id	sid	tid	token	lemma	pos
ma-11	1	1	2023	2023	NUM
ma-11	1	2	ada	ada	PROPN
ma-11	1	3	academica	academica	PROPN
ma-11	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-11	1	5	.	.	PUNCT
ma-11	2	1	j.	j.	PROPN
ma-11	2	2	math	math	PROPN
ma-11	2	3	.	.	PUNCT
ma-11	3	1	anal	anal	ADJ
ma-11	3	2	.	.	PUNCT
ma-11	4	1	3	3	NUM
ma-11	4	2	(	(	PUNCT
ma-11	4	3	2023	2023	NUM
ma-11	4	4	)	)	PUNCT
ma-11	4	5	12doi	12doi	NUM
ma-11	4	6	:	:	PUNCT
ma-11	4	7	10.28924	10.28924	NUM
ma-11	4	8	/	/	SYM
ma-11	4	9	ada	ada	PROPN
ma-11	4	10	/	/	SYM
ma-11	4	11	ma.3.12	ma.3.12	PROPN
ma-11	4	12	coordination	coordination	NOUN
ma-11	4	13	of	of	ADP
ma-11	4	14	classical	classical	ADJ
ma-11	4	15	and	and	CCONJ
ma-11	4	16	dynamic	dynamic	ADJ
ma-11	4	17	inequalities	inequality	NOUN
ma-11	4	18	complying	comply	VERB
ma-11	4	19	on	on	ADP
ma-11	4	20	time	time	NOUN
ma-11	4	21	scales	scale	NOUN
ma-11	4	22	muhammad	muhammad	PROPN
ma-11	4	23	jibril	jibril	PROPN
ma-11	4	24	shahab	shahab	PROPN
ma-11	4	25	sahir	sahir	PROPN
ma-11	4	26	department	department	PROPN
ma-11	4	27	of	of	ADP
ma-11	4	28	mathematics	mathematics	PROPN
ma-11	4	29	and	and	CCONJ
ma-11	4	30	statistics	statistic	NOUN
ma-11	4	31	,	,	PUNCT
ma-11	4	32	the	the	DET
ma-11	4	33	university	university	NOUN
ma-11	4	34	of	of	ADP
ma-11	4	35	lahore	lahore	PROPN
ma-11	4	36	,	,	PUNCT
ma-11	4	37	lahore	lahore	NOUN
ma-11	4	38	,	,	PUNCT
ma-11	4	39	pakistan	pakistan	PROPN
ma-11	4	40	correspondence	correspondence	NOUN
ma-11	4	41	:	:	PUNCT
ma-11	4	42	jibrielshahab@gmail.com	jibrielshahab@gmail.com	X
ma-11	5	1	abstract	abstract	PROPN
ma-11	5	2	.	.	PUNCT
ma-11	6	1	in	in	ADP
ma-11	6	2	this	this	DET
ma-11	6	3	research	research	NOUN
ma-11	6	4	article	article	NOUN
ma-11	6	5	,	,	PUNCT
ma-11	6	6	we	we	PRON
ma-11	6	7	present	present	VERB
ma-11	6	8	extensions	extension	NOUN
ma-11	6	9	of	of	ADP
ma-11	6	10	some	some	DET
ma-11	6	11	classical	classical	ADJ
ma-11	6	12	inequalities	inequality	NOUN
ma-11	6	13	such	such	ADJ
ma-11	6	14	asschweitzer	asschweitzer	NOUN
ma-11	6	15	,	,	PUNCT
ma-11	6	16	pólya	pólya	NOUN
ma-11	6	17	–	–	PUNCT
ma-11	6	18	szegö	szegö	ADJ
ma-11	6	19	,	,	PUNCT
ma-11	6	20	kantorovich	kantorovich	PROPN
ma-11	6	21	and	and	CCONJ
ma-11	6	22	greub	greub	PROPN
ma-11	6	23	–	–	PUNCT
ma-11	6	24	rheinboldt	rheinboldt	ADJ
ma-11	6	25	inequalities	inequality	NOUN
ma-11	6	26	of	of	ADP
ma-11	6	27	fractional	fractional	ADJ
ma-11	6	28	calculus	calculus	NOUN
ma-11	6	29	ontime	ontime	NOUN
ma-11	6	30	scales	scale	NOUN
ma-11	6	31	.	.	PUNCT
ma-11	7	1	to	to	PART
ma-11	7	2	investigate	investigate	VERB
ma-11	7	3	generalizations	generalization	NOUN
ma-11	7	4	of	of	ADP
ma-11	7	5	such	such	ADJ
ma-11	7	6	types	type	NOUN
ma-11	7	7	of	of	ADP
ma-11	7	8	classical	classical	ADJ
ma-11	7	9	inequalities	inequality	NOUN
ma-11	7	10	,	,	PUNCT
ma-11	7	11	we	we	PRON
ma-11	7	12	use	use	VERB
ma-11	7	13	the	the	DET
ma-11	7	14	timescales	timescale	NOUN
ma-11	7	15	riemann	riemann	PROPN
ma-11	7	16	–	–	PUNCT
ma-11	7	17	liouville	liouville	VERB
ma-11	7	18	type	type	NOUN
ma-11	7	19	fractional	fractional	ADJ
ma-11	7	20	integrals	integral	NOUN
ma-11	7	21	.	.	PUNCT
ma-11	8	1	we	we	PRON
ma-11	8	2	explore	explore	VERB
ma-11	8	3	dynamic	dynamic	ADJ
ma-11	8	4	inequalities	inequality	NOUN
ma-11	8	5	on	on	ADP
ma-11	8	6	delta	delta	PROPN
ma-11	8	7	calcu	calcu	NOUN
ma-11	8	8	-	-	PUNCT
ma-11	8	9	lus	lus	NOUN
ma-11	8	10	and	and	CCONJ
ma-11	8	11	their	their	PRON
ma-11	8	12	symmetric	symmetric	ADJ
ma-11	8	13	nabla	nabla	NOUN
ma-11	8	14	versions	version	NOUN
ma-11	8	15	.	.	PUNCT
ma-11	9	1	a	a	DET
ma-11	9	2	time	time	NOUN
ma-11	9	3	scale	scale	NOUN
ma-11	9	4	is	be	AUX
ma-11	9	5	an	an	DET
ma-11	9	6	arbitrary	arbitrary	ADJ
ma-11	9	7	nonempty	nonempty	NOUN
ma-11	9	8	closed	close	VERB
ma-11	9	9	subset	subset	NOUN
ma-11	9	10	of	of	ADP
ma-11	9	11	thereal	thereal	NOUN
ma-11	9	12	numbers	number	NOUN
ma-11	9	13	.	.	PUNCT
ma-11	10	1	the	the	DET
ma-11	10	2	theory	theory	NOUN
ma-11	10	3	of	of	ADP
ma-11	10	4	time	time	NOUN
ma-11	10	5	scales	scale	NOUN
ma-11	10	6	is	be	AUX
ma-11	10	7	applied	apply	VERB
ma-11	10	8	to	to	PART
ma-11	10	9	combine	combine	VERB
ma-11	10	10	results	result	NOUN
ma-11	10	11	in	in	ADP
ma-11	10	12	one	one	NUM
ma-11	10	13	comprehensive	comprehensive	ADJ
ma-11	10	14	form.the	form.the	DET
ma-11	10	15	calculus	calculus	NOUN
ma-11	10	16	of	of	ADP
ma-11	10	17	time	time	NOUN
ma-11	10	18	scales	scale	NOUN
ma-11	10	19	unifies	unify	VERB
ma-11	10	20	and	and	CCONJ
ma-11	10	21	extends	extend	VERB
ma-11	10	22	continuous	continuous	ADJ
ma-11	10	23	versions	version	NOUN
ma-11	10	24	and	and	CCONJ
ma-11	10	25	their	their	PRON
ma-11	10	26	discrete	discrete	ADJ
ma-11	10	27	and	and	CCONJ
ma-11	10	28	quantumanalogues	quantumanalogue	NOUN
ma-11	10	29	.	.	PUNCT
ma-11	11	1	by	by	ADP
ma-11	11	2	using	use	VERB
ma-11	11	3	the	the	DET
ma-11	11	4	calculus	calculus	NOUN
ma-11	11	5	of	of	ADP
ma-11	11	6	time	time	NOUN
ma-11	11	7	scales	scale	NOUN
ma-11	11	8	,	,	PUNCT
ma-11	11	9	results	result	NOUN
ma-11	11	10	are	be	AUX
ma-11	11	11	presented	present	VERB
ma-11	11	12	in	in	ADP
ma-11	11	13	more	more	ADJ
ma-11	11	14	general	general	ADJ
ma-11	11	15	form	form	NOUN
ma-11	11	16	.	.	PUNCT
ma-11	12	1	thishybrid	thishybrid	NOUN
ma-11	12	2	theory	theory	NOUN
ma-11	12	3	is	be	AUX
ma-11	12	4	also	also	ADV
ma-11	12	5	widely	widely	ADV
ma-11	12	6	applied	apply	VERB
ma-11	12	7	on	on	ADP
ma-11	12	8	dynamic	dynamic	ADJ
ma-11	12	9	inequalities	inequality	NOUN
ma-11	12	10	.	.	PUNCT
ma-11	13	1	1	1	X
ma-11	13	2	.	.	X
ma-11	13	3	introduction	introduction	NOUN
ma-11	13	4	the	the	DET
ma-11	13	5	calculus	calculus	NOUN
ma-11	13	6	of	of	ADP
ma-11	13	7	time	time	NOUN
ma-11	13	8	scales	scale	NOUN
ma-11	13	9	was	be	AUX
ma-11	13	10	initiated	initiate	VERB
ma-11	13	11	by	by	ADP
ma-11	13	12	stefan	stefan	PROPN
ma-11	13	13	hilger	hilger	PROPN
ma-11	14	1	[	[	X
ma-11	14	2	11	11	NUM
ma-11	14	3	]	]	PUNCT
ma-11	14	4	.	.	PUNCT
ma-11	15	1	the	the	DET
ma-11	15	2	three	three	NUM
ma-11	15	3	most	most	ADV
ma-11	15	4	popular	popular	ADJ
ma-11	15	5	examplesof	examplesof	NOUN
ma-11	15	6	calculus	calculus	NOUN
ma-11	15	7	on	on	ADP
ma-11	15	8	time	time	NOUN
ma-11	15	9	scales	scale	NOUN
ma-11	15	10	are	be	AUX
ma-11	15	11	differential	differential	ADJ
ma-11	15	12	calculus	calculus	NOUN
ma-11	15	13	,	,	PUNCT
ma-11	15	14	difference	difference	NOUN
ma-11	15	15	calculus	calculus	NOUN
ma-11	15	16	,	,	PUNCT
ma-11	15	17	and	and	CCONJ
ma-11	15	18	quantum	quantum	NOUN
ma-11	15	19	calculus	calculus	NOUN
ma-11	15	20	,	,	PUNCT
ma-11	15	21	i.e.	i.e.	X
ma-11	15	22	,when	,when	PUNCT
ma-11	15	23	t	t	NOUN
ma-11	15	24	=	=	SYM
ma-11	15	25	r	r	PROPN
ma-11	15	26	,	,	PUNCT
ma-11	15	27	t	t	NOUN
ma-11	15	28	=	=	SYM
ma-11	15	29	n	n	PROPN
ma-11	15	30	and	and	CCONJ
ma-11	15	31	t	t	NOUN
ma-11	15	32	=	=	PUNCT
ma-11	15	33	qn0	qn0	PROPN
ma-11	15	34	=	=	X
ma-11	15	35	{	{	PUNCT
ma-11	15	36	qt	qt	NOUN
ma-11	15	37	:	:	PUNCT
ma-11	15	38	t	t	PROPN
ma-11	15	39	∈	∈	PROPN
ma-11	15	40	n0	n0	PROPN
ma-11	15	41	}	}	PUNCT
ma-11	15	42	where	where	SCONJ
ma-11	15	43	q	q	X
ma-11	15	44	>	>	X
ma-11	15	45	1	1	X
ma-11	15	46	.	.	PUNCT
ma-11	16	1	the	the	DET
ma-11	16	2	time	time	NOUN
ma-11	16	3	scales	scale	VERB
ma-11	16	4	calculus	calculus	NOUN
ma-11	16	5	is	be	AUX
ma-11	16	6	studiedas	studieda	VERB
ma-11	16	7	delta	delta	NOUN
ma-11	16	8	calculus	calculus	NOUN
ma-11	16	9	,	,	PUNCT
ma-11	16	10	nabla	nabla	NOUN
ma-11	16	11	calculus	calculus	NOUN
ma-11	16	12	and	and	CCONJ
ma-11	16	13	diamond	diamond	NOUN
ma-11	16	14	-	-	PUNCT
ma-11	16	15	α	α	NOUN
ma-11	16	16	calculus	calculus	NOUN
ma-11	16	17	.	.	PUNCT
ma-11	17	1	during	during	ADP
ma-11	17	2	the	the	DET
ma-11	17	3	last	last	ADJ
ma-11	17	4	two	two	NUM
ma-11	17	5	decades	decade	NOUN
ma-11	17	6	,	,	PUNCT
ma-11	17	7	manyresearchers	manyresearcher	NOUN
ma-11	17	8	investigated	investigate	VERB
ma-11	17	9	several	several	ADJ
ma-11	17	10	dynamic	dynamic	ADJ
ma-11	17	11	inequalities	inequality	NOUN
ma-11	18	1	[	[	X
ma-11	18	2	1–4	1–4	NUM
ma-11	18	3	,	,	PUNCT
ma-11	18	4	7	7	NUM
ma-11	18	5	,	,	PUNCT
ma-11	18	6	16–18	16–18	NUM
ma-11	18	7	]	]	PUNCT
ma-11	18	8	.	.	PUNCT
ma-11	19	1	the	the	DET
ma-11	19	2	basic	basic	ADJ
ma-11	19	3	work	work	NOUN
ma-11	19	4	on	on	ADP
ma-11	19	5	dynamicinequalities	dynamicinequalitie	NOUN
ma-11	19	6	is	be	AUX
ma-11	19	7	done	do	VERB
ma-11	19	8	by	by	ADP
ma-11	19	9	ravi	ravi	PROPN
ma-11	19	10	agarwal	agarwal	PROPN
ma-11	19	11	,	,	PUNCT
ma-11	19	12	george	george	PROPN
ma-11	19	13	anastassiou	anastassiou	PROPN
ma-11	19	14	,	,	PUNCT
ma-11	19	15	martin	martin	PROPN
ma-11	19	16	bohner	bohner	PROPN
ma-11	19	17	,	,	PUNCT
ma-11	19	18	allan	allan	PROPN
ma-11	19	19	peterson	peterson	PROPN
ma-11	19	20	,	,	PUNCT
ma-11	19	21	donalo’regan	donalo’regan	NOUN
ma-11	19	22	,	,	PUNCT
ma-11	19	23	samir	samir	PROPN
ma-11	19	24	saker	saker	PROPN
ma-11	19	25	and	and	CCONJ
ma-11	19	26	many	many	ADJ
ma-11	19	27	other	other	ADJ
ma-11	19	28	authors.there	authors.there	PROPN
ma-11	19	29	have	have	AUX
ma-11	19	30	been	be	AUX
ma-11	19	31	recent	recent	ADJ
ma-11	19	32	achievements	achievement	NOUN
ma-11	19	33	of	of	ADP
ma-11	19	34	the	the	DET
ma-11	19	35	theory	theory	NOUN
ma-11	19	36	and	and	CCONJ
ma-11	19	37	applications	application	NOUN
ma-11	19	38	of	of	ADP
ma-11	19	39	dynamic	dynamic	ADJ
ma-11	19	40	inequalities	inequality	NOUN
ma-11	19	41	ontime	ontime	NOUN
ma-11	19	42	scales	scale	NOUN
ma-11	19	43	.	.	PUNCT
ma-11	20	1	from	from	ADP
ma-11	20	2	the	the	DET
ma-11	20	3	theoretical	theoretical	ADJ
ma-11	20	4	point	point	NOUN
ma-11	20	5	of	of	ADP
ma-11	20	6	view	view	NOUN
ma-11	20	7	,	,	PUNCT
ma-11	20	8	the	the	DET
ma-11	20	9	study	study	NOUN
ma-11	20	10	provides	provide	VERB
ma-11	20	11	a	a	DET
ma-11	20	12	harmonious	harmonious	ADJ
ma-11	20	13	reconciliation	reconciliation	NOUN
ma-11	20	14	andextension	andextension	NOUN
ma-11	20	15	of	of	ADP
ma-11	20	16	commonly	commonly	ADV
ma-11	20	17	known	know	VERB
ma-11	20	18	differential	differential	NOUN
ma-11	20	19	,	,	PUNCT
ma-11	20	20	difference	difference	NOUN
ma-11	20	21	and	and	CCONJ
ma-11	20	22	quantum	quantum	NOUN
ma-11	20	23	equations	equation	NOUN
ma-11	20	24	.	.	PUNCT
ma-11	21	1	moreover	moreover	ADV
ma-11	21	2	,	,	PUNCT
ma-11	21	3	it	it	PRON
ma-11	21	4	is	be	AUX
ma-11	21	5	animportant	animportant	ADJ
ma-11	21	6	tool	tool	NOUN
ma-11	21	7	in	in	ADP
ma-11	21	8	many	many	ADJ
ma-11	21	9	computational	computational	ADJ
ma-11	21	10	,	,	PUNCT
ma-11	21	11	biological	biological	ADJ
ma-11	21	12	,	,	PUNCT
ma-11	21	13	economical	economical	ADJ
ma-11	21	14	and	and	CCONJ
ma-11	21	15	numerical	numerical	ADJ
ma-11	21	16	applications.in	applications.in	PRON
ma-11	21	17	this	this	DET
ma-11	21	18	paper	paper	NOUN
ma-11	21	19	,	,	PUNCT
ma-11	21	20	it	it	PRON
ma-11	21	21	is	be	AUX
ma-11	21	22	assumed	assume	VERB
ma-11	21	23	that	that	SCONJ
ma-11	21	24	all	all	DET
ma-11	21	25	considerable	considerable	ADJ
ma-11	21	26	integrals	integral	NOUN
ma-11	21	27	exist	exist	VERB
ma-11	21	28	and	and	CCONJ
ma-11	21	29	are	be	AUX
ma-11	21	30	finite	finite	ADJ
ma-11	21	31	and	and	CCONJ
ma-11	21	32	t	t	PROPN
ma-11	21	33	is	be	AUX
ma-11	21	34	a	a	DET
ma-11	21	35	timescale	timescale	NOUN
ma-11	21	36	,	,	PUNCT
ma-11	21	37	a	a	DET
ma-11	21	38	,	,	PUNCT
ma-11	21	39	b	b	PROPN
ma-11	21	40	∈	∈	PROPN
ma-11	21	41	t	t	NOUN
ma-11	21	42	with	with	ADP
ma-11	21	43	a	a	DET
ma-11	21	44	<	<	X
ma-11	21	45	b	b	NOUN
ma-11	21	46	and	and	CCONJ
ma-11	21	47	an	an	DET
ma-11	21	48	interval	interval	NOUN
ma-11	21	49	[	[	X
ma-11	21	50	a	a	X
ma-11	21	51	,	,	PUNCT
ma-11	21	52	b]t	b]t	NOUN
ma-11	21	53	means	mean	VERB
ma-11	21	54	the	the	DET
ma-11	21	55	intersection	intersection	NOUN
ma-11	21	56	of	of	ADP
ma-11	21	57	a	a	DET
ma-11	21	58	real	real	ADJ
ma-11	21	59	interval	interval	NOUN
ma-11	21	60	with	with	ADP
ma-11	21	61	thegiven	thegiven	ADJ
ma-11	21	62	time	time	NOUN
ma-11	21	63	scale	scale	NOUN
ma-11	21	64	.	.	PUNCT
ma-11	22	1	received	receive	VERB
ma-11	22	2	:	:	PUNCT
ma-11	22	3	22	22	NUM
ma-11	22	4	aug	aug	PROPN
ma-11	22	5	2021	2021	NUM
ma-11	22	6	.	.	PUNCT
ma-11	23	1	key	key	ADJ
ma-11	23	2	words	word	NOUN
ma-11	23	3	and	and	CCONJ
ma-11	23	4	phrases	phrase	NOUN
ma-11	23	5	.	.	PUNCT
ma-11	24	1	time	time	NOUN
ma-11	24	2	scales	scale	NOUN
ma-11	24	3	;	;	PUNCT
ma-11	24	4	fractional	fractional	ADJ
ma-11	24	5	riemann	riemann	PROPN
ma-11	24	6	–	–	PUNCT
ma-11	24	7	liouville	liouville	VERB
ma-11	24	8	integral	integral	ADJ
ma-11	24	9	;	;	PUNCT
ma-11	24	10	schweitzer	schweitzer	NOUN
ma-11	24	11	,	,	PUNCT
ma-11	24	12	pólya	pólya	NOUN
ma-11	24	13	–	–	PUNCT
ma-11	24	14	szegö	szegö	ADJ
ma-11	24	15	,	,	PUNCT
ma-11	24	16	kantorovichand	kantorovichand	ADJ
ma-11	24	17	greub	greub	PROPN
ma-11	24	18	–	–	PUNCT
ma-11	24	19	rheinboldt	rheinboldt	ADJ
ma-11	24	20	inequalities	inequality	NOUN
ma-11	24	21	.	.	PUNCT
ma-11	25	1	1	1	NUM
ma-11	25	2	https://adac.ee	https://adac.ee	PROPN
ma-11	25	3	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	NUM
ma-11	25	4	eur	eur	NOUN
ma-11	25	5	.	.	PUNCT
ma-11	26	1	j.	j.	PROPN
ma-11	26	2	math	math	PROPN
ma-11	26	3	.	.	PUNCT
ma-11	27	1	anal	anal	PROPN
ma-11	27	2	.	.	PUNCT
ma-11	28	1	10.28924	10.28924	NUM
ma-11	28	2	/	/	SYM
ma-11	28	3	ada	ada	PROPN
ma-11	28	4	/	/	SYM
ma-11	28	5	ma.3.12	ma.3.12	PROPN
ma-11	28	6	22	22	NUM
ma-11	28	7	.	.	PUNCT
ma-11	29	1	preliminaries	preliminary	NOUN
ma-11	29	2	we	we	PRON
ma-11	29	3	need	need	VERB
ma-11	29	4	here	here	ADV
ma-11	29	5	basic	basic	ADJ
ma-11	29	6	concepts	concept	NOUN
ma-11	29	7	of	of	ADP
ma-11	29	8	delta	delta	PROPN
ma-11	29	9	calculus	calculus	NOUN
ma-11	29	10	.	.	PUNCT
ma-11	30	1	the	the	DET
ma-11	30	2	results	result	NOUN
ma-11	30	3	of	of	ADP
ma-11	30	4	delta	delta	NOUN
ma-11	30	5	calculus	calculus	NOUN
ma-11	30	6	are	be	AUX
ma-11	30	7	adopted	adopt	VERB
ma-11	30	8	frommonographs	frommonograph	NOUN
ma-11	30	9	[	[	X
ma-11	30	10	7	7	NUM
ma-11	30	11	,	,	PUNCT
ma-11	30	12	8].for	8].for	ADP
ma-11	30	13	t	t	PROPN
ma-11	30	14	∈	∈	PROPN
ma-11	30	15	t	t	PROPN
ma-11	30	16	,	,	PUNCT
ma-11	30	17	the	the	DET
ma-11	30	18	forward	forward	ADJ
ma-11	30	19	jump	jump	NOUN
ma-11	30	20	operator	operator	NOUN
ma-11	30	21	σ	σ	NOUN
ma-11	30	22	:	:	PUNCT
ma-11	30	23	t→	t→	DET
ma-11	30	24	t	t	PROPN
ma-11	30	25	is	be	AUX
ma-11	30	26	defined	define	VERB
ma-11	30	27	by	by	ADP
ma-11	30	28	σ(t	σ(t	PROPN
ma-11	30	29	)	)	PUNCT
ma-11	30	30	:	:	PUNCT
ma-11	31	1	=	=	PUNCT
ma-11	31	2	inf{s	inf{s	PROPN
ma-11	31	3	∈	∈	PROPN
ma-11	31	4	t	t	NOUN
ma-11	31	5	:	:	PUNCT
ma-11	31	6	s	s	X
ma-11	31	7	>	>	X
ma-11	31	8	t	t	PROPN
ma-11	31	9	}	}	PUNCT
ma-11	31	10	.	.	PUNCT
ma-11	32	1	the	the	DET
ma-11	32	2	mapping	mapping	NOUN
ma-11	32	3	µ	µ	X
ma-11	32	4	:	:	PUNCT
ma-11	32	5	t	t	NOUN
ma-11	32	6	→	→	SYM
ma-11	32	7	r+	r+	X
ma-11	32	8	0	0	NUM
ma-11	32	9	=	=	PUNCT
ma-11	33	1	[	[	X
ma-11	33	2	0,+∞	0,+∞	NUM
ma-11	33	3	)	)	PUNCT
ma-11	33	4	such	such	ADJ
ma-11	33	5	that	that	SCONJ
ma-11	33	6	µ(t	µ(t	ADJ
ma-11	33	7	)	)	PUNCT
ma-11	33	8	:	:	PUNCT
ma-11	34	1	=	=	SYM
ma-11	34	2	σ(t	σ(t	PROPN
ma-11	34	3	)	)	PUNCT
ma-11	35	1	−	−	PROPN
ma-11	35	2	t	t	PROPN
ma-11	35	3	is	be	AUX
ma-11	35	4	called	call	VERB
ma-11	35	5	the	the	DET
ma-11	35	6	forward	forward	ADJ
ma-11	35	7	graininessfunction	graininessfunction	NOUN
ma-11	35	8	.	.	PUNCT
ma-11	36	1	the	the	DET
ma-11	36	2	backward	backward	ADJ
ma-11	36	3	jump	jump	NOUN
ma-11	36	4	operator	operator	NOUN
ma-11	36	5	ρ	ρ	NOUN
ma-11	36	6	:	:	PUNCT
ma-11	36	7	t→	t→	DET
ma-11	36	8	t	t	PROPN
ma-11	36	9	is	be	AUX
ma-11	36	10	defined	define	VERB
ma-11	36	11	by	by	ADP
ma-11	36	12	ρ(t	ρ(t	NUM
ma-11	36	13	)	)	PUNCT
ma-11	36	14	:	:	PUNCT
ma-11	37	1	=	=	PUNCT
ma-11	37	2	sup{s	sup{s	PROPN
ma-11	37	3	∈	∈	PROPN
ma-11	37	4	t	t	NOUN
ma-11	37	5	:	:	PUNCT
ma-11	37	6	s	s	X
ma-11	37	7	<	<	X
ma-11	37	8	t	t	PROPN
ma-11	37	9	}	}	PUNCT
ma-11	37	10	.	.	PUNCT
ma-11	38	1	the	the	DET
ma-11	38	2	mapping	mapping	NOUN
ma-11	38	3	ν	ν	NOUN
ma-11	38	4	:	:	PUNCT
ma-11	38	5	t→	t→	X
ma-11	38	6	r+	r+	VERB
ma-11	38	7	0	0	PUNCT
ma-11	39	1	=	=	PUNCT
ma-11	40	1	[	[	X
ma-11	40	2	0,+∞	0,+∞	NUM
ma-11	40	3	)	)	PUNCT
ma-11	40	4	such	such	ADJ
ma-11	40	5	that	that	SCONJ
ma-11	40	6	ν(t	ν(t	NOUN
ma-11	40	7	)	)	PUNCT
ma-11	40	8	:	:	PUNCT
ma-11	40	9	=	=	PUNCT
ma-11	40	10	t	t	PROPN
ma-11	40	11	−	−	PROPN
ma-11	40	12	ρ(t	ρ(t	NUM
ma-11	40	13	)	)	PUNCT
ma-11	40	14	is	be	AUX
ma-11	40	15	called	call	VERB
ma-11	40	16	the	the	DET
ma-11	40	17	backward	backward	ADJ
ma-11	40	18	graininessfunction	graininessfunction	NOUN
ma-11	40	19	.	.	PUNCT
ma-11	41	1	if	if	SCONJ
ma-11	41	2	σ(t	σ(t	PROPN
ma-11	41	3	)	)	PUNCT
ma-11	41	4	>	>	X
ma-11	42	1	t	t	PROPN
ma-11	42	2	,	,	PUNCT
ma-11	42	3	we	we	PRON
ma-11	42	4	say	say	VERB
ma-11	42	5	that	that	SCONJ
ma-11	42	6	t	t	PROPN
ma-11	42	7	is	be	AUX
ma-11	42	8	right	right	ADV
ma-11	42	9	-	-	PUNCT
ma-11	42	10	scattered	scatter	VERB
ma-11	42	11	,	,	PUNCT
ma-11	42	12	while	while	SCONJ
ma-11	42	13	if	if	SCONJ
ma-11	42	14	ρ(t	ρ(t	NUM
ma-11	42	15	)	)	PUNCT
ma-11	42	16	<	<	X
ma-11	42	17	t	t	PROPN
ma-11	42	18	,	,	PUNCT
ma-11	42	19	we	we	PRON
ma-11	42	20	say	say	VERB
ma-11	42	21	that	that	SCONJ
ma-11	42	22	t	t	PROPN
ma-11	42	23	is	be	AUX
ma-11	42	24	left	leave	VERB
ma-11	42	25	-	-	PUNCT
ma-11	42	26	scattered	scatter	VERB
ma-11	42	27	.	.	PUNCT
ma-11	43	1	also	also	ADV
ma-11	43	2	,	,	PUNCT
ma-11	43	3	if	if	SCONJ
ma-11	43	4	t	t	PROPN
ma-11	43	5	<	<	X
ma-11	43	6	supt	supt	PROPN
ma-11	43	7	and	and	CCONJ
ma-11	43	8	σ(t	σ(t	PROPN
ma-11	43	9	)	)	PUNCT
ma-11	43	10	=	=	SYM
ma-11	43	11	t	t	PROPN
ma-11	43	12	,	,	PUNCT
ma-11	43	13	then	then	ADV
ma-11	43	14	t	t	PROPN
ma-11	43	15	is	be	AUX
ma-11	43	16	called	call	VERB
ma-11	43	17	right	right	ADV
ma-11	43	18	-	-	PUNCT
ma-11	43	19	dense	dense	ADJ
ma-11	43	20	,	,	PUNCT
ma-11	43	21	and	and	CCONJ
ma-11	43	22	if	if	SCONJ
ma-11	43	23	t	t	PROPN
ma-11	43	24	>	>	X
ma-11	43	25	inf	inf	PROPN
ma-11	43	26	t	t	PROPN
ma-11	43	27	and	and	CCONJ
ma-11	43	28	ρ(t	ρ(t	NUM
ma-11	43	29	)	)	PUNCT
ma-11	44	1	=	=	SYM
ma-11	44	2	t	t	PROPN
ma-11	44	3	,	,	PUNCT
ma-11	44	4	then	then	ADV
ma-11	44	5	t	t	PROPN
ma-11	44	6	is	be	AUX
ma-11	44	7	called	call	VERB
ma-11	44	8	left	left	ADJ
ma-11	44	9	-	-	PUNCT
ma-11	44	10	dense	dense	ADJ
ma-11	44	11	.	.	PUNCT
ma-11	45	1	if	if	SCONJ
ma-11	45	2	t	t	PROPN
ma-11	45	3	has	have	VERB
ma-11	45	4	a	a	DET
ma-11	45	5	left	left	ADV
ma-11	45	6	-	-	PUNCT
ma-11	45	7	scattered	scatter	VERB
ma-11	45	8	maximum	maximum	NOUN
ma-11	45	9	m	m	NOUN
ma-11	45	10	,	,	PUNCT
ma-11	45	11	then	then	ADV
ma-11	45	12	tk	tk	PROPN
ma-11	45	13	=	=	PROPN
ma-11	45	14	t	t	PROPN
ma-11	45	15	−	−	PROPN
ma-11	46	1	{	{	PUNCT
ma-11	47	1	m},otherwise	m},otherwise	ADJ
ma-11	47	2	tk	tk	NOUN
ma-11	47	3	=	=	PRON
ma-11	47	4	t.for	t.for	ADP
ma-11	47	5	a	a	DET
ma-11	47	6	function	function	NOUN
ma-11	48	1	f	f	NOUN
ma-11	48	2	:	:	PUNCT
ma-11	48	3	t→	t→	SYM
ma-11	48	4	r	r	X
ma-11	48	5	,	,	PUNCT
ma-11	48	6	the	the	DET
ma-11	48	7	delta	delta	NOUN
ma-11	48	8	derivative	derivative	ADJ
ma-11	48	9	f	f	PROPN
ma-11	48	10	∆	∆	PROPN
ma-11	48	11	is	be	AUX
ma-11	48	12	defined	define	VERB
ma-11	48	13	as	as	SCONJ
ma-11	48	14	follows	follow	VERB
ma-11	48	15	:	:	PUNCT
ma-11	48	16	let	let	VERB
ma-11	48	17	t	t	PROPN
ma-11	48	18	∈	∈	PROPN
ma-11	48	19	tk	tk	PROPN
ma-11	48	20	.	.	PUNCT
ma-11	49	1	if	if	SCONJ
ma-11	49	2	there	there	PRON
ma-11	49	3	exists	exist	VERB
ma-11	49	4	f	f	PROPN
ma-11	49	5	∆(t	∆(t	PROPN
ma-11	49	6	)	)	PUNCT
ma-11	49	7	∈	∈	PROPN
ma-11	49	8	r	r	NOUN
ma-11	49	9	such	such	ADJ
ma-11	49	10	that	that	PRON
ma-11	49	11	for	for	ADP
ma-11	49	12	all	all	DET
ma-11	49	13	ε	ε	PROPN
ma-11	49	14	>	>	X
ma-11	49	15	0	0	PROPN
ma-11	49	16	,	,	PUNCT
ma-11	49	17	there	there	PRON
ma-11	49	18	is	be	VERB
ma-11	49	19	a	a	DET
ma-11	49	20	neighborhood	neighborhood	NOUN
ma-11	49	21	u	u	NOUN
ma-11	49	22	of	of	ADP
ma-11	49	23	t	t	PROPN
ma-11	49	24	,	,	PUNCT
ma-11	49	25	such	such	ADJ
ma-11	49	26	that	that	SCONJ
ma-11	49	27	|f	|f	PROPN
ma-11	49	28	(	(	PUNCT
ma-11	49	29	σ(t))−	σ(t))−	ADP
ma-11	49	30	f	f	X
ma-11	49	31	(	(	PUNCT
ma-11	49	32	s)−	s)−	PROPN
ma-11	49	33	f	f	PROPN
ma-11	49	34	∆(t)(σ(t)−	∆(t)(σ(t)−	PROPN
ma-11	49	35	s)|	s)|	PROPN
ma-11	49	36	≤	≤	PROPN
ma-11	49	37	ε|σ(t)−	ε|σ(t)−	PROPN
ma-11	50	1	s|,for	s|,for	ADP
ma-11	50	2	all	all	DET
ma-11	50	3	s	s	PART
ma-11	50	4	∈	∈	PROPN
ma-11	50	5	u	u	NOUN
ma-11	50	6	,	,	PUNCT
ma-11	50	7	then	then	ADV
ma-11	50	8	f	f	PROPN
ma-11	50	9	is	be	AUX
ma-11	50	10	said	say	VERB
ma-11	50	11	to	to	PART
ma-11	50	12	be	be	AUX
ma-11	50	13	delta	delta	NOUN
ma-11	50	14	differentiable	differentiable	ADJ
ma-11	50	15	at	at	ADP
ma-11	50	16	t	t	PROPN
ma-11	50	17	,	,	PUNCT
ma-11	50	18	and	and	CCONJ
ma-11	50	19	f	f	PROPN
ma-11	50	20	∆(t	∆(t	PROPN
ma-11	50	21	)	)	PUNCT
ma-11	50	22	is	be	AUX
ma-11	50	23	called	call	VERB
ma-11	50	24	the	the	DET
ma-11	50	25	delta	delta	NOUN
ma-11	50	26	derivativeof	derivativeof	NOUN
ma-11	50	27	f	f	PROPN
ma-11	50	28	at	at	ADP
ma-11	50	29	t	t	PROPN
ma-11	50	30	.a	.a	PROPN
ma-11	51	1	function	function	PROPN
ma-11	51	2	f	f	NOUN
ma-11	51	3	:	:	PUNCT
ma-11	51	4	t→	t→	PUNCT
ma-11	51	5	r	r	NOUN
ma-11	51	6	is	be	AUX
ma-11	51	7	said	say	VERB
ma-11	51	8	to	to	PART
ma-11	51	9	be	be	AUX
ma-11	51	10	right	right	ADJ
ma-11	51	11	-	-	PUNCT
ma-11	51	12	dense	dense	ADJ
ma-11	51	13	continuous	continuous	ADJ
ma-11	51	14	(	(	PUNCT
ma-11	51	15	rd	rd	NOUN
ma-11	51	16	-	-	NOUN
ma-11	51	17	continuous	continuous	ADJ
ma-11	51	18	)	)	PUNCT
ma-11	51	19	,	,	PUNCT
ma-11	51	20	if	if	SCONJ
ma-11	51	21	it	it	PRON
ma-11	51	22	is	be	AUX
ma-11	51	23	continuous	continuous	ADJ
ma-11	51	24	ateach	ateach	NOUN
ma-11	51	25	right	right	ADJ
ma-11	51	26	-	-	PUNCT
ma-11	51	27	dense	dense	ADJ
ma-11	51	28	point	point	NOUN
ma-11	51	29	and	and	CCONJ
ma-11	51	30	there	there	PRON
ma-11	51	31	exists	exist	VERB
ma-11	51	32	a	a	DET
ma-11	51	33	finite	finite	ADJ
ma-11	51	34	left	left	ADV
ma-11	51	35	-	-	PUNCT
ma-11	51	36	sided	sided	ADJ
ma-11	51	37	limit	limit	NOUN
ma-11	51	38	at	at	ADP
ma-11	51	39	every	every	DET
ma-11	51	40	left	left	ADJ
ma-11	51	41	-	-	PUNCT
ma-11	51	42	dense	dense	ADJ
ma-11	51	43	point	point	NOUN
ma-11	51	44	.	.	PUNCT
ma-11	52	1	the	the	DET
ma-11	52	2	setof	setof	NOUN
ma-11	52	3	all	all	DET
ma-11	52	4	rd	rd	NOUN
ma-11	52	5	-	-	ADJ
ma-11	52	6	continuous	continuous	ADJ
ma-11	52	7	functions	function	NOUN
ma-11	52	8	is	be	AUX
ma-11	52	9	denoted	denote	VERB
ma-11	52	10	by	by	ADP
ma-11	52	11	crd(t	crd(t	PROPN
ma-11	52	12	,	,	PUNCT
ma-11	52	13	r).the	r).the	DET
ma-11	52	14	next	next	ADJ
ma-11	52	15	definition	definition	NOUN
ma-11	52	16	is	be	AUX
ma-11	52	17	given	give	VERB
ma-11	52	18	in	in	ADP
ma-11	52	19	[	[	X
ma-11	52	20	7	7	NUM
ma-11	52	21	,	,	PUNCT
ma-11	52	22	8	8	NUM
ma-11	52	23	]	]	PUNCT
ma-11	52	24	.	.	PUNCT
ma-11	53	1	definition	definition	NOUN
ma-11	53	2	2.1	2.1	NUM
ma-11	53	3	.	.	PUNCT
ma-11	54	1	a	a	DET
ma-11	54	2	function	function	NOUN
ma-11	54	3	f	f	NOUN
ma-11	54	4	:	:	PUNCT
ma-11	54	5	t	t	PROPN
ma-11	54	6	→	→	SYM
ma-11	54	7	r	r	NOUN
ma-11	54	8	is	be	AUX
ma-11	54	9	called	call	VERB
ma-11	54	10	a	a	DET
ma-11	54	11	delta	delta	NOUN
ma-11	54	12	antiderivative	antiderivative	ADJ
ma-11	54	13	of	of	ADP
ma-11	54	14	f	f	PROPN
ma-11	54	15	:	:	PUNCT
ma-11	54	16	t	t	PROPN
ma-11	54	17	→	→	SYM
ma-11	54	18	r	r	NOUN
ma-11	54	19	,	,	PUNCT
ma-11	54	20	provided	provide	VERB
ma-11	54	21	that	that	PRON
ma-11	54	22	f∆(t	f∆(t	NOUN
ma-11	54	23	)	)	PUNCT
ma-11	54	24	=	=	SYM
ma-11	54	25	f	f	PROPN
ma-11	54	26	(	(	PUNCT
ma-11	54	27	t	t	NOUN
ma-11	54	28	)	)	PUNCT
ma-11	54	29	holds	hold	VERB
ma-11	54	30	for	for	ADP
ma-11	54	31	all	all	DET
ma-11	54	32	t	t	NOUN
ma-11	54	33	∈	∈	PROPN
ma-11	54	34	tk	tk	PROPN
ma-11	54	35	.	.	PUNCT
ma-11	55	1	then	then	ADV
ma-11	55	2	the	the	DET
ma-11	55	3	delta	delta	NOUN
ma-11	55	4	integral	integral	ADJ
ma-11	55	5	of	of	ADP
ma-11	55	6	f	f	PROPN
ma-11	55	7	is	be	AUX
ma-11	55	8	defined	define	VERB
ma-11	55	9	by∫	by∫	PROPN
ma-11	55	10	b	b	PROPN
ma-11	55	11	a	a	DET
ma-11	55	12	f	f	PROPN
ma-11	55	13	(	(	PUNCT
ma-11	55	14	t)∆t	t)∆t	PROPN
ma-11	55	15	=	=	SYM
ma-11	55	16	f	f	PROPN
ma-11	55	17	(	(	PUNCT
ma-11	55	18	b)−	b)−	PROPN
ma-11	55	19	f	f	X
ma-11	55	20	(	(	PUNCT
ma-11	55	21	a	a	NOUN
ma-11	55	22	)	)	PUNCT
ma-11	55	23	.	.	PUNCT
ma-11	56	1	the	the	DET
ma-11	56	2	following	follow	VERB
ma-11	56	3	results	result	NOUN
ma-11	56	4	of	of	ADP
ma-11	56	5	nabla	nabla	NOUN
ma-11	56	6	calculus	calculus	NOUN
ma-11	56	7	are	be	AUX
ma-11	56	8	taken	take	VERB
ma-11	56	9	from	from	ADP
ma-11	56	10	[	[	PUNCT
ma-11	56	11	6–8].if	6–8].if	PROPN
ma-11	56	12	t	t	PROPN
ma-11	56	13	has	have	VERB
ma-11	56	14	a	a	DET
ma-11	56	15	right	right	ADV
ma-11	56	16	-	-	PUNCT
ma-11	56	17	scattered	scatter	VERB
ma-11	56	18	minimum	minimum	NOUN
ma-11	56	19	m	m	NOUN
ma-11	56	20	,	,	PUNCT
ma-11	56	21	then	then	ADV
ma-11	56	22	tk	tk	PROPN
ma-11	56	23	=	=	PROPN
ma-11	56	24	t	t	PROPN
ma-11	56	25	−	−	PROPN
ma-11	56	26	{	{	PUNCT
ma-11	56	27	m	m	PROPN
ma-11	56	28	}	}	PUNCT
ma-11	56	29	,	,	PUNCT
ma-11	56	30	otherwise	otherwise	ADV
ma-11	56	31	tk	tk	PROPN
ma-11	56	32	=	=	PROPN
ma-11	56	33	t.	t.	NOUN
ma-11	56	34	a	a	DET
ma-11	56	35	function	function	NOUN
ma-11	57	1	f	f	NOUN
ma-11	57	2	:	:	PUNCT
ma-11	57	3	tk	tk	PROPN
ma-11	57	4	→	→	SYM
ma-11	57	5	r	r	NOUN
ma-11	57	6	is	be	AUX
ma-11	57	7	called	call	VERB
ma-11	57	8	nabla	nabla	PROPN
ma-11	57	9	differentiable	differentiable	NOUN
ma-11	57	10	at	at	ADP
ma-11	57	11	t	t	PROPN
ma-11	57	12	∈	∈	PROPN
ma-11	57	13	tk	tk	PROPN
ma-11	57	14	,	,	PUNCT
ma-11	57	15	with	with	ADP
ma-11	57	16	nabla	nabla	PROPN
ma-11	57	17	derivative	derivative	PROPN
ma-11	57	18	f	f	PROPN
ma-11	57	19	∇(t	∇(t	PROPN
ma-11	57	20	)	)	PUNCT
ma-11	57	21	,	,	PUNCT
ma-11	57	22	if	if	SCONJ
ma-11	57	23	there	there	PRON
ma-11	57	24	exists	exist	VERB
ma-11	57	25	f	f	PROPN
ma-11	57	26	∇(t	∇(t	PROPN
ma-11	57	27	)	)	PUNCT
ma-11	57	28	∈	∈	PROPN
ma-11	57	29	r	r	NOUN
ma-11	57	30	such	such	ADJ
ma-11	57	31	that	that	SCONJ
ma-11	57	32	given	give	VERB
ma-11	57	33	any	any	DET
ma-11	57	34	ε	ε	PROPN
ma-11	57	35	>	>	X
ma-11	57	36	0	0	PROPN
ma-11	57	37	,	,	PUNCT
ma-11	57	38	there	there	PRON
ma-11	57	39	is	be	VERB
ma-11	57	40	a	a	DET
ma-11	57	41	neighborhood	neighborhood	NOUN
ma-11	57	42	v	v	NOUN
ma-11	57	43	of	of	ADP
ma-11	57	44	t	t	PROPN
ma-11	57	45	,	,	PUNCT
ma-11	57	46	such	such	ADJ
ma-11	57	47	that	that	SCONJ
ma-11	57	48	|f	|f	PROPN
ma-11	57	49	(	(	PUNCT
ma-11	57	50	ρ(t))−	ρ(t))−	PROPN
ma-11	57	51	f	f	X
ma-11	57	52	(	(	PUNCT
ma-11	57	53	s)−	s)−	PROPN
ma-11	57	54	f	f	PROPN
ma-11	57	55	∇(t)(ρ(t)−	∇(t)(ρ(t)−	PROPN
ma-11	57	56	s)|	s)|	PROPN
ma-11	57	57	≤	≤	NUM
ma-11	58	1	ε|ρ(t)−	ε|ρ(t)−	PROPN
ma-11	58	2	s|	s|	VERB
ma-11	58	3	,	,	PUNCT
ma-11	58	4	for	for	ADP
ma-11	58	5	all	all	DET
ma-11	58	6	s	s	PART
ma-11	58	7	∈	∈	NOUN
ma-11	58	8	v	v	NOUN
ma-11	58	9	.	.	PUNCT
ma-11	59	1	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	59	2	eur	eur	PROPN
ma-11	59	3	.	.	PUNCT
ma-11	60	1	j.	j.	PROPN
ma-11	60	2	math	math	PROPN
ma-11	60	3	.	.	PUNCT
ma-11	61	1	anal	anal	PROPN
ma-11	61	2	.	.	PUNCT
ma-11	62	1	10.28924	10.28924	NUM
ma-11	62	2	/	/	SYM
ma-11	62	3	ada	ada	PROPN
ma-11	62	4	/	/	SYM
ma-11	62	5	ma.3.12	ma.3.12	PROPN
ma-11	62	6	3a	3a	PROPN
ma-11	62	7	function	function	VERB
ma-11	62	8	f	f	NOUN
ma-11	62	9	:	:	PUNCT
ma-11	62	10	t→	t→	PUNCT
ma-11	62	11	r	r	NOUN
ma-11	62	12	is	be	AUX
ma-11	62	13	said	say	VERB
ma-11	62	14	to	to	PART
ma-11	62	15	be	be	AUX
ma-11	62	16	left	leave	VERB
ma-11	62	17	-	-	PUNCT
ma-11	62	18	dense	dense	ADJ
ma-11	62	19	continuous	continuous	ADJ
ma-11	62	20	(	(	PUNCT
ma-11	62	21	ld	ld	ADJ
ma-11	62	22	-	-	PUNCT
ma-11	62	23	continuous	continuous	ADJ
ma-11	62	24	)	)	PUNCT
ma-11	62	25	,	,	PUNCT
ma-11	62	26	provided	provide	VERB
ma-11	62	27	it	it	PRON
ma-11	62	28	is	be	AUX
ma-11	62	29	continuousat	continuousat	VERB
ma-11	62	30	all	all	DET
ma-11	62	31	left	left	ADJ
ma-11	62	32	-	-	PUNCT
ma-11	62	33	dense	dense	ADJ
ma-11	62	34	points	point	NOUN
ma-11	62	35	in	in	ADP
ma-11	62	36	t	t	PROPN
ma-11	62	37	and	and	CCONJ
ma-11	62	38	its	its	PRON
ma-11	62	39	right	right	ADJ
ma-11	62	40	-	-	PUNCT
ma-11	62	41	sided	side	VERB
ma-11	62	42	limits	limit	NOUN
ma-11	62	43	exist	exist	VERB
ma-11	62	44	(	(	PUNCT
ma-11	62	45	finite	finite	PROPN
ma-11	62	46	)	)	PUNCT
ma-11	62	47	at	at	ADP
ma-11	62	48	all	all	ADV
ma-11	62	49	right	right	ADJ
ma-11	62	50	-	-	PUNCT
ma-11	62	51	dense	dense	ADJ
ma-11	62	52	points	point	NOUN
ma-11	62	53	in	in	ADP
ma-11	62	54	t.the	t.the	DET
ma-11	62	55	set	set	NOUN
ma-11	62	56	of	of	ADP
ma-11	62	57	all	all	DET
ma-11	62	58	ld	ld	ADJ
ma-11	62	59	-	-	PUNCT
ma-11	62	60	continuous	continuous	ADJ
ma-11	62	61	functions	function	NOUN
ma-11	62	62	is	be	AUX
ma-11	62	63	denoted	denote	VERB
ma-11	62	64	by	by	ADP
ma-11	62	65	cld(t	cld(t	PROPN
ma-11	62	66	,	,	PUNCT
ma-11	62	67	r).the	r).the	DET
ma-11	62	68	next	next	ADJ
ma-11	62	69	definition	definition	NOUN
ma-11	62	70	is	be	AUX
ma-11	62	71	given	give	VERB
ma-11	62	72	in	in	ADP
ma-11	62	73	[	[	X
ma-11	62	74	6–8	6–8	NOUN
ma-11	62	75	]	]	PUNCT
ma-11	62	76	.	.	PUNCT
ma-11	63	1	definition	definition	NOUN
ma-11	63	2	2.2	2.2	NUM
ma-11	63	3	.	.	PUNCT
ma-11	64	1	a	a	DET
ma-11	64	2	function	function	NOUN
ma-11	64	3	g	g	NOUN
ma-11	64	4	:	:	PUNCT
ma-11	64	5	t→	t→	PUNCT
ma-11	64	6	r	r	NOUN
ma-11	64	7	is	be	AUX
ma-11	64	8	called	call	VERB
ma-11	64	9	a	a	DET
ma-11	64	10	nabla	nabla	NOUN
ma-11	64	11	antiderivative	antiderivative	NOUN
ma-11	64	12	of	of	ADP
ma-11	64	13	g	g	NOUN
ma-11	64	14	:	:	PUNCT
ma-11	64	15	t→	t→	X
ma-11	64	16	r	r	NOUN
ma-11	64	17	,	,	PUNCT
ma-11	64	18	provided	provide	VERB
ma-11	64	19	that	that	DET
ma-11	64	20	g∇(t	g∇(t	NOUN
ma-11	64	21	)	)	PUNCT
ma-11	64	22	=	=	SYM
ma-11	64	23	g(t	g(t	PROPN
ma-11	64	24	)	)	PUNCT
ma-11	64	25	holds	hold	VERB
ma-11	64	26	for	for	ADP
ma-11	64	27	all	all	DET
ma-11	64	28	t	t	NOUN
ma-11	64	29	∈	∈	PROPN
ma-11	64	30	tk	tk	PROPN
ma-11	64	31	.	.	PUNCT
ma-11	65	1	then	then	ADV
ma-11	65	2	the	the	DET
ma-11	65	3	nabla	nabla	NOUN
ma-11	65	4	integral	integral	ADJ
ma-11	65	5	of	of	ADP
ma-11	65	6	g	g	PROPN
ma-11	65	7	is	be	AUX
ma-11	65	8	defined	define	VERB
ma-11	65	9	by∫	by∫	PROPN
ma-11	65	10	b	b	PROPN
ma-11	65	11	a	a	DET
ma-11	65	12	g(t)∇t	g(t)∇t	NOUN
ma-11	65	13	=	=	SYM
ma-11	65	14	g(b)−	g(b)−	PROPN
ma-11	65	15	g(a	g(a	PROPN
ma-11	65	16	)	)	PUNCT
ma-11	65	17	.	.	PUNCT
ma-11	66	1	the	the	DET
ma-11	66	2	following	follow	VERB
ma-11	66	3	definition	definition	NOUN
ma-11	66	4	is	be	AUX
ma-11	66	5	taken	take	VERB
ma-11	66	6	from	from	ADP
ma-11	66	7	[	[	X
ma-11	66	8	2	2	NUM
ma-11	66	9	,	,	PUNCT
ma-11	66	10	4	4	NUM
ma-11	66	11	]	]	PUNCT
ma-11	66	12	.	.	PUNCT
ma-11	67	1	definition	definition	NOUN
ma-11	67	2	2.3	2.3	NUM
ma-11	67	3	.	.	PUNCT
ma-11	68	1	for	for	ADP
ma-11	68	2	α	α	PRON
ma-11	68	3	≥	≥	NOUN
ma-11	68	4	1	1	NUM
ma-11	68	5	,	,	PUNCT
ma-11	68	6	the	the	DET
ma-11	68	7	time	time	NOUN
ma-11	68	8	scale	scale	NOUN
ma-11	68	9	∆-riemann	∆-riemann	NOUN
ma-11	68	10	–	–	PUNCT
ma-11	68	11	liouville	liouville	NOUN
ma-11	68	12	type	type	NOUN
ma-11	68	13	fractional	fractional	ADJ
ma-11	68	14	integral	integral	ADJ
ma-11	68	15	for	for	ADP
ma-11	68	16	a	a	DET
ma-11	68	17	function	function	NOUN
ma-11	68	18	f	f	PROPN
ma-11	68	19	∈	∈	PROPN
ma-11	68	20	crd	crd	NOUN
ma-11	68	21	is	be	AUX
ma-11	68	22	defined	define	VERB
ma-11	68	23	by	by	ADP
ma-11	68	24	iαa	iαa	PROPN
ma-11	68	25	f	f	PROPN
ma-11	68	26	(	(	PUNCT
ma-11	68	27	t	t	PROPN
ma-11	68	28	)	)	PUNCT
ma-11	69	1	=	=	SYM
ma-11	69	2	∫	∫	PROPN
ma-11	69	3	t	t	PROPN
ma-11	69	4	a	a	DET
ma-11	69	5	hα−1(t	hα−1(t	PROPN
ma-11	69	6	,	,	PUNCT
ma-11	69	7	σ(τ))f	σ(τ))f	X
ma-11	69	8	(	(	PUNCT
ma-11	69	9	τ)∆τ	τ)∆τ	NUM
ma-11	69	10	,	,	PUNCT
ma-11	69	11	(	(	PUNCT
ma-11	69	12	1	1	X
ma-11	69	13	)	)	PUNCT
ma-11	69	14	which	which	PRON
ma-11	69	15	is	be	AUX
ma-11	69	16	an	an	DET
ma-11	69	17	integral	integral	ADJ
ma-11	69	18	on	on	ADP
ma-11	69	19	[	[	X
ma-11	69	20	a	a	DET
ma-11	69	21	,	,	PUNCT
ma-11	69	22	t)t	t)t	X
ma-11	69	23	,	,	PUNCT
ma-11	69	24	see	see	VERB
ma-11	69	25	[	[	X
ma-11	69	26	9	9	NUM
ma-11	69	27	]	]	PUNCT
ma-11	69	28	and	and	CCONJ
ma-11	69	29	hα	hα	X
ma-11	69	30	:	:	PUNCT
ma-11	69	31	t	t	PROPN
ma-11	69	32	×	×	PROPN
ma-11	69	33	t	t	PROPN
ma-11	69	34	→	→	SYM
ma-11	69	35	r	r	PROPN
ma-11	69	36	,	,	PUNCT
ma-11	69	37	α	α	PRON
ma-11	69	38	≥	≥	NOUN
ma-11	69	39	0	0	NUM
ma-11	69	40	are	be	AUX
ma-11	69	41	the	the	DET
ma-11	69	42	coordinate	coordinate	ADJ
ma-11	69	43	wiserd	wiserd	NOUN
ma-11	69	44	-	-	PUNCT
ma-11	69	45	continuous	continuous	ADJ
ma-11	69	46	functions	function	NOUN
ma-11	69	47	,	,	PUNCT
ma-11	69	48	such	such	ADJ
ma-11	69	49	that	that	SCONJ
ma-11	69	50	h0(t	h0(t	PROPN
ma-11	69	51	,	,	PUNCT
ma-11	69	52	s	s	PART
ma-11	69	53	)	)	PUNCT
ma-11	69	54	=	=	SYM
ma-11	69	55	1	1	NUM
ma-11	69	56	,	,	PUNCT
ma-11	69	57	hα+1(t	hα+1(t	PROPN
ma-11	69	58	,	,	PUNCT
ma-11	69	59	s	s	NOUN
ma-11	69	60	)	)	PUNCT
ma-11	69	61	=	=	SYM
ma-11	70	1	∫	∫	PROPN
ma-11	70	2	t	t	PROPN
ma-11	70	3	s	s	PROPN
ma-11	70	4	hα(τ	hα(τ	NUM
ma-11	70	5	,	,	PUNCT
ma-11	70	6	s)∆τ	s)∆τ	PROPN
ma-11	70	7	,	,	PUNCT
ma-11	70	8	∀s	∀s	PROPN
ma-11	70	9	,	,	PUNCT
ma-11	70	10	t	t	PROPN
ma-11	70	11	∈	∈	PROPN
ma-11	70	12	t.	t.	PROPN
ma-11	70	13	(	(	PUNCT
ma-11	70	14	2	2	NUM
ma-11	70	15	)	)	PUNCT
ma-11	70	16	notice	notice	NOUN
ma-11	70	17	that	that	SCONJ
ma-11	70	18	i1	i1	PROPN
ma-11	70	19	a	a	DET
ma-11	70	20	f	f	PROPN
ma-11	70	21	(	(	PUNCT
ma-11	70	22	t	t	PROPN
ma-11	70	23	)	)	PUNCT
ma-11	70	24	=	=	SYM
ma-11	71	1	∫	∫	PROPN
ma-11	71	2	t	t	PROPN
ma-11	71	3	a	a	DET
ma-11	71	4	f	f	PROPN
ma-11	71	5	(	(	PUNCT
ma-11	71	6	τ)∆τ	τ)∆τ	NUM
ma-11	71	7	,	,	PUNCT
ma-11	71	8	which	which	PRON
ma-11	71	9	is	be	AUX
ma-11	71	10	absolutely	absolutely	ADV
ma-11	71	11	continuous	continuous	ADJ
ma-11	71	12	in	in	ADP
ma-11	71	13	t	t	PROPN
ma-11	71	14	∈	∈	PROPN
ma-11	72	1	[	[	X
ma-11	72	2	a	a	DET
ma-11	72	3	,	,	PUNCT
ma-11	72	4	b]t	b]t	NOUN
ma-11	72	5	,	,	PUNCT
ma-11	72	6	see	see	VERB
ma-11	72	7	[	[	X
ma-11	72	8	9	9	NUM
ma-11	72	9	]	]	PUNCT
ma-11	72	10	.	.	PUNCT
ma-11	73	1	the	the	DET
ma-11	73	2	following	follow	VERB
ma-11	73	3	definition	definition	NOUN
ma-11	73	4	is	be	AUX
ma-11	73	5	taken	take	VERB
ma-11	73	6	from	from	ADP
ma-11	73	7	[	[	X
ma-11	73	8	3	3	NUM
ma-11	73	9	,	,	PUNCT
ma-11	73	10	4	4	NUM
ma-11	73	11	]	]	PUNCT
ma-11	73	12	.	.	PUNCT
ma-11	74	1	definition	definition	NOUN
ma-11	74	2	2.4	2.4	NUM
ma-11	74	3	.	.	PUNCT
ma-11	75	1	for	for	ADP
ma-11	75	2	α	α	PRON
ma-11	75	3	≥	≥	NOUN
ma-11	75	4	1	1	NUM
ma-11	75	5	,	,	PUNCT
ma-11	75	6	the	the	DET
ma-11	75	7	time	time	NOUN
ma-11	75	8	scale∇-riemann	scale∇-riemann	X
ma-11	75	9	–	–	PUNCT
ma-11	75	10	liouville	liouville	VERB
ma-11	75	11	type	type	NOUN
ma-11	75	12	fractional	fractional	ADJ
ma-11	75	13	integral	integral	ADJ
ma-11	75	14	for	for	ADP
ma-11	75	15	a	a	DET
ma-11	75	16	function	function	NOUN
ma-11	75	17	f	f	PROPN
ma-11	75	18	∈	∈	PROPN
ma-11	75	19	cld	cld	PROPN
ma-11	75	20	is	be	AUX
ma-11	75	21	defined	define	VERB
ma-11	75	22	by	by	ADP
ma-11	75	23	j	j	PROPN
ma-11	75	24	αa	αa	PROPN
ma-11	75	25	f	f	PROPN
ma-11	75	26	(	(	PUNCT
ma-11	75	27	t	t	PROPN
ma-11	75	28	)	)	PUNCT
ma-11	75	29	=	=	SYM
ma-11	76	1	∫	∫	PROPN
ma-11	76	2	t	t	PROPN
ma-11	76	3	a	a	DET
ma-11	76	4	ĥα−1(t	ĥα−1(t	PROPN
ma-11	76	5	,	,	PUNCT
ma-11	76	6	ρ(τ))f	ρ(τ))f	PROPN
ma-11	76	7	(	(	PUNCT
ma-11	76	8	τ)∇τ	τ)∇τ	NOUN
ma-11	76	9	,	,	PUNCT
ma-11	76	10	(	(	PUNCT
ma-11	76	11	3	3	X
ma-11	76	12	)	)	PUNCT
ma-11	76	13	which	which	PRON
ma-11	76	14	is	be	AUX
ma-11	76	15	an	an	DET
ma-11	76	16	integral	integral	ADJ
ma-11	76	17	on	on	ADP
ma-11	76	18	(	(	PUNCT
ma-11	76	19	a	a	PRON
ma-11	76	20	,	,	PUNCT
ma-11	76	21	t]t	t]t	NOUN
ma-11	76	22	,	,	PUNCT
ma-11	76	23	see	see	VERB
ma-11	76	24	[	[	X
ma-11	76	25	9	9	NUM
ma-11	76	26	]	]	PUNCT
ma-11	76	27	and	and	CCONJ
ma-11	76	28	ĥα	ĥα	NOUN
ma-11	76	29	:	:	PUNCT
ma-11	76	30	t	t	PROPN
ma-11	76	31	×	×	PROPN
ma-11	76	32	t	t	PROPN
ma-11	76	33	→	→	SYM
ma-11	76	34	r	r	PROPN
ma-11	76	35	,	,	PUNCT
ma-11	76	36	α	α	PRON
ma-11	76	37	≥	≥	NOUN
ma-11	76	38	0	0	NUM
ma-11	76	39	are	be	AUX
ma-11	76	40	the	the	DET
ma-11	76	41	coordinate	coordinate	ADJ
ma-11	76	42	wiseld	wiseld	NOUN
ma-11	76	43	-	-	PUNCT
ma-11	76	44	continuous	continuous	ADJ
ma-11	76	45	functions	function	NOUN
ma-11	76	46	,	,	PUNCT
ma-11	76	47	such	such	ADJ
ma-11	76	48	that	that	SCONJ
ma-11	76	49	ĥ0(t	ĥ0(t	PROPN
ma-11	76	50	,	,	PUNCT
ma-11	76	51	s	s	PART
ma-11	76	52	)	)	PUNCT
ma-11	76	53	=	=	SYM
ma-11	76	54	1	1	NUM
ma-11	76	55	,	,	PUNCT
ma-11	76	56	ĥα+1(t	ĥα+1(t	NUM
ma-11	76	57	,	,	PUNCT
ma-11	76	58	s	s	X
ma-11	76	59	)	)	PUNCT
ma-11	77	1	=	=	SYM
ma-11	77	2	∫	∫	PROPN
ma-11	78	1	t	t	PROPN
ma-11	78	2	s	s	PROPN
ma-11	78	3	ĥα(τ	ĥα(τ	PROPN
ma-11	78	4	,	,	PUNCT
ma-11	78	5	s)∇τ	s)∇τ	ADJ
ma-11	78	6	,	,	PUNCT
ma-11	78	7	∀s	∀s	PROPN
ma-11	78	8	,	,	PUNCT
ma-11	78	9	t	t	PROPN
ma-11	78	10	∈	∈	PROPN
ma-11	78	11	t.	t.	PROPN
ma-11	78	12	(	(	PUNCT
ma-11	78	13	4	4	NUM
ma-11	78	14	)	)	PUNCT
ma-11	78	15	notice	notice	NOUN
ma-11	78	16	that	that	SCONJ
ma-11	78	17	j	j	PROPN
ma-11	78	18	1	1	NUM
ma-11	78	19	a	a	DET
ma-11	78	20	f	f	PROPN
ma-11	78	21	(	(	PUNCT
ma-11	78	22	t	t	PROPN
ma-11	78	23	)	)	PUNCT
ma-11	78	24	=	=	SYM
ma-11	79	1	∫	∫	PROPN
ma-11	79	2	t	t	PROPN
ma-11	79	3	a	a	DET
ma-11	79	4	f	f	PROPN
ma-11	79	5	(	(	PUNCT
ma-11	79	6	τ)∇τ	τ)∇τ	PROPN
ma-11	79	7	,	,	PUNCT
ma-11	79	8	which	which	PRON
ma-11	79	9	is	be	AUX
ma-11	79	10	absolutely	absolutely	ADV
ma-11	79	11	continuous	continuous	ADJ
ma-11	79	12	in	in	ADP
ma-11	79	13	t	t	PROPN
ma-11	79	14	∈	∈	PROPN
ma-11	80	1	[	[	X
ma-11	80	2	a	a	DET
ma-11	80	3	,	,	PUNCT
ma-11	80	4	b]t	b]t	NOUN
ma-11	80	5	,	,	PUNCT
ma-11	80	6	see	see	VERB
ma-11	80	7	[	[	X
ma-11	80	8	9	9	NUM
ma-11	80	9	]	]	PUNCT
ma-11	80	10	.	.	PUNCT
ma-11	81	1	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	81	2	eur	eur	PROPN
ma-11	81	3	.	.	PUNCT
ma-11	82	1	j.	j.	PROPN
ma-11	82	2	math	math	PROPN
ma-11	82	3	.	.	PUNCT
ma-11	83	1	anal	anal	PROPN
ma-11	83	2	.	.	PUNCT
ma-11	84	1	10.28924	10.28924	NUM
ma-11	84	2	/	/	SYM
ma-11	84	3	ada	ada	PROPN
ma-11	84	4	/	/	SYM
ma-11	84	5	ma.3.12	ma.3.12	PROPN
ma-11	84	6	4we	4we	NOUN
ma-11	84	7	will	will	AUX
ma-11	84	8	generalize	generalize	VERB
ma-11	84	9	the	the	DET
ma-11	84	10	following	follow	VERB
ma-11	84	11	classical	classical	ADJ
ma-11	84	12	inequalities	inequality	NOUN
ma-11	84	13	[	[	X
ma-11	84	14	13	13	NUM
ma-11	84	15	]	]	PUNCT
ma-11	84	16	by	by	ADP
ma-11	84	17	using	use	VERB
ma-11	84	18	the	the	DET
ma-11	84	19	calculus	calculus	NOUN
ma-11	84	20	of	of	ADP
ma-11	84	21	time	time	NOUN
ma-11	84	22	scales.first	scales.first	ADV
ma-11	84	23	we	we	PRON
ma-11	84	24	consider	consider	VERB
ma-11	84	25	the	the	DET
ma-11	84	26	inequality	inequality	NOUN
ma-11	84	27	given	give	VERB
ma-11	84	28	by	by	ADP
ma-11	84	29	schweitzer	schweitzer	NOUN
ma-11	85	1	[	[	X
ma-11	85	2	19	19	NUM
ma-11	85	3	]	]	PUNCT
ma-11	85	4	such	such	ADJ
ma-11	85	5	that	that	SCONJ
ma-11	85	6	(	(	PUNCT
ma-11	85	7	1	1	NUM
ma-11	85	8	p	p	NOUN
ma-11	85	9	p∑	p∑	X
ma-11	85	10	k=1	k=1	PROPN
ma-11	85	11	xk	xk	PROPN
ma-11	85	12	)	)	PUNCT
ma-11	85	13	(	(	PUNCT
ma-11	85	14	1	1	NUM
ma-11	85	15	p	p	NOUN
ma-11	86	1	p∑	p∑	X
ma-11	86	2	k=1	k=1	NOUN
ma-11	86	3	1	1	NUM
ma-11	86	4	xk	xk	PROPN
ma-11	86	5	)	)	PUNCT
ma-11	86	6	≤	≤	NOUN
ma-11	87	1	(	(	PUNCT
ma-11	87	2	m	m	VERB
ma-11	87	3	+	+	ADJ
ma-11	87	4	m)2	m)2	PROPN
ma-11	87	5	4	4	NUM
ma-11	87	6	mm	mm	NOUN
ma-11	87	7	,	,	PUNCT
ma-11	87	8	(	(	PUNCT
ma-11	87	9	5	5	NUM
ma-11	87	10	)	)	PUNCT
ma-11	87	11	where	where	SCONJ
ma-11	87	12	0	0	NUM
ma-11	87	13	<	<	X
ma-11	87	14	m	m	VERB
ma-11	87	15	≤	≤	NUM
ma-11	87	16	xk	xk	PROPN
ma-11	87	17	≤	≤	PROPN
ma-11	87	18	m	m	VERB
ma-11	87	19	for	for	ADP
ma-11	87	20	k	k	PROPN
ma-11	87	21	=	=	SYM
ma-11	87	22	1	1	NUM
ma-11	87	23	,	,	PUNCT
ma-11	87	24	.	.	PUNCT
ma-11	87	25	.	.	PUNCT
ma-11	87	26	.	.	PUNCT
ma-11	88	1	,	,	PUNCT
ma-11	88	2	p.in	p.in	VERB
ma-11	88	3	the	the	DET
ma-11	88	4	same	same	ADJ
ma-11	88	5	paper	paper	NOUN
ma-11	88	6	,	,	PUNCT
ma-11	88	7	schweitzer	schweitzer	PROPN
ma-11	88	8	has	have	AUX
ma-11	88	9	also	also	ADV
ma-11	88	10	shown	show	VERB
ma-11	88	11	that	that	SCONJ
ma-11	88	12	if	if	SCONJ
ma-11	88	13	functions	function	NOUN
ma-11	88	14	y	y	PROPN
ma-11	88	15	7→	7→	NUM
ma-11	88	16	f	f	NOUN
ma-11	88	17	(	(	PUNCT
ma-11	88	18	y	y	NOUN
ma-11	88	19	)	)	PUNCT
ma-11	88	20	and	and	CCONJ
ma-11	88	21	y	y	PROPN
ma-11	88	22	7→	7→	PROPN
ma-11	88	23	1	1	NUM
ma-11	88	24	f	f	PROPN
ma-11	88	25	(	(	PUNCT
ma-11	88	26	y	y	NOUN
ma-11	88	27	)	)	PUNCT
ma-11	88	28	areintegrable	areintegrable	ADJ
ma-11	88	29	on	on	ADP
ma-11	88	30	[	[	X
ma-11	88	31	a	a	X
ma-11	88	32	,	,	PUNCT
ma-11	88	33	b	b	NOUN
ma-11	88	34	]	]	X
ma-11	88	35	and	and	CCONJ
ma-11	88	36	0	0	NUM
ma-11	88	37	<	<	X
ma-11	88	38	m	m	VERB
ma-11	88	39	≤	≤	ADJ
ma-11	88	40	f	f	X
ma-11	88	41	(	(	PUNCT
ma-11	88	42	y	y	NOUN
ma-11	88	43	)	)	PUNCT
ma-11	88	44	≤	≤	NUM
ma-11	88	45	m	m	VERB
ma-11	88	46	on	on	ADP
ma-11	88	47	[	[	X
ma-11	88	48	a	a	X
ma-11	88	49	,	,	PUNCT
ma-11	88	50	b	b	NOUN
ma-11	88	51	]	]	X
ma-11	88	52	,	,	PUNCT
ma-11	88	53	then∫	then∫	NOUN
ma-11	88	54	b	b	PROPN
ma-11	88	55	a	a	DET
ma-11	88	56	f	f	X
ma-11	88	57	(	(	PUNCT
ma-11	88	58	y)dy	y)dy	PROPN
ma-11	88	59	∫	∫	PROPN
ma-11	88	60	b	b	PROPN
ma-11	88	61	a	a	DET
ma-11	88	62	1	1	NUM
ma-11	88	63	f	f	NOUN
ma-11	88	64	(	(	PUNCT
ma-11	88	65	y	y	NOUN
ma-11	88	66	)	)	PUNCT
ma-11	88	67	dy	dy	NOUN
ma-11	88	68	≤	≤	NUM
ma-11	88	69	(	(	PUNCT
ma-11	88	70	m	m	VERB
ma-11	88	71	+	+	ADJ
ma-11	88	72	m)2	m)2	PROPN
ma-11	88	73	4	4	NUM
ma-11	88	74	mm	mm	NOUN
ma-11	88	75	(	(	PUNCT
ma-11	88	76	b	b	NOUN
ma-11	88	77	−	−	PROPN
ma-11	88	78	a)2	a)2	PROPN
ma-11	88	79	.	.	PUNCT
ma-11	89	1	(	(	PUNCT
ma-11	89	2	6	6	NUM
ma-11	89	3	)	)	PUNCT
ma-11	89	4	pólya	pólya	NOUN
ma-11	89	5	and	and	CCONJ
ma-11	89	6	szegö	szegö	NOUN
ma-11	90	1	[	[	X
ma-11	90	2	15	15	NUM
ma-11	90	3	]	]	PUNCT
ma-11	90	4	proved	prove	VERB
ma-11	90	5	that	that	SCONJ
ma-11	90	6	(	(	PUNCT
ma-11	90	7	p∑	p∑	X
ma-11	90	8	k=1	k=1	PUNCT
ma-11	91	1	x2	x2	INTJ
ma-11	91	2	k	k	PROPN
ma-11	91	3	)	)	PUNCT
ma-11	91	4	(	(	PUNCT
ma-11	91	5	p∑	p∑	X
ma-11	91	6	k=1	k=1	X
ma-11	92	1	y2	y2	PROPN
ma-11	92	2	k	k	PROPN
ma-11	92	3	)	)	PUNCT
ma-11	93	1	(	(	PUNCT
ma-11	93	2	p∑	p∑	NOUN
ma-11	93	3	k=1	k=1	X
ma-11	94	1	xkyk	xkyk	PROPN
ma-11	94	2	)	)	PUNCT
ma-11	94	3	2	2	NUM
ma-11	94	4	≤	≤	NUM
ma-11	94	5	√mn	√mn	PROPN
ma-11	94	6	mn	mn	PROPN
ma-11	95	1	+	+	CCONJ
ma-11	95	2	√	√	PROPN
ma-11	95	3	mn	mn	PROPN
ma-11	95	4	mn	mn	PROPN
ma-11	95	5	2	2	NUM
ma-11	95	6	2	2	ADV
ma-11	95	7	,	,	PUNCT
ma-11	95	8	(	(	PUNCT
ma-11	95	9	7	7	X
ma-11	95	10	)	)	PUNCT
ma-11	95	11	where	where	SCONJ
ma-11	95	12	0	0	NUM
ma-11	95	13	<	<	X
ma-11	95	14	m	m	VERB
ma-11	95	15	≤	≤	NUM
ma-11	95	16	xk	xk	X
ma-11	95	17	≤	≤	PROPN
ma-11	95	18	m	m	PROPN
ma-11	95	19	and	and	CCONJ
ma-11	95	20	0	0	NUM
ma-11	95	21	<	<	X
ma-11	95	22	n	n	CCONJ
ma-11	95	23	≤	≤	NUM
ma-11	95	24	yk	yk	NOUN
ma-11	95	25	≤	≤	NOUN
ma-11	95	26	n	n	CCONJ
ma-11	95	27	for	for	ADP
ma-11	95	28	k	k	PROPN
ma-11	95	29	=	=	SYM
ma-11	95	30	1	1	NUM
ma-11	95	31	,	,	PUNCT
ma-11	95	32	.	.	PUNCT
ma-11	95	33	.	.	PUNCT
ma-11	95	34	.	.	PUNCT
ma-11	96	1	,	,	PUNCT
ma-11	96	2	p.kantorovich	p.kantorovich	PUNCT
ma-11	97	1	[	[	X
ma-11	97	2	12	12	NUM
ma-11	97	3	]	]	PUNCT
ma-11	97	4	proved	prove	VERB
ma-11	97	5	that	that	SCONJ
ma-11	97	6	(	(	PUNCT
ma-11	97	7	p∑	p∑	NOUN
ma-11	97	8	k=1	k=1	X
ma-11	97	9	xky	xky	PROPN
ma-11	97	10	2	2	NUM
ma-11	97	11	k	k	NOUN
ma-11	97	12	)	)	PUNCT
ma-11	97	13	(	(	PUNCT
ma-11	97	14	p∑	p∑	X
ma-11	97	15	k=1	k=1	PROPN
ma-11	98	1	1	1	NUM
ma-11	98	2	xk	xk	INTJ
ma-11	98	3	y2	y2	PROPN
ma-11	98	4	k	k	PROPN
ma-11	98	5	)	)	PUNCT
ma-11	98	6	≤	≤	NOUN
ma-11	98	7	1	1	NUM
ma-11	98	8	4	4	NUM
ma-11	98	9	(	(	PUNCT
ma-11	98	10	√	√	NUM
ma-11	98	11	m	m	VERB
ma-11	98	12	m	m	VERB
ma-11	98	13	+	+	ADJ
ma-11	98	14	√	√	NUM
ma-11	98	15	m	m	VERB
ma-11	98	16	m	m	VERB
ma-11	98	17	)	)	PUNCT
ma-11	98	18	2	2	NUM
ma-11	98	19	(	(	PUNCT
ma-11	98	20	p∑	p∑	NOUN
ma-11	98	21	k=1	k=1	X
ma-11	99	1	y2	y2	INTJ
ma-11	99	2	k	k	PROPN
ma-11	99	3	)	)	PUNCT
ma-11	99	4	2	2	NUM
ma-11	99	5	,	,	PUNCT
ma-11	99	6	(	(	PUNCT
ma-11	99	7	8)	8)	NUM
ma-11	99	8	where	where	SCONJ
ma-11	99	9	0	0	NUM
ma-11	99	10	<	<	X
ma-11	99	11	m	m	VERB
ma-11	99	12	≤	≤	NUM
ma-11	99	13	xk	xk	X
ma-11	99	14	≤	≤	PROPN
ma-11	99	15	m	m	PROPN
ma-11	99	16	and	and	CCONJ
ma-11	99	17	yk	yk	PROPN
ma-11	99	18	∈	∈	PROPN
ma-11	99	19	r	r	NOUN
ma-11	99	20	for	for	ADP
ma-11	99	21	k	k	PROPN
ma-11	99	22	=	=	SYM
ma-11	99	23	1	1	NUM
ma-11	99	24	,	,	PUNCT
ma-11	99	25	.	.	PUNCT
ma-11	99	26	.	.	PUNCT
ma-11	100	1	.	.	PUNCT
ma-11	101	1	,	,	PUNCT
ma-11	102	1	p	p	X
ma-11	102	2	,	,	PUNCT
ma-11	102	3	and	and	CCONJ
ma-11	102	4	he	he	PRON
ma-11	102	5	pointed	point	VERB
ma-11	102	6	out	out	ADP
ma-11	102	7	that	that	SCONJ
ma-11	102	8	inequality	inequality	NOUN
ma-11	102	9	(	(	PUNCT
ma-11	102	10	8)	8)	NUM
ma-11	102	11	is	be	AUX
ma-11	102	12	aparticular	aparticular	ADJ
ma-11	102	13	case	case	NOUN
ma-11	102	14	of	of	ADP
ma-11	102	15	(	(	PUNCT
ma-11	102	16	7).greub	7).greub	NUM
ma-11	102	17	and	and	CCONJ
ma-11	102	18	rheinboldt	rheinboldt	ADJ
ma-11	103	1	[	[	PUNCT
ma-11	103	2	10	10	NUM
ma-11	103	3	]	]	PUNCT
ma-11	103	4	proved	prove	VERB
ma-11	103	5	that	that	SCONJ
ma-11	103	6	(	(	PUNCT
ma-11	103	7	p∑	p∑	X
ma-11	103	8	k=1	k=1	PUNCT
ma-11	104	1	x2	x2	INTJ
ma-11	105	1	k	k	PROPN
ma-11	105	2	z	z	PROPN
ma-11	105	3	2	2	NUM
ma-11	105	4	k	k	NOUN
ma-11	105	5	)	)	PUNCT
ma-11	105	6	(	(	PUNCT
ma-11	105	7	p∑	p∑	X
ma-11	105	8	k=1	k=1	X
ma-11	106	1	y2	y2	INTJ
ma-11	107	1	k	k	NOUN
ma-11	107	2	z	z	NOUN
ma-11	107	3	2	2	NUM
ma-11	107	4	k	k	NOUN
ma-11	107	5	)	)	PUNCT
ma-11	107	6	≤	≤	NOUN
ma-11	107	7	(	(	PUNCT
ma-11	107	8	mn	mn	NOUN
ma-11	107	9	+	+	PROPN
ma-11	107	10	mn)2	mn)2	PROPN
ma-11	107	11	4mnmn	4mnmn	NUM
ma-11	107	12	(	(	PUNCT
ma-11	107	13	p∑	p∑	X
ma-11	107	14	k=1	k=1	PROPN
ma-11	107	15	xkykz	xkykz	NOUN
ma-11	108	1	2	2	NUM
ma-11	108	2	k	k	NOUN
ma-11	108	3	)	)	PUNCT
ma-11	108	4	2	2	NUM
ma-11	108	5	,	,	PUNCT
ma-11	108	6	(	(	PUNCT
ma-11	108	7	9	9	X
ma-11	108	8	)	)	PUNCT
ma-11	109	1	where	where	SCONJ
ma-11	109	2	0	0	NUM
ma-11	109	3	<	<	X
ma-11	109	4	m	m	VERB
ma-11	109	5	≤	≤	NUM
ma-11	109	6	xk	xk	PROPN
ma-11	109	7	≤	≤	PROPN
ma-11	109	8	m	m	PROPN
ma-11	109	9	,	,	PUNCT
ma-11	109	10	0	0	PUNCT
ma-11	109	11	<	<	X
ma-11	109	12	n	n	PRON
ma-11	109	13	≤	≤	NUM
ma-11	109	14	yk	yk	NOUN
ma-11	109	15	≤	≤	NOUN
ma-11	109	16	n	n	CCONJ
ma-11	109	17	and	and	CCONJ
ma-11	109	18	zk	zk	PROPN
ma-11	109	19	∈	∈	PROPN
ma-11	109	20	r	r	NOUN
ma-11	109	21	for	for	ADP
ma-11	109	22	k	k	PROPN
ma-11	109	23	=	=	SYM
ma-11	109	24	1	1	NUM
ma-11	109	25	,	,	PUNCT
ma-11	109	26	.	.	PUNCT
ma-11	109	27	.	.	PUNCT
ma-11	109	28	.	.	PUNCT
ma-11	110	1	,	,	PUNCT
ma-11	110	2	p	p	NOUN
ma-11	110	3	with	with	ADP
ma-11	110	4	p∑	p∑	PROPN
ma-11	110	5	k=1	k=1	X
ma-11	111	1	z2	z2	PROPN
ma-11	111	2	k	k	PROPN
ma-11	112	1	<	<	X
ma-11	112	2	∞.	∞.	PROPN
ma-11	112	3	3	3	NUM
ma-11	112	4	.	.	PUNCT
ma-11	112	5	main	main	ADJ
ma-11	112	6	results	result	NOUN
ma-11	112	7	in	in	ADP
ma-11	112	8	order	order	NOUN
ma-11	112	9	to	to	PART
ma-11	112	10	present	present	VERB
ma-11	112	11	our	our	PRON
ma-11	112	12	main	main	ADJ
ma-11	112	13	results	result	NOUN
ma-11	112	14	,	,	PUNCT
ma-11	112	15	first	first	ADV
ma-11	112	16	we	we	PRON
ma-11	112	17	give	give	VERB
ma-11	112	18	a	a	DET
ma-11	112	19	simple	simple	ADJ
ma-11	112	20	proof	proof	NOUN
ma-11	112	21	for	for	ADP
ma-11	112	22	an	an	DET
ma-11	112	23	extension	extension	NOUN
ma-11	112	24	of	of	ADP
ma-11	112	25	pólya	pólya	NOUN
ma-11	112	26	–	–	PUNCT
ma-11	112	27	szegö’sinequality	szegö’sinequality	NOUN
ma-11	112	28	by	by	ADP
ma-11	112	29	using	use	VERB
ma-11	112	30	the	the	DET
ma-11	112	31	time	time	NOUN
ma-11	112	32	scale	scale	NOUN
ma-11	112	33	∆-riemann	∆-riemann	NOUN
ma-11	112	34	–	–	PUNCT
ma-11	112	35	liouville	liouville	VERB
ma-11	112	36	type	type	NOUN
ma-11	112	37	fractional	fractional	ADJ
ma-11	112	38	integral	integral	ADJ
ma-11	112	39	.	.	PUNCT
ma-11	113	1	theorem	theorem	NOUN
ma-11	113	2	3.1	3.1	NUM
ma-11	113	3	.	.	PUNCT
ma-11	114	1	let	let	VERB
ma-11	114	2	w	w	VERB
ma-11	114	3	,	,	PUNCT
ma-11	114	4	f	f	PROPN
ma-11	114	5	,	,	PUNCT
ma-11	114	6	g	g	PROPN
ma-11	114	7	∈	∈	PROPN
ma-11	114	8	crd	crd	NOUN
ma-11	114	9	(	(	PUNCT
ma-11	114	10	[	[	X
ma-11	114	11	a	a	PRON
ma-11	114	12	,	,	PUNCT
ma-11	114	13	b]t	b]t	NOUN
ma-11	114	14	,	,	PUNCT
ma-11	114	15	r−	r−	NOUN
ma-11	114	16	{	{	PUNCT
ma-11	114	17	0	0	NUM
ma-11	114	18	}	}	PUNCT
ma-11	114	19	)	)	PUNCT
ma-11	114	20	be	be	AUX
ma-11	114	21	∆-integrable	∆-integrable	ADJ
ma-11	114	22	functions	function	NOUN
ma-11	114	23	.	.	PUNCT
ma-11	115	1	assume	assume	VERB
ma-11	115	2	that	that	SCONJ
ma-11	115	3	there	there	PRON
ma-11	115	4	exist	exist	VERB
ma-11	115	5	four	four	NUM
ma-11	115	6	positive	positive	ADJ
ma-11	115	7	∆-integrable	∆-integrable	ADJ
ma-11	115	8	functions	function	NOUN
ma-11	115	9	f1	f1	NOUN
ma-11	115	10	,	,	PUNCT
ma-11	115	11	f2	f2	PROPN
ma-11	115	12	,	,	PUNCT
ma-11	115	13	g1	g1	PROPN
ma-11	115	14	and	and	CCONJ
ma-11	115	15	g2	g2	PROPN
ma-11	115	16	such	such	ADJ
ma-11	115	17	that	that	SCONJ
ma-11	115	18	:	:	PUNCT
ma-11	115	19	0	0	NUM
ma-11	115	20	<	<	X
ma-11	115	21	f1(y	f1(y	PROPN
ma-11	115	22	)	)	PUNCT
ma-11	115	23	≤	≤	NOUN
ma-11	115	24	|f	|f	PROPN
ma-11	116	1	(	(	PUNCT
ma-11	116	2	y)|	y)|	PROPN
ma-11	116	3	≤	≤	PROPN
ma-11	116	4	f2(y	f2(y	PROPN
ma-11	116	5	)	)	PUNCT
ma-11	116	6	<	<	X
ma-11	116	7	∞	∞	NUM
ma-11	116	8	and	and	CCONJ
ma-11	116	9	0	0	NUM
ma-11	116	10	<	<	X
ma-11	116	11	g1(y	g1(y	PROPN
ma-11	116	12	)	)	PUNCT
ma-11	116	13	≤	≤	PUNCT
ma-11	116	14	|g(y)|	|g(y)|	PROPN
ma-11	116	15	≤	≤	NOUN
ma-11	116	16	g2(y	g2(y	PROPN
ma-11	116	17	)	)	PUNCT
ma-11	116	18	<	<	X
ma-11	116	19	∞	∞	PROPN
ma-11	116	20	,	,	PUNCT
ma-11	116	21	(	(	PUNCT
ma-11	116	22	y	y	PROPN
ma-11	116	23	∈	∈	PROPN
ma-11	116	24	[	[	X
ma-11	116	25	a	a	X
ma-11	116	26	,	,	PUNCT
ma-11	116	27	x	x	X
ma-11	116	28	]	]	X
ma-11	116	29	t,∀x	t,∀x	X
ma-11	116	30	∈	∈	X
ma-11	117	1	[	[	X
ma-11	117	2	a	a	DET
ma-11	117	3	,	,	PUNCT
ma-11	117	4	b]t	b]t	NOUN
ma-11	117	5	)	)	PUNCT
ma-11	117	6	.	.	PUNCT
ma-11	118	1	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	118	2	eur	eur	PROPN
ma-11	118	3	.	.	PUNCT
ma-11	119	1	j.	j.	PROPN
ma-11	119	2	math	math	PROPN
ma-11	119	3	.	.	PUNCT
ma-11	120	1	anal	anal	PROPN
ma-11	120	2	.	.	PUNCT
ma-11	121	1	10.28924	10.28924	NUM
ma-11	121	2	/	/	SYM
ma-11	121	3	ada	ada	PROPN
ma-11	121	4	/	/	SYM
ma-11	121	5	ma.3.12	ma.3.12	PROPN
ma-11	121	6	5	5	NUM
ma-11	121	7	let	let	VERB
ma-11	121	8	α	α	PRON
ma-11	121	9	,	,	PUNCT
ma-11	121	10	β	β	X
ma-11	121	11	≥	≥	NUM
ma-11	121	12	1	1	NUM
ma-11	121	13	and	and	CCONJ
ma-11	121	14	hα−1	hα−1	NOUN
ma-11	121	15	(	(	PUNCT
ma-11	121	16	.	.	PUNCT
ma-11	121	17	,	,	PUNCT
ma-11	121	18	.	.	PUNCT
ma-11	121	19	)	)	PUNCT
ma-11	121	20	,	,	PUNCT
ma-11	121	21	hβ−1	hβ−1	PROPN
ma-11	121	22	(	(	PUNCT
ma-11	121	23	.	.	PUNCT
ma-11	121	24	,	,	PUNCT
ma-11	121	25	.	.	PUNCT
ma-11	121	26	)	)	PUNCT
ma-11	122	1	>	>	X
ma-11	122	2	0	0	X
ma-11	122	3	.	.	PUNCT
ma-11	123	1	then	then	ADV
ma-11	123	2	we	we	PRON
ma-11	123	3	have	have	VERB
ma-11	123	4	the	the	DET
ma-11	123	5	following	follow	VERB
ma-11	123	6	inequality	inequality	NOUN
ma-11	123	7	iαa	iαa	ADJ
ma-11	123	8	(	(	PUNCT
ma-11	123	9	(	(	PUNCT
ma-11	123	10	f1f2)(x)|w(x)|	f1f2)(x)|w(x)|	PROPN
ma-11	123	11	)	)	PUNCT
ma-11	123	12	iβa	iβa	NOUN
ma-11	123	13	(	(	PUNCT
ma-11	123	14	(	(	PUNCT
ma-11	123	15	g1g2)(x)|w(x)|	g1g2)(x)|w(x)|	NOUN
ma-11	123	16	)	)	PUNCT
ma-11	123	17	iαa	iαa	NOUN
ma-11	123	18	(	(	PUNCT
ma-11	123	19	|w(x)||f	|w(x)||f	X
ma-11	123	20	(	(	PUNCT
ma-11	123	21	x)|2	x)|2	PROPN
ma-11	123	22	)	)	PUNCT
ma-11	123	23	iβa	iβa	NOUN
ma-11	123	24	(	(	PUNCT
ma-11	123	25	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	123	26	)	)	PUNCT
ma-11	123	27	{	{	PUNCT
ma-11	123	28	iαa	iαa	PROPN
ma-11	123	29	(	(	PUNCT
ma-11	123	30	f1(x)|(wf	f1(x)|(wf	NOUN
ma-11	123	31	)	)	PUNCT
ma-11	123	32	(	(	PUNCT
ma-11	123	33	x)|	x)|	PROPN
ma-11	123	34	)	)	PUNCT
ma-11	123	35	iβa	iβa	NOUN
ma-11	123	36	(	(	PUNCT
ma-11	123	37	g1(x)|(wg)(x)|	g1(x)|(wg)(x)|	PROPN
ma-11	123	38	)	)	PUNCT
ma-11	124	1	+	+	CCONJ
ma-11	124	2	iαa	iαa	ADJ
ma-11	124	3	(	(	PUNCT
ma-11	124	4	f2(x)|(wf	f2(x)|(wf	X
ma-11	124	5	)	)	PUNCT
ma-11	124	6	(	(	PUNCT
ma-11	124	7	x)|	x)|	PROPN
ma-11	124	8	)	)	PUNCT
ma-11	124	9	iβa	iβa	NOUN
ma-11	124	10	(	(	PUNCT
ma-11	124	11	g2(x)|(wg)(x)|	g2(x)|(wg)(x)|	PROPN
ma-11	124	12	)	)	PUNCT
ma-11	124	13	}	}	PUNCT
ma-11	124	14	2	2	NUM
ma-11	124	15	≤	≤	NUM
ma-11	124	16	1	1	NUM
ma-11	124	17	4	4	NUM
ma-11	124	18	.	.	PUNCT
ma-11	125	1	(	(	PUNCT
ma-11	125	2	10	10	NUM
ma-11	125	3	)	)	PUNCT
ma-11	125	4	proof	proof	NOUN
ma-11	125	5	.	.	PUNCT
ma-11	126	1	using	use	VERB
ma-11	126	2	the	the	DET
ma-11	126	3	given	give	VERB
ma-11	126	4	conditions	condition	NOUN
ma-11	126	5	,	,	PUNCT
ma-11	126	6	for	for	ADP
ma-11	126	7	y	y	PROPN
ma-11	126	8	,	,	PUNCT
ma-11	126	9	z	z	NOUN
ma-11	126	10	∈	∈	PROPN
ma-11	127	1	[	[	X
ma-11	127	2	a	a	X
ma-11	127	3	,	,	PUNCT
ma-11	127	4	x	x	X
ma-11	127	5	]	]	X
ma-11	127	6	t	t	PROPN
ma-11	127	7	,	,	PUNCT
ma-11	127	8	∀x	∀x	X
ma-11	127	9	∈	∈	PROPN
ma-11	127	10	[	[	X
ma-11	127	11	a	a	DET
ma-11	127	12	,	,	PUNCT
ma-11	127	13	b]t	b]t	NOUN
ma-11	127	14	,	,	PUNCT
ma-11	127	15	we	we	PRON
ma-11	127	16	have	have	VERB
ma-11	127	17	(	(	PUNCT
ma-11	127	18	f2(y	f2(y	PROPN
ma-11	127	19	)	)	PUNCT
ma-11	127	20	g1(z	g1(z	PROPN
ma-11	127	21	)	)	PUNCT
ma-11	127	22	−	−	PROPN
ma-11	127	23	|f	|f	PROPN
ma-11	128	1	(	(	PUNCT
ma-11	128	2	y)|	y)|	PROPN
ma-11	128	3	|g(z)|	|g(z)|	PROPN
ma-11	128	4	)	)	PUNCT
ma-11	128	5	≥	≥	NOUN
ma-11	128	6	0	0	NUM
ma-11	128	7	,	,	PUNCT
ma-11	128	8	and	and	CCONJ
ma-11	128	9	(	(	PUNCT
ma-11	128	10	|f	|f	PROPN
ma-11	128	11	(	(	PUNCT
ma-11	128	12	y)|	y)|	PROPN
ma-11	128	13	|g(z)|	|g(z)|	PROPN
ma-11	128	14	−	−	PROPN
ma-11	128	15	f1(y	f1(y	NUM
ma-11	128	16	)	)	PUNCT
ma-11	128	17	g2(z	g2(z	PROPN
ma-11	128	18	)	)	PUNCT
ma-11	128	19	)	)	PUNCT
ma-11	128	20	≥	≥	NOUN
ma-11	128	21	0	0	NUM
ma-11	128	22	,	,	PUNCT
ma-11	128	23	which	which	PRON
ma-11	128	24	imply	imply	VERB
ma-11	128	25	that	that	SCONJ
ma-11	128	26	(	(	PUNCT
ma-11	128	27	f1(y	f1(y	NUM
ma-11	128	28	)	)	PUNCT
ma-11	128	29	g2(z	g2(z	PROPN
ma-11	128	30	)	)	PUNCT
ma-11	128	31	+	+	NUM
ma-11	128	32	f2(y	f2(y	PROPN
ma-11	128	33	)	)	PUNCT
ma-11	128	34	g1(z	g1(z	PROPN
ma-11	128	35	)	)	PUNCT
ma-11	128	36	)	)	PUNCT
ma-11	128	37	|f	|f	PROPN
ma-11	129	1	(	(	PUNCT
ma-11	129	2	y)|	y)|	PROPN
ma-11	129	3	|g(z)|	|g(z)|	PROPN
ma-11	129	4	≥	≥	PRON
ma-11	129	5	|f	|f	PROPN
ma-11	129	6	(	(	PUNCT
ma-11	129	7	y)|2	y)|2	X
ma-11	129	8	|g(z)|2	|g(z)|2	PUNCT
ma-11	129	9	+	+	PROPN
ma-11	129	10	f1(y)f2(y	f1(y)f2(y	PROPN
ma-11	129	11	)	)	PUNCT
ma-11	129	12	g1(z)g2(z	g1(z)g2(z	PROPN
ma-11	129	13	)	)	PUNCT
ma-11	129	14	.	.	PUNCT
ma-11	130	1	multiplying	multiply	VERB
ma-11	130	2	both	both	DET
ma-11	130	3	sides	side	NOUN
ma-11	130	4	by	by	ADP
ma-11	130	5	g1(z)g2(z)|g(z)|2	g1(z)g2(z)|g(z)|2	PROPN
ma-11	130	6	,	,	PUNCT
ma-11	130	7	we	we	PRON
ma-11	130	8	have	have	VERB
ma-11	130	9	f1(y)g1(z)|f	f1(y)g1(z)|f	NOUN
ma-11	130	10	(	(	PUNCT
ma-11	130	11	y)g(z)|+	y)g(z)|+	PROPN
ma-11	130	12	f2(y)g2(z)|f	f2(y)g2(z)|f	NOUN
ma-11	130	13	(	(	PUNCT
ma-11	130	14	y)g(z)|	y)g(z)|	PROPN
ma-11	130	15	≥	≥	NUM
ma-11	130	16	g1(z)g2(z)|f	g1(z)g2(z)|f	NOUN
ma-11	130	17	(	(	PUNCT
ma-11	130	18	y)|2	y)|2	X
ma-11	130	19	+	+	CCONJ
ma-11	130	20	f1(y)f2(y)|g(z)|2	f1(y)f2(y)|g(z)|2	NUM
ma-11	130	21	.	.	PUNCT
ma-11	131	1	(	(	PUNCT
ma-11	131	2	11	11	NUM
ma-11	131	3	)	)	PUNCT
ma-11	131	4	multiplying	multiply	VERB
ma-11	131	5	both	both	DET
ma-11	131	6	sides	side	NOUN
ma-11	131	7	of	of	ADP
ma-11	131	8	(	(	PUNCT
ma-11	131	9	11	11	NUM
ma-11	131	10	)	)	PUNCT
ma-11	131	11	by	by	ADP
ma-11	131	12	hα−1(x	hα−1(x	PROPN
ma-11	131	13	,	,	PUNCT
ma-11	131	14	σ(y))|w(y)|hβ−1(x	σ(y))|w(y)|hβ−1(x	NOUN
ma-11	131	15	,	,	PUNCT
ma-11	131	16	σ(z))|w(z)|	σ(z))|w(z)|	ADJ
ma-11	131	17	and	and	CCONJ
ma-11	131	18	double	double	ADJ
ma-11	131	19	integratingover	integratingover	NOUN
ma-11	131	20	y	y	PROPN
ma-11	131	21	and	and	CCONJ
ma-11	131	22	z	z	PROPN
ma-11	131	23	from	from	ADP
ma-11	131	24	a	a	DET
ma-11	131	25	to	to	ADP
ma-11	131	26	x	x	PRON
ma-11	131	27	,	,	PUNCT
ma-11	131	28	respectively	respectively	ADV
ma-11	131	29	,	,	PUNCT
ma-11	131	30	we	we	PRON
ma-11	131	31	have	have	VERB
ma-11	131	32	iαa	iαa	NOUN
ma-11	131	33	(	(	PUNCT
ma-11	131	34	f1(x)|w(x)f	f1(x)|w(x)f	PROPN
ma-11	131	35	(	(	PUNCT
ma-11	131	36	x)|	x)|	PROPN
ma-11	131	37	)	)	PUNCT
ma-11	131	38	iβa	iβa	NOUN
ma-11	131	39	(	(	PUNCT
ma-11	131	40	g1(x)|w(x)g(x)|	g1(x)|w(x)g(x)|	NOUN
ma-11	131	41	)	)	PUNCT
ma-11	131	42	+	+	CCONJ
ma-11	131	43	iαa	iαa	ADJ
ma-11	131	44	(	(	PUNCT
ma-11	131	45	f2(x)|w(x)f	f2(x)|w(x)f	NOUN
ma-11	131	46	(	(	PUNCT
ma-11	131	47	x)|	x)|	PROPN
ma-11	131	48	)	)	PUNCT
ma-11	131	49	iβa	iβa	NOUN
ma-11	131	50	(	(	PUNCT
ma-11	131	51	g2(x)|w(x)g(x)|	g2(x)|w(x)g(x)|	NOUN
ma-11	131	52	)	)	PUNCT
ma-11	131	53	≥	≥	NOUN
ma-11	131	54	iαa	iαa	NOUN
ma-11	131	55	(	(	PUNCT
ma-11	131	56	|w(x)||f	|w(x)||f	X
ma-11	131	57	(	(	PUNCT
ma-11	131	58	x)|2	x)|2	PROPN
ma-11	131	59	)	)	PUNCT
ma-11	131	60	iβa	iβa	NOUN
ma-11	131	61	(	(	PUNCT
ma-11	131	62	g1(x)g2(x)|w(x)|	g1(x)g2(x)|w(x)|	NOUN
ma-11	131	63	)	)	PUNCT
ma-11	132	1	+	+	CCONJ
ma-11	132	2	iαa	iαa	ADJ
ma-11	132	3	(	(	PUNCT
ma-11	132	4	f1(x)f2(x)|w(x)|	f1(x)f2(x)|w(x)|	NOUN
ma-11	132	5	)	)	PUNCT
ma-11	132	6	iβa	iβa	NOUN
ma-11	132	7	(	(	PUNCT
ma-11	132	8	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	132	9	)	)	PUNCT
ma-11	132	10	.	.	PUNCT
ma-11	133	1	(	(	PUNCT
ma-11	133	2	12	12	X
ma-11	133	3	)	)	PUNCT
ma-11	133	4	applying	apply	VERB
ma-11	133	5	the	the	DET
ma-11	133	6	am	am	NOUN
ma-11	133	7	-	-	PUNCT
ma-11	133	8	gm	gm	PROPN
ma-11	133	9	inequality	inequality	NOUN
ma-11	133	10	√ζη	√ζη	NOUN
ma-11	133	11	≤	≤	X
ma-11	133	12	ζ+η	ζ+η	NUM
ma-11	133	13	2	2	NUM
ma-11	133	14	,	,	PUNCT
ma-11	133	15	ζ	ζ	NOUN
ma-11	133	16	≥	≥	NOUN
ma-11	133	17	0	0	NUM
ma-11	133	18	,	,	PUNCT
ma-11	133	19	η	η	PROPN
ma-11	133	20	≥	≥	PROPN
ma-11	133	21	0	0	NUM
ma-11	133	22	,	,	PUNCT
ma-11	133	23	the	the	DET
ma-11	133	24	inequality	inequality	NOUN
ma-11	133	25	(	(	PUNCT
ma-11	133	26	12	12	NUM
ma-11	133	27	)	)	PUNCT
ma-11	133	28	takes	take	VERB
ma-11	133	29	the	the	DET
ma-11	133	30	form	form	NOUN
ma-11	133	31	iαa	iαa	ADJ
ma-11	133	32	(	(	PUNCT
ma-11	133	33	f1(x)|w(x)f	f1(x)|w(x)f	PROPN
ma-11	133	34	(	(	PUNCT
ma-11	133	35	x)|	x)|	PROPN
ma-11	133	36	)	)	PUNCT
ma-11	133	37	iβa	iβa	NOUN
ma-11	133	38	(	(	PUNCT
ma-11	133	39	g1(x)|w(x)g(x)|	g1(x)|w(x)g(x)|	NOUN
ma-11	133	40	)	)	PUNCT
ma-11	134	1	+	+	CCONJ
ma-11	134	2	iαa	iαa	ADJ
ma-11	134	3	(	(	PUNCT
ma-11	134	4	f2(x)|w(x)f	f2(x)|w(x)f	NOUN
ma-11	134	5	(	(	PUNCT
ma-11	134	6	x)|	x)|	PROPN
ma-11	134	7	)	)	PUNCT
ma-11	134	8	iβa	iβa	NOUN
ma-11	134	9	(	(	PUNCT
ma-11	134	10	g2(x)|w(x)g(x)|	g2(x)|w(x)g(x)|	NOUN
ma-11	134	11	)	)	PUNCT
ma-11	134	12	≥	≥	NOUN
ma-11	134	13	2	2	NUM
ma-11	134	14	√	√	NOUN
ma-11	134	15	iαa	iαa	NOUN
ma-11	134	16	(	(	PUNCT
ma-11	134	17	|w(x)||f	|w(x)||f	X
ma-11	134	18	(	(	PUNCT
ma-11	134	19	x)|2	x)|2	PROPN
ma-11	134	20	)	)	PUNCT
ma-11	134	21	iβa	iβa	NOUN
ma-11	134	22	(	(	PUNCT
ma-11	134	23	g1(x)g2(x)|w(x)|	g1(x)g2(x)|w(x)|	NOUN
ma-11	134	24	)	)	PUNCT
ma-11	134	25	iαa	iαa	NOUN
ma-11	134	26	(	(	PUNCT
ma-11	134	27	f1(x)f2(x)|w(x)|	f1(x)f2(x)|w(x)|	NOUN
ma-11	134	28	)	)	PUNCT
ma-11	134	29	iβa	iβa	NOUN
ma-11	134	30	(	(	PUNCT
ma-11	134	31	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	134	32	)	)	PUNCT
ma-11	134	33	.	.	PUNCT
ma-11	135	1	(	(	PUNCT
ma-11	135	2	13	13	NUM
ma-11	135	3	)	)	PUNCT
ma-11	135	4	inequality	inequality	NOUN
ma-11	135	5	(	(	PUNCT
ma-11	135	6	13	13	NUM
ma-11	135	7	)	)	PUNCT
ma-11	135	8	directly	directly	ADV
ma-11	135	9	yields	yield	VERB
ma-11	135	10	inequality	inequality	NOUN
ma-11	135	11	(	(	PUNCT
ma-11	135	12	10	10	NUM
ma-11	135	13	)	)	PUNCT
ma-11	135	14	.	.	PUNCT
ma-11	136	1	the	the	DET
ma-11	136	2	proof	proof	NOUN
ma-11	136	3	of	of	ADP
ma-11	136	4	theorem	theorem	ADJ
ma-11	136	5	3.1	3.1	NUM
ma-11	136	6	is	be	AUX
ma-11	136	7	completed	complete	VERB
ma-11	136	8	.	.	PUNCT
ma-11	137	1	�	�	PROPN
ma-11	137	2	now	now	ADV
ma-11	137	3	,	,	PUNCT
ma-11	137	4	we	we	PRON
ma-11	137	5	give	give	VERB
ma-11	137	6	an	an	DET
ma-11	137	7	extension	extension	NOUN
ma-11	137	8	of	of	ADP
ma-11	137	9	pólya	pólya	NOUN
ma-11	137	10	–	–	PUNCT
ma-11	137	11	szegö	szegö	NOUN
ma-11	137	12	’s	’s	PART
ma-11	137	13	inequality	inequality	NOUN
ma-11	137	14	by	by	ADP
ma-11	137	15	using	use	VERB
ma-11	137	16	the	the	DET
ma-11	137	17	time	time	NOUN
ma-11	137	18	scale	scale	NOUN
ma-11	137	19	∇-riemann	∇-riemann	PROPN
ma-11	137	20	–	–	PUNCT
ma-11	137	21	liouville	liouville	NOUN
ma-11	137	22	type	type	NOUN
ma-11	137	23	fractional	fractional	ADJ
ma-11	137	24	integral	integral	ADJ
ma-11	137	25	.	.	PUNCT
ma-11	138	1	theorem	theorem	NOUN
ma-11	138	2	3.2	3.2	NUM
ma-11	138	3	.	.	PUNCT
ma-11	139	1	let	let	VERB
ma-11	139	2	w	w	NOUN
ma-11	139	3	,	,	PUNCT
ma-11	139	4	f	f	PROPN
ma-11	139	5	,	,	PUNCT
ma-11	139	6	g	g	PROPN
ma-11	139	7	∈	∈	PROPN
ma-11	139	8	cld	cld	NOUN
ma-11	139	9	(	(	PUNCT
ma-11	139	10	[	[	X
ma-11	139	11	a	a	PRON
ma-11	139	12	,	,	PUNCT
ma-11	139	13	b]t	b]t	NOUN
ma-11	139	14	,	,	PUNCT
ma-11	139	15	r−	r−	NOUN
ma-11	139	16	{	{	PUNCT
ma-11	139	17	0	0	NUM
ma-11	139	18	}	}	PUNCT
ma-11	139	19	)	)	PUNCT
ma-11	139	20	be	be	AUX
ma-11	139	21	∇-integrable	∇-integrable	ADJ
ma-11	139	22	functions	function	NOUN
ma-11	139	23	.	.	PUNCT
ma-11	140	1	assume	assume	VERB
ma-11	140	2	that	that	SCONJ
ma-11	140	3	there	there	PRON
ma-11	140	4	exist	exist	VERB
ma-11	140	5	four	four	NUM
ma-11	140	6	positive	positive	ADJ
ma-11	140	7	∇-integrable	∇-integrable	ADJ
ma-11	140	8	functions	function	NOUN
ma-11	140	9	f1	f1	NOUN
ma-11	140	10	,	,	PUNCT
ma-11	140	11	f2	f2	PROPN
ma-11	140	12	,	,	PUNCT
ma-11	140	13	g1	g1	PROPN
ma-11	140	14	and	and	CCONJ
ma-11	140	15	g2	g2	PROPN
ma-11	140	16	such	such	ADJ
ma-11	140	17	that	that	SCONJ
ma-11	140	18	:	:	PUNCT
ma-11	140	19	0	0	NUM
ma-11	140	20	<	<	X
ma-11	140	21	f1(y	f1(y	PROPN
ma-11	140	22	)	)	PUNCT
ma-11	140	23	≤	≤	NOUN
ma-11	140	24	|f	|f	PROPN
ma-11	141	1	(	(	PUNCT
ma-11	141	2	y)|	y)|	PROPN
ma-11	141	3	≤	≤	PROPN
ma-11	141	4	f2(y	f2(y	PROPN
ma-11	141	5	)	)	PUNCT
ma-11	141	6	<	<	X
ma-11	141	7	∞	∞	NUM
ma-11	141	8	and	and	CCONJ
ma-11	141	9	0	0	NUM
ma-11	141	10	<	<	X
ma-11	141	11	g1(y	g1(y	PROPN
ma-11	141	12	)	)	PUNCT
ma-11	141	13	≤	≤	PUNCT
ma-11	141	14	|g(y)|	|g(y)|	PROPN
ma-11	141	15	≤	≤	NOUN
ma-11	141	16	g2(y	g2(y	PROPN
ma-11	141	17	)	)	PUNCT
ma-11	141	18	<	<	X
ma-11	141	19	∞	∞	PROPN
ma-11	141	20	,	,	PUNCT
ma-11	141	21	(	(	PUNCT
ma-11	141	22	y	y	PROPN
ma-11	141	23	∈	∈	PROPN
ma-11	141	24	[	[	X
ma-11	141	25	a	a	X
ma-11	141	26	,	,	PUNCT
ma-11	141	27	x	x	X
ma-11	141	28	]	]	X
ma-11	141	29	t,∀x	t,∀x	X
ma-11	141	30	∈	∈	X
ma-11	142	1	[	[	X
ma-11	142	2	a	a	DET
ma-11	142	3	,	,	PUNCT
ma-11	142	4	b]t	b]t	NOUN
ma-11	142	5	)	)	PUNCT
ma-11	142	6	.	.	PUNCT
ma-11	143	1	let	let	VERB
ma-11	143	2	α	α	PRON
ma-11	143	3	,	,	PUNCT
ma-11	143	4	β	β	X
ma-11	143	5	≥	≥	NUM
ma-11	143	6	1	1	NUM
ma-11	143	7	and	and	CCONJ
ma-11	143	8	ĥα−1	ĥα−1	NOUN
ma-11	143	9	(	(	PUNCT
ma-11	143	10	.	.	PUNCT
ma-11	143	11	,	,	PUNCT
ma-11	143	12	.	.	PUNCT
ma-11	143	13	)	)	PUNCT
ma-11	143	14	,	,	PUNCT
ma-11	143	15	ĥβ−1	ĥβ−1	PROPN
ma-11	143	16	(	(	PUNCT
ma-11	143	17	.	.	PUNCT
ma-11	143	18	,	,	PUNCT
ma-11	143	19	.	.	PUNCT
ma-11	143	20	)	)	PUNCT
ma-11	144	1	>	>	X
ma-11	144	2	0	0	X
ma-11	144	3	.	.	PUNCT
ma-11	145	1	then	then	ADV
ma-11	145	2	we	we	PRON
ma-11	145	3	have	have	VERB
ma-11	145	4	the	the	DET
ma-11	145	5	following	follow	VERB
ma-11	145	6	inequality	inequality	NOUN
ma-11	145	7	j	j	PROPN
ma-11	145	8	αa	αa	INTJ
ma-11	145	9	(	(	PUNCT
ma-11	145	10	(	(	PUNCT
ma-11	145	11	f1f2)(x)|w(x)|)j	f1f2)(x)|w(x)|)j	PROPN
ma-11	145	12	βa	βa	INTJ
ma-11	145	13	(	(	PUNCT
ma-11	145	14	(	(	PUNCT
ma-11	145	15	g1g2)(x)|w(x)|)j	g1g2)(x)|w(x)|)j	NOUN
ma-11	145	16	αa	αa	PROPN
ma-11	145	17	(	(	PUNCT
ma-11	145	18	|w(x)||f	|w(x)||f	X
ma-11	145	19	(	(	PUNCT
ma-11	145	20	x)|2	x)|2	PROPN
ma-11	145	21	)	)	PUNCT
ma-11	146	1	j	j	PROPN
ma-11	146	2	βa	βa	INTJ
ma-11	146	3	(	(	PUNCT
ma-11	146	4	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	146	5	)	)	PUNCT
ma-11	146	6	{	{	PUNCT
ma-11	147	1	j	j	PROPN
ma-11	147	2	αa	αa	INTJ
ma-11	147	3	(	(	PUNCT
ma-11	147	4	f1(x)|(wf	f1(x)|(wf	NOUN
ma-11	147	5	)	)	PUNCT
ma-11	147	6	(	(	PUNCT
ma-11	147	7	x)|)j	x)|)j	PROPN
ma-11	147	8	βa	βa	INTJ
ma-11	147	9	(	(	PUNCT
ma-11	147	10	g1(x)|(wg)(x)|	g1(x)|(wg)(x)|	PROPN
ma-11	147	11	)	)	PUNCT
ma-11	147	12	+	+	CCONJ
ma-11	147	13	j	j	PROPN
ma-11	147	14	αa	αa	INTJ
ma-11	147	15	(	(	PUNCT
ma-11	147	16	f2(x)|(wf	f2(x)|(wf	PROPN
ma-11	147	17	)	)	PUNCT
ma-11	147	18	(	(	PUNCT
ma-11	147	19	x)|)j	x)|)j	PROPN
ma-11	147	20	βa	βa	INTJ
ma-11	147	21	(	(	PUNCT
ma-11	147	22	g2(x)|(wg)(x)|	g2(x)|(wg)(x)|	PROPN
ma-11	147	23	)	)	PUNCT
ma-11	147	24	}	}	PUNCT
ma-11	147	25	2	2	NUM
ma-11	147	26	≤	≤	NUM
ma-11	147	27	1	1	NUM
ma-11	147	28	4	4	NUM
ma-11	147	29	.	.	PUNCT
ma-11	148	1	(	(	PUNCT
ma-11	148	2	14	14	NUM
ma-11	148	3	)	)	PUNCT
ma-11	148	4	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	148	5	eur	eur	PROPN
ma-11	148	6	.	.	PUNCT
ma-11	149	1	j.	j.	PROPN
ma-11	149	2	math	math	PROPN
ma-11	149	3	.	.	PUNCT
ma-11	150	1	anal	anal	PROPN
ma-11	150	2	.	.	PUNCT
ma-11	151	1	10.28924	10.28924	NUM
ma-11	151	2	/	/	SYM
ma-11	151	3	ada	ada	PROPN
ma-11	151	4	/	/	SYM
ma-11	151	5	ma.3.12	ma.3.12	PROPN
ma-11	151	6	6	6	NUM
ma-11	151	7	proof	proof	NOUN
ma-11	151	8	.	.	PUNCT
ma-11	152	1	similar	similar	ADJ
ma-11	152	2	to	to	ADP
ma-11	152	3	the	the	DET
ma-11	152	4	proof	proof	NOUN
ma-11	152	5	of	of	ADP
ma-11	152	6	theorem	theorem	ADJ
ma-11	152	7	3.1	3.1	NUM
ma-11	152	8	.	.	PUNCT
ma-11	152	9	�	�	PROPN
ma-11	152	10	corollary	corollary	ADJ
ma-11	152	11	3.3	3.3	NUM
ma-11	152	12	.	.	PUNCT
ma-11	153	1	let	let	VERB
ma-11	153	2	w	w	VERB
ma-11	153	3	,	,	PUNCT
ma-11	153	4	f	f	PROPN
ma-11	153	5	,	,	PUNCT
ma-11	153	6	g	g	PROPN
ma-11	153	7	∈	∈	PROPN
ma-11	153	8	crd	crd	NOUN
ma-11	153	9	(	(	PUNCT
ma-11	153	10	[	[	X
ma-11	153	11	a	a	PRON
ma-11	153	12	,	,	PUNCT
ma-11	153	13	b]t	b]t	NOUN
ma-11	153	14	,	,	PUNCT
ma-11	153	15	r−	r−	NOUN
ma-11	153	16	{	{	PUNCT
ma-11	153	17	0	0	NUM
ma-11	153	18	}	}	PUNCT
ma-11	153	19	)	)	PUNCT
ma-11	153	20	be	be	AUX
ma-11	153	21	∆-integrable	∆-integrable	ADJ
ma-11	153	22	functions	function	NOUN
ma-11	153	23	such	such	ADJ
ma-11	153	24	that	that	SCONJ
ma-11	153	25	0	0	NUM
ma-11	153	26	<	<	X
ma-11	153	27	m	m	VERB
ma-11	153	28	≤	≤	ADJ
ma-11	153	29	|f	|f	PROPN
ma-11	154	1	(	(	PUNCT
ma-11	154	2	y)|	y)|	NOUN
ma-11	154	3	≤	≤	VERB
ma-11	154	4	m	m	VERB
ma-11	154	5	<	<	X
ma-11	154	6	∞	∞	PROPN
ma-11	154	7	and	and	CCONJ
ma-11	154	8	0	0	NUM
ma-11	154	9	<	<	X
ma-11	154	10	n	n	PRON
ma-11	154	11	≤	≤	X
ma-11	154	12	|g(y)|	|g(y)|	PROPN
ma-11	154	13	≤	≤	NOUN
ma-11	154	14	n	n	CCONJ
ma-11	154	15	<	<	X
ma-11	154	16	∞	∞	NUM
ma-11	154	17	on	on	ADP
ma-11	154	18	the	the	DET
ma-11	154	19	set	set	NOUN
ma-11	154	20	[	[	X
ma-11	154	21	a	a	X
ma-11	154	22	,	,	PUNCT
ma-11	154	23	x	x	X
ma-11	154	24	]	]	X
ma-11	154	25	t	t	PROPN
ma-11	154	26	,	,	PUNCT
ma-11	154	27	∀x	∀x	X
ma-11	154	28	∈	∈	PROPN
ma-11	155	1	[	[	X
ma-11	155	2	a	a	PRON
ma-11	155	3	,	,	PUNCT
ma-11	155	4	b]t	b]t	NOUN
ma-11	155	5	.	.	PUNCT
ma-11	156	1	let	let	VERB
ma-11	156	2	α	α	PRON
ma-11	156	3	,	,	PUNCT
ma-11	156	4	β	β	X
ma-11	156	5	≥	≥	NUM
ma-11	156	6	1	1	NUM
ma-11	156	7	and	and	CCONJ
ma-11	156	8	hα−1	hα−1	NOUN
ma-11	156	9	(	(	PUNCT
ma-11	156	10	.	.	PUNCT
ma-11	156	11	,	,	PUNCT
ma-11	156	12	.	.	PUNCT
ma-11	156	13	)	)	PUNCT
ma-11	156	14	,	,	PUNCT
ma-11	156	15	hβ−1	hβ−1	PROPN
ma-11	156	16	(	(	PUNCT
ma-11	156	17	.	.	PUNCT
ma-11	156	18	,	,	PUNCT
ma-11	156	19	.	.	PUNCT
ma-11	156	20	)	)	PUNCT
ma-11	157	1	>	>	X
ma-11	157	2	0	0	X
ma-11	157	3	.	.	PUNCT
ma-11	158	1	then	then	ADV
ma-11	158	2	we	we	PRON
ma-11	158	3	have	have	VERB
ma-11	158	4	the	the	DET
ma-11	158	5	following	follow	VERB
ma-11	158	6	inequality	inequality	NOUN
ma-11	158	7	iαa	iαa	ADJ
ma-11	158	8	(	(	PUNCT
ma-11	158	9	|w(x)|	|w(x)|	PROPN
ma-11	158	10	)	)	PUNCT
ma-11	158	11	iβa	iβa	NOUN
ma-11	158	12	(	(	PUNCT
ma-11	158	13	|w(x)|	|w(x)|	PROPN
ma-11	158	14	)	)	PUNCT
ma-11	158	15	iαa	iαa	NOUN
ma-11	158	16	(	(	PUNCT
ma-11	158	17	|w(x)||f	|w(x)||f	X
ma-11	158	18	(	(	PUNCT
ma-11	158	19	x)|2	x)|2	PROPN
ma-11	158	20	)	)	PUNCT
ma-11	158	21	iβa	iβa	NOUN
ma-11	158	22	(	(	PUNCT
ma-11	158	23	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	158	24	)	)	PUNCT
ma-11	158	25	{	{	PUNCT
ma-11	158	26	iαa	iαa	PROPN
ma-11	158	27	(	(	PUNCT
ma-11	158	28	|(wf	|(wf	X
ma-11	158	29	)	)	PUNCT
ma-11	158	30	(	(	PUNCT
ma-11	158	31	x)|	x)|	PROPN
ma-11	158	32	)	)	PUNCT
ma-11	158	33	iβa	iβa	NOUN
ma-11	158	34	(	(	PUNCT
ma-11	158	35	|(wg)(x)|	|(wg)(x)|	PROPN
ma-11	158	36	)	)	PUNCT
ma-11	158	37	}	}	PUNCT
ma-11	158	38	2	2	NUM
ma-11	158	39	≤	≤	NUM
ma-11	158	40	1	1	NUM
ma-11	158	41	4	4	NUM
ma-11	158	42	(	(	PUNCT
ma-11	158	43	√	√	PROPN
ma-11	158	44	mn	mn	PROPN
ma-11	158	45	mn	mn	PROPN
ma-11	158	46	+	+	CCONJ
ma-11	158	47	√	√	PROPN
ma-11	158	48	mn	mn	PROPN
ma-11	158	49	mn	mn	PROPN
ma-11	158	50	)	)	PUNCT
ma-11	158	51	2	2	NUM
ma-11	158	52	.	.	PUNCT
ma-11	159	1	(	(	PUNCT
ma-11	159	2	15	15	NUM
ma-11	159	3	)	)	PUNCT
ma-11	159	4	proof	proof	NOUN
ma-11	159	5	.	.	PUNCT
ma-11	160	1	putting	put	VERB
ma-11	160	2	f1	f1	NOUN
ma-11	160	3	=	=	SYM
ma-11	160	4	m	m	PROPN
ma-11	160	5	,	,	PUNCT
ma-11	160	6	f2	f2	PROPN
ma-11	160	7	=	=	SYM
ma-11	160	8	m	m	PROPN
ma-11	160	9	,	,	PUNCT
ma-11	160	10	g1	g1	PROPN
ma-11	160	11	=	=	SYM
ma-11	160	12	n	n	PROPN
ma-11	160	13	and	and	CCONJ
ma-11	160	14	g2	g2	PROPN
ma-11	160	15	=	=	SYM
ma-11	161	1	n	n	PROPN
ma-11	161	2	in	in	ADP
ma-11	161	3	theorem	theorem	NOUN
ma-11	161	4	3.1	3.1	NUM
ma-11	161	5	,	,	PUNCT
ma-11	161	6	we	we	PRON
ma-11	161	7	get	get	VERB
ma-11	161	8	the	the	DET
ma-11	161	9	inequality	inequality	NOUN
ma-11	161	10	(	(	PUNCT
ma-11	161	11	15	15	NUM
ma-11	161	12	)	)	PUNCT
ma-11	161	13	.	.	PUNCT
ma-11	162	1	�	�	PROPN
ma-11	162	2	corollary	corollary	NOUN
ma-11	162	3	3.4	3.4	NUM
ma-11	162	4	.	.	PUNCT
ma-11	163	1	let	let	VERB
ma-11	163	2	w	w	VERB
ma-11	163	3	,	,	PUNCT
ma-11	163	4	f	f	PROPN
ma-11	163	5	,	,	PUNCT
ma-11	163	6	g	g	PROPN
ma-11	163	7	∈	∈	PROPN
ma-11	163	8	cld	cld	NOUN
ma-11	163	9	(	(	PUNCT
ma-11	163	10	[	[	X
ma-11	163	11	a	a	PRON
ma-11	163	12	,	,	PUNCT
ma-11	163	13	b]t	b]t	NOUN
ma-11	163	14	,	,	PUNCT
ma-11	163	15	r−	r−	NOUN
ma-11	163	16	{	{	PUNCT
ma-11	163	17	0	0	NUM
ma-11	163	18	}	}	PUNCT
ma-11	163	19	)	)	PUNCT
ma-11	163	20	be	be	AUX
ma-11	163	21	∇-integrable	∇-integrable	ADJ
ma-11	163	22	functions	function	NOUN
ma-11	163	23	such	such	ADJ
ma-11	163	24	that	that	SCONJ
ma-11	163	25	0	0	NUM
ma-11	163	26	<	<	X
ma-11	163	27	m	m	VERB
ma-11	163	28	≤	≤	ADJ
ma-11	163	29	|f	|f	PROPN
ma-11	164	1	(	(	PUNCT
ma-11	164	2	y)|	y)|	NOUN
ma-11	164	3	≤	≤	VERB
ma-11	164	4	m	m	VERB
ma-11	164	5	<	<	X
ma-11	164	6	∞	∞	PROPN
ma-11	164	7	and	and	CCONJ
ma-11	164	8	0	0	NUM
ma-11	164	9	<	<	X
ma-11	164	10	n	n	PRON
ma-11	164	11	≤	≤	X
ma-11	164	12	|g(y)|	|g(y)|	PROPN
ma-11	164	13	≤	≤	NOUN
ma-11	164	14	n	n	CCONJ
ma-11	164	15	<	<	X
ma-11	164	16	∞	∞	NUM
ma-11	164	17	on	on	ADP
ma-11	164	18	the	the	DET
ma-11	164	19	set	set	NOUN
ma-11	164	20	[	[	X
ma-11	164	21	a	a	X
ma-11	164	22	,	,	PUNCT
ma-11	164	23	x	x	X
ma-11	164	24	]	]	X
ma-11	164	25	t	t	PROPN
ma-11	164	26	,	,	PUNCT
ma-11	164	27	∀x	∀x	X
ma-11	164	28	∈	∈	PROPN
ma-11	165	1	[	[	X
ma-11	165	2	a	a	PRON
ma-11	165	3	,	,	PUNCT
ma-11	165	4	b]t	b]t	NOUN
ma-11	165	5	.	.	PUNCT
ma-11	166	1	let	let	VERB
ma-11	166	2	α	α	PRON
ma-11	166	3	,	,	PUNCT
ma-11	166	4	β	β	X
ma-11	166	5	≥	≥	NUM
ma-11	166	6	1	1	NUM
ma-11	166	7	and	and	CCONJ
ma-11	166	8	ĥα−1	ĥα−1	NOUN
ma-11	166	9	(	(	PUNCT
ma-11	166	10	.	.	PUNCT
ma-11	166	11	,	,	PUNCT
ma-11	166	12	.	.	PUNCT
ma-11	166	13	)	)	PUNCT
ma-11	166	14	,	,	PUNCT
ma-11	166	15	ĥβ−1	ĥβ−1	PROPN
ma-11	166	16	(	(	PUNCT
ma-11	166	17	.	.	PUNCT
ma-11	166	18	,	,	PUNCT
ma-11	166	19	.	.	PUNCT
ma-11	166	20	)	)	PUNCT
ma-11	167	1	>	>	X
ma-11	167	2	0	0	X
ma-11	167	3	.	.	PUNCT
ma-11	168	1	then	then	ADV
ma-11	168	2	we	we	PRON
ma-11	168	3	have	have	VERB
ma-11	168	4	the	the	DET
ma-11	168	5	following	follow	VERB
ma-11	168	6	inequality	inequality	NOUN
ma-11	168	7	j	j	PROPN
ma-11	168	8	αa	αa	PROPN
ma-11	168	9	(	(	PUNCT
ma-11	168	10	|w(x)|)j	|w(x)|)j	NOUN
ma-11	168	11	βa	βa	X
ma-11	168	12	(	(	PUNCT
ma-11	168	13	|w(x)|)j	|w(x)|)j	NOUN
ma-11	168	14	αa	αa	PROPN
ma-11	168	15	(	(	PUNCT
ma-11	168	16	|w(x)||f	|w(x)||f	X
ma-11	168	17	(	(	PUNCT
ma-11	168	18	x)|2	x)|2	PROPN
ma-11	168	19	)	)	PUNCT
ma-11	169	1	j	j	PROPN
ma-11	169	2	βa	βa	INTJ
ma-11	169	3	(	(	PUNCT
ma-11	169	4	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	169	5	)	)	PUNCT
ma-11	169	6	{	{	PUNCT
ma-11	170	1	j	j	PROPN
ma-11	170	2	αa	αa	INTJ
ma-11	170	3	(	(	PUNCT
ma-11	170	4	|(wf	|(wf	PROPN
ma-11	170	5	)	)	PUNCT
ma-11	170	6	(	(	PUNCT
ma-11	170	7	x)|)j	x)|)j	PROPN
ma-11	170	8	βa	βa	INTJ
ma-11	170	9	(	(	PUNCT
ma-11	170	10	|(wg)(x)|	|(wg)(x)|	PROPN
ma-11	170	11	)	)	PUNCT
ma-11	170	12	}	}	PUNCT
ma-11	170	13	2	2	NUM
ma-11	170	14	≤	≤	NUM
ma-11	170	15	1	1	NUM
ma-11	170	16	4	4	NUM
ma-11	170	17	(	(	PUNCT
ma-11	170	18	√	√	PROPN
ma-11	170	19	mn	mn	PROPN
ma-11	170	20	mn	mn	PROPN
ma-11	170	21	+	+	CCONJ
ma-11	170	22	√	√	PROPN
ma-11	170	23	mn	mn	PROPN
ma-11	170	24	mn	mn	PROPN
ma-11	170	25	)	)	PUNCT
ma-11	170	26	2	2	NUM
ma-11	170	27	.	.	PUNCT
ma-11	171	1	(	(	PUNCT
ma-11	171	2	16	16	NUM
ma-11	171	3	)	)	PUNCT
ma-11	171	4	proof	proof	NOUN
ma-11	171	5	.	.	PUNCT
ma-11	172	1	similar	similar	ADJ
ma-11	172	2	to	to	ADP
ma-11	172	3	the	the	DET
ma-11	172	4	proof	proof	NOUN
ma-11	172	5	of	of	ADP
ma-11	172	6	corollary	corollary	ADJ
ma-11	172	7	3.3	3.3	NUM
ma-11	172	8	.	.	PUNCT
ma-11	173	1	�	�	PROPN
ma-11	173	2	remark	remark	VERB
ma-11	173	3	3.1	3.1	NUM
ma-11	173	4	.	.	PUNCT
ma-11	174	1	let	let	VERB
ma-11	174	2	t	t	NOUN
ma-11	174	3	=	=	SYM
ma-11	174	4	r	r	PROPN
ma-11	174	5	,	,	PUNCT
ma-11	174	6	α	α	NOUN
ma-11	174	7	,	,	PUNCT
ma-11	174	8	β	β	X
ma-11	174	9	>	>	X
ma-11	174	10	0	0	PROPN
ma-11	174	11	,	,	PUNCT
ma-11	174	12	a	a	DET
ma-11	174	13	=	=	SYM
ma-11	174	14	0	0	NUM
ma-11	174	15	,	,	PUNCT
ma-11	174	16	x	x	X
ma-11	174	17	>	>	X
ma-11	174	18	0	0	NUM
ma-11	174	19	,	,	PUNCT
ma-11	174	20	w	w	PROPN
ma-11	174	21	≡	≡	PROPN
ma-11	174	22	1	1	NUM
ma-11	174	23	,	,	PUNCT
ma-11	174	24	f	f	PROPN
ma-11	174	25	>	>	X
ma-11	174	26	0	0	PROPN
ma-11	174	27	and	and	CCONJ
ma-11	174	28	g	g	PROPN
ma-11	174	29	>	>	X
ma-11	174	30	0	0	PROPN
ma-11	174	31	.	.	PUNCT
ma-11	175	1	then	then	ADV
ma-11	175	2	(	(	PUNCT
ma-11	175	3	10	10	NUM
ma-11	175	4	)	)	PUNCT
ma-11	175	5	reduces	reduce	VERB
ma-11	175	6	to	to	PART
ma-11	175	7	iα0	iα0	PROPN
ma-11	175	8	(	(	PUNCT
ma-11	175	9	(	(	PUNCT
ma-11	175	10	f1f2)(x	f1f2)(x	PROPN
ma-11	175	11	)	)	PUNCT
ma-11	175	12	)	)	PUNCT
ma-11	176	1	iβ0	iβ0	NOUN
ma-11	176	2	(	(	PUNCT
ma-11	176	3	(	(	PUNCT
ma-11	176	4	g1g2)(x	g1g2)(x	NOUN
ma-11	176	5	)	)	PUNCT
ma-11	176	6	)	)	PUNCT
ma-11	177	1	iα0	iα0	PROPN
ma-11	177	2	(	(	PUNCT
ma-11	177	3	f	f	PROPN
ma-11	177	4	2(x	2(x	NUM
ma-11	177	5	)	)	PUNCT
ma-11	177	6	)	)	PUNCT
ma-11	178	1	iβ0	iβ0	NOUN
ma-11	178	2	(	(	PUNCT
ma-11	178	3	g2(x	g2(x	NOUN
ma-11	178	4	)	)	PUNCT
ma-11	178	5	)	)	PUNCT
ma-11	178	6	{	{	PUNCT
ma-11	178	7	iα0	iα0	PUNCT
ma-11	178	8	(	(	PUNCT
ma-11	178	9	(	(	PUNCT
ma-11	178	10	f1f	f1f	NOUN
ma-11	178	11	)	)	PUNCT
ma-11	178	12	(	(	PUNCT
ma-11	178	13	x	x	NOUN
ma-11	178	14	)	)	PUNCT
ma-11	178	15	)	)	PUNCT
ma-11	178	16	iβ0	iβ0	NOUN
ma-11	178	17	(	(	PUNCT
ma-11	178	18	(	(	PUNCT
ma-11	178	19	g1g)(x	g1g)(x	PROPN
ma-11	178	20	)	)	PUNCT
ma-11	178	21	)	)	PUNCT
ma-11	179	1	+	+	CCONJ
ma-11	179	2	iα0	iα0	INTJ
ma-11	179	3	(	(	PUNCT
ma-11	179	4	(	(	PUNCT
ma-11	179	5	f2f	f2f	NOUN
ma-11	179	6	)	)	PUNCT
ma-11	179	7	(	(	PUNCT
ma-11	179	8	x	x	NOUN
ma-11	179	9	)	)	PUNCT
ma-11	179	10	)	)	PUNCT
ma-11	179	11	iβ0	iβ0	NOUN
ma-11	179	12	(	(	PUNCT
ma-11	179	13	(	(	PUNCT
ma-11	179	14	g2g)(x	g2g)(x	PROPN
ma-11	179	15	)	)	PUNCT
ma-11	179	16	)	)	PUNCT
ma-11	179	17	}	}	PUNCT
ma-11	179	18	2	2	NUM
ma-11	179	19	≤	≤	NUM
ma-11	179	20	1	1	NUM
ma-11	179	21	4	4	NUM
ma-11	179	22	,	,	PUNCT
ma-11	179	23	(	(	PUNCT
ma-11	179	24	17	17	NUM
ma-11	179	25	)	)	PUNCT
ma-11	179	26	as	as	SCONJ
ma-11	179	27	given	give	VERB
ma-11	179	28	in	in	ADP
ma-11	179	29	[	[	NOUN
ma-11	179	30	14	14	NUM
ma-11	179	31	]	]	PUNCT
ma-11	179	32	.	.	PUNCT
ma-11	180	1	theorem	theorem	ADJ
ma-11	180	2	3.5	3.5	NUM
ma-11	180	3	.	.	PUNCT
ma-11	181	1	let	let	VERB
ma-11	181	2	w	w	VERB
ma-11	181	3	,	,	PUNCT
ma-11	181	4	f	f	PROPN
ma-11	181	5	,	,	PUNCT
ma-11	181	6	g	g	PROPN
ma-11	181	7	∈	∈	PROPN
ma-11	181	8	crd	crd	NOUN
ma-11	181	9	(	(	PUNCT
ma-11	181	10	[	[	X
ma-11	181	11	a	a	PRON
ma-11	181	12	,	,	PUNCT
ma-11	181	13	b]t	b]t	NOUN
ma-11	181	14	,	,	PUNCT
ma-11	181	15	r−	r−	NOUN
ma-11	181	16	{	{	PUNCT
ma-11	181	17	0	0	NUM
ma-11	181	18	}	}	PUNCT
ma-11	181	19	)	)	PUNCT
ma-11	181	20	be	be	AUX
ma-11	181	21	∆-integrable	∆-integrable	ADJ
ma-11	181	22	functions	function	NOUN
ma-11	181	23	.	.	PUNCT
ma-11	182	1	assume	assume	VERB
ma-11	182	2	that	that	SCONJ
ma-11	182	3	there	there	PRON
ma-11	182	4	exist	exist	VERB
ma-11	182	5	four	four	NUM
ma-11	182	6	positive	positive	ADJ
ma-11	182	7	∆-integrable	∆-integrable	ADJ
ma-11	182	8	functions	function	NOUN
ma-11	182	9	f1	f1	NOUN
ma-11	182	10	,	,	PUNCT
ma-11	182	11	f2	f2	PROPN
ma-11	182	12	,	,	PUNCT
ma-11	182	13	g1	g1	PROPN
ma-11	182	14	and	and	CCONJ
ma-11	182	15	g2	g2	PROPN
ma-11	182	16	such	such	ADJ
ma-11	182	17	that	that	SCONJ
ma-11	182	18	:	:	PUNCT
ma-11	182	19	0	0	NUM
ma-11	182	20	<	<	X
ma-11	182	21	f1(y	f1(y	PROPN
ma-11	182	22	)	)	PUNCT
ma-11	182	23	≤	≤	NOUN
ma-11	182	24	|f	|f	PROPN
ma-11	183	1	(	(	PUNCT
ma-11	183	2	y)|	y)|	PROPN
ma-11	183	3	≤	≤	PROPN
ma-11	183	4	f2(y	f2(y	PROPN
ma-11	183	5	)	)	PUNCT
ma-11	183	6	<	<	X
ma-11	183	7	∞	∞	NUM
ma-11	183	8	and	and	CCONJ
ma-11	183	9	0	0	NUM
ma-11	183	10	<	<	X
ma-11	183	11	g1(y	g1(y	PROPN
ma-11	183	12	)	)	PUNCT
ma-11	183	13	≤	≤	PUNCT
ma-11	183	14	|g(y)|	|g(y)|	PROPN
ma-11	183	15	≤	≤	NOUN
ma-11	183	16	g2(y	g2(y	PROPN
ma-11	183	17	)	)	PUNCT
ma-11	183	18	<	<	X
ma-11	183	19	∞	∞	PROPN
ma-11	183	20	,	,	PUNCT
ma-11	183	21	(	(	PUNCT
ma-11	183	22	y	y	PROPN
ma-11	183	23	∈	∈	PROPN
ma-11	183	24	[	[	X
ma-11	183	25	a	a	X
ma-11	183	26	,	,	PUNCT
ma-11	183	27	x	x	X
ma-11	183	28	]	]	X
ma-11	183	29	t,∀x	t,∀x	X
ma-11	183	30	∈	∈	X
ma-11	184	1	[	[	X
ma-11	184	2	a	a	DET
ma-11	184	3	,	,	PUNCT
ma-11	184	4	b]t	b]t	NOUN
ma-11	184	5	)	)	PUNCT
ma-11	184	6	.	.	PUNCT
ma-11	185	1	let	let	VERB
ma-11	185	2	α	α	PRON
ma-11	185	3	,	,	PUNCT
ma-11	185	4	β	β	X
ma-11	185	5	≥	≥	NUM
ma-11	185	6	1	1	NUM
ma-11	185	7	and	and	CCONJ
ma-11	185	8	hα−1	hα−1	NOUN
ma-11	185	9	(	(	PUNCT
ma-11	185	10	.	.	PUNCT
ma-11	185	11	,	,	PUNCT
ma-11	185	12	.	.	PUNCT
ma-11	185	13	)	)	PUNCT
ma-11	185	14	,	,	PUNCT
ma-11	185	15	hβ−1	hβ−1	PROPN
ma-11	185	16	(	(	PUNCT
ma-11	185	17	.	.	PUNCT
ma-11	185	18	,	,	PUNCT
ma-11	185	19	.	.	PUNCT
ma-11	185	20	)	)	PUNCT
ma-11	186	1	>	>	X
ma-11	186	2	0	0	X
ma-11	186	3	.	.	PUNCT
ma-11	187	1	then	then	ADV
ma-11	187	2	we	we	PRON
ma-11	187	3	have	have	VERB
ma-11	187	4	the	the	DET
ma-11	187	5	following	follow	VERB
ma-11	187	6	inequality	inequality	NOUN
ma-11	187	7	iαa	iαa	ADJ
ma-11	187	8	(	(	PUNCT
ma-11	187	9	|w(x)||f	|w(x)||f	X
ma-11	187	10	(	(	PUNCT
ma-11	187	11	x)|2	x)|2	PROPN
ma-11	187	12	)	)	PUNCT
ma-11	187	13	iβa	iβa	NOUN
ma-11	187	14	(	(	PUNCT
ma-11	187	15	|w(x)||g(x)|2	|w(x)||g(x)|2	X
ma-11	187	16	)	)	PUNCT
ma-11	187	17	≤	≤	NOUN
ma-11	187	18	iαa	iαa	NOUN
ma-11	187	19	(	(	PUNCT
ma-11	187	20	f2(x	f2(x	PROPN
ma-11	187	21	)	)	PUNCT
ma-11	187	22	g1(x	g1(x	NOUN
ma-11	187	23	)	)	PUNCT
ma-11	187	24	|(wf	|(wf	PROPN
ma-11	187	25	g)(x)|	g)(x)|	PROPN
ma-11	187	26	)	)	PUNCT
ma-11	187	27	iβa	iβa	NOUN
ma-11	187	28	(	(	PUNCT
ma-11	187	29	g2(x	g2(x	NOUN
ma-11	187	30	)	)	PUNCT
ma-11	187	31	f1(x	f1(x	NUM
ma-11	187	32	)	)	PUNCT
ma-11	188	1	|(wf	|(wf	PROPN
ma-11	188	2	g)(x)|	g)(x)|	PROPN
ma-11	188	3	)	)	PUNCT
ma-11	188	4	.	.	PUNCT
ma-11	189	1	(	(	PUNCT
ma-11	189	2	18	18	NUM
ma-11	189	3	)	)	PUNCT
ma-11	189	4	proof	proof	NOUN
ma-11	189	5	.	.	PUNCT
ma-11	190	1	using	use	VERB
ma-11	190	2	the	the	DET
ma-11	190	3	given	give	VERB
ma-11	190	4	condition	condition	NOUN
ma-11	190	5	,	,	PUNCT
ma-11	190	6	for	for	ADP
ma-11	190	7	y	y	PROPN
ma-11	190	8	∈	∈	PROPN
ma-11	190	9	[	[	X
ma-11	190	10	a	a	X
ma-11	190	11	,	,	PUNCT
ma-11	190	12	x	x	X
ma-11	190	13	]	]	X
ma-11	190	14	t	t	PROPN
ma-11	190	15	,	,	PUNCT
ma-11	190	16	∀x	∀x	X
ma-11	190	17	∈	∈	PROPN
ma-11	190	18	[	[	X
ma-11	190	19	a	a	DET
ma-11	190	20	,	,	PUNCT
ma-11	190	21	b]t	b]t	NOUN
ma-11	190	22	,	,	PUNCT
ma-11	190	23	we	we	PRON
ma-11	190	24	have	have	VERB
ma-11	190	25	|f	|f	PROPN
ma-11	190	26	(	(	PUNCT
ma-11	190	27	y)|2	y)|2	PROPN
ma-11	190	28	≤	≤	NUM
ma-11	190	29	f2(y	f2(y	PROPN
ma-11	190	30	)	)	PUNCT
ma-11	190	31	g1(y	g1(y	PROPN
ma-11	190	32	)	)	PUNCT
ma-11	190	33	|f	|f	PROPN
ma-11	191	1	(	(	PUNCT
ma-11	191	2	y)g(y)|	y)g(y)|	PROPN
ma-11	191	3	.	.	PUNCT
ma-11	192	1	multiplying	multiply	VERB
ma-11	192	2	both	both	DET
ma-11	192	3	sides	side	NOUN
ma-11	192	4	of	of	ADP
ma-11	192	5	the	the	DET
ma-11	192	6	last	last	ADJ
ma-11	192	7	inequality	inequality	NOUN
ma-11	192	8	by	by	ADP
ma-11	192	9	hα−1(x	hα−1(x	PROPN
ma-11	192	10	,	,	PUNCT
ma-11	192	11	σ(y))|w(y)|	σ(y))|w(y)|	PROPN
ma-11	192	12	and	and	CCONJ
ma-11	192	13	integrating	integrate	VERB
ma-11	192	14	over	over	ADP
ma-11	192	15	y	y	PROPN
ma-11	192	16	from	from	ADP
ma-11	192	17	a	a	DET
ma-11	192	18	to	to	PART
ma-11	192	19	x	x	SYM
ma-11	192	20	,	,	PUNCT
ma-11	192	21	we	we	PRON
ma-11	192	22	have∫	have∫	VERB
ma-11	192	23	x	x	DET
ma-11	192	24	a	a	DET
ma-11	192	25	hα−1(x	hα−1(x	PROPN
ma-11	192	26	,	,	PUNCT
ma-11	192	27	σ(y))|w(y)||f	σ(y))|w(y)||f	PROPN
ma-11	192	28	(	(	PUNCT
ma-11	192	29	y)|2∆y	y)|2∆y	PROPN
ma-11	192	30	≤	≤	PROPN
ma-11	192	31	∫	∫	PROPN
ma-11	192	32	x	x	X
ma-11	192	33	a	a	DET
ma-11	192	34	hα−1(x	hα−1(x	NOUN
ma-11	192	35	,	,	PUNCT
ma-11	192	36	σ(y	σ(y	PROPN
ma-11	192	37	)	)	PUNCT
ma-11	192	38	)	)	PUNCT
ma-11	192	39	f2(y	f2(y	PROPN
ma-11	192	40	)	)	PUNCT
ma-11	192	41	g1(y	g1(y	PROPN
ma-11	192	42	)	)	PUNCT
ma-11	192	43	|w(y)||f	|w(y)||f	NOUN
ma-11	192	44	(	(	PUNCT
ma-11	192	45	y)g(y)|∆y	y)g(y)|∆y	NOUN
ma-11	192	46	.	.	PUNCT
ma-11	193	1	(	(	PUNCT
ma-11	193	2	19	19	NUM
ma-11	193	3	)	)	PUNCT
ma-11	193	4	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	193	5	eur	eur	PROPN
ma-11	193	6	.	.	PUNCT
ma-11	194	1	j.	j.	PROPN
ma-11	194	2	math	math	PROPN
ma-11	194	3	.	.	PUNCT
ma-11	195	1	anal	anal	PROPN
ma-11	195	2	.	.	PUNCT
ma-11	196	1	10.28924	10.28924	NUM
ma-11	196	2	/	/	SYM
ma-11	196	3	ada	ada	PROPN
ma-11	196	4	/	/	SYM
ma-11	196	5	ma.3.12	ma.3.12	NOUN
ma-11	196	6	7the	7the	PROPN
ma-11	196	7	inequality	inequality	NOUN
ma-11	196	8	(	(	PUNCT
ma-11	196	9	19	19	NUM
ma-11	196	10	)	)	PUNCT
ma-11	196	11	takes	take	VERB
ma-11	196	12	the	the	DET
ma-11	196	13	form	form	NOUN
ma-11	196	14	iαa	iαa	ADJ
ma-11	196	15	(	(	PUNCT
ma-11	196	16	|w(x)||f	|w(x)||f	X
ma-11	196	17	(	(	PUNCT
ma-11	196	18	x)|2	x)|2	PROPN
ma-11	196	19	)	)	PUNCT
ma-11	196	20	≤	≤	ADJ
ma-11	196	21	iαa	iαa	NOUN
ma-11	196	22	(	(	PUNCT
ma-11	196	23	f2(x	f2(x	NOUN
ma-11	196	24	)	)	PUNCT
ma-11	196	25	g1(x	g1(x	NOUN
ma-11	196	26	)	)	PUNCT
ma-11	196	27	|w(x)||f	|w(x)||f	NOUN
ma-11	196	28	(	(	PUNCT
ma-11	196	29	x)g(x)|	x)g(x)|	PROPN
ma-11	196	30	)	)	PUNCT
ma-11	196	31	.	.	PUNCT
ma-11	197	1	(	(	PUNCT
ma-11	197	2	20	20	NUM
ma-11	197	3	)	)	PUNCT
ma-11	197	4	similarly	similarly	ADV
ma-11	197	5	,	,	PUNCT
ma-11	197	6	we	we	PRON
ma-11	197	7	have	have	VERB
ma-11	197	8	that	that	DET
ma-11	197	9	iβa	iβa	NOUN
ma-11	197	10	(	(	PUNCT
ma-11	197	11	|w(x)||g(x)|2	|w(x)||g(x)|2	X
ma-11	197	12	)	)	PUNCT
ma-11	197	13	≤	≤	PROPN
ma-11	197	14	iβa	iβa	NOUN
ma-11	197	15	(	(	PUNCT
ma-11	197	16	g2(x	g2(x	NOUN
ma-11	197	17	)	)	PUNCT
ma-11	197	18	f1(x	f1(x	NOUN
ma-11	197	19	)	)	PUNCT
ma-11	197	20	|w(x)||f	|w(x)||f	NOUN
ma-11	197	21	(	(	PUNCT
ma-11	197	22	x)g(x)|	x)g(x)|	PROPN
ma-11	197	23	)	)	PUNCT
ma-11	197	24	.	.	PUNCT
ma-11	198	1	(	(	PUNCT
ma-11	198	2	21	21	NUM
ma-11	198	3	)	)	PUNCT
ma-11	198	4	multiplying	multiplying	NOUN
ma-11	198	5	(	(	PUNCT
ma-11	198	6	20	20	NUM
ma-11	198	7	)	)	PUNCT
ma-11	198	8	and	and	CCONJ
ma-11	198	9	(	(	PUNCT
ma-11	198	10	21	21	NUM
ma-11	198	11	)	)	PUNCT
ma-11	198	12	,	,	PUNCT
ma-11	198	13	we	we	PRON
ma-11	198	14	get	get	VERB
ma-11	198	15	the	the	DET
ma-11	198	16	desired	desire	VERB
ma-11	198	17	inequality	inequality	NOUN
ma-11	198	18	(	(	PUNCT
ma-11	198	19	18	18	NUM
ma-11	198	20	)	)	PUNCT
ma-11	198	21	.	.	PUNCT
ma-11	199	1	�	�	PROPN
ma-11	199	2	theorem	theorem	VERB
ma-11	199	3	3.6	3.6	NUM
ma-11	199	4	.	.	PUNCT
ma-11	200	1	let	let	VERB
ma-11	200	2	w	w	VERB
ma-11	200	3	,	,	PUNCT
ma-11	200	4	f	f	PROPN
ma-11	200	5	,	,	PUNCT
ma-11	200	6	g	g	PROPN
ma-11	200	7	∈	∈	PROPN
ma-11	200	8	cld	cld	NOUN
ma-11	200	9	(	(	PUNCT
ma-11	200	10	[	[	X
ma-11	200	11	a	a	PRON
ma-11	200	12	,	,	PUNCT
ma-11	200	13	b]t	b]t	NOUN
ma-11	200	14	,	,	PUNCT
ma-11	200	15	r−	r−	NOUN
ma-11	200	16	{	{	PUNCT
ma-11	200	17	0	0	NUM
ma-11	200	18	}	}	PUNCT
ma-11	200	19	)	)	PUNCT
ma-11	200	20	be	be	AUX
ma-11	200	21	∇-integrable	∇-integrable	ADJ
ma-11	200	22	functions	function	NOUN
ma-11	200	23	.	.	PUNCT
ma-11	201	1	assume	assume	VERB
ma-11	201	2	that	that	SCONJ
ma-11	201	3	there	there	PRON
ma-11	201	4	exist	exist	VERB
ma-11	201	5	four	four	NUM
ma-11	201	6	positive	positive	ADJ
ma-11	201	7	∇-integrable	∇-integrable	ADJ
ma-11	201	8	functions	function	NOUN
ma-11	201	9	f1	f1	NOUN
ma-11	201	10	,	,	PUNCT
ma-11	201	11	f2	f2	PROPN
ma-11	201	12	,	,	PUNCT
ma-11	201	13	g1	g1	PROPN
ma-11	201	14	and	and	CCONJ
ma-11	201	15	g2	g2	PROPN
ma-11	201	16	such	such	ADJ
ma-11	201	17	that	that	SCONJ
ma-11	201	18	:	:	PUNCT
ma-11	201	19	0	0	NUM
ma-11	201	20	<	<	X
ma-11	201	21	f1(y	f1(y	PROPN
ma-11	201	22	)	)	PUNCT
ma-11	201	23	≤	≤	NOUN
ma-11	201	24	|f	|f	PROPN
ma-11	202	1	(	(	PUNCT
ma-11	202	2	y)|	y)|	PROPN
ma-11	202	3	≤	≤	PROPN
ma-11	202	4	f2(y	f2(y	PROPN
ma-11	202	5	)	)	PUNCT
ma-11	202	6	<	<	X
ma-11	202	7	∞	∞	NUM
ma-11	202	8	and	and	CCONJ
ma-11	202	9	0	0	NUM
ma-11	202	10	<	<	X
ma-11	202	11	g1(y	g1(y	PROPN
ma-11	202	12	)	)	PUNCT
ma-11	202	13	≤	≤	PUNCT
ma-11	202	14	|g(y)|	|g(y)|	PROPN
ma-11	202	15	≤	≤	NOUN
ma-11	202	16	g2(y	g2(y	PROPN
ma-11	202	17	)	)	PUNCT
ma-11	202	18	<	<	X
ma-11	202	19	∞	∞	PROPN
ma-11	202	20	,	,	PUNCT
ma-11	202	21	(	(	PUNCT
ma-11	202	22	y	y	PROPN
ma-11	202	23	∈	∈	PROPN
ma-11	202	24	[	[	X
ma-11	202	25	a	a	X
ma-11	202	26	,	,	PUNCT
ma-11	202	27	x	x	X
ma-11	202	28	]	]	X
ma-11	202	29	t,∀x	t,∀x	X
ma-11	202	30	∈	∈	X
ma-11	203	1	[	[	X
ma-11	203	2	a	a	DET
ma-11	203	3	,	,	PUNCT
ma-11	203	4	b]t	b]t	NOUN
ma-11	203	5	)	)	PUNCT
ma-11	203	6	.	.	PUNCT
ma-11	204	1	let	let	VERB
ma-11	204	2	α	α	PRON
ma-11	204	3	,	,	PUNCT
ma-11	204	4	β	β	X
ma-11	204	5	≥	≥	NUM
ma-11	204	6	1	1	NUM
ma-11	204	7	and	and	CCONJ
ma-11	204	8	ĥα−1	ĥα−1	NOUN
ma-11	204	9	(	(	PUNCT
ma-11	204	10	.	.	PUNCT
ma-11	204	11	,	,	PUNCT
ma-11	204	12	.	.	PUNCT
ma-11	204	13	)	)	PUNCT
ma-11	204	14	,	,	PUNCT
ma-11	204	15	ĥβ−1	ĥβ−1	PROPN
ma-11	204	16	(	(	PUNCT
ma-11	204	17	.	.	PUNCT
ma-11	204	18	,	,	PUNCT
ma-11	204	19	.	.	PUNCT
ma-11	204	20	)	)	PUNCT
ma-11	205	1	>	>	X
ma-11	205	2	0	0	X
ma-11	205	3	.	.	PUNCT
ma-11	206	1	then	then	ADV
ma-11	206	2	we	we	PRON
ma-11	206	3	have	have	VERB
ma-11	206	4	the	the	DET
ma-11	206	5	following	follow	VERB
ma-11	206	6	inequality	inequality	NOUN
ma-11	206	7	j	j	PROPN
ma-11	206	8	αa	αa	INTJ
ma-11	206	9	(	(	PUNCT
ma-11	206	10	|w(x)||f	|w(x)||f	X
ma-11	206	11	(	(	PUNCT
ma-11	206	12	x)|2	x)|2	PROPN
ma-11	206	13	)	)	PUNCT
ma-11	207	1	j	j	PROPN
ma-11	207	2	βa	βa	INTJ
ma-11	207	3	(	(	PUNCT
ma-11	207	4	|w(x)||g(x)|2	|w(x)||g(x)|2	X
ma-11	207	5	)	)	PUNCT
ma-11	207	6	≤	≤	NUM
ma-11	208	1	j	j	PROPN
ma-11	208	2	αa	αa	PROPN
ma-11	208	3	(	(	PUNCT
ma-11	208	4	f2(x	f2(x	NOUN
ma-11	208	5	)	)	PUNCT
ma-11	208	6	g1(x	g1(x	NOUN
ma-11	208	7	)	)	PUNCT
ma-11	209	1	|(wf	|(wf	PROPN
ma-11	209	2	g)(x)|	g)(x)|	PROPN
ma-11	209	3	)	)	PUNCT
ma-11	210	1	j	j	PROPN
ma-11	210	2	βa	βa	INTJ
ma-11	210	3	(	(	PUNCT
ma-11	210	4	g2(x	g2(x	NOUN
ma-11	210	5	)	)	PUNCT
ma-11	210	6	f1(x	f1(x	NUM
ma-11	210	7	)	)	PUNCT
ma-11	210	8	|(wf	|(wf	PROPN
ma-11	210	9	g)(x)|	g)(x)|	PROPN
ma-11	210	10	)	)	PUNCT
ma-11	210	11	.(22	.(22	PUNCT
ma-11	210	12	)	)	PUNCT
ma-11	211	1	proof	proof	NOUN
ma-11	211	2	.	.	PUNCT
ma-11	212	1	similar	similar	ADJ
ma-11	212	2	to	to	ADP
ma-11	212	3	the	the	DET
ma-11	212	4	proof	proof	NOUN
ma-11	212	5	of	of	ADP
ma-11	212	6	theorem	theorem	ADJ
ma-11	212	7	3.5	3.5	NUM
ma-11	212	8	.	.	PUNCT
ma-11	213	1	�	�	PROPN
ma-11	213	2	corollary	corollary	PROPN
ma-11	213	3	3.7	3.7	NUM
ma-11	213	4	.	.	PUNCT
ma-11	214	1	let	let	VERB
ma-11	214	2	w	w	VERB
ma-11	214	3	,	,	PUNCT
ma-11	214	4	f	f	PROPN
ma-11	214	5	,	,	PUNCT
ma-11	214	6	g	g	PROPN
ma-11	214	7	∈	∈	PROPN
ma-11	214	8	crd	crd	NOUN
ma-11	214	9	(	(	PUNCT
ma-11	214	10	[	[	X
ma-11	214	11	a	a	PRON
ma-11	214	12	,	,	PUNCT
ma-11	214	13	b]t	b]t	NOUN
ma-11	214	14	,	,	PUNCT
ma-11	214	15	r−	r−	NOUN
ma-11	214	16	{	{	PUNCT
ma-11	214	17	0	0	NUM
ma-11	214	18	}	}	PUNCT
ma-11	214	19	)	)	PUNCT
ma-11	214	20	be	be	AUX
ma-11	214	21	∆-integrable	∆-integrable	ADJ
ma-11	214	22	functions	function	NOUN
ma-11	214	23	.	.	PUNCT
ma-11	215	1	assume	assume	VERB
ma-11	215	2	that	that	SCONJ
ma-11	215	3	there	there	PRON
ma-11	215	4	exist	exist	VERB
ma-11	215	5	four	four	NUM
ma-11	215	6	positive	positive	ADJ
ma-11	215	7	constants	constant	NOUN
ma-11	215	8	m	m	PROPN
ma-11	215	9	,	,	PUNCT
ma-11	215	10	m	m	VERB
ma-11	215	11	,	,	PUNCT
ma-11	215	12	n	n	PROPN
ma-11	215	13	and	and	CCONJ
ma-11	215	14	n	n	CCONJ
ma-11	216	1	such	such	ADJ
ma-11	216	2	that	that	SCONJ
ma-11	216	3	0	0	NUM
ma-11	216	4	<	<	X
ma-11	216	5	m	m	VERB
ma-11	216	6	≤	≤	ADJ
ma-11	216	7	|f	|f	PROPN
ma-11	217	1	(	(	PUNCT
ma-11	217	2	y)|	y)|	NOUN
ma-11	217	3	≤	≤	VERB
ma-11	217	4	m	m	VERB
ma-11	217	5	<	<	X
ma-11	217	6	∞	∞	PROPN
ma-11	217	7	and	and	CCONJ
ma-11	217	8	0	0	NUM
ma-11	217	9	<	<	X
ma-11	217	10	n	n	PRON
ma-11	217	11	≤	≤	X
ma-11	217	12	|g(y)|	|g(y)|	PROPN
ma-11	217	13	≤	≤	NOUN
ma-11	217	14	n	n	CCONJ
ma-11	217	15	<	<	X
ma-11	217	16	∞	∞	NUM
ma-11	217	17	on	on	ADP
ma-11	217	18	the	the	DET
ma-11	217	19	set	set	NOUN
ma-11	217	20	[	[	X
ma-11	217	21	a	a	X
ma-11	217	22	,	,	PUNCT
ma-11	217	23	x	x	X
ma-11	217	24	]	]	X
ma-11	217	25	t	t	PROPN
ma-11	217	26	,	,	PUNCT
ma-11	217	27	∀x	∀x	X
ma-11	217	28	∈	∈	PROPN
ma-11	218	1	[	[	X
ma-11	218	2	a	a	PRON
ma-11	218	3	,	,	PUNCT
ma-11	218	4	b]t	b]t	NOUN
ma-11	218	5	.	.	PUNCT
ma-11	219	1	let	let	VERB
ma-11	219	2	α	α	PRON
ma-11	219	3	,	,	PUNCT
ma-11	219	4	β	β	X
ma-11	219	5	≥	≥	NUM
ma-11	219	6	1	1	NUM
ma-11	219	7	and	and	CCONJ
ma-11	219	8	hα−1	hα−1	NOUN
ma-11	219	9	(	(	PUNCT
ma-11	219	10	.	.	PUNCT
ma-11	219	11	,	,	PUNCT
ma-11	219	12	.	.	PUNCT
ma-11	219	13	)	)	PUNCT
ma-11	219	14	,	,	PUNCT
ma-11	219	15	hβ−1	hβ−1	PROPN
ma-11	219	16	(	(	PUNCT
ma-11	219	17	.	.	PUNCT
ma-11	219	18	,	,	PUNCT
ma-11	219	19	.	.	PUNCT
ma-11	219	20	)	)	PUNCT
ma-11	220	1	>	>	X
ma-11	220	2	0	0	X
ma-11	220	3	.	.	PUNCT
ma-11	221	1	then	then	ADV
ma-11	221	2	we	we	PRON
ma-11	221	3	have	have	VERB
ma-11	221	4	the	the	DET
ma-11	221	5	following	follow	VERB
ma-11	221	6	inequality	inequality	NOUN
ma-11	221	7	iαa	iαa	ADJ
ma-11	221	8	(	(	PUNCT
ma-11	221	9	|w(x)||f	|w(x)||f	X
ma-11	221	10	(	(	PUNCT
ma-11	221	11	x)|2	x)|2	PROPN
ma-11	221	12	)	)	PUNCT
ma-11	221	13	iβa	iβa	NOUN
ma-11	221	14	(	(	PUNCT
ma-11	221	15	|w(x)||g(x)|2	|w(x)||g(x)|2	X
ma-11	221	16	)	)	PUNCT
ma-11	221	17	iαa	iαa	PROPN
ma-11	221	18	(	(	PUNCT
ma-11	221	19	|(wf	|(wf	PROPN
ma-11	221	20	g)(x)|	g)(x)|	PROPN
ma-11	221	21	)	)	PUNCT
ma-11	221	22	iβa	iβa	NOUN
ma-11	221	23	(	(	PUNCT
ma-11	221	24	|(wf	|(wf	PROPN
ma-11	221	25	g)(x)|	g)(x)|	PROPN
ma-11	221	26	)	)	PUNCT
ma-11	221	27	≤	≤	PROPN
ma-11	222	1	mn	mn	PROPN
ma-11	222	2	mn	mn	PROPN
ma-11	222	3	.	.	PUNCT
ma-11	223	1	(	(	PUNCT
ma-11	223	2	23	23	X
ma-11	223	3	)	)	PUNCT
ma-11	223	4	proof	proof	NOUN
ma-11	223	5	.	.	PUNCT
ma-11	224	1	putting	put	VERB
ma-11	224	2	f1	f1	NOUN
ma-11	224	3	=	=	SYM
ma-11	224	4	m	m	PROPN
ma-11	224	5	,	,	PUNCT
ma-11	224	6	f2	f2	PROPN
ma-11	224	7	=	=	SYM
ma-11	224	8	m	m	PROPN
ma-11	224	9	,	,	PUNCT
ma-11	224	10	g1	g1	PROPN
ma-11	224	11	=	=	SYM
ma-11	224	12	n	n	PROPN
ma-11	224	13	and	and	CCONJ
ma-11	224	14	g2	g2	PROPN
ma-11	224	15	=	=	SYM
ma-11	225	1	n	n	PROPN
ma-11	225	2	in	in	ADP
ma-11	225	3	theorem	theorem	NOUN
ma-11	225	4	3.5	3.5	NUM
ma-11	225	5	,	,	PUNCT
ma-11	225	6	we	we	PRON
ma-11	225	7	get	get	VERB
ma-11	225	8	the	the	DET
ma-11	225	9	desired	desire	VERB
ma-11	225	10	inequality	inequality	NOUN
ma-11	225	11	.	.	PUNCT
ma-11	226	1	�	�	PROPN
ma-11	226	2	corollary	corollary	ADJ
ma-11	226	3	3.8	3.8	NUM
ma-11	226	4	.	.	PUNCT
ma-11	227	1	let	let	VERB
ma-11	227	2	w	w	VERB
ma-11	227	3	,	,	PUNCT
ma-11	227	4	f	f	PROPN
ma-11	227	5	,	,	PUNCT
ma-11	227	6	g	g	PROPN
ma-11	227	7	∈	∈	PROPN
ma-11	227	8	cld	cld	NOUN
ma-11	227	9	(	(	PUNCT
ma-11	227	10	[	[	X
ma-11	227	11	a	a	PRON
ma-11	227	12	,	,	PUNCT
ma-11	227	13	b]t	b]t	NOUN
ma-11	227	14	,	,	PUNCT
ma-11	227	15	r−	r−	NOUN
ma-11	227	16	{	{	PUNCT
ma-11	227	17	0	0	NUM
ma-11	227	18	}	}	PUNCT
ma-11	227	19	)	)	PUNCT
ma-11	227	20	be	be	AUX
ma-11	227	21	∇-integrable	∇-integrable	ADJ
ma-11	227	22	functions	function	NOUN
ma-11	227	23	.	.	PUNCT
ma-11	228	1	assume	assume	VERB
ma-11	228	2	that	that	SCONJ
ma-11	228	3	there	there	PRON
ma-11	228	4	exist	exist	VERB
ma-11	228	5	four	four	NUM
ma-11	228	6	positive	positive	ADJ
ma-11	228	7	constants	constant	NOUN
ma-11	228	8	m	m	PROPN
ma-11	228	9	,	,	PUNCT
ma-11	228	10	m	m	VERB
ma-11	228	11	,	,	PUNCT
ma-11	228	12	n	n	PROPN
ma-11	228	13	and	and	CCONJ
ma-11	228	14	n	n	CCONJ
ma-11	229	1	such	such	ADJ
ma-11	229	2	that	that	SCONJ
ma-11	229	3	0	0	NUM
ma-11	229	4	<	<	X
ma-11	229	5	m	m	VERB
ma-11	229	6	≤	≤	ADJ
ma-11	229	7	|f	|f	PROPN
ma-11	230	1	(	(	PUNCT
ma-11	230	2	y)|	y)|	NOUN
ma-11	230	3	≤	≤	VERB
ma-11	230	4	m	m	VERB
ma-11	230	5	<	<	X
ma-11	230	6	∞	∞	PROPN
ma-11	230	7	and	and	CCONJ
ma-11	230	8	0	0	NUM
ma-11	230	9	<	<	X
ma-11	230	10	n	n	PRON
ma-11	230	11	≤	≤	X
ma-11	230	12	|g(y)|	|g(y)|	PROPN
ma-11	230	13	≤	≤	NOUN
ma-11	230	14	n	n	CCONJ
ma-11	230	15	<	<	X
ma-11	230	16	∞	∞	NUM
ma-11	230	17	on	on	ADP
ma-11	230	18	the	the	DET
ma-11	230	19	set	set	NOUN
ma-11	230	20	[	[	X
ma-11	230	21	a	a	X
ma-11	230	22	,	,	PUNCT
ma-11	230	23	x	x	X
ma-11	230	24	]	]	X
ma-11	230	25	t	t	PROPN
ma-11	230	26	,	,	PUNCT
ma-11	230	27	∀x	∀x	X
ma-11	230	28	∈	∈	PROPN
ma-11	231	1	[	[	X
ma-11	231	2	a	a	PRON
ma-11	231	3	,	,	PUNCT
ma-11	231	4	b]t	b]t	NOUN
ma-11	231	5	.	.	PUNCT
ma-11	232	1	let	let	VERB
ma-11	232	2	α	α	PRON
ma-11	232	3	,	,	PUNCT
ma-11	232	4	β	β	X
ma-11	232	5	≥	≥	NUM
ma-11	232	6	1	1	NUM
ma-11	232	7	and	and	CCONJ
ma-11	232	8	ĥα−1	ĥα−1	NOUN
ma-11	232	9	(	(	PUNCT
ma-11	232	10	.	.	PUNCT
ma-11	232	11	,	,	PUNCT
ma-11	232	12	.	.	PUNCT
ma-11	232	13	)	)	PUNCT
ma-11	232	14	,	,	PUNCT
ma-11	232	15	ĥβ−1	ĥβ−1	PROPN
ma-11	232	16	(	(	PUNCT
ma-11	232	17	.	.	PUNCT
ma-11	232	18	,	,	PUNCT
ma-11	232	19	.	.	PUNCT
ma-11	232	20	)	)	PUNCT
ma-11	233	1	>	>	X
ma-11	233	2	0	0	X
ma-11	233	3	.	.	PUNCT
ma-11	234	1	then	then	ADV
ma-11	234	2	we	we	PRON
ma-11	234	3	have	have	VERB
ma-11	234	4	the	the	DET
ma-11	234	5	following	follow	VERB
ma-11	234	6	inequality	inequality	NOUN
ma-11	234	7	j	j	PROPN
ma-11	234	8	αa	αa	INTJ
ma-11	234	9	(	(	PUNCT
ma-11	234	10	|w(x)||f	|w(x)||f	X
ma-11	234	11	(	(	PUNCT
ma-11	234	12	x)|2	x)|2	PROPN
ma-11	234	13	)	)	PUNCT
ma-11	235	1	j	j	PROPN
ma-11	235	2	βa	βa	INTJ
ma-11	235	3	(	(	PUNCT
ma-11	235	4	|w(x)||g(x)|2	|w(x)||g(x)|2	X
ma-11	235	5	)	)	PUNCT
ma-11	236	1	j	j	PROPN
ma-11	236	2	αa	αa	INTJ
ma-11	236	3	(	(	PUNCT
ma-11	236	4	|(wf	|(wf	NUM
ma-11	236	5	g)(x)|)j	g)(x)|)j	PROPN
ma-11	236	6	βa	βa	SYM
ma-11	236	7	(	(	PUNCT
ma-11	236	8	|(wf	|(wf	PROPN
ma-11	236	9	g)(x)|	g)(x)|	PROPN
ma-11	236	10	)	)	PUNCT
ma-11	236	11	≤	≤	PROPN
ma-11	236	12	mn	mn	PROPN
ma-11	236	13	mn	mn	PROPN
ma-11	236	14	.	.	PUNCT
ma-11	237	1	(	(	PUNCT
ma-11	237	2	24	24	NUM
ma-11	237	3	)	)	PUNCT
ma-11	237	4	proof	proof	NOUN
ma-11	237	5	.	.	PUNCT
ma-11	238	1	similar	similar	ADJ
ma-11	238	2	to	to	ADP
ma-11	238	3	the	the	DET
ma-11	238	4	proof	proof	NOUN
ma-11	238	5	of	of	ADP
ma-11	238	6	corollary	corollary	ADJ
ma-11	238	7	3.7	3.7	NUM
ma-11	238	8	.	.	PUNCT
ma-11	239	1	�	�	PROPN
ma-11	239	2	remark	remark	VERB
ma-11	239	3	3.2	3.2	NUM
ma-11	239	4	.	.	PUNCT
ma-11	240	1	let	let	VERB
ma-11	240	2	t	t	NOUN
ma-11	240	3	=	=	SYM
ma-11	240	4	r	r	PROPN
ma-11	240	5	,	,	PUNCT
ma-11	240	6	α	α	NOUN
ma-11	240	7	>	>	X
ma-11	240	8	0	0	PROPN
ma-11	240	9	,	,	PUNCT
ma-11	240	10	α	α	X
ma-11	240	11	=	=	SYM
ma-11	240	12	β	β	X
ma-11	240	13	,	,	PUNCT
ma-11	240	14	a	a	DET
ma-11	240	15	=	=	SYM
ma-11	240	16	0	0	NUM
ma-11	240	17	,	,	PUNCT
ma-11	240	18	x	x	X
ma-11	240	19	>	>	X
ma-11	240	20	0	0	NUM
ma-11	240	21	,	,	PUNCT
ma-11	240	22	w	w	PROPN
ma-11	240	23	≡	≡	PROPN
ma-11	240	24	1	1	NUM
ma-11	240	25	,	,	PUNCT
ma-11	240	26	f	f	PROPN
ma-11	240	27	>	>	X
ma-11	240	28	0	0	PROPN
ma-11	240	29	and	and	CCONJ
ma-11	240	30	g	g	PROPN
ma-11	240	31	>	>	X
ma-11	240	32	0	0	PROPN
ma-11	240	33	.	.	PUNCT
ma-11	241	1	then	then	ADV
ma-11	241	2	(	(	PUNCT
ma-11	241	3	23	23	X
ma-11	241	4	)	)	PUNCT
ma-11	241	5	reducesto	reducesto	NOUN
ma-11	241	6	iα0	iα0	PROPN
ma-11	241	7	(	(	PUNCT
ma-11	241	8	f	f	PROPN
ma-11	241	9	2(x	2(x	NUM
ma-11	241	10	)	)	PUNCT
ma-11	241	11	)	)	PUNCT
ma-11	242	1	iα0	iα0	PROPN
ma-11	242	2	(	(	PUNCT
ma-11	242	3	g2(x	g2(x	NOUN
ma-11	242	4	)	)	PUNCT
ma-11	242	5	)	)	PUNCT
ma-11	242	6	{	{	PUNCT
ma-11	242	7	iα0	iα0	PROPN
ma-11	242	8	(	(	PUNCT
ma-11	242	9	f	f	PROPN
ma-11	242	10	(	(	PUNCT
ma-11	242	11	x)g(x	x)g(x	PROPN
ma-11	242	12	)	)	PUNCT
ma-11	242	13	)	)	PUNCT
ma-11	242	14	}	}	PUNCT
ma-11	242	15	2	2	NUM
ma-11	242	16	≤	≤	NUM
ma-11	242	17	mn	mn	PROPN
ma-11	242	18	mn	mn	PROPN
ma-11	242	19	.	.	PUNCT
ma-11	243	1	(	(	PUNCT
ma-11	243	2	25	25	NUM
ma-11	243	3	)	)	PUNCT
ma-11	243	4	inequality	inequality	NOUN
ma-11	243	5	(	(	PUNCT
ma-11	243	6	25	25	NUM
ma-11	243	7	)	)	PUNCT
ma-11	243	8	may	may	AUX
ma-11	243	9	be	be	AUX
ma-11	243	10	found	find	VERB
ma-11	243	11	in	in	ADP
ma-11	243	12	[	[	X
ma-11	243	13	5	5	NUM
ma-11	243	14	]	]	PUNCT
ma-11	243	15	.	.	PUNCT
ma-11	244	1	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	244	2	eur	eur	PROPN
ma-11	244	3	.	.	PUNCT
ma-11	245	1	j.	j.	PROPN
ma-11	245	2	math	math	PROPN
ma-11	245	3	.	.	PUNCT
ma-11	246	1	anal	anal	PROPN
ma-11	246	2	.	.	PUNCT
ma-11	247	1	10.28924	10.28924	NUM
ma-11	247	2	/	/	SYM
ma-11	247	3	ada	ada	PROPN
ma-11	247	4	/	/	SYM
ma-11	247	5	ma.3.12	ma.3.12	PROPN
ma-11	247	6	8	8	NUM
ma-11	247	7	theorem	theorem	VERB
ma-11	247	8	3.9	3.9	NUM
ma-11	247	9	.	.	PUNCT
ma-11	248	1	let	let	VERB
ma-11	248	2	w	w	VERB
ma-11	248	3	,	,	PUNCT
ma-11	248	4	f	f	PROPN
ma-11	248	5	,	,	PUNCT
ma-11	248	6	g	g	PROPN
ma-11	248	7	∈	∈	PROPN
ma-11	248	8	crd	crd	NOUN
ma-11	248	9	(	(	PUNCT
ma-11	248	10	[	[	X
ma-11	248	11	a	a	PRON
ma-11	248	12	,	,	PUNCT
ma-11	248	13	b]t	b]t	NOUN
ma-11	248	14	,	,	PUNCT
ma-11	248	15	r−	r−	NOUN
ma-11	248	16	{	{	PUNCT
ma-11	248	17	0	0	NUM
ma-11	248	18	}	}	PUNCT
ma-11	248	19	)	)	PUNCT
ma-11	248	20	be	be	AUX
ma-11	248	21	∆-integrable	∆-integrable	ADJ
ma-11	248	22	functions	function	NOUN
ma-11	248	23	.	.	PUNCT
ma-11	249	1	assume	assume	VERB
ma-11	249	2	that	that	SCONJ
ma-11	249	3	there	there	PRON
ma-11	249	4	exist	exist	VERB
ma-11	249	5	four	four	NUM
ma-11	249	6	positive	positive	ADJ
ma-11	249	7	∆-integrable	∆-integrable	ADJ
ma-11	249	8	functions	function	NOUN
ma-11	249	9	f1	f1	NOUN
ma-11	249	10	,	,	PUNCT
ma-11	249	11	f2	f2	PROPN
ma-11	249	12	,	,	PUNCT
ma-11	249	13	g1	g1	PROPN
ma-11	249	14	and	and	CCONJ
ma-11	249	15	g2	g2	PROPN
ma-11	249	16	such	such	ADJ
ma-11	249	17	that	that	SCONJ
ma-11	249	18	:	:	PUNCT
ma-11	249	19	0	0	NUM
ma-11	249	20	<	<	X
ma-11	249	21	f1(y	f1(y	PROPN
ma-11	249	22	)	)	PUNCT
ma-11	249	23	≤	≤	NOUN
ma-11	249	24	|f	|f	PROPN
ma-11	250	1	(	(	PUNCT
ma-11	250	2	y)|	y)|	PROPN
ma-11	250	3	≤	≤	PROPN
ma-11	250	4	f2(y	f2(y	PROPN
ma-11	250	5	)	)	PUNCT
ma-11	250	6	<	<	X
ma-11	250	7	∞	∞	NUM
ma-11	250	8	and	and	CCONJ
ma-11	250	9	0	0	NUM
ma-11	250	10	<	<	X
ma-11	250	11	g1(y	g1(y	PROPN
ma-11	250	12	)	)	PUNCT
ma-11	250	13	≤	≤	PUNCT
ma-11	250	14	|g(y)|	|g(y)|	PROPN
ma-11	250	15	≤	≤	NOUN
ma-11	250	16	g2(y	g2(y	PROPN
ma-11	250	17	)	)	PUNCT
ma-11	250	18	<	<	X
ma-11	250	19	∞	∞	PROPN
ma-11	250	20	,	,	PUNCT
ma-11	250	21	(	(	PUNCT
ma-11	250	22	y	y	PROPN
ma-11	250	23	∈	∈	PROPN
ma-11	250	24	[	[	X
ma-11	250	25	a	a	X
ma-11	250	26	,	,	PUNCT
ma-11	250	27	x	x	X
ma-11	250	28	]	]	X
ma-11	250	29	t,∀x	t,∀x	X
ma-11	250	30	∈	∈	X
ma-11	251	1	[	[	X
ma-11	251	2	a	a	DET
ma-11	251	3	,	,	PUNCT
ma-11	251	4	b]t	b]t	NOUN
ma-11	251	5	)	)	PUNCT
ma-11	251	6	.	.	PUNCT
ma-11	252	1	let	let	VERB
ma-11	252	2	α	α	PRON
ma-11	252	3	≥	≥	NUM
ma-11	252	4	1	1	NUM
ma-11	252	5	and	and	CCONJ
ma-11	252	6	hα−1	hα−1	NOUN
ma-11	252	7	(	(	PUNCT
ma-11	252	8	.	.	PUNCT
ma-11	252	9	,	,	PUNCT
ma-11	252	10	.	.	PUNCT
ma-11	252	11	)	)	PUNCT
ma-11	253	1	>	>	X
ma-11	253	2	0	0	X
ma-11	253	3	.	.	PUNCT
ma-11	254	1	then	then	ADV
ma-11	254	2	we	we	PRON
ma-11	254	3	have	have	VERB
ma-11	254	4	the	the	DET
ma-11	254	5	following	follow	VERB
ma-11	254	6	inequality	inequality	NOUN
ma-11	254	7	iαa	iαa	NOUN
ma-11	254	8	(	(	PUNCT
ma-11	254	9	g1(x)g2(x)|w(x)||f	g1(x)g2(x)|w(x)||f	NOUN
ma-11	254	10	(	(	PUNCT
ma-11	254	11	x)|2	x)|2	PROPN
ma-11	254	12	)	)	PUNCT
ma-11	254	13	iαa	iαa	PROPN
ma-11	254	14	(	(	PUNCT
ma-11	254	15	f1(x)f2(x)|w(x)||g(x)|2	f1(x)f2(x)|w(x)||g(x)|2	PROPN
ma-11	254	16	)	)	PUNCT
ma-11	254	17	{	{	PUNCT
ma-11	254	18	iαa	iαa	NOUN
ma-11	254	19	(	(	PUNCT
ma-11	254	20	(	(	PUNCT
ma-11	254	21	f1(x)g1(x	f1(x)g1(x	ADJ
ma-11	254	22	)	)	PUNCT
ma-11	254	23	+	+	NUM
ma-11	254	24	f2(x)g2(x	f2(x)g2(x	NOUN
ma-11	254	25	)	)	PUNCT
ma-11	254	26	)	)	PUNCT
ma-11	254	27	|w(x)||f	|w(x)||f	NOUN
ma-11	254	28	(	(	PUNCT
ma-11	254	29	x)g(x)|)}2	x)g(x)|)}2	SYM
ma-11	254	30	≤	≤	NUM
ma-11	254	31	1	1	NUM
ma-11	254	32	4	4	NUM
ma-11	254	33	.	.	PUNCT
ma-11	255	1	(	(	PUNCT
ma-11	255	2	26	26	NUM
ma-11	255	3	)	)	PUNCT
ma-11	255	4	proof	proof	NOUN
ma-11	255	5	.	.	PUNCT
ma-11	256	1	using	use	VERB
ma-11	256	2	the	the	DET
ma-11	256	3	given	give	VERB
ma-11	256	4	conditions	condition	NOUN
ma-11	256	5	,	,	PUNCT
ma-11	256	6	for	for	ADP
ma-11	256	7	y	y	PROPN
ma-11	256	8	∈	∈	PROPN
ma-11	257	1	[	[	X
ma-11	257	2	a	a	X
ma-11	257	3	,	,	PUNCT
ma-11	257	4	x	x	X
ma-11	257	5	]	]	X
ma-11	257	6	t	t	PROPN
ma-11	257	7	,	,	PUNCT
ma-11	257	8	∀x	∀x	X
ma-11	257	9	∈	∈	PROPN
ma-11	257	10	[	[	X
ma-11	257	11	a	a	DET
ma-11	257	12	,	,	PUNCT
ma-11	257	13	b]t	b]t	NOUN
ma-11	257	14	,	,	PUNCT
ma-11	257	15	we	we	PRON
ma-11	257	16	have	have	VERB
ma-11	257	17	(	(	PUNCT
ma-11	257	18	f2(y	f2(y	NOUN
ma-11	257	19	)	)	PUNCT
ma-11	257	20	g1(y	g1(y	PROPN
ma-11	257	21	)	)	PUNCT
ma-11	258	1	−	−	PROPN
ma-11	259	1	|f	|f	PROPN
ma-11	260	1	(	(	PUNCT
ma-11	260	2	y)|	y)|	PROPN
ma-11	260	3	|g(y)|	|g(y)|	PROPN
ma-11	260	4	)	)	PUNCT
ma-11	260	5	≥	≥	NOUN
ma-11	260	6	0	0	NUM
ma-11	260	7	,	,	PUNCT
ma-11	260	8	and	and	CCONJ
ma-11	260	9	(	(	PUNCT
ma-11	260	10	|f	|f	PROPN
ma-11	260	11	(	(	PUNCT
ma-11	260	12	y)|	y)|	PROPN
ma-11	260	13	|g(y)|	|g(y)|	PROPN
ma-11	260	14	−	−	PROPN
ma-11	260	15	f1(y	f1(y	PROPN
ma-11	260	16	)	)	PUNCT
ma-11	260	17	g2(y	g2(y	PROPN
ma-11	260	18	)	)	PUNCT
ma-11	260	19	)	)	PUNCT
ma-11	260	20	≥	≥	NOUN
ma-11	260	21	0	0	X
ma-11	260	22	.	.	PUNCT
ma-11	261	1	multiplying	multiply	VERB
ma-11	261	2	the	the	DET
ma-11	261	3	last	last	ADJ
ma-11	261	4	two	two	NUM
ma-11	261	5	inequalities	inequality	NOUN
ma-11	261	6	,	,	PUNCT
ma-11	261	7	we	we	PRON
ma-11	261	8	have	have	VERB
ma-11	261	9	(	(	PUNCT
ma-11	261	10	f2(y	f2(y	NOUN
ma-11	261	11	)	)	PUNCT
ma-11	261	12	g1(y	g1(y	PROPN
ma-11	261	13	)	)	PUNCT
ma-11	262	1	−	−	PROPN
ma-11	263	1	|f	|f	PROPN
ma-11	264	1	(	(	PUNCT
ma-11	264	2	y)|	y)|	PROPN
ma-11	264	3	|g(y)|	|g(y)|	PROPN
ma-11	264	4	)	)	PUNCT
ma-11	264	5	(	(	PUNCT
ma-11	264	6	|f	|f	PROPN
ma-11	264	7	(	(	PUNCT
ma-11	264	8	y)|	y)|	PROPN
ma-11	264	9	|g(y)|	|g(y)|	PROPN
ma-11	264	10	−	−	PROPN
ma-11	264	11	f1(y	f1(y	PROPN
ma-11	264	12	)	)	PUNCT
ma-11	264	13	g2(y	g2(y	PROPN
ma-11	264	14	)	)	PUNCT
ma-11	264	15	)	)	PUNCT
ma-11	264	16	≥	≥	NOUN
ma-11	264	17	0	0	NUM
ma-11	264	18	,	,	PUNCT
ma-11	264	19	which	which	PRON
ma-11	264	20	implies	imply	VERB
ma-11	264	21	(	(	PUNCT
ma-11	264	22	f1(y	f1(y	NUM
ma-11	264	23	)	)	PUNCT
ma-11	264	24	g2(y	g2(y	NOUN
ma-11	264	25	)	)	PUNCT
ma-11	264	26	+	+	CCONJ
ma-11	264	27	f2(y	f2(y	PROPN
ma-11	264	28	)	)	PUNCT
ma-11	264	29	g1(y	g1(y	PROPN
ma-11	264	30	)	)	PUNCT
ma-11	264	31	)	)	PUNCT
ma-11	264	32	|f	|f	PROPN
ma-11	265	1	(	(	PUNCT
ma-11	265	2	y)|	y)|	PROPN
ma-11	265	3	|g(y)|	|g(y)|	PROPN
ma-11	265	4	≥	≥	PRON
ma-11	265	5	|f	|f	PROPN
ma-11	265	6	(	(	PUNCT
ma-11	265	7	y)|2	y)|2	NOUN
ma-11	265	8	|g(y)|2	|g(y)|2	NUM
ma-11	265	9	+	+	SYM
ma-11	265	10	f1(y)f2(y	f1(y)f2(y	PROPN
ma-11	265	11	)	)	PUNCT
ma-11	265	12	g1(y)g2(y	g1(y)g2(y	PROPN
ma-11	265	13	)	)	PUNCT
ma-11	265	14	.	.	PUNCT
ma-11	266	1	multiplying	multiply	VERB
ma-11	266	2	both	both	DET
ma-11	266	3	sides	side	NOUN
ma-11	266	4	by	by	ADP
ma-11	266	5	g1(y)g2(y)|g(y)|2	g1(y)g2(y)|g(y)|2	PROPN
ma-11	266	6	,	,	PUNCT
ma-11	266	7	we	we	PRON
ma-11	266	8	have	have	VERB
ma-11	266	9	f1(y)g1(y)|f	f1(y)g1(y)|f	NOUN
ma-11	266	10	(	(	PUNCT
ma-11	266	11	y)g(y)|+	y)g(y)|+	NOUN
ma-11	266	12	f2(y)g2(y)|f	f2(y)g2(y)|f	VERB
ma-11	266	13	(	(	PUNCT
ma-11	266	14	y)g(y)|	y)g(y)|	PROPN
ma-11	266	15	≥	≥	PRON
ma-11	266	16	g1(y)g2(y)|f	g1(y)g2(y)|f	NOUN
ma-11	266	17	(	(	PUNCT
ma-11	266	18	y)|2	y)|2	NOUN
ma-11	266	19	+	+	CCONJ
ma-11	266	20	f1(y)f2(y)|g(y)|2	f1(y)f2(y)|g(y)|2	PROPN
ma-11	266	21	.	.	PUNCT
ma-11	267	1	(	(	PUNCT
ma-11	267	2	27	27	NUM
ma-11	267	3	)	)	PUNCT
ma-11	267	4	multiplying	multiply	VERB
ma-11	267	5	both	both	DET
ma-11	267	6	sides	side	NOUN
ma-11	267	7	of	of	ADP
ma-11	267	8	(	(	PUNCT
ma-11	267	9	27	27	NUM
ma-11	267	10	)	)	PUNCT
ma-11	267	11	by	by	ADP
ma-11	267	12	hα−1(x	hα−1(x	PROPN
ma-11	267	13	,	,	PUNCT
ma-11	267	14	σ(y))|w(y)|	σ(y))|w(y)|	PROPN
ma-11	267	15	and	and	CCONJ
ma-11	267	16	integrating	integrate	VERB
ma-11	267	17	over	over	ADP
ma-11	267	18	y	y	PROPN
ma-11	267	19	from	from	ADP
ma-11	267	20	a	a	DET
ma-11	267	21	to	to	PART
ma-11	267	22	x	x	SYM
ma-11	267	23	,	,	PUNCT
ma-11	267	24	we	we	PRON
ma-11	267	25	have	have	VERB
ma-11	267	26	iαa	iαa	ADJ
ma-11	267	27	(	(	PUNCT
ma-11	267	28	(	(	PUNCT
ma-11	267	29	f1(x)g1(x	f1(x)g1(x	ADJ
ma-11	267	30	)	)	PUNCT
ma-11	267	31	+	+	NUM
ma-11	267	32	f2(x)g2(x	f2(x)g2(x	NOUN
ma-11	267	33	)	)	PUNCT
ma-11	267	34	)	)	PUNCT
ma-11	267	35	|w(x)||f	|w(x)||f	PROPN
ma-11	267	36	(	(	PUNCT
ma-11	267	37	x)g(x)|	x)g(x)|	PROPN
ma-11	267	38	)	)	PUNCT
ma-11	267	39	≥	≥	NOUN
ma-11	267	40	iαa	iαa	NOUN
ma-11	267	41	(	(	PUNCT
ma-11	267	42	g1(x)g2(x)|w(x)||f	g1(x)g2(x)|w(x)||f	NOUN
ma-11	267	43	(	(	PUNCT
ma-11	267	44	x)|2	x)|2	PROPN
ma-11	267	45	)	)	PUNCT
ma-11	268	1	+	+	CCONJ
ma-11	268	2	iαa	iαa	ADJ
ma-11	268	3	(	(	PUNCT
ma-11	268	4	f1(x)f2(x)|w(x)||g(x)|2	f1(x)f2(x)|w(x)||g(x)|2	NOUN
ma-11	268	5	)	)	PUNCT
ma-11	268	6	.	.	PUNCT
ma-11	269	1	(	(	PUNCT
ma-11	269	2	28	28	X
ma-11	269	3	)	)	PUNCT
ma-11	269	4	applying	apply	VERB
ma-11	269	5	the	the	DET
ma-11	269	6	am	am	NOUN
ma-11	269	7	-	-	PUNCT
ma-11	269	8	gm	gm	NOUN
ma-11	269	9	inequality	inequality	NOUN
ma-11	269	10	,	,	PUNCT
ma-11	269	11	we	we	PRON
ma-11	269	12	get	get	VERB
ma-11	269	13	iαa	iαa	ADJ
ma-11	269	14	(	(	PUNCT
ma-11	269	15	(	(	PUNCT
ma-11	269	16	f1(x)g1(x	f1(x)g1(x	ADJ
ma-11	269	17	)	)	PUNCT
ma-11	269	18	+	+	NUM
ma-11	269	19	f2(x)g2(x	f2(x)g2(x	NOUN
ma-11	269	20	)	)	PUNCT
ma-11	269	21	)	)	PUNCT
ma-11	269	22	|w(x)||f	|w(x)||f	PROPN
ma-11	269	23	(	(	PUNCT
ma-11	269	24	x)g(x)|	x)g(x)|	PROPN
ma-11	269	25	)	)	PUNCT
ma-11	269	26	≥	≥	NOUN
ma-11	269	27	2	2	NUM
ma-11	269	28	√	√	NOUN
ma-11	269	29	iαa	iαa	NOUN
ma-11	269	30	(	(	PUNCT
ma-11	269	31	g1(x)g2(x)|w(x)||f	g1(x)g2(x)|w(x)||f	NOUN
ma-11	269	32	(	(	PUNCT
ma-11	269	33	x)|2	x)|2	PROPN
ma-11	269	34	)	)	PUNCT
ma-11	269	35	iαa	iαa	PROPN
ma-11	269	36	(	(	PUNCT
ma-11	269	37	f1(x)f2(x)|w(x)||g(x)|2	f1(x)f2(x)|w(x)||g(x)|2	PROPN
ma-11	269	38	)	)	PUNCT
ma-11	269	39	.	.	PUNCT
ma-11	270	1	(	(	PUNCT
ma-11	270	2	29	29	NUM
ma-11	270	3	)	)	PUNCT
ma-11	270	4	analogously	analogously	ADV
ma-11	270	5	,	,	PUNCT
ma-11	270	6	we	we	PRON
ma-11	270	7	have	have	VERB
ma-11	270	8	that	that	DET
ma-11	270	9	iαa	iαa	ADJ
ma-11	270	10	(	(	PUNCT
ma-11	270	11	g1(x)g2(x)|w(x)||f	g1(x)g2(x)|w(x)||f	NOUN
ma-11	270	12	(	(	PUNCT
ma-11	270	13	x)|2	x)|2	PROPN
ma-11	270	14	)	)	PUNCT
ma-11	270	15	iαa	iαa	PROPN
ma-11	270	16	(	(	PUNCT
ma-11	270	17	f1(x)f2(x)|w(x)||g(x)|2	f1(x)f2(x)|w(x)||g(x)|2	NOUN
ma-11	270	18	)	)	PUNCT
ma-11	270	19	≤	≤	NOUN
ma-11	270	20	1	1	NUM
ma-11	270	21	4	4	NUM
ma-11	270	22	{	{	PUNCT
ma-11	270	23	iαa	iαa	NOUN
ma-11	270	24	(	(	PUNCT
ma-11	270	25	(	(	PUNCT
ma-11	270	26	f1(x)g1(x	f1(x)g1(x	ADJ
ma-11	270	27	)	)	PUNCT
ma-11	270	28	+	+	NUM
ma-11	270	29	f2(x)g2(x	f2(x)g2(x	NOUN
ma-11	270	30	)	)	PUNCT
ma-11	270	31	)	)	PUNCT
ma-11	270	32	|w(x)||f	|w(x)||f	NOUN
ma-11	270	33	(	(	PUNCT
ma-11	270	34	x)g(x)|)}2	x)g(x)|)}2	ADJ
ma-11	270	35	.	.	PUNCT
ma-11	271	1	(	(	PUNCT
ma-11	271	2	30	30	NUM
ma-11	271	3	)	)	PUNCT
ma-11	271	4	this	this	PRON
ma-11	271	5	directly	directly	ADV
ma-11	271	6	yields	yield	VERB
ma-11	271	7	the	the	DET
ma-11	271	8	desired	desire	VERB
ma-11	271	9	inequality	inequality	NOUN
ma-11	271	10	(	(	PUNCT
ma-11	271	11	26	26	NUM
ma-11	271	12	)	)	PUNCT
ma-11	271	13	.	.	PUNCT
ma-11	272	1	�	�	PROPN
ma-11	272	2	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	272	3	eur	eur	PROPN
ma-11	272	4	.	.	PUNCT
ma-11	273	1	j.	j.	PROPN
ma-11	273	2	math	math	PROPN
ma-11	273	3	.	.	PUNCT
ma-11	274	1	anal	anal	PROPN
ma-11	274	2	.	.	PUNCT
ma-11	275	1	10.28924	10.28924	NUM
ma-11	275	2	/	/	SYM
ma-11	275	3	ada	ada	PROPN
ma-11	275	4	/	/	SYM
ma-11	275	5	ma.3.12	ma.3.12	PROPN
ma-11	275	6	9	9	NUM
ma-11	275	7	theorem	theorem	VERB
ma-11	275	8	3.10	3.10	NUM
ma-11	275	9	.	.	PUNCT
ma-11	276	1	let	let	VERB
ma-11	276	2	w	w	VERB
ma-11	276	3	,	,	PUNCT
ma-11	276	4	f	f	PROPN
ma-11	276	5	,	,	PUNCT
ma-11	276	6	g	g	PROPN
ma-11	276	7	∈	∈	PROPN
ma-11	276	8	cld	cld	NOUN
ma-11	276	9	(	(	PUNCT
ma-11	276	10	[	[	X
ma-11	276	11	a	a	PRON
ma-11	276	12	,	,	PUNCT
ma-11	276	13	b]t	b]t	NOUN
ma-11	276	14	,	,	PUNCT
ma-11	276	15	r−	r−	NOUN
ma-11	276	16	{	{	PUNCT
ma-11	276	17	0	0	NUM
ma-11	276	18	}	}	PUNCT
ma-11	276	19	)	)	PUNCT
ma-11	276	20	be	be	AUX
ma-11	276	21	∇-integrable	∇-integrable	ADJ
ma-11	276	22	functions	function	NOUN
ma-11	276	23	.	.	PUNCT
ma-11	277	1	assume	assume	VERB
ma-11	277	2	that	that	SCONJ
ma-11	277	3	there	there	PRON
ma-11	277	4	exist	exist	VERB
ma-11	277	5	four	four	NUM
ma-11	277	6	positive	positive	ADJ
ma-11	277	7	∇-integrable	∇-integrable	ADJ
ma-11	277	8	functions	function	NOUN
ma-11	277	9	f1	f1	NOUN
ma-11	277	10	,	,	PUNCT
ma-11	277	11	f2	f2	PROPN
ma-11	277	12	,	,	PUNCT
ma-11	277	13	g1	g1	PROPN
ma-11	277	14	and	and	CCONJ
ma-11	277	15	g2	g2	PROPN
ma-11	277	16	such	such	ADJ
ma-11	277	17	that	that	SCONJ
ma-11	277	18	:	:	PUNCT
ma-11	277	19	0	0	NUM
ma-11	277	20	<	<	X
ma-11	277	21	f1(y	f1(y	PROPN
ma-11	277	22	)	)	PUNCT
ma-11	277	23	≤	≤	NOUN
ma-11	277	24	|f	|f	PROPN
ma-11	278	1	(	(	PUNCT
ma-11	278	2	y)|	y)|	PROPN
ma-11	278	3	≤	≤	PROPN
ma-11	278	4	f2(y	f2(y	PROPN
ma-11	278	5	)	)	PUNCT
ma-11	278	6	<	<	X
ma-11	278	7	∞	∞	NUM
ma-11	278	8	and	and	CCONJ
ma-11	278	9	0	0	NUM
ma-11	278	10	<	<	X
ma-11	278	11	g1(y	g1(y	PROPN
ma-11	278	12	)	)	PUNCT
ma-11	278	13	≤	≤	PUNCT
ma-11	278	14	|g(y)|	|g(y)|	PROPN
ma-11	278	15	≤	≤	NOUN
ma-11	278	16	g2(y	g2(y	PROPN
ma-11	278	17	)	)	PUNCT
ma-11	278	18	<	<	X
ma-11	278	19	∞	∞	PROPN
ma-11	278	20	,	,	PUNCT
ma-11	278	21	(	(	PUNCT
ma-11	278	22	y	y	PROPN
ma-11	278	23	∈	∈	PROPN
ma-11	278	24	[	[	X
ma-11	278	25	a	a	X
ma-11	278	26	,	,	PUNCT
ma-11	278	27	x	x	X
ma-11	278	28	]	]	X
ma-11	278	29	t,∀x	t,∀x	X
ma-11	278	30	∈	∈	X
ma-11	279	1	[	[	X
ma-11	279	2	a	a	DET
ma-11	279	3	,	,	PUNCT
ma-11	279	4	b]t	b]t	NOUN
ma-11	279	5	)	)	PUNCT
ma-11	279	6	.	.	PUNCT
ma-11	280	1	let	let	VERB
ma-11	280	2	α	α	PRON
ma-11	280	3	≥	≥	NUM
ma-11	280	4	1	1	NUM
ma-11	280	5	and	and	CCONJ
ma-11	280	6	ĥα−1	ĥα−1	NOUN
ma-11	280	7	(	(	PUNCT
ma-11	280	8	.	.	PUNCT
ma-11	280	9	,	,	PUNCT
ma-11	280	10	.	.	PUNCT
ma-11	280	11	)	)	PUNCT
ma-11	281	1	>	>	X
ma-11	281	2	0	0	X
ma-11	281	3	.	.	PUNCT
ma-11	282	1	then	then	ADV
ma-11	282	2	we	we	PRON
ma-11	282	3	have	have	VERB
ma-11	282	4	the	the	DET
ma-11	282	5	following	follow	VERB
ma-11	282	6	inequality	inequality	NOUN
ma-11	282	7	j	j	PROPN
ma-11	282	8	αa	αa	INTJ
ma-11	282	9	(	(	PUNCT
ma-11	282	10	g1(x)g2(x)|w(x)||f	g1(x)g2(x)|w(x)||f	NOUN
ma-11	282	11	(	(	PUNCT
ma-11	282	12	x)|2	x)|2	PROPN
ma-11	282	13	)	)	PUNCT
ma-11	283	1	j	j	PROPN
ma-11	283	2	αa	αa	INTJ
ma-11	283	3	(	(	PUNCT
ma-11	283	4	f1(x)f2(x)|w(x)||g(x)|2	f1(x)f2(x)|w(x)||g(x)|2	PROPN
ma-11	283	5	)	)	PUNCT
ma-11	283	6	{	{	PUNCT
ma-11	283	7	j	j	NOUN
ma-11	283	8	αa	αa	INTJ
ma-11	283	9	(	(	PUNCT
ma-11	283	10	(	(	PUNCT
ma-11	283	11	f1(x)g1(x	f1(x)g1(x	PROPN
ma-11	283	12	)	)	PUNCT
ma-11	283	13	+	+	NUM
ma-11	283	14	f2(x)g2(x	f2(x)g2(x	NOUN
ma-11	283	15	)	)	PUNCT
ma-11	283	16	)	)	PUNCT
ma-11	283	17	|w(x)||f	|w(x)||f	NOUN
ma-11	283	18	(	(	PUNCT
ma-11	283	19	x)g(x)|)}2	x)g(x)|)}2	SYM
ma-11	283	20	≤	≤	NUM
ma-11	283	21	1	1	NUM
ma-11	283	22	4	4	NUM
ma-11	283	23	.	.	PUNCT
ma-11	284	1	(	(	PUNCT
ma-11	284	2	31	31	NUM
ma-11	284	3	)	)	PUNCT
ma-11	284	4	proof	proof	NOUN
ma-11	284	5	.	.	PUNCT
ma-11	285	1	similar	similar	ADJ
ma-11	285	2	to	to	ADP
ma-11	285	3	the	the	DET
ma-11	285	4	proof	proof	NOUN
ma-11	285	5	of	of	ADP
ma-11	285	6	theorem	theorem	ADJ
ma-11	285	7	3.9	3.9	NUM
ma-11	285	8	.	.	PUNCT
ma-11	285	9	�	�	PROPN
ma-11	285	10	remark	remark	AUX
ma-11	285	11	3.3	3.3	NUM
ma-11	285	12	.	.	PUNCT
ma-11	286	1	let	let	VERB
ma-11	286	2	t	t	NOUN
ma-11	286	3	=	=	SYM
ma-11	286	4	r	r	PROPN
ma-11	286	5	,	,	PUNCT
ma-11	286	6	α	α	NOUN
ma-11	286	7	>	>	X
ma-11	286	8	0	0	PROPN
ma-11	286	9	,	,	PUNCT
ma-11	286	10	a	a	DET
ma-11	286	11	=	=	SYM
ma-11	286	12	0	0	NUM
ma-11	286	13	,	,	PUNCT
ma-11	286	14	x	x	X
ma-11	286	15	>	>	X
ma-11	286	16	0	0	NUM
ma-11	286	17	,	,	PUNCT
ma-11	286	18	w	w	PROPN
ma-11	286	19	≡	≡	PROPN
ma-11	286	20	1	1	NUM
ma-11	286	21	,	,	PUNCT
ma-11	286	22	f	f	PROPN
ma-11	286	23	>	>	X
ma-11	286	24	0	0	PROPN
ma-11	286	25	and	and	CCONJ
ma-11	286	26	g	g	PROPN
ma-11	286	27	>	>	X
ma-11	286	28	0	0	PROPN
ma-11	286	29	.	.	PUNCT
ma-11	287	1	then	then	ADV
ma-11	287	2	(	(	PUNCT
ma-11	287	3	26	26	NUM
ma-11	287	4	)	)	PUNCT
ma-11	287	5	reduces	reduce	VERB
ma-11	287	6	to	to	PART
ma-11	287	7	iα0	iα0	PROPN
ma-11	287	8	(	(	PUNCT
ma-11	287	9	g1(x)g2(x)f	g1(x)g2(x)f	PROPN
ma-11	287	10	2(x	2(x	NUM
ma-11	287	11	)	)	PUNCT
ma-11	287	12	)	)	PUNCT
ma-11	288	1	iα0	iα0	PROPN
ma-11	288	2	(	(	PUNCT
ma-11	288	3	f1(x)f2(x)g2(x	f1(x)f2(x)g2(x	PROPN
ma-11	288	4	)	)	PUNCT
ma-11	288	5	)	)	PUNCT
ma-11	288	6	{	{	PUNCT
ma-11	288	7	iα0	iα0	PUNCT
ma-11	288	8	(	(	PUNCT
ma-11	288	9	(	(	PUNCT
ma-11	288	10	f1(x)g1(x	f1(x)g1(x	PROPN
ma-11	288	11	)	)	PUNCT
ma-11	288	12	+	+	NUM
ma-11	288	13	f2(x)g2(x	f2(x)g2(x	NOUN
ma-11	288	14	)	)	PUNCT
ma-11	288	15	)	)	PUNCT
ma-11	289	1	f	f	PROPN
ma-11	289	2	(	(	PUNCT
ma-11	289	3	x)g(x	x)g(x	PROPN
ma-11	289	4	)	)	PUNCT
ma-11	289	5	)	)	PUNCT
ma-11	289	6	}	}	PUNCT
ma-11	289	7	2	2	NUM
ma-11	289	8	≤	≤	NUM
ma-11	289	9	1	1	NUM
ma-11	289	10	4	4	NUM
ma-11	289	11	.	.	PUNCT
ma-11	290	1	(	(	PUNCT
ma-11	290	2	32	32	NUM
ma-11	290	3	)	)	PUNCT
ma-11	290	4	inequality	inequality	NOUN
ma-11	290	5	(	(	PUNCT
ma-11	290	6	32	32	NUM
ma-11	290	7	)	)	PUNCT
ma-11	290	8	may	may	AUX
ma-11	290	9	be	be	AUX
ma-11	290	10	found	find	VERB
ma-11	290	11	in	in	ADP
ma-11	290	12	[	[	X
ma-11	290	13	14	14	NUM
ma-11	290	14	]	]	PUNCT
ma-11	290	15	.	.	PUNCT
ma-11	291	1	corollary	corollary	ADJ
ma-11	291	2	3.11	3.11	NUM
ma-11	291	3	.	.	PUNCT
ma-11	292	1	let	let	VERB
ma-11	292	2	w	w	VERB
ma-11	292	3	,	,	PUNCT
ma-11	292	4	f	f	PROPN
ma-11	292	5	,	,	PUNCT
ma-11	292	6	g	g	PROPN
ma-11	292	7	∈	∈	PROPN
ma-11	292	8	crd	crd	NOUN
ma-11	292	9	(	(	PUNCT
ma-11	292	10	[	[	X
ma-11	292	11	a	a	PRON
ma-11	292	12	,	,	PUNCT
ma-11	292	13	b]t	b]t	NOUN
ma-11	292	14	,	,	PUNCT
ma-11	292	15	r−	r−	NOUN
ma-11	292	16	{	{	PUNCT
ma-11	292	17	0	0	NUM
ma-11	292	18	}	}	PUNCT
ma-11	292	19	)	)	PUNCT
ma-11	292	20	be	be	AUX
ma-11	292	21	∆-integrable	∆-integrable	ADJ
ma-11	292	22	functions	function	NOUN
ma-11	292	23	.	.	PUNCT
ma-11	293	1	assume	assume	VERB
ma-11	293	2	that	that	SCONJ
ma-11	293	3	there	there	PRON
ma-11	293	4	exist	exist	VERB
ma-11	293	5	four	four	NUM
ma-11	293	6	positive	positive	ADJ
ma-11	293	7	constants	constant	NOUN
ma-11	293	8	m	m	PROPN
ma-11	293	9	,	,	PUNCT
ma-11	293	10	m	m	VERB
ma-11	293	11	,	,	PUNCT
ma-11	293	12	n	n	PROPN
ma-11	293	13	and	and	CCONJ
ma-11	293	14	n	n	CCONJ
ma-11	294	1	such	such	ADJ
ma-11	294	2	that	that	SCONJ
ma-11	294	3	0	0	NUM
ma-11	294	4	<	<	X
ma-11	294	5	m	m	VERB
ma-11	294	6	≤	≤	ADJ
ma-11	294	7	|f	|f	PROPN
ma-11	295	1	(	(	PUNCT
ma-11	295	2	y)|	y)|	NOUN
ma-11	295	3	≤	≤	VERB
ma-11	295	4	m	m	VERB
ma-11	295	5	<	<	X
ma-11	295	6	∞	∞	PROPN
ma-11	295	7	and	and	CCONJ
ma-11	295	8	0	0	NUM
ma-11	295	9	<	<	X
ma-11	295	10	n	n	PRON
ma-11	295	11	≤	≤	X
ma-11	295	12	|g(y)|	|g(y)|	PROPN
ma-11	295	13	≤	≤	NOUN
ma-11	295	14	n	n	CCONJ
ma-11	295	15	<	<	X
ma-11	295	16	∞	∞	NUM
ma-11	295	17	on	on	ADP
ma-11	295	18	the	the	DET
ma-11	295	19	set	set	NOUN
ma-11	295	20	[	[	X
ma-11	295	21	a	a	X
ma-11	295	22	,	,	PUNCT
ma-11	295	23	x	x	X
ma-11	295	24	]	]	X
ma-11	295	25	t	t	PROPN
ma-11	295	26	,	,	PUNCT
ma-11	295	27	∀x	∀x	X
ma-11	295	28	∈	∈	PROPN
ma-11	296	1	[	[	X
ma-11	296	2	a	a	PRON
ma-11	296	3	,	,	PUNCT
ma-11	296	4	b]t	b]t	NOUN
ma-11	296	5	.	.	PUNCT
ma-11	297	1	let	let	VERB
ma-11	297	2	α	α	PRON
ma-11	297	3	≥	≥	NUM
ma-11	297	4	1	1	NUM
ma-11	297	5	and	and	CCONJ
ma-11	297	6	hα−1	hα−1	NOUN
ma-11	297	7	(	(	PUNCT
ma-11	297	8	.	.	PUNCT
ma-11	297	9	,	,	PUNCT
ma-11	297	10	.	.	PUNCT
ma-11	297	11	)	)	PUNCT
ma-11	298	1	>	>	X
ma-11	298	2	0	0	X
ma-11	298	3	.	.	PUNCT
ma-11	299	1	then	then	ADV
ma-11	299	2	we	we	PRON
ma-11	299	3	have	have	VERB
ma-11	299	4	the	the	DET
ma-11	299	5	following	follow	VERB
ma-11	299	6	inequality	inequality	NOUN
ma-11	299	7	iαa	iαa	ADJ
ma-11	299	8	(	(	PUNCT
ma-11	299	9	|w(x)||f	|w(x)||f	X
ma-11	299	10	(	(	PUNCT
ma-11	299	11	x)|2	x)|2	PROPN
ma-11	299	12	)	)	PUNCT
ma-11	299	13	iαa	iαa	PROPN
ma-11	299	14	(	(	PUNCT
ma-11	299	15	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	299	16	)	)	PUNCT
ma-11	299	17	{	{	PUNCT
ma-11	299	18	iαa	iαa	NOUN
ma-11	299	19	(	(	PUNCT
ma-11	299	20	|w(x)||f	|w(x)||f	X
ma-11	299	21	(	(	PUNCT
ma-11	299	22	x)g(x)|)}2	x)g(x)|)}2	SYM
ma-11	299	23	≤	≤	NUM
ma-11	299	24	1	1	NUM
ma-11	299	25	4	4	NUM
ma-11	299	26	(	(	PUNCT
ma-11	299	27	√	√	PROPN
ma-11	299	28	mn	mn	PROPN
ma-11	299	29	mn	mn	PROPN
ma-11	299	30	+	+	CCONJ
ma-11	299	31	√	√	PROPN
ma-11	299	32	mn	mn	PROPN
ma-11	299	33	mn	mn	PROPN
ma-11	299	34	)	)	PUNCT
ma-11	299	35	2	2	NUM
ma-11	299	36	.	.	PUNCT
ma-11	300	1	(	(	PUNCT
ma-11	300	2	33	33	NUM
ma-11	300	3	)	)	PUNCT
ma-11	300	4	proof	proof	NOUN
ma-11	300	5	.	.	PUNCT
ma-11	301	1	putting	put	VERB
ma-11	301	2	f1	f1	NOUN
ma-11	301	3	=	=	SYM
ma-11	301	4	m	m	PROPN
ma-11	301	5	,	,	PUNCT
ma-11	301	6	f2	f2	PROPN
ma-11	301	7	=	=	SYM
ma-11	301	8	m	m	PROPN
ma-11	301	9	,	,	PUNCT
ma-11	301	10	g1	g1	PROPN
ma-11	301	11	=	=	SYM
ma-11	301	12	n	n	PROPN
ma-11	301	13	and	and	CCONJ
ma-11	301	14	g2	g2	PROPN
ma-11	301	15	=	=	SYM
ma-11	302	1	n	n	PROPN
ma-11	302	2	in	in	ADP
ma-11	302	3	theorem	theorem	NOUN
ma-11	302	4	3.9	3.9	NUM
ma-11	302	5	,	,	PUNCT
ma-11	302	6	we	we	PRON
ma-11	302	7	get	get	VERB
ma-11	302	8	the	the	DET
ma-11	302	9	desired	desire	VERB
ma-11	302	10	inequality(33	inequality(33	PROPN
ma-11	302	11	)	)	PUNCT
ma-11	302	12	.	.	PUNCT
ma-11	303	1	�	�	PROPN
ma-11	303	2	corollary	corollary	PROPN
ma-11	303	3	3.12	3.12	NUM
ma-11	303	4	.	.	PUNCT
ma-11	304	1	let	let	VERB
ma-11	304	2	w	w	VERB
ma-11	304	3	,	,	PUNCT
ma-11	304	4	f	f	PROPN
ma-11	304	5	,	,	PUNCT
ma-11	304	6	g	g	PROPN
ma-11	304	7	∈	∈	PROPN
ma-11	304	8	cld	cld	NOUN
ma-11	304	9	(	(	PUNCT
ma-11	304	10	[	[	X
ma-11	304	11	a	a	PRON
ma-11	304	12	,	,	PUNCT
ma-11	304	13	b]t	b]t	NOUN
ma-11	304	14	,	,	PUNCT
ma-11	304	15	r−	r−	NOUN
ma-11	304	16	{	{	PUNCT
ma-11	304	17	0	0	NUM
ma-11	304	18	}	}	PUNCT
ma-11	304	19	)	)	PUNCT
ma-11	304	20	be	be	AUX
ma-11	304	21	∇-integrable	∇-integrable	ADJ
ma-11	304	22	functions	function	NOUN
ma-11	304	23	.	.	PUNCT
ma-11	305	1	assume	assume	VERB
ma-11	305	2	that	that	SCONJ
ma-11	305	3	there	there	PRON
ma-11	305	4	exist	exist	VERB
ma-11	305	5	four	four	NUM
ma-11	305	6	positive	positive	ADJ
ma-11	305	7	constants	constant	NOUN
ma-11	305	8	m	m	PROPN
ma-11	305	9	,	,	PUNCT
ma-11	305	10	m	m	VERB
ma-11	305	11	,	,	PUNCT
ma-11	305	12	n	n	PROPN
ma-11	305	13	and	and	CCONJ
ma-11	305	14	n	n	CCONJ
ma-11	306	1	such	such	ADJ
ma-11	306	2	that	that	SCONJ
ma-11	306	3	0	0	NUM
ma-11	306	4	<	<	X
ma-11	306	5	m	m	VERB
ma-11	306	6	≤	≤	ADJ
ma-11	306	7	|f	|f	PROPN
ma-11	307	1	(	(	PUNCT
ma-11	307	2	y)|	y)|	NOUN
ma-11	307	3	≤	≤	VERB
ma-11	307	4	m	m	VERB
ma-11	307	5	<	<	X
ma-11	307	6	∞	∞	PROPN
ma-11	307	7	and	and	CCONJ
ma-11	307	8	0	0	NUM
ma-11	307	9	<	<	X
ma-11	307	10	n	n	PRON
ma-11	307	11	≤	≤	X
ma-11	307	12	|g(y)|	|g(y)|	PROPN
ma-11	307	13	≤	≤	NOUN
ma-11	307	14	n	n	CCONJ
ma-11	307	15	<	<	X
ma-11	307	16	∞	∞	NUM
ma-11	307	17	on	on	ADP
ma-11	307	18	the	the	DET
ma-11	307	19	set	set	NOUN
ma-11	307	20	[	[	X
ma-11	307	21	a	a	X
ma-11	307	22	,	,	PUNCT
ma-11	307	23	x	x	X
ma-11	307	24	]	]	X
ma-11	307	25	t	t	PROPN
ma-11	307	26	,	,	PUNCT
ma-11	307	27	∀x	∀x	X
ma-11	307	28	∈	∈	PROPN
ma-11	308	1	[	[	X
ma-11	308	2	a	a	PRON
ma-11	308	3	,	,	PUNCT
ma-11	308	4	b]t	b]t	NOUN
ma-11	308	5	.	.	PUNCT
ma-11	309	1	let	let	VERB
ma-11	309	2	α	α	PRON
ma-11	309	3	≥	≥	NUM
ma-11	309	4	1	1	NUM
ma-11	309	5	and	and	CCONJ
ma-11	309	6	ĥα−1	ĥα−1	NOUN
ma-11	309	7	(	(	PUNCT
ma-11	309	8	.	.	PUNCT
ma-11	309	9	,	,	PUNCT
ma-11	309	10	.	.	PUNCT
ma-11	309	11	)	)	PUNCT
ma-11	310	1	>	>	X
ma-11	310	2	0	0	X
ma-11	310	3	.	.	PUNCT
ma-11	311	1	then	then	ADV
ma-11	311	2	we	we	PRON
ma-11	311	3	have	have	VERB
ma-11	311	4	the	the	DET
ma-11	311	5	following	follow	VERB
ma-11	311	6	inequality	inequality	NOUN
ma-11	311	7	j	j	PROPN
ma-11	311	8	αa	αa	INTJ
ma-11	311	9	(	(	PUNCT
ma-11	311	10	|w(x)||f	|w(x)||f	X
ma-11	311	11	(	(	PUNCT
ma-11	311	12	x)|2	x)|2	PROPN
ma-11	311	13	)	)	PUNCT
ma-11	312	1	j	j	PROPN
ma-11	312	2	αa	αa	INTJ
ma-11	312	3	(	(	PUNCT
ma-11	312	4	|w(x)||g(x)|2	|w(x)||g(x)|2	NUM
ma-11	312	5	)	)	PUNCT
ma-11	312	6	{	{	PUNCT
ma-11	312	7	j	j	PROPN
ma-11	312	8	αa	αa	INTJ
ma-11	312	9	(	(	PUNCT
ma-11	312	10	|w(x)||f	|w(x)||f	X
ma-11	312	11	(	(	PUNCT
ma-11	312	12	x)g(x)|)}2	x)g(x)|)}2	SYM
ma-11	312	13	≤	≤	NUM
ma-11	312	14	1	1	NUM
ma-11	312	15	4	4	NUM
ma-11	312	16	(	(	PUNCT
ma-11	312	17	√	√	PROPN
ma-11	312	18	mn	mn	PROPN
ma-11	312	19	mn	mn	PROPN
ma-11	312	20	+	+	CCONJ
ma-11	312	21	√	√	PROPN
ma-11	312	22	mn	mn	PROPN
ma-11	312	23	mn	mn	PROPN
ma-11	312	24	)	)	PUNCT
ma-11	312	25	2	2	NUM
ma-11	312	26	.	.	PUNCT
ma-11	313	1	(	(	PUNCT
ma-11	313	2	34	34	NUM
ma-11	313	3	)	)	PUNCT
ma-11	313	4	proof	proof	NOUN
ma-11	313	5	.	.	PUNCT
ma-11	314	1	similar	similar	ADJ
ma-11	314	2	to	to	ADP
ma-11	314	3	the	the	DET
ma-11	314	4	proof	proof	NOUN
ma-11	314	5	of	of	ADP
ma-11	314	6	corollary	corollary	ADJ
ma-11	314	7	3.11	3.11	NUM
ma-11	314	8	.	.	PUNCT
ma-11	315	1	�	�	PROPN
ma-11	315	2	remark	remark	VERB
ma-11	315	3	3.4	3.4	NUM
ma-11	315	4	.	.	PUNCT
ma-11	316	1	we	we	PRON
ma-11	316	2	have	have	VERB
ma-11	316	3	the	the	DET
ma-11	316	4	following:(i	following:(i	NOUN
ma-11	316	5	)	)	PUNCT
ma-11	316	6	let	let	VERB
ma-11	316	7	α	α	NOUN
ma-11	316	8	=	=	SYM
ma-11	316	9	1	1	NUM
ma-11	316	10	,	,	PUNCT
ma-11	317	1	t	t	NOUN
ma-11	317	2	=	=	SYM
ma-11	317	3	z	z	PROPN
ma-11	317	4	,	,	PUNCT
ma-11	317	5	a	a	DET
ma-11	317	6	=	=	SYM
ma-11	317	7	1	1	NUM
ma-11	317	8	,	,	PUNCT
ma-11	317	9	x	x	PUNCT
ma-11	318	1	=	=	SYM
ma-11	318	2	b	b	X
ma-11	318	3	=	=	SYM
ma-11	319	1	p	p	NOUN
ma-11	320	1	+	+	ADJ
ma-11	320	2	1	1	NUM
ma-11	320	3	,	,	PUNCT
ma-11	320	4	xk	xk	X
ma-11	320	5	>	>	X
ma-11	320	6	0	0	PROPN
ma-11	320	7	,	,	PUNCT
ma-11	320	8	w(k	w(k	NOUN
ma-11	320	9	)	)	PUNCT
ma-11	320	10	=	=	PUNCT
ma-11	321	1	wk	wk	NOUN
ma-11	321	2	=	=	SYM
ma-11	321	3	1	1	NUM
ma-11	321	4	xk	xk	PROPN
ma-11	321	5	,	,	PUNCT
ma-11	321	6	f	f	PROPN
ma-11	321	7	(	(	PUNCT
ma-11	321	8	k	k	NOUN
ma-11	321	9	)	)	PUNCT
ma-11	321	10	=	=	SYM
ma-11	321	11	xk	xk	PROPN
ma-11	321	12	for	for	ADP
ma-11	321	13	k	k	PROPN
ma-11	321	14	=	=	SYM
ma-11	321	15	1	1	NUM
ma-11	321	16	,	,	PUNCT
ma-11	321	17	.	.	PUNCT
ma-11	321	18	.	.	PUNCT
ma-11	321	19	.	.	PUNCT
ma-11	322	1	,	,	PUNCT
ma-11	322	2	p	p	NOUN
ma-11	322	3	and	and	CCONJ
ma-11	322	4	n	n	CCONJ
ma-11	322	5	=	=	SYM
ma-11	322	6	g	g	PROPN
ma-11	322	7	=	=	SYM
ma-11	322	8	n	n	NOUN
ma-11	322	9	=	=	SYM
ma-11	322	10	1	1	X
ma-11	322	11	.	.	PUNCT
ma-11	322	12	then	then	ADV
ma-11	322	13	inequality	inequality	NOUN
ma-11	322	14	(	(	PUNCT
ma-11	322	15	33	33	NUM
ma-11	322	16	)	)	PUNCT
ma-11	322	17	reduces	reduce	VERB
ma-11	322	18	to	to	ADP
ma-11	322	19	inequality	inequality	NOUN
ma-11	322	20	(	(	PUNCT
ma-11	322	21	5).(ii	5).(ii	NUM
ma-11	322	22	)	)	PUNCT
ma-11	322	23	let	let	VERB
ma-11	322	24	α	α	NOUN
ma-11	322	25	=	=	SYM
ma-11	322	26	1	1	NUM
ma-11	322	27	,	,	PUNCT
ma-11	322	28	t	t	NOUN
ma-11	323	1	=	=	SYM
ma-11	323	2	r	r	NOUN
ma-11	323	3	,	,	PUNCT
ma-11	323	4	x	x	X
ma-11	323	5	=	=	SYM
ma-11	323	6	b	b	PROPN
ma-11	323	7	,	,	PUNCT
ma-11	323	8	0	0	PUNCT
ma-11	323	9	<	<	X
ma-11	323	10	m	m	VERB
ma-11	323	11	≤	≤	ADJ
ma-11	323	12	f	f	X
ma-11	323	13	(	(	PUNCT
ma-11	323	14	y	y	NOUN
ma-11	323	15	)	)	PUNCT
ma-11	323	16	≤	≤	NUM
ma-11	324	1	m	m	VERB
ma-11	324	2	<	<	X
ma-11	324	3	∞	∞	PROPN
ma-11	324	4	,	,	PUNCT
ma-11	324	5	w(y	w(y	NOUN
ma-11	324	6	)	)	PUNCT
ma-11	324	7	=	=	SYM
ma-11	324	8	1	1	NUM
ma-11	324	9	f	f	X
ma-11	324	10	(	(	PUNCT
ma-11	324	11	y	y	NOUN
ma-11	324	12	)	)	PUNCT
ma-11	324	13	on	on	ADP
ma-11	324	14	[	[	X
ma-11	324	15	a	a	X
ma-11	324	16	,	,	PUNCT
ma-11	324	17	b	b	NOUN
ma-11	324	18	]	]	X
ma-11	324	19	and	and	CCONJ
ma-11	324	20	n	n	CCONJ
ma-11	324	21	=	=	SYM
ma-11	324	22	g	g	PROPN
ma-11	324	23	=	=	SYM
ma-11	324	24	n	n	NOUN
ma-11	324	25	=	=	SYM
ma-11	324	26	1	1	X
ma-11	324	27	.	.	PUNCT
ma-11	324	28	then	then	ADV
ma-11	324	29	inequality	inequality	NOUN
ma-11	324	30	(	(	PUNCT
ma-11	324	31	33	33	NUM
ma-11	324	32	)	)	PUNCT
ma-11	324	33	reduces	reduce	VERB
ma-11	324	34	to	to	ADP
ma-11	324	35	inequality	inequality	NOUN
ma-11	324	36	(	(	PUNCT
ma-11	324	37	6).(iii	6).(iii	NUM
ma-11	324	38	)	)	PUNCT
ma-11	324	39	let	let	VERB
ma-11	324	40	α	α	NOUN
ma-11	324	41	=	=	SYM
ma-11	324	42	1	1	NUM
ma-11	324	43	,	,	PUNCT
ma-11	324	44	t	t	NOUN
ma-11	325	1	=	=	SYM
ma-11	325	2	z	z	PROPN
ma-11	325	3	,	,	PUNCT
ma-11	325	4	a	a	DET
ma-11	325	5	=	=	SYM
ma-11	325	6	1	1	NUM
ma-11	325	7	,	,	PUNCT
ma-11	325	8	x	x	PUNCT
ma-11	326	1	=	=	SYM
ma-11	326	2	b	b	X
ma-11	326	3	=	=	SYM
ma-11	327	1	p	p	NOUN
ma-11	328	1	+	+	ADJ
ma-11	328	2	1	1	NUM
ma-11	328	3	,	,	PUNCT
ma-11	328	4	w	w	PROPN
ma-11	328	5	≡	≡	PROPN
ma-11	328	6	1	1	NUM
ma-11	328	7	,	,	PUNCT
ma-11	328	8	f	f	PROPN
ma-11	328	9	(	(	PUNCT
ma-11	328	10	k	k	NOUN
ma-11	328	11	)	)	PUNCT
ma-11	328	12	=	=	SYM
ma-11	328	13	xk	xk	X
ma-11	328	14	>	>	X
ma-11	328	15	0	0	PUNCT
ma-11	328	16	and	and	CCONJ
ma-11	328	17	g(k	g(k	NOUN
ma-11	328	18	)	)	PUNCT
ma-11	328	19	=	=	SYM
ma-11	328	20	yk	yk	PROPN
ma-11	328	21	>	>	X
ma-11	328	22	0	0	PUNCT
ma-11	329	1	for	for	ADP
ma-11	329	2	k	k	PROPN
ma-11	329	3	=	=	SYM
ma-11	329	4	1	1	NUM
ma-11	329	5	,	,	PUNCT
ma-11	329	6	.	.	PUNCT
ma-11	329	7	.	.	PUNCT
ma-11	329	8	.	.	PUNCT
ma-11	330	1	,	,	PUNCT
ma-11	330	2	p.	p.	NOUN
ma-11	330	3	then	then	ADV
ma-11	330	4	inequality	inequality	NOUN
ma-11	330	5	(	(	PUNCT
ma-11	330	6	33	33	NUM
ma-11	330	7	)	)	PUNCT
ma-11	330	8	reduces	reduce	VERB
ma-11	330	9	to	to	ADP
ma-11	330	10	inequality	inequality	NOUN
ma-11	330	11	(	(	PUNCT
ma-11	330	12	7	7	NUM
ma-11	330	13	)	)	PUNCT
ma-11	330	14	.	.	PUNCT
ma-11	331	1	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	331	2	eur	eur	PROPN
ma-11	331	3	.	.	PUNCT
ma-11	332	1	j.	j.	PROPN
ma-11	332	2	math	math	PROPN
ma-11	332	3	.	.	PUNCT
ma-11	333	1	anal	anal	PROPN
ma-11	333	2	.	.	PUNCT
ma-11	334	1	10.28924	10.28924	NUM
ma-11	334	2	/	/	SYM
ma-11	334	3	ada	ada	PROPN
ma-11	334	4	/	/	SYM
ma-11	334	5	ma.3.12	ma.3.12	PROPN
ma-11	334	6	10(iv	10(iv	NUM
ma-11	334	7	)	)	PUNCT
ma-11	334	8	let	let	VERB
ma-11	334	9	α	α	NOUN
ma-11	334	10	=	=	SYM
ma-11	334	11	1	1	NUM
ma-11	334	12	,	,	PUNCT
ma-11	334	13	t	t	NOUN
ma-11	335	1	=	=	SYM
ma-11	335	2	z	z	PROPN
ma-11	335	3	,	,	PUNCT
ma-11	335	4	a	a	DET
ma-11	335	5	=	=	SYM
ma-11	335	6	1	1	NUM
ma-11	335	7	,	,	PUNCT
ma-11	335	8	x	x	PUNCT
ma-11	336	1	=	=	SYM
ma-11	336	2	b	b	X
ma-11	336	3	=	=	SYM
ma-11	337	1	p	p	NOUN
ma-11	338	1	+	+	ADJ
ma-11	338	2	1	1	NUM
ma-11	338	3	,	,	PUNCT
ma-11	338	4	xk	xk	X
ma-11	338	5	>	>	X
ma-11	338	6	0	0	PROPN
ma-11	338	7	,	,	PUNCT
ma-11	338	8	yk	yk	PROPN
ma-11	338	9	∈	∈	PROPN
ma-11	338	10	r	r	PROPN
ma-11	338	11	,	,	PUNCT
ma-11	338	12	w(k	w(k	NOUN
ma-11	338	13	)	)	PUNCT
ma-11	338	14	=	=	PUNCT
ma-11	339	1	wk	wk	NOUN
ma-11	339	2	=	=	SYM
ma-11	339	3	1	1	NUM
ma-11	339	4	xk	xk	NOUN
ma-11	339	5	y2	y2	PROPN
ma-11	340	1	k	k	PROPN
ma-11	340	2	,	,	PUNCT
ma-11	340	3	f	f	PROPN
ma-11	340	4	(	(	PUNCT
ma-11	340	5	k	k	NOUN
ma-11	340	6	)	)	PUNCT
ma-11	340	7	=	=	SYM
ma-11	340	8	xk	xk	PROPN
ma-11	340	9	for	for	ADP
ma-11	340	10	k	k	PROPN
ma-11	340	11	=	=	SYM
ma-11	340	12	1	1	NUM
ma-11	340	13	,	,	PUNCT
ma-11	340	14	.	.	PUNCT
ma-11	340	15	.	.	PUNCT
ma-11	340	16	.	.	PUNCT
ma-11	341	1	,	,	PUNCT
ma-11	341	2	p	p	NOUN
ma-11	341	3	and	and	CCONJ
ma-11	341	4	n	n	CCONJ
ma-11	341	5	=	=	SYM
ma-11	341	6	g	g	PROPN
ma-11	341	7	=	=	SYM
ma-11	341	8	n	n	NOUN
ma-11	341	9	=	=	SYM
ma-11	341	10	1	1	X
ma-11	341	11	.	.	PUNCT
ma-11	341	12	then	then	ADV
ma-11	341	13	inequality	inequality	NOUN
ma-11	341	14	(	(	PUNCT
ma-11	341	15	33	33	NUM
ma-11	341	16	)	)	PUNCT
ma-11	341	17	reduces	reduce	VERB
ma-11	341	18	to	to	ADP
ma-11	341	19	inequality	inequality	NOUN
ma-11	341	20	(	(	PUNCT
ma-11	341	21	8).(v	8).(v	NUM
ma-11	341	22	)	)	PUNCT
ma-11	341	23	let	let	VERB
ma-11	341	24	α	α	NOUN
ma-11	341	25	=	=	SYM
ma-11	341	26	1	1	NUM
ma-11	341	27	,	,	PUNCT
ma-11	341	28	t	t	NOUN
ma-11	342	1	=	=	SYM
ma-11	342	2	z	z	PROPN
ma-11	342	3	,	,	PUNCT
ma-11	342	4	a	a	DET
ma-11	342	5	=	=	SYM
ma-11	342	6	1	1	NUM
ma-11	342	7	,	,	PUNCT
ma-11	342	8	x	x	PUNCT
ma-11	343	1	=	=	SYM
ma-11	343	2	b	b	X
ma-11	343	3	=	=	SYM
ma-11	344	1	p	p	NOUN
ma-11	345	1	+	+	ADJ
ma-11	345	2	1	1	NUM
ma-11	345	3	,	,	PUNCT
ma-11	345	4	zk	zk	PROPN
ma-11	345	5	∈	∈	PROPN
ma-11	345	6	r	r	PROPN
ma-11	345	7	,	,	PUNCT
ma-11	345	8	w(k	w(k	NOUN
ma-11	345	9	)	)	PUNCT
ma-11	345	10	=	=	PUNCT
ma-11	345	11	wk	wk	NOUN
ma-11	345	12	=	=	PROPN
ma-11	345	13	z2	z2	PROPN
ma-11	345	14	k	k	PROPN
ma-11	345	15	,	,	PUNCT
ma-11	345	16	f	f	PROPN
ma-11	345	17	(	(	PUNCT
ma-11	345	18	k	k	NOUN
ma-11	345	19	)	)	PUNCT
ma-11	345	20	=	=	SYM
ma-11	345	21	xk	xk	X
ma-11	345	22	>	>	X
ma-11	345	23	0	0	PUNCT
ma-11	345	24	and	and	CCONJ
ma-11	345	25	g(k	g(k	NOUN
ma-11	345	26	)	)	PUNCT
ma-11	345	27	=	=	SYM
ma-11	345	28	yk	yk	PROPN
ma-11	345	29	>	>	X
ma-11	345	30	0	0	PUNCT
ma-11	345	31	for	for	ADP
ma-11	345	32	k	k	PROPN
ma-11	345	33	=	=	SYM
ma-11	345	34	1	1	NUM
ma-11	345	35	,	,	PUNCT
ma-11	345	36	.	.	PUNCT
ma-11	345	37	.	.	PUNCT
ma-11	345	38	.	.	PUNCT
ma-11	346	1	,	,	PUNCT
ma-11	346	2	p.	p.	NOUN
ma-11	346	3	then	then	ADV
ma-11	346	4	inequality	inequality	NOUN
ma-11	346	5	(	(	PUNCT
ma-11	346	6	33	33	NUM
ma-11	346	7	)	)	PUNCT
ma-11	346	8	reduces	reduce	VERB
ma-11	346	9	to	to	ADP
ma-11	346	10	inequality	inequality	NOUN
ma-11	346	11	(	(	PUNCT
ma-11	346	12	9	9	NUM
ma-11	346	13	)	)	PUNCT
ma-11	346	14	.	.	PUNCT
ma-11	347	1	4	4	X
ma-11	347	2	.	.	X
ma-11	347	3	conclusion	conclusion	VERB
ma-11	347	4	the	the	DET
ma-11	347	5	subject	subject	NOUN
ma-11	347	6	of	of	ADP
ma-11	347	7	dynamic	dynamic	ADJ
ma-11	347	8	inequalities	inequality	NOUN
ma-11	347	9	on	on	ADP
ma-11	347	10	time	time	NOUN
ma-11	347	11	scales	scale	NOUN
ma-11	347	12	has	have	AUX
ma-11	347	13	become	become	VERB
ma-11	347	14	a	a	DET
ma-11	347	15	crucial	crucial	ADJ
ma-11	347	16	field	field	NOUN
ma-11	347	17	of	of	ADP
ma-11	347	18	pure	pure	ADJ
ma-11	347	19	and	and	CCONJ
ma-11	347	20	appliedmathematics	appliedmathematic	NOUN
ma-11	347	21	.	.	PUNCT
ma-11	348	1	many	many	ADJ
ma-11	348	2	researchers	researcher	NOUN
ma-11	348	3	developed	develop	VERB
ma-11	348	4	interesting	interesting	ADJ
ma-11	348	5	results	result	NOUN
ma-11	348	6	concerning	concern	VERB
ma-11	348	7	fractional	fractional	ADJ
ma-11	348	8	calculus	calculus	NOUN
ma-11	348	9	on	on	ADP
ma-11	348	10	timescales	timescale	NOUN
ma-11	348	11	.	.	PUNCT
ma-11	349	1	due	due	ADP
ma-11	349	2	to	to	ADP
ma-11	349	3	utility	utility	NOUN
ma-11	349	4	of	of	ADP
ma-11	349	5	dynamic	dynamic	ADJ
ma-11	349	6	inequalities	inequality	NOUN
ma-11	349	7	in	in	ADP
ma-11	349	8	many	many	ADJ
ma-11	349	9	branches	branch	NOUN
ma-11	349	10	of	of	ADP
ma-11	349	11	mathematics	mathematic	NOUN
ma-11	349	12	,	,	PUNCT
ma-11	349	13	this	this	DET
ma-11	349	14	field	field	NOUN
ma-11	349	15	is	be	AUX
ma-11	349	16	given	give	VERB
ma-11	349	17	aprominent	aprominent	NOUN
ma-11	349	18	importance	importance	NOUN
ma-11	349	19	.	.	PUNCT
ma-11	350	1	this	this	DET
ma-11	350	2	field	field	NOUN
ma-11	350	3	has	have	VERB
ma-11	350	4	a	a	DET
ma-11	350	5	wide	wide	ADJ
ma-11	350	6	scope	scope	NOUN
ma-11	350	7	.	.	PUNCT
ma-11	351	1	recently	recently	ADV
ma-11	351	2	,	,	PUNCT
ma-11	351	3	interesting	interesting	ADJ
ma-11	351	4	results	result	NOUN
ma-11	351	5	have	have	AUX
ma-11	351	6	obtained	obtain	VERB
ma-11	351	7	byusing	byusing	NOUN
ma-11	351	8	specht	specht	PROPN
ma-11	351	9	’s	’s	PART
ma-11	351	10	ratio	ratio	NOUN
ma-11	351	11	and	and	CCONJ
ma-11	351	12	kantorovich	kantorovich	PROPN
ma-11	351	13	’s	’s	PART
ma-11	351	14	ratio	ratio	NOUN
ma-11	351	15	on	on	ADP
ma-11	351	16	time	time	NOUN
ma-11	351	17	scales	scale	NOUN
ma-11	351	18	as	as	SCONJ
ma-11	351	19	given	give	VERB
ma-11	351	20	in	in	ADP
ma-11	351	21	[	[	X
ma-11	351	22	18	18	NUM
ma-11	351	23	]	]	PUNCT
ma-11	351	24	.	.	PUNCT
ma-11	352	1	by	by	ADP
ma-11	352	2	using	use	VERB
ma-11	352	3	these	these	DET
ma-11	352	4	ratios	ratio	NOUN
ma-11	352	5	,	,	PUNCT
ma-11	352	6	we	we	PRON
ma-11	352	7	can	can	AUX
ma-11	352	8	explore	explore	VERB
ma-11	352	9	further	further	ADJ
ma-11	352	10	results.dynamic	results.dynamic	ADJ
ma-11	352	11	inequalities	inequality	NOUN
ma-11	352	12	may	may	AUX
ma-11	352	13	be	be	AUX
ma-11	352	14	extended	extend	VERB
ma-11	352	15	by	by	ADP
ma-11	352	16	applying	apply	VERB
ma-11	352	17	other	other	ADJ
ma-11	352	18	techniques	technique	NOUN
ma-11	352	19	such	such	ADJ
ma-11	352	20	as	as	ADP
ma-11	352	21	diamond	diamond	NOUN
ma-11	352	22	-	-	PUNCT
ma-11	352	23	α	α	NOUN
ma-11	352	24	inte	inte	NOUN
ma-11	352	25	-	-	PUNCT
ma-11	352	26	gral	gral	NOUN
ma-11	352	27	,	,	PUNCT
ma-11	352	28	which	which	PRON
ma-11	352	29	is	be	AUX
ma-11	352	30	defined	define	VERB
ma-11	352	31	as	as	ADP
ma-11	352	32	a	a	DET
ma-11	352	33	linear	linear	ADJ
ma-11	352	34	operator	operator	NOUN
ma-11	352	35	of	of	ADP
ma-11	352	36	delta	delta	PROPN
ma-11	352	37	and	and	CCONJ
ma-11	352	38	nabla	nabla	NOUN
ma-11	352	39	integrals	integral	NOUN
ma-11	352	40	on	on	ADP
ma-11	352	41	time	time	NOUN
ma-11	352	42	scales	scale	NOUN
ma-11	352	43	.	.	PUNCT
ma-11	353	1	quantumcalculus	quantumcalculus	NOUN
ma-11	353	2	,	,	PUNCT
ma-11	353	3	α	α	X
ma-11	353	4	,	,	PUNCT
ma-11	353	5	β	β	ADJ
ma-11	353	6	-	-	ADJ
ma-11	353	7	symmetric	symmetric	ADJ
ma-11	353	8	quantum	quantum	NOUN
ma-11	353	9	calculus	calculus	NOUN
ma-11	353	10	,	,	PUNCT
ma-11	353	11	functional	functional	ADJ
ma-11	353	12	generalization	generalization	NOUN
ma-11	353	13	,	,	PUNCT
ma-11	353	14	fractional	fractional	ADJ
ma-11	353	15	derivatives	derivative	NOUN
ma-11	353	16	and	and	CCONJ
ma-11	353	17	n	n	CCONJ
ma-11	353	18	-	-	PUNCT
ma-11	353	19	tuple	tuple	NOUN
ma-11	353	20	diamond	diamond	NOUN
ma-11	353	21	-	-	PUNCT
ma-11	353	22	alpha	alpha	NOUN
ma-11	353	23	integral	integral	ADJ
ma-11	353	24	are	be	AUX
ma-11	353	25	some	some	DET
ma-11	353	26	other	other	ADJ
ma-11	353	27	developed	develop	VERB
ma-11	353	28	techniques	technique	NOUN
ma-11	353	29	and	and	CCONJ
ma-11	353	30	we	we	PRON
ma-11	353	31	will	will	AUX
ma-11	353	32	continue	continue	VERB
ma-11	353	33	them	they	PRON
ma-11	353	34	toinvestigate	toinvestigate	VERB
ma-11	353	35	other	other	ADJ
ma-11	353	36	dynamic	dynamic	ADJ
ma-11	353	37	inequalities	inequality	NOUN
ma-11	353	38	in	in	ADP
ma-11	353	39	future	future	ADJ
ma-11	353	40	research	research	NOUN
ma-11	353	41	.	.	PUNCT
ma-11	354	1	references	reference	NOUN
ma-11	354	2	[	[	X
ma-11	354	3	1	1	NUM
ma-11	354	4	]	]	X
ma-11	354	5	r.p	r.p	PROPN
ma-11	354	6	.	.	PROPN
ma-11	354	7	agarwal	agarwal	PROPN
ma-11	354	8	,	,	PUNCT
ma-11	354	9	d.	d.	PROPN
ma-11	354	10	o’regan	o’regan	PROPN
ma-11	354	11	,	,	PUNCT
ma-11	354	12	s.h	s.h	PROPN
ma-11	354	13	.	.	PROPN
ma-11	354	14	saker	saker	PROPN
ma-11	354	15	,	,	PUNCT
ma-11	354	16	dynamic	dynamic	ADJ
ma-11	354	17	inequalities	inequality	NOUN
ma-11	354	18	on	on	ADP
ma-11	354	19	time	time	NOUN
ma-11	354	20	scales	scale	NOUN
ma-11	354	21	,	,	PUNCT
ma-11	354	22	springer	springer	NOUN
ma-11	354	23	,	,	PUNCT
ma-11	354	24	cham	cham	PROPN
ma-11	354	25	,	,	PUNCT
ma-11	354	26	switzerland	switzerland	PROPN
ma-11	354	27	,	,	PUNCT
ma-11	354	28	2014	2014	NUM
ma-11	354	29	.	.	PUNCT
ma-11	355	1	https://doi.org/10.1007/978-3-319-11002-8.[2	https://doi.org/10.1007/978-3-319-11002-8.[2	NOUN
ma-11	355	2	]	]	X
ma-11	355	3	g.a	g.a	PROPN
ma-11	355	4	.	.	PROPN
ma-11	355	5	anastassiou	anastassiou	PROPN
ma-11	355	6	,	,	PUNCT
ma-11	355	7	principles	principle	NOUN
ma-11	355	8	of	of	ADP
ma-11	355	9	delta	delta	NOUN
ma-11	355	10	fractional	fractional	ADJ
ma-11	355	11	calculus	calculus	NOUN
ma-11	355	12	on	on	ADP
ma-11	355	13	time	time	NOUN
ma-11	355	14	scales	scale	NOUN
ma-11	355	15	and	and	CCONJ
ma-11	355	16	inequalities	inequality	NOUN
ma-11	355	17	,	,	PUNCT
ma-11	355	18	math	math	NOUN
ma-11	355	19	.	.	PUNCT
ma-11	356	1	comp	comp	PROPN
ma-11	356	2	.	.	PUNCT
ma-11	356	3	model	model	PROPN
ma-11	356	4	.	.	PUNCT
ma-11	357	1	52(2010	52(2010	NUM
ma-11	357	2	)	)	PUNCT
ma-11	357	3	556–566	556–566	NUM
ma-11	357	4	.	.	PUNCT
ma-11	358	1	https://doi.org/10.1016/j.mcm.2010.03.055.[3	https://doi.org/10.1016/j.mcm.2010.03.055.[3	NOUN
ma-11	358	2	]	]	X
ma-11	358	3	g.a	g.a	PROPN
ma-11	358	4	.	.	PROPN
ma-11	358	5	anastassiou	anastassiou	PROPN
ma-11	358	6	,	,	PUNCT
ma-11	358	7	foundations	foundation	NOUN
ma-11	358	8	of	of	ADP
ma-11	358	9	nabla	nabla	NOUN
ma-11	358	10	fractional	fractional	ADJ
ma-11	358	11	calculus	calculus	NOUN
ma-11	358	12	on	on	ADP
ma-11	358	13	time	time	NOUN
ma-11	358	14	scales	scale	NOUN
ma-11	358	15	and	and	CCONJ
ma-11	358	16	inequalities	inequality	NOUN
ma-11	358	17	,	,	PUNCT
ma-11	358	18	comp	comp	PROPN
ma-11	358	19	.	.	PUNCT
ma-11	358	20	math	math	PROPN
ma-11	358	21	.	.	PUNCT
ma-11	359	1	appl	appl	PROPN
ma-11	359	2	.	.	PUNCT
ma-11	360	1	59(2010	59(2010	NOUN
ma-11	360	2	)	)	PUNCT
ma-11	360	3	3750–3762	3750–3762	NUM
ma-11	360	4	.	.	PUNCT
ma-11	361	1	https://doi.org/10.1016/j.camwa.2010.03.072.[4	https://doi.org/10.1016/j.camwa.2010.03.072.[4	PROPN
ma-11	361	2	]	]	X
ma-11	361	3	g.a	g.a	PROPN
ma-11	361	4	.	.	PROPN
ma-11	361	5	anastassiou	anastassiou	PROPN
ma-11	361	6	,	,	PUNCT
ma-11	361	7	integral	integral	ADJ
ma-11	361	8	operator	operator	NOUN
ma-11	361	9	inequalities	inequality	NOUN
ma-11	361	10	on	on	ADP
ma-11	361	11	time	time	NOUN
ma-11	361	12	scales	scale	NOUN
ma-11	361	13	,	,	PUNCT
ma-11	361	14	int	int	NOUN
ma-11	361	15	.	.	PUNCT
ma-11	362	1	j.	j.	PROPN
ma-11	362	2	diff	diff	PROPN
ma-11	362	3	.	.	PUNCT
ma-11	363	1	equ	equ	PROPN
ma-11	363	2	.	.	PROPN
ma-11	363	3	7	7	NUM
ma-11	363	4	(	(	PUNCT
ma-11	363	5	2012	2012	NUM
ma-11	363	6	)	)	PUNCT
ma-11	363	7	111–137.[5	111–137.[5	NUM
ma-11	363	8	]	]	X
ma-11	363	9	a.	a.	NOUN
ma-11	363	10	anber	anber	PROPN
ma-11	363	11	,	,	PUNCT
ma-11	363	12	z.	z.	PROPN
ma-11	363	13	dahmani	dahmani	PROPN
ma-11	363	14	,	,	PUNCT
ma-11	363	15	new	new	ADJ
ma-11	363	16	integral	integral	ADJ
ma-11	363	17	results	result	NOUN
ma-11	363	18	using	use	VERB
ma-11	363	19	pólya	pólya	NOUN
ma-11	363	20	–	–	PUNCT
ma-11	363	21	szegö	szegö	ADJ
ma-11	363	22	inequality	inequality	NOUN
ma-11	363	23	,	,	PUNCT
ma-11	363	24	acta	acta	PROPN
ma-11	363	25	comment	comment	NOUN
ma-11	363	26	.	.	PUNCT
ma-11	364	1	univ	univ	PROPN
ma-11	364	2	.	.	PUNCT
ma-11	364	3	tart	tart	PROPN
ma-11	364	4	.	.	PUNCT
ma-11	365	1	math	math	NOUN
ma-11	365	2	.	.	PUNCT
ma-11	366	1	17(2013	17(2013	NUM
ma-11	366	2	)	)	PUNCT
ma-11	367	1	171–178	171–178	NUM
ma-11	367	2	.	.	PUNCT
ma-11	368	1	https://doi.org/10.12697/acutm.2013.17.15.[6	https://doi.org/10.12697/acutm.2013.17.15.[6	X
ma-11	368	2	]	]	X
ma-11	368	3	d.	d.	PROPN
ma-11	368	4	anderson	anderson	PROPN
ma-11	368	5	,	,	PUNCT
ma-11	368	6	j.	j.	PROPN
ma-11	368	7	bullock	bullock	PROPN
ma-11	368	8	,	,	PUNCT
ma-11	368	9	l.	l.	PROPN
ma-11	368	10	erbe	erbe	PROPN
ma-11	368	11	,	,	PUNCT
ma-11	368	12	a.	a.	NOUN
ma-11	368	13	peterson	peterson	PROPN
ma-11	368	14	,	,	PUNCT
ma-11	368	15	h.	h.	PROPN
ma-11	368	16	tran	tran	PROPN
ma-11	368	17	,	,	PUNCT
ma-11	368	18	nabla	nabla	ADJ
ma-11	368	19	dynamic	dynamic	ADJ
ma-11	368	20	equations	equation	NOUN
ma-11	368	21	on	on	ADP
ma-11	368	22	time	time	NOUN
ma-11	368	23	scales	scale	NOUN
ma-11	368	24	,	,	PUNCT
ma-11	368	25	pan	pan	NOUN
ma-11	368	26	-	-	NOUN
ma-11	368	27	amer	amer	NOUN
ma-11	368	28	.	.	PUNCT
ma-11	369	1	math.j	math.j	PROPN
ma-11	369	2	.	.	PROPN
ma-11	369	3	13	13	NUM
ma-11	369	4	(	(	PUNCT
ma-11	369	5	2003	2003	NUM
ma-11	369	6	)	)	PUNCT
ma-11	369	7	1–47.[7	1–47.[7	NUM
ma-11	369	8	]	]	X
ma-11	369	9	m.	m.	NOUN
ma-11	369	10	bohner	bohner	NOUN
ma-11	369	11	,	,	PUNCT
ma-11	369	12	a.	a.	NOUN
ma-11	369	13	peterson	peterson	PROPN
ma-11	369	14	,	,	PUNCT
ma-11	369	15	dynamic	dynamic	ADJ
ma-11	369	16	equations	equation	NOUN
ma-11	369	17	on	on	ADP
ma-11	369	18	time	time	NOUN
ma-11	369	19	scales	scale	NOUN
ma-11	369	20	,	,	PUNCT
ma-11	369	21	birkhäuser	birkhäuser	PROPN
ma-11	369	22	boston	boston	PROPN
ma-11	369	23	,	,	PUNCT
ma-11	369	24	inc	inc	PROPN
ma-11	369	25	.	.	PROPN
ma-11	369	26	,	,	PUNCT
ma-11	369	27	boston	boston	PROPN
ma-11	369	28	,	,	PUNCT
ma-11	369	29	ma	ma	PROPN
ma-11	369	30	,	,	PUNCT
ma-11	369	31	2001.[8	2001.[8	NUM
ma-11	369	32	]	]	X
ma-11	369	33	m.	m.	NOUN
ma-11	369	34	bohner	bohner	NOUN
ma-11	369	35	,	,	PUNCT
ma-11	369	36	a.	a.	PROPN
ma-11	369	37	peterson	peterson	PROPN
ma-11	369	38	,	,	PUNCT
ma-11	369	39	advances	advance	VERB
ma-11	369	40	in	in	ADP
ma-11	369	41	dynamic	dynamic	ADJ
ma-11	369	42	equations	equation	NOUN
ma-11	369	43	on	on	ADP
ma-11	369	44	time	time	NOUN
ma-11	369	45	scales	scale	NOUN
ma-11	369	46	,	,	PUNCT
ma-11	369	47	birkhäuser	birkhäuser	PROPN
ma-11	369	48	boston	boston	PROPN
ma-11	369	49	,	,	PUNCT
ma-11	369	50	boston	boston	PROPN
ma-11	369	51	,	,	PUNCT
ma-11	369	52	ma	ma	PROPN
ma-11	369	53	,	,	PUNCT
ma-11	369	54	2003	2003	NUM
ma-11	369	55	.	.	PUNCT
ma-11	370	1	https://doi.org/10.1007/978-0-8176-8230-9.[9	https://doi.org/10.1007/978-0-8176-8230-9.[9	NOUN
ma-11	370	2	]	]	PUNCT
ma-11	370	3	m.	m.	PROPN
ma-11	370	4	bohner	bohner	NOUN
ma-11	370	5	,	,	PUNCT
ma-11	370	6	h.	h.	PROPN
ma-11	370	7	luo	luo	PROPN
ma-11	370	8	,	,	PUNCT
ma-11	370	9	singular	singular	ADJ
ma-11	370	10	second	second	ADJ
ma-11	370	11	-	-	PUNCT
ma-11	370	12	order	order	NOUN
ma-11	370	13	multipoint	multipoint	NOUN
ma-11	370	14	dynamic	dynamic	ADJ
ma-11	370	15	boundary	boundary	ADJ
ma-11	370	16	value	value	NOUN
ma-11	370	17	problems	problem	NOUN
ma-11	370	18	with	with	ADP
ma-11	370	19	mixed	mixed	ADJ
ma-11	370	20	derivatives	derivative	NOUN
ma-11	370	21	,	,	PUNCT
ma-11	370	22	adv.diff	adv.diff	PROPN
ma-11	370	23	.	.	PUNCT
ma-11	371	1	equ	equ	PROPN
ma-11	371	2	.	.	PUNCT
ma-11	372	1	(	(	PUNCT
ma-11	372	2	2006	2006	NUM
ma-11	372	3	)	)	PUNCT
ma-11	372	4	1–15	1–15	NUM
ma-11	372	5	.	.	PUNCT
ma-11	373	1	https://doi.org/10.1155/ade/2006/54989.[10	https://doi.org/10.1155/ade/2006/54989.[10	X
ma-11	373	2	]	]	X
ma-11	373	3	w.	w.	PROPN
ma-11	373	4	greub	greub	PROPN
ma-11	373	5	,	,	PUNCT
ma-11	373	6	w.	w.	PROPN
ma-11	373	7	rheinboldt	rheinboldt	PROPN
ma-11	373	8	,	,	PUNCT
ma-11	373	9	on	on	ADP
ma-11	373	10	a	a	DET
ma-11	373	11	generalization	generalization	NOUN
ma-11	373	12	of	of	ADP
ma-11	373	13	an	an	DET
ma-11	373	14	inequality	inequality	NOUN
ma-11	373	15	of	of	ADP
ma-11	373	16	l.v	l.v	PROPN
ma-11	373	17	.	.	PROPN
ma-11	373	18	kantorovich	kantorovich	PROPN
ma-11	373	19	,	,	PUNCT
ma-11	373	20	proc	proc	PROPN
ma-11	373	21	.	.	PUNCT
ma-11	374	1	amer	amer	PROPN
ma-11	374	2	.	.	PUNCT
ma-11	374	3	math	math	PROPN
ma-11	374	4	.	.	PUNCT
ma-11	375	1	soc	soc	PROPN
ma-11	375	2	.	.	PUNCT
ma-11	376	1	10	10	NUM
ma-11	376	2	(	(	PUNCT
ma-11	376	3	1959)407–415	1959)407–415	NUM
ma-11	376	4	.	.	PUNCT
ma-11	377	1	https://doi.org/10.1090/s0002-9939-1959-0105028-3.[11	https://doi.org/10.1090/s0002-9939-1959-0105028-3.[11	PROPN
ma-11	377	2	]	]	PUNCT
ma-11	377	3	s.	s.	PROPN
ma-11	377	4	hilger	hilger	PROPN
ma-11	377	5	,	,	PUNCT
ma-11	377	6	ein	ein	PROPN
ma-11	377	7	maβkettenkalkül	maβkettenkalkül	PROPN
ma-11	377	8	mit	mit	PROPN
ma-11	377	9	anwendung	anwendung	PROPN
ma-11	377	10	auf	auf	PROPN
ma-11	377	11	zentrumsmannigfaltigkeiten	zentrumsmannigfaltigkeiten	PROPN
ma-11	377	12	,	,	PUNCT
ma-11	377	13	ph.d	ph.d	PROPN
ma-11	377	14	.	.	PUNCT
ma-11	378	1	thesis	thesis	NOUN
ma-11	378	2	,	,	PUNCT
ma-11	378	3	universität	universität	ADJ
ma-11	378	4	würzburg,1988	würzburg,1988	NOUN
ma-11	378	5	.	.	PUNCT
ma-11	379	1	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	PROPN
ma-11	379	2	https://doi.org/10.1007/978-3-319-11002-8	https://doi.org/10.1007/978-3-319-11002-8	PROPN
ma-11	379	3	https://doi.org/10.1016/j.mcm.2010.03.055	https://doi.org/10.1016/j.mcm.2010.03.055	NOUN
ma-11	379	4	https://doi.org/10.1016/j.camwa.2010.03.072	https://doi.org/10.1016/j.camwa.2010.03.072	NOUN
ma-11	379	5	https://doi.org/10.12697/acutm.2013.17.15	https://doi.org/10.12697/acutm.2013.17.15	PRON
ma-11	379	6	https://doi.org/10.1007/978-0-8176-8230-9	https://doi.org/10.1007/978-0-8176-8230-9	VERB
ma-11	379	7	https://doi.org/10.1155/ade/2006/54989	https://doi.org/10.1155/ade/2006/54989	ADV
ma-11	379	8	https://doi.org/10.1090/s0002-9939-1959-0105028-3	https://doi.org/10.1090/s0002-9939-1959-0105028-3	PROPN
ma-11	379	9	eur	eur	PROPN
ma-11	379	10	.	.	PUNCT
ma-11	380	1	j.	j.	PROPN
ma-11	380	2	math	math	PROPN
ma-11	380	3	.	.	PUNCT
ma-11	381	1	anal	anal	PROPN
ma-11	381	2	.	.	PUNCT
ma-11	382	1	10.28924	10.28924	NUM
ma-11	382	2	/	/	SYM
ma-11	382	3	ada	ada	PROPN
ma-11	382	4	/	/	SYM
ma-11	382	5	ma.3.12	ma.3.12	PROPN
ma-11	382	6	11	11	NUM
ma-11	383	1	[	[	X
ma-11	383	2	12	12	NUM
ma-11	383	3	]	]	X
ma-11	383	4	l.v	l.v	PROPN
ma-11	383	5	.	.	PROPN
ma-11	383	6	kantorovich	kantorovich	PROPN
ma-11	383	7	,	,	PUNCT
ma-11	383	8	functional	functional	ADJ
ma-11	383	9	analysis	analysis	NOUN
ma-11	383	10	and	and	CCONJ
ma-11	383	11	applied	apply	VERB
ma-11	383	12	mathematics	mathematic	NOUN
ma-11	383	13	(	(	PUNCT
ma-11	383	14	russian	russian	PROPN
ma-11	383	15	)	)	PUNCT
ma-11	383	16	,	,	PUNCT
ma-11	383	17	uspehi	uspehi	PROPN
ma-11	383	18	mat	mat	PROPN
ma-11	383	19	.	.	PUNCT
ma-11	384	1	nauk	nauk	PROPN
ma-11	384	2	(	(	PUNCT
ma-11	384	3	n.s	n.s	PROPN
ma-11	384	4	.	.	PROPN
ma-11	384	5	)	)	PUNCT
ma-11	384	6	.	.	PUNCT
ma-11	385	1	3	3	NUM
ma-11	385	2	(	(	PUNCT
ma-11	385	3	1948	1948	NUM
ma-11	385	4	)	)	PUNCT
ma-11	385	5	89–185(in	89–185(in	NUM
ma-11	385	6	particular	particular	ADJ
ma-11	385	7	,	,	PUNCT
ma-11	385	8	pp	pp	ADJ
ma-11	385	9	.	.	PUNCT
ma-11	386	1	142–144	142–144	NUM
ma-11	386	2	)	)	PUNCT
ma-11	387	1	[	[	X
ma-11	387	2	also	also	ADV
ma-11	387	3	translated	translate	VERB
ma-11	387	4	from	from	ADP
ma-11	387	5	russian	russian	PROPN
ma-11	387	6	into	into	ADP
ma-11	387	7	english	english	PROPN
ma-11	387	8	by	by	ADP
ma-11	387	9	c.d	c.d	PROPN
ma-11	387	10	.	.	PROPN
ma-11	387	11	benster	benster	PROPN
ma-11	387	12	,	,	PUNCT
ma-11	387	13	nat	nat	PROPN
ma-11	387	14	.	.	PUNCT
ma-11	388	1	bur	bur	PROPN
ma-11	388	2	.	.	PROPN
ma-11	388	3	standards	standard	NOUN
ma-11	388	4	rep.no	rep.no	PROPN
ma-11	388	5	.	.	PUNCT
ma-11	388	6	1509	1509	NUM
ma-11	388	7	.	.	PUNCT
ma-11	389	1	1952	1952	NUM
ma-11	389	2	,	,	PUNCT
ma-11	389	3	202	202	NUM
ma-11	389	4	pp	pp	NOUN
ma-11	389	5	.	.	PUNCT
ma-11	390	1	(	(	PUNCT
ma-11	390	2	in	in	ADP
ma-11	390	3	particular	particular	ADJ
ma-11	390	4	,	,	PUNCT
ma-11	390	5	pp	pp	ADV
ma-11	390	6	.	.	PUNCT
ma-11	391	1	106–109)].[13	106–109)].[13	NUM
ma-11	391	2	]	]	X
ma-11	391	3	d.s	d.s	PROPN
ma-11	391	4	.	.	PROPN
ma-11	391	5	mitrinović	mitrinović	PROPN
ma-11	391	6	,	,	PUNCT
ma-11	391	7	analytic	analytic	ADJ
ma-11	391	8	inequalities	inequality	NOUN
ma-11	391	9	,	,	PUNCT
ma-11	391	10	springer	springer	NOUN
ma-11	391	11	-	-	PUNCT
ma-11	391	12	verlag	verlag	PROPN
ma-11	391	13	,	,	PUNCT
ma-11	391	14	berlin	berlin	PROPN
ma-11	391	15	,	,	PUNCT
ma-11	391	16	1970	1970	NUM
ma-11	391	17	.	.	PUNCT
ma-11	392	1	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
ma-11	392	2	978	978	NUM
ma-11	392	3	-	-	SYM
ma-11	392	4	3	3	NUM
ma-11	392	5	-	-	PUNCT
ma-11	392	6	642	642	NUM
ma-11	392	7	-	-	PUNCT
ma-11	392	8	99970	99970	NUM
ma-11	392	9	-	-	SYM
ma-11	392	10	3.[14	3.[14	NUM
ma-11	392	11	]	]	X
ma-11	392	12	s.k	s.k	PROPN
ma-11	392	13	.	.	PROPN
ma-11	392	14	ntouyas	ntouyas	PROPN
ma-11	392	15	,	,	PUNCT
ma-11	392	16	p.	p.	PROPN
ma-11	392	17	agarwal	agarwal	PROPN
ma-11	392	18	,	,	PUNCT
ma-11	392	19	j.	j.	PROPN
ma-11	392	20	tariboon	tariboon	PROPN
ma-11	392	21	,	,	PUNCT
ma-11	392	22	on	on	ADP
ma-11	392	23	pólya	pólya	NOUN
ma-11	392	24	–	–	PUNCT
ma-11	392	25	szegö	szegö	ADJ
ma-11	392	26	and	and	CCONJ
ma-11	392	27	chebyshev	chebyshev	PROPN
ma-11	392	28	types	type	NOUN
ma-11	392	29	inequalities	inequality	NOUN
ma-11	392	30	involving	involve	VERB
ma-11	392	31	the	the	DET
ma-11	392	32	riemann	riemann	PROPN
ma-11	392	33	–	–	PUNCT
ma-11	392	34	liouville	liouville	VERB
ma-11	392	35	fractional	fractional	ADJ
ma-11	392	36	integral	integral	ADJ
ma-11	392	37	operators	operator	NOUN
ma-11	392	38	,	,	PUNCT
ma-11	392	39	j.	j.	PROPN
ma-11	392	40	math	math	PROPN
ma-11	392	41	.	.	PUNCT
ma-11	393	1	ineq	ineq	PROPN
ma-11	393	2	.	.	PUNCT
ma-11	394	1	10	10	NUM
ma-11	394	2	(	(	PUNCT
ma-11	394	3	2016	2016	NUM
ma-11	394	4	)	)	PUNCT
ma-11	394	5	491–504	491–504	PROPN
ma-11	394	6	.	.	PUNCT
ma-11	395	1	https://doi.org/10.7153/jmi-10-38.[15	https://doi.org/10.7153/jmi-10-38.[15	PROPN
ma-11	395	2	]	]	X
ma-11	395	3	g.	g.	PROPN
ma-11	395	4	pólya	pólya	PROPN
ma-11	395	5	,	,	PUNCT
ma-11	395	6	g.	g.	PROPN
ma-11	395	7	szegö	szegö	PROPN
ma-11	395	8	,	,	PUNCT
ma-11	395	9	aufgaben	aufgaben	PROPN
ma-11	395	10	und	und	VERB
ma-11	395	11	lehrsätze	lehrsätze	PROPN
ma-11	395	12	aus	aus	PROPN
ma-11	395	13	der	der	PROPN
ma-11	395	14	analysis	analysis	NOUN
ma-11	395	15	,	,	PUNCT
ma-11	395	16	berlin	berlin	PROPN
ma-11	395	17	.	.	PROPN
ma-11	395	18	1	1	NUM
ma-11	395	19	(	(	PUNCT
ma-11	395	20	1925	1925	NUM
ma-11	395	21	)	)	PUNCT
ma-11	395	22	213–214	213–214	NUM
ma-11	395	23	.	.	PUNCT
ma-11	396	1	https://doi.org/10	https://doi.org/10	PROPN
ma-11	396	2	.	.	PUNCT
ma-11	397	1	1007/978	1007/978	NUM
ma-11	397	2	-	-	SYM
ma-11	397	3	3	3	NUM
ma-11	397	4	-	-	PUNCT
ma-11	397	5	662	662	NUM
ma-11	397	6	-	-	SYM
ma-11	397	7	38381	38381	NUM
ma-11	397	8	-	-	PUNCT
ma-11	397	9	0.[16	0.[16	NOUN
ma-11	397	10	]	]	X
ma-11	397	11	m.j.s	m.j.s	PROPN
ma-11	397	12	.	.	PROPN
ma-11	397	13	sahir	sahir	PROPN
ma-11	397	14	,	,	PUNCT
ma-11	397	15	formation	formation	NOUN
ma-11	397	16	of	of	ADP
ma-11	397	17	versions	version	NOUN
ma-11	397	18	of	of	ADP
ma-11	397	19	some	some	DET
ma-11	397	20	dynamic	dynamic	ADJ
ma-11	397	21	inequalities	inequality	NOUN
ma-11	397	22	unified	unify	VERB
ma-11	397	23	on	on	ADP
ma-11	397	24	time	time	NOUN
ma-11	397	25	scale	scale	NOUN
ma-11	397	26	calculus	calculus	NOUN
ma-11	397	27	,	,	PUNCT
ma-11	397	28	ural	ural	ADJ
ma-11	397	29	math	math	NOUN
ma-11	397	30	.	.	PUNCT
ma-11	398	1	j.	j.	PROPN
ma-11	398	2	4(2018	4(2018	PROPN
ma-11	398	3	)	)	PUNCT
ma-11	398	4	88–98	88–98	NUM
ma-11	398	5	.	.	PUNCT
ma-11	399	1	https://doi.org/10.15826/umj.2018.2.010.[17	https://doi.org/10.15826/umj.2018.2.010.[17	X
ma-11	399	2	]	]	X
ma-11	399	3	m.j.s	m.j.s	PROPN
ma-11	399	4	.	.	PROPN
ma-11	399	5	sahir	sahir	PROPN
ma-11	399	6	,	,	PUNCT
ma-11	399	7	symmetry	symmetry	NOUN
ma-11	399	8	of	of	ADP
ma-11	399	9	classical	classical	ADJ
ma-11	399	10	and	and	CCONJ
ma-11	399	11	extended	extended	ADJ
ma-11	399	12	dynamic	dynamic	ADJ
ma-11	399	13	inequalities	inequality	NOUN
ma-11	399	14	unified	unify	VERB
ma-11	399	15	on	on	ADP
ma-11	399	16	time	time	NOUN
ma-11	399	17	scale	scale	NOUN
ma-11	399	18	calculus	calculus	NOUN
ma-11	399	19	,	,	PUNCT
ma-11	399	20	turk	turk	PROPN
ma-11	399	21	.	.	PUNCT
ma-11	400	1	j.	j.	PROPN
ma-11	400	2	ineq.2	ineq.2	PROPN
ma-11	400	3	(	(	PUNCT
ma-11	400	4	2018	2018	NUM
ma-11	400	5	)	)	PUNCT
ma-11	400	6	11–22.[18	11–22.[18	NUM
ma-11	400	7	]	]	X
ma-11	400	8	m.j.s	m.j.s	PROPN
ma-11	400	9	.	.	PROPN
ma-11	400	10	sahir	sahir	PROPN
ma-11	400	11	,	,	PUNCT
ma-11	400	12	parity	parity	NOUN
ma-11	400	13	of	of	ADP
ma-11	400	14	classical	classical	ADJ
ma-11	400	15	and	and	CCONJ
ma-11	400	16	dynamic	dynamic	ADJ
ma-11	400	17	inequalities	inequality	NOUN
ma-11	400	18	magnified	magnify	VERB
ma-11	400	19	on	on	ADP
ma-11	400	20	time	time	NOUN
ma-11	400	21	scales	scale	NOUN
ma-11	400	22	,	,	PUNCT
ma-11	400	23	bull	bull	NOUN
ma-11	400	24	.	.	PUNCT
ma-11	401	1	int	int	NOUN
ma-11	401	2	.	.	PUNCT
ma-11	402	1	math	math	NOUN
ma-11	402	2	.	.	PUNCT
ma-11	403	1	virtual	virtual	ADJ
ma-11	403	2	inst	inst	NOUN
ma-11	403	3	.	.	PUNCT
ma-11	404	1	10(2020	10(2020	X
ma-11	404	2	)	)	PUNCT
ma-11	404	3	369–380.[19	369–380.[19	NOUN
ma-11	404	4	]	]	X
ma-11	405	1	p.	p.	NOUN
ma-11	405	2	schweitzer	schweitzer	PROPN
ma-11	405	3	,	,	PUNCT
ma-11	405	4	an	an	DET
ma-11	405	5	inequality	inequality	NOUN
ma-11	405	6	concerning	concern	VERB
ma-11	405	7	the	the	DET
ma-11	405	8	arithmetic	arithmetic	ADJ
ma-11	405	9	mean	mean	NOUN
ma-11	405	10	(	(	PUNCT
ma-11	405	11	hungarian	hungarian	PROPN
ma-11	405	12	)	)	PUNCT
ma-11	405	13	,	,	PUNCT
ma-11	405	14	math	math	NOUN
ma-11	405	15	.	.	PUNCT
ma-11	406	1	phys	phy	NOUN
ma-11	406	2	.	.	PUNCT
ma-11	407	1	lapok	lapok	NOUN
ma-11	407	2	.	.	PUNCT
ma-11	408	1	23	23	NUM
ma-11	408	2	(	(	PUNCT
ma-11	408	3	1914	1914	NUM
ma-11	408	4	)	)	PUNCT
ma-11	409	1	257–261	257–261	NUM
ma-11	409	2	.	.	PUNCT
ma-11	410	1	https://doi.org/10.28924/ada/ma.3.12	https://doi.org/10.28924/ada/ma.3.12	NUM
ma-11	410	2	https://doi.org/10.1007/978-3-642-99970-3	https://doi.org/10.1007/978-3-642-99970-3	PROPN
ma-11	410	3	https://doi.org/10.1007/978-3-642-99970-3	https://doi.org/10.1007/978-3-642-99970-3	NOUN
ma-11	410	4	https://doi.org/10.7153/jmi-10-38	https://doi.org/10.7153/jmi-10-38	NOUN
ma-11	410	5	https://doi.org/10.1007/978-3-662-38381-0	https://doi.org/10.1007/978-3-662-38381-0	VERB
ma-11	410	6	https://doi.org/10.1007/978-3-662-38381-0	https://doi.org/10.1007/978-3-662-38381-0	X
ma-11	410	7	https://doi.org/10.15826/umj.2018.2.010	https://doi.org/10.15826/umj.2018.2.010	NOUN
ma-11	410	8	1	1	NUM
ma-11	410	9	.	.	PUNCT
ma-11	410	10	introduction	introduction	NOUN
ma-11	410	11	2	2	NUM
ma-11	410	12	.	.	PUNCT
ma-11	410	13	preliminaries	preliminary	NOUN
ma-11	410	14	3	3	NUM
ma-11	410	15	.	.	X
ma-11	410	16	main	main	ADJ
ma-11	410	17	results	result	NOUN
ma-11	410	18	4	4	NUM
ma-11	410	19	.	.	PUNCT
ma-11	410	20	conclusion	conclusion	NOUN
ma-11	410	21	references	reference	NOUN