	3	,	,	PUNCT
ma-317	58	4	we	we	PRON
ma-317	58	5	providethe	providethe	VERB
ma-317	58	6	following	follow	VERB
ma-317	58	7	definitions	definition	NOUN
ma-317	58	8	which	which	PRON
ma-317	58	9	shall	shall	AUX
ma-317	58	10	pave	pave	VERB
ma-317	58	11	the	the	DET
ma-317	58	12	way	way	NOUN
ma-317	58	13	for	for	SCONJ
ma-317	58	14	us	we	PRON
ma-317	58	15	to	to	PART
ma-317	58	16	prove	prove	VERB
ma-317	58	17	our	our	PRON
ma-317	58	18	results	result	NOUN
ma-317	58	19	.	.	PUNCT
ma-317	59	1	throughout	throughout	ADP
ma-317	59	2	thispaper	thispaper	NOUN
ma-317	59	3	,	,	PUNCT
ma-317	59	4	n	n	X
ma-317	59	5	=	=	PUNCT
ma-317	59	6	{	{	PUNCT
ma-317	59	7	1	1	NUM
ma-317	59	8	,	,	PUNCT
ma-317	59	9	2	2	NUM
ma-317	59	10	,	,	PUNCT
ma-317	59	11	3	3	NUM
ma-317	59	12	,	,	PUNCT
ma-317	59	13	.	.	PUNCT
ma-317	59	14	.	.	PUNCT
ma-317	59	15	.	.	PUNCT
ma-317	60	1	}	}	PUNCT
ma-317	61	1	and	and	CCONJ
ma-317	61	2	n0	n0	NUM
ma-317	61	3	=	=	SYM
ma-317	61	4	{	{	PUNCT
ma-317	61	5	0	0	NUM
ma-317	61	6	,	,	PUNCT
ma-317	61	7	1	1	NUM
ma-317	61	8	,	,	PUNCT
ma-317	61	9	2	2	NUM
ma-317	61	10	,	,	PUNCT
ma-317	61	11	3	3	NUM
ma-317	61	12	,	,	PUNCT
ma-317	61	13	.	.	PUNCT
ma-317	61	14	.	.	PUNCT
ma-317	61	15	.	.	PUNCT
ma-317	62	1	}	}	PUNCT
ma-317	62	2	.	.	PUNCT
ma-317	63	1	definition	definition	NOUN
ma-317	63	2	1.1	1.1	NUM
ma-317	63	3	.	.	PUNCT
ma-317	64	1	a	a	DET
ma-317	64	2	real	real	ADV
ma-317	64	3	-	-	PUNCT
ma-317	64	4	valued	value	VERB
ma-317	64	5	function	function	NOUN
ma-317	64	6	k	k	PROPN
ma-317	64	7	defined	define	VERB
ma-317	64	8	on	on	ADP
ma-317	64	9	an	an	DET
ma-317	64	10	interval	interval	NOUN
ma-317	64	11	i	i	PRON
ma-317	64	12	⊆	⊆	NUM
ma-317	64	13	r	r	NOUN
ma-317	64	14	is	be	AUX
ma-317	64	15	said	say	VERB
ma-317	64	16	to	to	PART
ma-317	64	17	be	be	AUX
ma-317	64	18	convex	convex	ADJ
ma-317	64	19	on	on	ADP
ma-317	64	20	i	i	PRON
ma-317	64	21	iff	iff	PROPN
ma-317	64	22	k	k	PROPN
ma-317	64	23	(	(	PUNCT
ma-317	64	24	x	x	SYM
ma-317	64	25	u	u	PROPN
ma-317	64	26	+	+	NOUN
ma-317	64	27	y	y	PROPN
ma-317	64	28	v	v	NOUN
ma-317	64	29	)	)	PUNCT
ma-317	64	30	≤	≤	NUM
ma-317	64	31	k(x	k(x	PROPN
ma-317	64	32	)	)	PUNCT
ma-317	64	33	u	u	NOUN
ma-317	64	34	+	+	PROPN
ma-317	64	35	k(y	k(y	PROPN
ma-317	64	36	)	)	PUNCT
ma-317	64	37	v	v	NOUN
ma-317	64	38	(	(	PUNCT
ma-317	64	39	10	10	NUM
ma-317	64	40	)	)	PUNCT
ma-317	64	41	holds	hold	VERB
ma-317	64	42	for	for	ADP
ma-317	64	43	all	all	DET
ma-317	64	44	x	x	NOUN
ma-317	64	45	,	,	PUNCT
ma-317	64	46	y	y	PROPN
ma-317	64	47	∈	∈	PROPN
ma-317	64	48	i	i	PRON
ma-317	64	49	and	and	CCONJ
ma-317	64	50	u	u	X
ma-317	64	51	>	>	X
ma-317	64	52	1	1	NUM
ma-317	64	53	,	,	PUNCT
ma-317	64	54	v	v	PART
ma-317	64	55	>	>	X
ma-317	64	56	1	1	NUM
ma-317	64	57	such	such	ADJ
ma-317	64	58	that	that	SCONJ
ma-317	64	59	1	1	NUM
ma-317	64	60	u	u	NOUN
ma-317	64	61	+	+	NOUN
ma-317	64	62	1	1	NUM
ma-317	64	63	v	v	NOUN
ma-317	64	64	=	=	SYM
ma-317	64	65	1	1	NUM
ma-317	64	66	.	.	PUNCT
ma-317	64	67	equivalently	equivalently	ADV
ma-317	64	68	,	,	PUNCT
ma-317	64	69	k	k	PROPN
ma-317	64	70	is	be	AUX
ma-317	64	71	said	say	VERB
ma-317	64	72	to	to	PART
ma-317	64	73	be	be	AUX
ma-317	64	74	convexon	convexon	ADJ
ma-317	64	75	i	i	PRON
ma-317	64	76	iff	iff	VERB
ma-317	64	77	k′′(z	k′′(z	NOUN
ma-317	64	78	)	)	PUNCT
ma-317	64	79	≥	≥	X
ma-317	64	80	0	0	NUM
ma-317	64	81	(	(	PUNCT
ma-317	64	82	11	11	NUM
ma-317	64	83	)	)	PUNCT
ma-317	64	84	for	for	ADP
ma-317	64	85	all	all	DET
ma-317	64	86	z	z	NOUN
ma-317	64	87	∈	∈	NOUN
ma-317	64	88	i	i	PRON
ma-317	64	89	.	.	PUNCT
ma-317	65	1	if	if	SCONJ
ma-317	65	2	the	the	DET
ma-317	65	3	inequalities	inequality	NOUN
ma-317	65	4	(	(	PUNCT
ma-317	65	5	10	10	NUM
ma-317	65	6	)	)	PUNCT
ma-317	65	7	and	and	CCONJ
ma-317	65	8	(	(	PUNCT
ma-317	65	9	11	11	NUM
ma-317	65	10	)	)	PUNCT
ma-317	65	11	are	be	AUX
ma-317	65	12	reversed	reverse	VERB
ma-317	65	13	,	,	PUNCT
ma-317	65	14	then	then	ADV
ma-317	65	15	k	k	PROPN
ma-317	65	16	is	be	AUX
ma-317	65	17	said	say	VERB
ma-317	65	18	to	to	PART
ma-317	65	19	be	be	AUX
ma-317	65	20	concave	concave	VERB
ma-317	65	21	on	on	ADP
ma-317	65	22	i	i	PRON
ma-317	65	23	.	.	PUNCT
ma-317	66	1	definition	definition	NOUN
ma-317	66	2	1.2	1.2	NUM
ma-317	66	3	.	.	PUNCT
ma-317	67	1	a	a	DET
ma-317	67	2	positive	positive	ADJ
ma-317	67	3	real	real	ADV
ma-317	67	4	-	-	PUNCT
ma-317	67	5	valued	value	VERB
ma-317	67	6	function	function	NOUN
ma-317	67	7	k	k	PROPN
ma-317	67	8	defined	define	VERB
ma-317	67	9	on	on	ADP
ma-317	67	10	an	an	DET
ma-317	67	11	interval	interval	NOUN
ma-317	67	12	i	i	PRON
ma-317	67	13	⊆	⊆	NUM
ma-317	67	14	r	r	NOUN
ma-317	67	15	is	be	AUX
ma-317	67	16	said	say	VERB
ma-317	67	17	to	to	PART
ma-317	67	18	belogarithmically	belogarithmically	ADV
ma-317	67	19	convex	convex	VERB
ma-317	67	20	on	on	ADP
ma-317	67	21	i	i	PRON
ma-317	67	22	iff	iff	PROPN
ma-317	67	23	k	k	PROPN
ma-317	67	24	(	(	PUNCT
ma-317	67	25	x	x	SYM
ma-317	67	26	u	u	PROPN
ma-317	67	27	+	+	NOUN
ma-317	67	28	y	y	PROPN
ma-317	67	29	v	v	NOUN
ma-317	67	30	)	)	PUNCT
ma-317	67	31	≤	≤	NOUN
ma-317	68	1	[	[	X
ma-317	68	2	k(x	k(x	PROPN
ma-317	68	3	)	)	PUNCT
ma-317	68	4	]	]	PUNCT
ma-317	68	5	1	1	NUM
ma-317	68	6	u	u	NOUN
ma-317	68	7	[	[	X
ma-317	68	8	k(y	k(y	PROPN
ma-317	68	9	)	)	PUNCT
ma-317	68	10	]	]	PUNCT
ma-317	68	11	1	1	NUM
ma-317	68	12	v	v	X
ma-317	68	13	(	(	PUNCT
ma-317	68	14	12	12	NUM
ma-317	68	15	)	)	PUNCT
ma-317	68	16	holds	hold	VERB
ma-317	68	17	for	for	ADP
ma-317	68	18	all	all	DET
ma-317	68	19	x	x	NOUN
ma-317	68	20	,	,	PUNCT
ma-317	68	21	y	y	PROPN
ma-317	68	22	∈	∈	PROPN
ma-317	68	23	i	i	PRON
ma-317	68	24	and	and	CCONJ
ma-317	68	25	u	u	X
ma-317	68	26	>	>	X
ma-317	68	27	1	1	NUM
ma-317	68	28	,	,	PUNCT
ma-317	68	29	v	v	PART
ma-317	68	30	>	>	X
ma-317	68	31	1	1	NUM
ma-317	68	32	such	such	ADJ
ma-317	68	33	that	that	SCONJ
ma-317	68	34	1	1	NUM
ma-317	68	35	u	u	NOUN
ma-317	68	36	+	+	NOUN
ma-317	68	37	1	1	NUM
ma-317	68	38	v	v	NOUN
ma-317	68	39	=	=	SYM
ma-317	68	40	1	1	NUM
ma-317	68	41	.	.	PUNCT
ma-317	68	42	equivalently	equivalently	ADV
ma-317	68	43	,	,	PUNCT
ma-317	68	44	k	k	PROPN
ma-317	68	45	is	be	AUX
ma-317	68	46	said	say	VERB
ma-317	68	47	to	to	PART
ma-317	68	48	belogarithmically	belogarithmically	ADV
ma-317	68	49	convex	convex	VERB
ma-317	68	50	on	on	ADP
ma-317	68	51	i	i	PROPN
ma-317	68	52	iff	iff	PROPN
ma-317	69	1	[	[	X
ma-317	69	2	lnk(z)]′′	lnk(z)]′′	ADP
ma-317	69	3	≥	≥	NOUN
ma-317	69	4	0	0	NUM
ma-317	69	5	(	(	PUNCT
ma-317	69	6	13	13	NUM
ma-317	69	7	)	)	PUNCT
ma-317	69	8	for	for	ADP
ma-317	69	9	all	all	DET
ma-317	69	10	z	z	NOUN
ma-317	69	11	∈	∈	NOUN
ma-317	69	12	i	i	PRON
ma-317	69	13	.	.	PUNCT
ma-317	70	1	if	if	SCONJ
ma-317	70	2	the	the	DET
ma-317	70	3	inequalities	inequality	NOUN
ma-317	70	4	(	(	PUNCT
ma-317	70	5	12	12	NUM
ma-317	70	6	)	)	PUNCT
ma-317	70	7	and	and	CCONJ
ma-317	70	8	(	(	PUNCT
ma-317	70	9	13	13	NUM
ma-317	70	10	)	)	PUNCT
ma-317	70	11	are	be	AUX
ma-317	70	12	reversed	reverse	VERB
ma-317	70	13	,	,	PUNCT
ma-317	70	14	then	then	ADV
ma-317	70	15	k	k	PROPN
ma-317	70	16	is	be	AUX
ma-317	70	17	said	say	VERB
ma-317	70	18	to	to	PART
ma-317	70	19	be	be	AUX
ma-317	70	20	logarithmicallyconcave	logarithmicallyconcave	NOUN
ma-317	70	21	on	on	ADP
ma-317	70	22	i	i	PRON
ma-317	70	23	.	.	PUNCT
ma-317	71	1	definition	definition	NOUN
ma-317	71	2	1.3	1.3	NUM
ma-317	71	3	(	(	PUNCT
ma-317	71	4	[	[	X
ma-317	71	5	13	13	NUM
ma-317	71	6	]	]	NUM
ma-317	71	7	)	)	PUNCT
ma-317	71	8	.	.	PUNCT
ma-317	72	1	a	a	DET
ma-317	72	2	real	real	ADV
ma-317	72	3	-	-	PUNCT
ma-317	72	4	valued	value	VERB
ma-317	72	5	function	function	NOUN
ma-317	72	6	k	k	PROPN
ma-317	72	7	defined	define	VERB
ma-317	72	8	on	on	ADP
ma-317	72	9	an	an	DET
ma-317	72	10	interval	interval	NOUN
ma-317	72	11	i	i	PRON
ma-317	72	12	⊆	⊆	NUM
ma-317	72	13	r	r	NOUN
ma-317	72	14	is	be	AUX
ma-317	72	15	said	say	VERB
ma-317	72	16	to	to	PART
ma-317	72	17	be	be	AUX
ma-317	72	18	completelymonotonic	completelymonotonic	ADJ
ma-317	72	19	on	on	ADP
ma-317	72	20	i	i	PROPN
ma-317	72	21	iff	iff	PROPN
ma-317	72	22	(	(	PUNCT
ma-317	72	23	−1)nk(n)(z	−1)nk(n)(z	PROPN
ma-317	72	24	)	)	PUNCT
ma-317	72	25	≥	≥	X
ma-317	72	26	0	0	NUM
ma-317	72	27	holds	hold	VERB
ma-317	72	28	for	for	ADP
ma-317	72	29	all	all	DET
ma-317	72	30	z	z	NOUN
ma-317	72	31	∈	∈	NOUN
ma-317	73	1	i	i	PRON
ma-317	73	2	and	and	CCONJ
ma-317	73	3	n	n	PRON
ma-317	73	4	∈	∈	PROPN
ma-317	73	5	n0	n0	PROPN
ma-317	73	6	.	.	PUNCT
ma-317	74	1	definition	definition	NOUN
ma-317	74	2	1.4	1.4	NUM
ma-317	74	3	(	(	PUNCT
ma-317	74	4	[	[	X
ma-317	74	5	4	4	NUM
ma-317	74	6	]	]	NUM
ma-317	74	7	)	)	PUNCT
ma-317	74	8	.	.	PUNCT
ma-317	75	1	a	a	DET
ma-317	75	2	real	real	ADV
ma-317	75	3	-	-	PUNCT
ma-317	75	4	valued	value	VERB
ma-317	75	5	function	function	NOUN
ma-317	75	6	k	k	PROPN
ma-317	75	7	defined	define	VERB
ma-317	75	8	on	on	ADP
ma-317	75	9	an	an	DET
ma-317	75	10	interval	interval	NOUN
ma-317	75	11	i	i	PRON
ma-317	75	12	⊆	⊆	NUM
ma-317	75	13	r	r	NOUN
ma-317	75	14	is	be	AUX
ma-317	75	15	said	say	VERB
ma-317	75	16	to	to	PART
ma-317	75	17	be	be	AUX
ma-317	75	18	subadditiveon	subadditiveon	NOUN
ma-317	75	19	i	i	PRON
ma-317	75	20	iff	iff	VERB
ma-317	75	21	k(x	k(x	PROPN
ma-317	75	22	+	+	PROPN
ma-317	75	23	y	y	NOUN
ma-317	75	24	)	)	PUNCT
ma-317	75	25	≤	≤	NUM
ma-317	75	26	k(x	k(x	PROPN
ma-317	75	27	)	)	PUNCT
ma-317	76	1	+	+	NOUN
ma-317	76	2	k(y	k(y	X
ma-317	76	3	)	)	PUNCT
ma-317	76	4	holds	hold	VERB
ma-317	76	5	for	for	ADP
ma-317	76	6	all	all	DET
ma-317	76	7	x	x	NOUN
ma-317	76	8	,	,	PUNCT
ma-317	76	9	y	y	PROPN
ma-317	76	10	∈	∈	PROPN
ma-317	76	11	i	i	PRON
ma-317	76	12	.	.	PUNCT
ma-317	77	1	if	if	SCONJ
ma-317	77	2	the	the	DET
ma-317	77	3	inequality	inequality	NOUN
ma-317	77	4	is	be	AUX
ma-317	77	5	reversed	reverse	VERB
ma-317	77	6	,	,	PUNCT
ma-317	77	7	then	then	ADV
ma-317	77	8	k	k	PROPN
ma-317	77	9	is	be	AUX
ma-317	77	10	said	say	VERB
ma-317	77	11	to	to	PART
ma-317	77	12	be	be	AUX
ma-317	77	13	superadditive	superadditive	ADJ
ma-317	77	14	on	on	ADP
ma-317	77	15	i	i	PRON
ma-317	77	16	.	.	PUNCT
ma-317	78	1	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	PROPN
ma-317	78	2	eur	eur	PROPN
ma-317	78	3	.	.	PUNCT
ma-317	79	1	j.	j.	PROPN
ma-317	79	2	math	math	PROPN
ma-317	79	3	.	.	PUNCT
ma-317	80	1	anal	anal	PROPN
ma-317	80	2	.	.	PUNCT
ma-317	81	1	10.28924	10.28924	NUM
ma-317	81	2	/	/	SYM
ma-317	81	3	ada	ada	PROPN
ma-317	81	4	/	/	SYM
ma-317	81	5	ma.5.13	ma.5.13	PROPN
ma-317	81	6	4	4	NUM
ma-317	81	7	definition	definition	NOUN
ma-317	81	8	1.5	1.5	NUM
ma-317	81	9	(	(	PUNCT
ma-317	81	10	[	[	X
ma-317	81	11	4	4	NUM
ma-317	81	12	]	]	NUM
ma-317	81	13	)	)	PUNCT
ma-317	81	14	.	.	PUNCT
ma-317	82	1	a	a	DET
ma-317	82	2	real	real	ADV
ma-317	82	3	-	-	PUNCT
ma-317	82	4	valued	value	VERB
ma-317	82	5	function	function	NOUN
ma-317	82	6	k	k	PROPN
ma-317	82	7	defined	define	VERB
ma-317	82	8	on	on	ADP
ma-317	82	9	an	an	DET
ma-317	82	10	interval	interval	NOUN
ma-317	82	11	i	i	PRON
ma-317	82	12	⊆	⊆	NUM
ma-317	82	13	r	r	NOUN
ma-317	82	14	is	be	AUX
ma-317	82	15	said	say	VERB
ma-317	82	16	to	to	PART
ma-317	82	17	be	be	AUX
ma-317	82	18	starshapedon	starshapedon	VERB
ma-317	82	19	i	i	PRON
ma-317	82	20	iff	iff	PROPN
ma-317	82	21	k(αz	k(αz	PROPN
ma-317	82	22	)	)	PUNCT
ma-317	82	23	≤	≤	NOUN
ma-317	82	24	αk(z),for	αk(z),for	ADP
ma-317	82	25	all	all	DET
ma-317	82	26	z	z	NOUN
ma-317	82	27	∈	∈	NOUN
ma-317	83	1	i	i	PRON
ma-317	83	2	and	and	CCONJ
ma-317	83	3	α	α	PRON
ma-317	83	4	∈	∈	PROPN
ma-317	84	1	[	[	X
ma-317	84	2	0	0	NUM
ma-317	84	3	,	,	PUNCT
ma-317	84	4	1	1	NUM
ma-317	84	5	]	]	PUNCT
ma-317	84	6	.	.	PUNCT
ma-317	85	1	2	2	X
ma-317	85	2	.	.	NOUN
ma-317	85	3	results	result	NOUN
ma-317	85	4	and	and	CCONJ
ma-317	85	5	discussion	discussion	NOUN
ma-317	85	6	beginning	begin	VERB
ma-317	85	7	with	with	ADP
ma-317	85	8	the	the	DET
ma-317	85	9	following	follow	VERB
ma-317	85	10	lemmas	lemmas	PROPN
ma-317	85	11	,	,	PUNCT
ma-317	85	12	we	we	PRON
ma-317	85	13	now	now	ADV
ma-317	85	14	present	present	VERB
ma-317	85	15	our	our	PRON
ma-317	85	16	findings	finding	NOUN
ma-317	85	17	in	in	ADP
ma-317	85	18	this	this	DET
ma-317	85	19	section	section	NOUN
ma-317	85	20	.	.	PUNCT
ma-317	86	1	lemma	lemma	PROPN
ma-317	86	2	2.1	2.1	NUM
ma-317	86	3	.	.	PUNCT
ma-317	87	1	for	for	ADP
ma-317	87	2	t	t	PROPN
ma-317	87	3	>	>	X
ma-317	87	4	0	0	PROPN
ma-317	87	5	,	,	PUNCT
ma-317	87	6	the	the	DET
ma-317	87	7	inequality	inequality	NOUN
ma-317	87	8	1	1	NUM
ma-317	87	9	t2	t2	NOUN
ma-317	87	10	−	−	PROPN
ma-317	87	11	1	1	NUM
ma-317	87	12	12	12	NUM
ma-317	87	13	<	<	X
ma-317	87	14	e−t	e−t	NOUN
ma-317	87	15	(	(	PUNCT
ma-317	87	16	1−	1−	NUM
ma-317	87	17	e−t)2	e−t)2	NOUN
ma-317	87	18	<	<	SYM
ma-317	87	19	1	1	NUM
ma-317	87	20	t2	t2	NOUN
ma-317	87	21	(	(	PUNCT
ma-317	87	22	14	14	NUM
ma-317	87	23	)	)	PUNCT
ma-317	87	24	holds	hold	VERB
ma-317	87	25	.	.	PUNCT
ma-317	88	1	proof	proof	NOUN
ma-317	88	2	.	.	PUNCT
ma-317	89	1	see	see	VERB
ma-317	89	2	theorem	theorem	ADJ
ma-317	89	3	2	2	NUM
ma-317	89	4	of	of	ADP
ma-317	89	5	[	[	X
ma-317	89	6	8	8	NUM
ma-317	89	7	]	]	PUNCT
ma-317	89	8	.	.	PUNCT
ma-317	90	1	�	�	PROPN
ma-317	90	2	lemma	lemma	PROPN
ma-317	90	3	2.2	2.2	NUM
ma-317	90	4	.	.	PUNCT
ma-317	91	1	for	for	ADP
ma-317	91	2	t	t	PROPN
ma-317	91	3	>	>	X
ma-317	91	4	0	0	PROPN
ma-317	91	5	,	,	PUNCT
ma-317	91	6	the	the	DET
ma-317	91	7	function	function	NOUN
ma-317	91	8	p(t	p(t	NOUN
ma-317	91	9	)	)	PUNCT
ma-317	92	1	=	=	SYM
ma-317	92	2	1	1	NUM
ma-317	92	3	t2	t2	NOUN
ma-317	92	4	−	−	PROPN
ma-317	92	5	e−t	e−t	NOUN
ma-317	92	6	(	(	PUNCT
ma-317	92	7	1−	1−	NUM
ma-317	92	8	e−t)2	e−t)2	PROPN
ma-317	92	9	(	(	PUNCT
ma-317	92	10	15	15	NUM
ma-317	92	11	)	)	PUNCT
ma-317	92	12	is	be	AUX
ma-317	92	13	strictly	strictly	ADV
ma-317	92	14	decreasing	decrease	VERB
ma-317	92	15	.	.	PUNCT
ma-317	93	1	proof	proof	NOUN
ma-317	93	2	.	.	PUNCT
ma-317	94	1	see	see	VERB
ma-317	94	2	theorem	theorem	NOUN
ma-317	94	3	1	1	NUM
ma-317	94	4	of	of	ADP
ma-317	94	5	[	[	X
ma-317	94	6	8	8	NUM
ma-317	94	7	]	]	PUNCT
ma-317	94	8	or	or	CCONJ
ma-317	94	9	theorem	theorem	VERB
ma-317	94	10	1.1	1.1	NUM
ma-317	94	11	of	of	ADP
ma-317	94	12	[	[	X
ma-317	94	13	9	9	NUM
ma-317	94	14	]	]	PUNCT
ma-317	94	15	.	.	PUNCT
ma-317	95	1	�	�	PROPN
ma-317	95	2	theorem	theorem	VERB
ma-317	95	3	2.3	2.3	NUM
ma-317	95	4	.	.	PUNCT
ma-317	96	1	for	for	ADP
ma-317	96	2	t	t	PROPN
ma-317	96	3	>	>	X
ma-317	96	4	0	0	PROPN
ma-317	96	5	,	,	PUNCT
ma-317	96	6	let	let	VERB
ma-317	96	7	a(t	a(t	VERB
ma-317	96	8	)	)	PUNCT
ma-317	96	9	be	be	AUX
ma-317	96	10	defined	define	VERB
ma-317	96	11	as	as	ADP
ma-317	96	12	a(t	a(t	NOUN
ma-317	96	13	)	)	PUNCT
ma-317	97	1	=	=	SYM
ma-317	97	2	1	1	NUM
ma-317	97	3	t	t	NOUN
ma-317	97	4	−	−	NUM
ma-317	97	5	1	1	NUM
ma-317	97	6	et	et	NOUN
ma-317	97	7	−	−	NOUN
ma-317	97	8	1	1	NUM
ma-317	97	9	−	−	NOUN
ma-317	97	10	1	1	NUM
ma-317	97	11	2	2	NUM
ma-317	97	12	+	+	NUM
ma-317	97	13	t	t	PROPN
ma-317	97	14	12	12	NUM
ma-317	97	15	.	.	PUNCT
ma-317	98	1	(	(	PUNCT
ma-317	98	2	16	16	NUM
ma-317	98	3	)	)	PUNCT
ma-317	98	4	=	=	SYM
ma-317	98	5	t	t	PROPN
ma-317	98	6	12	12	NUM
ma-317	98	7	+	+	CCONJ
ma-317	98	8	1	1	NUM
ma-317	98	9	t	t	NOUN
ma-317	98	10	−	−	NUM
ma-317	98	11	1	1	NUM
ma-317	98	12	2	2	NUM
ma-317	98	13	coth	coth	NOUN
ma-317	98	14	(	(	PUNCT
ma-317	98	15	t	t	PROPN
ma-317	98	16	2	2	NUM
ma-317	98	17	)	)	PUNCT
ma-317	98	18	.	.	PUNCT
ma-317	99	1	(	(	PUNCT
ma-317	99	2	17	17	NUM
ma-317	99	3	)	)	PUNCT
ma-317	99	4	then	then	ADV
ma-317	99	5	:	:	PUNCT
ma-317	99	6	(	(	PUNCT
ma-317	99	7	a	a	X
ma-317	99	8	)	)	PUNCT
ma-317	99	9	a(t	a(t	NOUN
ma-317	99	10	)	)	PUNCT
ma-317	99	11	is	be	AUX
ma-317	99	12	strictly	strictly	ADV
ma-317	99	13	increasing	increase	VERB
ma-317	99	14	.	.	PUNCT
ma-317	100	1	(	(	PUNCT
ma-317	100	2	b	b	X
ma-317	100	3	)	)	PUNCT
ma-317	100	4	a(t	a(t	NOUN
ma-317	100	5	)	)	PUNCT
ma-317	100	6	is	be	AUX
ma-317	100	7	positive	positive	ADJ
ma-317	100	8	.	.	PUNCT
ma-317	101	1	(	(	PUNCT
ma-317	101	2	c	c	X
ma-317	101	3	)	)	PUNCT
ma-317	101	4	a(t	a(t	NOUN
ma-317	101	5	)	)	PUNCT
ma-317	101	6	is	be	AUX
ma-317	101	7	strictly	strictly	ADV
ma-317	101	8	convex	convex	ADJ
ma-317	101	9	.	.	PUNCT
ma-317	102	1	proof	proof	NOUN
ma-317	102	2	.	.	PUNCT
ma-317	103	1	by	by	ADP
ma-317	103	2	l’hopital	l’hopital	PROPN
ma-317	103	3	’s	’s	PART
ma-317	103	4	rule	rule	NOUN
ma-317	103	5	and	and	CCONJ
ma-317	103	6	simple	simple	ADJ
ma-317	103	7	computation	computation	NOUN
ma-317	103	8	,	,	PUNCT
ma-317	103	9	we	we	PRON
ma-317	103	10	have	have	VERB
ma-317	103	11	lim	lim	NOUN
ma-317	103	12	t→0	t→0	PUNCT
ma-317	103	13	a(t	a(t	VERB
ma-317	103	14	)	)	PUNCT
ma-317	104	1	=	=	SYM
ma-317	104	2	0	0	PUNCT
ma-317	104	3	and	and	CCONJ
ma-317	104	4	lim	lim	PROPN
ma-317	104	5	t→∞	t→∞	X
ma-317	104	6	a(t	a(t	NOUN
ma-317	104	7	)	)	PUNCT
ma-317	105	1	=	=	SYM
ma-317	105	2	∞.	∞.	PROPN
ma-317	105	3	then	then	ADV
ma-317	105	4	by	by	ADP
ma-317	105	5	making	make	VERB
ma-317	105	6	use	use	NOUN
ma-317	105	7	of	of	ADP
ma-317	105	8	the	the	DET
ma-317	105	9	left	left	ADJ
ma-317	105	10	-	-	PUNCT
ma-317	105	11	hand	hand	NOUN
ma-317	105	12	side	side	NOUN
ma-317	105	13	of	of	ADP
ma-317	105	14	(	(	PUNCT
ma-317	105	15	14	14	NUM
ma-317	105	16	)	)	PUNCT
ma-317	105	17	,	,	PUNCT
ma-317	105	18	we	we	PRON
ma-317	105	19	obtain	obtain	VERB
ma-317	105	20	a′(t	a′(t	ADP
ma-317	105	21	)	)	PUNCT
ma-317	105	22	=	=	SYM
ma-317	105	23	et	et	PROPN
ma-317	105	24	(	(	PUNCT
ma-317	105	25	et	et	NOUN
ma-317	105	26	−	−	PROPN
ma-317	105	27	1)2	1)2	NUM
ma-317	105	28	−	−	PROPN
ma-317	105	29	1	1	NUM
ma-317	105	30	t2	t2	NOUN
ma-317	105	31	+	+	CCONJ
ma-317	105	32	1	1	NUM
ma-317	105	33	12	12	NUM
ma-317	105	34	=	=	SYM
ma-317	105	35	e−t	e−t	NOUN
ma-317	105	36	(	(	PUNCT
ma-317	105	37	e−t	e−t	NOUN
ma-317	105	38	−	−	PROPN
ma-317	105	39	1)2	1)2	NUM
ma-317	105	40	−	−	PROPN
ma-317	105	41	1	1	NUM
ma-317	105	42	t2	t2	NOUN
ma-317	105	43	+	+	CCONJ
ma-317	105	44	1	1	NUM
ma-317	105	45	12	12	NUM
ma-317	105	46	>	>	SYM
ma-317	105	47	0	0	NUM
ma-317	105	48	.	.	PUNCT
ma-317	106	1	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	NUM
ma-317	106	2	eur	eur	PROPN
ma-317	106	3	.	.	PUNCT
ma-317	107	1	j.	j.	PROPN
ma-317	107	2	math	math	PROPN
ma-317	107	3	.	.	PUNCT
ma-317	108	1	anal	anal	PROPN
ma-317	108	2	.	.	PUNCT
ma-317	109	1	10.28924	10.28924	NUM
ma-317	109	2	/	/	SYM
ma-317	109	3	ada	ada	PROPN
ma-317	109	4	/	/	SYM
ma-317	109	5	ma.5.13	ma.5.13	PROPN
ma-317	109	6	5hence	5hence	NUM
ma-317	109	7	a(t	a(t	NOUN
ma-317	109	8	)	)	PUNCT
ma-317	109	9	is	be	AUX
ma-317	109	10	strictly	strictly	ADV
ma-317	109	11	increasing	increase	VERB
ma-317	109	12	and	and	CCONJ
ma-317	109	13	that	that	PRON
ma-317	109	14	completes	complete	VERB
ma-317	109	15	the	the	DET
ma-317	109	16	proof	proof	NOUN
ma-317	109	17	of	of	ADP
ma-317	109	18	(	(	PUNCT
ma-317	109	19	a	a	NOUN
ma-317	109	20	)	)	PUNCT
ma-317	109	21	.	.	PUNCT
ma-317	110	1	next	next	ADV
ma-317	110	2	,	,	PUNCT
ma-317	110	3	the	the	DET
ma-317	110	4	increasing	increase	VERB
ma-317	110	5	propertyof	propertyof	NOUN
ma-317	110	6	a(t	a(t	NOUN
ma-317	110	7	)	)	PUNCT
ma-317	110	8	implies	imply	VERB
ma-317	110	9	that	that	SCONJ
ma-317	110	10	for	for	ADP
ma-317	110	11	t	t	PROPN
ma-317	110	12	>	>	X
ma-317	110	13	0	0	NUM
ma-317	110	14	,	,	PUNCT
ma-317	110	15	a(t	a(t	NOUN
ma-317	110	16	)	)	PUNCT
ma-317	110	17	>	>	X
ma-317	111	1	lim	lim	PROPN
ma-317	111	2	t→0	t→0	PUNCT
ma-317	111	3	a(t	a(t	PROPN
ma-317	111	4	)	)	PUNCT
ma-317	112	1	=	=	SYM
ma-317	112	2	0which	0which	PROPN
ma-317	112	3	completes	complete	VERB
ma-317	112	4	the	the	DET
ma-317	112	5	proof	proof	NOUN
ma-317	112	6	of	of	ADP
ma-317	112	7	(	(	PUNCT
ma-317	112	8	b	b	NOUN
ma-317	112	9	)	)	PUNCT
ma-317	112	10	.	.	PUNCT
ma-317	113	1	next	next	ADV
ma-317	113	2	,	,	PUNCT
ma-317	113	3	as	as	ADP
ma-317	113	4	a	a	DET
ma-317	113	5	result	result	NOUN
ma-317	113	6	of	of	ADP
ma-317	113	7	lemma	lemma	PROPN
ma-317	113	8	2.2	2.2	NUM
ma-317	113	9	,	,	PUNCT
ma-317	113	10	we	we	PRON
ma-317	113	11	have	have	VERB
ma-317	113	12	a′′(t	a′′(t	NOUN
ma-317	113	13	)	)	PUNCT
ma-317	114	1	=	=	SYM
ma-317	114	2	−	−	PROPN
ma-317	114	3	(	(	PUNCT
ma-317	114	4	1	1	NUM
ma-317	114	5	t2	t2	NOUN
ma-317	114	6	−	−	PROPN
ma-317	114	7	e−t	e−t	NOUN
ma-317	114	8	(	(	PUNCT
ma-317	114	9	1−	1−	NUM
ma-317	114	10	e−t)2	e−t)2	NOUN
ma-317	114	11	)	)	PUNCT
ma-317	114	12	′	′	NUM
ma-317	115	1	=	=	SYM
ma-317	115	2	−p	−p	ADJ
ma-317	115	3	′(t	′(t	NOUN
ma-317	115	4	)	)	PUNCT
ma-317	115	5	>	>	X
ma-317	115	6	0	0	NUM
ma-317	115	7	which	which	PRON
ma-317	115	8	completes	complete	VERB
ma-317	115	9	the	the	DET
ma-317	115	10	proof	proof	NOUN
ma-317	115	11	of	of	ADP
ma-317	115	12	(	(	PUNCT
ma-317	115	13	c	c	NOUN
ma-317	115	14	)	)	PUNCT
ma-317	115	15	.	.	PUNCT
ma-317	116	1	�	�	PROPN
ma-317	116	2	remark	remark	VERB
ma-317	116	3	2.4	2.4	NUM
ma-317	116	4	.	.	PUNCT
ma-317	117	1	theorem	theorem	VERB
ma-317	117	2	2.3	2.3	NUM
ma-317	117	3	(	(	PUNCT
ma-317	117	4	c	c	NOUN
ma-317	117	5	)	)	PUNCT
ma-317	117	6	implies	imply	VERB
ma-317	117	7	that	that	SCONJ
ma-317	117	8	θ(z	θ(z	NOUN
ma-317	117	9	)	)	PUNCT
ma-317	117	10	is	be	AUX
ma-317	117	11	positive	positive	ADJ
ma-317	117	12	.	.	PUNCT
ma-317	118	1	corollary	corollary	ADJ
ma-317	118	2	2.5	2.5	NUM
ma-317	118	3	.	.	PUNCT
ma-317	119	1	for	for	ADP
ma-317	119	2	t	t	PROPN
ma-317	119	3	>	>	X
ma-317	119	4	0	0	PROPN
ma-317	119	5	,	,	PUNCT
ma-317	119	6	the	the	DET
ma-317	119	7	inequality	inequality	NOUN
ma-317	119	8	e−t	e−t	NOUN
ma-317	119	9	(	(	PUNCT
ma-317	119	10	1−	1−	NUM
ma-317	119	11	e−t)2	e−t)2	NOUN
ma-317	119	12	<	<	X
ma-317	119	13	2	2	NUM
ma-317	119	14	t3	t3	NOUN
ma-317	119	15	(	(	PUNCT
ma-317	119	16	18	18	NUM
ma-317	119	17	)	)	PUNCT
ma-317	119	18	holds	hold	VERB
ma-317	119	19	.	.	PUNCT
ma-317	120	1	proof	proof	NOUN
ma-317	120	2	.	.	PUNCT
ma-317	121	1	the	the	DET
ma-317	121	2	convexity	convexity	NOUN
ma-317	121	3	of	of	ADP
ma-317	121	4	a(t	a(t	NOUN
ma-317	121	5	)	)	PUNCT
ma-317	121	6	implies	imply	VERB
ma-317	121	7	that	that	SCONJ
ma-317	121	8	a′′(t	a′′(t	NOUN
ma-317	121	9	)	)	PUNCT
ma-317	121	10	=	=	SYM
ma-317	121	11	2	2	NUM
ma-317	121	12	t3	t3	NOUN
ma-317	121	13	+	+	X
ma-317	121	14	et	et	PROPN
ma-317	121	15	(	(	PUNCT
ma-317	121	16	et	et	NOUN
ma-317	121	17	−	−	PROPN
ma-317	121	18	1)2	1)2	NUM
ma-317	121	19	−	−	PROPN
ma-317	121	20	2e2	2e2	PROPN
ma-317	121	21	t	t	PROPN
ma-317	121	22	(	(	PUNCT
ma-317	121	23	et	et	NOUN
ma-317	121	24	−	−	PROPN
ma-317	121	25	1)3	1)3	PROPN
ma-317	121	26	>	>	X
ma-317	121	27	0	0	PROPN
ma-317	121	28	.	.	PUNCT
ma-317	122	1	this	this	DET
ma-317	122	2	simplifies	simplifie	NOUN
ma-317	122	3	to	to	ADP
ma-317	122	4	2	2	NUM
ma-317	122	5	t3	t3	NOUN
ma-317	122	6	(	(	PUNCT
ma-317	122	7	et	et	NOUN
ma-317	122	8	−	−	PROPN
ma-317	122	9	1)3	1)3	PROPN
ma-317	122	10	−	−	PROPN
ma-317	122	11	et(et	et(et	NOUN
ma-317	122	12	−	−	PROPN
ma-317	122	13	1	1	NUM
ma-317	122	14	)	)	PUNCT
ma-317	122	15	>	>	X
ma-317	123	1	0which	0which	PROPN
ma-317	123	2	further	further	ADJ
ma-317	123	3	simplifies	simplifie	NOUN
ma-317	123	4	to	to	ADP
ma-317	123	5	2	2	NUM
ma-317	123	6	t3	t3	NOUN
ma-317	123	7	>	>	X
ma-317	123	8	et	et	PROPN
ma-317	123	9	(	(	PUNCT
ma-317	123	10	et	et	NOUN
ma-317	123	11	−	−	PROPN
ma-317	123	12	1)2	1)2	NUM
ma-317	123	13	=	=	SYM
ma-317	123	14	e−t	e−t	NOUN
ma-317	123	15	(	(	PUNCT
ma-317	123	16	1−	1−	NUM
ma-317	123	17	e−t)2	e−t)2	NOUN
ma-317	123	18	.	.	PUNCT
ma-317	124	1	this	this	PRON
ma-317	124	2	completes	complete	VERB
ma-317	124	3	the	the	DET
ma-317	124	4	proof	proof	NOUN
ma-317	124	5	.	.	PUNCT
ma-317	125	1	�	�	PROPN
ma-317	125	2	remark	remark	VERB
ma-317	125	3	2.6	2.6	NUM
ma-317	125	4	.	.	PUNCT
ma-317	126	1	we	we	PRON
ma-317	126	2	note	note	VERB
ma-317	126	3	that	that	SCONJ
ma-317	126	4	1	1	NUM
ma-317	126	5	t2	t2	NOUN
ma-317	126	6	−	−	PROPN
ma-317	126	7	2	2	NUM
ma-317	126	8	t3	t3	NOUN
ma-317	126	9	<	<	X
ma-317	126	10	0	0	PUNCT
ma-317	127	1	if	if	SCONJ
ma-317	127	2	0	0	NUM
ma-317	127	3	<	<	X
ma-317	127	4	t	t	X
ma-317	127	5	<	<	X
ma-317	127	6	2	2	NUM
ma-317	127	7	and	and	CCONJ
ma-317	127	8	1	1	NUM
ma-317	127	9	t2	t2	NOUN
ma-317	127	10	−	−	PROPN
ma-317	127	11	2	2	NUM
ma-317	127	12	t3	t3	NOUN
ma-317	127	13	>	>	X
ma-317	127	14	0	0	PUNCT
ma-317	128	1	if	if	SCONJ
ma-317	128	2	t	t	PROPN
ma-317	128	3	>	>	X
ma-317	128	4	2	2	NUM
ma-317	128	5	.	.	PUNCT
ma-317	129	1	thus	thus	ADV
ma-317	129	2	,	,	PUNCT
ma-317	129	3	the	the	DET
ma-317	129	4	upperbound	upperbound	PROPN
ma-317	129	5	in	in	ADP
ma-317	129	6	(	(	PUNCT
ma-317	129	7	14	14	NUM
ma-317	129	8	)	)	PUNCT
ma-317	129	9	is	be	AUX
ma-317	129	10	better	well	ADJ
ma-317	129	11	that	that	SCONJ
ma-317	129	12	upper	upper	ADV
ma-317	129	13	bound	bind	VERB
ma-317	129	14	in	in	ADP
ma-317	129	15	(	(	PUNCT
ma-317	129	16	18	18	NUM
ma-317	129	17	)	)	PUNCT
ma-317	129	18	if	if	SCONJ
ma-317	129	19	0	0	NUM
ma-317	129	20	<	<	X
ma-317	129	21	t	t	X
ma-317	129	22	<	<	X
ma-317	129	23	2	2	NUM
ma-317	129	24	and	and	CCONJ
ma-317	129	25	the	the	DET
ma-317	129	26	upper	upper	ADJ
ma-317	129	27	bound	bind	VERB
ma-317	129	28	in	in	ADP
ma-317	129	29	(	(	PUNCT
ma-317	129	30	18	18	NUM
ma-317	129	31	)	)	PUNCT
ma-317	129	32	is	be	AUX
ma-317	129	33	betterthan	betterthan	ADJ
ma-317	129	34	the	the	DET
ma-317	129	35	upper	upper	ADJ
ma-317	129	36	bound	bind	VERB
ma-317	129	37	in	in	ADP
ma-317	129	38	(	(	PUNCT
ma-317	129	39	14	14	NUM
ma-317	129	40	)	)	PUNCT
ma-317	129	41	if	if	SCONJ
ma-317	129	42	t	t	PROPN
ma-317	129	43	>	>	X
ma-317	129	44	2	2	X
ma-317	129	45	.	.	PUNCT
ma-317	130	1	theorem	theorem	NOUN
ma-317	130	2	2.7	2.7	NUM
ma-317	130	3	.	.	PUNCT
ma-317	131	1	the	the	DET
ma-317	131	2	function	function	NOUN
ma-317	131	3	θ(z	θ(z	NOUN
ma-317	131	4	)	)	PUNCT
ma-317	131	5	is	be	AUX
ma-317	131	6	completely	completely	ADV
ma-317	131	7	monotonic	monotonic	ADJ
ma-317	131	8	on	on	ADP
ma-317	131	9	(	(	PUNCT
ma-317	131	10	0,∞	0,∞	NUM
ma-317	131	11	)	)	PUNCT
ma-317	131	12	.	.	PUNCT
ma-317	132	1	proof	proof	NOUN
ma-317	132	2	.	.	PUNCT
ma-317	133	1	differentiating	differentiate	VERB
ma-317	133	2	r	r	NOUN
ma-317	133	3	number	number	NOUN
ma-317	133	4	of	of	ADP
ma-317	133	5	times	time	NOUN
ma-317	133	6	of	of	ADP
ma-317	133	7	(	(	PUNCT
ma-317	133	8	9	9	X
ma-317	133	9	)	)	PUNCT
ma-317	133	10	gives	give	VERB
ma-317	133	11	θ(r)(z	θ(r)(z	VERB
ma-317	133	12	)	)	PUNCT
ma-317	133	13	=	=	PRON
ma-317	133	14	(	(	PUNCT
ma-317	134	1	−1)r	−1)r	X
ma-317	134	2	∫	∫	NOUN
ma-317	134	3	∞	∞	PROPN
ma-317	134	4	0	0	NUM
ma-317	134	5	a(t)tr−2e−ztdt	a(t)tr−2e−ztdt	X
ma-317	134	6	(	(	PUNCT
ma-317	134	7	19	19	NUM
ma-317	134	8	)	)	PUNCT
ma-317	134	9	where	where	SCONJ
ma-317	134	10	r	r	NOUN
ma-317	134	11	∈	∈	PROPN
ma-317	134	12	n0	n0	NOUN
ma-317	134	13	and	and	CCONJ
ma-317	134	14	θ(0)(z	θ(0)(z	NOUN
ma-317	134	15	)	)	PUNCT
ma-317	134	16	=	=	SYM
ma-317	134	17	θ(z	θ(z	NOUN
ma-317	134	18	)	)	PUNCT
ma-317	134	19	.	.	PUNCT
ma-317	135	1	this	this	PRON
ma-317	135	2	implies	imply	VERB
ma-317	135	3	that	that	SCONJ
ma-317	135	4	(	(	PUNCT
ma-317	135	5	−1)rθ(r)(z	−1)rθ(r)(z	PROPN
ma-317	135	6	)	)	PUNCT
ma-317	135	7	=	=	PUNCT
ma-317	135	8	(	(	PUNCT
ma-317	135	9	−1)2r	−1)2r	NOUN
ma-317	135	10	∫	∫	PROPN
ma-317	135	11	∞	∞	PROPN
ma-317	135	12	0	0	NUM
ma-317	135	13	a(t)tr−2e−ztdt	a(t)tr−2e−ztdt	VERB
ma-317	136	1	=	=	X
ma-317	136	2	∫	∫	PROPN
ma-317	136	3	∞	∞	PROPN
ma-317	136	4	0	0	NUM
ma-317	136	5	a(t)tr−2e−ztdt	a(t)tr−2e−ztdt	X
ma-317	136	6	>	>	X
ma-317	136	7	0	0	X
ma-317	136	8	.	.	PUNCT
ma-317	137	1	this	this	PRON
ma-317	137	2	completes	complete	VERB
ma-317	137	3	the	the	DET
ma-317	137	4	proof	proof	NOUN
ma-317	137	5	.	.	PUNCT
ma-317	138	1	�	�	PROPN
ma-317	138	2	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	NUM
ma-317	138	3	eur	eur	PROPN
ma-317	138	4	.	.	PUNCT
ma-317	139	1	j.	j.	PROPN
ma-317	139	2	math	math	PROPN
ma-317	139	3	.	.	PUNCT
ma-317	140	1	anal	anal	PROPN
ma-317	140	2	.	.	PUNCT
ma-317	141	1	10.28924	10.28924	NUM
ma-317	141	2	/	/	SYM
ma-317	141	3	ada	ada	PROPN
ma-317	141	4	/	/	SYM
ma-317	141	5	ma.5.13	ma.5.13	PROPN
ma-317	141	6	6	6	NUM
ma-317	141	7	remark	remark	NOUN
ma-317	141	8	2.8	2.8	NUM
ma-317	141	9	.	.	PUNCT
ma-317	142	1	the	the	DET
ma-317	142	2	representation	representation	NOUN
ma-317	142	3	(	(	PUNCT
ma-317	142	4	19	19	NUM
ma-317	142	5	)	)	PUNCT
ma-317	142	6	implies	imply	VERB
ma-317	142	7	that	that	SCONJ
ma-317	142	8	the	the	DET
ma-317	142	9	function	function	NOUN
ma-317	142	10	θ(z	θ(z	NOUN
ma-317	142	11	)	)	PUNCT
ma-317	142	12	is	be	AUX
ma-317	142	13	decreasing	decrease	VERB
ma-317	142	14	and	and	CCONJ
ma-317	142	15	convex	convex	VERB
ma-317	142	16	on	on	ADP
ma-317	142	17	(	(	PUNCT
ma-317	142	18	0,∞	0,∞	NOUN
ma-317	142	19	)	)	PUNCT
ma-317	142	20	.	.	PUNCT
ma-317	143	1	theorem	theorem	VERB
ma-317	143	2	2.9	2.9	NUM
ma-317	143	3	.	.	PUNCT
ma-317	144	1	let	let	VERB
ma-317	144	2	r	r	PRON
ma-317	144	3	∈	∈	PROPN
ma-317	144	4	n0	n0	X
ma-317	144	5	be	be	AUX
ma-317	144	6	even	even	ADV
ma-317	144	7	.	.	PUNCT
ma-317	145	1	then	then	ADV
ma-317	145	2	the	the	DET
ma-317	145	3	function	function	NOUN
ma-317	145	4	θ(r)(z	θ(r)(z	VERB
ma-317	145	5	)	)	PUNCT
ma-317	145	6	is	be	AUX
ma-317	145	7	logarithmically	logarithmically	ADV
ma-317	145	8	convex	convex	ADJ
ma-317	145	9	on	on	ADP
ma-317	145	10	(	(	PUNCT
ma-317	145	11	0,∞	0,∞	NUM
ma-317	145	12	)	)	PUNCT
ma-317	145	13	.	.	PUNCT
ma-317	146	1	that	that	PRON
ma-317	146	2	is	be	AUX
ma-317	146	3	,	,	PUNCT
ma-317	146	4	the	the	DET
ma-317	146	5	inequality	inequality	NOUN
ma-317	146	6	θ(r	θ(r	X
ma-317	146	7	)	)	PUNCT
ma-317	146	8	(	(	PUNCT
ma-317	146	9	x	x	SYM
ma-317	146	10	u	u	NOUN
ma-317	146	11	+	+	NOUN
ma-317	146	12	y	y	PROPN
ma-317	146	13	v	v	NOUN
ma-317	146	14	)	)	PUNCT
ma-317	146	15	≤	≤	NOUN
ma-317	146	16	[	[	PUNCT
ma-317	146	17	θ(r)(x	θ(r)(x	NUM
ma-317	146	18	)	)	PUNCT
ma-317	146	19	]	]	PUNCT
ma-317	146	20	1	1	NUM
ma-317	146	21	u	u	NOUN
ma-317	146	22	[	[	PUNCT
ma-317	146	23	θ(r)(y	θ(r)(y	ADJ
ma-317	146	24	)	)	PUNCT
ma-317	146	25	]	]	PUNCT
ma-317	146	26	1	1	NUM
ma-317	146	27	v	v	X
ma-317	146	28	(	(	PUNCT
ma-317	146	29	20	20	NUM
ma-317	146	30	)	)	PUNCT
ma-317	146	31	holds	hold	VERB
ma-317	146	32	for	for	ADP
ma-317	146	33	x	x	PUNCT
ma-317	146	34	>	>	X
ma-317	146	35	0	0	PROPN
ma-317	146	36	,	,	PUNCT
ma-317	146	37	y	y	PROPN
ma-317	146	38	>	>	X
ma-317	146	39	0	0	PROPN
ma-317	146	40	,	,	PUNCT
ma-317	146	41	u	u	NOUN
ma-317	146	42	>	>	X
ma-317	146	43	1	1	NUM
ma-317	146	44	,	,	PUNCT
ma-317	146	45	v	v	PART
ma-317	146	46	>	>	SYM
ma-317	146	47	1	1	NUM
ma-317	146	48	and	and	CCONJ
ma-317	146	49	1	1	NUM
ma-317	146	50	u	u	NOUN
ma-317	146	51	+	+	NOUN
ma-317	146	52	1	1	NUM
ma-317	146	53	v	v	NOUN
ma-317	146	54	=	=	SYM
ma-317	146	55	1	1	NUM
ma-317	146	56	.	.	PUNCT
ma-317	147	1	proof	proof	NOUN
ma-317	147	2	.	.	PUNCT
ma-317	148	1	let	let	VERB
ma-317	148	2	r	r	PRON
ma-317	148	3	∈	∈	PROPN
ma-317	148	4	n0	n0	X
ma-317	148	5	be	be	AUX
ma-317	148	6	an	an	DET
ma-317	148	7	even	even	ADJ
ma-317	148	8	number	number	NOUN
ma-317	148	9	.	.	PUNCT
ma-317	149	1	then	then	ADV
ma-317	149	2	,	,	PUNCT
ma-317	149	3	by	by	ADP
ma-317	149	4	using	use	VERB
ma-317	149	5	(	(	PUNCT
ma-317	149	6	19	19	NUM
ma-317	149	7	)	)	PUNCT
ma-317	149	8	and	and	CCONJ
ma-317	149	9	holder	holder	NOUN
ma-317	149	10	’s	’s	PART
ma-317	149	11	inequality	inequality	NOUN
ma-317	149	12	for	for	ADP
ma-317	149	13	integrals	integral	NOUN
ma-317	149	14	,	,	PUNCT
ma-317	149	15	we	we	PRON
ma-317	149	16	obtain	obtain	VERB
ma-317	149	17	θ(r	θ(r	NOUN
ma-317	149	18	)	)	PUNCT
ma-317	150	1	(	(	PUNCT
ma-317	150	2	x	x	SYM
ma-317	150	3	u	u	NOUN
ma-317	150	4	+	+	NOUN
ma-317	150	5	y	y	PROPN
ma-317	150	6	v	v	NOUN
ma-317	150	7	)	)	PUNCT
ma-317	150	8	=	=	SYM
ma-317	151	1	∫	∫	PROPN
ma-317	151	2	∞	∞	PROPN
ma-317	151	3	0	0	PROPN
ma-317	151	4	a(t)tr−2e−	a(t)tr−2e−	PROPN
ma-317	151	5	(	(	PUNCT
ma-317	151	6	x	x	SYM
ma-317	151	7	u	u	PROPN
ma-317	151	8	+	+	NOUN
ma-317	151	9	y	y	PROPN
ma-317	151	10	v	v	X
ma-317	151	11	)	)	PUNCT
ma-317	151	12	tdt	tdt	PROPN
ma-317	151	13	=	=	SYM
ma-317	151	14	∫	∫	PROPN
ma-317	151	15	∞	∞	PROPN
ma-317	151	16	0	0	PROPN
ma-317	151	17	(	(	PUNCT
ma-317	151	18	a(t)tr−2	a(t)tr−2	PROPN
ma-317	151	19	)	)	PUNCT
ma-317	151	20	1	1	NUM
ma-317	151	21	u	u	NOUN
ma-317	151	22	+	+	NOUN
ma-317	151	23	1	1	NUM
ma-317	151	24	v	v	ADP
ma-317	151	25	e−	e−	PROPN
ma-317	151	26	(	(	PUNCT
ma-317	151	27	x	x	SYM
ma-317	151	28	u	u	PROPN
ma-317	151	29	+	+	NOUN
ma-317	151	30	y	y	PROPN
ma-317	151	31	v	v	X
ma-317	151	32	)	)	PUNCT
ma-317	151	33	tdt	tdt	PROPN
ma-317	151	34	=	=	SYM
ma-317	151	35	∫	∫	PROPN
ma-317	151	36	∞	∞	PROPN
ma-317	151	37	0	0	NUM
ma-317	151	38	a(t	a(t	NOUN
ma-317	151	39	)	)	PUNCT
ma-317	151	40	1	1	NUM
ma-317	151	41	u	u	NOUN
ma-317	151	42	t	t	PROPN
ma-317	151	43	r−2	r−2	PROPN
ma-317	151	44	u	u	NOUN
ma-317	151	45	e	e	X
ma-317	151	46	−xt	−xt	ADP
ma-317	151	47	u	u	PRON
ma-317	151	48	a(t	a(t	PROPN
ma-317	151	49	)	)	PUNCT
ma-317	151	50	1	1	NUM
ma-317	151	51	v	v	ADP
ma-317	151	52	t	t	X
ma-317	151	53	r−2	r−2	PROPN
ma-317	151	54	v	v	NOUN
ma-317	151	55	e	e	X
ma-317	151	56	−yt	−yt	X
ma-317	151	57	v	v	X
ma-317	151	58	dt	dt	NOUN
ma-317	151	59	≤	≤	PROPN
ma-317	151	60	(	(	PUNCT
ma-317	151	61	∫	∫	PROPN
ma-317	151	62	∞	∞	PROPN
ma-317	151	63	0	0	NUM
ma-317	152	1	a(t)tr−2e−xtdt	a(t)tr−2e−xtdt	CCONJ
ma-317	152	2	)	)	PUNCT
ma-317	152	3	1	1	NUM
ma-317	152	4	u	u	NOUN
ma-317	152	5	(	(	PUNCT
ma-317	152	6	∫	∫	PROPN
ma-317	152	7	∞	∞	PROPN
ma-317	152	8	0	0	NUM
ma-317	152	9	a(t)tr−2e−ytdt	a(t)tr−2e−ytdt	CCONJ
ma-317	152	10	)	)	PUNCT
ma-317	152	11	1	1	NUM
ma-317	152	12	v	v	NOUN
ma-317	152	13	=	=	X
ma-317	152	14	[	[	PUNCT
ma-317	152	15	θ(r)(x	θ(r)(x	NUM
ma-317	152	16	)	)	PUNCT
ma-317	152	17	]	]	PUNCT
ma-317	152	18	1	1	NUM
ma-317	152	19	u	u	NOUN
ma-317	152	20	[	[	PUNCT
ma-317	152	21	θ(r)(y	θ(r)(y	ADJ
ma-317	152	22	)	)	PUNCT
ma-317	152	23	]	]	PUNCT
ma-317	152	24	1	1	NUM
ma-317	152	25	v	v	NOUN
ma-317	152	26	which	which	PRON
ma-317	152	27	completes	complete	VERB
ma-317	152	28	the	the	DET
ma-317	152	29	proof	proof	NOUN
ma-317	152	30	.	.	PUNCT
ma-317	153	1	�	�	PROPN
ma-317	153	2	remark	remark	VERB
ma-317	153	3	2.10	2.10	NUM
ma-317	153	4	.	.	PUNCT
ma-317	154	1	the	the	DET
ma-317	154	2	particular	particular	ADJ
ma-317	154	3	case	case	NOUN
ma-317	154	4	where	where	SCONJ
ma-317	154	5	r	r	NOUN
ma-317	154	6	=	=	SYM
ma-317	154	7	0	0	NUM
ma-317	154	8	in	in	ADP
ma-317	154	9	theorem	theorem	ADJ
ma-317	154	10	2.9	2.9	NUM
ma-317	154	11	proves	prove	VERB
ma-317	154	12	that	that	SCONJ
ma-317	154	13	θ(z	θ(z	NOUN
ma-317	154	14	)	)	PUNCT
ma-317	154	15	is	be	AUX
ma-317	154	16	logarithmicallyconvex	logarithmicallyconvex	NOUN
ma-317	154	17	on	on	ADP
ma-317	154	18	(	(	PUNCT
ma-317	154	19	0,∞	0,∞	NUM
ma-317	154	20	)	)	PUNCT
ma-317	154	21	.	.	PUNCT
ma-317	155	1	remark	remark	PROPN
ma-317	155	2	2.11	2.11	NUM
ma-317	155	3	.	.	PUNCT
ma-317	156	1	theorem	theorem	VERB
ma-317	156	2	2.9	2.9	NUM
ma-317	156	3	shows	show	VERB
ma-317	156	4	that	that	SCONJ
ma-317	156	5	,	,	PUNCT
ma-317	156	6	for	for	ADP
ma-317	156	7	even	even	ADV
ma-317	156	8	r	r	PROPN
ma-317	156	9	∈	∈	PROPN
ma-317	156	10	n0	n0	PROPN
ma-317	156	11	,	,	PUNCT
ma-317	156	12	the	the	DET
ma-317	156	13	function	function	NOUN
ma-317	156	14	t	t	PROPN
ma-317	156	15	(	(	PUNCT
ma-317	156	16	z	z	NOUN
ma-317	156	17	)	)	PUNCT
ma-317	156	18	=	=	SYM
ma-317	156	19	θ(r+1)(z	θ(r+1)(z	NOUN
ma-317	156	20	)	)	PUNCT
ma-317	156	21	θ(r)(z	θ(r)(z	VERB
ma-317	156	22	)	)	PUNCT
ma-317	156	23	(	(	PUNCT
ma-317	156	24	21	21	NUM
ma-317	156	25	)	)	PUNCT
ma-317	156	26	is	be	AUX
ma-317	156	27	increasing	increase	VERB
ma-317	156	28	on	on	ADP
ma-317	156	29	(	(	PUNCT
ma-317	156	30	0,∞	0,∞	NOUN
ma-317	156	31	)	)	PUNCT
ma-317	156	32	.	.	PUNCT
ma-317	157	1	corollary	corollary	ADJ
ma-317	157	2	2.12	2.12	NUM
ma-317	157	3	.	.	PUNCT
ma-317	158	1	let	let	VERB
ma-317	158	2	r	r	PRON
ma-317	158	3	∈	∈	PROPN
ma-317	158	4	n0	n0	NOUN
ma-317	158	5	be	be	AUX
ma-317	158	6	even	even	ADV
ma-317	158	7	and	and	CCONJ
ma-317	158	8	p	p	X
ma-317	158	9	∈	∈	PROPN
ma-317	158	10	(	(	PUNCT
ma-317	158	11	0	0	NUM
ma-317	158	12	,	,	PUNCT
ma-317	158	13	1	1	NUM
ma-317	158	14	]	]	PUNCT
ma-317	158	15	.	.	PUNCT
ma-317	159	1	then	then	ADV
ma-317	159	2	the	the	DET
ma-317	159	3	function	function	NOUN
ma-317	159	4	b(z	b(z	NOUN
ma-317	159	5	)	)	PUNCT
ma-317	159	6	=	=	SYM
ma-317	159	7	θ(r)(pz	θ(r)(pz	X
ma-317	159	8	)	)	PUNCT
ma-317	160	1	[	[	X
ma-317	160	2	θ(r)(z)]p	θ(r)(z)]p	X
ma-317	160	3	decreasing	decrease	VERB
ma-317	160	4	on	on	ADP
ma-317	160	5	(	(	PUNCT
ma-317	160	6	0,∞	0,∞	NOUN
ma-317	160	7	)	)	PUNCT
ma-317	160	8	.	.	PUNCT
ma-317	161	1	consequently	consequently	ADV
ma-317	161	2	,	,	PUNCT
ma-317	161	3	for	for	ADP
ma-317	161	4	0	0	NUM
ma-317	161	5	<	<	X
ma-317	161	6	x	x	PUNCT
ma-317	161	7	≤	≤	PROPN
ma-317	161	8	y	y	PROPN
ma-317	161	9	,	,	PUNCT
ma-317	161	10	the	the	DET
ma-317	161	11	inequality	inequality	NOUN
ma-317	161	12	(	(	PUNCT
ma-317	161	13	θ(r)(y	θ(r)(y	ADJ
ma-317	161	14	)	)	PUNCT
ma-317	161	15	θ(r)(x	θ(r)(x	NUM
ma-317	161	16	)	)	PUNCT
ma-317	161	17	)	)	PUNCT
ma-317	161	18	p	p	NOUN
ma-317	161	19	≥	≥	NOUN
ma-317	161	20	θ(r)(py	θ(r)(py	PROPN
ma-317	161	21	)	)	PUNCT
ma-317	161	22	θ(r)(px	θ(r)(px	NUM
ma-317	161	23	)	)	PUNCT
ma-317	161	24	(	(	PUNCT
ma-317	161	25	22	22	NUM
ma-317	161	26	)	)	PUNCT
ma-317	161	27	holds	hold	VERB
ma-317	161	28	.	.	PUNCT
ma-317	162	1	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	NUM
ma-317	162	2	eur	eur	PROPN
ma-317	162	3	.	.	PUNCT
ma-317	163	1	j.	j.	PROPN
ma-317	163	2	math	math	PROPN
ma-317	163	3	.	.	PUNCT
ma-317	164	1	anal	anal	PROPN
ma-317	164	2	.	.	PUNCT
ma-317	165	1	10.28924	10.28924	NUM
ma-317	165	2	/	/	SYM
ma-317	165	3	ada	ada	PROPN
ma-317	165	4	/	/	SYM
ma-317	165	5	ma.5.13	ma.5.13	PROPN
ma-317	165	6	7	7	NUM
ma-317	165	7	proof	proof	NOUN
ma-317	165	8	.	.	PUNCT
ma-317	166	1	logarithmic	logarithmic	ADJ
ma-317	166	2	differentiation	differentiation	NOUN
ma-317	166	3	of	of	ADP
ma-317	166	4	b(z	b(z	NOUN
ma-317	166	5	)	)	PUNCT
ma-317	166	6	and	and	CCONJ
ma-317	166	7	applying	apply	VERB
ma-317	166	8	the	the	DET
ma-317	166	9	increasing	increase	VERB
ma-317	166	10	property	property	NOUN
ma-317	166	11	of	of	ADP
ma-317	166	12	t	t	PROPN
ma-317	166	13	(	(	PUNCT
ma-317	166	14	z	z	NOUN
ma-317	166	15	)	)	PUNCT
ma-317	166	16	gives	give	VERB
ma-317	166	17	b′(z	b′(z	NOUN
ma-317	166	18	)	)	PUNCT
ma-317	166	19	b(z	b(z	NOUN
ma-317	166	20	)	)	PUNCT
ma-317	167	1	=	=	SYM
ma-317	167	2	p	p	NOUN
ma-317	167	3	θ(r+1)(pz	θ(r+1)(pz	PROPN
ma-317	167	4	)	)	PUNCT
ma-317	167	5	θ(r)(pz	θ(r)(pz	NUM
ma-317	167	6	)	)	PUNCT
ma-317	168	1	−	−	PROPN
ma-317	168	2	p	p	NOUN
ma-317	168	3	θ(r+1)(z	θ(r+1)(z	NOUN
ma-317	168	4	)	)	PUNCT
ma-317	168	5	θ(r)(z	θ(r)(z	VERB
ma-317	168	6	)	)	PUNCT
ma-317	169	1	=	=	SYM
ma-317	169	2	p	p	X
ma-317	169	3	[	[	PUNCT
ma-317	169	4	θ(r+1)(pz	θ(r+1)(pz	PROPN
ma-317	169	5	)	)	PUNCT
ma-317	169	6	θ(r)(pz	θ(r)(pz	NUM
ma-317	169	7	)	)	PUNCT
ma-317	169	8	−	−	PROPN
ma-317	169	9	θ(r+1)(z	θ(r+1)(z	NOUN
ma-317	169	10	)	)	PUNCT
ma-317	169	11	θ(r)(z	θ(r)(z	VERB
ma-317	169	12	)	)	PUNCT
ma-317	169	13	]	]	PUNCT
ma-317	170	1	≤	≤	NUM
ma-317	170	2	0	0	NUM
ma-317	170	3	.	.	PUNCT
ma-317	171	1	hence	hence	ADV
ma-317	171	2	b(z	b(z	NOUN
ma-317	171	3	)	)	PUNCT
ma-317	171	4	is	be	AUX
ma-317	171	5	decreasing	decrease	VERB
ma-317	171	6	.	.	PUNCT
ma-317	172	1	consequently	consequently	ADV
ma-317	172	2	,	,	PUNCT
ma-317	172	3	for	for	ADP
ma-317	172	4	0	0	NUM
ma-317	172	5	<	<	X
ma-317	172	6	x	x	PUNCT
ma-317	172	7	≤	≤	NOUN
ma-317	172	8	y	y	PROPN
ma-317	172	9	,	,	PUNCT
ma-317	172	10	we	we	PRON
ma-317	172	11	have	have	AUX
ma-317	172	12	b(x	b(x	VERB
ma-317	172	13	)	)	PUNCT
ma-317	172	14	≥	≥	X
ma-317	172	15	b(y	b(y	PROPN
ma-317	172	16	)	)	PUNCT
ma-317	172	17	which	which	PRON
ma-317	172	18	whenrearranged	whenrearrange	VERB
ma-317	172	19	gives	give	VERB
ma-317	172	20	(	(	PUNCT
ma-317	172	21	22	22	NUM
ma-317	172	22	)	)	PUNCT
ma-317	172	23	.	.	PUNCT
ma-317	173	1	�	�	PROPN
ma-317	173	2	remark	remark	VERB
ma-317	173	3	2.13	2.13	NUM
ma-317	173	4	.	.	PUNCT
ma-317	174	1	if	if	SCONJ
ma-317	174	2	p	p	PROPN
ma-317	174	3	≥	≥	PUNCT
ma-317	174	4	1	1	NUM
ma-317	174	5	in	in	ADP
ma-317	174	6	corollary	corollary	ADJ
ma-317	174	7	2.12	2.12	NUM
ma-317	174	8	,	,	PUNCT
ma-317	174	9	then	then	ADV
ma-317	174	10	the	the	DET
ma-317	174	11	reverse	reverse	ADJ
ma-317	174	12	cases	case	NOUN
ma-317	174	13	of	of	ADP
ma-317	174	14	the	the	DET
ma-317	174	15	conclusions	conclusion	NOUN
ma-317	174	16	are	be	AUX
ma-317	174	17	obtained	obtain	VERB
ma-317	174	18	.	.	PUNCT
ma-317	175	1	theorem	theorem	VERB
ma-317	175	2	2.14	2.14	NUM
ma-317	175	3	.	.	PUNCT
ma-317	176	1	let	let	VERB
ma-317	176	2	r	r	NOUN
ma-317	176	3	∈	∈	PROPN
ma-317	176	4	n0	n0	NOUN
ma-317	176	5	and	and	CCONJ
ma-317	176	6	s	s	PROPN
ma-317	176	7	∈	∈	PROPN
ma-317	176	8	n0	n0	PROPN
ma-317	176	9	.	.	PUNCT
ma-317	177	1	then	then	ADV
ma-317	177	2	the	the	DET
ma-317	177	3	inequality∣∣∣θ	inequality∣∣∣θ	PROPN
ma-317	177	4	(	(	PUNCT
ma-317	177	5	r	r	NOUN
ma-317	177	6	u	u	NOUN
ma-317	177	7	+	+	X
ma-317	177	8	s	s	NOUN
ma-317	177	9	v	v	NOUN
ma-317	177	10	)	)	PUNCT
ma-317	177	11	(	(	PUNCT
ma-317	177	12	x	x	SYM
ma-317	177	13	u	u	NOUN
ma-317	177	14	+	+	NOUN
ma-317	177	15	y	y	PROPN
ma-317	177	16	v	v	NOUN
ma-317	177	17	)	)	PUNCT
ma-317	177	18	∣∣∣	∣∣∣	ADJ
ma-317	177	19	≤	≤	NUM
ma-317	177	20	∣∣∣θ(r)(x	∣∣∣θ(r)(x	NUM
ma-317	177	21	)	)	PUNCT
ma-317	177	22	∣∣∣	∣∣∣	NOUN
ma-317	177	23	1	1	NUM
ma-317	177	24	u	u	NOUN
ma-317	177	25	∣∣∣θ(s)(y	∣∣∣θ(s)(y	NUM
ma-317	177	26	)	)	PUNCT
ma-317	177	27	∣∣∣	∣∣∣	NOUN
ma-317	177	28	1	1	NUM
ma-317	177	29	v	v	NOUN
ma-317	177	30	(	(	PUNCT
ma-317	177	31	23	23	NUM
ma-317	177	32	)	)	PUNCT
ma-317	177	33	holds	hold	VERB
ma-317	177	34	for	for	ADP
ma-317	177	35	x	x	PUNCT
ma-317	177	36	>	>	X
ma-317	177	37	0	0	PROPN
ma-317	177	38	,	,	PUNCT
ma-317	177	39	y	y	PROPN
ma-317	177	40	>	>	X
ma-317	177	41	0	0	PROPN
ma-317	177	42	,	,	PUNCT
ma-317	177	43	u	u	NOUN
ma-317	177	44	>	>	X
ma-317	177	45	1	1	NUM
ma-317	177	46	,	,	PUNCT
ma-317	177	47	v	v	PART
ma-317	177	48	>	>	SYM
ma-317	177	49	1	1	NUM
ma-317	177	50	and	and	CCONJ
ma-317	177	51	1	1	NUM
ma-317	177	52	u	u	NOUN
ma-317	177	53	+	+	NOUN
ma-317	177	54	1	1	NUM
ma-317	177	55	v	v	NOUN
ma-317	177	56	=	=	SYM
ma-317	177	57	1	1	NUM
ma-317	177	58	.	.	PUNCT
ma-317	178	1	proof	proof	NOUN
ma-317	178	2	.	.	PUNCT
ma-317	179	1	by	by	ADP
ma-317	179	2	using	use	VERB
ma-317	179	3	(	(	PUNCT
ma-317	179	4	19	19	NUM
ma-317	179	5	)	)	PUNCT
ma-317	179	6	and	and	CCONJ
ma-317	179	7	holder	holder	NOUN
ma-317	179	8	’s	’s	PART
ma-317	179	9	inequality	inequality	NOUN
ma-317	179	10	for	for	ADP
ma-317	179	11	integrals	integral	NOUN
ma-317	179	12	,	,	PUNCT
ma-317	179	13	we	we	PRON
ma-317	179	14	obtain∣∣∣θ	obtain∣∣∣θ	VERB
ma-317	179	15	(	(	PUNCT
ma-317	179	16	r	r	NOUN
ma-317	179	17	u	u	NOUN
ma-317	179	18	+	+	X
ma-317	179	19	s	s	NOUN
ma-317	179	20	v	v	NOUN
ma-317	179	21	)	)	PUNCT
ma-317	179	22	(	(	PUNCT
ma-317	179	23	x	x	SYM
ma-317	179	24	u	u	NOUN
ma-317	179	25	+	+	NOUN
ma-317	179	26	y	y	PROPN
ma-317	179	27	v	v	NOUN
ma-317	179	28	)	)	PUNCT
ma-317	179	29	∣∣∣	∣∣∣	NOUN
ma-317	180	1	=	=	SYM
ma-317	180	2	∫	∫	PROPN
ma-317	180	3	∞	∞	PROPN
ma-317	180	4	0	0	X
ma-317	181	1	a(t)t	a(t)t	NOUN
ma-317	181	2	(	(	PUNCT
ma-317	181	3	r	r	NOUN
ma-317	181	4	u	u	NOUN
ma-317	181	5	+	+	X
ma-317	181	6	s	s	NOUN
ma-317	181	7	v	v	NOUN
ma-317	181	8	)	)	PUNCT
ma-317	181	9	−2e−	−2e−	PROPN
ma-317	181	10	(	(	PUNCT
ma-317	181	11	x	x	SYM
ma-317	181	12	u	u	PROPN
ma-317	181	13	+	+	NOUN
ma-317	181	14	y	y	PROPN
ma-317	181	15	v	v	X
ma-317	181	16	)	)	PUNCT
ma-317	181	17	tdt	tdt	PROPN
ma-317	181	18	=	=	SYM
ma-317	181	19	∫	∫	PROPN
ma-317	181	20	∞	∞	PROPN
ma-317	181	21	0	0	NUM
ma-317	181	22	a(t	a(t	NOUN
ma-317	181	23	)	)	PUNCT
ma-317	181	24	(	(	PUNCT
ma-317	181	25	1	1	NUM
ma-317	181	26	u	u	NOUN
ma-317	181	27	+	+	NOUN
ma-317	181	28	1	1	NUM
ma-317	181	29	v	v	NOUN
ma-317	181	30	)	)	PUNCT
ma-317	181	31	t	t	PROPN
ma-317	181	32	(	(	PUNCT
ma-317	181	33	r	r	NOUN
ma-317	181	34	u	u	NOUN
ma-317	181	35	+	+	X
ma-317	181	36	s	s	NOUN
ma-317	181	37	v	v	NOUN
ma-317	181	38	)	)	PUNCT
ma-317	181	39	−2	−2	NOUN
ma-317	181	40	(	(	PUNCT
ma-317	181	41	1	1	NUM
ma-317	181	42	u	u	NOUN
ma-317	181	43	+	+	NOUN
ma-317	181	44	1	1	NUM
ma-317	181	45	v	v	NOUN
ma-317	181	46	)	)	PUNCT
ma-317	181	47	e−	e−	PROPN
ma-317	181	48	(	(	PUNCT
ma-317	181	49	x	x	SYM
ma-317	181	50	u	u	PROPN
ma-317	181	51	+	+	NOUN
ma-317	181	52	y	y	PROPN
ma-317	181	53	v	v	X
ma-317	181	54	)	)	PUNCT
ma-317	181	55	tdt	tdt	PROPN
ma-317	181	56	=	=	SYM
ma-317	181	57	∫	∫	PROPN
ma-317	182	1	∞	∞	PROPN
ma-317	182	2	0	0	NUM
ma-317	182	3	a(t	a(t	NOUN
ma-317	182	4	)	)	PUNCT
ma-317	182	5	1	1	NUM
ma-317	182	6	u	u	NOUN
ma-317	182	7	t	t	PROPN
ma-317	182	8	r−2	r−2	PROPN
ma-317	182	9	u	u	NOUN
ma-317	182	10	e	e	X
ma-317	182	11	−xt	−xt	ADP
ma-317	182	12	u	u	PRON
ma-317	182	13	a(t	a(t	PROPN
ma-317	182	14	)	)	PUNCT
ma-317	182	15	1	1	NUM
ma-317	182	16	v	v	ADP
ma-317	182	17	t	t	NOUN
ma-317	182	18	s−2	s−2	PROPN
ma-317	182	19	v	v	NUM
ma-317	182	20	e	e	X
ma-317	182	21	−yt	−yt	X
ma-317	182	22	v	v	X
ma-317	182	23	dt	dt	NOUN
ma-317	182	24	≤	≤	PROPN
ma-317	182	25	(	(	PUNCT
ma-317	182	26	∫	∫	PROPN
ma-317	182	27	∞	∞	PROPN
ma-317	182	28	0	0	NUM
ma-317	183	1	a(t)tr−2e−xtdt	a(t)tr−2e−xtdt	CCONJ
ma-317	183	2	)	)	PUNCT
ma-317	183	3	1	1	NUM
ma-317	183	4	u	u	NOUN
ma-317	183	5	(	(	PUNCT
ma-317	183	6	∫	∫	PROPN
ma-317	183	7	∞	∞	PROPN
ma-317	183	8	0	0	NUM
ma-317	183	9	a(t)ts−2e−ytdt	a(t)ts−2e−ytdt	ADJ
ma-317	183	10	)	)	PUNCT
ma-317	183	11	1	1	NUM
ma-317	183	12	v	v	NOUN
ma-317	183	13	=	=	X
ma-317	183	14	[	[	PUNCT
ma-317	183	15	θ(r)(x	θ(r)(x	NUM
ma-317	183	16	)	)	PUNCT
ma-317	183	17	]	]	PUNCT
ma-317	183	18	1	1	NUM
ma-317	183	19	u	u	NOUN
ma-317	183	20	[	[	PUNCT
ma-317	183	21	θ(s)(y	θ(s)(y	PROPN
ma-317	183	22	)	)	PUNCT
ma-317	183	23	]	]	PUNCT
ma-317	183	24	1	1	NUM
ma-317	183	25	v	v	NOUN
ma-317	183	26	which	which	PRON
ma-317	183	27	completes	complete	VERB
ma-317	183	28	the	the	DET
ma-317	183	29	proof	proof	NOUN
ma-317	183	30	.	.	PUNCT
ma-317	184	1	�	�	PROPN
ma-317	184	2	remark	remark	VERB
ma-317	184	3	2.15	2.15	NUM
ma-317	184	4	.	.	PUNCT
ma-317	185	1	if	if	SCONJ
ma-317	185	2	r	r	NOUN
ma-317	185	3	=	=	SYM
ma-317	185	4	s	s	X
ma-317	185	5	in	in	ADP
ma-317	185	6	theorem	theorem	NOUN
ma-317	185	7	2.14	2.14	NUM
ma-317	185	8	,	,	PUNCT
ma-317	185	9	then	then	ADV
ma-317	185	10	we	we	PRON
ma-317	185	11	obtain∣∣∣θ(r	obtain∣∣∣θ(r	NOUN
ma-317	185	12	)	)	PUNCT
ma-317	186	1	(	(	PUNCT
ma-317	186	2	x	x	SYM
ma-317	186	3	u	u	NOUN
ma-317	186	4	+	+	NOUN
ma-317	186	5	y	y	PROPN
ma-317	186	6	v	v	NOUN
ma-317	186	7	)	)	PUNCT
ma-317	186	8	∣∣∣	∣∣∣	ADJ
ma-317	186	9	≤	≤	NUM
ma-317	186	10	∣∣∣θ(r)(x	∣∣∣θ(r)(x	NUM
ma-317	186	11	)	)	PUNCT
ma-317	186	12	∣∣∣	∣∣∣	NOUN
ma-317	186	13	1	1	NUM
ma-317	186	14	u	u	NOUN
ma-317	186	15	∣∣∣θ(r)(y	∣∣∣θ(r)(y	NUM
ma-317	186	16	)	)	PUNCT
ma-317	186	17	∣∣∣	∣∣∣	NOUN
ma-317	186	18	1	1	NUM
ma-317	186	19	v	v	NOUN
ma-317	186	20	(	(	PUNCT
ma-317	186	21	24	24	NUM
ma-317	186	22	)	)	PUNCT
ma-317	186	23	which	which	PRON
ma-317	186	24	shows	show	VERB
ma-317	186	25	that	that	SCONJ
ma-317	186	26	,	,	PUNCT
ma-317	186	27	for	for	ADP
ma-317	186	28	r	r	PROPN
ma-317	186	29	∈	∈	PROPN
ma-317	186	30	n0	n0	PROPN
ma-317	186	31	,	,	PUNCT
ma-317	186	32	the	the	DET
ma-317	186	33	function	function	NOUN
ma-317	186	34	∣∣θ(r)(z	∣∣θ(r)(z	NOUN
ma-317	186	35	)	)	PUNCT
ma-317	186	36	∣∣	∣∣	NUM
ma-317	186	37	is	be	AUX
ma-317	186	38	logarithmically	logarithmically	ADV
ma-317	186	39	convex	convex	ADJ
ma-317	186	40	on	on	ADP
ma-317	186	41	(	(	PUNCT
ma-317	186	42	0,∞	0,∞	NUM
ma-317	186	43	)	)	PUNCT
ma-317	186	44	.	.	PUNCT
ma-317	187	1	remark	remark	VERB
ma-317	187	2	2.16	2.16	NUM
ma-317	187	3	.	.	PUNCT
ma-317	188	1	if	if	SCONJ
ma-317	188	2	r	r	NOUN
ma-317	188	3	=	=	PUNCT
ma-317	188	4	k	k	NOUN
ma-317	188	5	−	−	PROPN
ma-317	188	6	1	1	NUM
ma-317	188	7	,	,	PUNCT
ma-317	188	8	s	s	PART
ma-317	188	9	=	=	SYM
ma-317	188	10	k	k	PROPN
ma-317	189	1	+	+	PROPN
ma-317	189	2	1	1	NUM
ma-317	189	3	,	,	PUNCT
ma-317	189	4	u	u	NOUN
ma-317	189	5	=	=	SYM
ma-317	189	6	v	v	NOUN
ma-317	189	7	=	=	SYM
ma-317	189	8	2	2	NUM
ma-317	189	9	and	and	CCONJ
ma-317	189	10	x	x	X
ma-317	190	1	=	=	PUNCT
ma-317	190	2	y	y	PROPN
ma-317	190	3	=	=	SYM
ma-317	190	4	z	z	PROPN
ma-317	190	5	in	in	ADP
ma-317	190	6	theorem	theorem	NOUN
ma-317	190	7	2.14	2.14	NUM
ma-317	190	8	,	,	PUNCT
ma-317	190	9	then	then	ADV
ma-317	190	10	we	we	PRON
ma-317	190	11	obtainthe	obtainthe	VERB
ma-317	190	12	turan	turan	NOUN
ma-317	190	13	-	-	PUNCT
ma-317	190	14	type	type	NOUN
ma-317	190	15	inequality	inequality	NOUN
ma-317	190	16	∣∣∣θ(k	∣∣∣θ(k	NOUN
ma-317	190	17	)	)	PUNCT
ma-317	190	18	(	(	PUNCT
ma-317	190	19	z	z	NOUN
ma-317	190	20	)	)	PUNCT
ma-317	190	21	∣∣∣2	∣∣∣2	NOUN
ma-317	190	22	≤	≤	NOUN
ma-317	190	23	∣∣∣θ(k−1)(z	∣∣∣θ(k−1)(z	PROPN
ma-317	190	24	)	)	PUNCT
ma-317	190	25	∣∣∣	∣∣∣	NOUN
ma-317	190	26	∣∣∣θ(k+1)(z	∣∣∣θ(k+1)(z	NOUN
ma-317	190	27	)	)	PUNCT
ma-317	190	28	∣∣∣	∣∣∣	NOUN
ma-317	190	29	.	.	PUNCT
ma-317	191	1	(	(	PUNCT
ma-317	191	2	25	25	NUM
ma-317	191	3	)	)	PUNCT
ma-317	191	4	where	where	SCONJ
ma-317	191	5	k	k	PROPN
ma-317	191	6	∈	∈	PROPN
ma-317	191	7	n0	n0	PROPN
ma-317	191	8	.	.	PUNCT
ma-317	192	1	this	this	PRON
ma-317	192	2	is	be	AUX
ma-317	192	3	equivalent	equivalent	ADJ
ma-317	192	4	to∣∣∣θ(k+1	to∣∣∣θ(k+1	NOUN
ma-317	192	5	)	)	PUNCT
ma-317	192	6	(	(	PUNCT
ma-317	192	7	z	z	NOUN
ma-317	192	8	)	)	PUNCT
ma-317	192	9	∣∣∣2	∣∣∣2	NOUN
ma-317	192	10	≤	≤	NUM
ma-317	192	11	∣∣∣θ(k)(z	∣∣∣θ(k)(z	VERB
ma-317	192	12	)	)	PUNCT
ma-317	192	13	∣∣∣	∣∣∣	ADJ
ma-317	192	14	∣∣∣θ(k+2)(z	∣∣∣θ(k+2)(z	NUM
ma-317	192	15	)	)	PUNCT
ma-317	192	16	∣∣∣	∣∣∣	NOUN
ma-317	192	17	(	(	PUNCT
ma-317	192	18	26	26	NUM
ma-317	192	19	)	)	PUNCT
ma-317	192	20	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	NUM
ma-317	192	21	eur	eur	NOUN
ma-317	192	22	.	.	PUNCT
ma-317	193	1	j.	j.	PROPN
ma-317	193	2	math	math	PROPN
ma-317	193	3	.	.	PUNCT
ma-317	194	1	anal	anal	PROPN
ma-317	194	2	.	.	PUNCT
ma-317	195	1	10.28924	10.28924	NUM
ma-317	195	2	/	/	SYM
ma-317	195	3	ada	ada	PROPN
ma-317	195	4	/	/	SYM
ma-317	195	5	ma.5.13	ma.5.13	PROPN
ma-317	195	6	8where	8where	NUM
ma-317	195	7	k	k	PROPN
ma-317	195	8	∈	∈	PROPN
ma-317	195	9	n0	n0	PROPN
ma-317	195	10	.	.	PUNCT
ma-317	196	1	lemma	lemma	PROPN
ma-317	196	2	2.17	2.17	NUM
ma-317	196	3	.	.	PUNCT
ma-317	197	1	the	the	DET
ma-317	197	2	function	function	NOUN
ma-317	197	3	θ′(z	θ′(z	CCONJ
ma-317	197	4	)	)	PUNCT
ma-317	197	5	is	be	AUX
ma-317	197	6	increasing	increase	VERB
ma-317	197	7	on	on	ADP
ma-317	197	8	(	(	PUNCT
ma-317	197	9	0,∞	0,∞	NUM
ma-317	197	10	)	)	PUNCT
ma-317	197	11	.	.	PUNCT
ma-317	198	1	proof	proof	NOUN
ma-317	198	2	.	.	PUNCT
ma-317	199	1	this	this	PRON
ma-317	199	2	follows	follow	VERB
ma-317	199	3	directly	directly	ADV
ma-317	199	4	from	from	ADP
ma-317	199	5	(	(	PUNCT
ma-317	199	6	19	19	NUM
ma-317	199	7	)	)	PUNCT
ma-317	199	8	since	since	SCONJ
ma-317	199	9	(	(	PUNCT
ma-317	199	10	θ′(z))′	θ′(z))′	PROPN
ma-317	199	11	=	=	PUNCT
ma-317	199	12	θ′′(z	θ′′(z	ADV
ma-317	199	13	)	)	PUNCT
ma-317	199	14	>	>	X
ma-317	199	15	0	0	X
ma-317	199	16	.	.	PUNCT
ma-317	199	17	�	�	PROPN
ma-317	199	18	theorem	theorem	VERB
ma-317	199	19	2.18	2.18	NUM
ma-317	199	20	.	.	PUNCT
ma-317	200	1	the	the	DET
ma-317	200	2	function	function	NOUN
ma-317	200	3	−θ(z	−θ(z	ADV
ma-317	200	4	)	)	PUNCT
ma-317	200	5	is	be	AUX
ma-317	200	6	starshaped	starshape	VERB
ma-317	200	7	on	on	ADP
ma-317	200	8	(	(	PUNCT
ma-317	200	9	0,∞	0,∞	NUM
ma-317	200	10	)	)	PUNCT
ma-317	200	11	.	.	PUNCT
ma-317	201	1	that	that	PRON
ma-317	201	2	is	be	AUX
ma-317	201	3	,	,	PUNCT
ma-317	201	4	the	the	DET
ma-317	201	5	inequality	inequality	NOUN
ma-317	201	6	−θ(αz	−θ(αz	PRON
ma-317	201	7	)	)	PUNCT
ma-317	201	8	≤	≤	NUM
ma-317	201	9	−αθ(z	−αθ(z	NOUN
ma-317	201	10	)	)	PUNCT
ma-317	201	11	(	(	PUNCT
ma-317	201	12	27	27	NUM
ma-317	201	13	)	)	PUNCT
ma-317	201	14	holds	hold	VERB
ma-317	201	15	for	for	ADP
ma-317	201	16	α	α	PRON
ma-317	201	17	∈	∈	PROPN
ma-317	202	1	[	[	X
ma-317	202	2	0	0	NUM
ma-317	202	3	,	,	PUNCT
ma-317	202	4	1	1	NUM
ma-317	202	5	]	]	PUNCT
ma-317	202	6	and	and	CCONJ
ma-317	202	7	z	z	NOUN
ma-317	202	8	∈	∈	PROPN
ma-317	202	9	(	(	PUNCT
ma-317	202	10	0,∞	0,∞	NOUN
ma-317	202	11	)	)	PUNCT
ma-317	202	12	.	.	PUNCT
ma-317	203	1	proof	proof	NOUN
ma-317	203	2	.	.	PUNCT
ma-317	204	1	let	let	VERB
ma-317	204	2	k(z	k(z	PRON
ma-317	204	3	)	)	PUNCT
ma-317	205	1	=	=	SYM
ma-317	205	2	θ(αz)−	θ(αz)−	X
ma-317	205	3	αθ(z	αθ(z	NUM
ma-317	205	4	)	)	PUNCT
ma-317	205	5	for	for	ADP
ma-317	205	6	α	α	PRON
ma-317	205	7	∈	∈	PROPN
ma-317	206	1	[	[	X
ma-317	206	2	0	0	NUM
ma-317	206	3	,	,	PUNCT
ma-317	206	4	1	1	NUM
ma-317	206	5	]	]	PUNCT
ma-317	206	6	and	and	CCONJ
ma-317	206	7	z	z	NOUN
ma-317	206	8	∈	∈	PROPN
ma-317	206	9	(	(	PUNCT
ma-317	206	10	0,∞	0,∞	NOUN
ma-317	206	11	)	)	PUNCT
ma-317	206	12	.	.	PUNCT
ma-317	207	1	then	then	ADV
ma-317	207	2	k′(z	k′(z	AUX
ma-317	207	3	)	)	PUNCT
ma-317	207	4	=	=	SYM
ma-317	207	5	αθ′(αz)−	αθ′(αz)−	ADJ
ma-317	207	6	αθ′(z	αθ′(z	NOUN
ma-317	207	7	)	)	PUNCT
ma-317	207	8	=	=	SYM
ma-317	207	9	α	α	PROPN
ma-317	207	10	[	[	PUNCT
ma-317	207	11	θ′(αz)−θ′(z	θ′(αz)−θ′(z	NOUN
ma-317	207	12	)	)	PUNCT
ma-317	207	13	]	]	PUNCT
ma-317	208	1	<	<	X
ma-317	208	2	0	0	PUNCT
ma-317	208	3	since	since	SCONJ
ma-317	208	4	θ′(z	θ′(z	PROPN
ma-317	208	5	)	)	PUNCT
ma-317	208	6	is	be	AUX
ma-317	208	7	increasing	increase	VERB
ma-317	208	8	.	.	PUNCT
ma-317	209	1	hence	hence	ADV
ma-317	209	2	k(z	k(z	PROPN
ma-317	209	3	)	)	PUNCT
ma-317	209	4	is	be	AUX
ma-317	209	5	decreasing	decrease	VERB
ma-317	209	6	.	.	PUNCT
ma-317	210	1	then	then	ADV
ma-317	210	2	for	for	ADP
ma-317	210	3	z	z	PROPN
ma-317	210	4	∈	∈	PROPN
ma-317	210	5	(	(	PUNCT
ma-317	210	6	0,∞	0,∞	NOUN
ma-317	210	7	)	)	PUNCT
ma-317	210	8	,	,	PUNCT
ma-317	210	9	we	we	PRON
ma-317	210	10	have	have	VERB
ma-317	210	11	k(z	k(z	PROPN
ma-317	210	12	)	)	PUNCT
ma-317	210	13	≥	≥	PROPN
ma-317	210	14	lim	lim	PROPN
ma-317	210	15	z→∞	z→∞	PROPN
ma-317	210	16	k(z	k(z	PROPN
ma-317	210	17	)	)	PUNCT
ma-317	211	1	=	=	SYM
ma-317	211	2	0	0	NUM
ma-317	211	3	which	which	PRON
ma-317	211	4	implies	imply	VERB
ma-317	211	5	that	that	PRON
ma-317	211	6	θ(αz	θ(αz	NUM
ma-317	211	7	)	)	PUNCT
ma-317	211	8	≥	≥	NOUN
ma-317	212	1	αθ(z).this	αθ(z).this	NOUN
ma-317	212	2	then	then	ADV
ma-317	212	3	gives	give	VERB
ma-317	212	4	rise	rise	NOUN
ma-317	212	5	to	to	ADP
ma-317	212	6	the	the	DET
ma-317	212	7	inequality	inequality	NOUN
ma-317	212	8	(	(	PUNCT
ma-317	212	9	27	27	NUM
ma-317	212	10	)	)	PUNCT
ma-317	212	11	and	and	CCONJ
ma-317	212	12	that	that	PRON
ma-317	212	13	completes	complete	VERB
ma-317	212	14	the	the	DET
ma-317	212	15	proof	proof	NOUN
ma-317	212	16	.	.	PUNCT
ma-317	213	1	�	�	PROPN
ma-317	213	2	theorem	theorem	VERB
ma-317	213	3	2.19	2.19	NUM
ma-317	213	4	.	.	PUNCT
ma-317	214	1	let	let	VERB
ma-317	214	2	r	r	NOUN
ma-317	214	3	∈	∈	PROPN
ma-317	214	4	n0	n0	PROPN
ma-317	214	5	.	.	PUNCT
ma-317	215	1	then	then	ADV
ma-317	215	2	θ(r)(z	θ(r)(z	VERB
ma-317	215	3	)	)	PUNCT
ma-317	215	4	is	be	AUX
ma-317	215	5	strictly	strictly	ADV
ma-317	215	6	subadditive	subadditive	ADJ
ma-317	215	7	if	if	SCONJ
ma-317	215	8	r	r	NOUN
ma-317	215	9	is	be	AUX
ma-317	215	10	even	even	ADV
ma-317	215	11	and	and	CCONJ
ma-317	215	12	θ(r)(z	θ(r)(z	VERB
ma-317	215	13	)	)	PUNCT
ma-317	216	1	is	be	AUX
ma-317	216	2	strictly	strictly	ADV
ma-317	216	3	superadditive	superadditive	ADJ
ma-317	216	4	if	if	SCONJ
ma-317	216	5	r	r	NOUN
ma-317	216	6	is	be	AUX
ma-317	216	7	odd	odd	ADJ
ma-317	216	8	.	.	PUNCT
ma-317	217	1	that	that	PRON
ma-317	217	2	is	be	AUX
ma-317	217	3	,	,	PUNCT
ma-317	217	4	for	for	ADP
ma-317	217	5	x	x	X
ma-317	217	6	,	,	PUNCT
ma-317	217	7	y	y	PROPN
ma-317	217	8	∈	∈	PROPN
ma-317	217	9	(	(	PUNCT
ma-317	217	10	0,∞	0,∞	NOUN
ma-317	217	11	)	)	PUNCT
ma-317	217	12	,	,	PUNCT
ma-317	217	13	it	it	PRON
ma-317	217	14	holds	hold	VERB
ma-317	217	15	that	that	PRON
ma-317	217	16	θ(r)(x	θ(r)(x	NUM
ma-317	217	17	+	+	NUM
ma-317	217	18	y	y	X
ma-317	217	19	)	)	PUNCT
ma-317	217	20	<	<	X
ma-317	217	21	θ(r)(x	θ(r)(x	NUM
ma-317	217	22	)	)	PUNCT
ma-317	217	23	+	+	CCONJ
ma-317	218	1	θ(r)(y	θ(r)(y	NUM
ma-317	218	2	)	)	PUNCT
ma-317	218	3	(	(	PUNCT
ma-317	218	4	28	28	NUM
ma-317	218	5	)	)	PUNCT
ma-317	218	6	if	if	SCONJ
ma-317	218	7	r	r	NOUN
ma-317	218	8	is	be	AUX
ma-317	218	9	even	even	ADV
ma-317	218	10	,	,	PUNCT
ma-317	218	11	and	and	CCONJ
ma-317	218	12	θ(r)(x	θ(r)(x	NUM
ma-317	218	13	+	+	CCONJ
ma-317	218	14	y	y	X
ma-317	218	15	)	)	PUNCT
ma-317	218	16	>	>	PUNCT
ma-317	218	17	θ(r)(x	θ(r)(x	NUM
ma-317	218	18	)	)	PUNCT
ma-317	218	19	+	+	CCONJ
ma-317	218	20	θ(r)(y	θ(r)(y	NUM
ma-317	218	21	)	)	PUNCT
ma-317	218	22	(	(	PUNCT
ma-317	218	23	29	29	NUM
ma-317	218	24	)	)	PUNCT
ma-317	218	25	if	if	SCONJ
ma-317	218	26	r	r	NOUN
ma-317	218	27	is	be	AUX
ma-317	218	28	odd	odd	ADJ
ma-317	218	29	.	.	PUNCT
ma-317	219	1	proof	proof	NOUN
ma-317	219	2	.	.	PUNCT
ma-317	220	1	let	let	VERB
ma-317	220	2	u(x	u(x	NOUN
ma-317	220	3	,	,	PUNCT
ma-317	220	4	y	y	NOUN
ma-317	220	5	)	)	PUNCT
ma-317	220	6	=	=	PUNCT
ma-317	221	1	θ(r)(x	θ(r)(x	PUNCT
ma-317	221	2	+	+	NUM
ma-317	221	3	y	y	NOUN
ma-317	221	4	)	)	PUNCT
ma-317	221	5	−θ(r)(x	−θ(r)(x	NOUN
ma-317	221	6	)	)	PUNCT
ma-317	221	7	−θ(r)(y	−θ(r)(y	NUM
ma-317	221	8	)	)	PUNCT
ma-317	221	9	.	.	PUNCT
ma-317	222	1	with	with	ADP
ma-317	222	2	no	no	DET
ma-317	222	3	loss	loss	NOUN
ma-317	222	4	of	of	ADP
ma-317	222	5	generality	generality	NOUN
ma-317	222	6	,	,	PUNCT
ma-317	222	7	let	let	VERB
ma-317	222	8	y	y	PRON
ma-317	222	9	be	be	AUX
ma-317	222	10	fixed.then	fixed.then	ADV
ma-317	222	11	by	by	ADP
ma-317	222	12	differentiating	differentiate	VERB
ma-317	222	13	with	with	ADP
ma-317	222	14	respect	respect	NOUN
ma-317	222	15	to	to	ADP
ma-317	222	16	x	x	PRON
ma-317	222	17	,	,	PUNCT
ma-317	222	18	and	and	CCONJ
ma-317	222	19	using	use	VERB
ma-317	222	20	(	(	PUNCT
ma-317	222	21	19	19	NUM
ma-317	222	22	)	)	PUNCT
ma-317	222	23	,	,	PUNCT
ma-317	222	24	we	we	PRON
ma-317	222	25	have	have	VERB
ma-317	222	26	u	u	NOUN
ma-317	222	27	′(x	′(x	NOUN
ma-317	222	28	,	,	PUNCT
ma-317	222	29	y	y	NOUN
ma-317	222	30	)	)	PUNCT
ma-317	222	31	=	=	SYM
ma-317	222	32	θ(r+1)(x	θ(r+1)(x	NOUN
ma-317	222	33	+	+	CCONJ
ma-317	222	34	y)−θ(r+1)(x	y)−θ(r+1)(x	NOUN
ma-317	222	35	)	)	PUNCT
ma-317	222	36	(	(	PUNCT
ma-317	223	1	−1)r+1	−1)r+1	INTJ
ma-317	223	2	∫	∫	PROPN
ma-317	223	3	∞	∞	PROPN
ma-317	223	4	0	0	NUM
ma-317	224	1	a(t)tr−1e−(x+y)tdt	a(t)tr−1e−(x+y)tdt	PROPN
ma-317	224	2	−	−	PROPN
ma-317	224	3	(	(	PUNCT
ma-317	224	4	−1)r+1	−1)r+1	INTJ
ma-317	224	5	∫	∫	PROPN
ma-317	224	6	∞	∞	PROPN
ma-317	224	7	0	0	NUM
ma-317	224	8	a(t)tr−1e−xtdt	a(t)tr−1e−xtdt	NOUN
ma-317	224	9	=	=	PUNCT
ma-317	224	10	(	(	PUNCT
ma-317	224	11	−1)r+1	−1)r+1	INTJ
ma-317	224	12	∫	∫	PROPN
ma-317	224	13	∞	∞	NUM
ma-317	224	14	0	0	NUM
ma-317	225	1	a(t)tr−1	a(t)tr−1	PROPN
ma-317	225	2	[	[	PUNCT
ma-317	225	3	e−(x+y)t	e−(x+y)t	NOUN
ma-317	225	4	−	−	NOUN
ma-317	225	5	e−xt	e−xt	X
ma-317	225	6	]	]	PUNCT
ma-317	225	7	dt	dt	PUNCT
ma-317	225	8	:	:	PUNCT
ma-317	225	9	=	=	SYM
ma-317	225	10	v(x	v(x	PROPN
ma-317	225	11	,	,	PUNCT
ma-317	225	12	y	y	PROPN
ma-317	225	13	)	)	PUNCT
ma-317	225	14	https://doi.org/10.28924/ada/ma.5.13	https://doi.org/10.28924/ada/ma.5.13	PROPN
ma-317	225	15	eur	eur	PROPN
ma-317	225	16	.	.	PUNCT
ma-317	226	1	j.	j.	PROPN
ma-317	226	2	math	math	PROPN
ma-317	226	3	.	.	PUNCT
ma-317	227	1	anal	anal	PROPN
ma-317	227	2	.	.	PUNCT
ma-317	228	1	10.28924	10.28924	NUM
ma-317	228	2	/	/	SYM
ma-317	228	3	ada	ada	PROPN
ma-317	228	4	/	/	SYM
ma-317	228	5	ma.5.13	ma.5.13	PROPN
ma-317	228	6	9suppose	9suppose	NUM
ma-317	228	7	that	that	PRON
ma-317	228	8	r	r	NOUN
ma-317	228	9	is	be	AUX
ma-317	228	10	even	even	ADV
ma-317	228	11	.	.	PUNCT
ma-317	229	1	then	then	ADV
ma-317	229	2	v(x	v(x	PROPN
ma-317	229	3	,	,	PUNCT
ma-317	229	4	y	y	PROPN
ma-317	229	5	)	)	PUNCT
ma-317	229	6	>	>	X
ma-317	229	7	0	0	X
ma-317	229	8	.	.	PUNCT
ma-317	230	1	this	this	PRON
ma-317	230	2	implies	imply	VERB
ma-317	230	3	that	that	SCONJ
ma-317	230	4	u(x	u(x	NOUN
ma-317	230	5	,	,	PUNCT
ma-317	230	6	y	y	NOUN
ma-317	230	7	)	)	PUNCT
ma-317	230	8	is	be	AUX
ma-317	230	9	increasing	increase	VERB
ma-317	230	10	in	in	ADP
ma-317	230	11	terms	term	NOUN
ma-317	230	12	of	of	ADP
ma-317	230	13	x	x	PUNCT
ma-317	230	14	.hence	.hence	NOUN
ma-317	230	15	,	,	PUNCT
ma-317	230	16	for	for	ADP
ma-317	230	17	x	x	PROPN
ma-317	230	18	∈	∈	PROPN
ma-317	230	19	(	(	PUNCT
ma-317	230	20	0,∞	0,∞	NOUN
ma-317	230	21	)	)	PUNCT
ma-317	230	22	,	,	PUNCT
ma-317	230	23	we	we	PRON
ma-317	230	24	have	have	VERB
ma-317	230	25	u(x	u(x	NOUN
ma-317	230	26	,	,	PUNCT
ma-317	230	27	y	y	NOUN
ma-317	230	28	)	)	PUNCT
ma-317	231	1	<	<	X
ma-317	231	2	lim	lim	PROPN
ma-317	231	3	x→∞	x→∞	NUM
ma-317	231	4	u(x	u(x	PROPN
ma-317	231	5	,	,	PUNCT
ma-317	231	6	y	y	NOUN
ma-317	231	7	)	)	PUNCT
ma-317	231	8	=	=	SYM
ma-317	231	9	−θ(r)(y	−θ(r)(y	NUM
ma-317	231	10	)	)	PUNCT
ma-317	231	11	<	<	X
ma-317	231	12	0	0	NUM
ma-317	231	13	which	which	PRON
ma-317	231	14	gives	give	VERB
ma-317	231	15	rise	rise	NOUN
ma-317	231	16	to	to	ADP
ma-317	231	17	the	the	DET
ma-317	231	18	inequality	inequality	NOUN
ma-317	231	19	(	(	PUNCT
ma-317	231	20	28	28	NUM
ma-317	231	21	)	)	PUNCT
ma-317	231	22	.	.	PUNCT
ma-317	232	1	likewise	likewise	ADV
ma-317	232	2	,	,	PUNCT
ma-317	232	3	suppose	suppose	VERB
ma-317	232	4	that	that	SCONJ
ma-317	232	5	r	r	NOUN
ma-317	232	6	is	be	AUX
ma-317	232	7	odd	odd	ADJ
ma-317	232	8	.	.	PUNCT
ma-317	233	1	then	then	ADV
ma-317	233	2	v(x	v(x	PROPN
ma-317	233	3	,	,	PUNCT
ma-317	233	4	y	y	NOUN
ma-317	233	5	)	)	PUNCT
ma-317	233	6	<	<	X
ma-317	233	7	0	0	X
ma-317	233	8	.	.	PUNCT
ma-317	233	9	thisimplies	thisimplie	NOUN
ma-317	233	10	that	that	SCONJ
ma-317	233	11	u(x	u(x	VERB
ma-317	233	12	,	,	PUNCT
ma-317	233	13	y	y	NOUN
ma-317	233	14	)	)	PUNCT
ma-317	233	15	is	be	AUX
ma-317	233	16	decreasing	decrease	VERB
ma-317	233	17	in	in	ADP
ma-317	233	18	terms	term	NOUN
ma-317	233	19	of	of	ADP
ma-317	233	20	x	x	X
ma-317	233	21	.	.	PUNCT
ma-317	234	1	hence	hence	ADV
ma-317	234	2	,	,	PUNCT
ma-317	234	3	for	for	ADP
ma-317	234	4	x	x	PROPN
ma-317	234	5	∈	∈	PROPN
ma-317	234	6	(	(	PUNCT
ma-317	234	7	0,∞	0,∞	NOUN
ma-317	234	8	)	)	PUNCT
ma-317	234	9	,	,	PUNCT
ma-317	234	10	we	we	PRON
ma-317	234	11	have	have	VERB
ma-317	234	12	u(x	u(x	NOUN
ma-317	234	13	,	,	PUNCT
ma-317	234	14	y	y	NOUN
ma-317	234	15	)	)	PUNCT
ma-317	234	16	>	>	X
ma-317	235	1	lim	lim	PROPN
ma-317	235	2	x→∞	x→∞	NUM
ma-317	235	3	u(x	u(x	PROPN
ma-317	235	4	,	,	PUNCT
ma-317	235	5	y	y	NOUN
ma-317	235	6	)	)	PUNCT
ma-317	235	7	=	=	SYM
ma-317	235	8	−θ(r)(y	−θ(r)(y	NUM
ma-317	235	9	)	)	PUNCT
ma-317	235	10	>	>	X
ma-317	235	11	0	0	NUM
ma-317	235	12	which	which	PRON
ma-317	235	13	gives	give	VERB
ma-317	235	14	rise	rise	NOUN
ma-317	235	15	to	to	ADP
ma-317	235	16	the	the	DET
ma-317	235	17	inequality	inequality	NOUN
ma-317	235	18	(	(	PUNCT
ma-317	235	19	29	29	NUM
ma-317	235	20	)	)	PUNCT
ma-317	235	21	.	.	PUNCT
ma-317	236	1	this	this	PRON
ma-317	236	2	completes	complete	VERB
ma-317	236	3	the	the	DET
ma-317	236	4	proof	proof	NOUN
ma-317	236	5	.	.	PUNCT
ma-317	237	1	�	�	PROPN
ma-317	237	2	remark	remark	VERB
ma-317	237	3	2.20	2.20	NUM
ma-317	237	4	.	.	PUNCT
ma-317	238	1	the	the	DET
ma-317	238	2	particular	particular	ADJ
ma-317	238	3	case	case	NOUN
ma-317	238	4	where	where	SCONJ
ma-317	238	5	r	r	NOUN
ma-317	238	6	=	=	SYM
ma-317	238	7	0	0	NUM
ma-317	238	8	in	in	ADP
ma-317	238	9	theorem	theorem	NOUN
ma-317	238	10	2.19	2.19	NUM
ma-317	238	11	,	,	PUNCT
ma-317	238	12	shows	show	VERB
ma-317	238	13	that	that	SCONJ
ma-317	238	14	the	the	DET
ma-317	238	15	function	function	NOUN
ma-317	238	16	θ(z	θ(z	NOUN
ma-317	238	17	)	)	PUNCT
ma-317	238	18	isstrictly	isstrictly	ADV
ma-317	238	19	subadditive	subadditive	ADJ
ma-317	238	20	on	on	ADP
ma-317	238	21	(	(	PUNCT
ma-317	238	22	0,∞	0,∞	NUM
ma-317	238	23	)	)	PUNCT
ma-317	238	24	.	.	PUNCT
ma-317	239	1	references	reference	NOUN
ma-317	239	2	[	[	X
ma-317	239	3	1	1	X
ma-317	239	4	]	]	PUNCT
ma-317	239	5	v.	v.	CCONJ
ma-317	239	6	s.	s.	PROPN
ma-317	239	7	adamchik	adamchik	PROPN
ma-317	239	8	,	,	PUNCT
ma-317	239	9	contributions	contribution	NOUN
ma-317	239	10	to	to	ADP
ma-317	239	11	the	the	DET
ma-317	239	12	theory	theory	NOUN
ma-317	239	13	of	of	ADP
ma-317	239	14	the	the	DET
ma-317	239	15	barnes	barnes	PROPN
ma-317	239	16	function	function	PROPN
ma-317	239	17	,	,	PUNCT
ma-317	239	18	int	int	PROPN
ma-317	239	19	.	.	PUNCT
ma-317	240	1	j.	j.	PROPN
ma-317	240	2	math	math	PROPN
ma-317	240	3	.	.	PUNCT
ma-317	241	1	comput	comput	NOUN
ma-317	241	2	.	.	PUNCT
ma-317	242	1	sci	sci	PROPN
ma-317	242	2	.	.	PROPN
ma-317	242	3	9	9	NUM
ma-317	242	4	(	(	PUNCT
ma-317	242	5	2014	2014	NUM
ma-317	242	6	)	)	PUNCT
ma-317	242	7	,	,	PUNCT
ma-317	242	8	11	11	NUM
ma-317	242	9	-	-	SYM
ma-317	242	10	30	30	NUM
ma-317	242	11	.	.	PUNCT
ma-317	243	1	https://future-in-tech.net/9.1/r-adamchikcontributions.pdf.[2	https://future-in-tech.net/9.1/r-adamchikcontributions.pdf.[2	ADV
ma-317	243	2	]	]	PUNCT
ma-317	244	1	t.	t.	PROPN
ma-317	244	2	batbold	batbold	PROPN
ma-317	244	3	,	,	PUNCT
ma-317	244	4	some	some	DET
ma-317	244	5	remarks	remark	NOUN
ma-317	244	6	on	on	ADP
ma-317	244	7	results	result	NOUN
ma-317	244	8	of	of	ADP
ma-317	244	9	mortici	mortici	NOUN
ma-317	244	10	,	,	PUNCT
ma-317	244	11	kragujevac	kragujevac	PROPN
ma-317	244	12	j.	j.	PROPN
ma-317	244	13	math	math	PROPN
ma-317	244	14	.	.	PUNCT
ma-317	245	1	36	36	NUM
ma-317	245	2	(	(	PUNCT
ma-317	245	3	2012	2012	NUM
ma-317	245	4	)	)	PUNCT
ma-317	245	5	,	,	PUNCT
ma-317	245	6	73	73	NUM
ma-317	245	7	-	-	SYM
ma-317	245	8	76	76	NUM
ma-317	245	9	.	.	PUNCT
ma-317	246	1	https://imi.pmf.kg.ac	https://imi.pmf.kg.ac	PROPN
ma-317	246	2	.	.	PUNCT
ma-317	246	3	rs	rs	PROPN
ma-317	246	4	/	/	SYM
ma-317	246	5	kjm	kjm	VERB
ma-317	246	6	/	/	SYM
ma-317	246	7	pub/13476258401367_kjom3601	pub/13476258401367_kjom3601	NOUN
ma-317	246	8	-	-	NOUN
ma-317	246	9	08.pdf.[3	08.pdf.[3	NOUN
ma-317	246	10	]	]	PUNCT
ma-317	246	11	p.	p.	NOUN
ma-317	246	12	k.	k.	PROPN
ma-317	247	1	bhandari	bhandari	PROPN
ma-317	247	2	and	and	CCONJ
ma-317	247	3	s.	s.	PROPN
ma-317	247	4	k.	k.	PROPN
ma-317	247	5	bissu	bissu	PROPN
ma-317	247	6	,	,	PUNCT
ma-317	247	7	on	on	ADP
ma-317	247	8	some	some	DET
ma-317	247	9	inequalities	inequality	NOUN
ma-317	247	10	involving	involve	VERB
ma-317	247	11	turan	turan	NOUN
ma-317	247	12	-	-	PUNCT
ma-317	247	13	type	type	NOUN
ma-317	247	14	inequalities	inequality	NOUN
ma-317	247	15	,	,	PUNCT
ma-317	247	16	cogent	cogent	ADJ
ma-317	247	17	math	math	NOUN
ma-317	247	18	.	.	PUNCT
ma-317	248	1	3	3	NUM
ma-317	248	2	(	(	PUNCT
ma-317	248	3	2016),1130678	2016),1130678	NUM
ma-317	248	4	.	.	PUNCT
ma-317	249	1	https://doi.org/10.1080/23311835.2015.1130678.[4	https://doi.org/10.1080/23311835.2015.1130678.[4	NUM
ma-317	249	2	]	]	X
ma-317	249	3	a.	a.	NOUN
ma-317	249	4	m.	m.	NOUN
ma-317	249	5	bruckner	bruckner	PROPN
ma-317	249	6	and	and	CCONJ
ma-317	249	7	e.	e.	PROPN
ma-317	249	8	ostrow	ostrow	PROPN
ma-317	249	9	,	,	PUNCT
ma-317	249	10	some	some	DET
ma-317	249	11	function	function	NOUN
ma-317	249	12	classes	class	NOUN
ma-317	249	13	related	relate	VERB
ma-317	249	14	to	to	ADP
ma-317	249	15	the	the	DET
ma-317	249	16	class	class	NOUN
ma-317	249	17	of	of	ADP
ma-317	249	18	convex	convex	NOUN
ma-317	249	19	functions	function	NOUN
ma-317	249	20	,	,	PUNCT
ma-317	250	1	pac	pac	PROPN
ma-317	250	2	.	.	PUNCT
ma-317	250	3	j.	j.	PROPN
ma-317	250	4	math	math	PROPN
ma-317	250	5	.	.	PUNCT
ma-317	251	1	12(1962	12(1962	NUM
ma-317	251	2	)	)	PUNCT
ma-317	251	3	,	,	PUNCT
ma-317	251	4	1203	1203	NUM
ma-317	251	5	-	-	SYM
ma-317	251	6	1215	1215	NUM
ma-317	251	7	.	.	PUNCT
ma-317	252	1	https://doi.org/10.2140/pjm.1962.12.1203.[5	https://doi.org/10.2140/pjm.1962.12.1203.[5	SYM
ma-317	252	2	]	]	X
ma-317	252	3	c	c	X
ma-317	252	4	-	-	PUNCT
ma-317	252	5	p.	p.	NOUN
ma-317	252	6	chen	chen	PROPN
ma-317	252	7	and	and	CCONJ
ma-317	252	8	f.	f.	PROPN
ma-317	252	9	qi	qi	PROPN
ma-317	252	10	,	,	PUNCT
ma-317	252	11	completely	completely	ADV
ma-317	252	12	monotonic	monotonic	ADJ
ma-317	252	13	functions	function	NOUN
ma-317	252	14	related	relate	VERB
ma-317	252	15	to	to	ADP
ma-317	252	16	the	the	DET
ma-317	252	17	gamma	gamma	NOUN
ma-317	252	18	functions	function	NOUN
ma-317	252	19	,	,	PUNCT
ma-317	252	20	rgmia	rgmia	NOUN
ma-317	252	21	res	re	NOUN
ma-317	252	22	.	.	PUNCT
ma-317	252	23	rep	rep	PROPN
ma-317	252	24	.	.	PROPN
ma-317	252	25	coll	coll	PROPN
ma-317	252	26	.	.	PUNCT
ma-317	253	1	8(2005	8(2005	NUM
ma-317	253	2	)	)	PUNCT
ma-317	253	3	,	,	PUNCT
ma-317	253	4	3	3	X
ma-317	253	5	.	.	X
ma-317	253	6	https://rgmia.org/papers/v8n2/comp-monoto-property.pdf.[6	https://rgmia.org/papers/v8n2/comp-monoto-property.pdf.[6	PROPN
ma-317	253	7	]	]	X
ma-317	253	8	j.	j.	PROPN
ma-317	253	9	choi	choi	PROPN
ma-317	253	10	,	,	PUNCT
ma-317	253	11	some	some	DET
ma-317	253	12	mathematical	mathematical	ADJ
ma-317	253	13	constants	constant	NOUN
ma-317	253	14	,	,	PUNCT
ma-317	253	15	appl	appl	PROPN
ma-317	253	16	.	.	PROPN
ma-317	253	17	math	math	PROPN
ma-317	253	18	.	.	PUNCT
ma-317	254	1	comput	comput	NOUN
ma-317	254	2	.	.	PUNCT
ma-317	255	1	187	187	NUM
ma-317	255	2	(	(	PUNCT
ma-317	255	3	2007	2007	NUM
ma-317	255	4	)	)	PUNCT
ma-317	255	5	,	,	PUNCT
ma-317	255	6	122	122	NUM
ma-317	255	7	-	-	SYM
ma-317	255	8	140	140	NUM
ma-317	255	9	.	.	PUNCT
ma-317	256	1	https://doi.org/10.1016/j	https://doi.org/10.1016/j	NOUN
ma-317	256	2	.	.	PUNCT
ma-317	257	1	amc.2006.08.091.[7	amc.2006.08.091.[7	PROPN
ma-317	257	2	]	]	PUNCT
ma-317	257	3	s.	s.	PROPN
ma-317	257	4	guo	guo	PROPN
ma-317	257	5	and	and	CCONJ
ma-317	257	6	f.	f.	PROPN
ma-317	257	7	qi	qi	PROPN
ma-317	257	8	,	,	PUNCT
ma-317	257	9	a	a	DET
ma-317	257	10	class	class	NOUN
ma-317	257	11	of	of	ADP
ma-317	257	12	completely	completely	ADV
ma-317	257	13	monotonic	monotonic	ADJ
ma-317	257	14	functions	function	NOUN
ma-317	257	15	related	relate	VERB
ma-317	257	16	to	to	ADP
ma-317	257	17	the	the	DET
ma-317	257	18	remainder	remainder	NOUN
ma-317	257	19	of	of	ADP
ma-317	257	20	binet	binet	NOUN
ma-317	257	21	’s	’s	PART
ma-317	257	22	formula	formula	NOUN
ma-317	257	23	withapplications	withapplication	NOUN
ma-317	257	24	,	,	PUNCT
ma-317	257	25	tamsui	tamsui	PROPN
ma-317	257	26	oxf	oxf	PROPN
ma-317	257	27	.	.	PUNCT
ma-317	258	1	j.	j.	PROPN
ma-317	258	2	math	math	PROPN
ma-317	258	3	.	.	PUNCT
ma-317	259	1	sci	sci	PROPN
ma-317	259	2	.	.	PROPN
ma-317	260	1	25	25	NUM
ma-317	260	2	(	(	PUNCT
ma-317	260	3	2009	2009	NUM
ma-317	260	4	)	)	PUNCT
ma-317	260	5	,	,	PUNCT
ma-317	260	6	9	9	NUM
ma-317	260	7	-	-	SYM
ma-317	260	8	14	14	NUM
ma-317	260	9	.	.	PUNCT
ma-317	261	1	https://ibi.au.edu.tw/var/file/18/1018/img/1838/	https://ibi.au.edu.tw/var/file/18/1018/img/1838/	PROPN
ma-317	261	2	25(1)-2	25(1)-2	NOUN
ma-317	261	3	-	-	SYM
ma-317	261	4	009.pdf.[8	009.pdf.[8	X
ma-317	261	5	]	]	X
ma-317	261	6	a	a	X
ma-317	261	7	-	-	PUNCT
ma-317	261	8	q.	q.	NOUN
ma-317	261	9	liu	liu	PROPN
ma-317	261	10	,	,	PUNCT
ma-317	261	11	g	g	PROPN
ma-317	261	12	-	-	PUNCT
ma-317	261	13	f.	f.	PROPN
ma-317	261	14	li	li	PROPN
ma-317	261	15	,	,	PUNCT
ma-317	261	16	b	b	X
ma-317	261	17	-	-	PUNCT
ma-317	261	18	n.	n.	ADJ
ma-317	261	19	guo	guo	PROPN
ma-317	261	20	and	and	CCONJ
ma-317	261	21	f.	f.	PROPN
ma-317	261	22	qi	qi	PROPN
ma-317	261	23	,	,	PUNCT
ma-317	261	24	monotonicity	monotonicity	NOUN
ma-317	261	25	and	and	CCONJ
ma-317	261	26	logarithmic	logarithmic	ADJ
ma-317	261	27	concavity	concavity	NOUN
ma-317	261	28	of	of	ADP
ma-317	261	29	two	two	NUM
ma-317	261	30	functions	function	NOUN
ma-317	261	31	involving	involve	VERB
ma-317	261	32	exponentialfunction	exponentialfunction	NOUN
ma-317	261	33	,	,	PUNCT
ma-317	261	34	int	int	NOUN
ma-317	261	35	.	.	PUNCT
ma-317	262	1	j.	j.	PROPN
ma-317	262	2	math	math	PROPN
ma-317	262	3	.	.	PUNCT
ma-317	263	1	ed	ed	NOUN
ma-317	263	2	.	.	PUNCT
ma-317	264	1	sci	sci	PROPN
ma-317	264	2	.	.	PUNCT
ma-317	264	3	tech	tech	PROPN
ma-317	264	4	.	.	PUNCT
ma-317	265	1	39	39	NUM
ma-317	265	2	(	(	PUNCT
ma-317	265	3	2008	2008	NUM
ma-317	265	4	)	)	PUNCT
ma-317	265	5	,	,	PUNCT
ma-317	265	6	686	686	NUM
ma-317	265	7	-	-	SYM
ma-317	265	8	691	691	NUM
ma-317	265	9	.	.	PUNCT
ma-317	266	1	https://doi.org/10.1080/00207390801986841.[9	https://doi.org/10.1080/00207390801986841.[9	X
ma-317	266	2	]	]	X
ma-317	266	3	k.	k.	PROPN
ma-317	266	4	nantomah	nantomah	PROPN
ma-317	266	5	,	,	PUNCT
ma-317	266	6	alternative	alternative	ADJ
ma-317	266	7	proof	proof	NOUN
ma-317	266	8	of	of	ADP
ma-317	266	9	a	a	DET
ma-317	266	10	monotonicity	monotonicity	NOUN
ma-317	266	11	property	property	NOUN
ma-317	266	12	of	of	ADP
ma-317	266	13	certain	certain	ADJ
ma-317	266	14	function	function	NOUN
ma-317	266	15	,	,	PUNCT
ma-317	266	16	math	math	NOUN
ma-317	266	17	.	.	PUNCT
ma-317	267	1	anal	anal	PROPN
ma-317	267	2	.	.	PUNCT
ma-317	268	1	contemp	contemp	NOUN
ma-317	268	2	.	.	PUNCT
ma-317	269	1	appl	appl	PROPN
ma-317	269	2	.	.	PROPN
ma-317	270	1	5	5	NUM
ma-317	270	2	(	(	PUNCT
ma-317	270	3	2023),65	2023),65	NUM
ma-317	270	4	-	-	SYM
ma-317	270	5	68	68	NUM
ma-317	270	6	.	.	PUNCT
ma-317	271	1	https://doi.org/10.30495/maca.2023.1987128.1067.[10	https://doi.org/10.30495/maca.2023.1987128.1067.[10	X
ma-317	271	2	]	]	X
ma-317	271	3	y.	y.	NOUN
ma-317	271	4	a.	a.	NOUN
ma-317	271	5	neretin	neretin	PROPN
ma-317	271	6	,	,	PUNCT
ma-317	271	7	the	the	DET
ma-317	271	8	double	double	ADJ
ma-317	271	9	gamma	gamma	NOUN
ma-317	271	10	function	function	NOUN
ma-317	271	11	and	and	CCONJ
ma-317	271	12	vladimir	vladimir	PROPN
ma-317	271	13	alekseevsky	alekseevsky	PROPN
ma-317	271	14	,	,	PUNCT
ma-317	271	15	arxiv:2402.07740	arxiv:2402.07740	PROPN
ma-317	271	16	.	.	PUNCT
ma-317	272	1	https://arxiv.org/pdf/	https://arxiv.org/pdf/	PROPN
ma-317	272	2	2402.07740.[11	2402.07740.[11	PROPN
ma-317	272	3	]	]	PUNCT
ma-317	272	4	f.	f.	PROPN
ma-317	272	5	qi	qi	PROPN
ma-317	272	6	and	and	CCONJ
ma-317	272	7	b	b	X
ma-317	272	8	-	-	PUNCT
ma-317	272	9	n.	n.	ADJ
ma-317	272	10	guo	guo	PROPN
ma-317	272	11	,	,	PUNCT
ma-317	272	12	some	some	DET
ma-317	272	13	properties	property	NOUN
ma-317	272	14	of	of	ADP
ma-317	272	15	extended	extended	ADJ
ma-317	272	16	remainder	remainder	NOUN
ma-317	272	17	of	of	ADP
ma-317	272	18	binet	binet	NOUN
ma-317	272	19	’s	’s	PART
ma-317	272	20	first	first	ADJ
ma-317	272	21	formula	formula	NOUN
ma-317	272	22	for	for	ADP
ma-317	272	23	logarithm	logarithm	NOUN
ma-317	272	24	of	of	ADP
ma-317	272	25	gammafunction	gammafunction	NOUN
ma-317	272	26	,	,	PUNCT
ma-317	272	27	math	math	NOUN
ma-317	272	28	.	.	PUNCT
ma-317	273	1	slovaca	slovaca	PROPN
ma-317	273	2	,	,	PUNCT
ma-317	273	3	60	60	NUM
ma-317	273	4	(	(	PUNCT
ma-317	273	5	2010	2010	NUM
ma-317	273	6	)	)	PUNCT
ma-317	273	7	,	,	PUNCT
ma-317	273	8	461	461	NUM
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ma-317	273	10	470	470	NUM
ma-317	273	11	.	.	PUNCT
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ma-317	274	13	some	some	DET
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ma-317	274	18	barnes	barnes	PROPN
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ma-317	274	34	.	.	PUNCT
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ma-317	276	8	,	,	PUNCT
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ma-317	276	10	university	university	PROPN
ma-317	276	11	press	press	PROPN
ma-317	276	12	,	,	PUNCT
ma-317	276	13	london	london	PROPN
ma-317	276	14	,	,	PUNCT
ma-317	276	15	1941	1941	NUM
ma-317	276	16	.	.	PUNCT
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ma-317	279	8	https://doi.org/10.30495/maca.2023.1987128.1067	https://doi.org/10.30495/maca.2023.1987128.1067	NUM
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ma-317	279	15	1	1	X
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ma-317	279	18	2	2	NUM
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ma-317	279	20	results	result	NOUN
ma-317	279	21	and	and	CCONJ
ma-317	279	22	discussion	discussion	NOUN
ma-317	279	23	references	reference	NOUN