id	sid	tid	token	lemma	pos
cana-534	1	1	communications	communication	NOUN
cana-534	1	2	on	on	ADP
cana-534	1	3	applied	apply	VERB
cana-534	1	4	nonlinear	nonlinear	ADJ
cana-534	1	5	analysis	analysis	NOUN
cana-534	1	6	issn	issn	NOUN
cana-534	1	7	:	:	PUNCT
cana-534	1	8	1074	1074	NUM
cana-534	1	9	-	-	PUNCT
cana-534	1	10	133x	133x	NUM
cana-534	1	11	vol	vol	NOUN
cana-534	1	12	31	31	NUM
cana-534	1	13	no	no	NOUN
cana-534	1	14	.	.	NOUN
cana-534	1	15	2	2	NUM
cana-534	1	16	(	(	PUNCT
cana-534	1	17	2024	2024	NUM
cana-534	1	18	)	)	PUNCT
cana-534	1	19	197	197	NUM
cana-534	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	1	21	on	on	ADP
cana-534	1	22	a	a	DET
cana-534	1	23	certain	certain	ADJ
cana-534	1	24	subclass	subclass	NOUN
cana-534	1	25	of	of	ADP
cana-534	1	26	analytic	analytic	ADJ
cana-534	1	27	functions	function	NOUN
cana-534	1	28	defined	define	VERB
cana-534	1	29	by	by	ADP
cana-534	1	30	bessel	bessel	NOUN
cana-534	1	31	functions	function	NOUN
cana-534	1	32	katterapalle	katterapalle	PROPN
cana-534	1	33	sridevi𝟏	sridevi𝟏	NOUN
cana-534	1	34	,	,	PUNCT
cana-534	1	35	pasunoori	pasunoori	ADJ
cana-534	1	36	srinivasulu𝟐	srinivasulu𝟐	NOUN
cana-534	1	37	1	1	NUM
cana-534	1	38	department	department	NOUN
cana-534	1	39	of	of	ADP
cana-534	1	40	mathematics	mathematic	NOUN
cana-534	1	41	,	,	PUNCT
cana-534	1	42	dr.b.r.ambedkar	dr.b.r.ambedkar	PROPN
cana-534	1	43	open	open	PROPN
cana-534	1	44	university	university	PROPN
cana-534	1	45	,	,	PUNCT
cana-534	1	46	hyderabad	hyderabad	PROPN
cana-534	1	47	500	500	NUM
cana-534	1	48	033	033	NUM
cana-534	1	49	,	,	PUNCT
cana-534	1	50	t.s	t.s	PROPN
cana-534	1	51	,	,	PUNCT
cana-534	1	52	india	india	PROPN
cana-534	1	53	.	.	PUNCT
cana-534	2	1	e	e	X
cana-534	2	2	-	-	NOUN
cana-534	2	3	mail	mail	NOUN
cana-534	2	4	:	:	PUNCT
cana-534	2	5	sridevidrk18@gmail.com	sridevidrk18@gmail.com	X
cana-534	2	6	2	2	NUM
cana-534	2	7	department	department	NOUN
cana-534	2	8	of	of	ADP
cana-534	2	9	mathematics	mathematic	NOUN
cana-534	2	10	,	,	PUNCT
cana-534	2	11	dr.b.r.ambedkar	dr.b.r.ambedkar	PROPN
cana-534	2	12	open	open	PROPN
cana-534	2	13	university	university	PROPN
cana-534	2	14	,	,	PUNCT
cana-534	2	15	hyderabad	hyderabad	PROPN
cana-534	2	16	500	500	NUM
cana-534	2	17	033	033	NUM
cana-534	2	18	,	,	PUNCT
cana-534	2	19	t.s	t.s	PROPN
cana-534	2	20	,	,	PUNCT
cana-534	2	21	india	india	PROPN
cana-534	2	22	.	.	PUNCT
cana-534	3	1	e	e	X
cana-534	3	2	-	-	NOUN
cana-534	3	3	mail	mail	NOUN
cana-534	3	4	:	:	PUNCT
cana-534	3	5	srinivasrgukt1203@gmail.com	srinivasrgukt1203@gmail.com	X
cana-534	3	6	article	article	NOUN
cana-534	3	7	history	history	NOUN
cana-534	3	8	:	:	PUNCT
cana-534	3	9	received	receive	VERB
cana-534	3	10	:	:	PUNCT
cana-534	3	11	22	22	NUM
cana-534	3	12	-	-	SYM
cana-534	3	13	01	01	NUM
cana-534	3	14	-	-	PUNCT
cana-534	3	15	2024	2024	NUM
cana-534	3	16	revised	revise	VERB
cana-534	3	17	:	:	PUNCT
cana-534	3	18	08	08	NUM
cana-534	3	19	-	-	PUNCT
cana-534	3	20	04	04	NUM
cana-534	3	21	-	-	PUNCT
cana-534	3	22	2024	2024	NUM
cana-534	3	23	accepted	accept	VERB
cana-534	3	24	:	:	PUNCT
cana-534	3	25	29	29	NUM
cana-534	3	26	-	-	PUNCT
cana-534	3	27	04	04	NUM
cana-534	3	28	-	-	PUNCT
cana-534	3	29	2024	2024	NUM
cana-534	3	30	abstract	abstract	NOUN
cana-534	3	31	:	:	PUNCT
cana-534	3	32	in	in	ADP
cana-534	3	33	this	this	DET
cana-534	3	34	work	work	NOUN
cana-534	3	35	,	,	PUNCT
cana-534	3	36	we	we	PRON
cana-534	3	37	introduce	introduce	VERB
cana-534	3	38	and	and	CCONJ
cana-534	3	39	investigate	investigate	VERB
cana-534	3	40	a	a	DET
cana-534	3	41	new	new	ADJ
cana-534	3	42	subclass	subclass	NOUN
cana-534	3	43	of	of	ADP
cana-534	3	44	analytic	analytic	ADJ
cana-534	3	45	functions	function	NOUN
cana-534	3	46	in	in	ADP
cana-534	3	47	the	the	DET
cana-534	3	48	open	open	ADJ
cana-534	3	49	unit	unit	NOUN
cana-534	3	50	disc	disc	VERB
cana-534	3	51	𝑈	𝑈	PROPN
cana-534	3	52	with	with	ADP
cana-534	3	53	negative	negative	ADJ
cana-534	3	54	coefficients	coefficient	NOUN
cana-534	3	55	.	.	PUNCT
cana-534	4	1	the	the	DET
cana-534	4	2	object	object	NOUN
cana-534	4	3	of	of	ADP
cana-534	4	4	the	the	DET
cana-534	4	5	present	present	ADJ
cana-534	4	6	paper	paper	NOUN
cana-534	4	7	is	be	AUX
cana-534	4	8	to	to	PART
cana-534	4	9	determine	determine	VERB
cana-534	4	10	the	the	DET
cana-534	4	11	coefficient	coefficient	NOUN
cana-534	4	12	estimates	estimate	NOUN
cana-534	4	13	,	,	PUNCT
cana-534	4	14	extreme	extreme	ADJ
cana-534	4	15	points	point	NOUN
cana-534	4	16	,	,	PUNCT
cana-534	4	17	integral	integral	ADJ
cana-534	4	18	means	mean	NOUN
cana-534	4	19	inequalities	inequality	NOUN
cana-534	4	20	and	and	CCONJ
cana-534	4	21	subordination	subordination	NOUN
cana-534	4	22	results	result	NOUN
cana-534	4	23	for	for	ADP
cana-534	4	24	this	this	DET
cana-534	4	25	class	class	NOUN
cana-534	4	26	.	.	PUNCT
cana-534	5	1	keywords	keyword	NOUN
cana-534	5	2	:	:	PUNCT
cana-534	5	3	analytic	analytic	ADJ
cana-534	5	4	function	function	NOUN
cana-534	5	5	,	,	PUNCT
cana-534	5	6	uniformly	uniformly	ADV
cana-534	5	7	starlike	starlike	NOUN
cana-534	5	8	function	function	NOUN
cana-534	5	9	,	,	PUNCT
cana-534	5	10	coefficient	coefficient	NOUN
cana-534	5	11	estimate	estimate	NOUN
cana-534	5	12	,	,	PUNCT
cana-534	5	13	subordination	subordination	NOUN
cana-534	5	14	.	.	PUNCT
cana-534	6	1	1.introduction	1.introduction	NUM
cana-534	6	2	:	:	PUNCT
cana-534	6	3	let	let	VERB
cana-534	6	4	𝐴	𝐴	PROPN
cana-534	6	5	be	be	AUX
cana-534	6	6	the	the	DET
cana-534	6	7	class	class	NOUN
cana-534	6	8	of	of	ADP
cana-534	6	9	functions	function	NOUN
cana-534	6	10	𝑓	𝑓	PRON
cana-534	6	11	normalized	normalize	VERB
cana-534	6	12	by	by	ADP
cana-534	6	13	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	6	14	)	)	PUNCT
cana-534	6	15	=	=	PUNCT
cana-534	6	16	𝑧	𝑧	PROPN
cana-534	7	1	+	+	CCONJ
cana-534	7	2	∑	∑	PROPN
cana-534	7	3	𝑎𝑛	𝑎𝑛	PROPN
cana-534	7	4	∞	∞	NUM
cana-534	7	5	𝑛=2	𝑛=2	NOUN
cana-534	7	6	𝑧𝑛	𝑧𝑛	INTJ
cana-534	7	7	(	(	PUNCT
cana-534	7	8	1.1	1.1	NUM
cana-534	7	9	)	)	PUNCT
cana-534	7	10	and	and	CCONJ
cana-534	7	11	𝑇	𝑇	PROPN
cana-534	7	12	denote	denote	VERB
cana-534	7	13	the	the	DET
cana-534	7	14	class	class	NOUN
cana-534	7	15	of	of	ADP
cana-534	7	16	functions	function	NOUN
cana-534	7	17	in	in	ADP
cana-534	7	18	the	the	DET
cana-534	7	19	form	form	NOUN
cana-534	7	20	of	of	ADP
cana-534	7	21	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	7	22	)	)	PUNCT
cana-534	8	1	=	=	PUNCT
cana-534	8	2	𝑧	𝑧	PRON
cana-534	8	3	−	−	NOUN
cana-534	8	4	∑	∑	PROPN
cana-534	8	5	𝑎𝑛	𝑎𝑛	PROPN
cana-534	8	6	∞	∞	PROPN
cana-534	8	7	𝑛=2	𝑛=2	PROPN
cana-534	8	8	𝑧𝑛	𝑧𝑛	NOUN
cana-534	8	9	,	,	PUNCT
cana-534	8	10	(	(	PUNCT
cana-534	8	11	𝑎𝑛	𝑎𝑛	NOUN
cana-534	8	12	≥	≥	NOUN
cana-534	8	13	0	0	NUM
cana-534	8	14	)	)	PUNCT
cana-534	8	15	(	(	PUNCT
cana-534	8	16	1.2	1.2	NUM
cana-534	8	17	)	)	PUNCT
cana-534	8	18	,	,	PUNCT
cana-534	8	19	which	which	PRON
cana-534	8	20	are	be	AUX
cana-534	8	21	analytic	analytic	ADJ
cana-534	8	22	in	in	ADP
cana-534	8	23	the	the	DET
cana-534	8	24	open	open	ADJ
cana-534	8	25	unit	unit	NOUN
cana-534	8	26	disk	disk	NOUN
cana-534	8	27	𝑈	𝑈	NOUN
cana-534	8	28	=	=	PUNCT
cana-534	8	29	{	{	PUNCT
cana-534	8	30	𝑧	𝑧	NOUN
cana-534	8	31	:	:	PUNCT
cana-534	8	32	𝑧	𝑧	PRON
cana-534	8	33	∈	∈	PROPN
cana-534	8	34	𝒞	𝒞	NOUN
cana-534	8	35	and	and	CCONJ
cana-534	8	36	|𝑧|	|𝑧|	VERB
cana-534	8	37	<	<	X
cana-534	8	38	1	1	NUM
cana-534	8	39	}	}	PUNCT
cana-534	8	40	.	.	PUNCT
cana-534	9	1	this	this	DET
cana-534	9	2	subclass	subclass	NOUN
cana-534	9	3	was	be	AUX
cana-534	9	4	given	give	VERB
cana-534	9	5	in	in	ADP
cana-534	9	6	.	.	PUNCT
cana-534	10	1	let	let	VERB
cana-534	10	2	𝑇∗(𝛼	𝑇∗(𝛼	ADJ
cana-534	10	3	)	)	PUNCT
cana-534	10	4	and	and	CCONJ
cana-534	10	5	𝐶(𝛼	𝐶(𝛼	X
cana-534	10	6	)	)	PUNCT
cana-534	10	7	be	be	AUX
cana-534	10	8	indicate	indicate	VERB
cana-534	10	9	starlike	starlike	NOUN
cana-534	10	10	and	and	CCONJ
cana-534	10	11	convex	convex	NOUN
cana-534	10	12	functions	function	NOUN
cana-534	10	13	of	of	ADP
cana-534	10	14	order	order	NOUN
cana-534	10	15	𝛼	𝛼	NOUN
cana-534	10	16	,	,	PUNCT
cana-534	10	17	(	(	PUNCT
cana-534	10	18	0	0	NUM
cana-534	10	19	≤	≤	NUM
cana-534	10	20	𝛼	𝛼	X
cana-534	10	21	<	<	X
cana-534	10	22	1	1	NUM
cana-534	10	23	)	)	PUNCT
cana-534	10	24	,	,	PUNCT
cana-534	10	25	respectively	respectively	ADV
cana-534	10	26	.	.	PUNCT
cana-534	11	1	the	the	DET
cana-534	11	2	classes	class	NOUN
cana-534	11	3	𝑈𝐶𝑉(𝛼	𝑈𝐶𝑉(𝛼	NUM
cana-534	11	4	,	,	PUNCT
cana-534	11	5	𝜎	𝜎	NOUN
cana-534	11	6	)	)	PUNCT
cana-534	11	7	consists	consist	VERB
cana-534	11	8	of	of	ADP
cana-534	11	9	uniform	uniform	ADJ
cana-534	11	10	𝜎	𝜎	PROPN
cana-534	11	11	−convex	−convex	NOUN
cana-534	11	12	functions	function	NOUN
cana-534	11	13	of	of	ADP
cana-534	11	14	order	order	NOUN
cana-534	11	15	𝛼	𝛼	NOUN
cana-534	11	16	and	and	CCONJ
cana-534	11	17	𝑆𝑃(𝛼	𝑆𝑃(𝛼	NOUN
cana-534	11	18	,	,	PUNCT
cana-534	11	19	𝜎	𝜎	X
cana-534	11	20	)	)	PUNCT
cana-534	11	21	consists	consist	VERB
cana-534	11	22	parabolic	parabolic	NOUN
cana-534	11	23	𝜎	𝜎	PROPN
cana-534	11	24	−	−	PROPN
cana-534	11	25	starlike	starlike	NOUN
cana-534	11	26	functions	function	NOUN
cana-534	11	27	of	of	ADP
cana-534	11	28	order	order	NOUN
cana-534	11	29	𝛼	𝛼	NOUN
cana-534	11	30	,	,	PUNCT
cana-534	11	31	−1	−1	NOUN
cana-534	11	32	<	<	X
cana-534	11	33	𝛼	𝛼	X
cana-534	11	34	≤	≤	NUM
cana-534	11	35	1	1	NUM
cana-534	11	36	,	,	PUNCT
cana-534	11	37	𝜎	𝜎	PRON
cana-534	11	38	≥	≥	NOUN
cana-534	11	39	0	0	NUM
cana-534	11	40	,	,	PUNCT
cana-534	11	41	generalizes	generalize	VERB
cana-534	11	42	the	the	DET
cana-534	11	43	class	class	NOUN
cana-534	11	44	𝑈𝐶𝑉	𝑈𝐶𝑉	PROPN
cana-534	11	45	and	and	CCONJ
cana-534	11	46	𝑆𝑃	𝑆𝑃	PROPN
cana-534	11	47	respectively	respectively	ADV
cana-534	11	48	,	,	PUNCT
cana-534	11	49	were	be	AUX
cana-534	11	50	given	give	VERB
cana-534	11	51	in	in	ADP
cana-534	11	52	such	such	ADJ
cana-534	11	53	that	that	SCONJ
cana-534	11	54	𝑈𝐶𝑉(𝛼	𝑈𝐶𝑉(𝛼	PROPN
cana-534	11	55	,	,	PUNCT
cana-534	11	56	𝜎	𝜎	NOUN
cana-534	11	57	)	)	PUNCT
cana-534	11	58	=	=	SYM
cana-534	11	59	{	{	PUNCT
cana-534	11	60	𝑓	𝑓	PROPN
cana-534	11	61	∈	∈	PROPN
cana-534	11	62	𝐴	𝐴	PROPN
cana-534	11	63	:	:	PUNCT
cana-534	12	1	𝑅𝑒	𝑅𝑒	VERB
cana-534	12	2	{	{	PUNCT
cana-534	12	3	1	1	NUM
cana-534	12	4	+	+	NUM
cana-534	12	5	𝑧𝑓″(𝑧	𝑧𝑓″(𝑧	NOUN
cana-534	12	6	)	)	PUNCT
cana-534	12	7	𝑓′(𝑧	𝑓′(𝑧	NOUN
cana-534	12	8	)	)	PUNCT
cana-534	12	9	−	−	ADP
cana-534	12	10	𝛼	𝛼	X
cana-534	12	11	}	}	PUNCT
cana-534	12	12	>	>	X
cana-534	12	13	𝜎	𝜎	PROPN
cana-534	12	14	{	{	PUNCT
cana-534	12	15	𝑧𝑓″(𝑧	𝑧𝑓″(𝑧	NOUN
cana-534	12	16	)	)	PUNCT
cana-534	12	17	𝑓′(𝑧	𝑓′(𝑧	NOUN
cana-534	12	18	)	)	PUNCT
cana-534	12	19	}	}	PUNCT
cana-534	12	20	,	,	PUNCT
cana-534	12	21	𝑧	𝑧	PROPN
cana-534	12	22	∈	∈	PROPN
cana-534	12	23	𝑈	𝑈	PROPN
cana-534	12	24	}	}	PUNCT
cana-534	12	25	(	(	PUNCT
cana-534	12	26	1.3	1.3	NUM
cana-534	12	27	)	)	PUNCT
cana-534	12	28	and	and	CCONJ
cana-534	12	29	𝑆𝑃(𝛼	𝑆𝑃(𝛼	PROPN
cana-534	12	30	,	,	PUNCT
cana-534	12	31	𝜎	𝜎	X
cana-534	12	32	)	)	PUNCT
cana-534	12	33	=	=	SYM
cana-534	12	34	{	{	PUNCT
cana-534	12	35	𝑓	𝑓	PROPN
cana-534	12	36	∈	∈	PROPN
cana-534	12	37	𝐴	𝐴	PROPN
cana-534	12	38	:	:	PUNCT
cana-534	13	1	𝑅𝑒	𝑅𝑒	PROPN
cana-534	13	2	{	{	PUNCT
cana-534	13	3	𝑧𝑓′(𝑧	𝑧𝑓′(𝑧	PROPN
cana-534	13	4	)	)	PUNCT
cana-534	13	5	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	13	6	)	)	PUNCT
cana-534	13	7	−	−	PUNCT
cana-534	13	8	𝛼	𝛼	X
cana-534	13	9	}	}	PUNCT
cana-534	13	10	>	>	X
cana-534	13	11	𝜎	𝜎	PROPN
cana-534	13	12	{	{	PUNCT
cana-534	13	13	𝑧𝑓′(𝑧	𝑧𝑓′(𝑧	PROPN
cana-534	13	14	)	)	PUNCT
cana-534	13	15	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	13	16	)	)	PUNCT
cana-534	13	17	−	−	ADP
cana-534	14	1	1	1	NUM
cana-534	14	2	}	}	PUNCT
cana-534	14	3	,	,	PUNCT
cana-534	14	4	𝑧	𝑧	PROPN
cana-534	14	5	∈	∈	PROPN
cana-534	14	6	𝑈	𝑈	PROPN
cana-534	14	7	}	}	PUNCT
cana-534	14	8	.	.	PUNCT
cana-534	15	1	(	(	PUNCT
cana-534	15	2	1.4	1.4	NUM
cana-534	15	3	)	)	PUNCT
cana-534	15	4	it	it	PRON
cana-534	15	5	is	be	AUX
cana-534	15	6	obvious	obvious	ADJ
cana-534	15	7	from	from	ADP
cana-534	15	8	(	(	PUNCT
cana-534	15	9	1.3	1.3	NUM
cana-534	15	10	)	)	PUNCT
cana-534	15	11	and	and	CCONJ
cana-534	15	12	(	(	PUNCT
cana-534	15	13	1.4	1.4	NUM
cana-534	15	14	)	)	PUNCT
cana-534	16	1	that	that	PRON
cana-534	16	2	𝑓	𝑓	DET
cana-534	16	3	∈	∈	PROPN
cana-534	16	4	𝑈𝐶𝑉(𝛼	𝑈𝐶𝑉(𝛼	NUM
cana-534	16	5	,	,	PUNCT
cana-534	16	6	𝜎	𝜎	NOUN
cana-534	16	7	)	)	PUNCT
cana-534	16	8	if	if	SCONJ
cana-534	16	9	and	and	CCONJ
cana-534	16	10	only	only	ADV
cana-534	16	11	if	if	SCONJ
cana-534	16	12	𝑧𝑓′(𝑧	𝑧𝑓′(𝑧	PROPN
cana-534	16	13	)	)	PUNCT
cana-534	16	14	∈	∈	PROPN
cana-534	16	15	𝑆𝑃(𝛼	𝑆𝑃(𝛼	PROPN
cana-534	16	16	,	,	PUNCT
cana-534	16	17	𝜎	𝜎	NOUN
cana-534	16	18	)	)	PUNCT
cana-534	17	1	.	.	PUNCT
cana-534	18	1	some	some	DET
cana-534	18	2	interesting	interesting	ADJ
cana-534	18	3	situations	situation	NOUN
cana-534	18	4	of	of	ADP
cana-534	18	5	the	the	DET
cana-534	18	6	class	class	NOUN
cana-534	18	7	of	of	ADP
cana-534	18	8	starlike	starlike	NOUN
cana-534	18	9	and	and	CCONJ
cana-534	18	10	convex	convex	NOUN
cana-534	18	11	of	of	ADP
cana-534	18	12	order	order	NOUN
cana-534	18	13	𝛼	𝛼	PRON
cana-534	18	14	associated	associate	VERB
cana-534	18	15	with	with	ADP
cana-534	18	16	bessel	bessel	NOUN
cana-534	18	17	functions	function	NOUN
cana-534	18	18	(	(	PUNCT
cana-534	18	19	as	as	ADP
cana-534	18	20	hypergeometric	hypergeometric	ADJ
cana-534	18	21	function	function	NOUN
cana-534	18	22	)	)	PUNCT
cana-534	18	23	,	,	PUNCT
cana-534	18	24	finding	find	VERB
cana-534	18	25	condition	condition	NOUN
cana-534	18	26	on	on	ADP
cana-534	18	27	the	the	DET
cana-534	18	28	triple	triple	ADJ
cana-534	18	29	𝑝	𝑝	PROPN
cana-534	18	30	,	,	PUNCT
cana-534	18	31	𝑏	𝑏	PROPN
cana-534	18	32	and	and	CCONJ
cana-534	18	33	𝑐	𝑐	ADP
cana-534	18	34	such	such	ADJ
cana-534	18	35	that	that	SCONJ
cana-534	18	36	the	the	DET
cana-534	18	37	function	function	NOUN
cana-534	18	38	𝑢𝑝,𝑏,𝑐	𝑢𝑝,𝑏,𝑐	NOUN
cana-534	18	39	is	be	AUX
cana-534	18	40	starlike	starlike	NOUN
cana-534	18	41	and	and	CCONJ
cana-534	18	42	convex	convex	NOUN
cana-534	18	43	of	of	ADP
cana-534	18	44	order	order	NOUN
cana-534	18	45	𝛼	𝛼	NOUN
cana-534	18	46	and	and	CCONJ
cana-534	18	47	finding	find	VERB
cana-534	18	48	conditions	condition	NOUN
cana-534	18	49	on	on	ADP
cana-534	18	50	the	the	DET
cana-534	18	51	parameters	parameter	NOUN
cana-534	18	52	for	for	ADP
cana-534	18	53	which	which	PRON
cana-534	18	54	the	the	DET
cana-534	18	55	gaussian	gaussian	ADJ
cana-534	18	56	hypergeometric	hypergeometric	ADJ
cana-534	18	57	functions	function	NOUN
cana-534	18	58	belong	belong	VERB
cana-534	18	59	to	to	ADP
cana-534	18	60	the	the	DET
cana-534	18	61	various	various	ADJ
cana-534	18	62	classes	class	NOUN
cana-534	18	63	of	of	ADP
cana-534	18	64	functions	function	NOUN
cana-534	18	65	have	have	AUX
cana-534	18	66	discussed	discuss	VERB
cana-534	18	67	in	in	ADP
cana-534	18	68	the	the	DET
cana-534	18	69	references	reference	NOUN
cana-534	18	70	[	[	X
cana-534	18	71	1,2,4,9,10	1,2,4,9,10	NOUN
cana-534	18	72	]	]	PUNCT
cana-534	18	73	.	.	PUNCT
cana-534	19	1	let	let	VERB
cana-534	19	2	us	we	PRON
cana-534	19	3	take	take	VERB
cana-534	19	4	into	into	ADP
cana-534	19	5	consideration	consideration	NOUN
cana-534	19	6	second	second	ADJ
cana-534	19	7	order	order	NOUN
cana-534	19	8	linear	linear	VERB
cana-534	19	9	homogenous	homogenous	ADJ
cana-534	19	10	differential	differential	NOUN
cana-534	19	11	equation	equation	NOUN
cana-534	19	12	(	(	PUNCT
cana-534	19	13	see	see	VERB
cana-534	19	14	[	[	X
cana-534	19	15	3	3	NUM
cana-534	19	16	]	]	PUNCT
cana-534	19	17	)	)	PUNCT
cana-534	19	18	.	.	PUNCT
cana-534	20	1	communications	communication	NOUN
cana-534	20	2	on	on	ADP
cana-534	20	3	applied	apply	VERB
cana-534	20	4	nonlinear	nonlinear	ADJ
cana-534	20	5	analysis	analysis	NOUN
cana-534	20	6	issn	issn	NOUN
cana-534	20	7	:	:	PUNCT
cana-534	20	8	1074	1074	NUM
cana-534	20	9	-	-	PUNCT
cana-534	20	10	133x	133x	NUM
cana-534	20	11	vol	vol	NOUN
cana-534	20	12	31	31	NUM
cana-534	20	13	no	no	NOUN
cana-534	20	14	.	.	NOUN
cana-534	20	15	2	2	NUM
cana-534	20	16	(	(	PUNCT
cana-534	20	17	2024	2024	NUM
cana-534	20	18	)	)	PUNCT
cana-534	20	19	198	198	NUM
cana-534	20	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	20	21	𝑧2𝜔″(𝑧	𝑧2𝜔″(𝑧	ADV
cana-534	20	22	)	)	PUNCT
cana-534	21	1	+	+	CCONJ
cana-534	21	2	𝑏𝑧𝜔′(𝑧	𝑏𝑧𝜔′(𝑧	NOUN
cana-534	21	3	)	)	PUNCT
cana-534	22	1	+	+	CCONJ
cana-534	23	1	[	[	X
cana-534	23	2	𝑐𝑧2	𝑐𝑧2	INTJ
cana-534	23	3	−	−	PROPN
cana-534	23	4	𝑝2	𝑝2	NOUN
cana-534	23	5	+	+	CCONJ
cana-534	23	6	(	(	PUNCT
cana-534	23	7	1	1	NUM
cana-534	23	8	−	−	PROPN
cana-534	23	9	𝑏)𝑝]𝜔(𝑧	𝑏)𝑝]𝜔(𝑧	PROPN
cana-534	23	10	)	)	PUNCT
cana-534	23	11	=	=	SYM
cana-534	23	12	0	0	NUM
cana-534	23	13	,	,	PUNCT
cana-534	23	14	(	(	PUNCT
cana-534	23	15	𝑝	𝑝	NOUN
cana-534	23	16	,	,	PUNCT
cana-534	23	17	𝑏	𝑏	PROPN
cana-534	23	18	,	,	PUNCT
cana-534	23	19	𝑐	𝑐	PROPN
cana-534	23	20	∈	∈	PROPN
cana-534	23	21	𝒞	𝒞	PROPN
cana-534	23	22	)	)	PUNCT
cana-534	23	23	.	.	PUNCT
cana-534	24	1	(	(	PUNCT
cana-534	24	2	1.5	1.5	NUM
cana-534	24	3	)	)	PUNCT
cana-534	24	4	as	as	ADP
cana-534	24	5	a	a	DET
cana-534	24	6	particular	particular	ADJ
cana-534	24	7	solition	solition	NOUN
cana-534	24	8	of	of	ADP
cana-534	24	9	(	(	PUNCT
cana-534	24	10	1.5	1.5	NUM
cana-534	24	11	)	)	PUNCT
cana-534	24	12	generalized	generalized	ADJ
cana-534	24	13	bessel	bessel	ADJ
cana-534	24	14	function	function	NOUN
cana-534	24	15	of	of	ADP
cana-534	24	16	the	the	DET
cana-534	24	17	first	first	ADJ
cana-534	24	18	kind	kind	NOUN
cana-534	24	19	of	of	ADP
cana-534	24	20	order	order	NOUN
cana-534	24	21	𝑝	𝑝	NOUN
cana-534	24	22	,	,	PUNCT
cana-534	24	23	is	be	AUX
cana-534	24	24	defined	define	VERB
cana-534	24	25	in	in	ADP
cana-534	24	26	as	as	ADP
cana-534	24	27	following	follow	VERB
cana-534	24	28	:	:	PUNCT
cana-534	24	29	𝜔(𝑧	𝜔(𝑧	ADJ
cana-534	24	30	)	)	PUNCT
cana-534	24	31	=	=	SYM
cana-534	25	1	𝜔𝑝,𝑏,𝑐(𝑧	𝜔𝑝,𝑏,𝑐(𝑧	X
cana-534	25	2	)	)	PUNCT
cana-534	26	1	=	=	SYM
cana-534	26	2	∑	∑	PUNCT
cana-534	26	3	(	(	PUNCT
cana-534	26	4	−1)𝑛𝑐𝑛	−1)𝑛𝑐𝑛	PROPN
cana-534	26	5	𝑛	𝑛	PROPN
cana-534	26	6	!	!	PUNCT
cana-534	26	7	𝛤	𝛤	PROPN
cana-534	26	8	(	(	PUNCT
cana-534	26	9	𝑝	𝑝	PROPN
cana-534	26	10	+	+	NOUN
cana-534	26	11	𝑛	𝑛	VERB
cana-534	26	12	+	+	CCONJ
cana-534	26	13	𝑏	𝑏	PROPN
cana-534	26	14	+	+	CCONJ
cana-534	26	15	1	1	NUM
cana-534	26	16	2	2	NUM
cana-534	26	17	)	)	PUNCT
cana-534	26	18	∞	∞	NUM
cana-534	26	19	𝑛=0	𝑛=0	NOUN
cana-534	26	20	(	(	PUNCT
cana-534	26	21	𝑧	𝑧	PROPN
cana-534	26	22	2	2	NUM
cana-534	26	23	)	)	PUNCT
cana-534	26	24	2𝑛+𝑝	2𝑛+𝑝	NUM
cana-534	26	25	,	,	PUNCT
cana-534	26	26	𝑧	𝑧	PROPN
cana-534	26	27	∈	∈	PROPN
cana-534	26	28	𝐶	𝐶	PROPN
cana-534	26	29	,	,	PUNCT
cana-534	26	30	(	(	PUNCT
cana-534	26	31	1.6	1.6	NUM
cana-534	26	32	)	)	PUNCT
cana-534	26	33	where	where	SCONJ
cana-534	26	34	𝛤	𝛤	PRON
cana-534	26	35	stands	stand	VERB
cana-534	26	36	for	for	ADP
cana-534	26	37	the	the	DET
cana-534	26	38	euler	euler	PROPN
cana-534	26	39	gamma	gamma	PROPN
cana-534	26	40	function	function	NOUN
cana-534	26	41	and	and	CCONJ
cana-534	26	42	𝜏	𝜏	NOUN
cana-534	26	43	=	=	SYM
cana-534	26	44	𝑝	𝑝	PROPN
cana-534	26	45	+	+	CCONJ
cana-534	26	46	𝑏+1	𝑏+1	PROPN
cana-534	26	47	2	2	NUM
cana-534	26	48	∉	∉	X
cana-534	26	49	𝑍0	𝑍0	PROPN
cana-534	26	50	=	=	PUNCT
cana-534	26	51	{	{	PUNCT
cana-534	26	52	0	0	NUM
cana-534	26	53	,	,	PUNCT
cana-534	26	54	−1	−1	NOUN
cana-534	26	55	,	,	PUNCT
cana-534	26	56	−2	−2	NOUN
cana-534	26	57	,	,	PUNCT
cana-534	26	58	⋯	⋯	PROPN
cana-534	26	59	}	}	PUNCT
cana-534	26	60	.	.	PUNCT
cana-534	27	1	though	though	SCONJ
cana-534	27	2	the	the	DET
cana-534	27	3	series	series	NOUN
cana-534	27	4	given	give	VERB
cana-534	27	5	in	in	ADP
cana-534	27	6	(	(	PUNCT
cana-534	27	7	1.6	1.6	NUM
cana-534	27	8	)	)	PUNCT
cana-534	27	9	is	be	AUX
cana-534	27	10	convergent	convergent	NOUN
cana-534	27	11	everywhere	everywhere	ADV
cana-534	27	12	,	,	PUNCT
cana-534	27	13	the	the	DET
cana-534	27	14	function	function	NOUN
cana-534	27	15	𝜔𝑝,𝑏,𝑐	𝜔𝑝,𝑏,𝑐	ADV
cana-534	27	16	is	be	AUX
cana-534	27	17	not	not	PART
cana-534	27	18	univalent	univalent	ADJ
cana-534	27	19	in	in	ADP
cana-534	27	20	𝑈.	𝑈.	PROPN
cana-534	27	21	specially	specially	ADV
cana-534	27	22	,	,	PUNCT
cana-534	27	23	choosing	choose	VERB
cana-534	27	24	𝑏	𝑏	NOUN
cana-534	27	25	=	=	SYM
cana-534	27	26	𝑐	𝑐	NOUN
cana-534	27	27	=	=	SYM
cana-534	27	28	1	1	NUM
cana-534	27	29	in	in	ADP
cana-534	27	30	(	(	PUNCT
cana-534	27	31	1.6	1.6	NUM
cana-534	27	32	)	)	PUNCT
cana-534	27	33	,	,	PUNCT
cana-534	27	34	we	we	PRON
cana-534	27	35	get	get	VERB
cana-534	27	36	bessel	bessel	ADJ
cana-534	27	37	function	function	NOUN
cana-534	27	38	of	of	ADP
cana-534	27	39	the	the	DET
cana-534	27	40	first	first	ADJ
cana-534	27	41	kind	kind	NOUN
cana-534	27	42	of	of	ADP
cana-534	27	43	order	order	NOUN
cana-534	27	44	𝑝	𝑝	NOUN
cana-534	27	45	given	give	VERB
cana-534	27	46	in	in	ADP
cana-534	27	47	as	as	ADP
cana-534	27	48	𝐽𝑝(𝑧	𝐽𝑝(𝑧	NOUN
cana-534	27	49	)	)	PUNCT
cana-534	27	50	=	=	PUNCT
cana-534	28	1	∑	∑	PUNCT
cana-534	28	2	(	(	PUNCT
cana-534	28	3	−1)𝑛	−1)𝑛	X
cana-534	28	4	𝑛	𝑛	PROPN
cana-534	28	5	!	!	PUNCT
cana-534	29	1	𝛤(𝑝	𝛤(𝑝	ADP
cana-534	29	2	+	+	NUM
cana-534	29	3	𝑛	𝑛	VERB
cana-534	29	4	+	+	NOUN
cana-534	29	5	1	1	NUM
cana-534	29	6	)	)	PUNCT
cana-534	29	7	∞	∞	NUM
cana-534	30	1	𝑛=0	𝑛=0	NOUN
cana-534	30	2	(	(	PUNCT
cana-534	30	3	𝑧	𝑧	PROPN
cana-534	30	4	2	2	NUM
cana-534	30	5	)	)	PUNCT
cana-534	30	6	2𝑛+𝑝	2𝑛+𝑝	NUM
cana-534	30	7	,	,	PUNCT
cana-534	30	8	𝑧	𝑧	PROPN
cana-534	30	9	∈	∈	PROPN
cana-534	30	10	𝐶.	𝐶.	PROPN
cana-534	30	11	(	(	PUNCT
cana-534	30	12	1.7	1.7	NUM
cana-534	30	13	)	)	PUNCT
cana-534	30	14	choosing	choose	VERB
cana-534	30	15	𝑏	𝑏	NOUN
cana-534	30	16	=	=	SYM
cana-534	30	17	1	1	NUM
cana-534	30	18	and	and	CCONJ
cana-534	30	19	𝑐	𝑐	NOUN
cana-534	30	20	=	=	SYM
cana-534	30	21	−1	−1	NOUN
cana-534	30	22	in	in	ADV
cana-534	30	23	(	(	PUNCT
cana-534	30	24	1.6	1.6	NUM
cana-534	30	25	)	)	PUNCT
cana-534	30	26	,	,	PUNCT
cana-534	30	27	we	we	PRON
cana-534	30	28	get	get	VERB
cana-534	30	29	the	the	DET
cana-534	30	30	modified	modify	VERB
cana-534	30	31	bessel	bessel	NOUN
cana-534	30	32	function	function	NOUN
cana-534	30	33	of	of	ADP
cana-534	30	34	the	the	DET
cana-534	30	35	first	first	ADJ
cana-534	30	36	kind	kind	ADJ
cana-534	30	37	order	order	NOUN
cana-534	30	38	of	of	ADP
cana-534	30	39	𝑝	𝑝	NOUN
cana-534	30	40	given	give	VERB
cana-534	30	41	in	in	ADP
cana-534	30	42	as	as	ADP
cana-534	30	43	𝐼𝑝(𝑧	𝐼𝑝(𝑧	NOUN
cana-534	30	44	)	)	PUNCT
cana-534	30	45	=	=	PUNCT
cana-534	31	1	∑	∑	PROPN
cana-534	31	2	1	1	NUM
cana-534	31	3	𝑛!𝛤(𝑝+𝑛+1	𝑛!𝛤(𝑝+𝑛+1	PROPN
cana-534	31	4	)	)	PUNCT
cana-534	31	5	∞	∞	NUM
cana-534	32	1	𝑛=0	𝑛=0	NOUN
cana-534	32	2	(	(	PUNCT
cana-534	32	3	𝑧	𝑧	PROPN
cana-534	32	4	2	2	NUM
cana-534	32	5	)	)	PUNCT
cana-534	32	6	2𝑛+𝑝	2𝑛+𝑝	NUM
cana-534	32	7	,	,	PUNCT
cana-534	32	8	𝑧	𝑧	PROPN
cana-534	32	9	∈	∈	PROPN
cana-534	32	10	𝐶.	𝐶.	PROPN
cana-534	32	11	(	(	PUNCT
cana-534	32	12	1.8	1.8	NUM
cana-534	32	13	)	)	PUNCT
cana-534	32	14	further	far	ADV
cana-534	32	15	choosing	choose	VERB
cana-534	32	16	𝑏	𝑏	NOUN
cana-534	32	17	=	=	SYM
cana-534	32	18	2	2	NUM
cana-534	32	19	and	and	CCONJ
cana-534	32	20	𝑐	𝑐	NOUN
cana-534	32	21	=	=	SYM
cana-534	32	22	1	1	NUM
cana-534	32	23	in	in	ADP
cana-534	32	24	(	(	PUNCT
cana-534	32	25	1.6	1.6	NUM
cana-534	32	26	)	)	PUNCT
cana-534	32	27	,	,	PUNCT
cana-534	32	28	the	the	DET
cana-534	32	29	functions	function	NOUN
cana-534	32	30	𝜔𝑝,𝑏,𝑐	𝜔𝑝,𝑏,𝑐	ADV
cana-534	32	31	reduces	reduce	VERB
cana-534	32	32	to	to	ADP
cana-534	32	33	√2	√2	PROPN
cana-534	32	34	𝑗𝑝	𝑗𝑝	PROPN
cana-534	32	35	√𝜋	√𝜋	NOUN
cana-534	32	36	,	,	PUNCT
cana-534	32	37	where	where	SCONJ
cana-534	32	38	𝑗𝑝	𝑗𝑝	PROPN
cana-534	32	39	is	be	AUX
cana-534	32	40	the	the	DET
cana-534	32	41	spherical	spherical	ADJ
cana-534	32	42	bessel	bessel	NOUN
cana-534	32	43	function	function	NOUN
cana-534	32	44	of	of	ADP
cana-534	32	45	the	the	DET
cana-534	32	46	first	first	ADJ
cana-534	32	47	kind	kind	NOUN
cana-534	32	48	of	of	ADP
cana-534	32	49	order	order	NOUN
cana-534	32	50	𝑝	𝑝	ADP
cana-534	32	51	,	,	PUNCT
cana-534	32	52	given	give	VERB
cana-534	32	53	in	in	ADP
cana-534	32	54	as	as	ADP
cana-534	32	55	𝑗𝑝(𝑧	𝑗𝑝(𝑧	NUM
cana-534	32	56	)	)	PUNCT
cana-534	33	1	=	=	SYM
cana-534	34	1	√	√	NUM
cana-534	34	2	𝜋	𝜋	NOUN
cana-534	34	3	2	2	NUM
cana-534	34	4	∑	∑	PUNCT
cana-534	34	5	(	(	PUNCT
cana-534	34	6	−1)𝑛	−1)𝑛	X
cana-534	34	7	𝑛	𝑛	PROPN
cana-534	35	1	!	!	PUNCT
cana-534	35	2	𝛤	𝛤	PROPN
cana-534	35	3	(	(	PUNCT
cana-534	35	4	𝑝	𝑝	PROPN
cana-534	35	5	+	+	NOUN
cana-534	35	6	𝑛	𝑛	PROPN
cana-534	35	7	+	+	CCONJ
cana-534	35	8	3	3	NUM
cana-534	35	9	2	2	NUM
cana-534	35	10	)	)	PUNCT
cana-534	35	11	∞	∞	NUM
cana-534	35	12	𝑛=0	𝑛=0	NOUN
cana-534	35	13	(	(	PUNCT
cana-534	35	14	𝑧	𝑧	PROPN
cana-534	35	15	2	2	NUM
cana-534	35	16	)	)	PUNCT
cana-534	35	17	2𝑛+𝑝	2𝑛+𝑝	NUM
cana-534	35	18	,	,	PUNCT
cana-534	35	19	𝑧	𝑧	PROPN
cana-534	35	20	∈	∈	PROPN
cana-534	35	21	𝐶.	𝐶.	PROPN
cana-534	35	22	(	(	PUNCT
cana-534	35	23	1.9	1.9	NUM
cana-534	35	24	)	)	PUNCT
cana-534	35	25	the	the	DET
cana-534	35	26	function	function	NOUN
cana-534	35	27	𝜗𝑝,𝑏,𝑐	𝜗𝑝,𝑏,𝑐	NOUN
cana-534	35	28	is	be	AUX
cana-534	35	29	defined	define	VERB
cana-534	35	30	in	in	ADP
cana-534	35	31	as	as	ADP
cana-534	35	32	𝜗𝑝,𝑏,𝑐(𝑧	𝜗𝑝,𝑏,𝑐(𝑧	NOUN
cana-534	35	33	)	)	PUNCT
cana-534	36	1	=	=	SYM
cana-534	36	2	2𝑝𝛤	2𝑝𝛤	NUM
cana-534	36	3	(	(	PUNCT
cana-534	36	4	𝑝	𝑝	NOUN
cana-534	36	5	+	+	NUM
cana-534	36	6	𝑏	𝑏	PROPN
cana-534	36	7	+	+	CCONJ
cana-534	36	8	1	1	NUM
cana-534	36	9	2	2	NUM
cana-534	36	10	)	)	PUNCT
cana-534	36	11	𝑧1−	𝑧1−	PROPN
cana-534	36	12	𝑝	𝑝	PROPN
cana-534	36	13	2𝜔𝑝,𝑏,𝑐(√𝑧	2𝜔𝑝,𝑏,𝑐(√𝑧	NUM
cana-534	36	14	)	)	PUNCT
cana-534	36	15	(	(	PUNCT
cana-534	36	16	1.10	1.10	NUM
cana-534	36	17	)	)	PUNCT
cana-534	36	18	in	in	ADP
cana-534	36	19	terms	term	NOUN
cana-534	36	20	of	of	ADP
cana-534	36	21	generalized	generalized	ADJ
cana-534	36	22	bessel	bessel	NOUN
cana-534	36	23	function	function	NOUN
cana-534	36	24	𝜔𝑝,𝑏,𝑐.	𝜔𝑝,𝑏,𝑐.	NOUN
cana-534	36	25	by	by	ADP
cana-534	36	26	the	the	DET
cana-534	36	27	help	help	NOUN
cana-534	36	28	of	of	ADP
cana-534	36	29	pochhammer	pochhammer	NOUN
cana-534	36	30	symbol	symbol	NOUN
cana-534	36	31	,	,	PUNCT
cana-534	36	32	gamma	gamma	NOUN
cana-534	36	33	function	function	NOUN
cana-534	36	34	is	be	AUX
cana-534	36	35	defined	define	VERB
cana-534	36	36	as	as	ADP
cana-534	36	37	and	and	CCONJ
cana-534	36	38	we	we	PRON
cana-534	36	39	get	get	VERB
cana-534	36	40	𝜗𝑝,𝑏,𝑐	𝜗𝑝,𝑏,𝑐	NOUN
cana-534	36	41	given	give	VERB
cana-534	36	42	in	in	ADP
cana-534	36	43	(	(	PUNCT
cana-534	36	44	1.10	1.10	NUM
cana-534	36	45	)	)	PUNCT
cana-534	36	46	as	as	ADP
cana-534	36	47	𝜗𝑝,𝑏,𝑐(𝑧	𝜗𝑝,𝑏,𝑐(𝑧	VERB
cana-534	36	48	)	)	PUNCT
cana-534	37	1	=	=	PUNCT
cana-534	37	2	𝑧	𝑧	PROPN
cana-534	37	3	+	+	NOUN
cana-534	37	4	∑	∑	PROPN
cana-534	37	5	(	(	PUNCT
cana-534	37	6	−𝑐)𝑛	−𝑐)𝑛	NOUN
cana-534	37	7	4𝑛(𝜏)𝑛𝑛	4𝑛(𝜏)𝑛𝑛	NOUN
cana-534	37	8	!	!	PUNCT
cana-534	38	1	∞	∞	NUM
cana-534	38	2	𝑛=1	𝑛=1	NOUN
cana-534	38	3	,	,	PUNCT
cana-534	38	4	(	(	PUNCT
cana-534	38	5	1.11	1.11	NUM
cana-534	38	6	)	)	PUNCT
cana-534	39	1	where	where	SCONJ
cana-534	39	2	𝜏	𝜏	NOUN
cana-534	39	3	=	=	SYM
cana-534	39	4	𝑝	𝑝	PROPN
cana-534	39	5	+	+	CCONJ
cana-534	39	6	𝑏+1	𝑏+1	PROPN
cana-534	39	7	2	2	NUM
cana-534	39	8	∉	∉	X
cana-534	39	9	𝑍0	𝑍0	PROPN
cana-534	39	10	and	and	CCONJ
cana-534	39	11	𝑁	𝑁	PROPN
cana-534	39	12	=	=	SYM
cana-534	39	13	{	{	PUNCT
cana-534	39	14	1,2,3	1,2,3	NUM
cana-534	39	15	,	,	PUNCT
cana-534	39	16	⋯	⋯	VERB
cana-534	39	17	}	}	PUNCT
cana-534	39	18	.	.	PUNCT
cana-534	40	1	we	we	PRON
cana-534	40	2	will	will	AUX
cana-534	40	3	write	write	VERB
cana-534	40	4	𝜗𝜏,𝑐(𝑧	𝜗𝜏,𝑐(𝑧	NOUN
cana-534	40	5	)	)	PUNCT
cana-534	40	6	=	=	PUNCT
cana-534	40	7	𝜗𝑝,𝑏,𝑐(𝑧	𝜗𝑝,𝑏,𝑐(𝑧	NOUN
cana-534	40	8	)	)	PUNCT
cana-534	40	9	for	for	ADP
cana-534	40	10	convenience	convenience	NOUN
cana-534	40	11	.	.	PUNCT
cana-534	41	1	now	now	ADV
cana-534	41	2	,	,	PUNCT
cana-534	41	3	we	we	PRON
cana-534	41	4	consider	consider	VERB
cana-534	41	5	𝑆𝜏	𝑆𝜏	PROPN
cana-534	41	6	𝑐	𝑐	PROPN
cana-534	41	7	operator	operator	NOUN
cana-534	41	8	given	give	VERB
cana-534	41	9	as	as	ADP
cana-534	41	10	communications	communication	NOUN
cana-534	41	11	on	on	ADP
cana-534	41	12	applied	apply	VERB
cana-534	41	13	nonlinear	nonlinear	ADJ
cana-534	41	14	analysis	analysis	NOUN
cana-534	41	15	issn	issn	NOUN
cana-534	41	16	:	:	PUNCT
cana-534	41	17	1074	1074	NUM
cana-534	41	18	-	-	PUNCT
cana-534	41	19	133x	133x	NUM
cana-534	41	20	vol	vol	NOUN
cana-534	41	21	31	31	NUM
cana-534	41	22	no	no	NOUN
cana-534	41	23	.	.	NOUN
cana-534	41	24	2	2	NUM
cana-534	41	25	(	(	PUNCT
cana-534	41	26	2024	2024	NUM
cana-534	41	27	)	)	PUNCT
cana-534	42	1	199	199	NUM
cana-534	42	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	42	3	𝑆𝜏	𝑆𝜏	PROPN
cana-534	42	4	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	42	5	)	)	PUNCT
cana-534	42	6	=	=	SYM
cana-534	42	7	𝜗𝜏,𝑐(𝑧	𝜗𝜏,𝑐(𝑧	NUM
cana-534	42	8	)	)	PUNCT
cana-534	42	9	∗	∗	NOUN
cana-534	42	10	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	42	11	)	)	PUNCT
cana-534	42	12	=	=	SYM
cana-534	42	13	𝑧	𝑧	PROPN
cana-534	42	14	+	+	NOUN
cana-534	42	15	∑	∑	PROPN
cana-534	42	16	(	(	PUNCT
cana-534	42	17	−𝑐)𝑛𝑎𝑛+1	−𝑐)𝑛𝑎𝑛+1	PROPN
cana-534	42	18	4𝑛(𝜏)𝑛𝑛	4𝑛(𝜏)𝑛𝑛	NOUN
cana-534	42	19	!	!	PUNCT
cana-534	43	1	∞	∞	NUM
cana-534	43	2	𝑛=1	𝑛=1	NOUN
cana-534	43	3	𝑧𝑛+1	𝑧𝑛+1	NOUN
cana-534	43	4	=	=	SYM
cana-534	43	5	𝑧	𝑧	PROPN
cana-534	44	1	+	+	NOUN
cana-534	44	2	∑	∑	PROPN
cana-534	44	3	(	(	PUNCT
cana-534	44	4	−𝑐)𝑛−1𝑎𝑛	−𝑐)𝑛−1𝑎𝑛	NOUN
cana-534	44	5	4𝑛−1(𝜏)𝑛−1(𝑛	4𝑛−1(𝜏)𝑛−1(𝑛	ADV
cana-534	44	6	−	−	NOUN
cana-534	44	7	1	1	NUM
cana-534	44	8	)	)	PUNCT
cana-534	44	9	!	!	PUNCT
cana-534	45	1	∞	∞	NUM
cana-534	46	1	𝑛=2	𝑛=2	AUX
cana-534	46	2	𝑧𝑛	𝑧𝑛	ADP
cana-534	46	3	=	=	PUNCT
cana-534	46	4	𝑧	𝑧	PROPN
cana-534	47	1	+	+	PUNCT
cana-534	47	2	∑	∑	PROPN
cana-534	47	3	𝐸	𝐸	PROPN
cana-534	47	4	∞	∞	ADJ
cana-534	47	5	𝑛=2	𝑛=2	PROPN
cana-534	47	6	(	(	PUNCT
cana-534	47	7	𝑐	𝑐	NOUN
cana-534	47	8	,	,	PUNCT
cana-534	47	9	𝜏	𝜏	NOUN
cana-534	47	10	,	,	PUNCT
cana-534	47	11	𝑛)𝑎𝑛𝑧𝑛	𝑛)𝑎𝑛𝑧𝑛	ADV
cana-534	47	12	where	where	SCONJ
cana-534	47	13	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	47	14	,	,	PUNCT
cana-534	47	15	𝜏	𝜏	NOUN
cana-534	47	16	,	,	PUNCT
cana-534	47	17	𝑛	𝑛	NOUN
cana-534	47	18	)	)	PUNCT
cana-534	47	19	=	=	PUNCT
cana-534	47	20	(	(	PUNCT
cana-534	47	21	−𝑐)𝑛−1	−𝑐)𝑛−1	NOUN
cana-534	48	1	4𝑛−1(𝜏)𝑛−1(𝑛	4𝑛−1(𝜏)𝑛−1(𝑛	ADV
cana-534	48	2	−	−	NOUN
cana-534	48	3	1	1	NUM
cana-534	48	4	)	)	PUNCT
cana-534	48	5	!	!	PUNCT
cana-534	48	6	,	,	PUNCT
cana-534	49	1	𝜏	𝜏	X
cana-534	49	2	=	=	SYM
cana-534	49	3	(	(	PUNCT
cana-534	49	4	𝑝	𝑝	PROPN
cana-534	49	5	+	+	NUM
cana-534	49	6	𝑏	𝑏	PROPN
cana-534	49	7	+	+	CCONJ
cana-534	49	8	1	1	NUM
cana-534	49	9	2	2	NUM
cana-534	49	10	)	)	PUNCT
cana-534	49	11	≠	≠	PROPN
cana-534	49	12	0	0	NUM
cana-534	49	13	,	,	PUNCT
cana-534	49	14	−1	−1	NOUN
cana-534	49	15	,	,	PUNCT
cana-534	49	16	−2	−2	PROPN
cana-534	49	17	,	,	PUNCT
cana-534	49	18	⋯.	⋯.	PROPN
cana-534	49	19	(	(	PUNCT
cana-534	49	20	1.12	1.12	NUM
cana-534	49	21	)	)	PUNCT
cana-534	49	22	for	for	ADP
cana-534	49	23	𝛼	𝛼	PRON
cana-534	49	24	≥	≥	NOUN
cana-534	49	25	0	0	NUM
cana-534	49	26	,	,	PUNCT
cana-534	49	27	0	0	NUM
cana-534	49	28	≤	≤	NOUN
cana-534	49	29	𝛽	𝛽	NOUN
cana-534	49	30	<	<	X
cana-534	49	31	1	1	NUM
cana-534	49	32	,	,	PUNCT
cana-534	49	33	we	we	PRON
cana-534	49	34	set	set	VERB
cana-534	49	35	𝑆𝜏	𝑆𝜏	PROPN
cana-534	49	36	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	49	37	,	,	PUNCT
cana-534	49	38	𝛽	𝛽	NOUN
cana-534	49	39	)	)	PUNCT
cana-534	49	40	be	be	VERB
cana-534	49	41	the	the	DET
cana-534	49	42	subclass	subclass	NOUN
cana-534	49	43	of	of	ADP
cana-534	49	44	𝐴	𝐴	PROPN
cana-534	49	45	consisting	consist	VERB
cana-534	49	46	of	of	ADP
cana-534	49	47	functions	function	NOUN
cana-534	49	48	of	of	ADP
cana-534	49	49	the	the	DET
cana-534	49	50	form	form	NOUN
cana-534	49	51	(	(	PUNCT
cana-534	49	52	1.1	1.1	NUM
cana-534	49	53	)	)	PUNCT
cana-534	49	54	and	and	CCONJ
cana-534	49	55	satisfy	satisfy	VERB
cana-534	50	1	𝑅𝑒	𝑅𝑒	PROPN
cana-534	50	2	(	(	PUNCT
cana-534	50	3	𝑆𝜏	𝑆𝜏	PROPN
cana-534	50	4	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	50	5	)	)	PUNCT
cana-534	50	6	𝑧	𝑧	PRON
cana-534	50	7	)	)	PUNCT
cana-534	50	8	≥	≥	NOUN
cana-534	50	9	𝛼	𝛼	PROPN
cana-534	50	10	|(𝑆𝜏	|(𝑆𝜏	PROPN
cana-534	50	11	𝑐𝑓(𝑧))′	𝑐𝑓(𝑧))′	NUM
cana-534	50	12	−	−	PROPN
cana-534	51	1	𝑆𝜏	𝑆𝜏	PROPN
cana-534	51	2	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	51	3	)	)	PUNCT
cana-534	51	4	𝑧	𝑧	PROPN
cana-534	52	1	|	|	NOUN
cana-534	52	2	+	+	CCONJ
cana-534	52	3	𝛽	𝛽	PROPN
cana-534	52	4	(	(	PUNCT
cana-534	52	5	1.13	1.13	NUM
cana-534	52	6	)	)	PUNCT
cana-534	52	7	where	where	SCONJ
cana-534	52	8	𝑆𝜏	𝑆𝜏	PROPN
cana-534	52	9	𝑐𝑓(𝑧	𝑐𝑓(𝑧	NOUN
cana-534	52	10	)	)	PUNCT
cana-534	52	11	is	be	AUX
cana-534	52	12	given	give	VERB
cana-534	52	13	by	by	ADP
cana-534	52	14	(	(	PUNCT
cana-534	52	15	1.12	1.12	NUM
cana-534	52	16	)	)	PUNCT
cana-534	52	17	.	.	PUNCT
cana-534	53	1	we	we	PRON
cana-534	53	2	further	far	ADV
cana-534	53	3	let	let	VERB
cana-534	53	4	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	53	5	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	53	6	,	,	PUNCT
cana-534	53	7	𝛽	𝛽	NOUN
cana-534	53	8	)	)	PUNCT
cana-534	53	9	=	=	PUNCT
cana-534	54	1	𝑆𝜏	𝑆𝜏	PROPN
cana-534	54	2	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	54	3	,	,	PUNCT
cana-534	54	4	𝛽	𝛽	NOUN
cana-534	54	5	)	)	PUNCT
cana-534	54	6	∩	∩	X
cana-534	54	7	𝑇.	𝑇.	PROPN
cana-534	54	8	in	in	ADP
cana-534	54	9	this	this	DET
cana-534	54	10	paper	paper	NOUN
cana-534	54	11	,	,	PUNCT
cana-534	54	12	we	we	PRON
cana-534	54	13	obtain	obtain	VERB
cana-534	54	14	coefficient	coefficient	NOUN
cana-534	54	15	inequalities	inequality	NOUN
cana-534	54	16	,	,	PUNCT
cana-534	54	17	extreme	extreme	ADJ
cana-534	54	18	points	point	NOUN
cana-534	54	19	,	,	PUNCT
cana-534	54	20	integral	integral	ADJ
cana-534	54	21	means	mean	NOUN
cana-534	54	22	inequalities	inequality	NOUN
cana-534	54	23	for	for	ADP
cana-534	54	24	the	the	DET
cana-534	54	25	functions	function	NOUN
cana-534	54	26	in	in	ADP
cana-534	54	27	the	the	DET
cana-534	54	28	class	class	NOUN
cana-534	54	29	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	54	30	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	54	31	,	,	PUNCT
cana-534	54	32	𝛽	𝛽	NOUN
cana-534	54	33	)	)	PUNCT
cana-534	54	34	and	and	CCONJ
cana-534	54	35	also	also	ADV
cana-534	54	36	subordination	subordination	NOUN
cana-534	54	37	results	result	NOUN
cana-534	54	38	for	for	ADP
cana-534	54	39	the	the	DET
cana-534	54	40	class	class	NOUN
cana-534	54	41	of	of	ADP
cana-534	54	42	function	function	NOUN
cana-534	54	43	𝑓	𝑓	DET
cana-534	54	44	∈	∈	NOUN
cana-534	54	45	𝑆𝜏	𝑆𝜏	PROPN
cana-534	54	46	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	54	47	,	,	PUNCT
cana-534	54	48	𝛽	𝛽	NOUN
cana-534	54	49	)	)	PUNCT
cana-534	54	50	.	.	PUNCT
cana-534	55	1	2	2	X
cana-534	55	2	.	.	X
cana-534	55	3	coefficient	coefficient	NOUN
cana-534	55	4	estimates	estimate	NOUN
cana-534	55	5	theorem	theorem	VERB
cana-534	55	6	2.1	2.1	NUM
cana-534	55	7	.	.	PUNCT
cana-534	56	1	the	the	DET
cana-534	56	2	function	function	NOUN
cana-534	56	3	𝑓	𝑓	PRON
cana-534	56	4	defined	define	VERB
cana-534	56	5	by	by	ADP
cana-534	56	6	(	(	PUNCT
cana-534	56	7	1.1	1.1	NUM
cana-534	56	8	)	)	PUNCT
cana-534	56	9	is	be	AUX
cana-534	56	10	in	in	ADP
cana-534	56	11	the	the	DET
cana-534	56	12	class	class	NOUN
cana-534	56	13	𝑆𝜏	𝑆𝜏	PROPN
cana-534	56	14	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	56	15	,	,	PUNCT
cana-534	56	16	𝛽	𝛽	NOUN
cana-534	56	17	)	)	PUNCT
cana-534	56	18	if	if	SCONJ
cana-534	56	19	∑	∑	PROPN
cana-534	56	20	[	[	PUNCT
cana-534	56	21	1	1	NUM
cana-534	56	22	+	+	NUM
cana-534	56	23	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	56	24	−	−	PROPN
cana-534	56	25	1)]∞	1)]∞	NUM
cana-534	56	26	𝑛=2	𝑛=2	PROPN
cana-534	56	27	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	56	28	,	,	PUNCT
cana-534	56	29	𝜏	𝜏	NOUN
cana-534	56	30	,	,	PUNCT
cana-534	56	31	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	ADV
cana-534	56	32	≤	≤	NUM
cana-534	56	33	1	1	NUM
cana-534	56	34	−	−	PROPN
cana-534	56	35	𝛽	𝛽	NOUN
cana-534	56	36	,	,	PUNCT
cana-534	56	37	(	(	PUNCT
cana-534	56	38	2.1	2.1	NUM
cana-534	56	39	)	)	PUNCT
cana-534	56	40	where	where	SCONJ
cana-534	56	41	𝛼	𝛼	X
cana-534	56	42	≥	≥	NOUN
cana-534	56	43	0,0	0,0	NUM
cana-534	56	44	≤	≤	NUM
cana-534	56	45	𝛽	𝛽	NOUN
cana-534	56	46	<	<	X
cana-534	56	47	1	1	NUM
cana-534	56	48	and	and	CCONJ
cana-534	56	49	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	56	50	,	,	PUNCT
cana-534	56	51	𝜏	𝜏	NOUN
cana-534	56	52	,	,	PUNCT
cana-534	56	53	𝑛	𝑛	NOUN
cana-534	56	54	)	)	PUNCT
cana-534	56	55	is	be	AUX
cana-534	56	56	given	give	VERB
cana-534	56	57	by	by	ADP
cana-534	56	58	(	(	PUNCT
cana-534	56	59	1.12	1.12	NUM
cana-534	56	60	)	)	PUNCT
cana-534	56	61	.	.	PUNCT
cana-534	57	1	proof	proof	NOUN
cana-534	57	2	.	.	PUNCT
cana-534	58	1	it	it	PRON
cana-534	58	2	suffices	suffice	VERB
cana-534	58	3	to	to	PART
cana-534	58	4	show	show	VERB
cana-534	58	5	that	that	SCONJ
cana-534	58	6	𝛼	𝛼	ADP
cana-534	58	7	|(𝑆𝜏	|(𝑆𝜏	PROPN
cana-534	58	8	𝑐𝑓(𝑧))′	𝑐𝑓(𝑧))′	NUM
cana-534	58	9	−	−	PROPN
cana-534	59	1	𝑆𝜏	𝑆𝜏	PROPN
cana-534	59	2	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	59	3	)	)	PUNCT
cana-534	59	4	𝑧	𝑧	PRON
cana-534	60	1	|	|	ADV
cana-534	60	2	−	−	ADP
cana-534	60	3	𝑅𝑒	𝑅𝑒	PROPN
cana-534	60	4	{	{	PUNCT
cana-534	60	5	𝑆𝜏	𝑆𝜏	PROPN
cana-534	60	6	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	60	7	)	)	PUNCT
cana-534	60	8	𝑧	𝑧	DET
cana-534	60	9	−	−	NUM
cana-534	60	10	1	1	NUM
cana-534	60	11	}	}	PUNCT
cana-534	60	12	≤	≤	NUM
cana-534	60	13	1	1	NUM
cana-534	60	14	−	−	NOUN
cana-534	60	15	𝛽.	𝛽.	NOUN
cana-534	61	1	we	we	PRON
cana-534	61	2	have	have	VERB
cana-534	61	3	𝛼	𝛼	X
cana-534	61	4	|(𝑆𝜏	|(𝑆𝜏	NOUN
cana-534	61	5	𝑐𝑓(𝑧))′	𝑐𝑓(𝑧))′	NUM
cana-534	62	1	−	−	PROPN
cana-534	63	1	𝑆𝜏	𝑆𝜏	PROPN
cana-534	63	2	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	63	3	)	)	PUNCT
cana-534	63	4	𝑧	𝑧	PRON
cana-534	64	1	|	|	ADV
cana-534	64	2	−	−	ADP
cana-534	64	3	𝑅𝑒	𝑅𝑒	PROPN
cana-534	64	4	{	{	PUNCT
cana-534	64	5	𝑆𝜏	𝑆𝜏	PROPN
cana-534	64	6	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	64	7	)	)	PUNCT
cana-534	64	8	𝑧	𝑧	PRON
cana-534	64	9	−	−	NUM
cana-534	64	10	1	1	NUM
cana-534	64	11	}	}	PUNCT
cana-534	64	12	≤	≤	NOUN
cana-534	64	13	𝛼	𝛼	NUM
cana-534	64	14	|	|	ADV
cana-534	64	15	∑	∑	PUNCT
cana-534	64	16	(	(	PUNCT
cana-534	64	17	𝑛	𝑛	PRON
cana-534	64	18	−	−	PROPN
cana-534	64	19	1)∞	1)∞	PROPN
cana-534	64	20	𝑛=2	𝑛=2	PROPN
cana-534	64	21	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	64	22	,	,	PUNCT
cana-534	64	23	𝜏	𝜏	NOUN
cana-534	64	24	,	,	PUNCT
cana-534	64	25	𝑛)𝑎𝑛𝑧𝑛	𝑛)𝑎𝑛𝑧𝑛	ADV
cana-534	64	26	𝑧	𝑧	NOUN
cana-534	65	1	|	|	ADV
cana-534	65	2	+	+	CCONJ
cana-534	65	3	|	|	ADV
cana-534	65	4	∑	∑	ADP
cana-534	65	5	𝐸∞	𝐸∞	ADJ
cana-534	65	6	𝑛=2	𝑛=2	PROPN
cana-534	65	7	(	(	PUNCT
cana-534	65	8	𝑐	𝑐	PROPN
cana-534	65	9	,	,	PUNCT
cana-534	65	10	𝜏	𝜏	NOUN
cana-534	65	11	,	,	PUNCT
cana-534	65	12	𝑛)𝑎𝑛𝑧𝑛	𝑛)𝑎𝑛𝑧𝑛	ADV
cana-534	65	13	𝑧	𝑧	PRON
cana-534	65	14	|	|	ADV
cana-534	65	15	≤	≤	X
cana-534	65	16	𝛼	𝛼	NOUN
cana-534	65	17	∑(𝑛	∑(𝑛	NOUN
cana-534	65	18	−	−	NOUN
cana-534	65	19	1	1	NUM
cana-534	65	20	)	)	PUNCT
cana-534	65	21	∞	∞	PROPN
cana-534	65	22	𝑛=2	𝑛=2	PROPN
cana-534	65	23	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	65	24	,	,	PUNCT
cana-534	65	25	𝜏	𝜏	NOUN
cana-534	65	26	,	,	PUNCT
cana-534	65	27	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	PRON
cana-534	65	28	+	+	CCONJ
cana-534	65	29	∑	∑	PROPN
cana-534	65	30	𝐸	𝐸	PROPN
cana-534	65	31	∞	∞	ADJ
cana-534	65	32	𝑛=2	𝑛=2	PROPN
cana-534	65	33	(	(	PUNCT
cana-534	65	34	𝑐	𝑐	NOUN
cana-534	65	35	,	,	PUNCT
cana-534	65	36	𝜏	𝜏	NOUN
cana-534	65	37	,	,	PUNCT
cana-534	65	38	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	PRON
cana-534	65	39	=	=	PUNCT
cana-534	65	40	∑[1	∑[1	NUM
cana-534	65	41	+	+	NUM
cana-534	65	42	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	65	43	−	−	NOUN
cana-534	65	44	1	1	NUM
cana-534	65	45	)	)	PUNCT
cana-534	65	46	]	]	PUNCT
cana-534	66	1	∞	∞	NUM
cana-534	66	2	𝑛=2	𝑛=2	PROPN
cana-534	66	3	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	66	4	,	,	PUNCT
cana-534	66	5	𝜏	𝜏	NOUN
cana-534	66	6	,	,	PUNCT
cana-534	66	7	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	PROPN
cana-534	66	8	.	.	PUNCT
cana-534	67	1	the	the	DET
cana-534	67	2	last	last	ADJ
cana-534	67	3	expression	expression	NOUN
cana-534	67	4	is	be	AUX
cana-534	67	5	bounded	bound	VERB
cana-534	67	6	above	above	ADV
cana-534	67	7	by	by	ADP
cana-534	67	8	(	(	PUNCT
cana-534	67	9	1	1	NUM
cana-534	67	10	−	−	PROPN
cana-534	67	11	𝛽	𝛽	NOUN
cana-534	67	12	)	)	PUNCT
cana-534	67	13	if	if	SCONJ
cana-534	67	14	∑[1	∑[1	NUM
cana-534	67	15	+	+	ADJ
cana-534	67	16	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	67	17	−	−	NOUN
cana-534	67	18	1	1	NUM
cana-534	67	19	)	)	PUNCT
cana-534	67	20	]	]	PUNCT
cana-534	68	1	∞	∞	NUM
cana-534	68	2	𝑛=2	𝑛=2	PROPN
cana-534	68	3	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	68	4	,	,	PUNCT
cana-534	68	5	𝜏	𝜏	NOUN
cana-534	68	6	,	,	PUNCT
cana-534	68	7	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	ADV
cana-534	68	8	≤	≤	NUM
cana-534	68	9	1	1	NUM
cana-534	68	10	−	−	NOUN
cana-534	68	11	𝛽	𝛽	NOUN
cana-534	68	12	communications	communication	NOUN
cana-534	68	13	on	on	ADP
cana-534	68	14	applied	apply	VERB
cana-534	68	15	nonlinear	nonlinear	ADJ
cana-534	68	16	analysis	analysis	NOUN
cana-534	68	17	issn	issn	NOUN
cana-534	68	18	:	:	PUNCT
cana-534	68	19	1074	1074	NUM
cana-534	68	20	-	-	PUNCT
cana-534	68	21	133x	133x	NUM
cana-534	68	22	vol	vol	NOUN
cana-534	68	23	31	31	NUM
cana-534	68	24	no	no	NOUN
cana-534	68	25	.	.	NOUN
cana-534	68	26	2	2	NUM
cana-534	68	27	(	(	PUNCT
cana-534	68	28	2024	2024	NUM
cana-534	68	29	)	)	PUNCT
cana-534	68	30	200	200	NUM
cana-534	68	31	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	68	32	and	and	CCONJ
cana-534	68	33	the	the	DET
cana-534	68	34	proof	proof	NOUN
cana-534	68	35	of	of	ADP
cana-534	68	36	theorem	theorem	NOUN
cana-534	68	37	is	be	AUX
cana-534	68	38	completed	complete	VERB
cana-534	68	39	.	.	PUNCT
cana-534	69	1	in	in	ADP
cana-534	69	2	the	the	DET
cana-534	69	3	following	following	NOUN
cana-534	69	4	theorem	theorem	NOUN
cana-534	69	5	,	,	PUNCT
cana-534	69	6	we	we	PRON
cana-534	69	7	obtain	obtain	VERB
cana-534	69	8	necessary	necessary	ADJ
cana-534	69	9	and	and	CCONJ
cana-534	69	10	sufficient	sufficient	ADJ
cana-534	69	11	conditions	condition	NOUN
cana-534	69	12	for	for	ADP
cana-534	69	13	functions	function	NOUN
cana-534	69	14	in	in	ADP
cana-534	69	15	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	69	16	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	69	17	,	,	PUNCT
cana-534	69	18	𝛽	𝛽	NOUN
cana-534	69	19	)	)	PUNCT
cana-534	69	20	.	.	PUNCT
cana-534	70	1	theorem	theorem	VERB
cana-534	70	2	2.2	2.2	NUM
cana-534	70	3	.	.	PUNCT
cana-534	71	1	for	for	ADP
cana-534	71	2	𝛼	𝛼	PRON
cana-534	71	3	≥	≥	NOUN
cana-534	71	4	0,0	0,0	NUM
cana-534	71	5	≤	≤	NUM
cana-534	71	6	𝛽	𝛽	NOUN
cana-534	71	7	<	<	X
cana-534	71	8	1	1	NUM
cana-534	71	9	,	,	PUNCT
cana-534	71	10	a	a	DET
cana-534	71	11	function	function	NOUN
cana-534	71	12	𝑓	𝑓	PRON
cana-534	71	13	of	of	ADP
cana-534	71	14	the	the	DET
cana-534	71	15	form	form	NOUN
cana-534	71	16	(	(	PUNCT
cana-534	71	17	1.2	1.2	NUM
cana-534	71	18	)	)	PUNCT
cana-534	71	19	to	to	PART
cana-534	71	20	be	be	AUX
cana-534	71	21	in	in	ADP
cana-534	71	22	the	the	DET
cana-534	71	23	class	class	NOUN
cana-534	71	24	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	71	25	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	71	26	,	,	PUNCT
cana-534	71	27	𝛽	𝛽	NOUN
cana-534	71	28	)	)	PUNCT
cana-534	71	29	if	if	SCONJ
cana-534	71	30	and	and	CCONJ
cana-534	71	31	only	only	ADV
cana-534	71	32	if	if	SCONJ
cana-534	71	33	∑[1	∑[1	NUM
cana-534	71	34	+	+	ADJ
cana-534	71	35	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	71	36	−	−	NOUN
cana-534	71	37	1	1	NUM
cana-534	71	38	)	)	PUNCT
cana-534	71	39	]	]	PUNCT
cana-534	72	1	∞	∞	NUM
cana-534	72	2	𝑛=2	𝑛=2	PROPN
cana-534	72	3	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	72	4	,	,	PUNCT
cana-534	72	5	𝜏	𝜏	NOUN
cana-534	72	6	,	,	PUNCT
cana-534	72	7	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	ADV
cana-534	72	8	≤	≤	NUM
cana-534	72	9	1	1	NUM
cana-534	72	10	−	−	NOUN
cana-534	72	11	𝛽.	𝛽.	NOUN
cana-534	72	12	proof	proof	NOUN
cana-534	72	13	.	.	PUNCT
cana-534	73	1	suppose	suppose	VERB
cana-534	73	2	𝑓(𝑧	𝑓(𝑧	NOUN
cana-534	73	3	)	)	PUNCT
cana-534	73	4	of	of	ADP
cana-534	73	5	the	the	DET
cana-534	73	6	form	form	NOUN
cana-534	73	7	(	(	PUNCT
cana-534	73	8	1.2	1.2	NUM
cana-534	73	9	)	)	PUNCT
cana-534	73	10	is	be	AUX
cana-534	73	11	in	in	ADP
cana-534	73	12	the	the	DET
cana-534	73	13	class	class	NOUN
cana-534	73	14	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	73	15	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	73	16	,	,	PUNCT
cana-534	73	17	𝛽	𝛽	NOUN
cana-534	73	18	)	)	PUNCT
cana-534	73	19	.	.	PUNCT
cana-534	74	1	then	then	ADV
cana-534	74	2	𝑅𝑒	𝑅𝑒	PROPN
cana-534	74	3	{	{	PUNCT
cana-534	74	4	𝑆𝜏	𝑆𝜏	PROPN
cana-534	74	5	𝑐𝑓(𝑧	𝑐𝑓(𝑧	NOUN
cana-534	74	6	)	)	PUNCT
cana-534	74	7	𝑧	𝑧	X
cana-534	74	8	}	}	PUNCT
cana-534	74	9	−	−	PROPN
cana-534	74	10	𝛼	𝛼	X
cana-534	74	11	|(𝑆𝜏	|(𝑆𝜏	NOUN
cana-534	74	12	𝑐𝑓(𝑧))′	𝑐𝑓(𝑧))′	NUM
cana-534	74	13	−	−	PROPN
cana-534	75	1	𝑆𝜏	𝑆𝜏	PROPN
cana-534	75	2	𝑐𝑓(𝑧	𝑐𝑓(𝑧	PROPN
cana-534	75	3	)	)	PUNCT
cana-534	75	4	𝑧	𝑧	PRON
cana-534	76	1	|	|	ADV
cana-534	76	2	≥	≥	NOUN
cana-534	76	3	𝛽.	𝛽.	NOUN
cana-534	77	1	equivalently	equivalently	ADV
cana-534	77	2	𝑅𝑒	𝑅𝑒	VERB
cana-534	77	3	[	[	SYM
cana-534	77	4	1	1	NUM
cana-534	77	5	−	−	NUM
cana-534	77	6	∑	∑	PUNCT
cana-534	77	7	𝐸	𝐸	PROPN
cana-534	77	8	∞	∞	ADJ
cana-534	77	9	𝑛=2	𝑛=2	PROPN
cana-534	77	10	(	(	PUNCT
cana-534	77	11	𝑐	𝑐	NOUN
cana-534	77	12	,	,	PUNCT
cana-534	77	13	𝜏	𝜏	NOUN
cana-534	77	14	,	,	PUNCT
cana-534	77	15	𝑛)|𝑎𝑛|𝑧𝑛−1	𝑛)|𝑎𝑛|𝑧𝑛−1	NOUN
cana-534	77	16	]	]	PUNCT
cana-534	77	17	−	−	PUNCT
cana-534	77	18	𝛼	𝛼	PRON
cana-534	78	1	[	[	X
cana-534	78	2	∑(𝑛	∑(𝑛	X
cana-534	78	3	−	−	ADP
cana-534	78	4	1	1	NUM
cana-534	78	5	)	)	PUNCT
cana-534	78	6	∞	∞	PROPN
cana-534	78	7	𝑛=2	𝑛=2	PROPN
cana-534	78	8	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	78	9	,	,	PUNCT
cana-534	78	10	𝜏	𝜏	NOUN
cana-534	78	11	,	,	PUNCT
cana-534	78	12	𝑛)𝑎𝑛𝑧𝑛−1	𝑛)𝑎𝑛𝑧𝑛−1	PROPN
cana-534	78	13	]	]	X
cana-534	78	14	≥	≥	X
cana-534	78	15	𝛽.	𝛽.	NOUN
cana-534	78	16	letting	let	VERB
cana-534	78	17	𝑧	𝑧	PRON
cana-534	78	18	to	to	PART
cana-534	78	19	be	be	AUX
cana-534	78	20	real	real	ADJ
cana-534	78	21	values	value	NOUN
cana-534	78	22	and	and	CCONJ
cana-534	78	23	as	as	ADP
cana-534	78	24	|𝑧|	|𝑧|	PROPN
cana-534	78	25	→	→	SYM
cana-534	78	26	1	1	NUM
cana-534	78	27	,	,	PUNCT
cana-534	78	28	we	we	PRON
cana-534	78	29	have	have	VERB
cana-534	78	30	1	1	NUM
cana-534	78	31	−	−	NOUN
cana-534	78	32	∑	∑	PUNCT
cana-534	78	33	𝐸	𝐸	PROPN
cana-534	79	1	∞	∞	ADJ
cana-534	79	2	𝑛=2	𝑛=2	PROPN
cana-534	79	3	(	(	PUNCT
cana-534	79	4	𝑐	𝑐	NOUN
cana-534	79	5	,	,	PUNCT
cana-534	79	6	𝜏	𝜏	NOUN
cana-534	79	7	,	,	PUNCT
cana-534	79	8	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	PROPN
cana-534	79	9	−	−	NOUN
cana-534	79	10	𝛼	𝛼	INTJ
cana-534	79	11	∑(𝑛	∑(𝑛	NOUN
cana-534	80	1	−	−	NOUN
cana-534	80	2	1	1	NUM
cana-534	80	3	)	)	PUNCT
cana-534	80	4	∞	∞	PROPN
cana-534	80	5	𝑛=2	𝑛=2	PROPN
cana-534	80	6	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	80	7	,	,	PUNCT
cana-534	80	8	𝜏	𝜏	NOUN
cana-534	80	9	,	,	PUNCT
cana-534	80	10	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	PRON
cana-534	80	11	≥	≥	NOUN
cana-534	80	12	𝛽	𝛽	NOUN
cana-534	80	13	which	which	PRON
cana-534	80	14	implies	imply	VERB
cana-534	80	15	∑[1	∑[1	NUM
cana-534	80	16	+	+	NUM
cana-534	80	17	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	80	18	−	−	NOUN
cana-534	80	19	1	1	NUM
cana-534	80	20	)	)	PUNCT
cana-534	80	21	]	]	PUNCT
cana-534	81	1	∞	∞	NUM
cana-534	81	2	𝑛=2	𝑛=2	PROPN
cana-534	81	3	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	81	4	,	,	PUNCT
cana-534	81	5	𝜏	𝜏	NOUN
cana-534	81	6	,	,	PUNCT
cana-534	81	7	𝑛)|𝑎𝑛|	𝑛)|𝑎𝑛|	ADV
cana-534	81	8	≤	≤	NUM
cana-534	81	9	1	1	NUM
cana-534	81	10	−	−	PROPN
cana-534	81	11	𝛽	𝛽	NOUN
cana-534	81	12	,	,	PUNCT
cana-534	81	13	where	where	SCONJ
cana-534	81	14	𝛼	𝛼	X
cana-534	81	15	≥	≥	NOUN
cana-534	81	16	0	0	NUM
cana-534	81	17	,	,	PUNCT
cana-534	81	18	0	0	NUM
cana-534	81	19	≤	≤	NOUN
cana-534	81	20	𝛽	𝛽	NOUN
cana-534	81	21	<	<	X
cana-534	81	22	1	1	NUM
cana-534	81	23	,	,	PUNCT
cana-534	81	24	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	81	25	,	,	PUNCT
cana-534	81	26	𝜏	𝜏	NOUN
cana-534	81	27	,	,	PUNCT
cana-534	81	28	𝑛	𝑛	NOUN
cana-534	81	29	)	)	PUNCT
cana-534	81	30	is	be	AUX
cana-534	81	31	given	give	VERB
cana-534	81	32	by	by	ADP
cana-534	81	33	(	(	PUNCT
cana-534	81	34	1.12	1.12	NUM
cana-534	81	35	)	)	PUNCT
cana-534	81	36	and	and	CCONJ
cana-534	81	37	the	the	DET
cana-534	81	38	sufficiency	sufficiency	NOUN
cana-534	81	39	follows	follow	VERB
cana-534	81	40	from	from	ADP
cana-534	81	41	theorem	theorem	ADJ
cana-534	81	42	2.1	2.1	NUM
cana-534	81	43	.	.	PUNCT
cana-534	82	1	corollary	corollary	ADJ
cana-534	82	2	2.3	2.3	NUM
cana-534	82	3	.	.	PUNCT
cana-534	83	1	if	if	SCONJ
cana-534	83	2	𝑓	𝑓	DET
cana-534	83	3	∈	∈	NOUN
cana-534	83	4	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	83	5	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	83	6	,	,	PUNCT
cana-534	83	7	𝛽	𝛽	NOUN
cana-534	83	8	)	)	PUNCT
cana-534	83	9	then	then	ADV
cana-534	83	10	𝑎𝑛	𝑎𝑛	VERB
cana-534	83	11	≤	≤	NUM
cana-534	83	12	1−𝛽	1−𝛽	NUM
cana-534	83	13	[	[	SYM
cana-534	83	14	1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛	1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛	NUM
cana-534	83	15	)	)	PUNCT
cana-534	83	16	.	.	PUNCT
cana-534	84	1	equality	equality	NOUN
cana-534	84	2	holds	hold	VERB
cana-534	84	3	for	for	ADP
cana-534	84	4	the	the	DET
cana-534	84	5	function	function	NOUN
cana-534	84	6	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	84	7	)	)	PUNCT
cana-534	84	8	=	=	PUNCT
cana-534	85	1	𝑧	𝑧	PRON
cana-534	85	2	−	−	NUM
cana-534	85	3	1−𝛽	1−𝛽	NUM
cana-534	86	1	[	[	SYM
cana-534	86	2	1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛	1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛	NUM
cana-534	86	3	)	)	PUNCT
cana-534	86	4	𝑧𝑛	𝑧𝑛	PROPN
cana-534	86	5	,	,	PUNCT
cana-534	86	6	𝛼	𝛼	X
cana-534	86	7	≥	≥	NOUN
cana-534	86	8	0	0	NUM
cana-534	86	9	,	,	PUNCT
cana-534	86	10	0	0	NUM
cana-534	86	11	≤	≤	NOUN
cana-534	86	12	𝛽	𝛽	NOUN
cana-534	86	13	<	<	X
cana-534	86	14	1	1	NUM
cana-534	86	15	,	,	PUNCT
cana-534	86	16	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	86	17	,	,	PUNCT
cana-534	86	18	𝜏	𝜏	NOUN
cana-534	86	19	,	,	PUNCT
cana-534	86	20	𝑛	𝑛	NOUN
cana-534	86	21	)	)	PUNCT
cana-534	86	22	is	be	AUX
cana-534	86	23	given	give	VERB
cana-534	86	24	by	by	ADP
cana-534	86	25	(	(	PUNCT
cana-534	86	26	1.12	1.12	NUM
cana-534	86	27	)	)	PUNCT
cana-534	86	28	.	.	PUNCT
cana-534	87	1	3	3	X
cana-534	87	2	.	.	X
cana-534	87	3	extreme	extreme	ADJ
cana-534	87	4	points	point	NOUN
cana-534	87	5	theorem	theorem	VERB
cana-534	87	6	3.1	3.1	NUM
cana-534	87	7	.	.	PUNCT
cana-534	88	1	let	let	VERB
cana-534	88	2	𝑓1(𝑧	𝑓1(𝑧	NOUN
cana-534	88	3	)	)	PUNCT
cana-534	88	4	=	=	SYM
cana-534	88	5	𝑧	𝑧	NOUN
cana-534	88	6	and	and	CCONJ
cana-534	88	7	𝑓𝑛(𝑧	𝑓𝑛(𝑧	NUM
cana-534	88	8	)	)	PUNCT
cana-534	89	1	=	=	SYM
cana-534	89	2	𝑧	𝑧	PRON
cana-534	89	3	−	−	NUM
cana-534	89	4	1−𝛽	1−𝛽	NUM
cana-534	90	1	[	[	SYM
cana-534	90	2	1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛	1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛	NUM
cana-534	90	3	)	)	PUNCT
cana-534	90	4	𝑧𝑛	𝑧𝑛	PROPN
cana-534	90	5	,	,	PUNCT
cana-534	90	6	𝑛	𝑛	DET
cana-534	90	7	≥	≥	NUM
cana-534	90	8	2	2	NUM
cana-534	90	9	for	for	ADP
cana-534	90	10	𝛼	𝛼	PRON
cana-534	90	11	≥	≥	NOUN
cana-534	90	12	0,0	0,0	NUM
cana-534	90	13	≤	≤	NUM
cana-534	90	14	𝛽	𝛽	NOUN
cana-534	90	15	<	<	X
cana-534	90	16	1	1	NUM
cana-534	90	17	,	,	PUNCT
cana-534	90	18	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	90	19	,	,	PUNCT
cana-534	90	20	𝜏	𝜏	NOUN
cana-534	90	21	,	,	PUNCT
cana-534	90	22	𝑛	𝑛	NOUN
cana-534	90	23	)	)	PUNCT
cana-534	90	24	is	be	AUX
cana-534	90	25	given	give	VERB
cana-534	90	26	by	by	ADP
cana-534	90	27	(	(	PUNCT
cana-534	90	28	1.12	1.12	NUM
cana-534	90	29	)	)	PUNCT
cana-534	90	30	then	then	ADV
cana-534	90	31	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	90	32	)	)	PUNCT
cana-534	90	33	is	be	AUX
cana-534	90	34	in	in	ADP
cana-534	90	35	the	the	DET
cana-534	90	36	class	class	NOUN
cana-534	90	37	𝐸(𝑐	𝐸(𝑐	PROPN
cana-534	90	38	,	,	PUNCT
cana-534	90	39	𝜏	𝜏	NOUN
cana-534	90	40	,	,	PUNCT
cana-534	90	41	𝑛	𝑛	NOUN
cana-534	90	42	)	)	PUNCT
cana-534	90	43	if	if	SCONJ
cana-534	91	1	and	and	CCONJ
cana-534	91	2	only	only	ADV
cana-534	91	3	if	if	SCONJ
cana-534	91	4	it	it	PRON
cana-534	91	5	can	can	AUX
cana-534	91	6	be	be	AUX
cana-534	91	7	expressed	express	VERB
cana-534	91	8	in	in	ADP
cana-534	91	9	the	the	DET
cana-534	91	10	form	form	NOUN
cana-534	91	11	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	91	12	)	)	PUNCT
cana-534	91	13	=	=	SYM
cana-534	91	14	∑	∑	PUNCT
cana-534	91	15	𝜆𝑛	𝜆𝑛	PROPN
cana-534	91	16	∞	∞	PROPN
cana-534	91	17	𝑛=1	𝑛=1	NOUN
cana-534	91	18	𝑓𝑛(𝑧	𝑓𝑛(𝑧	ADV
cana-534	91	19	)	)	PUNCT
cana-534	91	20	,	,	PUNCT
cana-534	91	21	where	where	SCONJ
cana-534	91	22	𝜆𝑛	𝜆𝑛	NOUN
cana-534	91	23	and	and	CCONJ
cana-534	91	24	∑	∑	ADP
cana-534	91	25	𝜆𝑛	𝜆𝑛	PROPN
cana-534	91	26	∞	∞	NUM
cana-534	91	27	𝑛=1	𝑛=1	NOUN
cana-534	91	28	=	=	SYM
cana-534	91	29	1	1	X
cana-534	91	30	.	.	PUNCT
cana-534	91	31	communications	communication	NOUN
cana-534	91	32	on	on	ADP
cana-534	91	33	applied	apply	VERB
cana-534	91	34	nonlinear	nonlinear	ADJ
cana-534	91	35	analysis	analysis	NOUN
cana-534	91	36	issn	issn	NOUN
cana-534	91	37	:	:	PUNCT
cana-534	91	38	1074	1074	NUM
cana-534	91	39	-	-	PUNCT
cana-534	91	40	133x	133x	NUM
cana-534	91	41	vol	vol	NOUN
cana-534	91	42	31	31	NUM
cana-534	91	43	no	no	NOUN
cana-534	91	44	.	.	NOUN
cana-534	91	45	2	2	NUM
cana-534	91	46	(	(	PUNCT
cana-534	91	47	2024	2024	NUM
cana-534	91	48	)	)	PUNCT
cana-534	92	1	201	201	NUM
cana-534	92	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	92	3	proof	proof	NOUN
cana-534	92	4	.	.	PUNCT
cana-534	93	1	if	if	SCONJ
cana-534	93	2	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	93	3	)	)	PUNCT
cana-534	93	4	=	=	SYM
cana-534	94	1	∑	∑	PUNCT
cana-534	94	2	𝜆𝑛	𝜆𝑛	PROPN
cana-534	94	3	∞	∞	PROPN
cana-534	94	4	𝑛=1	𝑛=1	NOUN
cana-534	94	5	𝑓𝑛(𝑧	𝑓𝑛(𝑧	ADV
cana-534	94	6	)	)	PUNCT
cana-534	94	7	with	with	ADP
cana-534	94	8	𝜆𝑛	𝜆𝑛	PROPN
cana-534	94	9	≥	≥	NOUN
cana-534	94	10	0	0	NUM
cana-534	94	11	and	and	CCONJ
cana-534	94	12	∑	∑	ADP
cana-534	94	13	𝜆𝑛	𝜆𝑛	PROPN
cana-534	94	14	∞	∞	NUM
cana-534	94	15	𝑛=1	𝑛=1	NOUN
cana-534	94	16	=	=	SYM
cana-534	94	17	1	1	X
cana-534	94	18	.	.	PUNCT
cana-534	94	19	then	then	ADV
cana-534	94	20	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	94	21	)	)	PUNCT
cana-534	94	22	=	=	SYM
cana-534	94	23	∑	∑	PUNCT
cana-534	94	24	𝜆𝑛	𝜆𝑛	PROPN
cana-534	94	25	∞	∞	PROPN
cana-534	94	26	𝑛=1	𝑛=1	NOUN
cana-534	94	27	𝑓𝑛(𝑧	𝑓𝑛(𝑧	ADV
cana-534	94	28	)	)	PUNCT
cana-534	95	1	=	=	SYM
cana-534	95	2	𝜆1𝑓1(𝑧	𝜆1𝑓1(𝑧	VERB
cana-534	95	3	)	)	PUNCT
cana-534	95	4	+	+	CCONJ
cana-534	95	5	∑	∑	PUNCT
cana-534	95	6	𝜆𝑛	𝜆𝑛	PROPN
cana-534	95	7	∞	∞	NUM
cana-534	95	8	𝑛=2	𝑛=2	NOUN
cana-534	95	9	𝑓𝑛(𝑧	𝑓𝑛(𝑧	PUNCT
cana-534	95	10	)	)	PUNCT
cana-534	95	11	=	=	PUNCT
cana-534	95	12	(	(	PUNCT
cana-534	95	13	1	1	NUM
cana-534	95	14	−	−	NOUN
cana-534	95	15	∑	∑	PUNCT
cana-534	95	16	𝜆𝑛	𝜆𝑛	PROPN
cana-534	95	17	∞	∞	PROPN
cana-534	95	18	𝑛=2	𝑛=2	NOUN
cana-534	95	19	)	)	PUNCT
cana-534	95	20	𝑧	𝑧	PROPN
cana-534	96	1	+	+	NOUN
cana-534	96	2	∑	∑	PROPN
cana-534	96	3	[	[	X
cana-534	96	4	𝜆𝑛	𝜆𝑛	X
cana-534	96	5	(	(	PUNCT
cana-534	96	6	𝑧	𝑧	PROPN
cana-534	96	7	−	−	NUM
cana-534	96	8	1	1	NUM
cana-534	96	9	−	−	PROPN
cana-534	96	10	𝛽	𝛽	NOUN
cana-534	96	11	[	[	X
cana-534	96	12	1	1	NUM
cana-534	96	13	+	+	NUM
cana-534	96	14	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	96	15	−	−	ADP
cana-534	97	1	1)]𝐸(𝑐	1)]𝐸(𝑐	NUM
cana-534	97	2	,	,	PUNCT
cana-534	97	3	𝜏	𝜏	NOUN
cana-534	97	4	,	,	PUNCT
cana-534	97	5	𝑛	𝑛	NOUN
cana-534	97	6	)	)	PUNCT
cana-534	97	7	𝑧𝑛	𝑧𝑛	PROPN
cana-534	97	8	)	)	PUNCT
cana-534	97	9	]	]	PUNCT
cana-534	98	1	∞	∞	NUM
cana-534	98	2	𝑛=2	𝑛=2	PUNCT
cana-534	98	3	=	=	SYM
cana-534	98	4	𝑧	𝑧	PRON
cana-534	98	5	−	−	NOUN
cana-534	98	6	∑	∑	PROPN
cana-534	98	7	1	1	NUM
cana-534	98	8	−	−	NOUN
cana-534	98	9	𝛽	𝛽	NOUN
cana-534	98	10	[	[	X
cana-534	98	11	1	1	NUM
cana-534	98	12	+	+	NUM
cana-534	98	13	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	98	14	−	−	ADP
cana-534	99	1	1)]𝐸(𝑐	1)]𝐸(𝑐	NUM
cana-534	99	2	,	,	PUNCT
cana-534	99	3	𝜏	𝜏	NOUN
cana-534	99	4	,	,	PUNCT
cana-534	99	5	𝑛	𝑛	ADJ
cana-534	99	6	)	)	PUNCT
cana-534	99	7	∞	∞	NUM
cana-534	99	8	𝑛=2	𝑛=2	PROPN
cana-534	99	9	𝑧𝑛.	𝑧𝑛.	NOUN
cana-534	99	10	now	now	ADV
cana-534	99	11	∑	∑	PUNCT
cana-534	99	12	[	[	X
cana-534	99	13	1	1	NUM
cana-534	99	14	+	+	NUM
cana-534	99	15	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	99	16	−	−	ADP
cana-534	100	1	1)]𝐸(𝑐	1)]𝐸(𝑐	NUM
cana-534	100	2	,	,	PUNCT
cana-534	100	3	𝜏	𝜏	NOUN
cana-534	100	4	,	,	PUNCT
cana-534	100	5	𝑛	𝑛	NOUN
cana-534	100	6	)	)	PUNCT
cana-534	100	7	1	1	NUM
cana-534	100	8	−	−	PROPN
cana-534	100	9	𝛽	𝛽	NOUN
cana-534	100	10	∞	∞	NUM
cana-534	100	11	𝑛=2	𝑛=2	PROPN
cana-534	100	12	1	1	NUM
cana-534	100	13	−	−	NOUN
cana-534	100	14	𝛽	𝛽	NOUN
cana-534	100	15	[	[	X
cana-534	100	16	1	1	NUM
cana-534	100	17	+	+	NUM
cana-534	100	18	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	100	19	−	−	ADP
cana-534	100	20	1)]𝐸(𝑐	1)]𝐸(𝑐	NUM
cana-534	100	21	,	,	PUNCT
cana-534	100	22	𝜏	𝜏	NOUN
cana-534	100	23	,	,	PUNCT
cana-534	100	24	𝑛	𝑛	NOUN
cana-534	100	25	)	)	PUNCT
cana-534	100	26	𝜆𝑛	𝜆𝑛	NOUN
cana-534	100	27	=	=	PUNCT
cana-534	100	28	∑	∑	PUNCT
cana-534	100	29	𝜆𝑛	𝜆𝑛	PROPN
cana-534	100	30	∞	∞	NUM
cana-534	100	31	𝑛=2	𝑛=2	NOUN
cana-534	100	32	=	=	SYM
cana-534	100	33	1	1	NUM
cana-534	100	34	−	−	NOUN
cana-534	100	35	𝜆1	𝜆1	NOUN
cana-534	100	36	≤	≤	NOUN
cana-534	100	37	1	1	NUM
cana-534	100	38	.	.	PUNCT
cana-534	101	1	then	then	ADV
cana-534	101	2	𝑓	𝑓	DET
cana-534	101	3	∈	∈	NOUN
cana-534	101	4	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	101	5	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	101	6	,	,	PUNCT
cana-534	101	7	𝛽	𝛽	NOUN
cana-534	101	8	)	)	PUNCT
cana-534	101	9	.	.	PUNCT
cana-534	102	1	conversely	conversely	ADV
cana-534	102	2	suppose	suppose	VERB
cana-534	102	3	that	that	SCONJ
cana-534	102	4	𝑓	𝑓	DET
cana-534	102	5	∈	∈	PROPN
cana-534	102	6	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	102	7	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	102	8	,	,	PUNCT
cana-534	102	9	𝛽	𝛽	NOUN
cana-534	102	10	)	)	PUNCT
cana-534	102	11	.	.	PUNCT
cana-534	103	1	then	then	ADV
cana-534	103	2	corollary	corollary	ADJ
cana-534	103	3	2.3	2.3	NUM
cana-534	103	4	gives	give	VERB
cana-534	103	5	𝑎𝑛	𝑎𝑛	NOUN
cana-534	103	6	≤	≤	NUM
cana-534	103	7	1	1	NUM
cana-534	103	8	−	−	NOUN
cana-534	103	9	𝛽	𝛽	NOUN
cana-534	103	10	[	[	X
cana-534	103	11	1	1	NUM
cana-534	103	12	+	+	NUM
cana-534	103	13	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	103	14	−	−	ADP
cana-534	103	15	1)]𝐸(𝑐	1)]𝐸(𝑐	NUM
cana-534	103	16	,	,	PUNCT
cana-534	103	17	𝜏	𝜏	NOUN
cana-534	103	18	,	,	PUNCT
cana-534	103	19	𝑛	𝑛	NOUN
cana-534	103	20	)	)	PUNCT
cana-534	103	21	,	,	PUNCT
cana-534	103	22	𝑛	𝑛	DET
cana-534	103	23	≥	≥	NUM
cana-534	103	24	2	2	NUM
cana-534	103	25	set	set	NOUN
cana-534	103	26	𝜆𝑛	𝜆𝑛	NOUN
cana-534	104	1	=	=	PUNCT
cana-534	105	1	[	[	X
cana-534	105	2	1	1	NUM
cana-534	105	3	+	+	NUM
cana-534	105	4	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	105	5	−	−	ADP
cana-534	106	1	1)]𝐸(𝑐	1)]𝐸(𝑐	NUM
cana-534	106	2	,	,	PUNCT
cana-534	106	3	𝜏	𝜏	NOUN
cana-534	106	4	,	,	PUNCT
cana-534	106	5	𝑛	𝑛	NOUN
cana-534	106	6	)	)	PUNCT
cana-534	106	7	1	1	NUM
cana-534	106	8	−	−	PROPN
cana-534	106	9	𝛽	𝛽	PROPN
cana-534	106	10	𝑎𝑛	𝑎𝑛	PROPN
cana-534	106	11	,	,	PUNCT
cana-534	106	12	𝑛	𝑛	DET
cana-534	106	13	≥	≥	NOUN
cana-534	106	14	2	2	NUM
cana-534	106	15	where	where	SCONJ
cana-534	106	16	𝜆𝑛	𝜆𝑛	NOUN
cana-534	106	17	=	=	SYM
cana-534	106	18	1	1	NUM
cana-534	106	19	−	−	NOUN
cana-534	106	20	∑	∑	PUNCT
cana-534	106	21	𝜆𝑛	𝜆𝑛	PROPN
cana-534	106	22	∞	∞	PROPN
cana-534	106	23	𝑛=2	𝑛=2	PROPN
cana-534	106	24	.	.	PUNCT
cana-534	107	1	then	then	ADV
cana-534	107	2	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	107	3	)	)	PUNCT
cana-534	108	1	=	=	PUNCT
cana-534	108	2	𝑧	𝑧	PRON
cana-534	108	3	−	−	NOUN
cana-534	108	4	∑	∑	PROPN
cana-534	108	5	𝑎𝑛	𝑎𝑛	PROPN
cana-534	108	6	∞	∞	NUM
cana-534	108	7	𝑛=2	𝑛=2	NOUN
cana-534	108	8	𝑧𝑛	𝑧𝑛	ADP
cana-534	108	9	=	=	PUNCT
cana-534	108	10	𝑧	𝑧	ADJ
cana-534	108	11	−	−	NOUN
cana-534	108	12	∑	∑	PUNCT
cana-534	108	13	𝜆𝑛	𝜆𝑛	PROPN
cana-534	108	14	∞	∞	PROPN
cana-534	108	15	𝑛=2	𝑛=2	PROPN
cana-534	108	16	1	1	NUM
cana-534	108	17	−	−	NOUN
cana-534	108	18	𝛽	𝛽	NOUN
cana-534	108	19	[	[	X
cana-534	108	20	1	1	NUM
cana-534	108	21	+	+	NUM
cana-534	108	22	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	108	23	−	−	ADP
cana-534	108	24	1)]𝐸(𝑐	1)]𝐸(𝑐	NUM
cana-534	108	25	,	,	PUNCT
cana-534	108	26	𝜏	𝜏	NOUN
cana-534	108	27	,	,	PUNCT
cana-534	108	28	𝑛	𝑛	NOUN
cana-534	108	29	)	)	PUNCT
cana-534	108	30	=	=	SYM
cana-534	108	31	𝑧	𝑧	NOUN
cana-534	108	32	−	−	PROPN
cana-534	109	1	[	[	X
cana-534	109	2	1	1	NUM
cana-534	109	3	−	−	NOUN
cana-534	109	4	∑	∑	PUNCT
cana-534	109	5	𝜆𝑛	𝜆𝑛	PROPN
cana-534	109	6	∞	∞	PROPN
cana-534	109	7	𝑛=2	𝑛=2	PROPN
cana-534	109	8	]	]	PUNCT
cana-534	110	1	+	+	CCONJ
cana-534	110	2	∑	∑	PUNCT
cana-534	110	3	𝜆𝑛	𝜆𝑛	PROPN
cana-534	110	4	∞	∞	NUM
cana-534	110	5	𝑛=2	𝑛=2	NOUN
cana-534	110	6	𝑓𝑛(𝑧	𝑓𝑛(𝑧	PUNCT
cana-534	110	7	)	)	PUNCT
cana-534	111	1	=	=	SYM
cana-534	111	2	𝜆1𝑓1(𝑧	𝜆1𝑓1(𝑧	VERB
cana-534	111	3	)	)	PUNCT
cana-534	111	4	+	+	CCONJ
cana-534	111	5	∑	∑	PUNCT
cana-534	111	6	𝜆𝑛	𝜆𝑛	PROPN
cana-534	111	7	∞	∞	NUM
cana-534	111	8	𝑛=2	𝑛=2	NOUN
cana-534	111	9	𝑓𝑛(𝑧	𝑓𝑛(𝑧	PUNCT
cana-534	111	10	)	)	PUNCT
cana-534	111	11	=	=	SYM
cana-534	111	12	∑	∑	PUNCT
cana-534	111	13	𝜆𝑛	𝜆𝑛	PROPN
cana-534	111	14	∞	∞	PROPN
cana-534	111	15	𝑛=1	𝑛=1	NOUN
cana-534	111	16	𝑓𝑛(𝑧	𝑓𝑛(𝑧	ADV
cana-534	111	17	)	)	PUNCT
cana-534	111	18	.	.	PUNCT
cana-534	112	1	the	the	DET
cana-534	112	2	poof	poof	NOUN
cana-534	112	3	of	of	ADP
cana-534	112	4	theorem	theorem	NOUN
cana-534	112	5	is	be	AUX
cana-534	112	6	completed	complete	VERB
cana-534	112	7	.	.	PUNCT
cana-534	113	1	communications	communication	NOUN
cana-534	113	2	on	on	ADP
cana-534	113	3	applied	apply	VERB
cana-534	113	4	nonlinear	nonlinear	ADJ
cana-534	113	5	analysis	analysis	NOUN
cana-534	113	6	issn	issn	NOUN
cana-534	113	7	:	:	PUNCT
cana-534	113	8	1074	1074	NUM
cana-534	113	9	-	-	PUNCT
cana-534	113	10	133x	133x	NUM
cana-534	113	11	vol	vol	NOUN
cana-534	113	12	31	31	NUM
cana-534	113	13	no	no	NOUN
cana-534	113	14	.	.	NOUN
cana-534	113	15	2	2	NUM
cana-534	113	16	(	(	PUNCT
cana-534	113	17	2024	2024	NUM
cana-534	113	18	)	)	PUNCT
cana-534	113	19	202	202	NUM
cana-534	113	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	113	21	4	4	X
cana-534	113	22	.	.	PUNCT
cana-534	113	23	integral	integral	ADJ
cana-534	113	24	means	mean	NOUN
cana-534	113	25	inequalities	inequality	NOUN
cana-534	113	26	definition	definition	NOUN
cana-534	113	27	4.1	4.1	NUM
cana-534	113	28	.	.	PUNCT
cana-534	114	1	(	(	PUNCT
cana-534	114	2	subordination	subordination	NOUN
cana-534	114	3	principle	principle	NOUN
cana-534	114	4	)	)	PUNCT
cana-534	114	5	for	for	ADP
cana-534	114	6	analytic	analytic	ADJ
cana-534	114	7	function	function	NOUN
cana-534	114	8	𝑔	𝑔	NOUN
cana-534	114	9	and	and	CCONJ
cana-534	114	10	ℎ	ℎ	NOUN
cana-534	114	11	with	with	ADP
cana-534	114	12	𝑔(0	𝑔(0	PROPN
cana-534	114	13	)	)	PUNCT
cana-534	114	14	=	=	SYM
cana-534	114	15	ℎ(0	ℎ(0	PROPN
cana-534	114	16	)	)	PUNCT
cana-534	114	17	,	,	PUNCT
cana-534	114	18	𝑔	𝑔	PROPN
cana-534	114	19	is	be	AUX
cana-534	114	20	said	say	VERB
cana-534	114	21	to	to	PART
cana-534	114	22	be	be	AUX
cana-534	114	23	subordinate	subordinate	ADJ
cana-534	114	24	to	to	ADP
cana-534	114	25	ℎ	ℎ	PROPN
cana-534	114	26	,	,	PUNCT
cana-534	114	27	denoted	denote	VERB
cana-534	114	28	by	by	ADP
cana-534	114	29	𝑔	𝑔	PROPN
cana-534	114	30	≺	≺	NOUN
cana-534	114	31	ℎ	ℎ	PART
cana-534	114	32	if	if	SCONJ
cana-534	114	33	there	there	PRON
cana-534	114	34	exists	exist	VERB
cana-534	114	35	an	an	DET
cana-534	114	36	analytic	analytic	ADJ
cana-534	114	37	function	function	NOUN
cana-534	114	38	𝜔	𝜔	ADP
cana-534	114	39	such	such	ADJ
cana-534	114	40	that	that	DET
cana-534	114	41	𝜔(0	𝜔(0	NOUN
cana-534	114	42	)	)	PUNCT
cana-534	114	43	=	=	SYM
cana-534	114	44	0	0	NUM
cana-534	114	45	,	,	PUNCT
cana-534	114	46	|𝜔(𝑧)|	|𝜔(𝑧)|	ADJ
cana-534	114	47	<	<	X
cana-534	114	48	1	1	NUM
cana-534	114	49	and	and	CCONJ
cana-534	114	50	𝑔(𝑧	𝑔(𝑧	NOUN
cana-534	114	51	)	)	PUNCT
cana-534	114	52	=	=	SYM
cana-534	114	53	ℎ(𝜔(𝑧	ℎ(𝜔(𝑧	PROPN
cana-534	114	54	)	)	PUNCT
cana-534	114	55	)	)	PUNCT
cana-534	114	56	,	,	PUNCT
cana-534	114	57	for	for	ADP
cana-534	114	58	all	all	DET
cana-534	114	59	𝑧	𝑧	PRON
cana-534	114	60	∈	∈	PROPN
cana-534	114	61	𝑈.	𝑈.	PROPN
cana-534	114	62	lemma	lemma	PROPN
cana-534	114	63	4.2	4.2	NUM
cana-534	114	64	.	.	PUNCT
cana-534	115	1	[	[	X
cana-534	115	2	6	6	NUM
cana-534	115	3	]	]	PUNCT
cana-534	115	4	if	if	SCONJ
cana-534	115	5	the	the	DET
cana-534	115	6	function	function	NOUN
cana-534	115	7	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	115	8	)	)	PUNCT
cana-534	115	9	and	and	CCONJ
cana-534	115	10	𝑔(𝑧	𝑔(𝑧	NOUN
cana-534	115	11	)	)	PUNCT
cana-534	115	12	are	be	AUX
cana-534	115	13	analytic	analytic	ADJ
cana-534	115	14	in	in	ADP
cana-534	115	15	𝑈	𝑈	PROPN
cana-534	115	16	with	with	ADP
cana-534	115	17	𝑔(𝑧	𝑔(𝑧	NOUN
cana-534	115	18	)	)	PUNCT
cana-534	115	19	≺	≺	NOUN
cana-534	115	20	ℎ(𝑧	ℎ(𝑧	PROPN
cana-534	115	21	)	)	PUNCT
cana-534	115	22	then	then	ADV
cana-534	115	23	∫	∫	PROPN
cana-534	115	24	|𝑔(𝑟𝑒𝑖𝜃)|	|𝑔(𝑟𝑒𝑖𝜃)|	PROPN
cana-534	115	25	𝑝2𝜋	𝑝2𝜋	VERB
cana-534	115	26	0	0	NUM
cana-534	115	27	𝑑𝜃	𝑑𝜃	ADP
cana-534	115	28	≤	≤	NUM
cana-534	115	29	∫	∫	PROPN
cana-534	115	30	|𝑓(𝑟𝑒𝑖𝜃)|	|𝑓(𝑟𝑒𝑖𝜃)|	PROPN
cana-534	115	31	𝑝2𝜋	𝑝2𝜋	PROPN
cana-534	115	32	0	0	NUM
cana-534	115	33	𝑑𝜃	𝑑𝜃	PROPN
cana-534	115	34	(	(	PUNCT
cana-534	115	35	0	0	NUM
cana-534	115	36	≤	≤	NUM
cana-534	115	37	𝑟	𝑟	NOUN
cana-534	115	38	<	<	X
cana-534	115	39	1	1	NUM
cana-534	115	40	,	,	PUNCT
cana-534	115	41	𝑝	𝑝	NOUN
cana-534	115	42	>	>	NOUN
cana-534	115	43	0	0	NUM
cana-534	115	44	)	)	PUNCT
cana-534	115	45	.	.	PUNCT
cana-534	116	1	theorem	theorem	VERB
cana-534	116	2	4.3	4.3	NUM
cana-534	116	3	.	.	PUNCT
cana-534	117	1	suppose	suppose	VERB
cana-534	117	2	𝑓	𝑓	DET
cana-534	117	3	∈	∈	PROPN
cana-534	117	4	𝑇𝑆𝜏	𝑇𝑆𝜏	NOUN
cana-534	117	5	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	117	6	,	,	PUNCT
cana-534	117	7	𝛽	𝛽	NOUN
cana-534	117	8	)	)	PUNCT
cana-534	117	9	,	,	PUNCT
cana-534	117	10	𝑝	𝑝	NOUN
cana-534	117	11	>	>	X
cana-534	117	12	0	0	NUM
cana-534	117	13	,	,	PUNCT
cana-534	117	14	𝛼	𝛼	X
cana-534	117	15	≥	≥	NOUN
cana-534	117	16	0,0	0,0	NUM
cana-534	117	17	≤	≤	NUM
cana-534	117	18	𝛽	𝛽	NOUN
cana-534	117	19	<	<	X
cana-534	117	20	1	1	NUM
cana-534	117	21	and	and	CCONJ
cana-534	117	22	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	117	23	)	)	PUNCT
cana-534	117	24	is	be	AUX
cana-534	117	25	defined	define	VERB
cana-534	117	26	by	by	ADP
cana-534	117	27	𝑓2(𝑧	𝑓2(𝑧	NOUN
cana-534	117	28	)	)	PUNCT
cana-534	117	29	=	=	SYM
cana-534	117	30	𝑧	𝑧	PRON
cana-534	117	31	−	−	NUM
cana-534	117	32	1−𝛽	1−𝛽	NUM
cana-534	117	33	(	(	PUNCT
cana-534	117	34	1+𝛼)𝐸(𝑐,𝜏,𝑛	1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	117	35	)	)	PUNCT
cana-534	117	36	.	.	PUNCT
cana-534	118	1	then	then	ADV
cana-534	118	2	for	for	ADP
cana-534	118	3	𝑧	𝑧	PRON
cana-534	118	4	=	=	SYM
cana-534	118	5	𝑟𝑒𝑖𝜃	𝑟𝑒𝑖𝜃	NOUN
cana-534	118	6	,	,	PUNCT
cana-534	118	7	0	0	NUM
cana-534	118	8	≤	≤	NUM
cana-534	119	1	𝑟	𝑟	X
cana-534	119	2	<	<	X
cana-534	119	3	1	1	NUM
cana-534	119	4	,	,	PUNCT
cana-534	119	5	∫	∫	PROPN
cana-534	119	6	|𝑓(𝑧)|𝑝2𝜋	|𝑓(𝑧)|𝑝2𝜋	NOUN
cana-534	119	7	0	0	NUM
cana-534	119	8	𝑑𝜃	𝑑𝜃	PROPN
cana-534	119	9	≤	≤	NUM
cana-534	119	10	∫	∫	PROPN
cana-534	119	11	|𝑓2(𝑧)|𝑝2𝜋	|𝑓2(𝑧)|𝑝2𝜋	NOUN
cana-534	119	12	0	0	NUM
cana-534	119	13	𝑑𝜃	𝑑𝜃	PROPN
cana-534	119	14	(	(	PUNCT
cana-534	119	15	4.1	4.1	NUM
cana-534	119	16	)	)	PUNCT
cana-534	119	17	proof	proof	NOUN
cana-534	119	18	.	.	PUNCT
cana-534	120	1	for	for	ADP
cana-534	120	2	𝑓(𝑧	𝑓(𝑧	NUM
cana-534	120	3	)	)	PUNCT
cana-534	120	4	=	=	SYM
cana-534	120	5	𝑧	𝑧	PRON
cana-534	120	6	−	−	PROPN
cana-534	120	7	∑	∑	PROPN
cana-534	120	8	|𝑎𝑛|∞	|𝑎𝑛|∞	NUM
cana-534	120	9	𝑛=2	𝑛=2	PROPN
cana-534	120	10	𝑧𝑛	𝑧𝑛	PROPN
cana-534	120	11	,	,	PUNCT
cana-534	120	12	(	(	PUNCT
cana-534	120	13	4.1	4.1	NUM
cana-534	120	14	)	)	PUNCT
cana-534	120	15	is	be	AUX
cana-534	120	16	equivalent	equivalent	ADJ
cana-534	120	17	to	to	ADP
cana-534	120	18	proving	prove	VERB
cana-534	120	19	that	that	SCONJ
cana-534	120	20	∫	∫	PROPN
cana-534	120	21	|𝑧	|𝑧	NOUN
cana-534	120	22	−	−	PROPN
cana-534	120	23	∑|𝑎𝑛|	∑|𝑎𝑛|	NOUN
cana-534	120	24	∞	∞	PROPN
cana-534	120	25	𝑛=2	𝑛=2	NOUN
cana-534	120	26	𝑧𝑛|	𝑧𝑛|	VERB
cana-534	120	27	𝑝2𝜋	𝑝2𝜋	NOUN
cana-534	120	28	0	0	NUM
cana-534	120	29	𝑑𝜃	𝑑𝜃	ADP
cana-534	121	1	≤	≤	NUM
cana-534	121	2	∫	∫	PROPN
cana-534	121	3	|𝑧	|𝑧	NOUN
cana-534	121	4	−	−	PROPN
cana-534	121	5	1	1	NUM
cana-534	121	6	−	−	PROPN
cana-534	121	7	𝛽	𝛽	NOUN
cana-534	121	8	1	1	NUM
cana-534	121	9	+	+	NUM
cana-534	121	10	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	121	11	,	,	PUNCT
cana-534	121	12	𝜏	𝜏	NOUN
cana-534	121	13	,	,	PUNCT
cana-534	121	14	𝑛	𝑛	NOUN
cana-534	121	15	)	)	PUNCT
cana-534	121	16	|	|	ADV
cana-534	121	17	𝑝	𝑝	ADP
cana-534	121	18	2𝜋	2𝜋	NOUN
cana-534	121	19	0	0	PUNCT
cana-534	122	1	𝑑𝜃	𝑑𝜃	ADP
cana-534	122	2	,	,	PUNCT
cana-534	122	3	(	(	PUNCT
cana-534	122	4	𝑝	𝑝	NOUN
cana-534	122	5	>	>	PUNCT
cana-534	122	6	0	0	NUM
cana-534	122	7	)	)	PUNCT
cana-534	122	8	.	.	PUNCT
cana-534	123	1	by	by	ADP
cana-534	123	2	applying	apply	VERB
cana-534	123	3	little	little	ADJ
cana-534	123	4	wood	wood	NOUN
cana-534	123	5	’s	’s	PART
cana-534	123	6	subordination	subordination	NOUN
cana-534	123	7	theorem	theorem	NOUN
cana-534	123	8	(	(	PUNCT
cana-534	123	9	lemma	lemma	PROPN
cana-534	123	10	4.2	4.2	NUM
cana-534	123	11	)	)	PUNCT
cana-534	123	12	,	,	PUNCT
cana-534	123	13	it	it	PRON
cana-534	123	14	would	would	AUX
cana-534	123	15	be	be	AUX
cana-534	123	16	sufficient	sufficient	ADJ
cana-534	123	17	to	to	PART
cana-534	123	18	show	show	VERB
cana-534	123	19	that	that	SCONJ
cana-534	124	1	1	1	NUM
cana-534	124	2	−	−	NUM
cana-534	124	3	∑|𝑎𝑛|	∑|𝑎𝑛|	NOUN
cana-534	124	4	∞	∞	PROPN
cana-534	124	5	𝑛=2	𝑛=2	PROPN
cana-534	124	6	𝑧𝑛−1	𝑧𝑛−1	NOUN
cana-534	124	7	≺	≺	NOUN
cana-534	124	8	1	1	NUM
cana-534	124	9	−	−	NUM
cana-534	124	10	1	1	NUM
cana-534	124	11	−	−	PROPN
cana-534	124	12	𝛽	𝛽	NOUN
cana-534	124	13	1	1	NUM
cana-534	124	14	+	+	NUM
cana-534	124	15	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	124	16	,	,	PUNCT
cana-534	124	17	𝜏	𝜏	NOUN
cana-534	124	18	,	,	PUNCT
cana-534	124	19	𝑛	𝑛	NOUN
cana-534	124	20	)	)	PUNCT
cana-534	124	21	𝑧.	𝑧.	NOUN
cana-534	124	22	(	(	PUNCT
cana-534	124	23	4.2	4.2	NUM
cana-534	124	24	)	)	PUNCT
cana-534	124	25	setting	set	VERB
cana-534	124	26	1	1	NUM
cana-534	124	27	−	−	NOUN
cana-534	124	28	∑|𝑎𝑛|	∑|𝑎𝑛|	DET
cana-534	124	29	∞	∞	PROPN
cana-534	124	30	𝑛=2	𝑛=2	PROPN
cana-534	124	31	𝑧𝑛−1	𝑧𝑛−1	NOUN
cana-534	124	32	≺	≺	NOUN
cana-534	124	33	1	1	NUM
cana-534	124	34	−	−	NUM
cana-534	124	35	1	1	NUM
cana-534	124	36	−	−	PROPN
cana-534	124	37	𝛽	𝛽	NOUN
cana-534	124	38	1	1	NUM
cana-534	124	39	+	+	NUM
cana-534	124	40	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	124	41	,	,	PUNCT
cana-534	124	42	𝜏	𝜏	NOUN
cana-534	124	43	,	,	PUNCT
cana-534	124	44	𝑛	𝑛	NOUN
cana-534	124	45	)	)	PUNCT
cana-534	124	46	𝜔(𝑧	𝜔(𝑧	VERB
cana-534	124	47	)	)	PUNCT
cana-534	124	48	.	.	PUNCT
cana-534	125	1	we	we	PRON
cana-534	125	2	have	have	AUX
cana-534	125	3	𝜔(𝑧	𝜔(𝑧	VERB
cana-534	125	4	)	)	PUNCT
cana-534	125	5	=	=	SYM
cana-534	125	6	1+𝛼)𝐸(𝑐,𝜏,𝑛	1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	125	7	)	)	PUNCT
cana-534	125	8	1−𝛽	1−𝛽	NUM
cana-534	125	9	∑	∑	PUNCT
cana-534	125	10	𝑎𝑛	𝑎𝑛	PROPN
cana-534	125	11	∞	∞	NUM
cana-534	125	12	𝑛=2	𝑛=2	NOUN
cana-534	125	13	𝑧𝑛−1	𝑧𝑛−1	NOUN
cana-534	125	14	and	and	CCONJ
cana-534	125	15	and	and	CCONJ
cana-534	125	16	𝜔(𝑧	𝜔(𝑧	VERB
cana-534	125	17	)	)	PUNCT
cana-534	125	18	is	be	AUX
cana-534	125	19	analytic	analytic	ADJ
cana-534	125	20	in	in	ADP
cana-534	125	21	𝑈	𝑈	PROPN
cana-534	125	22	with	with	ADP
cana-534	125	23	𝜔(0	𝜔(0	NOUN
cana-534	125	24	)	)	PUNCT
cana-534	125	25	=	=	SYM
cana-534	126	1	0	0	X
cana-534	126	2	.	.	PUNCT
cana-534	127	1	moreover	moreover	ADV
cana-534	127	2	it	it	PRON
cana-534	127	3	suffices	suffice	VERB
cana-534	127	4	to	to	PART
cana-534	127	5	prove	prove	VERB
cana-534	127	6	that	that	SCONJ
cana-534	127	7	𝜔(𝑧	𝜔(𝑧	VERB
cana-534	127	8	)	)	PUNCT
cana-534	127	9	satisfies	satisfy	VERB
cana-534	127	10	|𝜔(𝑧)|	|𝜔(𝑧)|	ADJ
cana-534	127	11	<	<	X
cana-534	127	12	1	1	NUM
cana-534	127	13	,	,	PUNCT
cana-534	127	14	𝑧	𝑧	DET
cana-534	127	15	∈	∈	PROPN
cana-534	127	16	𝑈.	𝑈.	NOUN
cana-534	127	17	now	now	ADV
cana-534	127	18	|𝜔(𝑧)|	|𝜔(𝑧)|	PROPN
cana-534	127	19	=	=	SYM
cana-534	127	20	|∑	|∑	X
cana-534	127	21	(	(	PUNCT
cana-534	127	22	1	1	NUM
cana-534	127	23	+	+	CCONJ
cana-534	127	24	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	127	25	,	,	PUNCT
cana-534	127	26	𝜏	𝜏	NOUN
cana-534	127	27	,	,	PUNCT
cana-534	127	28	𝑛	𝑛	NOUN
cana-534	127	29	)	)	PUNCT
cana-534	127	30	1	1	NUM
cana-534	127	31	−	−	PROPN
cana-534	127	32	𝛽	𝛽	NOUN
cana-534	127	33	∞	∞	PROPN
cana-534	127	34	𝑛=2	𝑛=2	X
cana-534	127	35	𝑎𝑛𝑧𝑛−1|	𝑎𝑛𝑧𝑛−1|	NOUN
cana-534	127	36	≤	≤	NUM
cana-534	127	37	|𝑧|	|𝑧|	PROPN
cana-534	127	38	∑	∑	PUNCT
cana-534	127	39	(	(	PUNCT
cana-534	127	40	1	1	NUM
cana-534	127	41	+	+	CCONJ
cana-534	127	42	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	127	43	,	,	PUNCT
cana-534	127	44	𝜏	𝜏	NOUN
cana-534	127	45	,	,	PUNCT
cana-534	127	46	𝑛	𝑛	NOUN
cana-534	127	47	)	)	PUNCT
cana-534	127	48	1	1	NUM
cana-534	127	49	−	−	PROPN
cana-534	127	50	𝛽	𝛽	NOUN
cana-534	127	51	∞	∞	PROPN
cana-534	127	52	𝑛=2	𝑛=2	NOUN
cana-534	127	53	|𝑎𝑛|	|𝑎𝑛|	PRON
cana-534	127	54	(	(	PUNCT
cana-534	127	55	4.3	4.3	NUM
cana-534	127	56	)	)	PUNCT
cana-534	127	57	≤	≤	NOUN
cana-534	127	58	|𝑧|	|𝑧|	PROPN
cana-534	127	59	<	<	X
cana-534	127	60	1	1	NUM
cana-534	127	61	.	.	PUNCT
cana-534	127	62	thus	thus	ADV
cana-534	127	63	is	be	AUX
cana-534	127	64	view	view	NOUN
cana-534	127	65	of	of	ADP
cana-534	127	66	the	the	DET
cana-534	127	67	inequality	inequality	NOUN
cana-534	127	68	(	(	PUNCT
cana-534	127	69	4.3	4.3	NUM
cana-534	127	70	)	)	PUNCT
cana-534	127	71	the	the	DET
cana-534	127	72	subordination	subordination	NOUN
cana-534	127	73	(	(	PUNCT
cana-534	127	74	4.2	4.2	NUM
cana-534	127	75	)	)	PUNCT
cana-534	127	76	follows	follow	VERB
cana-534	127	77	,	,	PUNCT
cana-534	127	78	which	which	PRON
cana-534	127	79	proves	prove	VERB
cana-534	127	80	the	the	DET
cana-534	127	81	theorem	theorem	NOUN
cana-534	127	82	.	.	PROPN
cana-534	127	83	5	5	X
cana-534	127	84	.	.	X
cana-534	127	85	subordination	subordination	NOUN
cana-534	127	86	results	result	VERB
cana-534	127	87	definition	definition	NOUN
cana-534	127	88	5.1	5.1	NUM
cana-534	127	89	.	.	PUNCT
cana-534	128	1	(	(	PUNCT
cana-534	128	2	subordination	subordination	NOUN
cana-534	128	3	factor	factor	NOUN
cana-534	128	4	sequence	sequence	NOUN
cana-534	128	5	)	)	PUNCT
cana-534	128	6	a	a	DET
cana-534	128	7	sequence	sequence	NOUN
cana-534	128	8	{	{	PUNCT
cana-534	128	9	𝑏𝑛	𝑏𝑛	NOUN
cana-534	128	10	}	}	PUNCT
cana-534	128	11	𝑛=2	𝑛=2	PROPN
cana-534	128	12	∞	∞	NUM
cana-534	128	13	of	of	ADP
cana-534	128	14	complex	complex	ADJ
cana-534	128	15	numbers	number	NOUN
cana-534	128	16	is	be	AUX
cana-534	128	17	said	say	VERB
cana-534	128	18	to	to	PART
cana-534	128	19	be	be	AUX
cana-534	128	20	a	a	DET
cana-534	128	21	communications	communication	NOUN
cana-534	128	22	on	on	ADP
cana-534	128	23	applied	apply	VERB
cana-534	128	24	nonlinear	nonlinear	ADJ
cana-534	128	25	analysis	analysis	NOUN
cana-534	128	26	issn	issn	NOUN
cana-534	128	27	:	:	PUNCT
cana-534	128	28	1074	1074	NUM
cana-534	128	29	-	-	PUNCT
cana-534	128	30	133x	133x	NUM
cana-534	128	31	vol	vol	NOUN
cana-534	128	32	31	31	NUM
cana-534	128	33	no	no	NOUN
cana-534	128	34	.	.	NOUN
cana-534	128	35	2	2	NUM
cana-534	128	36	(	(	PUNCT
cana-534	128	37	2024	2024	NUM
cana-534	128	38	)	)	PUNCT
cana-534	128	39	203	203	NUM
cana-534	128	40	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	128	41	subordinating	subordinate	VERB
cana-534	128	42	sequence	sequence	NOUN
cana-534	128	43	if	if	SCONJ
cana-534	128	44	,	,	PUNCT
cana-534	128	45	whenever	whenever	SCONJ
cana-534	128	46	𝑓(𝑧	𝑓(𝑧	ADV
cana-534	128	47	)	)	PUNCT
cana-534	128	48	=	=	PUNCT
cana-534	129	1	∑	∑	PUNCT
cana-534	129	2	𝑎𝑛	𝑎𝑛	PROPN
cana-534	129	3	∞	∞	PROPN
cana-534	129	4	𝑛=2	𝑛=2	PROPN
cana-534	129	5	𝑧𝑛	𝑧𝑛	NOUN
cana-534	129	6	,	,	PUNCT
cana-534	129	7	𝑎1	𝑎1	NOUN
cana-534	129	8	=	=	SYM
cana-534	129	9	1	1	NUM
cana-534	129	10	is	be	AUX
cana-534	129	11	regular	regular	ADJ
cana-534	129	12	,	,	PUNCT
cana-534	129	13	univalent	univalent	ADJ
cana-534	129	14	and	and	CCONJ
cana-534	129	15	convex	convex	NOUN
cana-534	129	16	in	in	ADP
cana-534	129	17	𝑈	𝑈	PROPN
cana-534	129	18	,	,	PUNCT
cana-534	129	19	we	we	PRON
cana-534	129	20	have	have	VERB
cana-534	129	21	∑	∑	PROPN
cana-534	129	22	𝑏𝑛	𝑏𝑛	ADP
cana-534	129	23	∞	∞	PROPN
cana-534	129	24	𝑛=1	𝑛=1	NOUN
cana-534	129	25	𝑎𝑛𝑧𝑛	𝑎𝑛𝑧𝑛	PROPN
cana-534	129	26	≺	≺	NOUN
cana-534	129	27	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	129	28	)	)	PUNCT
cana-534	129	29	,	,	PUNCT
cana-534	129	30	𝑧	𝑧	PROPN
cana-534	129	31	∈	∈	PROPN
cana-534	129	32	𝑈.	𝑈.	NOUN
cana-534	129	33	theorem	theorem	VERB
cana-534	129	34	5.2	5.2	NUM
cana-534	129	35	.	.	PUNCT
cana-534	130	1	[	[	X
cana-534	130	2	11	11	NUM
cana-534	130	3	]	]	PUNCT
cana-534	130	4	the	the	DET
cana-534	130	5	sequence	sequence	NOUN
cana-534	130	6	{	{	PUNCT
cana-534	130	7	𝑏𝑛	𝑏𝑛	ADP
cana-534	130	8	}	}	PUNCT
cana-534	130	9	𝑛=2	𝑛=2	X
cana-534	130	10	∞	∞	PROPN
cana-534	130	11	is	be	AUX
cana-534	130	12	a	a	DET
cana-534	130	13	subordinating	subordinate	VERB
cana-534	130	14	factor	factor	NOUN
cana-534	130	15	sequence	sequence	NOUN
cana-534	130	16	if	if	SCONJ
cana-534	130	17	and	and	CCONJ
cana-534	130	18	only	only	ADV
cana-534	130	19	if	if	SCONJ
cana-534	130	20	𝑅𝑒{1	𝑅𝑒{1	ADP
cana-534	130	21	+	+	CCONJ
cana-534	130	22	2	2	NUM
cana-534	130	23	∑	∑	NOUN
cana-534	130	24	𝑏𝑛	𝑏𝑛	ADP
cana-534	130	25	∞	∞	PROPN
cana-534	130	26	𝑛=1	𝑛=1	NOUN
cana-534	130	27	𝑧𝑛	𝑧𝑛	ADP
cana-534	130	28	}	}	PUNCT
cana-534	130	29	>	>	X
cana-534	130	30	0	0	NUM
cana-534	130	31	,	,	PUNCT
cana-534	130	32	𝑧	𝑧	PRON
cana-534	130	33	∈	∈	PROPN
cana-534	130	34	𝑈.	𝑈.	NOUN
cana-534	130	35	theorem	theorem	VERB
cana-534	130	36	5.3	5.3	NUM
cana-534	130	37	.	.	PUNCT
cana-534	131	1	let	let	VERB
cana-534	131	2	𝑓	𝑓	DET
cana-534	131	3	∈	∈	NOUN
cana-534	131	4	𝑆𝜏	𝑆𝜏	PROPN
cana-534	131	5	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	131	6	,	,	PUNCT
cana-534	131	7	𝛽	𝛽	NOUN
cana-534	131	8	)	)	PUNCT
cana-534	131	9	and	and	CCONJ
cana-534	131	10	𝑔(𝑧	𝑔(𝑧	NOUN
cana-534	131	11	)	)	PUNCT
cana-534	131	12	any	any	DET
cana-534	131	13	function	function	NOUN
cana-534	131	14	in	in	ADP
cana-534	131	15	the	the	DET
cana-534	131	16	usual	usual	ADJ
cana-534	131	17	class	class	NOUN
cana-534	131	18	of	of	ADP
cana-534	131	19	convex	convex	PROPN
cana-534	131	20	function	function	NOUN
cana-534	131	21	𝐶.	𝐶.	PROPN
cana-534	131	22	then	then	ADV
cana-534	131	23	(	(	PUNCT
cana-534	131	24	1	1	NUM
cana-534	131	25	+	+	CCONJ
cana-534	131	26	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	131	27	,	,	PUNCT
cana-534	131	28	𝜏	𝜏	NOUN
cana-534	131	29	,	,	PUNCT
cana-534	131	30	𝑛	𝑛	NOUN
cana-534	131	31	)	)	PUNCT
cana-534	131	32	2(1	2(1	NUM
cana-534	131	33	−	−	ADP
cana-534	131	34	𝛽	𝛽	NOUN
cana-534	131	35	)	)	PUNCT
cana-534	132	1	+	+	CCONJ
cana-534	132	2	(	(	PUNCT
cana-534	132	3	1	1	NUM
cana-534	132	4	+	+	NUM
cana-534	132	5	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	132	6	,	,	PUNCT
cana-534	132	7	𝜏	𝜏	NOUN
cana-534	132	8	,	,	PUNCT
cana-534	132	9	𝑛	𝑛	NOUN
cana-534	132	10	)	)	PUNCT
cana-534	132	11	(	(	PUNCT
cana-534	132	12	𝑓	𝑓	DET
cana-534	132	13	∗	∗	NOUN
cana-534	132	14	𝑔)(𝑧	𝑔)(𝑧	NOUN
cana-534	132	15	)	)	PUNCT
cana-534	132	16	≺	≺	NOUN
cana-534	132	17	𝑔(𝑧	𝑔(𝑧	NOUN
cana-534	132	18	)	)	PUNCT
cana-534	132	19	(	(	PUNCT
cana-534	132	20	5.1	5.1	NUM
cana-534	132	21	)	)	PUNCT
cana-534	132	22	where	where	SCONJ
cana-534	132	23	𝛼	𝛼	X
cana-534	132	24	≥	≥	NOUN
cana-534	132	25	0,0	0,0	NUM
cana-534	132	26	≤	≤	NUM
cana-534	132	27	𝛽	𝛽	NOUN
cana-534	132	28	<	<	X
cana-534	132	29	1	1	NUM
cana-534	132	30	with	with	ADP
cana-534	132	31	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	132	32	,	,	PUNCT
cana-534	132	33	𝜏	𝜏	NOUN
cana-534	132	34	,	,	PUNCT
cana-534	132	35	𝑛	𝑛	NOUN
cana-534	132	36	)	)	PUNCT
cana-534	132	37	is	be	AUX
cana-534	132	38	given	give	VERB
cana-534	132	39	by	by	ADP
cana-534	132	40	(	(	PUNCT
cana-534	132	41	1.12	1.12	NUM
cana-534	132	42	)	)	PUNCT
cana-534	132	43	𝑅𝑒{𝑓(𝑧	𝑅𝑒{𝑓(𝑧	NUM
cana-534	132	44	)	)	PUNCT
cana-534	132	45	}	}	PUNCT
cana-534	132	46	>	>	X
cana-534	132	47	−	−	PROPN
cana-534	132	48	(	(	PUNCT
cana-534	132	49	1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	132	50	)	)	PUNCT
cana-534	132	51	(	(	PUNCT
cana-534	132	52	1+𝛼)𝐸(𝑐,𝜏,𝑛	1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	132	53	)	)	PUNCT
cana-534	132	54	,	,	PUNCT
cana-534	132	55	𝑧	𝑧	PROPN
cana-534	132	56	∈	∈	PROPN
cana-534	132	57	𝐸.	𝐸.	PROPN
cana-534	132	58	(	(	PUNCT
cana-534	132	59	5.2	5.2	NUM
cana-534	132	60	)	)	PUNCT
cana-534	132	61	the	the	DET
cana-534	132	62	constant	constant	ADJ
cana-534	132	63	(	(	PUNCT
cana-534	132	64	1+𝛼)𝐸(𝑐,𝜏,𝑛	1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	132	65	)	)	PUNCT
cana-534	132	66	2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	132	67	)	)	PUNCT
cana-534	132	68	is	be	AUX
cana-534	132	69	the	the	DET
cana-534	132	70	best	good	ADJ
cana-534	132	71	estimate	estimate	NOUN
cana-534	132	72	.	.	PUNCT
cana-534	133	1	proof	proof	NOUN
cana-534	133	2	.	.	PUNCT
cana-534	134	1	let	let	VERB
cana-534	134	2	𝑓	𝑓	DET
cana-534	134	3	∈	∈	NOUN
cana-534	134	4	𝑆𝜏	𝑆𝜏	PROPN
cana-534	134	5	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	134	6	,	,	PUNCT
cana-534	134	7	𝛽	𝛽	NOUN
cana-534	134	8	)	)	PUNCT
cana-534	134	9	and	and	CCONJ
cana-534	134	10	𝑔(𝑧	𝑔(𝑧	X
cana-534	134	11	)	)	PUNCT
cana-534	134	12	=	=	PUNCT
cana-534	135	1	𝑧	𝑧	PROPN
cana-534	136	1	+	+	CCONJ
cana-534	136	2	∑	∑	PUNCT
cana-534	136	3	𝑐𝑛	𝑐𝑛	PROPN
cana-534	136	4	∞	∞	PROPN
cana-534	136	5	𝑛=2	𝑛=2	PROPN
cana-534	136	6	𝑧𝑛	𝑧𝑛	ADP
cana-534	136	7	∈	∈	PROPN
cana-534	136	8	𝐶.	𝐶.	PROPN
cana-534	136	9	then	then	ADV
cana-534	136	10	(	(	PUNCT
cana-534	136	11	1	1	NUM
cana-534	136	12	+	+	CCONJ
cana-534	136	13	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	136	14	,	,	PUNCT
cana-534	136	15	𝜏	𝜏	NOUN
cana-534	136	16	,	,	PUNCT
cana-534	136	17	𝑛	𝑛	NOUN
cana-534	136	18	)	)	PUNCT
cana-534	136	19	2(1	2(1	NUM
cana-534	136	20	−	−	ADP
cana-534	136	21	𝛽	𝛽	NOUN
cana-534	136	22	)	)	PUNCT
cana-534	136	23	+	+	CCONJ
cana-534	136	24	(	(	PUNCT
cana-534	136	25	1	1	NUM
cana-534	136	26	+	+	NUM
cana-534	136	27	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	136	28	,	,	PUNCT
cana-534	136	29	𝜏	𝜏	NOUN
cana-534	136	30	,	,	PUNCT
cana-534	136	31	𝑛	𝑛	NOUN
cana-534	136	32	)	)	PUNCT
cana-534	136	33	(	(	PUNCT
cana-534	136	34	𝑓	𝑓	DET
cana-534	136	35	∗	∗	NOUN
cana-534	136	36	𝑔)(𝑧	𝑔)(𝑧	NOUN
cana-534	136	37	)	)	PUNCT
cana-534	136	38	=	=	PUNCT
cana-534	136	39	(	(	PUNCT
cana-534	136	40	1	1	NUM
cana-534	136	41	+	+	CCONJ
cana-534	136	42	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	136	43	,	,	PUNCT
cana-534	136	44	𝜏	𝜏	NOUN
cana-534	136	45	,	,	PUNCT
cana-534	136	46	𝑛	𝑛	NOUN
cana-534	136	47	)	)	PUNCT
cana-534	136	48	2(1	2(1	NUM
cana-534	136	49	−	−	ADP
cana-534	136	50	𝛽	𝛽	NOUN
cana-534	136	51	)	)	PUNCT
cana-534	136	52	+	+	CCONJ
cana-534	136	53	(	(	PUNCT
cana-534	136	54	1	1	NUM
cana-534	136	55	+	+	NUM
cana-534	136	56	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	136	57	,	,	PUNCT
cana-534	136	58	𝜏	𝜏	NOUN
cana-534	136	59	,	,	PUNCT
cana-534	136	60	𝑛	𝑛	NOUN
cana-534	136	61	)	)	PUNCT
cana-534	136	62	(	(	PUNCT
cana-534	136	63	𝑧	𝑧	PROPN
cana-534	136	64	+	+	CCONJ
cana-534	136	65	∑	∑	PUNCT
cana-534	136	66	𝑐𝑛	𝑐𝑛	PROPN
cana-534	136	67	∞	∞	PROPN
cana-534	136	68	𝑛=2	𝑛=2	PROPN
cana-534	136	69	𝑎𝑛𝑧𝑛	𝑎𝑛𝑧𝑛	NOUN
cana-534	136	70	)	)	PUNCT
cana-534	136	71	.	.	PUNCT
cana-534	137	1	then	then	ADV
cana-534	137	2	by	by	ADP
cana-534	137	3	definition	definition	NOUN
cana-534	137	4	5.1	5.1	NUM
cana-534	137	5	,	,	PUNCT
cana-534	137	6	the	the	DET
cana-534	137	7	subordination	subordination	NOUN
cana-534	137	8	result	result	VERB
cana-534	137	9	holds	hold	VERB
cana-534	137	10	true	true	ADJ
cana-534	137	11	if	if	SCONJ
cana-534	137	12	{	{	PUNCT
cana-534	137	13	(	(	PUNCT
cana-534	137	14	1+𝛼)𝐸(𝑐,𝜏,𝑛	1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	137	15	)	)	PUNCT
cana-534	137	16	2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	137	17	)	)	PUNCT
cana-534	137	18	}	}	PUNCT
cana-534	137	19	𝑛=1	𝑛=1	VERB
cana-534	137	20	∞	∞	NOUN
cana-534	137	21	is	be	AUX
cana-534	137	22	a	a	DET
cana-534	137	23	subordinating	subordinate	VERB
cana-534	137	24	factor	factor	NOUN
cana-534	137	25	sequence	sequence	NOUN
cana-534	137	26	with	with	ADP
cana-534	137	27	𝑎1	𝑎1	PROPN
cana-534	137	28	=	=	SYM
cana-534	137	29	1	1	X
cana-534	137	30	.	.	PUNCT
cana-534	138	1	in	in	ADP
cana-534	138	2	view	view	NOUN
cana-534	138	3	of	of	ADP
cana-534	138	4	theorem	theorem	NOUN
cana-534	138	5	5.2	5.2	NUM
cana-534	138	6	,	,	PUNCT
cana-534	138	7	this	this	PRON
cana-534	138	8	is	be	AUX
cana-534	138	9	equivalent	equivalent	ADJ
cana-534	138	10	to	to	ADP
cana-534	138	11	the	the	DET
cana-534	138	12	following	following	ADJ
cana-534	138	13	inequality	inequality	NOUN
cana-534	138	14	.	.	PUNCT
cana-534	139	1	𝑅𝑒	𝑅𝑒	VERB
cana-534	139	2	{	{	PUNCT
cana-534	139	3	1	1	NUM
cana-534	139	4	+	+	NUM
cana-534	139	5	∑	∑	PUNCT
cana-534	139	6	(	(	PUNCT
cana-534	139	7	1	1	NUM
cana-534	139	8	+	+	CCONJ
cana-534	139	9	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	139	10	,	,	PUNCT
cana-534	139	11	𝜏	𝜏	NOUN
cana-534	139	12	,	,	PUNCT
cana-534	139	13	𝑛	𝑛	NOUN
cana-534	139	14	)	)	PUNCT
cana-534	139	15	(	(	PUNCT
cana-534	139	16	1	1	NUM
cana-534	139	17	−	−	PROPN
cana-534	139	18	𝛽	𝛽	NOUN
cana-534	139	19	)	)	PUNCT
cana-534	139	20	+	+	CCONJ
cana-534	139	21	(	(	PUNCT
cana-534	139	22	1	1	NUM
cana-534	139	23	+	+	NUM
cana-534	139	24	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	139	25	,	,	PUNCT
cana-534	139	26	𝜏	𝜏	NOUN
cana-534	139	27	,	,	PUNCT
cana-534	139	28	𝑛	𝑛	ADJ
cana-534	139	29	)	)	PUNCT
cana-534	139	30	∞	∞	PROPN
cana-534	139	31	𝑛=1	𝑛=1	NOUN
cana-534	139	32	𝑎𝑛𝑧𝑛	𝑎𝑛𝑧𝑛	VERB
cana-534	139	33	}	}	PUNCT
cana-534	139	34	>	>	X
cana-534	139	35	0	0	NUM
cana-534	139	36	,	,	PUNCT
cana-534	139	37	𝑧	𝑧	DET
cana-534	139	38	∈	∈	PROPN
cana-534	139	39	𝑈.	𝑈.	NOUN
cana-534	139	40	(	(	PUNCT
cana-534	139	41	5.3	5.3	NUM
cana-534	139	42	)	)	PUNCT
cana-534	139	43	now	now	ADV
cana-534	139	44	for	for	ADP
cana-534	139	45	|𝑧|	|𝑧|	PRON
cana-534	139	46	=	=	SYM
cana-534	139	47	𝑟	𝑟	X
cana-534	139	48	<	<	X
cana-534	139	49	1	1	NUM
cana-534	139	50	,	,	PUNCT
cana-534	139	51	we	we	PRON
cana-534	139	52	have	have	AUX
cana-534	139	53	𝑅𝑒	𝑅𝑒	VERB
cana-534	139	54	{	{	PUNCT
cana-534	139	55	1	1	NUM
cana-534	139	56	+	+	NOUN
cana-534	139	57	∑	∑	PUNCT
cana-534	139	58	(	(	PUNCT
cana-534	139	59	1	1	NUM
cana-534	139	60	+	+	CCONJ
cana-534	139	61	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	139	62	,	,	PUNCT
cana-534	139	63	𝜏	𝜏	NOUN
cana-534	139	64	,	,	PUNCT
cana-534	139	65	𝑛	𝑛	NOUN
cana-534	139	66	)	)	PUNCT
cana-534	139	67	2(1	2(1	NUM
cana-534	139	68	−	−	ADP
cana-534	139	69	𝛽	𝛽	NOUN
cana-534	139	70	)	)	PUNCT
cana-534	139	71	+	+	CCONJ
cana-534	139	72	(	(	PUNCT
cana-534	139	73	1	1	NUM
cana-534	139	74	+	+	NUM
cana-534	139	75	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	139	76	,	,	PUNCT
cana-534	139	77	𝜏	𝜏	NOUN
cana-534	139	78	,	,	PUNCT
cana-534	139	79	𝑛	𝑛	ADJ
cana-534	139	80	)	)	PUNCT
cana-534	139	81	∞	∞	PROPN
cana-534	139	82	𝑛=1	𝑛=1	NOUN
cana-534	139	83	𝑎𝑛𝑧𝑛	𝑎𝑛𝑧𝑛	VERB
cana-534	139	84	}	}	PUNCT
cana-534	139	85	=	=	PUNCT
cana-534	140	1	𝑅𝑒	𝑅𝑒	VERB
cana-534	140	2	{	{	PUNCT
cana-534	140	3	1	1	NUM
cana-534	140	4	+	+	CCONJ
cana-534	140	5	(	(	PUNCT
cana-534	140	6	1	1	NUM
cana-534	140	7	+	+	NUM
cana-534	140	8	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	140	9	,	,	PUNCT
cana-534	140	10	𝜏	𝜏	NOUN
cana-534	140	11	,	,	PUNCT
cana-534	140	12	𝑛	𝑛	NOUN
cana-534	140	13	)	)	PUNCT
cana-534	140	14	(	(	PUNCT
cana-534	140	15	1	1	NUM
cana-534	140	16	−	−	PROPN
cana-534	140	17	𝛽	𝛽	NOUN
cana-534	140	18	)	)	PUNCT
cana-534	140	19	+	+	CCONJ
cana-534	140	20	(	(	PUNCT
cana-534	140	21	1	1	NUM
cana-534	140	22	+	+	NUM
cana-534	140	23	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	140	24	,	,	PUNCT
cana-534	140	25	𝜏	𝜏	NOUN
cana-534	140	26	,	,	PUNCT
cana-534	140	27	𝑛	𝑛	NOUN
cana-534	140	28	)	)	PUNCT
cana-534	140	29	𝑧	𝑧	VERB
cana-534	141	1	+	+	NOUN
cana-534	141	2	∑	∑	PUNCT
cana-534	141	3	(	(	PUNCT
cana-534	141	4	1	1	NUM
cana-534	141	5	+	+	NUM
cana-534	141	6	𝛼)∞	𝛼)∞	NOUN
cana-534	141	7	𝑛=2	𝑛=2	PROPN
cana-534	141	8	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	141	9	,	,	PUNCT
cana-534	141	10	𝜏	𝜏	NOUN
cana-534	141	11	,	,	PUNCT
cana-534	141	12	𝑛)𝑎𝑛𝑧𝑛	𝑛)𝑎𝑛𝑧𝑛	PROPN
cana-534	141	13	(	(	PUNCT
cana-534	141	14	1	1	NUM
cana-534	141	15	−	−	PROPN
cana-534	141	16	𝛽	𝛽	NOUN
cana-534	141	17	)	)	PUNCT
cana-534	141	18	+	+	CCONJ
cana-534	141	19	(	(	PUNCT
cana-534	141	20	1	1	NUM
cana-534	141	21	+	+	NUM
cana-534	141	22	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	141	23	,	,	PUNCT
cana-534	141	24	𝜏	𝜏	NOUN
cana-534	141	25	,	,	PUNCT
cana-534	141	26	𝑛	𝑛	NOUN
cana-534	141	27	)	)	PUNCT
cana-534	141	28	}	}	PUNCT
cana-534	141	29	≥	≥	VERB
cana-534	141	30	1	1	NUM
cana-534	141	31	−	−	PROPN
cana-534	141	32	(	(	PUNCT
cana-534	141	33	1	1	NUM
cana-534	141	34	+	+	CCONJ
cana-534	141	35	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	141	36	,	,	PUNCT
cana-534	141	37	𝜏	𝜏	NOUN
cana-534	141	38	,	,	PUNCT
cana-534	141	39	𝑛	𝑛	NOUN
cana-534	141	40	)	)	PUNCT
cana-534	141	41	(	(	PUNCT
cana-534	141	42	1	1	NUM
cana-534	141	43	−	−	PROPN
cana-534	141	44	𝛽	𝛽	NOUN
cana-534	141	45	)	)	PUNCT
cana-534	142	1	+	+	CCONJ
cana-534	142	2	(	(	PUNCT
cana-534	142	3	1	1	NUM
cana-534	142	4	+	+	NUM
cana-534	142	5	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	142	6	,	,	PUNCT
cana-534	142	7	𝜏	𝜏	NOUN
cana-534	142	8	,	,	PUNCT
cana-534	142	9	𝑛	𝑛	NOUN
cana-534	142	10	)	)	PUNCT
cana-534	142	11	𝑟	𝑟	NOUN
cana-534	142	12	−	−	NOUN
cana-534	142	13	∑	∑	PUNCT
cana-534	142	14	(	(	PUNCT
cana-534	142	15	1	1	NUM
cana-534	142	16	+	+	NUM
cana-534	142	17	𝛼)∞	𝛼)∞	NOUN
cana-534	142	18	𝑛=2	𝑛=2	PROPN
cana-534	142	19	𝐸(𝑐	𝐸(𝑐	NUM
cana-534	142	20	,	,	PUNCT
cana-534	142	21	𝜏	𝜏	NOUN
cana-534	142	22	,	,	PUNCT
cana-534	142	23	𝑛)𝑎𝑛𝑟𝑛	𝑛)𝑎𝑛𝑟𝑛	ADJ
cana-534	142	24	(	(	PUNCT
cana-534	142	25	1	1	NUM
cana-534	142	26	−	−	PROPN
cana-534	142	27	𝛽	𝛽	NOUN
cana-534	142	28	)	)	PUNCT
cana-534	143	1	+	+	CCONJ
cana-534	143	2	(	(	PUNCT
cana-534	143	3	1	1	NUM
cana-534	143	4	+	+	NUM
cana-534	143	5	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	143	6	,	,	PUNCT
cana-534	143	7	𝜏	𝜏	NOUN
cana-534	143	8	,	,	PUNCT
cana-534	143	9	𝑛	𝑛	NOUN
cana-534	143	10	)	)	PUNCT
cana-534	143	11	≥	≥	NOUN
cana-534	143	12	1	1	NUM
cana-534	143	13	−	−	PROPN
cana-534	143	14	(	(	PUNCT
cana-534	143	15	1	1	NUM
cana-534	143	16	+	+	CCONJ
cana-534	143	17	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	143	18	,	,	PUNCT
cana-534	143	19	𝜏	𝜏	NOUN
cana-534	143	20	,	,	PUNCT
cana-534	143	21	𝑛	𝑛	NOUN
cana-534	143	22	)	)	PUNCT
cana-534	143	23	(	(	PUNCT
cana-534	143	24	1	1	NUM
cana-534	143	25	−	−	PROPN
cana-534	143	26	𝛽	𝛽	NOUN
cana-534	143	27	)	)	PUNCT
cana-534	144	1	+	+	CCONJ
cana-534	144	2	(	(	PUNCT
cana-534	144	3	1	1	NUM
cana-534	144	4	+	+	NUM
cana-534	144	5	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	144	6	,	,	PUNCT
cana-534	144	7	𝜏	𝜏	NOUN
cana-534	144	8	,	,	PUNCT
cana-534	144	9	𝑛	𝑛	NOUN
cana-534	144	10	)	)	PUNCT
cana-534	144	11	𝑟	𝑟	NOUN
cana-534	144	12	−	−	PROPN
cana-534	144	13	1	1	NUM
cana-534	144	14	−	−	PROPN
cana-534	144	15	𝛽	𝛽	PROPN
cana-534	144	16	(	(	PUNCT
cana-534	144	17	1	1	NUM
cana-534	144	18	−	−	PROPN
cana-534	144	19	𝛽	𝛽	NOUN
cana-534	144	20	)	)	PUNCT
cana-534	144	21	+	+	CCONJ
cana-534	144	22	(	(	PUNCT
cana-534	144	23	1	1	NUM
cana-534	144	24	+	+	NUM
cana-534	144	25	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	144	26	,	,	PUNCT
cana-534	144	27	𝜏	𝜏	NOUN
cana-534	144	28	,	,	PUNCT
cana-534	144	29	𝑛	𝑛	NOUN
cana-534	144	30	)	)	PUNCT
cana-534	144	31	𝑟	𝑟	NOUN
cana-534	144	32	>	>	X
cana-534	144	33	0	0	X
cana-534	144	34	.	.	PUNCT
cana-534	144	35	communications	communication	NOUN
cana-534	144	36	on	on	ADP
cana-534	144	37	applied	apply	VERB
cana-534	144	38	nonlinear	nonlinear	ADJ
cana-534	144	39	analysis	analysis	NOUN
cana-534	144	40	issn	issn	NOUN
cana-534	144	41	:	:	PUNCT
cana-534	144	42	1074	1074	NUM
cana-534	144	43	-	-	PUNCT
cana-534	144	44	133x	133x	NUM
cana-534	144	45	vol	vol	NOUN
cana-534	144	46	31	31	NUM
cana-534	144	47	no	no	NOUN
cana-534	144	48	.	.	NOUN
cana-534	144	49	2	2	NUM
cana-534	144	50	(	(	PUNCT
cana-534	144	51	2024	2024	NUM
cana-534	144	52	)	)	PUNCT
cana-534	144	53	204	204	NUM
cana-534	144	54	https://internationalpubls.com	https://internationalpubls.com	X
cana-534	144	55	using	use	VERB
cana-534	144	56	(	(	PUNCT
cana-534	144	57	2.1	2.1	NUM
cana-534	144	58	)	)	PUNCT
cana-534	144	59	and	and	CCONJ
cana-534	144	60	the	the	DET
cana-534	144	61	fact	fact	NOUN
cana-534	144	62	that	that	SCONJ
cana-534	144	63	1	1	NUM
cana-534	144	64	+	+	NUM
cana-534	144	65	𝛼(𝑛	𝛼(𝑛	PROPN
cana-534	145	1	−	−	PROPN
cana-534	145	2	1)𝐸(𝑐	1)𝐸(𝑐	NUM
cana-534	145	3	,	,	PUNCT
cana-534	145	4	𝜏	𝜏	NOUN
cana-534	145	5	,	,	PUNCT
cana-534	145	6	𝑛	𝑛	NOUN
cana-534	145	7	)	)	PUNCT
cana-534	145	8	is	be	AUX
cana-534	145	9	increasing	increase	VERB
cana-534	145	10	function	function	NOUN
cana-534	145	11	for	for	ADP
cana-534	145	12	𝑛	𝑛	PRON
cana-534	145	13	≥	≥	NUM
cana-534	145	14	2	2	NUM
cana-534	145	15	.	.	PUNCT
cana-534	146	1	this	this	PRON
cana-534	146	2	proves	prove	VERB
cana-534	146	3	the	the	DET
cana-534	146	4	inequality	inequality	NOUN
cana-534	146	5	(	(	PUNCT
cana-534	146	6	5.3	5.3	NUM
cana-534	146	7	)	)	PUNCT
cana-534	146	8	and	and	CCONJ
cana-534	146	9	hence	hence	ADV
cana-534	146	10	also	also	ADV
cana-534	146	11	the	the	DET
cana-534	146	12	subordination	subordination	NOUN
cana-534	146	13	result	result	NOUN
cana-534	146	14	(	(	PUNCT
cana-534	146	15	5.1	5.1	NUM
cana-534	146	16	)	)	PUNCT
cana-534	146	17	asserted	assert	VERB
cana-534	146	18	by	by	ADP
cana-534	146	19	theorem	theorem	NOUN
cana-534	146	20	5.3	5.3	NUM
cana-534	146	21	.	.	PUNCT
cana-534	147	1	the	the	DET
cana-534	147	2	inequality	inequality	NOUN
cana-534	147	3	(	(	PUNCT
cana-534	147	4	5.2	5.2	NUM
cana-534	147	5	)	)	PUNCT
cana-534	147	6	follows	follow	VERB
cana-534	147	7	from	from	ADP
cana-534	147	8	(	(	PUNCT
cana-534	147	9	5.1	5.1	NUM
cana-534	147	10	)	)	PUNCT
cana-534	147	11	by	by	ADP
cana-534	147	12	taking	take	VERB
cana-534	147	13	𝑔(𝑧	𝑔(𝑧	NOUN
cana-534	147	14	)	)	PUNCT
cana-534	147	15	=	=	PUNCT
cana-534	148	1	𝑧	𝑧	PRON
cana-534	148	2	1	1	NUM
cana-534	148	3	−	−	NOUN
cana-534	148	4	𝑧	𝑧	NOUN
cana-534	148	5	=	=	X
cana-534	148	6	𝑧	𝑧	PROPN
cana-534	149	1	+	+	CCONJ
cana-534	149	2	∑	∑	PROPN
cana-534	149	3	𝑧𝑛	𝑧𝑛	ADP
cana-534	149	4	∞	∞	NUM
cana-534	149	5	𝑛=2	𝑛=2	X
cana-534	149	6	∈	∈	PROPN
cana-534	149	7	𝐶.	𝐶.	PROPN
cana-534	149	8	now	now	ADV
cana-534	149	9	we	we	PRON
cana-534	149	10	consider	consider	VERB
cana-534	149	11	the	the	DET
cana-534	149	12	function	function	NOUN
cana-534	149	13	𝑓(𝑧	𝑓(𝑧	PROPN
cana-534	149	14	)	)	PUNCT
cana-534	150	1	=	=	PUNCT
cana-534	150	2	𝑧	𝑧	DET
cana-534	150	3	−	−	NUM
cana-534	150	4	1−𝛽	1−𝛽	NUM
cana-534	150	5	(	(	PUNCT
cana-534	150	6	1+𝛼)𝐸(𝑐,𝜏,𝑛	1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	150	7	)	)	PUNCT
cana-534	150	8	𝑧2	𝑧2	NOUN
cana-534	150	9	,	,	PUNCT
cana-534	150	10	where	where	SCONJ
cana-534	150	11	𝛼	𝛼	X
cana-534	150	12	≥	≥	NOUN
cana-534	150	13	0,0	0,0	NUM
cana-534	150	14	≤	≤	NUM
cana-534	150	15	𝛽	𝛽	NOUN
cana-534	150	16	<	<	X
cana-534	150	17	1	1	NUM
cana-534	150	18	.	.	PUNCT
cana-534	151	1	clearly	clearly	ADV
cana-534	151	2	𝐹	𝐹	PROPN
cana-534	151	3	∈	∈	PROPN
cana-534	152	1	𝑆𝜏	𝑆𝜏	PROPN
cana-534	152	2	𝑐(𝛼	𝑐(𝛼	PROPN
cana-534	152	3	,	,	PUNCT
cana-534	152	4	𝛽	𝛽	NOUN
cana-534	152	5	)	)	PUNCT
cana-534	152	6	.	.	PUNCT
cana-534	153	1	for	for	ADP
cana-534	153	2	the	the	DET
cana-534	153	3	function	function	NOUN
cana-534	153	4	(	(	PUNCT
cana-534	153	5	5.1	5.1	NUM
cana-534	153	6	)	)	PUNCT
cana-534	153	7	becomes	become	VERB
cana-534	153	8	(	(	PUNCT
cana-534	153	9	1	1	NUM
cana-534	153	10	+	+	CCONJ
cana-534	153	11	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	153	12	,	,	PUNCT
cana-534	153	13	𝜏	𝜏	NOUN
cana-534	153	14	,	,	PUNCT
cana-534	153	15	𝑛	𝑛	NOUN
cana-534	153	16	)	)	PUNCT
cana-534	153	17	2(1	2(1	NUM
cana-534	153	18	−	−	ADP
cana-534	153	19	𝛽	𝛽	NOUN
cana-534	153	20	)	)	PUNCT
cana-534	154	1	+	+	CCONJ
cana-534	154	2	(	(	PUNCT
cana-534	154	3	1	1	NUM
cana-534	154	4	+	+	NUM
cana-534	154	5	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	154	6	,	,	PUNCT
cana-534	154	7	𝜏	𝜏	NOUN
cana-534	154	8	,	,	PUNCT
cana-534	154	9	𝑛	𝑛	NOUN
cana-534	154	10	)	)	PUNCT
cana-534	154	11	𝐹(𝑧	𝐹(𝑧	NUM
cana-534	154	12	)	)	PUNCT
cana-534	154	13	≺	≺	NOUN
cana-534	154	14	𝑧	𝑧	VERB
cana-534	154	15	1	1	NUM
cana-534	154	16	−	−	NOUN
cana-534	154	17	𝑧	𝑧	NOUN
cana-534	154	18	.	.	PUNCT
cana-534	155	1	it	it	PRON
cana-534	155	2	is	be	AUX
cana-534	155	3	easily	easily	ADV
cana-534	155	4	verified	verify	VERB
cana-534	155	5	that	that	SCONJ
cana-534	155	6	min𝑅𝑒	min𝑅𝑒	PROPN
cana-534	155	7	{	{	PUNCT
cana-534	155	8	(	(	PUNCT
cana-534	155	9	1	1	NUM
cana-534	155	10	+	+	CCONJ
cana-534	155	11	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	155	12	,	,	PUNCT
cana-534	155	13	𝜏	𝜏	NOUN
cana-534	155	14	,	,	PUNCT
cana-534	155	15	𝑛	𝑛	NOUN
cana-534	155	16	)	)	PUNCT
cana-534	155	17	2(1	2(1	NUM
cana-534	155	18	−	−	ADP
cana-534	155	19	𝛽	𝛽	NOUN
cana-534	155	20	)	)	PUNCT
cana-534	155	21	+	+	CCONJ
cana-534	155	22	(	(	PUNCT
cana-534	155	23	1	1	NUM
cana-534	155	24	+	+	NUM
cana-534	155	25	𝛼)𝐸(𝑐	𝛼)𝐸(𝑐	PROPN
cana-534	155	26	,	,	PUNCT
cana-534	155	27	𝜏	𝜏	NOUN
cana-534	155	28	,	,	PUNCT
cana-534	155	29	𝑛	𝑛	NOUN
cana-534	155	30	)	)	PUNCT
cana-534	155	31	𝐹(𝑧	𝐹(𝑧	NUM
cana-534	155	32	)	)	PUNCT
cana-534	155	33	}	}	PUNCT
cana-534	155	34	=	=	SYM
cana-534	155	35	−1	−1	NOUN
cana-534	155	36	2	2	NUM
cana-534	155	37	,	,	PUNCT
cana-534	155	38	𝑧	𝑧	DET
cana-534	155	39	∈	∈	PROPN
cana-534	155	40	𝑈.	𝑈.	NOUN
cana-534	155	41	this	this	PRON
cana-534	155	42	shows	show	VERB
cana-534	155	43	that	that	SCONJ
cana-534	155	44	the	the	DET
cana-534	155	45	constant	constant	ADJ
cana-534	155	46	(	(	PUNCT
cana-534	155	47	1+𝛼)𝐸(𝑐,𝜏,𝑛	1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	155	48	)	)	PUNCT
cana-534	155	49	2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛	NUM
cana-534	155	50	)	)	PUNCT
cana-534	155	51	𝐹(𝑧	𝐹(𝑧	NUM
cana-534	155	52	)	)	PUNCT
cana-534	155	53	≺	≺	NOUN
cana-534	155	54	𝑧	𝑧	VERB
cana-534	155	55	1−𝑧	1−𝑧	NUM
cana-534	155	56	is	be	AUX
cana-534	155	57	best	well	ADV
cana-534	155	58	possible	possible	ADJ
cana-534	155	59	.	.	PUNCT
cana-534	156	1	refrences	refrence	NOUN
cana-534	157	1	[	[	X
cana-534	157	2	1	1	NUM
cana-534	157	3	]	]	PUNCT
cana-534	157	4	a.	a.	NOUN
cana-534	157	5	baricz	baricz	NOUN
cana-534	157	6	,	,	PUNCT
cana-534	157	7	geometric	geometric	ADJ
cana-534	157	8	properties	property	NOUN
cana-534	157	9	of	of	ADP
cana-534	157	10	generalized	generalized	ADJ
cana-534	157	11	bessel	bessel	NOUN
cana-534	157	12	function	function	NOUN
cana-534	157	13	,	,	PUNCT
cana-534	157	14	publ	publ	PROPN
cana-534	157	15	.	.	PUNCT
cana-534	158	1	math	math	NOUN
cana-534	158	2	.	.	PUNCT
cana-534	159	1	debrecan	debrecan	PROPN
cana-534	159	2	,	,	PUNCT
cana-534	159	3	73	73	NUM
cana-534	159	4	(	(	PUNCT
cana-534	159	5	2008	2008	NUM
cana-534	159	6	)	)	PUNCT
cana-534	159	7	,	,	PUNCT
cana-534	159	8	155178	155178	NUM
cana-534	159	9	.	.	PUNCT
cana-534	160	1	[	[	X
cana-534	160	2	2	2	NUM
cana-534	160	3	]	]	PUNCT
cana-534	160	4	a.	a.	NOUN
cana-534	160	5	baricz	baricz	NOUN
cana-534	160	6	,	,	PUNCT
cana-534	160	7	generalized	generalized	ADJ
cana-534	160	8	bessel	bessel	NOUN
cana-534	160	9	functions	function	NOUN
cana-534	160	10	of	of	ADP
cana-534	160	11	the	the	DET
cana-534	160	12	first	first	ADJ
cana-534	160	13	kind	kind	NOUN
cana-534	160	14	,	,	PUNCT
cana-534	160	15	ph.d	ph.d	PROPN
cana-534	160	16	thesis	thesis	NOUN
cana-534	160	17	,	,	PUNCT
cana-534	160	18	babes	babes	NOUN
cana-534	160	19	-	-	PUNCT
cana-534	160	20	bolyai	bolyai	NOUN
cana-534	160	21	university	university	NOUN
cana-534	160	22	,	,	PUNCT
cana-534	160	23	clujnapoca	clujnapoca	NOUN
cana-534	160	24	,	,	PUNCT
cana-534	160	25	2008	2008	NUM
cana-534	160	26	.	.	PUNCT
cana-534	161	1	[	[	X
cana-534	161	2	3	3	NUM
cana-534	161	3	]	]	PUNCT
cana-534	161	4	a.	a.	NOUN
cana-534	161	5	baricz	baricz	NOUN
cana-534	161	6	,	,	PUNCT
cana-534	161	7	generalized	generalized	ADJ
cana-534	161	8	bessel	bessel	NOUN
cana-534	161	9	functions	function	NOUN
cana-534	161	10	of	of	ADP
cana-534	161	11	the	the	DET
cana-534	161	12	first	first	ADJ
cana-534	161	13	kind	kind	NOUN
cana-534	161	14	,	,	PUNCT
cana-534	161	15	lecture	lecture	NOUN
cana-534	161	16	notes	note	NOUN
cana-534	161	17	in	in	ADP
cana-534	161	18	math	math	NOUN
cana-534	161	19	.	.	PUNCT
cana-534	161	20	,	,	PUNCT
cana-534	161	21	1994	1994	NUM
cana-534	161	22	,	,	PUNCT
cana-534	161	23	springer	springer	NOUN
cana-534	161	24	,	,	PUNCT
cana-534	161	25	berlin	berlin	PROPN
cana-534	161	26	,	,	PUNCT
cana-534	161	27	2010	2010	NUM
cana-534	161	28	.	.	PUNCT
cana-534	162	1	[	[	X
cana-534	162	2	4	4	NUM
cana-534	162	3	]	]	PUNCT
cana-534	162	4	a.	a.	NOUN
cana-534	162	5	baricz	baricz	NOUN
cana-534	162	6	and	and	CCONJ
cana-534	162	7	b.	b.	PROPN
cana-534	162	8	a.	a.	PROPN
cana-534	162	9	frasin	frasin	PROPN
cana-534	162	10	,	,	PUNCT
cana-534	162	11	univalence	univalence	NOUN
cana-534	162	12	of	of	ADP
cana-534	162	13	integral	integral	ADJ
cana-534	162	14	operators	operator	NOUN
cana-534	162	15	involving	involve	VERB
cana-534	162	16	bessel	bessel	ADJ
cana-534	162	17	functions	function	NOUN
cana-534	162	18	,	,	PUNCT
cana-534	162	19	appl	appl	PROPN
cana-534	162	20	.	.	PROPN
cana-534	162	21	math	math	PROPN
cana-534	162	22	.	.	PUNCT
cana-534	163	1	lett	lett	PROPN
cana-534	163	2	.	.	PROPN
cana-534	163	3	,	,	PUNCT
cana-534	163	4	23	23	NUM
cana-534	163	5	(	(	PUNCT
cana-534	163	6	2010	2010	NUM
cana-534	163	7	)	)	PUNCT
cana-534	163	8	,	,	PUNCT
cana-534	163	9	371	371	NUM
cana-534	163	10	-	-	SYM
cana-534	163	11	376	376	NUM
cana-534	163	12	.	.	PUNCT
cana-534	164	1	[	[	X
cana-534	164	2	5	5	X
cana-534	164	3	]	]	PUNCT
cana-534	164	4	e.	e.	PROPN
cana-534	164	5	deniz	deniz	PROPN
cana-534	164	6	,	,	PUNCT
cana-534	164	7	h.	h.	PROPN
cana-534	164	8	orhan	orhan	PROPN
cana-534	164	9	and	and	CCONJ
cana-534	164	10	h.	h.	PROPN
cana-534	164	11	m.	m.	PROPN
cana-534	164	12	srivastava	srivastava	PROPN
cana-534	164	13	,	,	PUNCT
cana-534	164	14	some	some	DET
cana-534	164	15	sufficient	sufficient	ADJ
cana-534	164	16	conditions	condition	NOUN
cana-534	164	17	for	for	ADP
cana-534	164	18	univalence	univalence	NOUN
cana-534	164	19	of	of	ADP
cana-534	164	20	certain	certain	ADJ
cana-534	164	21	families	family	NOUN
cana-534	164	22	of	of	ADP
cana-534	164	23	integral	integral	ADJ
cana-534	164	24	operators	operator	NOUN
cana-534	164	25	involving	involve	VERB
cana-534	164	26	generalized	generalized	ADJ
cana-534	164	27	bessel	bessel	ADJ
cana-534	164	28	functions	function	NOUN
cana-534	164	29	,	,	PUNCT
cana-534	164	30	taiwan	taiwan	PROPN
cana-534	164	31	.	.	PUNCT
cana-534	165	1	j.	j.	PROPN
cana-534	165	2	math	math	PROPN
cana-534	165	3	.	.	PUNCT
cana-534	165	4	,	,	PUNCT
cana-534	165	5	15(2	15(2	NUM
cana-534	165	6	)	)	PUNCT
cana-534	165	7	(	(	PUNCT
cana-534	165	8	2011	2011	NUM
cana-534	165	9	)	)	PUNCT
cana-534	165	10	,	,	PUNCT
cana-534	165	11	883	883	NUM
cana-534	165	12	-	-	SYM
cana-534	165	13	917	917	NUM
cana-534	165	14	.	.	PUNCT
cana-534	166	1	[	[	X
cana-534	166	2	6	6	NUM
cana-534	166	3	]	]	X
cana-534	166	4	j.e	j.e	PROPN
cana-534	166	5	.	.	PROPN
cana-534	166	6	littlewood	littlewood	PROPN
cana-534	166	7	,	,	PUNCT
cana-534	166	8	on	on	ADP
cana-534	166	9	inequalities	inequality	NOUN
cana-534	166	10	in	in	ADP
cana-534	166	11	theory	theory	NOUN
cana-534	166	12	of	of	ADP
cana-534	166	13	functions	function	NOUN
cana-534	166	14	,	,	PUNCT
cana-534	166	15	proc	proc	NOUN
cana-534	166	16	.	.	PUNCT
cana-534	167	1	london	london	PROPN
cana-534	167	2	math	math	PROPN
cana-534	167	3	.	.	PUNCT
cana-534	168	1	soc	soc	PROPN
cana-534	168	2	.	.	PUNCT
cana-534	168	3	,	,	PUNCT
cana-534	168	4	23	23	NUM
cana-534	168	5	(	(	PUNCT
cana-534	168	6	1925	1925	NUM
cana-534	168	7	)	)	PUNCT
cana-534	168	8	,	,	PUNCT
cana-534	168	9	481	481	NUM
cana-534	168	10	-	-	SYM
cana-534	168	11	519	519	NUM
cana-534	168	12	.	.	PUNCT
cana-534	169	1	[	[	X
cana-534	169	2	7	7	NUM
cana-534	169	3	]	]	X
cana-534	169	4	f.	f.	PROPN
cana-534	169	5	,	,	PUNCT
cana-534	169	6	rønning	rønning	ADJ
cana-534	169	7	,	,	PUNCT
cana-534	169	8	uniformly	uniformly	ADV
cana-534	169	9	convex	convex	NOUN
cana-534	169	10	functions	function	NOUN
cana-534	169	11	and	and	CCONJ
cana-534	169	12	a	a	DET
cana-534	169	13	corresponding	corresponding	ADJ
cana-534	169	14	class	class	NOUN
cana-534	169	15	of	of	ADP
cana-534	169	16	starlike	starlike	NOUN
cana-534	169	17	functions	function	NOUN
cana-534	169	18	,	,	PUNCT
cana-534	169	19	proc	proc	NOUN
cana-534	169	20	.	.	PUNCT
cana-534	170	1	amer	amer	PROPN
cana-534	170	2	.	.	PUNCT
cana-534	170	3	math	math	PROPN
cana-534	170	4	.	.	PUNCT
cana-534	171	1	soc	soc	PROPN
cana-534	171	2	.	.	PROPN
cana-534	171	3	,	,	PUNCT
cana-534	171	4	118	118	NUM
cana-534	171	5	(	(	PUNCT
cana-534	171	6	1993	1993	NUM
cana-534	171	7	)	)	PUNCT
cana-534	171	8	,	,	PUNCT
cana-534	171	9	189	189	NUM
cana-534	171	10	-	-	SYM
cana-534	171	11	196	196	NUM
cana-534	171	12	.	.	PUNCT
cana-534	172	1	‘	'	PUNCT
cana-534	172	2	[	[	X
cana-534	172	3	8	8	NUM
cana-534	172	4	]	]	X
cana-534	172	5	h.	h.	PROPN
cana-534	172	6	silverman	silverman	PROPN
cana-534	172	7	,	,	PUNCT
cana-534	172	8	univalent	univalent	ADJ
cana-534	172	9	functions	function	NOUN
cana-534	172	10	with	with	ADP
cana-534	172	11	negative	negative	ADJ
cana-534	172	12	coefficients	coefficient	NOUN
cana-534	172	13	,	,	PUNCT
cana-534	172	14	proc	proc	NOUN
cana-534	172	15	.	.	PUNCT
cana-534	173	1	amer	amer	PROPN
cana-534	173	2	.	.	PUNCT
cana-534	173	3	math	math	PROPN
cana-534	173	4	.	.	PUNCT
cana-534	174	1	soc	soc	PROPN
cana-534	174	2	.	.	PROPN
cana-534	174	3	,	,	PUNCT
cana-534	174	4	51	51	NUM
cana-534	174	5	(	(	PUNCT
cana-534	174	6	1975	1975	NUM
cana-534	174	7	)	)	PUNCT
cana-534	174	8	109	109	NUM
cana-534	174	9	–	–	PUNCT
cana-534	174	10	116	116	NUM
cana-534	174	11	.	.	PUNCT
cana-534	175	1	[	[	X
cana-534	175	2	9	9	NUM
cana-534	175	3	]	]	PUNCT
cana-534	175	4	p.	p.	NOUN
cana-534	175	5	thirupathi	thirupathi	PROPN
cana-534	175	6	reddy	reddy	PROPN
cana-534	175	7	and	and	CCONJ
cana-534	175	8	b.	b.	PROPN
cana-534	175	9	venkateswarlu	venkateswarlu	PROPN
cana-534	175	10	,	,	PUNCT
cana-534	175	11	on	on	ADP
cana-534	175	12	a	a	DET
cana-534	175	13	certain	certain	ADJ
cana-534	175	14	subclass	subclass	NOUN
cana-534	175	15	of	of	ADP
cana-534	175	16	uniformly	uniformly	ADJ
cana-534	175	17	convex	convex	NOUN
cana-534	175	18	functions	function	NOUN
cana-534	175	19	defined	define	VERB
cana-534	175	20	by	by	ADP
cana-534	175	21	bessel	bessel	ADJ
cana-534	175	22	functions	function	NOUN
cana-534	175	23	,	,	PUNCT
cana-534	175	24	transylvanian	transylvanian	ADJ
cana-534	175	25	j.	j.	PROPN
cana-534	175	26	of	of	ADP
cana-534	175	27	math	math	NOUN
cana-534	175	28	.	.	PUNCT
cana-534	176	1	and	and	CCONJ
cana-534	176	2	mech	mech	NOUN
cana-534	176	3	.	.	PUNCT
cana-534	176	4	,	,	PUNCT
cana-534	177	1	10(1	10(1	NUM
cana-534	177	2	)	)	PUNCT
cana-534	177	3	(	(	PUNCT
cana-534	177	4	2018	2018	NUM
cana-534	177	5	)	)	PUNCT
cana-534	177	6	,	,	PUNCT
cana-534	177	7	43	43	NUM
cana-534	177	8	-	-	SYM
cana-534	177	9	49	49	NUM
cana-534	177	10	.	.	PUNCT
cana-534	178	1	[	[	X
cana-534	178	2	10	10	NUM
cana-534	178	3	]	]	X
cana-534	178	4	p.	p.	PROPN
cana-534	178	5	thirupathi	thirupathi	PROPN
cana-534	178	6	reddy	reddy	PROPN
cana-534	178	7	and	and	CCONJ
cana-534	178	8	b.	b.	PROPN
cana-534	178	9	venkateswarlu	venkateswarlu	PROPN
cana-534	178	10	,	,	PUNCT
cana-534	178	11	a	a	DET
cana-534	178	12	certain	certain	ADJ
cana-534	178	13	subclass	subclass	NOUN
cana-534	178	14	of	of	ADP
cana-534	178	15	uniformly	uniformly	ADV
cana-534	178	16	convex	convex	NOUN
cana-534	178	17	fucntions	fucntion	NOUN
cana-534	178	18	defined	define	VERB
cana-534	178	19	by	by	ADP
cana-534	178	20	bessel	bessel	ADJ
cana-534	178	21	functions	function	NOUN
cana-534	178	22	,	,	PUNCT
cana-534	178	23	proyecciones	proyeccione	VERB
cana-534	178	24	j.	j.	PROPN
cana-534	178	25	of	of	ADP
cana-534	178	26	math	math	PROPN
cana-534	178	27	.	.	PUNCT
cana-534	179	1	,	,	PUNCT
cana-534	179	2	38	38	NUM
cana-534	179	3	(	(	PUNCT
cana-534	179	4	4	4	NUM
cana-534	179	5	)	)	PUNCT
cana-534	179	6	(	(	PUNCT
cana-534	179	7	2019	2019	NUM
cana-534	179	8	)	)	PUNCT
cana-534	179	9	719731	719731	NUM
cana-534	179	10	.	.	PUNCT
cana-534	180	1	[	[	X
cana-534	180	2	11	11	NUM
cana-534	180	3	]	]	PUNCT
cana-534	180	4	h.	h.	PROPN
cana-534	180	5	s.	s.	PROPN
cana-534	180	6	wilf	wilf	PROPN
cana-534	180	7	,	,	PUNCT
cana-534	180	8	subordinating	subordinating	NOUN
cana-534	180	9	factor	factor	NOUN
cana-534	180	10	sequence	sequence	NOUN
cana-534	180	11	for	for	ADP
cana-534	180	12	convex	convex	NOUN
cana-534	180	13	maps	map	NOUN
cana-534	180	14	of	of	ADP
cana-534	180	15	the	the	DET
cana-534	180	16	unit	unit	NOUN
cana-534	180	17	circle	circle	NOUN
cana-534	180	18	,	,	PUNCT
cana-534	180	19	proc	proc	PROPN
cana-534	180	20	.	.	PUNCT
cana-534	181	1	amer	amer	PROPN
cana-534	181	2	.	.	PUNCT
cana-534	181	3	math	math	PROPN
cana-534	181	4	.	.	PUNCT
cana-534	182	1	soc	soc	PROPN
cana-534	182	2	.	.	PROPN
cana-534	182	3	,	,	PUNCT
cana-534	182	4	12	12	NUM
cana-534	182	5	(	(	PUNCT
cana-534	182	6	1961	1961	NUM
cana-534	182	7	)	)	PUNCT
cana-534	182	8	,	,	PUNCT
cana-534	182	9	689	689	NUM
cana-534	182	10	-	-	SYM
cana-534	182	11	693	693	NUM
cana-534	182	12	.	.	PUNCT
cana-534	183	1	[	[	X
cana-534	183	2	12	12	NUM
cana-534	183	3	]	]	PUNCT
cana-534	183	4	j.	j.	PROPN
cana-534	183	5	a.	a.	PROPN
cana-534	183	6	baker	baker	PROPN
cana-534	183	7	,	,	PUNCT
cana-534	183	8	j.	j.	PROPN
cana-534	183	9	lawrence	lawrence	PROPN
cana-534	183	10	,	,	PUNCT
cana-534	183	11	and	and	CCONJ
cana-534	183	12	f.	f.	PROPN
cana-534	183	13	zorzitto	zorzitto	PROPN
cana-534	183	14	,	,	PUNCT
cana-534	183	15	the	the	DET
cana-534	183	16	stability	stability	NOUN
cana-534	183	17	of	of	ADP
cana-534	183	18	the	the	DET
cana-534	183	19	equation	equation	NOUN
cana-534	183	20	f(x	f(x	PROPN
cana-534	183	21	+	+	CCONJ
cana-534	183	22	y	y	X
cana-534	183	23	)	)	PUNCT
cana-534	183	24	=	=	SYM
cana-534	183	25	f(x)f(y	f(x)f(y	NOUN
cana-534	183	26	)	)	PUNCT
cana-534	183	27	,	,	PUNCT
cana-534	183	28	proc	proc	PROPN
cana-534	183	29	.	.	PUNCT
cana-534	184	1	amer	amer	PROPN
cana-534	184	2	.	.	PUNCT
cana-534	184	3	math	math	PROPN
cana-534	184	4	.	.	PUNCT
cana-534	185	1	soc	soc	PROPN
cana-534	185	2	.	.	PUNCT
cana-534	186	1	74(1979	74(1979	NUM
cana-534	186	2	)	)	PUNCT
cana-534	186	3	,	,	PUNCT
cana-534	186	4	242	242	NUM
cana-534	186	5	-	-	SYM
cana-534	186	6	246	246	NUM
cana-534	186	7	.	.	PUNCT
cana-534	187	1	[	[	X
cana-534	187	2	13	13	NUM
cana-534	187	3	]	]	X
cana-534	187	4	d.	d.	PROPN
cana-534	187	5	h.	h.	PROPN
cana-534	187	6	hyers	hyers	PROPN
cana-534	187	7	,	,	PUNCT
cana-534	187	8	on	on	ADP
cana-534	187	9	the	the	DET
cana-534	187	10	stability	stability	NOUN
cana-534	187	11	of	of	ADP
cana-534	187	12	the	the	DET
cana-534	187	13	linear	linear	ADJ
cana-534	187	14	functional	functional	ADJ
cana-534	187	15	equation	equation	NOUN
cana-534	187	16	,	,	PUNCT
cana-534	187	17	proc	proc	NOUN
cana-534	187	18	.	.	PUNCT
cana-534	188	1	nat	nat	PROPN
cana-534	188	2	.	.	PUNCT
cana-534	189	1	acad	acad	PROPN
cana-534	189	2	.	.	PUNCT
cana-534	190	1	sci	sci	PROPN
cana-534	190	2	.	.	PUNCT
cana-534	190	3	u.s.a	u.s.a	PROPN
cana-534	190	4	.	.	PUNCT
cana-534	190	5	27(1941	27(1941	NUM
cana-534	190	6	)	)	PUNCT
cana-534	190	7	,	,	PUNCT
cana-534	190	8	222224	222224	NUM
cana-534	190	9	.	.	PUNCT
cana-534	191	1	[	[	X
cana-534	191	2	14	14	NUM
cana-534	191	3	]	]	X
cana-534	191	4	l.	l.	PROPN
cana-534	191	5	szekelyhidi	szekelyhidi	PROPN
cana-534	191	6	,	,	PUNCT
cana-534	191	7	on	on	ADP
cana-534	191	8	a	a	DET
cana-534	191	9	theorem	theorem	NOUN
cana-534	191	10	of	of	ADP
cana-534	191	11	baker	baker	PROPN
cana-534	191	12	,	,	PUNCT
cana-534	191	13	lawrence	lawrence	NOUN
cana-534	191	14	and	and	CCONJ
cana-534	191	15	zorzitto	zorzitto	NOUN
cana-534	191	16	,	,	PUNCT
cana-534	191	17	proc	proc	NOUN
cana-534	191	18	.	.	PUNCT
cana-534	192	1	amer	amer	PROPN
cana-534	192	2	.	.	PUNCT
cana-534	192	3	math	math	PROPN
cana-534	192	4	.	.	PUNCT
cana-534	193	1	soc	soc	PROPN
cana-534	193	2	.	.	PUNCT
cana-534	194	1	84(1982	84(1982	NUM
cana-534	194	2	)	)	PUNCT
cana-534	194	3	,	,	PUNCT
cana-534	194	4	95	95	NUM
cana-534	194	5	-	-	SYM
cana-534	194	6	96	96	NUM
cana-534	194	7	.	.	PUNCT
cana-534	195	1	[	[	X
cana-534	195	2	15	15	NUM
cana-534	195	3	]	]	X
cana-534	195	4	s.	s.	PROPN
cana-534	195	5	m.	m.	PROPN
cana-534	195	6	ulam	ulam	PROPN
cana-534	195	7	,	,	PUNCT
cana-534	195	8	problems	problem	NOUN
cana-534	195	9	in	in	ADP
cana-534	195	10	modern	modern	ADJ
cana-534	195	11	mathematics	mathematic	NOUN
cana-534	195	12	,	,	PUNCT
cana-534	195	13	science	science	NOUN
cana-534	195	14	editions	edition	NOUN
cana-534	195	15	,	,	PUNCT
cana-534	195	16	wiley	wiley	NOUN
cana-534	195	17	,	,	PUNCT
cana-534	195	18	new	new	PROPN
cana-534	195	19	york	york	PROPN
cana-534	195	20	,	,	PUNCT
cana-534	195	21	1960	1960	NUM
cana-534	195	22	.	.	PUNCT