id	sid	tid	token	lemma	pos
cana-510	1	1	communications	communication	NOUN
cana-510	1	2	on	on	ADP
cana-510	1	3	applied	apply	VERB
cana-510	1	4	nonlinear	nonlinear	ADJ
cana-510	1	5	analysis	analysis	NOUN
cana-510	1	6	issn	issn	NOUN
cana-510	1	7	:	:	PUNCT
cana-510	1	8	1074	1074	NUM
cana-510	1	9	-	-	PUNCT
cana-510	1	10	133x	133x	NUM
cana-510	1	11	vol	vol	NOUN
cana-510	1	12	31	31	NUM
cana-510	1	13	no	no	NOUN
cana-510	1	14	.	.	NOUN
cana-510	1	15	2	2	NUM
cana-510	1	16	(	(	PUNCT
cana-510	1	17	2024	2024	NUM
cana-510	1	18	)	)	PUNCT
cana-510	1	19	22	22	NUM
cana-510	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	1	21	some	some	DET
cana-510	1	22	applications	application	NOUN
cana-510	1	23	of	of	ADP
cana-510	1	24	fractional	fractional	ADJ
cana-510	1	25	derivative	derivative	ADJ
cana-510	1	26	operator	operator	NOUN
cana-510	1	27	b.d	b.d	PROPN
cana-510	1	28	.	.	PROPN
cana-510	1	29	karande1	karande1	PROPN
cana-510	1	30	,	,	PUNCT
cana-510	1	31	arun	arun	PROPN
cana-510	1	32	b.	b.	PROPN
cana-510	1	33	damkondwar2	damkondwar2	PROPN
cana-510	2	1	1department	1department	NUM
cana-510	2	2	of	of	ADP
cana-510	2	3	mathematics	mathematic	NOUN
cana-510	2	4	,	,	PUNCT
cana-510	2	5	udaygiri	udaygiri	PROPN
cana-510	2	6	mahavidyalaya	mahavidyalaya	PROPN
cana-510	2	7	,	,	PUNCT
cana-510	2	8	udgir	udgir	PROPN
cana-510	2	9	,	,	PUNCT
cana-510	2	10	dis	dis	PROPN
cana-510	2	11	:	:	PUNCT
cana-510	2	12	latur	latur	PROPN
cana-510	2	13	,	,	PUNCT
cana-510	2	14	m.s	m.s	PROPN
cana-510	2	15	,	,	PUNCT
cana-510	2	16	india	india	PROPN
cana-510	2	17	.	.	PUNCT
cana-510	3	1	e	e	X
cana-510	4	1	-	-	NOUN
cana-510	4	2	mail	mail	NOUN
cana-510	4	3	:	:	PUNCT
cana-510	4	4	bdkarande2@gmail.com	bdkarande2@gmail.com	PROPN
cana-510	4	5	2	2	NUM
cana-510	4	6	department	department	NOUN
cana-510	4	7	of	of	ADP
cana-510	4	8	mathematics	mathematic	NOUN
cana-510	4	9	,	,	PUNCT
cana-510	4	10	government	government	NOUN
cana-510	4	11	polytechnic	polytechnic	ADJ
cana-510	4	12	,	,	PUNCT
cana-510	4	13	hingoli	hingoli	NOUN
cana-510	4	14	431	431	NUM
cana-510	4	15	513	513	NUM
cana-510	4	16	,	,	PUNCT
cana-510	4	17	maharashtra	maharashtra	PROPN
cana-510	4	18	,	,	PUNCT
cana-510	4	19	india	india	PROPN
cana-510	4	20	.	.	PUNCT
cana-510	5	1	e	e	X
cana-510	5	2	-	-	NOUN
cana-510	5	3	mail	mail	NOUN
cana-510	5	4	:	:	PUNCT
cana-510	6	1	arundigras@gmail.com	arundigras@gmail.com	X
cana-510	6	2	coresponding	coresponde	VERB
cana-510	6	3	author	author	NOUN
cana-510	6	4	:	:	PUNCT
cana-510	6	5	arun	arun	PROPN
cana-510	6	6	b.	b.	PROPN
cana-510	6	7	damkondwar	damkondwar	PROPN
cana-510	6	8	article	article	NOUN
cana-510	6	9	history	history	NOUN
cana-510	6	10	:	:	PUNCT
cana-510	6	11	received	receive	VERB
cana-510	6	12	:	:	PUNCT
cana-510	6	13	18	18	NUM
cana-510	6	14	-	-	SYM
cana-510	6	15	01	01	NUM
cana-510	6	16	-	-	PUNCT
cana-510	6	17	2024	2024	NUM
cana-510	6	18	revised	revise	VERB
cana-510	6	19	:	:	PUNCT
cana-510	6	20	30	30	NUM
cana-510	6	21	-	-	SYM
cana-510	6	22	03	03	NUM
cana-510	6	23	-	-	PUNCT
cana-510	6	24	2024	2024	NUM
cana-510	6	25	accepted	accept	VERB
cana-510	6	26	:	:	PUNCT
cana-510	6	27	22	22	NUM
cana-510	6	28	-	-	PUNCT
cana-510	6	29	04	04	NUM
cana-510	6	30	-	-	PUNCT
cana-510	6	31	2024	2024	NUM
cana-510	6	32	abstract	abstract	NOUN
cana-510	6	33	:	:	PUNCT
cana-510	6	34	introduction	introduction	NOUN
cana-510	6	35	:	:	PUNCT
cana-510	6	36	the	the	DET
cana-510	6	37	aim	aim	NOUN
cana-510	6	38	of	of	ADP
cana-510	6	39	this	this	DET
cana-510	6	40	paper	paper	NOUN
cana-510	6	41	is	be	AUX
cana-510	6	42	to	to	PART
cana-510	6	43	introduce	introduce	VERB
cana-510	6	44	a	a	DET
cana-510	6	45	new	new	ADJ
cana-510	6	46	subclass	subclass	NOUN
cana-510	6	47	ts(ω	ts(ω	NUM
cana-510	6	48	,	,	PUNCT
cana-510	6	49	σ	σ	PROPN
cana-510	6	50	,	,	PUNCT
cana-510	6	51	ς	ς	PROPN
cana-510	6	52	,	,	PUNCT
cana-510	6	53	ϑ	ϑ	NOUN
cana-510	6	54	)	)	PUNCT
cana-510	6	55	of	of	ADP
cana-510	6	56	univalent	univalent	ADJ
cana-510	6	57	functions	function	NOUN
cana-510	6	58	with	with	ADP
cana-510	6	59	negative	negative	ADJ
cana-510	6	60	coefficients	coefficient	NOUN
cana-510	6	61	related	relate	VERB
cana-510	6	62	to	to	ADP
cana-510	6	63	fractional	fractional	ADJ
cana-510	6	64	derivative	derivative	ADJ
cana-510	6	65	operator	operator	NOUN
cana-510	6	66	in	in	ADP
cana-510	6	67	the	the	DET
cana-510	6	68	unit	unit	NOUN
cana-510	6	69	disk	disk	NOUN
cana-510	6	70	𝕌	𝕌	PROPN
cana-510	6	71	=	=	SYM
cana-510	6	72	{	{	PUNCT
cana-510	6	73	z	z	NOUN
cana-510	6	74	∈	∈	PROPN
cana-510	6	75	ℂ	ℂ	PROPN
cana-510	6	76	:	:	PUNCT
cana-510	6	77	|z|	|z|	VERB
cana-510	6	78	<	<	X
cana-510	6	79	1	1	NUM
cana-510	6	80	}	}	PUNCT
cana-510	6	81	.	.	PUNCT
cana-510	7	1	we	we	PRON
cana-510	7	2	obtain	obtain	VERB
cana-510	7	3	basic	basic	ADJ
cana-510	7	4	properties	property	NOUN
cana-510	7	5	like	like	ADP
cana-510	7	6	coefficient	coefficient	NOUN
cana-510	7	7	inequality	inequality	NOUN
cana-510	7	8	,	,	PUNCT
cana-510	7	9	distortion	distortion	NOUN
cana-510	7	10	and	and	CCONJ
cana-510	7	11	covering	covering	NOUN
cana-510	7	12	theorem	theorem	VERB
cana-510	7	13	,	,	PUNCT
cana-510	7	14	radii	radius	NOUN
cana-510	7	15	of	of	ADP
cana-510	7	16	starlikeness	starlikeness	NOUN
cana-510	7	17	,	,	PUNCT
cana-510	7	18	convexity	convexity	NOUN
cana-510	7	19	and	and	CCONJ
cana-510	7	20	close	close	NOUN
cana-510	7	21	-	-	PUNCT
cana-510	7	22	to	to	ADP
cana-510	7	23	-	-	PUNCT
cana-510	7	24	convexity	convexity	NOUN
cana-510	7	25	,	,	PUNCT
cana-510	7	26	extreme	extreme	ADJ
cana-510	7	27	points	point	NOUN
cana-510	7	28	,	,	PUNCT
cana-510	7	29	hadamard	hadamard	ADJ
cana-510	7	30	product	product	NOUN
cana-510	7	31	,	,	PUNCT
cana-510	7	32	and	and	CCONJ
cana-510	7	33	closure	closure	NOUN
cana-510	7	34	theorems	theorem	NOUN
cana-510	7	35	for	for	ADP
cana-510	7	36	functions	function	NOUN
cana-510	7	37	belonging	belong	VERB
cana-510	7	38	to	to	ADP
cana-510	7	39	our	our	PRON
cana-510	7	40	class	class	NOUN
cana-510	7	41	keywords	keyword	NOUN
cana-510	7	42	:	:	PUNCT
cana-510	7	43	univalent	univalent	ADJ
cana-510	7	44	,	,	PUNCT
cana-510	7	45	derivative	derivative	ADJ
cana-510	7	46	opertator	opertator	NOUN
cana-510	7	47	,	,	PUNCT
cana-510	7	48	starlike	starlike	NOUN
cana-510	7	49	,	,	PUNCT
cana-510	7	50	extreme	extreme	ADJ
cana-510	7	51	points	point	NOUN
cana-510	7	52	,	,	PUNCT
cana-510	7	53	hadamard	hadamard	ADJ
cana-510	7	54	product	product	NOUN
cana-510	7	55	.	.	PUNCT
cana-510	8	1	1	1	X
cana-510	8	2	.	.	X
cana-510	8	3	introduction	introduction	NOUN
cana-510	8	4	let	let	VERB
cana-510	8	5	𝐴	𝐴	PROPN
cana-510	8	6	signify	signify	VERB
cana-510	8	7	the	the	DET
cana-510	8	8	class	class	NOUN
cana-510	8	9	of	of	ADP
cana-510	8	10	all	all	DET
cana-510	8	11	functions	function	NOUN
cana-510	8	12	𝑢(𝑧	𝑢(𝑧	NOUN
cana-510	8	13	)	)	PUNCT
cana-510	8	14	of	of	ADP
cana-510	8	15	the	the	DET
cana-510	8	16	type	type	NOUN
cana-510	8	17	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	8	18	)	)	PUNCT
cana-510	8	19	=	=	PUNCT
cana-510	9	1	𝑧	𝑧	PROPN
cana-510	10	1	+	+	CCONJ
cana-510	10	2	∑	∑	PROPN
cana-510	10	3	𝑎𝑛	𝑎𝑛	PROPN
cana-510	10	4	∞	∞	NUM
cana-510	10	5	𝑛=2	𝑛=2	NOUN
cana-510	10	6	𝑧𝑛	𝑧𝑛	INTJ
cana-510	10	7	(	(	PUNCT
cana-510	10	8	1.1	1.1	NUM
cana-510	10	9	)	)	PUNCT
cana-510	10	10	in	in	ADP
cana-510	10	11	the	the	DET
cana-510	10	12	open	open	ADJ
cana-510	10	13	unit	unit	NOUN
cana-510	10	14	disc	disc	VERB
cana-510	10	15	𝕌	𝕌	PROPN
cana-510	10	16	=	=	SYM
cana-510	10	17	{	{	PUNCT
cana-510	10	18	𝑧	𝑧	NOUN
cana-510	10	19	∈	∈	PROPN
cana-510	10	20	ℂ	ℂ	PROPN
cana-510	10	21	:	:	PUNCT
cana-510	10	22	|𝑧|	|𝑧|	NOUN
cana-510	10	23	<	<	X
cana-510	10	24	1	1	NUM
cana-510	10	25	}	}	PUNCT
cana-510	10	26	.	.	PUNCT
cana-510	11	1	let	let	VERB
cana-510	11	2	𝑆	𝑆	PROPN
cana-510	11	3	be	be	AUX
cana-510	11	4	the	the	DET
cana-510	11	5	subclass	subclass	NOUN
cana-510	11	6	of	of	ADP
cana-510	11	7	𝐴	𝐴	PROPN
cana-510	11	8	consisting	consist	VERB
cana-510	11	9	of	of	ADP
cana-510	11	10	univalent	univalent	ADJ
cana-510	11	11	functions	function	NOUN
cana-510	11	12	and	and	CCONJ
cana-510	11	13	satisfy	satisfy	VERB
cana-510	11	14	the	the	DET
cana-510	11	15	following	follow	VERB
cana-510	11	16	usual	usual	ADJ
cana-510	11	17	normalization	normalization	NOUN
cana-510	11	18	condition	condition	NOUN
cana-510	11	19	𝑢(0	𝑢(0	PROPN
cana-510	11	20	)	)	PUNCT
cana-510	11	21	=	=	PUNCT
cana-510	12	1	𝑢′(0	𝑢′(0	NOUN
cana-510	12	2	)	)	PUNCT
cana-510	12	3	−	−	PROPN
cana-510	12	4	1	1	NUM
cana-510	12	5	=	=	SYM
cana-510	12	6	0	0	NUM
cana-510	12	7	.	.	PUNCT
cana-510	13	1	we	we	PRON
cana-510	13	2	denote	denote	VERB
cana-510	13	3	by	by	ADP
cana-510	13	4	𝑆	𝑆	PROPN
cana-510	13	5	the	the	DET
cana-510	13	6	subclass	subclass	NOUN
cana-510	13	7	of	of	ADP
cana-510	13	8	𝐴	𝐴	PROPN
cana-510	13	9	consisting	consist	VERB
cana-510	13	10	of	of	ADP
cana-510	13	11	functions	function	NOUN
cana-510	13	12	𝑢(𝑧	𝑢(𝑧	NOUN
cana-510	13	13	)	)	PUNCT
cana-510	13	14	which	which	PRON
cana-510	13	15	are	be	AUX
cana-510	13	16	all	all	ADV
cana-510	13	17	univalent	univalent	ADJ
cana-510	13	18	in	in	ADP
cana-510	13	19	𝕌.	𝕌.	PROPN
cana-510	13	20	a	a	DET
cana-510	13	21	function	function	NOUN
cana-510	13	22	𝑢	𝑢	PROPN
cana-510	13	23	∈	∈	PROPN
cana-510	13	24	𝐴	𝐴	PROPN
cana-510	13	25	is	be	AUX
cana-510	13	26	a	a	DET
cana-510	13	27	starlike	starlike	NOUN
cana-510	13	28	function	function	NOUN
cana-510	13	29	of	of	ADP
cana-510	13	30	the	the	DET
cana-510	13	31	order	order	NOUN
cana-510	13	32	𝑚	𝑚	NOUN
cana-510	13	33	,	,	PUNCT
cana-510	13	34	0	0	NUM
cana-510	13	35	≤	≤	NOUN
cana-510	13	36	𝑚	𝑚	X
cana-510	13	37	<	<	X
cana-510	13	38	1	1	NUM
cana-510	13	39	,	,	PUNCT
cana-510	13	40	if	if	SCONJ
cana-510	13	41	it	it	PRON
cana-510	13	42	satisfy	satisfy	VERB
cana-510	13	43	ℜ	ℜ	PROPN
cana-510	13	44	{	{	PUNCT
cana-510	13	45	𝑧𝑢′(𝑧	𝑧𝑢′(𝑧	NOUN
cana-510	13	46	)	)	PUNCT
cana-510	13	47	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	13	48	)	)	PUNCT
cana-510	13	49	}	}	PUNCT
cana-510	13	50	>	>	PUNCT
cana-510	13	51	𝑚	𝑚	X
cana-510	13	52	,	,	PUNCT
cana-510	13	53	𝑧	𝑧	DET
cana-510	13	54	∈	∈	PROPN
cana-510	13	55	𝕌.	𝕌.	PROPN
cana-510	13	56	(	(	PUNCT
cana-510	13	57	1.2	1.2	NUM
cana-510	13	58	)	)	PUNCT
cana-510	13	59	we	we	PRON
cana-510	13	60	denote	denote	VERB
cana-510	13	61	this	this	DET
cana-510	13	62	class	class	NOUN
cana-510	13	63	with	with	ADP
cana-510	13	64	𝑆∗(𝑚	𝑆∗(𝑚	PROPN
cana-510	13	65	)	)	PUNCT
cana-510	13	66	.	.	PUNCT
cana-510	14	1	a	a	DET
cana-510	14	2	function	function	NOUN
cana-510	14	3	𝑢	𝑢	PROPN
cana-510	14	4	∈	∈	PROPN
cana-510	14	5	𝐴	𝐴	PROPN
cana-510	14	6	is	be	AUX
cana-510	14	7	a	a	DET
cana-510	14	8	convex	convex	ADJ
cana-510	14	9	function	function	NOUN
cana-510	14	10	of	of	ADP
cana-510	14	11	the	the	DET
cana-510	14	12	order	order	NOUN
cana-510	14	13	𝑚	𝑚	NOUN
cana-510	14	14	,	,	PUNCT
cana-510	14	15	0	0	NUM
cana-510	14	16	≤	≤	NOUN
cana-510	14	17	𝑚	𝑚	X
cana-510	14	18	<	<	X
cana-510	14	19	1	1	NUM
cana-510	14	20	,	,	PUNCT
cana-510	14	21	if	if	SCONJ
cana-510	14	22	it	it	PRON
cana-510	14	23	fulfil	fulfil	VERB
cana-510	14	24	ℜ	ℜ	PROPN
cana-510	14	25	{	{	PUNCT
cana-510	14	26	1	1	NUM
cana-510	14	27	+	+	CCONJ
cana-510	14	28	𝑧𝑢″(𝑧	𝑧𝑢″(𝑧	NUM
cana-510	14	29	)	)	PUNCT
cana-510	14	30	𝑢′(𝑧	𝑢′(𝑧	X
cana-510	14	31	)	)	PUNCT
cana-510	14	32	}	}	PUNCT
cana-510	14	33	>	>	PUNCT
cana-510	15	1	𝑚	𝑚	X
cana-510	15	2	,	,	PUNCT
cana-510	15	3	𝑧	𝑧	DET
cana-510	15	4	∈	∈	PROPN
cana-510	15	5	𝕌.	𝕌.	PROPN
cana-510	15	6	(	(	PUNCT
cana-510	15	7	1.3	1.3	NUM
cana-510	15	8	)	)	PUNCT
cana-510	15	9	we	we	PRON
cana-510	15	10	denote	denote	VERB
cana-510	15	11	this	this	DET
cana-510	15	12	class	class	NOUN
cana-510	15	13	with	with	ADP
cana-510	15	14	𝐾(𝑚	𝐾(𝑚	NOUN
cana-510	15	15	)	)	PUNCT
cana-510	15	16	.	.	PUNCT
cana-510	16	1	note	note	VERB
cana-510	16	2	that	that	SCONJ
cana-510	16	3	𝑆∗(0	𝑆∗(0	NOUN
cana-510	16	4	)	)	PUNCT
cana-510	17	1	=	=	SYM
cana-510	17	2	𝑆∗	𝑆∗	PROPN
cana-510	17	3	and	and	CCONJ
cana-510	17	4	𝐾(0	𝐾(0	NUM
cana-510	17	5	)	)	PUNCT
cana-510	18	1	=	=	SYM
cana-510	18	2	𝐾	𝐾	PROPN
cana-510	18	3	are	be	AUX
cana-510	18	4	the	the	DET
cana-510	18	5	usual	usual	ADJ
cana-510	18	6	classes	class	NOUN
cana-510	18	7	of	of	ADP
cana-510	18	8	starlike	starlike	NOUN
cana-510	18	9	and	and	CCONJ
cana-510	18	10	convex	convex	NOUN
cana-510	18	11	functions	function	NOUN
cana-510	18	12	in	in	ADP
cana-510	18	13	𝕌	𝕌	PROPN
cana-510	18	14	respectively	respectively	ADV
cana-510	18	15	.	.	PUNCT
cana-510	19	1	let	let	VERB
cana-510	19	2	𝑇	𝑇	PROPN
cana-510	19	3	denote	denote	VERB
cana-510	19	4	the	the	DET
cana-510	19	5	class	class	NOUN
cana-510	19	6	of	of	ADP
cana-510	19	7	functions	function	NOUN
cana-510	19	8	analytic	analytic	ADJ
cana-510	19	9	in	in	ADP
cana-510	19	10	𝕌	𝕌	PROPN
cana-510	19	11	that	that	PRON
cana-510	19	12	are	be	AUX
cana-510	19	13	of	of	ADP
cana-510	19	14	the	the	DET
cana-510	19	15	form	form	NOUN
cana-510	19	16	communications	communication	NOUN
cana-510	19	17	on	on	ADP
cana-510	19	18	applied	apply	VERB
cana-510	19	19	nonlinear	nonlinear	ADJ
cana-510	19	20	analysis	analysis	NOUN
cana-510	19	21	issn	issn	NOUN
cana-510	19	22	:	:	PUNCT
cana-510	19	23	1074	1074	NUM
cana-510	19	24	-	-	PUNCT
cana-510	19	25	133x	133x	NUM
cana-510	19	26	vol	vol	NOUN
cana-510	19	27	31	31	NUM
cana-510	19	28	no	no	NOUN
cana-510	19	29	.	.	NOUN
cana-510	19	30	2	2	NUM
cana-510	19	31	(	(	PUNCT
cana-510	19	32	2024	2024	NUM
cana-510	19	33	)	)	PUNCT
cana-510	19	34	23	23	NUM
cana-510	19	35	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	19	36	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	19	37	)	)	PUNCT
cana-510	20	1	=	=	PUNCT
cana-510	20	2	𝑧	𝑧	PRON
cana-510	20	3	−	−	NOUN
cana-510	20	4	∑	∑	PROPN
cana-510	20	5	𝑎𝑛	𝑎𝑛	PROPN
cana-510	20	6	∞	∞	PROPN
cana-510	20	7	𝑛=2	𝑛=2	PROPN
cana-510	20	8	𝑧𝑛	𝑧𝑛	NOUN
cana-510	20	9	,	,	PUNCT
cana-510	20	10	𝑎𝑛	𝑎𝑛	PROPN
cana-510	20	11	≥	≥	NOUN
cana-510	20	12	0	0	PUNCT
cana-510	20	13	𝑧	𝑧	PRON
cana-510	20	14	∈	∈	PROPN
cana-510	20	15	𝕌	𝕌	PROPN
cana-510	20	16	(	(	PUNCT
cana-510	20	17	1.4	1.4	NUM
cana-510	20	18	)	)	PUNCT
cana-510	20	19	and	and	CCONJ
cana-510	20	20	let	let	VERB
cana-510	20	21	𝑇∗(𝑚	𝑇∗(𝑚	VERB
cana-510	20	22	)	)	PUNCT
cana-510	21	1	=	=	SYM
cana-510	21	2	𝑇	𝑇	PROPN
cana-510	21	3	∩	∩	NOUN
cana-510	21	4	𝑆∗(𝑚	𝑆∗(𝑚	NOUN
cana-510	21	5	)	)	PUNCT
cana-510	21	6	,	,	PUNCT
cana-510	21	7	𝐶(𝑚	𝐶(𝑚	ADP
cana-510	21	8	)	)	PUNCT
cana-510	21	9	=	=	SYM
cana-510	21	10	𝑇	𝑇	PROPN
cana-510	21	11	∩	∩	NOUN
cana-510	21	12	𝐾(𝑚	𝐾(𝑚	NOUN
cana-510	21	13	)	)	PUNCT
cana-510	21	14	.	.	PUNCT
cana-510	22	1	the	the	DET
cana-510	22	2	class	class	NOUN
cana-510	22	3	𝑇∗(𝑚	𝑇∗(𝑚	NOUN
cana-510	22	4	)	)	PUNCT
cana-510	22	5	and	and	CCONJ
cana-510	22	6	allied	allied	ADJ
cana-510	22	7	classes	class	NOUN
cana-510	22	8	possess	possess	VERB
cana-510	22	9	some	some	DET
cana-510	22	10	interesting	interesting	ADJ
cana-510	22	11	properties	property	NOUN
cana-510	22	12	and	and	CCONJ
cana-510	22	13	have	have	AUX
cana-510	22	14	been	be	AUX
cana-510	22	15	extensively	extensively	ADV
cana-510	22	16	studied	study	VERB
cana-510	22	17	by	by	ADP
cana-510	22	18	silverman	silverman	NOUN
cana-510	22	19	[	[	X
cana-510	22	20	14	14	NUM
cana-510	22	21	]	]	PUNCT
cana-510	22	22	.	.	PUNCT
cana-510	23	1	many	many	ADJ
cana-510	23	2	basically	basically	ADV
cana-510	23	3	equivalent	equivalent	ADJ
cana-510	23	4	definitions	definition	NOUN
cana-510	23	5	of	of	ADP
cana-510	23	6	fractional	fractional	ADJ
cana-510	23	7	computation	computation	NOUN
cana-510	23	8	have	have	AUX
cana-510	23	9	been	be	AUX
cana-510	23	10	given	give	VERB
cana-510	23	11	in	in	ADP
cana-510	23	12	literature	literature	NOUN
cana-510	23	13	(	(	PUNCT
cana-510	23	14	(	(	PUNCT
cana-510	23	15	cf.)e.g	cf.)e.g	NUM
cana-510	23	16	.	.	PUNCT
cana-510	23	17	,[13	,[13	PUNCT
cana-510	23	18	]	]	PUNCT
cana-510	24	1	and	and	CCONJ
cana-510	24	2	(	(	PUNCT
cana-510	24	3	[	[	X
cana-510	24	4	15	15	NUM
cana-510	24	5	]	]	PUNCT
cana-510	24	6	,	,	PUNCT
cana-510	24	7	p.	p.	NOUN
cana-510	24	8	45	45	NUM
cana-510	24	9	)	)	PUNCT
cana-510	24	10	)	)	PUNCT
cana-510	24	11	.	.	PUNCT
cana-510	25	1	we	we	PRON
cana-510	25	2	state	state	VERB
cana-510	25	3	the	the	DET
cana-510	25	4	following	follow	VERB
cana-510	25	5	definitions	definition	NOUN
cana-510	25	6	due	due	ADP
cana-510	25	7	to	to	ADP
cana-510	25	8	owa	owa	PROPN
cana-510	25	9	and	and	CCONJ
cana-510	25	10	srivastava	srivastava	PROPN
cana-510	25	11	[	[	X
cana-510	25	12	9	9	NUM
cana-510	25	13	]	]	PUNCT
cana-510	25	14	which	which	PRON
cana-510	25	15	have	have	AUX
cana-510	25	16	been	be	AUX
cana-510	25	17	used	use	VERB
cana-510	25	18	rather	rather	ADV
cana-510	25	19	frequently	frequently	ADV
cana-510	25	20	in	in	ADP
cana-510	25	21	the	the	DET
cana-510	25	22	theory	theory	NOUN
cana-510	25	23	of	of	ADP
cana-510	25	24	analytic	analytic	ADJ
cana-510	25	25	functions	function	NOUN
cana-510	25	26	(	(	PUNCT
cana-510	25	27	see	see	VERB
cana-510	25	28	also	also	ADV
cana-510	25	29	[	[	X
cana-510	25	30	4	4	NUM
cana-510	25	31	]	]	PUNCT
cana-510	25	32	)	)	PUNCT
cana-510	25	33	.	.	PUNCT
cana-510	26	1	definition	definition	NOUN
cana-510	26	2	1.1	1.1	NUM
cana-510	26	3	.	.	PUNCT
cana-510	27	1	the	the	DET
cana-510	27	2	fractional	fractional	ADJ
cana-510	27	3	integral	integral	NOUN
cana-510	27	4	of	of	ADP
cana-510	27	5	order	order	NOUN
cana-510	27	6	ϑ	ϑ	NOUN
cana-510	27	7	is	be	AUX
cana-510	27	8	defined	define	VERB
cana-510	27	9	,	,	PUNCT
cana-510	27	10	for	for	ADP
cana-510	27	11	a	a	DET
cana-510	27	12	function	function	NOUN
cana-510	27	13	u(z	u(z	NOUN
cana-510	27	14	)	)	PUNCT
cana-510	27	15	,	,	PUNCT
cana-510	27	16	by	by	ADP
cana-510	27	17	dz	dz	ADJ
cana-510	27	18	−ϑu(z	−ϑu(z	NOUN
cana-510	27	19	)	)	PUNCT
cana-510	27	20	=	=	SYM
cana-510	27	21	1	1	NUM
cana-510	27	22	ω(ϑ	ω(ϑ	NUM
cana-510	27	23	)	)	PUNCT
cana-510	27	24	∫	∫	PROPN
cana-510	27	25	f(ζ	f(ζ	PROPN
cana-510	27	26	)	)	PUNCT
cana-510	28	1	(	(	PUNCT
cana-510	28	2	z−ζ)1−ϑ	z−ζ)1−ϑ	NOUN
cana-510	28	3	z	z	NOUN
cana-510	28	4	0	0	NUM
cana-510	28	5	dζ	dζ	PROPN
cana-510	28	6	,	,	PUNCT
cana-510	28	7	(	(	PUNCT
cana-510	28	8	ϑ	ϑ	X
cana-510	28	9	>	>	X
cana-510	28	10	0	0	NUM
cana-510	28	11	)	)	PUNCT
cana-510	28	12	(	(	PUNCT
cana-510	28	13	1.5	1.5	NUM
cana-510	28	14	)	)	PUNCT
cana-510	28	15	and	and	CCONJ
cana-510	28	16	the	the	DET
cana-510	28	17	fractional	fractional	ADJ
cana-510	28	18	derivative	derivative	NOUN
cana-510	28	19	of	of	ADP
cana-510	28	20	order	order	NOUN
cana-510	28	21	ς	ς	PROPN
cana-510	28	22	is	be	AUX
cana-510	28	23	defined	define	VERB
cana-510	28	24	,	,	PUNCT
cana-510	28	25	for	for	ADP
cana-510	28	26	a	a	DET
cana-510	28	27	function	function	NOUN
cana-510	28	28	u(z	u(z	NOUN
cana-510	28	29	)	)	PUNCT
cana-510	28	30	,	,	PUNCT
cana-510	28	31	by	by	ADP
cana-510	28	32	dz	dz	NOUN
cana-510	28	33	ϑu(z	ϑu(z	NOUN
cana-510	28	34	)	)	PUNCT
cana-510	28	35	=	=	SYM
cana-510	28	36	1	1	NUM
cana-510	28	37	ω(1−ϑ	ω(1−ϑ	NOUN
cana-510	28	38	)	)	PUNCT
cana-510	28	39	d	d	X
cana-510	28	40	dz	dz	PROPN
cana-510	28	41	∫	∫	PROPN
cana-510	28	42	f(ζ	f(ζ	PROPN
cana-510	28	43	)	)	PUNCT
cana-510	28	44	(	(	PUNCT
cana-510	29	1	z−ζ)ϑ	z−ζ)ϑ	NOUN
cana-510	29	2	z	z	NOUN
cana-510	29	3	0	0	NUM
cana-510	29	4	dζ	dζ	PROPN
cana-510	29	5	,	,	PUNCT
cana-510	29	6	(	(	PUNCT
cana-510	29	7	0	0	NUM
cana-510	29	8	≤	≤	NOUN
cana-510	29	9	ϑ	ϑ	X
cana-510	29	10	<	<	X
cana-510	29	11	1	1	NUM
cana-510	29	12	)	)	PUNCT
cana-510	29	13	(	(	PUNCT
cana-510	29	14	1.6	1.6	NUM
cana-510	29	15	)	)	PUNCT
cana-510	29	16	where	where	SCONJ
cana-510	29	17	u(z	u(z	NOUN
cana-510	29	18	)	)	PUNCT
cana-510	29	19	is	be	AUX
cana-510	29	20	an	an	DET
cana-510	29	21	analytic	analytic	ADJ
cana-510	29	22	function	function	NOUN
cana-510	29	23	in	in	ADP
cana-510	29	24	a	a	DET
cana-510	29	25	simply	simply	ADV
cana-510	29	26	-	-	PUNCT
cana-510	29	27	connected	connect	VERB
cana-510	29	28	region	region	NOUN
cana-510	29	29	of	of	ADP
cana-510	29	30	the	the	DET
cana-510	29	31	z	z	NOUN
cana-510	29	32	-	-	NOUN
cana-510	29	33	plane	plane	NOUN
cana-510	29	34	containing	contain	VERB
cana-510	29	35	the	the	DET
cana-510	29	36	origin	origin	NOUN
cana-510	29	37	,	,	PUNCT
cana-510	29	38	and	and	CCONJ
cana-510	29	39	the	the	DET
cana-510	29	40	multiplicity	multiplicity	NOUN
cana-510	29	41	of	of	ADP
cana-510	29	42	(	(	PUNCT
cana-510	29	43	z	z	NOUN
cana-510	29	44	−	−	PROPN
cana-510	29	45	ζ)ϑ−1	ζ)ϑ−1	NOUN
cana-510	29	46	involved	involve	VERB
cana-510	29	47	in	in	ADP
cana-510	29	48	(	(	PUNCT
cana-510	29	49	1.5	1.5	NUM
cana-510	29	50	)	)	PUNCT
cana-510	29	51	(	(	PUNCT
cana-510	29	52	and	and	CCONJ
cana-510	29	53	that	that	PRON
cana-510	29	54	of	of	ADP
cana-510	29	55	(	(	PUNCT
cana-510	29	56	z	z	NOUN
cana-510	29	57	−	−	PROPN
cana-510	29	58	ζ)−ϑ	ζ)−ϑ	NOUN
cana-510	29	59	involved	involve	VERB
cana-510	29	60	in	in	ADP
cana-510	29	61	(	(	PUNCT
cana-510	29	62	1.6	1.6	NUM
cana-510	29	63	)	)	PUNCT
cana-510	29	64	is	be	AUX
cana-510	29	65	removed	remove	VERB
cana-510	29	66	by	by	ADP
cana-510	29	67	requiring	require	VERB
cana-510	29	68	log(z	log(z	NOUN
cana-510	29	69	−	−	NOUN
cana-510	29	70	ζ	ζ	NOUN
cana-510	29	71	)	)	PUNCT
cana-510	29	72	to	to	PART
cana-510	29	73	be	be	AUX
cana-510	29	74	real	real	ADJ
cana-510	29	75	when	when	SCONJ
cana-510	29	76	(	(	PUNCT
cana-510	29	77	z	z	NOUN
cana-510	29	78	−	−	PROPN
cana-510	29	79	ζ	ζ	NOUN
cana-510	29	80	)	)	PUNCT
cana-510	29	81	>	>	X
cana-510	30	1	0	0	X
cana-510	30	2	.	.	PUNCT
cana-510	30	3	definition	definition	NOUN
cana-510	30	4	1.2	1.2	NUM
cana-510	30	5	.	.	PUNCT
cana-510	31	1	under	under	ADP
cana-510	31	2	the	the	DET
cana-510	31	3	hypotheses	hypothesis	NOUN
cana-510	31	4	of	of	ADP
cana-510	31	5	definition1.1	definition1.1	NOUN
cana-510	31	6	,	,	PUNCT
cana-510	31	7	the	the	DET
cana-510	31	8	fractional	fractional	ADJ
cana-510	31	9	derivative	derivative	NOUN
cana-510	31	10	of	of	ADP
cana-510	31	11	order	order	NOUN
cana-510	31	12	n	n	NOUN
cana-510	31	13	+	+	X
cana-510	31	14	ϑ	ϑ	X
cana-510	31	15	is	be	AUX
cana-510	31	16	defined	define	VERB
cana-510	31	17	by	by	ADP
cana-510	31	18	dz	dz	NOUN
cana-510	31	19	n+ϑu(z	n+ϑu(z	NOUN
cana-510	31	20	)	)	PUNCT
cana-510	31	21	=	=	PRON
cana-510	32	1	dn	dn	ADP
cana-510	32	2	dzn	dzn	NOUN
cana-510	32	3	dz	dz	PROPN
cana-510	32	4	ϑu(z	ϑu(z	PROPN
cana-510	32	5	)	)	PUNCT
cana-510	32	6	,	,	PUNCT
cana-510	32	7	(	(	PUNCT
cana-510	32	8	0	0	NUM
cana-510	32	9	≤	≤	NOUN
cana-510	32	10	ϑ	ϑ	X
cana-510	32	11	<	<	X
cana-510	32	12	1	1	NUM
cana-510	32	13	;	;	PUNCT
cana-510	32	14	n	n	PRON
cana-510	32	15	∈	∈	PROPN
cana-510	32	16	n0	n0	X
cana-510	32	17	=	=	SYM
cana-510	32	18	n	n	PRON
cana-510	32	19	∪	∪	X
cana-510	32	20	{	{	PUNCT
cana-510	32	21	0	0	NUM
cana-510	32	22	}	}	PUNCT
cana-510	32	23	)	)	PUNCT
cana-510	32	24	.	.	PUNCT
cana-510	33	1	(	(	PUNCT
cana-510	33	2	1.7	1.7	NUM
cana-510	33	3	)	)	PUNCT
cana-510	33	4	with	with	ADP
cana-510	33	5	the	the	DET
cana-510	33	6	aid	aid	NOUN
cana-510	33	7	of	of	ADP
cana-510	33	8	the	the	DET
cana-510	33	9	above	above	ADJ
cana-510	33	10	definitions	definition	NOUN
cana-510	33	11	,	,	PUNCT
cana-510	33	12	owa	owa	PROPN
cana-510	33	13	and	and	CCONJ
cana-510	33	14	srivastava	srivastava	PROPN
cana-510	33	15	[	[	X
cana-510	33	16	9	9	NUM
cana-510	33	17	]	]	PUNCT
cana-510	33	18	defined	define	VERB
cana-510	33	19	the	the	DET
cana-510	33	20	fractional	fractional	ADJ
cana-510	33	21	operator	operator	NOUN
cana-510	33	22	ℐz	ℐz	PROPN
cana-510	33	23	ϑ	ϑ	X
cana-510	33	24	by	by	ADP
cana-510	33	25	𝒥𝑧	𝒥𝑧	PROPN
cana-510	33	26	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	33	27	)	)	PUNCT
cana-510	33	28	=	=	SYM
cana-510	33	29	𝜔(2	𝜔(2	PROPN
cana-510	33	30	−	−	PROPN
cana-510	33	31	𝜗)𝑧𝜗𝐷𝑧	𝜗)𝑧𝜗𝐷𝑧	PROPN
cana-510	33	32	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	33	33	)	)	PUNCT
cana-510	33	34	,	,	PUNCT
cana-510	33	35	(	(	PUNCT
cana-510	33	36	𝜗	𝜗	X
cana-510	33	37	≠	≠	PROPN
cana-510	33	38	2,3,4	2,3,4	NUM
cana-510	33	39	,	,	PUNCT
cana-510	33	40	⋯	⋯	NOUN
cana-510	33	41	)	)	PUNCT
cana-510	34	1	𝒥𝑧	𝒥𝑧	ADJ
cana-510	34	2	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	34	3	)	)	PUNCT
cana-510	34	4	=	=	SYM
cana-510	34	5	𝑧	𝑧	PROPN
cana-510	35	1	+	+	NOUN
cana-510	35	2	∑	∑	PUNCT
cana-510	35	3	𝛩	𝛩	PROPN
cana-510	35	4	∞	∞	PROPN
cana-510	35	5	𝑛=2	𝑛=2	PROPN
cana-510	35	6	(	(	PUNCT
cana-510	35	7	𝜗	𝜗	NOUN
cana-510	35	8	,	,	PUNCT
cana-510	35	9	𝑛)𝑎𝑛𝑧𝑛	𝑛)𝑎𝑛𝑧𝑛	PROPN
cana-510	35	10	(	(	PUNCT
cana-510	35	11	1.8	1.8	NUM
cana-510	35	12	)	)	PUNCT
cana-510	35	13	where	where	SCONJ
cana-510	35	14	𝛩(𝜗	𝛩(𝜗	NUM
cana-510	35	15	,	,	PUNCT
cana-510	35	16	𝑛	𝑛	NOUN
cana-510	35	17	)	)	PUNCT
cana-510	35	18	=	=	NOUN
cana-510	36	1	𝜔(𝑛	𝜔(𝑛	NOUN
cana-510	36	2	+	+	NUM
cana-510	36	3	1)𝜔(2	1)𝜔(2	NUM
cana-510	36	4	−	−	NOUN
cana-510	36	5	𝜗	𝜗	NOUN
cana-510	36	6	)	)	PUNCT
cana-510	36	7	𝜔(𝑛	𝜔(𝑛	PUNCT
cana-510	36	8	−	−	NOUN
cana-510	36	9	𝜗	𝜗	NOUN
cana-510	36	10	+	+	NOUN
cana-510	36	11	1	1	NUM
cana-510	36	12	)	)	PUNCT
cana-510	36	13	and	and	CCONJ
cana-510	36	14	𝛩(𝜗	𝛩(𝜗	NUM
cana-510	36	15	,	,	PUNCT
cana-510	36	16	2	2	NUM
cana-510	36	17	)	)	PUNCT
cana-510	36	18	=	=	SYM
cana-510	36	19	2	2	NUM
cana-510	36	20	(	(	PUNCT
cana-510	36	21	2	2	NUM
cana-510	36	22	−	−	NUM
cana-510	36	23	𝜗	𝜗	NOUN
cana-510	36	24	)	)	PUNCT
cana-510	36	25	.	.	PUNCT
cana-510	37	1	now	now	ADV
cana-510	37	2	,	,	PUNCT
cana-510	37	3	by	by	ADP
cana-510	37	4	making	make	VERB
cana-510	37	5	use	use	NOUN
cana-510	37	6	of	of	ADP
cana-510	37	7	the	the	DET
cana-510	37	8	linear	linear	ADJ
cana-510	37	9	operator	operator	NOUN
cana-510	37	10	𝒥𝑧	𝒥𝑧	PROPN
cana-510	37	11	𝜗𝑢	𝜗𝑢	NOUN
cana-510	37	12	,	,	PUNCT
cana-510	37	13	we	we	PRON
cana-510	37	14	define	define	VERB
cana-510	37	15	a	a	DET
cana-510	37	16	new	new	ADJ
cana-510	37	17	subclass	subclass	NOUN
cana-510	37	18	of	of	ADP
cana-510	37	19	functions	function	NOUN
cana-510	37	20	belonging	belong	VERB
cana-510	37	21	to	to	ADP
cana-510	37	22	the	the	DET
cana-510	37	23	class	class	NOUN
cana-510	37	24	𝐴.	𝐴.	PROPN
cana-510	37	25	definition	definition	NOUN
cana-510	37	26	1.3	1.3	NUM
cana-510	37	27	.	.	PUNCT
cana-510	38	1	for	for	ADP
cana-510	38	2	0	0	NUM
cana-510	38	3	≤	≤	NUM
cana-510	38	4	ω	ω	NOUN
cana-510	38	5	<	<	X
cana-510	38	6	1,0	1,0	NUM
cana-510	38	7	≤	≤	NUM
cana-510	38	8	σ	σ	NOUN
cana-510	38	9	<	<	X
cana-510	38	10	1,0	1,0	NUM
cana-510	38	11	<	<	X
cana-510	38	12	ς	ς	X
cana-510	38	13	<	<	X
cana-510	38	14	1	1	NUM
cana-510	38	15	,	,	PUNCT
cana-510	38	16	and	and	CCONJ
cana-510	38	17	0	0	NUM
cana-510	38	18	≤	≤	NOUN
cana-510	38	19	ϑ	ϑ	X
cana-510	38	20	<	<	X
cana-510	38	21	1	1	NUM
cana-510	38	22	,	,	PUNCT
cana-510	38	23	we	we	PRON
cana-510	38	24	let	let	VERB
cana-510	38	25	ts(ω	ts(ω	NOUN
cana-510	38	26	,	,	PUNCT
cana-510	38	27	σ	σ	PROPN
cana-510	38	28	,	,	PUNCT
cana-510	38	29	ς	ς	PROPN
cana-510	38	30	,	,	PUNCT
cana-510	38	31	ϑ	ϑ	NOUN
cana-510	38	32	)	)	PUNCT
cana-510	38	33	be	be	AUX
cana-510	38	34	the	the	DET
cana-510	38	35	subclass	subclass	NOUN
cana-510	38	36	of	of	ADP
cana-510	38	37	u	u	NOUN
cana-510	38	38	consisting	consist	VERB
cana-510	38	39	of	of	ADP
cana-510	38	40	functions	function	NOUN
cana-510	38	41	of	of	ADP
cana-510	38	42	the	the	DET
cana-510	38	43	form	form	NOUN
cana-510	38	44	(	(	PUNCT
cana-510	38	45	1.4	1.4	NUM
cana-510	38	46	)	)	PUNCT
cana-510	38	47	and	and	CCONJ
cana-510	38	48	its	its	PRON
cana-510	38	49	geometrical	geometrical	ADJ
cana-510	38	50	condition	condition	NOUN
cana-510	38	51	satisfy	satisfy	NOUN
cana-510	38	52	|	|	ADV
cana-510	38	53	𝜔((𝒥𝑧	𝜔((𝒥𝑧	X
cana-510	38	54	𝜗𝑢(𝑧))′−	𝜗𝑢(𝑧))′−	PROPN
cana-510	39	1	𝒥𝑧	𝒥𝑧	PROPN
cana-510	39	2	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	39	3	)	)	PUNCT
cana-510	39	4	𝑧	𝑧	PRON
cana-510	39	5	)	)	PUNCT
cana-510	39	6	𝜎(𝒥𝑧	𝜎(𝒥𝑧	PROPN
cana-510	39	7	𝜗𝑢(𝑧))′+(1−𝜔	𝜗𝑢(𝑧))′+(1−𝜔	PROPN
cana-510	39	8	)	)	PUNCT
cana-510	40	1	𝒥𝑧	𝒥𝑧	NOUN
cana-510	40	2	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	40	3	)	)	PUNCT
cana-510	40	4	𝑧	𝑧	PROPN
cana-510	41	1	|	|	ADV
cana-510	41	2	<	<	X
cana-510	41	3	ς	ς	PROPN
cana-510	41	4	,	,	PUNCT
cana-510	41	5	𝑧	𝑧	DET
cana-510	41	6	∈	∈	NOUN
cana-510	41	7	𝕌	𝕌	NOUN
cana-510	41	8	where	where	SCONJ
cana-510	41	9	𝒥𝑧	𝒥𝑧	ADJ
cana-510	41	10	𝜗	𝜗	NOUN
cana-510	41	11	,	,	PUNCT
cana-510	41	12	is	be	AUX
cana-510	41	13	given	give	VERB
cana-510	41	14	by	by	ADP
cana-510	41	15	(	(	PUNCT
cana-510	41	16	1.8	1.8	NUM
cana-510	41	17	)	)	PUNCT
cana-510	41	18	.	.	PUNCT
cana-510	42	1	communications	communication	NOUN
cana-510	42	2	on	on	ADP
cana-510	42	3	applied	apply	VERB
cana-510	42	4	nonlinear	nonlinear	ADJ
cana-510	42	5	analysis	analysis	NOUN
cana-510	42	6	issn	issn	NOUN
cana-510	42	7	:	:	PUNCT
cana-510	42	8	1074	1074	NUM
cana-510	42	9	-	-	PUNCT
cana-510	42	10	133x	133x	NUM
cana-510	42	11	vol	vol	NOUN
cana-510	42	12	31	31	NUM
cana-510	42	13	no	no	NOUN
cana-510	42	14	.	.	NOUN
cana-510	42	15	2	2	NUM
cana-510	42	16	(	(	PUNCT
cana-510	42	17	2024	2024	NUM
cana-510	42	18	)	)	PUNCT
cana-510	42	19	24	24	NUM
cana-510	42	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	42	21	2	2	NUM
cana-510	42	22	.	.	PUNCT
cana-510	42	23	coefficient	coefficient	NOUN
cana-510	42	24	inequality	inequality	NOUN
cana-510	42	25	in	in	ADP
cana-510	42	26	the	the	DET
cana-510	42	27	following	following	NOUN
cana-510	42	28	theorem	theorem	NOUN
cana-510	42	29	,	,	PUNCT
cana-510	42	30	we	we	PRON
cana-510	42	31	obtain	obtain	VERB
cana-510	42	32	a	a	DET
cana-510	42	33	necessary	necessary	ADJ
cana-510	42	34	and	and	CCONJ
cana-510	42	35	sufficient	sufficient	ADJ
cana-510	42	36	condition	condition	NOUN
cana-510	42	37	for	for	ADP
cana-510	42	38	function	function	NOUN
cana-510	42	39	to	to	PART
cana-510	42	40	be	be	AUX
cana-510	42	41	in	in	ADP
cana-510	42	42	the	the	DET
cana-510	42	43	class	class	NOUN
cana-510	42	44	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	42	45	,	,	PUNCT
cana-510	42	46	𝜎	𝜎	PROPN
cana-510	42	47	,	,	PUNCT
cana-510	42	48	ς	ς	PROPN
cana-510	42	49	,	,	PUNCT
cana-510	42	50	𝜗	𝜗	NOUN
cana-510	42	51	)	)	PUNCT
cana-510	42	52	.	.	PUNCT
cana-510	43	1	theorem	theorem	VERB
cana-510	43	2	2.1	2.1	NUM
cana-510	43	3	.	.	PUNCT
cana-510	44	1	let	let	VERB
cana-510	44	2	the	the	DET
cana-510	44	3	function	function	NOUN
cana-510	44	4	u	u	NOUN
cana-510	44	5	be	be	AUX
cana-510	44	6	defined	define	VERB
cana-510	44	7	by	by	ADP
cana-510	44	8	(	(	PUNCT
cana-510	44	9	1.4	1.4	NUM
cana-510	44	10	)	)	PUNCT
cana-510	44	11	.	.	PUNCT
cana-510	45	1	then	then	ADV
cana-510	45	2	u	u	PROPN
cana-510	45	3	∈	∈	PROPN
cana-510	45	4	ts(ω	ts(ω	NUM
cana-510	45	5	,	,	PUNCT
cana-510	45	6	σ	σ	PROPN
cana-510	45	7	,	,	PUNCT
cana-510	45	8	ς	ς	PROPN
cana-510	45	9	,	,	PUNCT
cana-510	45	10	ϑ	ϑ	NOUN
cana-510	45	11	)	)	PUNCT
cana-510	45	12	if	if	SCONJ
cana-510	45	13	and	and	CCONJ
cana-510	45	14	only	only	ADV
cana-510	45	15	if	if	SCONJ
cana-510	45	16	∑	∑	PROPN
cana-510	45	17	[	[	X
cana-510	45	18	𝜔(𝑛	𝜔(𝑛	X
cana-510	45	19	−	−	PROPN
cana-510	45	20	1	1	NUM
cana-510	45	21	)	)	PUNCT
cana-510	45	22	+	+	NOUN
cana-510	45	23	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	45	24	+	+	CCONJ
cana-510	45	25	1	1	NUM
cana-510	45	26	−	−	NOUN
cana-510	45	27	𝜔)]∞	𝜔)]∞	NOUN
cana-510	45	28	𝑛=2	𝑛=2	NOUN
cana-510	45	29	𝛩(𝑛	𝛩(𝑛	PROPN
cana-510	45	30	,	,	PUNCT
cana-510	45	31	𝜗)𝑎𝑛	𝜗)𝑎𝑛	PROPN
cana-510	45	32	≤	≤	PROPN
cana-510	45	33	ς(𝜎	ς(𝜎	PROPN
cana-510	45	34	+	+	CCONJ
cana-510	45	35	(	(	PUNCT
cana-510	45	36	1	1	NUM
cana-510	45	37	−	−	NOUN
cana-510	45	38	𝜔	𝜔	NOUN
cana-510	45	39	)	)	PUNCT
cana-510	45	40	)	)	PUNCT
cana-510	45	41	,	,	PUNCT
cana-510	45	42	(	(	PUNCT
cana-510	45	43	2.1	2.1	NUM
cana-510	45	44	)	)	PUNCT
cana-510	45	45	where	where	SCONJ
cana-510	45	46	0	0	NUM
cana-510	45	47	<	<	X
cana-510	45	48	ς	ς	X
cana-510	45	49	<	<	X
cana-510	45	50	1,0	1,0	NUM
cana-510	45	51	≤	≤	NUM
cana-510	45	52	𝜔	𝜔	PART
cana-510	45	53	<	<	X
cana-510	45	54	1,0	1,0	NUM
cana-510	45	55	≤	≤	NUM
cana-510	45	56	𝜎	𝜎	X
cana-510	45	57	<	<	X
cana-510	45	58	1	1	NUM
cana-510	45	59	,	,	PUNCT
cana-510	45	60	and	and	CCONJ
cana-510	45	61	0	0	NUM
cana-510	45	62	≤	≤	NUM
cana-510	45	63	𝜗	𝜗	NOUN
cana-510	45	64	<	<	X
cana-510	45	65	1	1	NUM
cana-510	45	66	.	.	PUNCT
cana-510	46	1	the	the	DET
cana-510	46	2	result	result	NOUN
cana-510	46	3	(	(	PUNCT
cana-510	46	4	2.1	2.1	NUM
cana-510	46	5	)	)	PUNCT
cana-510	46	6	is	be	AUX
cana-510	46	7	sharp	sharp	ADJ
cana-510	46	8	for	for	SCONJ
cana-510	46	9	the	the	DET
cana-510	46	10	function	function	NOUN
cana-510	46	11	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	46	12	)	)	PUNCT
cana-510	47	1	=	=	PUNCT
cana-510	47	2	𝑧	𝑧	DET
cana-510	47	3	−	−	PROPN
cana-510	47	4	ς(𝜎+(1−𝜔	ς(𝜎+(1−𝜔	NOUN
cana-510	47	5	)	)	PUNCT
cana-510	47	6	)	)	PUNCT
cana-510	48	1	[	[	X
cana-510	48	2	𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	NOUN
cana-510	48	3	)	)	PUNCT
cana-510	48	4	𝑧𝑛	𝑧𝑛	PROPN
cana-510	48	5	,	,	PUNCT
cana-510	48	6	𝑛	𝑛	DET
cana-510	48	7	≥	≥	NOUN
cana-510	48	8	2	2	NUM
cana-510	48	9	.	.	PUNCT
cana-510	48	10	proof	proof	NOUN
cana-510	48	11	.	.	PUNCT
cana-510	49	1	suppose	suppose	VERB
cana-510	49	2	that	that	SCONJ
cana-510	49	3	the	the	DET
cana-510	49	4	inequality	inequality	NOUN
cana-510	49	5	(	(	PUNCT
cana-510	49	6	2.1	2.1	NUM
cana-510	49	7	)	)	PUNCT
cana-510	49	8	holds	hold	VERB
cana-510	49	9	true	true	ADJ
cana-510	49	10	and	and	CCONJ
cana-510	49	11	|𝑧|	|𝑧|	ADJ
cana-510	49	12	=	=	SYM
cana-510	50	1	1	1	X
cana-510	50	2	.	.	PUNCT
cana-510	51	1	then	then	ADV
cana-510	51	2	we	we	PRON
cana-510	51	3	obtain	obtain	VERB
cana-510	51	4	|𝜔	|𝜔	X
cana-510	51	5	(	(	PUNCT
cana-510	51	6	(	(	PUNCT
cana-510	51	7	𝒥𝑧	𝒥𝑧	ADJ
cana-510	51	8	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	51	9	)	)	PUNCT
cana-510	51	10	)	)	PUNCT
cana-510	52	1	′	′	NUM
cana-510	53	1	−	−	PUNCT
cana-510	54	1	𝒥𝑧	𝒥𝑧	ADJ
cana-510	54	2	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	54	3	)	)	PUNCT
cana-510	54	4	𝑧	𝑧	X
cana-510	54	5	)	)	PUNCT
cana-510	54	6	|	|	ADV
cana-510	54	7	−	−	PROPN
cana-510	54	8	ς	ς	PROPN
cana-510	54	9	|𝜎	|𝜎	X
cana-510	54	10	(	(	PUNCT
cana-510	54	11	𝒥𝑧	𝒥𝑧	PROPN
cana-510	54	12	𝜗𝑢(𝑧))′	𝜗𝑢(𝑧))′	PROPN
cana-510	54	13	+	+	CCONJ
cana-510	54	14	(	(	PUNCT
cana-510	54	15	1	1	NUM
cana-510	54	16	−	−	NOUN
cana-510	54	17	𝜔	𝜔	NOUN
cana-510	54	18	)	)	PUNCT
cana-510	54	19	𝒥𝑧	𝒥𝑧	NOUN
cana-510	54	20	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	54	21	)	)	PUNCT
cana-510	54	22	𝑧	𝑧	NOUN
cana-510	54	23	)	)	PUNCT
cana-510	54	24	|	|	ADV
cana-510	54	25	=	=	SYM
cana-510	55	1	|−𝜔	|−𝜔	NOUN
cana-510	55	2	∑(𝑛	∑(𝑛	PUNCT
cana-510	56	1	−	−	ADP
cana-510	56	2	1	1	NUM
cana-510	56	3	)	)	PUNCT
cana-510	56	4	∞	∞	NUM
cana-510	56	5	𝑛=2	𝑛=2	PART
cana-510	56	6	𝛩(𝑛	𝛩(𝑛	ADP
cana-510	56	7	,	,	PUNCT
cana-510	56	8	𝜗)𝑎𝑛𝑧𝑛−1|	𝜗)𝑎𝑛𝑧𝑛−1|	PRON
cana-510	56	9	−ς	−ς	VERB
cana-510	56	10	|𝜎	|𝜎	X
cana-510	56	11	+	+	CCONJ
cana-510	56	12	(	(	PUNCT
cana-510	56	13	1	1	NUM
cana-510	56	14	−	−	NOUN
cana-510	56	15	𝜔	𝜔	NOUN
cana-510	56	16	)	)	PUNCT
cana-510	57	1	−	−	NOUN
cana-510	57	2	∑(𝑛𝜎	∑(𝑛𝜎	NOUN
cana-510	57	3	+	+	CCONJ
cana-510	57	4	1	1	NUM
cana-510	57	5	−	−	NOUN
cana-510	57	6	𝜔	𝜔	NUM
cana-510	57	7	)	)	PUNCT
cana-510	57	8	∞	∞	NUM
cana-510	57	9	𝑛=2	𝑛=2	X
cana-510	57	10	𝛩(𝑛	𝛩(𝑛	ADP
cana-510	57	11	,	,	PUNCT
cana-510	57	12	𝜗)𝑎𝑛𝑧𝑛−1|	𝜗)𝑎𝑛𝑧𝑛−1|	ADP
cana-510	57	13	≤	≤	ADJ
cana-510	57	14	∑[𝜔(𝑛	∑[𝜔(𝑛	PROPN
cana-510	57	15	−	−	PROPN
cana-510	57	16	1	1	NUM
cana-510	57	17	)	)	PUNCT
cana-510	57	18	+	+	NOUN
cana-510	57	19	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	57	20	+	+	CCONJ
cana-510	57	21	1	1	NUM
cana-510	57	22	−	−	NOUN
cana-510	57	23	𝜔	𝜔	NOUN
cana-510	57	24	)	)	PUNCT
cana-510	57	25	]	]	PUNCT
cana-510	58	1	∞	∞	NUM
cana-510	58	2	𝑛=2	𝑛=2	X
cana-510	58	3	𝛩(𝑛	𝛩(𝑛	ADP
cana-510	58	4	,	,	PUNCT
cana-510	58	5	𝜗)𝑎𝑛	𝜗)𝑎𝑛	PROPN
cana-510	58	6	−	−	PROPN
cana-510	58	7	ς(𝜎	ς(𝜎	PROPN
cana-510	58	8	+	+	CCONJ
cana-510	58	9	(	(	PUNCT
cana-510	58	10	1	1	NUM
cana-510	58	11	−	−	NOUN
cana-510	58	12	𝜔	𝜔	NOUN
cana-510	58	13	)	)	PUNCT
cana-510	58	14	)	)	PUNCT
cana-510	58	15	≤	≤	NUM
cana-510	58	16	0	0	NUM
cana-510	58	17	.	.	PUNCT
cana-510	59	1	hence	hence	ADV
cana-510	59	2	,	,	PUNCT
cana-510	59	3	by	by	ADP
cana-510	59	4	maximum	maximum	ADJ
cana-510	59	5	modulus	modulus	ADJ
cana-510	59	6	principle,𝑢	principle,𝑢	NUM
cana-510	59	7	∈	∈	PROPN
cana-510	59	8	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	59	9	,	,	PUNCT
cana-510	59	10	𝜎	𝜎	PROPN
cana-510	59	11	,	,	PUNCT
cana-510	59	12	ς	ς	PROPN
cana-510	59	13	,	,	PUNCT
cana-510	59	14	𝜗	𝜗	NOUN
cana-510	59	15	)	)	PUNCT
cana-510	59	16	.	.	PUNCT
cana-510	60	1	now	now	ADV
cana-510	60	2	assume	assume	VERB
cana-510	60	3	that	that	SCONJ
cana-510	60	4	𝑢	𝑢	PROPN
cana-510	60	5	∈	∈	PROPN
cana-510	60	6	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	60	7	,	,	PUNCT
cana-510	60	8	𝜎	𝜎	PROPN
cana-510	60	9	,	,	PUNCT
cana-510	60	10	ς	ς	PROPN
cana-510	60	11	,	,	PUNCT
cana-510	60	12	𝜗	𝜗	NOUN
cana-510	60	13	)	)	PUNCT
cana-510	60	14	so	so	SCONJ
cana-510	60	15	that	that	SCONJ
cana-510	60	16	|	|	ADV
cana-510	60	17	|	|	ADV
cana-510	60	18	𝜔	𝜔	VERB
cana-510	60	19	(	(	PUNCT
cana-510	60	20	(	(	PUNCT
cana-510	60	21	𝒥𝑧	𝒥𝑧	ADJ
cana-510	60	22	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	60	23	)	)	PUNCT
cana-510	60	24	)	)	PUNCT
cana-510	61	1	′	′	NUM
cana-510	62	1	−	−	PUNCT
cana-510	63	1	𝒥𝑧	𝒥𝑧	ADJ
cana-510	63	2	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	63	3	)	)	PUNCT
cana-510	63	4	𝑧	𝑧	PRON
cana-510	63	5	)	)	PUNCT
cana-510	63	6	𝜎(𝒥𝑧	𝜎(𝒥𝑧	X
cana-510	63	7	𝜗𝑢(𝑧))′	𝜗𝑢(𝑧))′	PROPN
cana-510	64	1	+	+	CCONJ
cana-510	64	2	(	(	PUNCT
cana-510	64	3	1	1	NUM
cana-510	64	4	−	−	NOUN
cana-510	64	5	𝜔	𝜔	NOUN
cana-510	64	6	)	)	PUNCT
cana-510	64	7	𝒥𝑧	𝒥𝑧	NOUN
cana-510	64	8	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	64	9	)	)	PUNCT
cana-510	64	10	𝑧	𝑧	PRON
cana-510	65	1	|	|	ADV
cana-510	65	2	|	|	ADV
cana-510	65	3	<	<	X
cana-510	65	4	ς	ς	PROPN
cana-510	65	5	,	,	PUNCT
cana-510	65	6	𝑧	𝑧	DET
cana-510	65	7	∈	∈	PROPN
cana-510	65	8	𝕌	𝕌	NOUN
cana-510	65	9	hence	hence	ADV
cana-510	65	10	|𝜔	|𝜔	X
cana-510	66	1	(	(	PUNCT
cana-510	66	2	(	(	PUNCT
cana-510	66	3	𝒥𝑧	𝒥𝑧	ADJ
cana-510	66	4	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	66	5	)	)	PUNCT
cana-510	66	6	)	)	PUNCT
cana-510	67	1	′	′	NUM
cana-510	68	1	−	−	PUNCT
cana-510	69	1	𝒥𝑧	𝒥𝑧	ADJ
cana-510	69	2	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	69	3	)	)	PUNCT
cana-510	69	4	𝑧	𝑧	X
cana-510	69	5	)	)	PUNCT
cana-510	70	1	|	|	ADV
cana-510	70	2	<	<	X
cana-510	70	3	ς	ς	X
cana-510	70	4	|𝜎	|𝜎	X
cana-510	70	5	(	(	PUNCT
cana-510	70	6	𝒥𝑧	𝒥𝑧	PROPN
cana-510	70	7	𝜗𝑢(𝑧))′	𝜗𝑢(𝑧))′	PROPN
cana-510	70	8	+	+	CCONJ
cana-510	70	9	(	(	PUNCT
cana-510	70	10	1	1	NUM
cana-510	70	11	−	−	NOUN
cana-510	70	12	𝜔	𝜔	NOUN
cana-510	70	13	)	)	PUNCT
cana-510	70	14	𝒥𝑧	𝒥𝑧	NOUN
cana-510	70	15	𝜗𝑢(𝑧	𝜗𝑢(𝑧	NOUN
cana-510	70	16	)	)	PUNCT
cana-510	70	17	𝑧	𝑧	NOUN
cana-510	70	18	)	)	PUNCT
cana-510	70	19	|	|	ADV
cana-510	70	20	.	.	PUNCT
cana-510	71	1	therefore	therefore	ADV
cana-510	71	2	,	,	PUNCT
cana-510	71	3	we	we	PRON
cana-510	71	4	get	get	VERB
cana-510	71	5	communications	communication	NOUN
cana-510	71	6	on	on	ADP
cana-510	71	7	applied	apply	VERB
cana-510	71	8	nonlinear	nonlinear	ADJ
cana-510	71	9	analysis	analysis	NOUN
cana-510	71	10	issn	issn	NOUN
cana-510	71	11	:	:	PUNCT
cana-510	71	12	1074	1074	NUM
cana-510	71	13	-	-	PUNCT
cana-510	71	14	133x	133x	NUM
cana-510	71	15	vol	vol	NOUN
cana-510	71	16	31	31	NUM
cana-510	71	17	no	no	NOUN
cana-510	71	18	.	.	NOUN
cana-510	71	19	2	2	NUM
cana-510	71	20	(	(	PUNCT
cana-510	71	21	2024	2024	NUM
cana-510	71	22	)	)	PUNCT
cana-510	71	23	25	25	NUM
cana-510	71	24	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	71	25	|−	|−	ADV
cana-510	71	26	∑	∑	ADP
cana-510	71	27	𝜔	𝜔	PROPN
cana-510	71	28	∞	∞	NUM
cana-510	71	29	𝑛=2	𝑛=2	PUNCT
cana-510	71	30	(	(	PUNCT
cana-510	71	31	𝑛	𝑛	DET
cana-510	71	32	−	−	PROPN
cana-510	71	33	1)𝛩(𝑛	1)𝛩(𝑛	NUM
cana-510	71	34	,	,	PUNCT
cana-510	71	35	𝜗)𝑎𝑛𝑧𝑛−1|	𝜗)𝑎𝑛𝑧𝑛−1|	PRON
cana-510	71	36	<	<	X
cana-510	71	37	ς	ς	PROPN
cana-510	71	38	|𝜎	|𝜎	X
cana-510	71	39	+	+	CCONJ
cana-510	71	40	(	(	PUNCT
cana-510	71	41	1	1	NUM
cana-510	71	42	−	−	NOUN
cana-510	71	43	𝜔	𝜔	NOUN
cana-510	71	44	)	)	PUNCT
cana-510	72	1	−	−	NOUN
cana-510	73	1	∑(𝑛𝜎	∑(𝑛𝜎	NOUN
cana-510	74	1	+	+	CCONJ
cana-510	74	2	1	1	NUM
cana-510	74	3	−	−	NOUN
cana-510	74	4	𝜔	𝜔	NUM
cana-510	74	5	)	)	PUNCT
cana-510	74	6	∞	∞	NUM
cana-510	74	7	𝑛=2	𝑛=2	PART
cana-510	74	8	𝛩(𝑛	𝛩(𝑛	ADP
cana-510	74	9	,	,	PUNCT
cana-510	74	10	𝜗)𝑎𝑛𝑧𝑛−1|	𝜗)𝑎𝑛𝑧𝑛−1|	X
cana-510	74	11	.	.	PUNCT
cana-510	75	1	thus	thus	ADV
cana-510	75	2	∑[𝜔(𝑛	∑[𝜔(𝑛	PROPN
cana-510	75	3	−	−	PROPN
cana-510	75	4	1	1	NUM
cana-510	75	5	)	)	PUNCT
cana-510	75	6	+	+	NOUN
cana-510	75	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	75	8	+	+	CCONJ
cana-510	75	9	1	1	NUM
cana-510	75	10	−	−	NOUN
cana-510	75	11	𝜔	𝜔	NOUN
cana-510	75	12	)	)	PUNCT
cana-510	75	13	]	]	PUNCT
cana-510	76	1	∞	∞	NUM
cana-510	76	2	𝑛=2	𝑛=2	X
cana-510	76	3	𝛩(𝑛	𝛩(𝑛	ADP
cana-510	76	4	,	,	PUNCT
cana-510	76	5	𝜗)𝑎𝑛	𝜗)𝑎𝑛	PROPN
cana-510	76	6	≤	≤	PROPN
cana-510	76	7	ς(𝜎	ς(𝜎	PROPN
cana-510	76	8	+	+	CCONJ
cana-510	76	9	(	(	PUNCT
cana-510	76	10	1	1	NUM
cana-510	76	11	−	−	NOUN
cana-510	76	12	𝜔	𝜔	NOUN
cana-510	76	13	)	)	PUNCT
cana-510	76	14	)	)	PUNCT
cana-510	77	1	and	and	CCONJ
cana-510	77	2	this	this	PRON
cana-510	77	3	completes	complete	VERB
cana-510	77	4	the	the	DET
cana-510	77	5	proof	proof	NOUN
cana-510	77	6	.	.	PUNCT
cana-510	78	1	corollary	corollary	ADJ
cana-510	78	2	2.1	2.1	NUM
cana-510	78	3	.	.	PUNCT
cana-510	79	1	let	let	VERB
cana-510	79	2	the	the	DET
cana-510	79	3	function	function	NOUN
cana-510	79	4	𝑢	𝑢	PROPN
cana-510	79	5	∈	∈	PROPN
cana-510	79	6	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	79	7	,	,	PUNCT
cana-510	79	8	𝜎	𝜎	PROPN
cana-510	79	9	,	,	PUNCT
cana-510	79	10	ς	ς	PROPN
cana-510	79	11	,	,	PUNCT
cana-510	79	12	𝜗).then	𝜗).then	PROPN
cana-510	79	13	𝑎𝑛	𝑎𝑛	VERB
cana-510	79	14	≤	≤	PROPN
cana-510	79	15	ς(𝜎+(1−𝜔	ς(𝜎+(1−𝜔	NOUN
cana-510	79	16	)	)	PUNCT
cana-510	79	17	)	)	PUNCT
cana-510	80	1	[	[	X
cana-510	80	2	𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	NOUN
cana-510	80	3	)	)	PUNCT
cana-510	80	4	𝑧𝑛	𝑧𝑛	PROPN
cana-510	80	5	,	,	PUNCT
cana-510	80	6	𝑛	𝑛	DET
cana-510	80	7	≥	≥	NOUN
cana-510	80	8	2	2	NUM
cana-510	80	9	.	.	NOUN
cana-510	80	10	3	3	NUM
cana-510	80	11	.	.	NOUN
cana-510	80	12	distortion	distortion	NOUN
cana-510	80	13	and	and	CCONJ
cana-510	80	14	covering	covering	NOUN
cana-510	80	15	theorem	theorem	NOUN
cana-510	80	16	we	we	PRON
cana-510	80	17	introduce	introduce	VERB
cana-510	80	18	the	the	DET
cana-510	80	19	growth	growth	NOUN
cana-510	80	20	and	and	CCONJ
cana-510	80	21	distortion	distortion	NOUN
cana-510	80	22	theorems	theorem	NOUN
cana-510	80	23	for	for	ADP
cana-510	80	24	the	the	DET
cana-510	80	25	functions	function	NOUN
cana-510	80	26	in	in	ADP
cana-510	80	27	the	the	DET
cana-510	80	28	class	class	NOUN
cana-510	80	29	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	80	30	,	,	PUNCT
cana-510	80	31	𝜎	𝜎	PROPN
cana-510	80	32	,	,	PUNCT
cana-510	80	33	ς	ς	PROPN
cana-510	80	34	,	,	PUNCT
cana-510	80	35	𝜗	𝜗	NOUN
cana-510	80	36	)	)	PUNCT
cana-510	80	37	theorem	theorem	VERB
cana-510	80	38	3.1	3.1	NUM
cana-510	80	39	.	.	PUNCT
cana-510	81	1	let	let	VERB
cana-510	81	2	the	the	DET
cana-510	81	3	function	function	NOUN
cana-510	81	4	𝑢	𝑢	PROPN
cana-510	81	5	∈	∈	PROPN
cana-510	81	6	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	81	7	,	,	PUNCT
cana-510	81	8	𝜎	𝜎	PROPN
cana-510	81	9	,	,	PUNCT
cana-510	81	10	ς	ς	PROPN
cana-510	81	11	,	,	PUNCT
cana-510	81	12	𝜗	𝜗	NOUN
cana-510	81	13	)	)	PUNCT
cana-510	81	14	.	.	PUNCT
cana-510	82	1	then	then	ADV
cana-510	82	2	|𝑧|	|𝑧|	PROPN
cana-510	82	3	−	−	PROPN
cana-510	83	1	ς(𝜎	ς(𝜎	PROPN
cana-510	84	1	+	+	CCONJ
cana-510	85	1	(	(	PUNCT
cana-510	85	2	1	1	NUM
cana-510	85	3	−	−	NOUN
cana-510	85	4	𝜔	𝜔	NOUN
cana-510	85	5	)	)	PUNCT
cana-510	85	6	)	)	PUNCT
cana-510	86	1	𝛩(2	𝛩(2	NUM
cana-510	86	2	,	,	PUNCT
cana-510	87	1	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	87	2	+	+	NUM
cana-510	87	3	ς(2𝜎	ς(2𝜎	NUM
cana-510	87	4	+	+	CCONJ
cana-510	87	5	1	1	NUM
cana-510	87	6	−	−	NOUN
cana-510	87	7	𝜔	𝜔	NOUN
cana-510	87	8	)	)	PUNCT
cana-510	87	9	]	]	PUNCT
cana-510	88	1	|𝑧|2	|𝑧|2	VERB
cana-510	88	2	≤	≤	PUNCT
cana-510	88	3	|𝑢(𝑧)|	|𝑢(𝑧)|	PROPN
cana-510	88	4	≤	≤	NUM
cana-510	88	5	|𝑧|	|𝑧|	NOUN
cana-510	88	6	+	+	PROPN
cana-510	88	7	ς(𝜎	ς(𝜎	PROPN
cana-510	88	8	+	+	CCONJ
cana-510	88	9	(	(	PUNCT
cana-510	88	10	1	1	NUM
cana-510	88	11	−	−	NOUN
cana-510	88	12	𝜔	𝜔	NOUN
cana-510	88	13	)	)	PUNCT
cana-510	88	14	)	)	PUNCT
cana-510	88	15	𝛩(2	𝛩(2	NUM
cana-510	88	16	,	,	PUNCT
cana-510	88	17	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	88	18	+	+	NUM
cana-510	88	19	ς(2𝜎	ς(2𝜎	NUM
cana-510	88	20	+	+	CCONJ
cana-510	88	21	1	1	NUM
cana-510	88	22	−	−	NOUN
cana-510	88	23	𝜔	𝜔	NOUN
cana-510	88	24	)	)	PUNCT
cana-510	88	25	]	]	PUNCT
cana-510	89	1	|𝑧|2	|𝑧|2	PROPN
cana-510	89	2	.	.	PUNCT
cana-510	90	1	the	the	DET
cana-510	90	2	result	result	NOUN
cana-510	90	3	is	be	AUX
cana-510	90	4	sharp	sharp	ADJ
cana-510	90	5	and	and	CCONJ
cana-510	90	6	attained	attain	VERB
cana-510	90	7	𝑢(𝑧	𝑢(𝑧	NOUN
cana-510	90	8	)	)	PUNCT
cana-510	91	1	=	=	PUNCT
cana-510	91	2	𝑧	𝑧	DET
cana-510	91	3	−	−	PROPN
cana-510	91	4	ς(𝜎+(1−𝜔	ς(𝜎+(1−𝜔	NOUN
cana-510	91	5	)	)	PUNCT
cana-510	91	6	)	)	PUNCT
cana-510	92	1	𝛩(2,𝜗)[𝜔+ς(2𝜎+1−𝜔	𝛩(2,𝜗)[𝜔+ς(2𝜎+1−𝜔	ADJ
cana-510	92	2	)	)	PUNCT
cana-510	92	3	]	]	PUNCT
cana-510	93	1	𝑧2	𝑧2	NOUN
cana-510	93	2	.	.	PUNCT
cana-510	93	3	proof	proof	NOUN
cana-510	93	4	.	.	PUNCT
cana-510	94	1	|𝑢(𝑧)|	|𝑢(𝑧)|	NOUN
cana-510	94	2	=	=	PUNCT
cana-510	94	3	|𝑧	|𝑧	NOUN
cana-510	94	4	−	−	NOUN
cana-510	94	5	∑	∑	PROPN
cana-510	94	6	𝑎𝑛	𝑎𝑛	PROPN
cana-510	94	7	∞	∞	NUM
cana-510	94	8	𝑛=2	𝑛=2	PROPN
cana-510	94	9	𝑧𝑛|	𝑧𝑛|	VERB
cana-510	94	10	≤	≤	NUM
cana-510	94	11	|𝑧|	|𝑧|	PROPN
cana-510	95	1	+	+	CCONJ
cana-510	95	2	∑	∑	PROPN
cana-510	95	3	𝑎𝑛	𝑎𝑛	PROPN
cana-510	95	4	∞	∞	NUM
cana-510	95	5	𝑛=2	𝑛=2	PROPN
cana-510	95	6	|𝑧|𝑛	|𝑧|𝑛	NOUN
cana-510	95	7	≤	≤	NUM
cana-510	95	8	|𝑧|	|𝑧|	PROPN
cana-510	95	9	+	+	CCONJ
cana-510	95	10	|𝑧|2	|𝑧|2	VERB
cana-510	95	11	∑	∑	PROPN
cana-510	95	12	𝑎𝑛	𝑎𝑛	PROPN
cana-510	95	13	∞	∞	NUM
cana-510	95	14	𝑛=2	𝑛=2	NOUN
cana-510	95	15	.	.	PUNCT
cana-510	96	1	by	by	ADP
cana-510	96	2	theorem	theorem	NOUN
cana-510	96	3	2.1	2.1	NUM
cana-510	96	4	,	,	PUNCT
cana-510	96	5	we	we	PRON
cana-510	96	6	get	get	VERB
cana-510	96	7	∑	∑	PUNCT
cana-510	96	8	𝑎𝑛	𝑎𝑛	PROPN
cana-510	96	9	∞	∞	NUM
cana-510	96	10	𝑛=2	𝑛=2	NOUN
cana-510	96	11	≤	≤	NOUN
cana-510	96	12	ς(𝜎	ς(𝜎	PROPN
cana-510	96	13	+	+	CCONJ
cana-510	96	14	(	(	PUNCT
cana-510	96	15	1	1	NUM
cana-510	96	16	−	−	NOUN
cana-510	96	17	𝜔	𝜔	NOUN
cana-510	96	18	)	)	PUNCT
cana-510	96	19	)	)	PUNCT
cana-510	97	1	[	[	X
cana-510	97	2	𝜔	𝜔	X
cana-510	97	3	+	+	NUM
cana-510	97	4	ς(2𝜎	ς(2𝜎	NUM
cana-510	97	5	+	+	CCONJ
cana-510	97	6	1	1	NUM
cana-510	97	7	−	−	NOUN
cana-510	97	8	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	97	9	,	,	PUNCT
cana-510	97	10	𝜗	𝜗	NOUN
cana-510	97	11	)	)	PUNCT
cana-510	97	12	(	(	PUNCT
cana-510	97	13	3.1	3.1	NUM
cana-510	97	14	)	)	PUNCT
cana-510	97	15	.	.	PUNCT
cana-510	98	1	communications	communication	NOUN
cana-510	98	2	on	on	ADP
cana-510	98	3	applied	apply	VERB
cana-510	98	4	nonlinear	nonlinear	ADJ
cana-510	98	5	analysis	analysis	NOUN
cana-510	98	6	issn	issn	NOUN
cana-510	98	7	:	:	PUNCT
cana-510	98	8	1074	1074	NUM
cana-510	98	9	-	-	PUNCT
cana-510	98	10	133x	133x	NUM
cana-510	98	11	vol	vol	NOUN
cana-510	98	12	31	31	NUM
cana-510	98	13	no	no	NOUN
cana-510	98	14	.	.	NOUN
cana-510	98	15	2	2	NUM
cana-510	98	16	(	(	PUNCT
cana-510	98	17	2024	2024	NUM
cana-510	98	18	)	)	PUNCT
cana-510	98	19	26	26	NUM
cana-510	98	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	98	21	thus	thus	ADV
cana-510	98	22	|𝑢(𝑧)|	|𝑢(𝑧)|	ADJ
cana-510	98	23	≤	≤	NUM
cana-510	98	24	|𝑧|	|𝑧|	NOUN
cana-510	98	25	+	+	PROPN
cana-510	98	26	ς(𝜎	ς(𝜎	PROPN
cana-510	98	27	+	+	CCONJ
cana-510	98	28	(	(	PUNCT
cana-510	98	29	1	1	NUM
cana-510	98	30	−	−	NOUN
cana-510	98	31	𝜔	𝜔	NOUN
cana-510	98	32	)	)	PUNCT
cana-510	98	33	)	)	PUNCT
cana-510	98	34	𝛩(2	𝛩(2	NUM
cana-510	98	35	,	,	PUNCT
cana-510	98	36	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	98	37	+	+	NUM
cana-510	98	38	ς(2𝜎	ς(2𝜎	NUM
cana-510	98	39	+	+	CCONJ
cana-510	98	40	1	1	NUM
cana-510	98	41	−	−	NOUN
cana-510	98	42	𝜔	𝜔	NOUN
cana-510	98	43	)	)	PUNCT
cana-510	98	44	]	]	PUNCT
cana-510	99	1	|𝑧|2	|𝑧|2	PROPN
cana-510	99	2	.	.	PUNCT
cana-510	100	1	also	also	ADV
cana-510	100	2	|𝑢(𝑧)|	|𝑢(𝑧)|	VERB
cana-510	100	3	≥	≥	NUM
cana-510	100	4	|𝑧|	|𝑧|	NOUN
cana-510	100	5	−	−	NOUN
cana-510	100	6	∑	∑	PROPN
cana-510	100	7	𝑎𝑛	𝑎𝑛	PROPN
cana-510	100	8	∞	∞	NUM
cana-510	100	9	𝑛=2	𝑛=2	PROPN
cana-510	100	10	|𝑧|𝑛	|𝑧|𝑛	NOUN
cana-510	100	11	≥	≥	NUM
cana-510	100	12	|𝑧|	|𝑧|	NOUN
cana-510	100	13	−	−	PROPN
cana-510	100	14	|𝑧|2	|𝑧|2	VERB
cana-510	100	15	∑	∑	PROPN
cana-510	100	16	𝑎𝑛	𝑎𝑛	PROPN
cana-510	100	17	∞	∞	PROPN
cana-510	100	18	𝑛=2	𝑛=2	NOUN
cana-510	100	19	≥	≥	NOUN
cana-510	100	20	|𝑧|	|𝑧|	NOUN
cana-510	100	21	−	−	PROPN
cana-510	101	1	ς(𝜎	ς(𝜎	PROPN
cana-510	101	2	+	+	CCONJ
cana-510	101	3	(	(	PUNCT
cana-510	101	4	1	1	NUM
cana-510	101	5	−	−	NOUN
cana-510	101	6	𝜔	𝜔	NOUN
cana-510	101	7	)	)	PUNCT
cana-510	101	8	)	)	PUNCT
cana-510	101	9	𝛩(2	𝛩(2	NUM
cana-510	101	10	,	,	PUNCT
cana-510	101	11	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	101	12	+	+	NUM
cana-510	101	13	ς(2𝜎	ς(2𝜎	NUM
cana-510	101	14	+	+	CCONJ
cana-510	101	15	1	1	NUM
cana-510	101	16	−	−	NOUN
cana-510	101	17	𝜔	𝜔	NOUN
cana-510	101	18	)	)	PUNCT
cana-510	101	19	]	]	PUNCT
cana-510	102	1	|𝑧|2	|𝑧|2	PROPN
cana-510	102	2	.	.	PUNCT
cana-510	103	1	theorem	theorem	VERB
cana-510	103	2	3.2	3.2	NUM
cana-510	103	3	.	.	PUNCT
cana-510	104	1	let	let	VERB
cana-510	104	2	𝑢	𝑢	PRON
cana-510	104	3	∈	∈	PROPN
cana-510	104	4	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	104	5	,	,	PUNCT
cana-510	104	6	𝜎	𝜎	PROPN
cana-510	104	7	,	,	PUNCT
cana-510	104	8	ς	ς	PROPN
cana-510	104	9	,	,	PUNCT
cana-510	104	10	𝜗	𝜗	NOUN
cana-510	104	11	)	)	PUNCT
cana-510	104	12	.	.	PUNCT
cana-510	105	1	then	then	ADV
cana-510	105	2	1	1	NUM
cana-510	105	3	−	−	PROPN
cana-510	105	4	2ς(𝜎	2ς(𝜎	NUM
cana-510	105	5	+	+	CCONJ
cana-510	105	6	(	(	PUNCT
cana-510	105	7	1	1	NUM
cana-510	105	8	−	−	NOUN
cana-510	105	9	𝜔	𝜔	NOUN
cana-510	105	10	)	)	PUNCT
cana-510	105	11	)	)	PUNCT
cana-510	105	12	𝛩(2	𝛩(2	NUM
cana-510	105	13	,	,	PUNCT
cana-510	105	14	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	105	15	+	+	NUM
cana-510	105	16	ς(2𝜎	ς(2𝜎	NUM
cana-510	105	17	+	+	CCONJ
cana-510	105	18	1	1	NUM
cana-510	105	19	−	−	NOUN
cana-510	105	20	𝜔	𝜔	NOUN
cana-510	105	21	)	)	PUNCT
cana-510	105	22	]	]	PUNCT
cana-510	105	23	|𝑧|	|𝑧|	PROPN
cana-510	105	24	≤	≤	NUM
cana-510	105	25	|𝑢′(𝑧)|	|𝑢′(𝑧)|	NUM
cana-510	105	26	≤	≤	NUM
cana-510	105	27	1	1	NUM
cana-510	105	28	+	+	NUM
cana-510	105	29	2ς(𝜎	2ς(𝜎	NUM
cana-510	105	30	+	+	CCONJ
cana-510	105	31	(	(	PUNCT
cana-510	105	32	1	1	NUM
cana-510	105	33	−	−	NOUN
cana-510	105	34	𝜔	𝜔	NOUN
cana-510	105	35	)	)	PUNCT
cana-510	105	36	)	)	PUNCT
cana-510	105	37	𝛩(2	𝛩(2	NUM
cana-510	105	38	,	,	PUNCT
cana-510	105	39	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	105	40	+	+	NUM
cana-510	105	41	ς(2𝜎	ς(2𝜎	NUM
cana-510	105	42	+	+	CCONJ
cana-510	105	43	1	1	NUM
cana-510	105	44	−	−	NOUN
cana-510	105	45	𝜔	𝜔	NOUN
cana-510	105	46	)	)	PUNCT
cana-510	105	47	]	]	PUNCT
cana-510	105	48	|𝑧|	|𝑧|	NOUN
cana-510	105	49	with	with	ADP
cana-510	105	50	equality	equality	NOUN
cana-510	105	51	for	for	ADP
cana-510	105	52	𝑢(𝑧	𝑢(𝑧	NOUN
cana-510	105	53	)	)	PUNCT
cana-510	105	54	=	=	PUNCT
cana-510	105	55	𝑧	𝑧	DET
cana-510	105	56	−	−	PROPN
cana-510	105	57	2ς(𝜎+(1−𝜔	2ς(𝜎+(1−𝜔	NUM
cana-510	105	58	)	)	PUNCT
cana-510	105	59	)	)	PUNCT
cana-510	105	60	𝛩(2,𝜗)[𝜔+ς(2𝜎+1−𝜔	𝛩(2,𝜗)[𝜔+ς(2𝜎+1−𝜔	ADJ
cana-510	105	61	)	)	PUNCT
cana-510	105	62	]	]	PUNCT
cana-510	106	1	𝑧2	𝑧2	NOUN
cana-510	106	2	.	.	PUNCT
cana-510	106	3	proof	proof	NOUN
cana-510	106	4	.	.	PUNCT
cana-510	107	1	notice	notice	VERB
cana-510	107	2	that	that	SCONJ
cana-510	107	3	𝛩(2	𝛩(2	NUM
cana-510	107	4	,	,	PUNCT
cana-510	107	5	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	107	6	+	+	NUM
cana-510	107	7	ς(2𝜎	ς(2𝜎	NUM
cana-510	107	8	+	+	CCONJ
cana-510	107	9	1	1	NUM
cana-510	107	10	−	−	NOUN
cana-510	107	11	𝜔	𝜔	NOUN
cana-510	107	12	)	)	PUNCT
cana-510	107	13	]	]	PUNCT
cana-510	107	14	∑	∑	PUNCT
cana-510	107	15	𝑛	𝑛	PRON
cana-510	107	16	∞	∞	NUM
cana-510	107	17	𝑛=2	𝑛=2	PROPN
cana-510	107	18	𝑎𝑛	𝑎𝑛	VERB
cana-510	107	19	≤	≤	NUM
cana-510	107	20	∑	∑	PUNCT
cana-510	107	21	𝑛	𝑛	DET
cana-510	107	22	∞	∞	NUM
cana-510	107	23	𝑛=2	𝑛=2	PROPN
cana-510	108	1	[	[	X
cana-510	108	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	108	3	−	−	PROPN
cana-510	108	4	1	1	NUM
cana-510	108	5	)	)	PUNCT
cana-510	108	6	+	+	NOUN
cana-510	108	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	108	8	+	+	CCONJ
cana-510	108	9	1	1	NUM
cana-510	108	10	−	−	NOUN
cana-510	108	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	ADP
cana-510	108	12	,	,	PUNCT
cana-510	108	13	𝜗)𝑎𝑛	𝜗)𝑎𝑛	PROPN
cana-510	108	14	≤	≤	PROPN
cana-510	108	15	ς(𝜎	ς(𝜎	PROPN
cana-510	108	16	+	+	CCONJ
cana-510	108	17	(	(	PUNCT
cana-510	108	18	1	1	NUM
cana-510	108	19	−	−	NOUN
cana-510	108	20	𝜔	𝜔	NOUN
cana-510	108	21	)	)	PUNCT
cana-510	108	22	)	)	PUNCT
cana-510	108	23	,	,	PUNCT
cana-510	108	24	(	(	PUNCT
cana-510	108	25	3.2	3.2	NUM
cana-510	108	26	)	)	PUNCT
cana-510	108	27	from	from	ADP
cana-510	108	28	theorem	theorem	ADJ
cana-510	108	29	2.1	2.1	NUM
cana-510	108	30	.	.	PUNCT
cana-510	109	1	thus	thus	ADV
cana-510	109	2	|𝑢′(𝑧)|	|𝑢′(𝑧)|	NUM
cana-510	109	3	=	=	SYM
cana-510	109	4	|1	|1	NUM
cana-510	109	5	−	−	NOUN
cana-510	109	6	∑	∑	PUNCT
cana-510	109	7	𝑛	𝑛	DET
cana-510	109	8	∞	∞	NUM
cana-510	109	9	𝑛=2	𝑛=2	X
cana-510	109	10	𝑎𝑛𝑧𝑛−1|	𝑎𝑛𝑧𝑛−1|	NOUN
cana-510	109	11	≤	≤	NOUN
cana-510	109	12	1	1	NUM
cana-510	109	13	+	+	CCONJ
cana-510	109	14	∑	∑	PROPN
cana-510	109	15	𝑛	𝑛	DET
cana-510	109	16	∞	∞	PROPN
cana-510	109	17	𝑛=2	𝑛=2	X
cana-510	109	18	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	NOUN
cana-510	109	19	≤	≤	NUM
cana-510	109	20	1	1	NUM
cana-510	109	21	+	+	NUM
cana-510	109	22	|𝑧|	|𝑧|	PROPN
cana-510	109	23	∑	∑	DET
cana-510	109	24	𝑛	𝑛	DET
cana-510	109	25	∞	∞	NUM
cana-510	109	26	𝑛=2	𝑛=2	NOUN
cana-510	109	27	𝑎𝑛	𝑎𝑛	VERB
cana-510	109	28	≤	≤	NUM
cana-510	109	29	1	1	NUM
cana-510	109	30	+	+	NUM
cana-510	109	31	|𝑧|	|𝑧|	PROPN
cana-510	109	32	2ς(𝜎	2ς(𝜎	NUM
cana-510	109	33	+	+	CCONJ
cana-510	109	34	(	(	PUNCT
cana-510	109	35	1	1	NUM
cana-510	109	36	−	−	NOUN
cana-510	109	37	𝜔	𝜔	NOUN
cana-510	109	38	)	)	PUNCT
cana-510	109	39	)	)	PUNCT
cana-510	109	40	𝛩(2	𝛩(2	NUM
cana-510	109	41	,	,	PUNCT
cana-510	109	42	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	109	43	+	+	NUM
cana-510	109	44	ς(2𝜎	ς(2𝜎	NUM
cana-510	109	45	+	+	CCONJ
cana-510	109	46	1	1	NUM
cana-510	109	47	−	−	NOUN
cana-510	109	48	𝜔	𝜔	NOUN
cana-510	109	49	)	)	PUNCT
cana-510	109	50	]	]	PUNCT
cana-510	109	51	.	.	PUNCT
cana-510	110	1	(	(	PUNCT
cana-510	110	2	3.3	3.3	NUM
cana-510	110	3	)	)	PUNCT
cana-510	110	4	communications	communication	NOUN
cana-510	110	5	on	on	ADP
cana-510	110	6	applied	apply	VERB
cana-510	110	7	nonlinear	nonlinear	ADJ
cana-510	110	8	analysis	analysis	NOUN
cana-510	110	9	issn	issn	NOUN
cana-510	110	10	:	:	PUNCT
cana-510	110	11	1074	1074	NUM
cana-510	110	12	-	-	PUNCT
cana-510	110	13	133x	133x	NUM
cana-510	110	14	vol	vol	NOUN
cana-510	110	15	31	31	NUM
cana-510	110	16	no	no	NOUN
cana-510	110	17	.	.	NOUN
cana-510	110	18	2	2	NUM
cana-510	110	19	(	(	PUNCT
cana-510	110	20	2024	2024	NUM
cana-510	110	21	)	)	PUNCT
cana-510	110	22	27	27	NUM
cana-510	110	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	110	24	on	on	ADP
cana-510	110	25	the	the	DET
cana-510	110	26	other	other	ADJ
cana-510	110	27	hand	hand	NOUN
cana-510	111	1	|𝑢′(𝑧)|	|𝑢′(𝑧)|	NOUN
cana-510	111	2	=	=	SYM
cana-510	111	3	|1	|1	NUM
cana-510	111	4	−	−	NOUN
cana-510	111	5	∑	∑	PUNCT
cana-510	111	6	𝑛	𝑛	DET
cana-510	111	7	∞	∞	NUM
cana-510	111	8	𝑛=2	𝑛=2	X
cana-510	111	9	𝑎𝑛𝑧𝑛−1|	𝑎𝑛𝑧𝑛−1|	NOUN
cana-510	111	10	≥	≥	NOUN
cana-510	111	11	1	1	NUM
cana-510	111	12	−	−	NOUN
cana-510	111	13	∑	∑	PUNCT
cana-510	111	14	𝑛	𝑛	DET
cana-510	111	15	∞	∞	PROPN
cana-510	111	16	𝑛=2	𝑛=2	X
cana-510	111	17	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	NUM
cana-510	111	18	≥	≥	NOUN
cana-510	111	19	1	1	NUM
cana-510	111	20	−	−	PROPN
cana-510	111	21	|𝑧|	|𝑧|	PROPN
cana-510	111	22	∑	∑	ADP
cana-510	111	23	𝑛	𝑛	DET
cana-510	111	24	∞	∞	NUM
cana-510	111	25	𝑛=2	𝑛=2	PROPN
cana-510	111	26	𝑎𝑛	𝑎𝑛	PRON
cana-510	111	27	≥	≥	NUM
cana-510	111	28	1	1	NUM
cana-510	111	29	−	−	PROPN
cana-510	111	30	|𝑧|	|𝑧|	PROPN
cana-510	111	31	2ς(𝜎	2ς(𝜎	NUM
cana-510	111	32	+	+	CCONJ
cana-510	112	1	(	(	PUNCT
cana-510	112	2	1	1	NUM
cana-510	112	3	−	−	NOUN
cana-510	112	4	𝜔	𝜔	NOUN
cana-510	112	5	)	)	PUNCT
cana-510	112	6	)	)	PUNCT
cana-510	113	1	𝛩(2	𝛩(2	NUM
cana-510	113	2	,	,	PUNCT
cana-510	114	1	𝜗)[𝜔	𝜗)[𝜔	VERB
cana-510	114	2	+	+	NUM
cana-510	114	3	ς(2𝜎	ς(2𝜎	NUM
cana-510	114	4	+	+	CCONJ
cana-510	114	5	1	1	NUM
cana-510	114	6	−	−	NOUN
cana-510	114	7	𝜔	𝜔	NOUN
cana-510	114	8	)	)	PUNCT
cana-510	114	9	]	]	PUNCT
cana-510	114	10	(	(	PUNCT
cana-510	114	11	3.4	3.4	NUM
cana-510	114	12	)	)	PUNCT
cana-510	114	13	.	.	PUNCT
cana-510	115	1	combining	combine	VERB
cana-510	115	2	(	(	PUNCT
cana-510	115	3	3.3	3.3	NUM
cana-510	115	4	)	)	PUNCT
cana-510	115	5	and	and	CCONJ
cana-510	115	6	(	(	PUNCT
cana-510	115	7	3.4	3.4	NUM
cana-510	115	8	)	)	PUNCT
cana-510	115	9	,	,	PUNCT
cana-510	115	10	we	we	PRON
cana-510	115	11	get	get	VERB
cana-510	115	12	the	the	DET
cana-510	115	13	result	result	NOUN
cana-510	115	14	.	.	PUNCT
cana-510	116	1	4	4	X
cana-510	116	2	.	.	X
cana-510	116	3	radii	radius	NOUN
cana-510	116	4	of	of	ADP
cana-510	116	5	starlikeness	starlikeness	NOUN
cana-510	116	6	,	,	PUNCT
cana-510	116	7	convexity	convexity	NOUN
cana-510	116	8	and	and	CCONJ
cana-510	116	9	close	close	NOUN
cana-510	116	10	-	-	PUNCT
cana-510	116	11	to	to	ADP
cana-510	116	12	-	-	PUNCT
cana-510	116	13	convexity	convexity	NOUN
cana-510	116	14	in	in	ADP
cana-510	116	15	the	the	DET
cana-510	116	16	following	follow	VERB
cana-510	116	17	theorems	theorem	NOUN
cana-510	116	18	,	,	PUNCT
cana-510	116	19	we	we	PRON
cana-510	116	20	obtain	obtain	VERB
cana-510	116	21	the	the	DET
cana-510	116	22	radii	radius	NOUN
cana-510	116	23	of	of	ADP
cana-510	116	24	starlikeness	starlikeness	NOUN
cana-510	116	25	,	,	PUNCT
cana-510	116	26	convexity	convexity	NOUN
cana-510	116	27	and	and	CCONJ
cana-510	116	28	close	close	NOUN
cana-510	116	29	-	-	PUNCT
cana-510	116	30	to	to	ADP
cana-510	116	31	-	-	PUNCT
cana-510	116	32	convexity	convexity	NOUN
cana-510	116	33	for	for	ADP
cana-510	116	34	the	the	DET
cana-510	116	35	class	class	NOUN
cana-510	116	36	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	116	37	,	,	PUNCT
cana-510	116	38	𝜎	𝜎	PROPN
cana-510	116	39	,	,	PUNCT
cana-510	116	40	ς	ς	PROPN
cana-510	116	41	,	,	PUNCT
cana-510	116	42	𝜗	𝜗	NOUN
cana-510	116	43	)	)	PUNCT
cana-510	116	44	.	.	PUNCT
cana-510	117	1	theorem	theorem	VERB
cana-510	117	2	4.1	4.1	NUM
cana-510	117	3	.	.	PUNCT
cana-510	118	1	let	let	VERB
cana-510	118	2	𝑢	𝑢	PRON
cana-510	118	3	∈	∈	PROPN
cana-510	118	4	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	118	5	,	,	PUNCT
cana-510	118	6	𝜎	𝜎	PROPN
cana-510	118	7	,	,	PUNCT
cana-510	118	8	ς	ς	PROPN
cana-510	118	9	,	,	PUNCT
cana-510	118	10	𝜗	𝜗	NOUN
cana-510	118	11	)	)	PUNCT
cana-510	118	12	.	.	PUNCT
cana-510	119	1	then	then	ADV
cana-510	119	2	𝑢	𝑢	PROPN
cana-510	119	3	is	be	AUX
cana-510	119	4	starlike	starlike	NOUN
cana-510	119	5	in	in	ADP
cana-510	119	6	|𝑧|	|𝑧|	PROPN
cana-510	119	7	<	<	X
cana-510	119	8	𝑅1	𝑅1	NOUN
cana-510	119	9	of	of	ADP
cana-510	119	10	order	order	NOUN
cana-510	119	11	𝛿	𝛿	ADJ
cana-510	119	12	,	,	PUNCT
cana-510	119	13	0	0	NUM
cana-510	119	14	≤	≤	NOUN
cana-510	119	15	𝛿	𝛿	PRON
cana-510	119	16	<	<	X
cana-510	119	17	1	1	NUM
cana-510	119	18	,	,	PUNCT
cana-510	119	19	where	where	SCONJ
cana-510	119	20	𝑅1	𝑅1	PROPN
cana-510	119	21	=	=	SYM
cana-510	119	22	inf	inf	PROPN
cana-510	119	23	𝑛	𝑛	PROPN
cana-510	119	24	{	{	PUNCT
cana-510	119	25	(	(	PUNCT
cana-510	119	26	1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗	1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗	NUM
cana-510	119	27	)	)	PUNCT
cana-510	119	28	(	(	PUNCT
cana-510	119	29	𝑛−𝛿)ς(𝜎+(1−𝜔	𝑛−𝛿)ς(𝜎+(1−𝜔	NOUN
cana-510	119	30	)	)	PUNCT
cana-510	119	31	)	)	PUNCT
cana-510	119	32	}	}	PUNCT
cana-510	119	33	1	1	NUM
cana-510	119	34	𝑛−1	𝑛−1	PROPN
cana-510	119	35	,	,	PUNCT
cana-510	119	36	𝑛	𝑛	PRON
cana-510	119	37	≥	≥	NUM
cana-510	119	38	2	2	NUM
cana-510	119	39	(	(	PUNCT
cana-510	119	40	4.1	4.1	NUM
cana-510	119	41	)	)	PUNCT
cana-510	119	42	.	.	PUNCT
cana-510	120	1	proof	proof	NOUN
cana-510	120	2	.	.	PUNCT
cana-510	121	1	𝑢	𝑢	PROPN
cana-510	121	2	is	be	AUX
cana-510	121	3	starlike	starlike	NOUN
cana-510	121	4	of	of	ADP
cana-510	121	5	order	order	NOUN
cana-510	121	6	𝛿	𝛿	ADJ
cana-510	121	7	,	,	PUNCT
cana-510	121	8	0	0	NUM
cana-510	121	9	≤	≤	NOUN
cana-510	121	10	𝛿	𝛿	PRON
cana-510	121	11	<	<	X
cana-510	121	12	1	1	NUM
cana-510	121	13	if	if	SCONJ
cana-510	121	14	ℜ	ℜ	ADV
cana-510	121	15	{	{	PUNCT
cana-510	121	16	𝑧𝑢′(𝑧	𝑧𝑢′(𝑧	NOUN
cana-510	121	17	)	)	PUNCT
cana-510	121	18	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	121	19	)	)	PUNCT
cana-510	121	20	}	}	PUNCT
cana-510	121	21	>	>	X
cana-510	121	22	𝛿.	𝛿.	NOUN
cana-510	122	1	thus	thus	ADV
cana-510	122	2	it	it	PRON
cana-510	122	3	is	be	AUX
cana-510	122	4	enough	enough	ADJ
cana-510	122	5	to	to	PART
cana-510	122	6	show	show	VERB
cana-510	122	7	that	that	SCONJ
cana-510	122	8	|	|	ADV
cana-510	122	9	𝑧𝑢′(𝑧	𝑧𝑢′(𝑧	ADJ
cana-510	122	10	)	)	PUNCT
cana-510	122	11	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	122	12	)	)	PUNCT
cana-510	122	13	−	−	PROPN
cana-510	123	1	1|	1|	NUM
cana-510	124	1	=	=	SYM
cana-510	125	1	|	|	ADV
cana-510	125	2	−	−	NOUN
cana-510	125	3	∑	∑	PROPN
cana-510	125	4	(	(	PUNCT
cana-510	125	5	𝑛	𝑛	PROPN
cana-510	125	6	−	−	PROPN
cana-510	125	7	1)∞	1)∞	NUM
cana-510	125	8	𝑛=2	𝑛=2	X
cana-510	125	9	𝑎𝑛𝑧𝑛−1	𝑎𝑛𝑧𝑛−1	PROPN
cana-510	125	10	1	1	NUM
cana-510	125	11	−	−	NOUN
cana-510	125	12	∑	∑	PUNCT
cana-510	125	13	𝑎𝑛	𝑎𝑛	PROPN
cana-510	125	14	∞	∞	NUM
cana-510	125	15	𝑛=2	𝑛=2	PROPN
cana-510	125	16	𝑧𝑛−1	𝑧𝑛−1	NOUN
cana-510	125	17	|	|	ADV
cana-510	125	18	≤	≤	NUM
cana-510	125	19	∑	∑	PUNCT
cana-510	125	20	(	(	PUNCT
cana-510	125	21	𝑛	𝑛	PRON
cana-510	125	22	−	−	PROPN
cana-510	125	23	1)∞	1)∞	NUM
cana-510	125	24	𝑛=2	𝑛=2	NOUN
cana-510	125	25	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	PROPN
cana-510	125	26	1	1	NUM
cana-510	125	27	−	−	NOUN
cana-510	125	28	∑	∑	PUNCT
cana-510	125	29	𝑎𝑛	𝑎𝑛	PROPN
cana-510	125	30	∞	∞	NUM
cana-510	125	31	𝑛=2	𝑛=2	X
cana-510	125	32	|𝑧|𝑛−1	|𝑧|𝑛−1	NOUN
cana-510	125	33	.	.	PUNCT
cana-510	126	1	(	(	PUNCT
cana-510	126	2	4.2	4.2	NUM
cana-510	126	3	)	)	PUNCT
cana-510	126	4	thus	thus	ADV
cana-510	126	5	|	|	ADV
cana-510	126	6	𝑧𝑢′(𝑧	𝑧𝑢′(𝑧	ADJ
cana-510	126	7	)	)	PUNCT
cana-510	126	8	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	126	9	)	)	PUNCT
cana-510	126	10	−	−	ADP
cana-510	126	11	1|	1|	NUM
cana-510	126	12	≤	≤	NUM
cana-510	126	13	1	1	NUM
cana-510	126	14	−	−	NOUN
cana-510	126	15	𝛿	𝛿	PROPN
cana-510	126	16	𝑖𝑓	𝑖𝑓	ADP
cana-510	126	17	∑	∑	PROPN
cana-510	126	18	(	(	PUNCT
cana-510	126	19	𝑛	𝑛	PRON
cana-510	126	20	−	−	PROPN
cana-510	126	21	𝛿	𝛿	NOUN
cana-510	126	22	)	)	PUNCT
cana-510	126	23	(	(	PUNCT
cana-510	126	24	1	1	NUM
cana-510	126	25	−	−	PROPN
cana-510	126	26	𝛿	𝛿	ADJ
cana-510	126	27	)	)	PUNCT
cana-510	126	28	∞	∞	NUM
cana-510	126	29	𝑛=2	𝑛=2	NOUN
cana-510	126	30	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	NOUN
cana-510	126	31	≤	≤	NUM
cana-510	126	32	1	1	NUM
cana-510	126	33	.	.	PUNCT
cana-510	126	34	hence	hence	ADV
cana-510	126	35	by	by	ADP
cana-510	126	36	theorem	theorem	NOUN
cana-510	126	37	2.1	2.1	NUM
cana-510	126	38	,	,	PUNCT
cana-510	126	39	(	(	PUNCT
cana-510	126	40	4.2	4.2	NUM
cana-510	126	41	)	)	PUNCT
cana-510	126	42	will	will	AUX
cana-510	126	43	be	be	AUX
cana-510	126	44	true	true	ADJ
cana-510	126	45	if	if	SCONJ
cana-510	126	46	𝑛	𝑛	PRON
cana-510	126	47	−	−	NOUN
cana-510	126	48	𝛿	𝛿	ADJ
cana-510	126	49	1	1	NUM
cana-510	126	50	−	−	NOUN
cana-510	126	51	𝛿	𝛿	DET
cana-510	126	52	|𝑧|𝑛−1	|𝑧|𝑛−1	X
cana-510	126	53	≤	≤	NOUN
cana-510	126	54	(	(	PUNCT
cana-510	126	55	𝜔(𝑛	𝜔(𝑛	PROPN
cana-510	126	56	−	−	PROPN
cana-510	126	57	1	1	NUM
cana-510	126	58	)	)	PUNCT
cana-510	127	1	+	+	NOUN
cana-510	127	2	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	127	3	+	+	CCONJ
cana-510	127	4	1	1	NUM
cana-510	127	5	−	−	NOUN
cana-510	127	6	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	127	7	,	,	PUNCT
cana-510	127	8	𝜗	𝜗	NOUN
cana-510	127	9	)	)	PUNCT
cana-510	127	10	ς(𝜎	ς(𝜎	NOUN
cana-510	127	11	+	+	CCONJ
cana-510	127	12	(	(	PUNCT
cana-510	127	13	1	1	NUM
cana-510	127	14	−	−	NOUN
cana-510	127	15	𝜔	𝜔	NOUN
cana-510	127	16	)	)	PUNCT
cana-510	127	17	communications	communication	NOUN
cana-510	127	18	on	on	ADP
cana-510	127	19	applied	apply	VERB
cana-510	127	20	nonlinear	nonlinear	ADJ
cana-510	127	21	analysis	analysis	NOUN
cana-510	127	22	issn	issn	NOUN
cana-510	127	23	:	:	PUNCT
cana-510	127	24	1074	1074	NUM
cana-510	127	25	-	-	PUNCT
cana-510	127	26	133x	133x	NUM
cana-510	127	27	vol	vol	NOUN
cana-510	127	28	31	31	NUM
cana-510	127	29	no	no	NOUN
cana-510	127	30	.	.	NOUN
cana-510	127	31	2	2	NUM
cana-510	127	32	(	(	PUNCT
cana-510	127	33	2024	2024	NUM
cana-510	127	34	)	)	PUNCT
cana-510	127	35	28	28	NUM
cana-510	127	36	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	127	37	or	or	CCONJ
cana-510	127	38	if	if	SCONJ
cana-510	127	39	|𝑧|	|𝑧|	PROPN
cana-510	127	40	≤	≤	X
cana-510	127	41	[	[	PUNCT
cana-510	127	42	(	(	PUNCT
cana-510	127	43	1	1	NUM
cana-510	127	44	−	−	NOUN
cana-510	127	45	𝛿)(𝜔(𝑛	𝛿)(𝜔(𝑛	PUNCT
cana-510	128	1	−	−	NOUN
cana-510	128	2	1	1	NUM
cana-510	128	3	)	)	PUNCT
cana-510	128	4	+	+	NOUN
cana-510	128	5	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	128	6	+	+	CCONJ
cana-510	128	7	1	1	NUM
cana-510	128	8	−	−	NOUN
cana-510	128	9	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	128	10	,	,	PUNCT
cana-510	128	11	𝜗	𝜗	NOUN
cana-510	128	12	)	)	PUNCT
cana-510	128	13	(	(	PUNCT
cana-510	128	14	𝑛	𝑛	DET
cana-510	128	15	−	−	NUM
cana-510	128	16	𝛿)ς(𝜎	𝛿)ς(𝜎	NOUN
cana-510	128	17	+	+	CCONJ
cana-510	128	18	(	(	PUNCT
cana-510	128	19	1	1	NUM
cana-510	128	20	−	−	NOUN
cana-510	128	21	𝜔	𝜔	NOUN
cana-510	128	22	)	)	PUNCT
cana-510	128	23	)	)	PUNCT
cana-510	129	1	]	]	PUNCT
cana-510	129	2	1	1	NUM
cana-510	129	3	𝑛−1	𝑛−1	NUM
cana-510	129	4	,	,	PUNCT
cana-510	129	5	𝑛	𝑛	PRON
cana-510	129	6	≥	≥	NOUN
cana-510	129	7	2	2	NUM
cana-510	129	8	.	.	PUNCT
cana-510	130	1	the	the	DET
cana-510	130	2	theorem	theorem	NOUN
cana-510	130	3	is	be	AUX
cana-510	130	4	proved	prove	VERB
cana-510	130	5	.	.	PUNCT
cana-510	131	1	theorem	theorem	VERB
cana-510	131	2	4.2	4.2	NUM
cana-510	131	3	.	.	PUNCT
cana-510	132	1	let	let	VERB
cana-510	132	2	𝑢	𝑢	PRON
cana-510	132	3	∈	∈	PROPN
cana-510	132	4	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	132	5	,	,	PUNCT
cana-510	132	6	𝜎	𝜎	PROPN
cana-510	132	7	,	,	PUNCT
cana-510	132	8	ς	ς	PROPN
cana-510	132	9	,	,	PUNCT
cana-510	132	10	𝜗	𝜗	NOUN
cana-510	132	11	)	)	PUNCT
cana-510	132	12	.	.	PUNCT
cana-510	133	1	then	then	ADV
cana-510	133	2	u	u	PRON
cana-510	133	3	is	be	AUX
cana-510	133	4	convex	convex	ADJ
cana-510	133	5	in	in	ADP
cana-510	133	6	|z|	|z|	NOUN
cana-510	133	7	<	<	X
cana-510	133	8	r2	r2	NOUN
cana-510	133	9	of	of	ADP
cana-510	133	10	order	order	NOUN
cana-510	133	11	δ	δ	PROPN
cana-510	133	12	,	,	PUNCT
cana-510	133	13	0	0	NUM
cana-510	133	14	≤	≤	NUM
cana-510	133	15	δ	δ	X
cana-510	133	16	<	<	X
cana-510	133	17	1	1	NUM
cana-510	133	18	,	,	PUNCT
cana-510	133	19	where	where	SCONJ
cana-510	133	20	𝑅2	𝑅2	NOUN
cana-510	133	21	=	=	SYM
cana-510	133	22	inf	inf	ADJ
cana-510	133	23	𝑛	𝑛	PROPN
cana-510	133	24	{	{	PUNCT
cana-510	133	25	(	(	PUNCT
cana-510	133	26	1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗	1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗	NUM
cana-510	133	27	)	)	PUNCT
cana-510	133	28	𝑛(𝑛−𝛿)ς(𝜎+(1−𝜔	𝑛(𝑛−𝛿)ς(𝜎+(1−𝜔	NOUN
cana-510	133	29	)	)	PUNCT
cana-510	133	30	)	)	PUNCT
cana-510	133	31	}	}	PUNCT
cana-510	133	32	1	1	NUM
cana-510	133	33	𝑛−1	𝑛−1	PROPN
cana-510	133	34	,	,	PUNCT
cana-510	133	35	𝑛	𝑛	DET
cana-510	133	36	≥	≥	NOUN
cana-510	133	37	2	2	NUM
cana-510	133	38	.	.	PUNCT
cana-510	134	1	(	(	PUNCT
cana-510	134	2	4.3	4.3	NUM
cana-510	134	3	)	)	PUNCT
cana-510	134	4	proof	proof	NOUN
cana-510	134	5	.	.	PUNCT
cana-510	135	1	𝑢	𝑢	PROPN
cana-510	135	2	is	be	AUX
cana-510	135	3	convex	convex	NOUN
cana-510	135	4	of	of	ADP
cana-510	135	5	order	order	NOUN
cana-510	135	6	𝛿	𝛿	ADJ
cana-510	135	7	,	,	PUNCT
cana-510	135	8	0	0	NUM
cana-510	135	9	≤	≤	NOUN
cana-510	136	1	𝛿	𝛿	DET
cana-510	136	2	<	<	X
cana-510	136	3	1	1	NUM
cana-510	136	4	if	if	SCONJ
cana-510	136	5	ℜ	ℜ	ADJ
cana-510	136	6	{	{	PUNCT
cana-510	136	7	1	1	NUM
cana-510	136	8	+	+	CCONJ
cana-510	136	9	𝑧𝑢″(𝑧	𝑧𝑢″(𝑧	NUM
cana-510	136	10	)	)	PUNCT
cana-510	136	11	𝑢′(𝑧	𝑢′(𝑧	X
cana-510	136	12	)	)	PUNCT
cana-510	136	13	}	}	PUNCT
cana-510	136	14	>	>	X
cana-510	136	15	𝛿.	𝛿.	NOUN
cana-510	136	16	thus	thus	ADV
cana-510	136	17	it	it	PRON
cana-510	136	18	is	be	AUX
cana-510	136	19	enough	enough	ADJ
cana-510	136	20	to	to	PART
cana-510	136	21	show	show	VERB
cana-510	136	22	that	that	SCONJ
cana-510	136	23	|	|	ADV
cana-510	136	24	𝑧𝑢″(𝑧	𝑧𝑢″(𝑧	NOUN
cana-510	136	25	)	)	PUNCT
cana-510	136	26	𝑢′(𝑧	𝑢′(𝑧	NUM
cana-510	136	27	)	)	PUNCT
cana-510	137	1	|	|	ADV
cana-510	137	2	=	=	SYM
cana-510	137	3	|	|	ADV
cana-510	137	4	−	−	NOUN
cana-510	137	5	∑	∑	ADP
cana-510	137	6	𝑛∞	𝑛∞	ADJ
cana-510	137	7	𝑛=2	𝑛=2	PROPN
cana-510	137	8	(	(	PUNCT
cana-510	137	9	𝑛	𝑛	PRON
cana-510	137	10	−	−	PROPN
cana-510	137	11	1)𝑎𝑛𝑧𝑛−1	1)𝑎𝑛𝑧𝑛−1	NUM
cana-510	137	12	1	1	NUM
cana-510	137	13	−	−	NOUN
cana-510	137	14	∑	∑	ADP
cana-510	137	15	𝑛∞	𝑛∞	ADJ
cana-510	137	16	𝑛=2	𝑛=2	X
cana-510	137	17	𝑎𝑛𝑧𝑛−1	𝑎𝑛𝑧𝑛−1	PROPN
cana-510	137	18	|	|	ADV
cana-510	137	19	≤	≤	NUM
cana-510	137	20	∑	∑	PUNCT
cana-510	137	21	𝑛∞	𝑛∞	ADJ
cana-510	137	22	𝑛=2	𝑛=2	NOUN
cana-510	137	23	(	(	PUNCT
cana-510	137	24	𝑛	𝑛	PROPN
cana-510	137	25	−	−	PROPN
cana-510	137	26	1)𝑎𝑛|𝑧|𝑛−1	1)𝑎𝑛|𝑧|𝑛−1	NUM
cana-510	137	27	1	1	NUM
cana-510	137	28	−	−	NOUN
cana-510	137	29	∑	∑	ADP
cana-510	137	30	𝑛∞	𝑛∞	ADJ
cana-510	137	31	𝑛=2	𝑛=2	NOUN
cana-510	137	32	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	NUM
cana-510	137	33	.	.	PUNCT
cana-510	138	1	(	(	PUNCT
cana-510	138	2	4.4	4.4	NUM
cana-510	138	3	)	)	PUNCT
cana-510	138	4	thus	thus	ADV
cana-510	138	5	|	|	ADV
cana-510	138	6	𝑧𝑢″(𝑧	𝑧𝑢″(𝑧	NOUN
cana-510	138	7	)	)	PUNCT
cana-510	138	8	𝑢′(𝑧	𝑢′(𝑧	PUNCT
cana-510	138	9	)	)	PUNCT
cana-510	139	1	|	|	ADV
cana-510	139	2	≤	≤	NUM
cana-510	139	3	1	1	NUM
cana-510	139	4	−	−	NOUN
cana-510	139	5	𝛿	𝛿	PROPN
cana-510	139	6	𝑖𝑓	𝑖𝑓	PRON
cana-510	139	7	∑	∑	PUNCT
cana-510	139	8	𝑛(𝑛	𝑛(𝑛	PROPN
cana-510	139	9	−	−	PROPN
cana-510	139	10	𝛿	𝛿	NOUN
cana-510	139	11	)	)	PUNCT
cana-510	139	12	(	(	PUNCT
cana-510	139	13	1	1	NUM
cana-510	139	14	−	−	PROPN
cana-510	139	15	𝛿	𝛿	ADJ
cana-510	139	16	)	)	PUNCT
cana-510	139	17	∞	∞	NUM
cana-510	139	18	𝑛=2	𝑛=2	NOUN
cana-510	139	19	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	NOUN
cana-510	139	20	≤	≤	NUM
cana-510	139	21	1	1	NUM
cana-510	139	22	.	.	PUNCT
cana-510	139	23	hence	hence	ADV
cana-510	139	24	by	by	ADP
cana-510	139	25	theorem	theorem	NOUN
cana-510	139	26	2.1	2.1	NUM
cana-510	139	27	,	,	PUNCT
cana-510	139	28	(	(	PUNCT
cana-510	139	29	4.4)will	4.4)will	PRON
cana-510	139	30	be	be	AUX
cana-510	139	31	true	true	ADJ
cana-510	139	32	if	if	SCONJ
cana-510	139	33	𝑛(𝑛	𝑛(𝑛	PROPN
cana-510	139	34	−	−	PROPN
cana-510	139	35	𝛿	𝛿	NOUN
cana-510	139	36	)	)	PUNCT
cana-510	139	37	1	1	NUM
cana-510	139	38	−	−	PROPN
cana-510	139	39	𝛿	𝛿	DET
cana-510	139	40	|𝑧|𝑛−1	|𝑧|𝑛−1	X
cana-510	139	41	≤	≤	NOUN
cana-510	139	42	(	(	PUNCT
cana-510	139	43	𝜔(𝑛	𝜔(𝑛	PROPN
cana-510	139	44	−	−	PROPN
cana-510	139	45	1	1	NUM
cana-510	139	46	)	)	PUNCT
cana-510	139	47	+	+	NOUN
cana-510	139	48	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	139	49	+	+	CCONJ
cana-510	139	50	1	1	NUM
cana-510	139	51	−	−	NOUN
cana-510	139	52	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	139	53	,	,	PUNCT
cana-510	139	54	𝜗	𝜗	NOUN
cana-510	139	55	)	)	PUNCT
cana-510	139	56	ς(𝜎	ς(𝜎	NOUN
cana-510	140	1	+	+	CCONJ
cana-510	140	2	(	(	PUNCT
cana-510	140	3	1	1	NUM
cana-510	140	4	−	−	NOUN
cana-510	140	5	𝜔	𝜔	NOUN
cana-510	140	6	)	)	PUNCT
cana-510	140	7	or	or	CCONJ
cana-510	140	8	if	if	SCONJ
cana-510	140	9	|𝑧|	|𝑧|	PROPN
cana-510	140	10	≤	≤	X
cana-510	140	11	[	[	PUNCT
cana-510	140	12	(	(	PUNCT
cana-510	140	13	1	1	NUM
cana-510	140	14	−	−	NOUN
cana-510	140	15	𝛿)(𝜔(𝑛	𝛿)(𝜔(𝑛	PUNCT
cana-510	140	16	−	−	NOUN
cana-510	140	17	1	1	NUM
cana-510	140	18	)	)	PUNCT
cana-510	141	1	+	+	NOUN
cana-510	141	2	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	141	3	+	+	CCONJ
cana-510	141	4	1	1	NUM
cana-510	141	5	−	−	NOUN
cana-510	141	6	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	141	7	,	,	PUNCT
cana-510	141	8	𝜗	𝜗	NOUN
cana-510	141	9	)	)	PUNCT
cana-510	141	10	𝑛(𝑛	𝑛(𝑛	PROPN
cana-510	141	11	−	−	NOUN
cana-510	141	12	𝛿)ς(𝜎	𝛿)ς(𝜎	NOUN
cana-510	141	13	+	+	CCONJ
cana-510	141	14	(	(	PUNCT
cana-510	141	15	1	1	NUM
cana-510	141	16	−	−	NOUN
cana-510	141	17	𝜔	𝜔	NOUN
cana-510	141	18	)	)	PUNCT
cana-510	141	19	)	)	PUNCT
cana-510	141	20	]	]	PUNCT
cana-510	141	21	1	1	NUM
cana-510	141	22	𝑛−1	𝑛−1	NUM
cana-510	141	23	,	,	PUNCT
cana-510	141	24	𝑛	𝑛	DET
cana-510	141	25	≥	≥	NOUN
cana-510	141	26	2	2	NUM
cana-510	141	27	.	.	PUNCT
cana-510	142	1	the	the	DET
cana-510	142	2	theorem	theorem	NOUN
cana-510	142	3	proved	prove	VERB
cana-510	142	4	.	.	PUNCT
cana-510	143	1	theorem	theorem	VERB
cana-510	143	2	4.3	4.3	NUM
cana-510	143	3	.	.	PUNCT
cana-510	144	1	let	let	VERB
cana-510	144	2	𝑢	𝑢	PRON
cana-510	144	3	∈	∈	PROPN
cana-510	144	4	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	144	5	,	,	PUNCT
cana-510	144	6	𝜎	𝜎	PROPN
cana-510	144	7	,	,	PUNCT
cana-510	144	8	ς	ς	PROPN
cana-510	144	9	,	,	PUNCT
cana-510	144	10	𝜗	𝜗	NOUN
cana-510	144	11	)	)	PUNCT
cana-510	144	12	.	.	PUNCT
cana-510	145	1	then	then	ADV
cana-510	145	2	u	u	PRON
cana-510	145	3	is	be	AUX
cana-510	145	4	close	close	ADJ
cana-510	145	5	-	-	PUNCT
cana-510	145	6	to	to	AUX
cana-510	145	7	-	-	PUNCT
cana-510	145	8	convex	convex	NOUN
cana-510	145	9	in	in	ADP
cana-510	145	10	|z|	|z|	NOUN
cana-510	145	11	<	<	X
cana-510	145	12	r3	r3	PROPN
cana-510	145	13	of	of	ADP
cana-510	145	14	order	order	NOUN
cana-510	145	15	δ	δ	PROPN
cana-510	145	16	,	,	PUNCT
cana-510	145	17	0	0	NUM
cana-510	145	18	≤	≤	NUM
cana-510	145	19	δ	δ	X
cana-510	145	20	<	<	X
cana-510	145	21	1	1	NUM
cana-510	145	22	,	,	PUNCT
cana-510	145	23	where	where	SCONJ
cana-510	145	24	𝑅3	𝑅3	PROPN
cana-510	145	25	=	=	SYM
cana-510	145	26	inf	inf	PROPN
cana-510	145	27	𝑛	𝑛	PROPN
cana-510	145	28	{	{	PUNCT
cana-510	145	29	(	(	PUNCT
cana-510	145	30	1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗	1−𝛿)(𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔))𝛩(𝑛,𝜗	NUM
cana-510	145	31	)	)	PUNCT
cana-510	145	32	𝑛ς(𝜎+(1−𝜔	𝑛ς(𝜎+(1−𝜔	PROPN
cana-510	145	33	)	)	PUNCT
cana-510	145	34	)	)	PUNCT
cana-510	145	35	}	}	PUNCT
cana-510	145	36	1	1	NUM
cana-510	145	37	𝑛−1	𝑛−1	PROPN
cana-510	145	38	,	,	PUNCT
cana-510	145	39	𝑛	𝑛	PRON
cana-510	145	40	≥	≥	NOUN
cana-510	145	41	2	2	NUM
cana-510	145	42	.	.	PUNCT
cana-510	145	43	(	(	PUNCT
cana-510	145	44	4.5	4.5	NUM
cana-510	145	45	)	)	PUNCT
cana-510	145	46	proof	proof	NOUN
cana-510	145	47	.	.	PUNCT
cana-510	146	1	𝑢	𝑢	NOUN
cana-510	146	2	is	be	AUX
cana-510	146	3	close	close	ADJ
cana-510	146	4	-	-	PUNCT
cana-510	146	5	to	to	ADP
cana-510	146	6	-	-	PUNCT
cana-510	146	7	convex	convex	NOUN
cana-510	146	8	of	of	ADP
cana-510	146	9	order	order	NOUN
cana-510	146	10	𝛿	𝛿	ADJ
cana-510	146	11	,	,	PUNCT
cana-510	146	12	0	0	NUM
cana-510	146	13	≤	≤	NOUN
cana-510	147	1	𝛿	𝛿	PRON
cana-510	147	2	<	<	X
cana-510	147	3	1	1	NUM
cana-510	147	4	if	if	SCONJ
cana-510	147	5	ℜ{𝑢′(𝑧	ℜ{𝑢′(𝑧	PROPN
cana-510	147	6	)	)	PUNCT
cana-510	147	7	}	}	PUNCT
cana-510	147	8	>	>	X
cana-510	147	9	𝛿.	𝛿.	NOUN
cana-510	147	10	thus	thus	ADV
cana-510	147	11	it	it	PRON
cana-510	147	12	is	be	AUX
cana-510	147	13	enough	enough	ADJ
cana-510	147	14	to	to	PART
cana-510	147	15	show	show	VERB
cana-510	147	16	that	that	SCONJ
cana-510	147	17	communications	communication	NOUN
cana-510	147	18	on	on	ADP
cana-510	147	19	applied	apply	VERB
cana-510	147	20	nonlinear	nonlinear	ADJ
cana-510	147	21	analysis	analysis	NOUN
cana-510	147	22	issn	issn	NOUN
cana-510	147	23	:	:	PUNCT
cana-510	147	24	1074	1074	NUM
cana-510	147	25	-	-	PUNCT
cana-510	147	26	133x	133x	NUM
cana-510	147	27	vol	vol	NOUN
cana-510	147	28	31	31	NUM
cana-510	147	29	no	no	NOUN
cana-510	147	30	.	.	NOUN
cana-510	147	31	2	2	NUM
cana-510	147	32	(	(	PUNCT
cana-510	147	33	2024	2024	NUM
cana-510	147	34	)	)	PUNCT
cana-510	147	35	29	29	NUM
cana-510	147	36	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	147	37	|𝑢′(𝑧	|𝑢′(𝑧	PROPN
cana-510	147	38	)	)	PUNCT
cana-510	147	39	−	−	PROPN
cana-510	147	40	1|	1|	NUM
cana-510	148	1	=	=	NOUN
cana-510	148	2	|−	|−	ADJ
cana-510	148	3	∑	∑	ADP
cana-510	148	4	𝑛	𝑛	DET
cana-510	148	5	∞	∞	NUM
cana-510	148	6	𝑛=2	𝑛=2	X
cana-510	148	7	𝑎𝑛𝑧𝑛−1|	𝑎𝑛𝑧𝑛−1|	NOUN
cana-510	148	8	≤	≤	NOUN
cana-510	148	9	∑	∑	PUNCT
cana-510	148	10	𝑛	𝑛	PRON
cana-510	148	11	∞	∞	NUM
cana-510	148	12	𝑛=2	𝑛=2	X
cana-510	148	13	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	NOUN
cana-510	148	14	.	.	PUNCT
cana-510	149	1	thus	thus	ADV
cana-510	149	2	|𝑢′(𝑧	|𝑢′(𝑧	CCONJ
cana-510	149	3	)	)	PUNCT
cana-510	149	4	−	−	PROPN
cana-510	149	5	1|	1|	NUM
cana-510	149	6	≤	≤	NUM
cana-510	149	7	1	1	NUM
cana-510	149	8	−	−	NOUN
cana-510	149	9	𝛿	𝛿	PROPN
cana-510	149	10	𝑖𝑓	𝑖𝑓	ADJ
cana-510	149	11	∑	∑	PROPN
cana-510	149	12	𝑛	𝑛	PROPN
cana-510	149	13	(	(	PUNCT
cana-510	149	14	1	1	NUM
cana-510	149	15	−	−	PROPN
cana-510	149	16	𝛿	𝛿	ADJ
cana-510	149	17	)	)	PUNCT
cana-510	149	18	∞	∞	NUM
cana-510	149	19	𝑛=2	𝑛=2	NOUN
cana-510	149	20	𝑎𝑛|𝑧|𝑛−1	𝑎𝑛|𝑧|𝑛−1	NOUN
cana-510	149	21	≤	≤	NUM
cana-510	149	22	1	1	NUM
cana-510	149	23	.	.	PUNCT
cana-510	150	1	(	(	PUNCT
cana-510	150	2	4.6	4.6	NUM
cana-510	150	3	)	)	PUNCT
cana-510	150	4	hence	hence	ADV
cana-510	150	5	by	by	ADP
cana-510	150	6	theorem	theorem	NOUN
cana-510	150	7	2.1	2.1	NUM
cana-510	150	8	,	,	PUNCT
cana-510	150	9	(	(	PUNCT
cana-510	150	10	4.6	4.6	NUM
cana-510	150	11	)	)	PUNCT
cana-510	150	12	will	will	AUX
cana-510	150	13	be	be	AUX
cana-510	150	14	true	true	ADJ
cana-510	150	15	if	if	SCONJ
cana-510	150	16	𝑛	𝑛	ADP
cana-510	150	17	1	1	NUM
cana-510	150	18	−	−	NOUN
cana-510	150	19	𝛿	𝛿	DET
cana-510	150	20	|𝑧|𝑛−1	|𝑧|𝑛−1	X
cana-510	150	21	≤	≤	NOUN
cana-510	150	22	(	(	PUNCT
cana-510	150	23	𝜔(𝑛	𝜔(𝑛	PROPN
cana-510	150	24	−	−	PROPN
cana-510	150	25	1	1	NUM
cana-510	150	26	)	)	PUNCT
cana-510	150	27	+	+	NOUN
cana-510	150	28	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	150	29	+	+	CCONJ
cana-510	150	30	1	1	NUM
cana-510	150	31	−	−	NOUN
cana-510	150	32	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	150	33	,	,	PUNCT
cana-510	150	34	𝜗	𝜗	NOUN
cana-510	150	35	)	)	PUNCT
cana-510	150	36	ς(𝜎	ς(𝜎	NOUN
cana-510	151	1	+	+	CCONJ
cana-510	151	2	(	(	PUNCT
cana-510	151	3	1	1	NUM
cana-510	151	4	−	−	NOUN
cana-510	151	5	𝜔	𝜔	NOUN
cana-510	151	6	)	)	PUNCT
cana-510	151	7	or	or	CCONJ
cana-510	151	8	if	if	SCONJ
cana-510	151	9	|𝑧|	|𝑧|	PROPN
cana-510	151	10	≤	≤	X
cana-510	151	11	[	[	PUNCT
cana-510	151	12	(	(	PUNCT
cana-510	151	13	1	1	NUM
cana-510	151	14	−	−	NOUN
cana-510	151	15	𝛿)(𝜔(𝑛	𝛿)(𝜔(𝑛	PUNCT
cana-510	151	16	−	−	NOUN
cana-510	151	17	1	1	NUM
cana-510	151	18	)	)	PUNCT
cana-510	152	1	+	+	NOUN
cana-510	152	2	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	152	3	+	+	CCONJ
cana-510	152	4	1	1	NUM
cana-510	152	5	−	−	NOUN
cana-510	152	6	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	152	7	,	,	PUNCT
cana-510	152	8	𝜗	𝜗	NOUN
cana-510	152	9	)	)	PUNCT
cana-510	152	10	𝑛ς(𝜎	𝑛ς(𝜎	PUNCT
cana-510	153	1	+	+	CCONJ
cana-510	153	2	(	(	PUNCT
cana-510	153	3	1	1	NUM
cana-510	153	4	−	−	NOUN
cana-510	153	5	𝜔	𝜔	NOUN
cana-510	153	6	)	)	PUNCT
cana-510	153	7	)	)	PUNCT
cana-510	154	1	]	]	PUNCT
cana-510	154	2	1	1	NUM
cana-510	154	3	𝑛−1	𝑛−1	NUM
cana-510	154	4	,	,	PUNCT
cana-510	154	5	𝑛	𝑛	PRON
cana-510	154	6	≥	≥	NOUN
cana-510	154	7	2	2	NUM
cana-510	154	8	.	.	PUNCT
cana-510	155	1	the	the	DET
cana-510	155	2	theorem	theorem	NOUN
cana-510	155	3	follows	follow	VERB
cana-510	155	4	.	.	PUNCT
cana-510	156	1	5	5	X
cana-510	156	2	.	.	PUNCT
cana-510	156	3	extreme	extreme	ADJ
cana-510	156	4	points	point	NOUN
cana-510	156	5	in	in	ADP
cana-510	156	6	the	the	DET
cana-510	156	7	following	following	NOUN
cana-510	156	8	theorem	theorem	NOUN
cana-510	156	9	,	,	PUNCT
cana-510	156	10	we	we	PRON
cana-510	156	11	obtain	obtain	VERB
cana-510	156	12	extreme	extreme	ADJ
cana-510	156	13	points	point	NOUN
cana-510	156	14	for	for	ADP
cana-510	156	15	the	the	DET
cana-510	156	16	class	class	NOUN
cana-510	156	17	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	156	18	,	,	PUNCT
cana-510	156	19	𝜎	𝜎	PROPN
cana-510	156	20	,	,	PUNCT
cana-510	156	21	ς	ς	PROPN
cana-510	156	22	,	,	PUNCT
cana-510	156	23	𝜗	𝜗	NOUN
cana-510	156	24	)	)	PUNCT
cana-510	156	25	.	.	PUNCT
cana-510	157	1	theorem	theorem	VERB
cana-510	157	2	5.1	5.1	NUM
cana-510	157	3	.	.	PUNCT
cana-510	158	1	let	let	VERB
cana-510	158	2	𝑢1(𝑧	𝑢1(𝑧	NOUN
cana-510	158	3	)	)	PUNCT
cana-510	158	4	=	=	SYM
cana-510	158	5	𝑧	𝑧	NOUN
cana-510	158	6	and	and	CCONJ
cana-510	158	7	𝑢𝑛(𝑧	𝑢𝑛(𝑧	ADV
cana-510	158	8	)	)	PUNCT
cana-510	159	1	=	=	SYM
cana-510	159	2	𝑧	𝑧	PRON
cana-510	159	3	−	−	NOUN
cana-510	159	4	𝜍(𝜎+(1−𝜔	𝜍(𝜎+(1−𝜔	NOUN
cana-510	159	5	)	)	PUNCT
cana-510	159	6	)	)	PUNCT
cana-510	160	1	[	[	X
cana-510	160	2	𝜔(𝑛−1)+𝜍(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	𝜔(𝑛−1)+𝜍(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	X
cana-510	160	3	)	)	PUNCT
cana-510	160	4	𝑧𝑛	𝑧𝑛	PROPN
cana-510	160	5	,	,	PUNCT
cana-510	160	6	for	for	ADP
cana-510	160	7	𝑛	𝑛	PRON
cana-510	160	8	=	=	SYM
cana-510	160	9	2,3	2,3	NUM
cana-510	160	10	,	,	PUNCT
cana-510	160	11	⋯.	⋯.	PROPN
cana-510	160	12	then	then	ADV
cana-510	160	13	𝑢	𝑢	PROPN
cana-510	160	14	∈	∈	PROPN
cana-510	160	15	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	160	16	,	,	PUNCT
cana-510	160	17	𝜎	𝜎	NOUN
cana-510	160	18	,	,	PUNCT
cana-510	160	19	𝜍	𝜍	NOUN
cana-510	160	20	,	,	PUNCT
cana-510	160	21	𝜗	𝜗	NOUN
cana-510	160	22	)	)	PUNCT
cana-510	160	23	if	if	SCONJ
cana-510	160	24	and	and	CCONJ
cana-510	160	25	only	only	ADV
cana-510	160	26	if	if	SCONJ
cana-510	160	27	it	it	PRON
cana-510	160	28	can	can	AUX
cana-510	160	29	be	be	AUX
cana-510	160	30	expressed	express	VERB
cana-510	160	31	in	in	ADP
cana-510	160	32	the	the	DET
cana-510	160	33	form	form	NOUN
cana-510	160	34	𝑢(𝑧	𝑢(𝑧	PROPN
cana-510	160	35	)	)	PUNCT
cana-510	160	36	=	=	PUNCT
cana-510	161	1	∑	∑	PUNCT
cana-510	161	2	𝜃𝑛	𝜃𝑛	NUM
cana-510	161	3	∞	∞	PROPN
cana-510	161	4	𝑛=1	𝑛=1	NOUN
cana-510	161	5	𝑢𝑛(𝑧	𝑢𝑛(𝑧	ADV
cana-510	161	6	)	)	PUNCT
cana-510	161	7	,	,	PUNCT
cana-510	161	8	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-510	161	9	𝜃𝑛	𝜃𝑛	X
cana-510	161	10	≥	≥	NOUN
cana-510	161	11	0	0	NUM
cana-510	161	12	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-510	161	13	∑	∑	PROPN
cana-510	161	14	𝜃𝑛	𝜃𝑛	PROPN
cana-510	161	15	∞	∞	NUM
cana-510	161	16	𝑛=1	𝑛=1	NOUN
cana-510	161	17	=	=	SYM
cana-510	161	18	1	1	X
cana-510	161	19	.	.	PUNCT
cana-510	162	1	proof	proof	NOUN
cana-510	162	2	.	.	PUNCT
cana-510	163	1	assume	assume	VERB
cana-510	163	2	that	that	SCONJ
cana-510	163	3	𝑢(𝑧	𝑢(𝑧	VERB
cana-510	163	4	)	)	PUNCT
cana-510	164	1	=	=	PUNCT
cana-510	164	2	∑	∑	PUNCT
cana-510	164	3	𝜃𝑛	𝜃𝑛	NUM
cana-510	164	4	∞	∞	PROPN
cana-510	164	5	𝑛=1	𝑛=1	NOUN
cana-510	164	6	𝑢𝑛(𝑧	𝑢𝑛(𝑧	ADV
cana-510	164	7	)	)	PUNCT
cana-510	164	8	,	,	PUNCT
cana-510	164	9	hence	hence	ADV
cana-510	164	10	we	we	PRON
cana-510	164	11	get	get	VERB
cana-510	164	12	𝑢(𝑧	𝑢(𝑧	VERB
cana-510	164	13	)	)	PUNCT
cana-510	165	1	=	=	PUNCT
cana-510	165	2	𝑧	𝑧	PRON
cana-510	165	3	−	−	PROPN
cana-510	165	4	∑	∑	PROPN
cana-510	165	5	ς(𝜎	ς(𝜎	PROPN
cana-510	165	6	+	+	CCONJ
cana-510	165	7	(	(	PUNCT
cana-510	165	8	1	1	NUM
cana-510	165	9	−	−	NOUN
cana-510	165	10	𝜔))𝜃𝑛	𝜔))𝜃𝑛	PROPN
cana-510	166	1	[	[	X
cana-510	166	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	166	3	−	−	PROPN
cana-510	166	4	1	1	NUM
cana-510	166	5	)	)	PUNCT
cana-510	166	6	+	+	NOUN
cana-510	166	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	166	8	+	+	CCONJ
cana-510	166	9	1	1	NUM
cana-510	166	10	−	−	NOUN
cana-510	166	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	166	12	,	,	PUNCT
cana-510	166	13	𝜗	𝜗	NOUN
cana-510	166	14	)	)	PUNCT
cana-510	166	15	∞	∞	NUM
cana-510	166	16	𝑛=2	𝑛=2	PROPN
cana-510	166	17	𝑧𝑛.	𝑧𝑛.	NOUN
cana-510	166	18	now	now	ADV
cana-510	166	19	,	,	PUNCT
cana-510	166	20	𝑢	𝑢	PROPN
cana-510	166	21	∈	∈	PROPN
cana-510	166	22	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	166	23	,	,	PUNCT
cana-510	166	24	𝜎	𝜎	PROPN
cana-510	166	25	,	,	PUNCT
cana-510	166	26	ς	ς	PROPN
cana-510	166	27	,	,	PUNCT
cana-510	166	28	𝜗	𝜗	NOUN
cana-510	166	29	)	)	PUNCT
cana-510	166	30	,	,	PUNCT
cana-510	166	31	since	since	SCONJ
cana-510	166	32	∑	∑	PROPN
cana-510	166	33	[	[	X
cana-510	166	34	𝜔(𝑛	𝜔(𝑛	X
cana-510	166	35	−	−	PROPN
cana-510	166	36	1	1	NUM
cana-510	166	37	)	)	PUNCT
cana-510	166	38	+	+	NOUN
cana-510	166	39	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	166	40	+	+	CCONJ
cana-510	166	41	1	1	NUM
cana-510	166	42	−	−	NOUN
cana-510	166	43	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	166	44	,	,	PUNCT
cana-510	166	45	𝜗	𝜗	NOUN
cana-510	166	46	)	)	PUNCT
cana-510	166	47	ς(𝜎	ς(𝜎	NOUN
cana-510	166	48	+	+	CCONJ
cana-510	166	49	(	(	PUNCT
cana-510	166	50	1	1	NUM
cana-510	166	51	−	−	NOUN
cana-510	166	52	𝜔	𝜔	NOUN
cana-510	166	53	)	)	PUNCT
cana-510	166	54	)	)	PUNCT
cana-510	167	1	∞	∞	NUM
cana-510	168	1	𝑛=2	𝑛=2	PROPN
cana-510	168	2	×	×	NOUN
cana-510	168	3	ς(𝜎	ς(𝜎	PROPN
cana-510	168	4	+	+	CCONJ
cana-510	168	5	(	(	PUNCT
cana-510	168	6	1	1	NUM
cana-510	168	7	−	−	NOUN
cana-510	168	8	𝜔))𝜃𝑛	𝜔))𝜃𝑛	PROPN
cana-510	169	1	[	[	X
cana-510	169	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	169	3	−	−	PROPN
cana-510	169	4	1	1	NUM
cana-510	169	5	)	)	PUNCT
cana-510	169	6	+	+	NOUN
cana-510	169	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	169	8	+	+	CCONJ
cana-510	169	9	1	1	NUM
cana-510	169	10	−	−	NOUN
cana-510	169	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	169	12	,	,	PUNCT
cana-510	169	13	𝜗	𝜗	NOUN
cana-510	169	14	)	)	PUNCT
cana-510	169	15	=	=	PUNCT
cana-510	169	16	∑	∑	PUNCT
cana-510	169	17	𝜃𝑛	𝜃𝑛	NOUN
cana-510	169	18	∞	∞	NUM
cana-510	169	19	𝑛=2	𝑛=2	X
cana-510	169	20	=	=	SYM
cana-510	169	21	1	1	NUM
cana-510	169	22	−	−	NOUN
cana-510	169	23	𝜃1	𝜃1	VERB
cana-510	169	24	≤	≤	ADJ
cana-510	169	25	1	1	NUM
cana-510	169	26	.	.	PUNCT
cana-510	170	1	communications	communication	NOUN
cana-510	170	2	on	on	ADP
cana-510	170	3	applied	apply	VERB
cana-510	170	4	nonlinear	nonlinear	ADJ
cana-510	170	5	analysis	analysis	NOUN
cana-510	170	6	issn	issn	NOUN
cana-510	170	7	:	:	PUNCT
cana-510	170	8	1074	1074	NUM
cana-510	170	9	-	-	PUNCT
cana-510	170	10	133x	133x	NUM
cana-510	170	11	vol	vol	NOUN
cana-510	170	12	31	31	NUM
cana-510	170	13	no	no	NOUN
cana-510	170	14	.	.	NOUN
cana-510	170	15	2	2	NUM
cana-510	170	16	(	(	PUNCT
cana-510	170	17	2024	2024	NUM
cana-510	170	18	)	)	PUNCT
cana-510	170	19	30	30	NUM
cana-510	170	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	170	21	conversely	conversely	ADV
cana-510	170	22	,	,	PUNCT
cana-510	170	23	suppose	suppose	VERB
cana-510	170	24	𝑢	𝑢	PRON
cana-510	170	25	∈	∈	PROPN
cana-510	170	26	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	170	27	,	,	PUNCT
cana-510	170	28	𝜎	𝜎	PROPN
cana-510	170	29	,	,	PUNCT
cana-510	170	30	ς	ς	PROPN
cana-510	170	31	,	,	PUNCT
cana-510	170	32	𝜗	𝜗	NOUN
cana-510	170	33	)	)	PUNCT
cana-510	170	34	.	.	PUNCT
cana-510	171	1	then	then	ADV
cana-510	171	2	we	we	PRON
cana-510	171	3	show	show	VERB
cana-510	171	4	that	that	SCONJ
cana-510	171	5	𝑢	𝑢	PRON
cana-510	171	6	can	can	AUX
cana-510	171	7	be	be	AUX
cana-510	171	8	written	write	VERB
cana-510	171	9	in	in	ADP
cana-510	171	10	the	the	DET
cana-510	171	11	form	form	NOUN
cana-510	171	12	∑	∑	PUNCT
cana-510	171	13	𝜃𝑛	𝜃𝑛	PROPN
cana-510	171	14	∞	∞	PROPN
cana-510	171	15	𝑛=1	𝑛=1	NOUN
cana-510	171	16	𝑢𝑛(𝑧	𝑢𝑛(𝑧	ADV
cana-510	171	17	)	)	PUNCT
cana-510	171	18	.	.	PUNCT
cana-510	172	1	now	now	ADV
cana-510	172	2	𝑢	𝑢	PROPN
cana-510	172	3	∈	∈	PROPN
cana-510	172	4	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	172	5	,	,	PUNCT
cana-510	172	6	𝜎	𝜎	PROPN
cana-510	172	7	,	,	PUNCT
cana-510	172	8	ς	ς	PROPN
cana-510	172	9	,	,	PUNCT
cana-510	172	10	𝜗	𝜗	NOUN
cana-510	172	11	)	)	PUNCT
cana-510	172	12	implies	imply	VERB
cana-510	172	13	from	from	ADP
cana-510	172	14	theorem	theorem	ADJ
cana-510	172	15	2.1	2.1	NUM
cana-510	172	16	,	,	PUNCT
cana-510	172	17	𝑎𝑛	𝑎𝑛	PROPN
cana-510	172	18	≤	≤	NOUN
cana-510	172	19	ς(𝜎	ς(𝜎	PROPN
cana-510	172	20	+	+	CCONJ
cana-510	172	21	(	(	PUNCT
cana-510	172	22	1	1	NUM
cana-510	172	23	−	−	NOUN
cana-510	172	24	𝜔	𝜔	NOUN
cana-510	172	25	)	)	PUNCT
cana-510	172	26	)	)	PUNCT
cana-510	173	1	[	[	X
cana-510	173	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	173	3	−	−	PROPN
cana-510	173	4	1	1	NUM
cana-510	173	5	)	)	PUNCT
cana-510	173	6	+	+	NOUN
cana-510	173	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	173	8	+	+	CCONJ
cana-510	173	9	1	1	NUM
cana-510	173	10	−	−	NOUN
cana-510	173	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	173	12	,	,	PUNCT
cana-510	173	13	𝜗	𝜗	NOUN
cana-510	173	14	)	)	PUNCT
cana-510	173	15	.	.	PUNCT
cana-510	174	1	setting	set	VERB
cana-510	174	2	𝜃𝑛	𝜃𝑛	NOUN
cana-510	174	3	=	=	PUNCT
cana-510	175	1	[	[	X
cana-510	175	2	𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	𝜔(𝑛−1)+ς(𝑛𝜎+1−𝜔)]𝛩(𝑛,𝜗	NOUN
cana-510	175	3	)	)	PUNCT
cana-510	175	4	ς(𝜎+(1−𝜔	ς(𝜎+(1−𝜔	NOUN
cana-510	175	5	)	)	PUNCT
cana-510	175	6	)	)	PUNCT
cana-510	176	1	𝑎𝑛	𝑎𝑛	PROPN
cana-510	176	2	,	,	PUNCT
cana-510	176	3	𝑛	𝑛	PRON
cana-510	176	4	=	=	SYM
cana-510	176	5	2,3	2,3	NUM
cana-510	176	6	,	,	PUNCT
cana-510	176	7	⋯	⋯	NOUN
cana-510	176	8	and	and	CCONJ
cana-510	176	9	𝜃1	𝜃1	VERB
cana-510	176	10	=	=	SYM
cana-510	176	11	1	1	NUM
cana-510	176	12	−	−	NOUN
cana-510	176	13	∑	∑	PROPN
cana-510	176	14	𝜃𝑛	𝜃𝑛	PROPN
cana-510	176	15	∞	∞	PROPN
cana-510	176	16	𝑛=2	𝑛=2	PROPN
cana-510	176	17	,	,	PUNCT
cana-510	176	18	we	we	PRON
cana-510	176	19	obtain	obtain	VERB
cana-510	176	20	𝑢(𝑧	𝑢(𝑧	NOUN
cana-510	176	21	)	)	PUNCT
cana-510	176	22	=	=	PUNCT
cana-510	177	1	∑	∑	PUNCT
cana-510	177	2	𝜃𝑛	𝜃𝑛	NUM
cana-510	177	3	∞	∞	PROPN
cana-510	177	4	𝑛=1	𝑛=1	NOUN
cana-510	177	5	𝑢𝑛(𝑧	𝑢𝑛(𝑧	ADV
cana-510	177	6	)	)	PUNCT
cana-510	177	7	.	.	PUNCT
cana-510	178	1	6	6	X
cana-510	178	2	.	.	X
cana-510	178	3	hadamard	hadamard	ADJ
cana-510	178	4	product	product	NOUN
cana-510	178	5	in	in	ADP
cana-510	178	6	the	the	DET
cana-510	178	7	following	following	NOUN
cana-510	178	8	theorem	theorem	NOUN
cana-510	178	9	,	,	PUNCT
cana-510	178	10	we	we	PRON
cana-510	178	11	obtain	obtain	VERB
cana-510	178	12	the	the	DET
cana-510	178	13	convolution	convolution	NOUN
cana-510	178	14	result	result	NOUN
cana-510	178	15	for	for	ADP
cana-510	178	16	functions	function	NOUN
cana-510	178	17	belongs	belong	VERB
cana-510	178	18	to	to	ADP
cana-510	178	19	the	the	DET
cana-510	178	20	class	class	NOUN
cana-510	178	21	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	178	22	,	,	PUNCT
cana-510	178	23	𝜎	𝜎	PROPN
cana-510	178	24	,	,	PUNCT
cana-510	178	25	ς	ς	PROPN
cana-510	178	26	,	,	PUNCT
cana-510	178	27	𝜗	𝜗	NOUN
cana-510	178	28	)	)	PUNCT
cana-510	178	29	.	.	PUNCT
cana-510	179	1	theorem	theorem	VERB
cana-510	179	2	6.1	6.1	NUM
cana-510	179	3	.	.	PUNCT
cana-510	180	1	let	let	VERB
cana-510	180	2	𝑢	𝑢	NOUN
cana-510	180	3	,	,	PUNCT
cana-510	180	4	𝑔	𝑔	PROPN
cana-510	180	5	∈	∈	PROPN
cana-510	180	6	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	180	7	,	,	PUNCT
cana-510	180	8	𝜎	𝜎	NOUN
cana-510	180	9	,	,	PUNCT
cana-510	180	10	𝜍	𝜍	NOUN
cana-510	180	11	,	,	PUNCT
cana-510	180	12	𝜗	𝜗	NOUN
cana-510	180	13	)	)	PUNCT
cana-510	180	14	.	.	PUNCT
cana-510	181	1	then	then	ADV
cana-510	181	2	𝑢	𝑢	X
cana-510	181	3	∗	∗	X
cana-510	181	4	𝑔	𝑔	PROPN
cana-510	181	5	∈	∈	PROPN
cana-510	181	6	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	181	7	,	,	PUNCT
cana-510	181	8	𝜎	𝜎	PROPN
cana-510	181	9	,	,	PUNCT
cana-510	181	10	𝜁	𝜁	PROPN
cana-510	181	11	,	,	PUNCT
cana-510	181	12	𝜗	𝜗	NOUN
cana-510	181	13	)	)	PUNCT
cana-510	181	14	for	for	ADP
cana-510	181	15	𝑢(𝑧	𝑢(𝑧	NOUN
cana-510	181	16	)	)	PUNCT
cana-510	181	17	=	=	PUNCT
cana-510	182	1	𝑧	𝑧	DET
cana-510	182	2	−	−	NOUN
cana-510	182	3	∑	∑	PROPN
cana-510	182	4	𝑎𝑛	𝑎𝑛	PROPN
cana-510	182	5	∞	∞	PROPN
cana-510	182	6	𝑛=2	𝑛=2	PROPN
cana-510	182	7	𝑧𝑛	𝑧𝑛	PROPN
cana-510	182	8	,	,	PUNCT
cana-510	182	9	𝑔(𝑧	𝑔(𝑧	PROPN
cana-510	182	10	)	)	PUNCT
cana-510	182	11	=	=	SYM
cana-510	182	12	𝑧	𝑧	PRON
cana-510	182	13	−	−	NOUN
cana-510	182	14	∑	∑	PROPN
cana-510	182	15	𝑏𝑛	𝑏𝑛	ADP
cana-510	182	16	∞	∞	PROPN
cana-510	182	17	𝑛=2	𝑛=2	ADP
cana-510	182	18	𝑧𝑛	𝑧𝑛	NOUN
cana-510	182	19	and	and	CCONJ
cana-510	182	20	(	(	PUNCT
cana-510	182	21	𝑢	𝑢	X
cana-510	182	22	∗	∗	NOUN
cana-510	182	23	𝑔)(𝑧	𝑔)(𝑧	NOUN
cana-510	182	24	)	)	PUNCT
cana-510	182	25	=	=	SYM
cana-510	182	26	𝑧	𝑧	PRON
cana-510	182	27	−	−	NOUN
cana-510	182	28	∑	∑	PROPN
cana-510	182	29	𝑎𝑛	𝑎𝑛	PROPN
cana-510	182	30	∞	∞	NUM
cana-510	182	31	𝑛=2	𝑛=2	NOUN
cana-510	182	32	𝑏𝑛𝑧𝑛	𝑏𝑛𝑧𝑛	NOUN
cana-510	182	33	,	,	PUNCT
cana-510	182	34	where	where	SCONJ
cana-510	182	35	𝜁	𝜁	PRON
cana-510	182	36	≥	≥	NOUN
cana-510	182	37	𝜍2(𝜎	𝜍2(𝜎	X
cana-510	182	38	+	+	CCONJ
cana-510	182	39	(	(	PUNCT
cana-510	182	40	1	1	NUM
cana-510	182	41	−	−	PROPN
cana-510	182	42	𝜔))𝜔(𝑛	𝜔))𝜔(𝑛	PROPN
cana-510	182	43	−	−	PROPN
cana-510	182	44	1	1	NUM
cana-510	182	45	)	)	PUNCT
cana-510	183	1	[	[	X
cana-510	183	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	183	3	−	−	PROPN
cana-510	183	4	1	1	NUM
cana-510	183	5	)	)	PUNCT
cana-510	183	6	+	+	NOUN
cana-510	183	7	𝜍(𝑛𝜎	𝜍(𝑛𝜎	NUM
cana-510	183	8	+	+	CCONJ
cana-510	183	9	1	1	NUM
cana-510	183	10	−	−	NOUN
cana-510	183	11	𝜔)]2𝛩(𝑛	𝜔)]2𝛩(𝑛	ADJ
cana-510	183	12	,	,	PUNCT
cana-510	183	13	𝜗	𝜗	NOUN
cana-510	183	14	)	)	PUNCT
cana-510	183	15	−	−	PUNCT
cana-510	184	1	𝜍2(𝜎	𝜍2(𝜎	X
cana-510	184	2	+	+	CCONJ
cana-510	184	3	(	(	PUNCT
cana-510	184	4	1	1	NUM
cana-510	184	5	−	−	PROPN
cana-510	184	6	𝜔))(𝑛𝜎	𝜔))(𝑛𝜎	NOUN
cana-510	184	7	+	+	NUM
cana-510	184	8	1	1	NUM
cana-510	184	9	−	−	NOUN
cana-510	184	10	𝜔	𝜔	NOUN
cana-510	184	11	)	)	PUNCT
cana-510	184	12	.	.	PUNCT
cana-510	185	1	proof	proof	NOUN
cana-510	185	2	.	.	PUNCT
cana-510	186	1	𝑢	𝑢	PROPN
cana-510	186	2	∈	∈	PROPN
cana-510	186	3	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	186	4	,	,	PUNCT
cana-510	186	5	𝜎	𝜎	PROPN
cana-510	186	6	,	,	PUNCT
cana-510	186	7	ς	ς	PROPN
cana-510	186	8	,	,	PUNCT
cana-510	186	9	𝜗	𝜗	NOUN
cana-510	186	10	)	)	PUNCT
cana-510	186	11	and	and	CCONJ
cana-510	186	12	so	so	ADV
cana-510	186	13	∑	∑	PROPN
cana-510	187	1	[	[	X
cana-510	187	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	187	3	−	−	PROPN
cana-510	187	4	1	1	NUM
cana-510	187	5	)	)	PUNCT
cana-510	187	6	+	+	NOUN
cana-510	187	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	187	8	+	+	CCONJ
cana-510	187	9	1	1	NUM
cana-510	187	10	−	−	NOUN
cana-510	187	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	187	12	,	,	PUNCT
cana-510	187	13	𝜗	𝜗	NOUN
cana-510	187	14	)	)	PUNCT
cana-510	187	15	ς(𝜎	ς(𝜎	NOUN
cana-510	187	16	+	+	CCONJ
cana-510	187	17	(	(	PUNCT
cana-510	187	18	1	1	NUM
cana-510	187	19	−	−	NOUN
cana-510	187	20	𝜔	𝜔	NOUN
cana-510	187	21	)	)	PUNCT
cana-510	187	22	)	)	PUNCT
cana-510	188	1	∞	∞	NUM
cana-510	188	2	𝑛=2	𝑛=2	X
cana-510	188	3	𝑎𝑛	𝑎𝑛	VERB
cana-510	188	4	≤	≤	NUM
cana-510	188	5	1	1	NUM
cana-510	188	6	,	,	PUNCT
cana-510	188	7	and	and	CCONJ
cana-510	188	8	∑	∑	PROPN
cana-510	189	1	[	[	X
cana-510	189	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	189	3	−	−	PROPN
cana-510	189	4	1	1	NUM
cana-510	189	5	)	)	PUNCT
cana-510	189	6	+	+	NOUN
cana-510	189	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	189	8	+	+	CCONJ
cana-510	189	9	1	1	NUM
cana-510	189	10	−	−	NOUN
cana-510	189	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	189	12	,	,	PUNCT
cana-510	189	13	𝜗	𝜗	NOUN
cana-510	189	14	)	)	PUNCT
cana-510	189	15	ς(𝜎	ς(𝜎	NOUN
cana-510	189	16	+	+	CCONJ
cana-510	189	17	(	(	PUNCT
cana-510	189	18	1	1	NUM
cana-510	189	19	−	−	NOUN
cana-510	189	20	𝜔	𝜔	NOUN
cana-510	189	21	)	)	PUNCT
cana-510	189	22	)	)	PUNCT
cana-510	190	1	∞	∞	NUM
cana-510	190	2	𝑛=2	𝑛=2	X
cana-510	190	3	𝑏𝑛	𝑏𝑛	ADP
cana-510	190	4	≤	≤	ADJ
cana-510	190	5	1	1	NUM
cana-510	190	6	.	.	PUNCT
cana-510	191	1	we	we	PRON
cana-510	191	2	have	have	VERB
cana-510	191	3	to	to	PART
cana-510	191	4	find	find	VERB
cana-510	191	5	the	the	DET
cana-510	191	6	smallest	small	ADJ
cana-510	191	7	number	number	NOUN
cana-510	191	8	𝜁	𝜁	PRON
cana-510	191	9	such	such	ADJ
cana-510	191	10	that	that	SCONJ
cana-510	191	11	∑	∑	PROPN
cana-510	192	1	[	[	X
cana-510	192	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	192	3	−	−	PROPN
cana-510	192	4	1	1	NUM
cana-510	192	5	)	)	PUNCT
cana-510	192	6	+	+	CCONJ
cana-510	192	7	𝜁(𝑛𝜎	𝜁(𝑛𝜎	VERB
cana-510	192	8	+	+	CCONJ
cana-510	192	9	1	1	NUM
cana-510	192	10	−	−	NOUN
cana-510	192	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	192	12	,	,	PUNCT
cana-510	192	13	𝜗	𝜗	NOUN
cana-510	192	14	)	)	PUNCT
cana-510	192	15	𝜁(𝜎	𝜁(𝜎	PROPN
cana-510	192	16	+	+	CCONJ
cana-510	192	17	(	(	PUNCT
cana-510	192	18	1	1	NUM
cana-510	192	19	−	−	NOUN
cana-510	192	20	𝜔	𝜔	NOUN
cana-510	192	21	)	)	PUNCT
cana-510	192	22	)	)	PUNCT
cana-510	193	1	∞	∞	NUM
cana-510	193	2	𝑛=2	𝑛=2	NOUN
cana-510	193	3	𝑎𝑛𝑏𝑛	𝑎𝑛𝑏𝑛	NOUN
cana-510	193	4	≤	≤	ADV
cana-510	193	5	1	1	NUM
cana-510	193	6	.	.	PUNCT
cana-510	193	7	by	by	ADP
cana-510	193	8	cauchy	cauchy	PROPN
cana-510	193	9	-	-	PUNCT
cana-510	193	10	schwarz	schwarz	PROPN
cana-510	193	11	inequality	inequality	PROPN
cana-510	193	12	∑	∑	PROPN
cana-510	194	1	[	[	X
cana-510	194	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	194	3	−	−	PROPN
cana-510	194	4	1	1	NUM
cana-510	194	5	)	)	PUNCT
cana-510	194	6	+	+	NOUN
cana-510	194	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	194	8	+	+	CCONJ
cana-510	194	9	1	1	NUM
cana-510	194	10	−	−	NOUN
cana-510	194	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	194	12	,	,	PUNCT
cana-510	194	13	𝜗	𝜗	NOUN
cana-510	194	14	)	)	PUNCT
cana-510	194	15	ς(𝜎	ς(𝜎	NOUN
cana-510	194	16	+	+	CCONJ
cana-510	194	17	(	(	PUNCT
cana-510	194	18	1	1	NUM
cana-510	194	19	−	−	NOUN
cana-510	194	20	𝜔	𝜔	NOUN
cana-510	194	21	)	)	PUNCT
cana-510	194	22	)	)	PUNCT
cana-510	195	1	∞	∞	NUM
cana-510	195	2	𝑛=2	𝑛=2	NOUN
cana-510	195	3	√𝑎𝑛𝑏𝑛	√𝑎𝑛𝑏𝑛	NOUN
cana-510	195	4	≤	≤	NUM
cana-510	195	5	1	1	NUM
cana-510	195	6	.	.	PUNCT
cana-510	196	1	(	(	PUNCT
cana-510	196	2	6.1	6.1	NUM
cana-510	196	3	)	)	PUNCT
cana-510	196	4	therefore	therefore	ADV
cana-510	196	5	it	it	PRON
cana-510	196	6	is	be	AUX
cana-510	196	7	enough	enough	ADJ
cana-510	196	8	to	to	PART
cana-510	196	9	show	show	VERB
cana-510	196	10	that	that	SCONJ
cana-510	196	11	communications	communication	NOUN
cana-510	196	12	on	on	ADP
cana-510	196	13	applied	apply	VERB
cana-510	196	14	nonlinear	nonlinear	ADJ
cana-510	196	15	analysis	analysis	NOUN
cana-510	196	16	issn	issn	NOUN
cana-510	196	17	:	:	PUNCT
cana-510	196	18	1074	1074	NUM
cana-510	196	19	-	-	PUNCT
cana-510	196	20	133x	133x	NUM
cana-510	196	21	vol	vol	NOUN
cana-510	196	22	31	31	NUM
cana-510	196	23	no	no	NOUN
cana-510	196	24	.	.	NOUN
cana-510	196	25	2	2	NUM
cana-510	196	26	(	(	PUNCT
cana-510	196	27	2024	2024	NUM
cana-510	196	28	)	)	PUNCT
cana-510	196	29	31	31	NUM
cana-510	196	30	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	197	1	[	[	X
cana-510	197	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	197	3	−	−	PROPN
cana-510	197	4	1	1	NUM
cana-510	197	5	)	)	PUNCT
cana-510	197	6	+	+	CCONJ
cana-510	197	7	𝜁(𝑛𝜎	𝜁(𝑛𝜎	VERB
cana-510	197	8	+	+	CCONJ
cana-510	197	9	1	1	NUM
cana-510	197	10	−	−	NOUN
cana-510	197	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	197	12	,	,	PUNCT
cana-510	197	13	𝜗	𝜗	NOUN
cana-510	197	14	)	)	PUNCT
cana-510	197	15	𝜁(𝜎	𝜁(𝜎	PROPN
cana-510	197	16	+	+	CCONJ
cana-510	197	17	(	(	PUNCT
cana-510	197	18	1	1	NUM
cana-510	197	19	−	−	NOUN
cana-510	197	20	𝜔	𝜔	NOUN
cana-510	197	21	)	)	PUNCT
cana-510	197	22	)	)	PUNCT
cana-510	197	23	𝑎𝑛𝑏𝑛	𝑎𝑛𝑏𝑛	NOUN
cana-510	197	24	≤	≤	NOUN
cana-510	198	1	[	[	X
cana-510	198	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	198	3	−	−	PROPN
cana-510	198	4	1	1	NUM
cana-510	198	5	)	)	PUNCT
cana-510	198	6	+	+	NOUN
cana-510	198	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	198	8	+	+	CCONJ
cana-510	198	9	1	1	NUM
cana-510	198	10	−	−	NOUN
cana-510	198	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	198	12	,	,	PUNCT
cana-510	198	13	𝜗	𝜗	NOUN
cana-510	198	14	)	)	PUNCT
cana-510	198	15	ς(𝜎	ς(𝜎	NOUN
cana-510	198	16	+	+	CCONJ
cana-510	198	17	(	(	PUNCT
cana-510	198	18	1	1	NUM
cana-510	198	19	−	−	NOUN
cana-510	198	20	𝜔	𝜔	NOUN
cana-510	198	21	)	)	PUNCT
cana-510	198	22	)	)	PUNCT
cana-510	198	23	√𝑎𝑛𝑏𝑛.	√𝑎𝑛𝑏𝑛.	NOUN
cana-510	198	24	that	that	PRON
cana-510	198	25	is	be	AUX
cana-510	198	26	√𝑎𝑛𝑏𝑛	√𝑎𝑛𝑏𝑛	NOUN
cana-510	198	27	≤	≤	NOUN
cana-510	199	1	[	[	X
cana-510	199	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	199	3	−	−	PROPN
cana-510	199	4	1	1	NUM
cana-510	199	5	)	)	PUNCT
cana-510	199	6	+	+	NOUN
cana-510	199	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	199	8	+	+	CCONJ
cana-510	199	9	1	1	NUM
cana-510	199	10	−	−	NOUN
cana-510	199	11	𝜔)]𝜁	𝜔)]𝜁	NOUN
cana-510	200	1	[	[	X
cana-510	200	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	200	3	−	−	PROPN
cana-510	200	4	1	1	X
cana-510	200	5	)	)	PUNCT
cana-510	200	6	+	+	CCONJ
cana-510	200	7	𝜁(𝑛𝜎	𝜁(𝑛𝜎	VERB
cana-510	200	8	+	+	CCONJ
cana-510	200	9	1	1	NUM
cana-510	200	10	−	−	NOUN
cana-510	200	11	𝜔)]ς	𝜔)]ς	NOUN
cana-510	200	12	.	.	PUNCT
cana-510	201	1	from	from	ADP
cana-510	201	2	(	(	PUNCT
cana-510	201	3	6.1	6.1	NUM
cana-510	201	4	)	)	PUNCT
cana-510	201	5	√𝑎𝑛𝑏𝑛	√𝑎𝑛𝑏𝑛	NOUN
cana-510	201	6	≤	≤	NOUN
cana-510	201	7	ς(𝜎	ς(𝜎	PROPN
cana-510	201	8	+	+	CCONJ
cana-510	201	9	(	(	PUNCT
cana-510	201	10	1	1	NUM
cana-510	201	11	−	−	NOUN
cana-510	201	12	𝜔	𝜔	NOUN
cana-510	201	13	)	)	PUNCT
cana-510	201	14	)	)	PUNCT
cana-510	202	1	[	[	X
cana-510	202	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	202	3	−	−	PROPN
cana-510	202	4	1	1	NUM
cana-510	202	5	)	)	PUNCT
cana-510	202	6	+	+	NOUN
cana-510	202	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	202	8	+	+	CCONJ
cana-510	202	9	1	1	NUM
cana-510	202	10	−	−	NOUN
cana-510	202	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	202	12	,	,	PUNCT
cana-510	202	13	𝜗	𝜗	NOUN
cana-510	202	14	)	)	PUNCT
cana-510	202	15	.	.	PUNCT
cana-510	203	1	thus	thus	ADV
cana-510	203	2	it	it	PRON
cana-510	203	3	is	be	AUX
cana-510	203	4	enough	enough	ADJ
cana-510	203	5	to	to	PART
cana-510	203	6	show	show	VERB
cana-510	203	7	that	that	SCONJ
cana-510	203	8	ς(𝜎	ς(𝜎	PROPN
cana-510	203	9	+	+	CCONJ
cana-510	203	10	(	(	PUNCT
cana-510	203	11	1	1	NUM
cana-510	203	12	−	−	NOUN
cana-510	203	13	𝜔	𝜔	NOUN
cana-510	203	14	)	)	PUNCT
cana-510	203	15	)	)	PUNCT
cana-510	204	1	[	[	X
cana-510	204	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	204	3	−	−	PROPN
cana-510	204	4	1	1	NUM
cana-510	204	5	)	)	PUNCT
cana-510	204	6	+	+	NOUN
cana-510	204	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	204	8	+	+	CCONJ
cana-510	204	9	1	1	NUM
cana-510	204	10	−	−	NOUN
cana-510	204	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	204	12	,	,	PUNCT
cana-510	204	13	𝜗	𝜗	NOUN
cana-510	204	14	)	)	PUNCT
cana-510	204	15	≤	≤	NOUN
cana-510	205	1	[	[	X
cana-510	205	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	205	3	−	−	PROPN
cana-510	205	4	1	1	NUM
cana-510	205	5	)	)	PUNCT
cana-510	205	6	+	+	NOUN
cana-510	205	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	205	8	+	+	CCONJ
cana-510	205	9	1	1	NUM
cana-510	205	10	−	−	NOUN
cana-510	205	11	𝜔)]𝜁	𝜔)]𝜁	NOUN
cana-510	206	1	[	[	X
cana-510	206	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	206	3	−	−	PROPN
cana-510	206	4	1	1	X
cana-510	206	5	)	)	PUNCT
cana-510	206	6	+	+	CCONJ
cana-510	206	7	𝜁(𝑛𝜎	𝜁(𝑛𝜎	VERB
cana-510	206	8	+	+	CCONJ
cana-510	206	9	1	1	NUM
cana-510	206	10	−	−	NOUN
cana-510	206	11	𝜔)]ς	𝜔)]ς	NOUN
cana-510	206	12	,	,	PUNCT
cana-510	206	13	which	which	PRON
cana-510	206	14	simplifies	simplify	VERB
cana-510	206	15	to	to	ADP
cana-510	206	16	𝜁	𝜁	DET
cana-510	206	17	≥	≥	NOUN
cana-510	206	18	ς2(𝜎	ς2(𝜎	NOUN
cana-510	206	19	+	+	CCONJ
cana-510	206	20	(	(	PUNCT
cana-510	206	21	1	1	NUM
cana-510	206	22	−	−	PROPN
cana-510	206	23	𝜔))𝜔(𝑛	𝜔))𝜔(𝑛	PROPN
cana-510	206	24	−	−	PROPN
cana-510	206	25	1	1	NUM
cana-510	206	26	)	)	PUNCT
cana-510	207	1	[	[	X
cana-510	207	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	207	3	−	−	PROPN
cana-510	207	4	1	1	NUM
cana-510	207	5	)	)	PUNCT
cana-510	207	6	+	+	NOUN
cana-510	207	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	207	8	+	+	CCONJ
cana-510	207	9	1	1	NUM
cana-510	207	10	−	−	NOUN
cana-510	207	11	𝜔)]2𝛩(𝑛	𝜔)]2𝛩(𝑛	ADJ
cana-510	207	12	,	,	PUNCT
cana-510	207	13	𝜗	𝜗	NOUN
cana-510	207	14	)	)	PUNCT
cana-510	207	15	−	−	NOUN
cana-510	208	1	ς2(𝜎	ς2(𝜎	NUM
cana-510	209	1	+	+	CCONJ
cana-510	209	2	(	(	PUNCT
cana-510	209	3	1	1	NUM
cana-510	209	4	−	−	PROPN
cana-510	209	5	𝜔))(𝑛𝜎	𝜔))(𝑛𝜎	NOUN
cana-510	209	6	+	+	NUM
cana-510	209	7	1	1	NUM
cana-510	209	8	−	−	NOUN
cana-510	209	9	𝜔	𝜔	NOUN
cana-510	209	10	)	)	PUNCT
cana-510	209	11	.	.	PUNCT
cana-510	210	1	7	7	X
cana-510	210	2	.	.	X
cana-510	210	3	closure	closure	NOUN
cana-510	210	4	theorems	theorem	NOUN
cana-510	210	5	we	we	PRON
cana-510	210	6	shall	shall	AUX
cana-510	210	7	prove	prove	VERB
cana-510	210	8	the	the	DET
cana-510	210	9	following	follow	VERB
cana-510	210	10	closure	closure	NOUN
cana-510	210	11	theorems	theorem	NOUN
cana-510	210	12	for	for	ADP
cana-510	210	13	the	the	DET
cana-510	210	14	class	class	NOUN
cana-510	210	15	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	210	16	,	,	PUNCT
cana-510	210	17	𝜎	𝜎	PROPN
cana-510	210	18	,	,	PUNCT
cana-510	210	19	ς	ς	PROPN
cana-510	210	20	,	,	PUNCT
cana-510	210	21	𝜗	𝜗	NOUN
cana-510	210	22	)	)	PUNCT
cana-510	210	23	.	.	PUNCT
cana-510	211	1	theorem	theorem	VERB
cana-510	211	2	7.1	7.1	NUM
cana-510	211	3	.	.	PUNCT
cana-510	212	1	let	let	VERB
cana-510	212	2	𝑢𝑗	𝑢𝑗	NOUN
cana-510	212	3	∈	∈	PROPN
cana-510	212	4	𝑇𝑆(𝜔	𝑇𝑆(𝜔	PROPN
cana-510	212	5	,	,	PUNCT
cana-510	212	6	𝜎	𝜎	PROPN
cana-510	212	7	,	,	PUNCT
cana-510	212	8	ς	ς	PROPN
cana-510	212	9	,	,	PUNCT
cana-510	212	10	𝜗	𝜗	NOUN
cana-510	212	11	)	)	PUNCT
cana-510	212	12	.	.	PUNCT
cana-510	212	13	)	)	PUNCT
cana-510	213	1	,	,	PUNCT
cana-510	213	2	j=1,2	j=1,2	PROPN
cana-510	213	3	,	,	PUNCT
cana-510	213	4	…	…	PUNCT
cana-510	213	5	then	then	ADV
cana-510	213	6	𝑔(𝑧	𝑔(𝑧	X
cana-510	213	7	)	)	PUNCT
cana-510	213	8	=	=	PUNCT
cana-510	214	1	∑	∑	PUNCT
cana-510	214	2	𝑐𝑗	𝑐𝑗	NUM
cana-510	214	3	𝑠	𝑠	X
cana-510	214	4	𝑗=1	𝑗=1	NOUN
cana-510	214	5	𝑢𝑗(𝑧	𝑢𝑗(𝑧	ADV
cana-510	214	6	)	)	PUNCT
cana-510	214	7	∈	∈	PROPN
cana-510	214	8	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	214	9	,	,	PUNCT
cana-510	214	10	𝜎	𝜎	PROPN
cana-510	214	11	,	,	PUNCT
cana-510	214	12	ς	ς	PROPN
cana-510	214	13	,	,	PUNCT
cana-510	214	14	𝜗	𝜗	NOUN
cana-510	214	15	)	)	PUNCT
cana-510	214	16	.	.	PUNCT
cana-510	215	1	for	for	ADP
cana-510	215	2	𝑢𝑗(𝑧	𝑢𝑗(𝑧	NOUN
cana-510	215	3	)	)	PUNCT
cana-510	215	4	=	=	SYM
cana-510	215	5	𝑧	𝑧	PRON
cana-510	215	6	−	−	PROPN
cana-510	215	7	∑	∑	PUNCT
cana-510	215	8	𝑎𝑛,𝑗	𝑎𝑛,𝑗	X
cana-510	215	9	∞	∞	PROPN
cana-510	215	10	𝑛=2	𝑛=2	PROPN
cana-510	215	11	𝑧𝑛	𝑧𝑛	PROPN
cana-510	215	12	,	,	PUNCT
cana-510	216	1	where	where	SCONJ
cana-510	216	2	∑	∑	ADV
cana-510	216	3	𝑐𝑗	𝑐𝑗	NOUN
cana-510	216	4	𝑠	𝑠	NOUN
cana-510	216	5	𝑗=1	𝑗=1	PUNCT
cana-510	216	6	=	=	SYM
cana-510	216	7	1	1	X
cana-510	216	8	.	.	PUNCT
cana-510	217	1	proof	proof	NOUN
cana-510	217	2	.	.	PUNCT
cana-510	218	1	𝑔(𝑧	𝑔(𝑧	X
cana-510	218	2	)	)	PUNCT
cana-510	218	3	=	=	PUNCT
cana-510	219	1	∑	∑	PUNCT
cana-510	219	2	𝑐𝑗	𝑐𝑗	NUM
cana-510	219	3	𝑠	𝑠	X
cana-510	219	4	𝑗=1	𝑗=1	NOUN
cana-510	219	5	𝑢𝑗(𝑧	𝑢𝑗(𝑧	PUNCT
cana-510	219	6	)	)	PUNCT
cana-510	220	1	=	=	SYM
cana-510	220	2	𝑧	𝑧	PRON
cana-510	220	3	−	−	NOUN
cana-510	220	4	∑	∑	INTJ
cana-510	220	5	∑	∑	PROPN
cana-510	220	6	𝑐𝑗	𝑐𝑗	NOUN
cana-510	220	7	𝑠	𝑠	NOUN
cana-510	220	8	𝑗=1	𝑗=1	PROPN
cana-510	220	9	∞	∞	NUM
cana-510	220	10	𝑛=2	𝑛=2	X
cana-510	220	11	𝑎𝑛,𝑗𝑧𝑛	𝑎𝑛,𝑗𝑧𝑛	NOUN
cana-510	220	12	=	=	SYM
cana-510	220	13	𝑧	𝑧	ADJ
cana-510	220	14	−	−	NOUN
cana-510	220	15	∑	∑	NOUN
cana-510	220	16	𝑒𝑛	𝑒𝑛	PROPN
cana-510	220	17	∞	∞	PROPN
cana-510	220	18	𝑛=2	𝑛=2	PROPN
cana-510	220	19	𝑧𝑛	𝑧𝑛	ADP
cana-510	220	20	,	,	PUNCT
cana-510	220	21	where	where	SCONJ
cana-510	220	22	𝑒𝑛	𝑒𝑛	ADP
cana-510	220	23	=	=	PUNCT
cana-510	220	24	∑	∑	PROPN
cana-510	220	25	𝑐𝑗	𝑐𝑗	INTJ
cana-510	220	26	𝑠	𝑠	INTJ
cana-510	220	27	𝑗=1	𝑗=1	NOUN
cana-510	220	28	𝑎𝑛,𝑗.	𝑎𝑛,𝑗.	ADJ
cana-510	220	29	thus	thus	ADV
cana-510	220	30	𝑔(𝑧	𝑔(𝑧	NOUN
cana-510	220	31	)	)	PUNCT
cana-510	220	32	∈	∈	NOUN
cana-510	220	33	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	220	34	,	,	PUNCT
cana-510	220	35	𝜎	𝜎	PROPN
cana-510	220	36	,	,	PUNCT
cana-510	220	37	ς	ς	PROPN
cana-510	220	38	,	,	PUNCT
cana-510	220	39	𝜗	𝜗	NOUN
cana-510	220	40	)	)	PUNCT
cana-510	220	41	if	if	SCONJ
cana-510	220	42	communications	communication	NOUN
cana-510	220	43	on	on	ADP
cana-510	220	44	applied	apply	VERB
cana-510	220	45	nonlinear	nonlinear	ADJ
cana-510	220	46	analysis	analysis	NOUN
cana-510	220	47	issn	issn	NOUN
cana-510	220	48	:	:	PUNCT
cana-510	220	49	1074	1074	NUM
cana-510	220	50	-	-	PUNCT
cana-510	220	51	133x	133x	NUM
cana-510	220	52	vol	vol	NOUN
cana-510	220	53	31	31	NUM
cana-510	220	54	no	no	NOUN
cana-510	220	55	.	.	NOUN
cana-510	220	56	2	2	NUM
cana-510	220	57	(	(	PUNCT
cana-510	220	58	2024	2024	NUM
cana-510	220	59	)	)	PUNCT
cana-510	220	60	32	32	NUM
cana-510	220	61	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	220	62	∑	∑	PUNCT
cana-510	221	1	[	[	X
cana-510	221	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	221	3	−	−	PROPN
cana-510	221	4	1	1	NUM
cana-510	221	5	)	)	PUNCT
cana-510	221	6	+	+	NOUN
cana-510	221	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	221	8	+	+	CCONJ
cana-510	221	9	1	1	NUM
cana-510	221	10	−	−	NOUN
cana-510	221	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	221	12	,	,	PUNCT
cana-510	221	13	𝜗	𝜗	NOUN
cana-510	221	14	)	)	PUNCT
cana-510	221	15	ς(𝜎	ς(𝜎	NOUN
cana-510	221	16	+	+	CCONJ
cana-510	221	17	(	(	PUNCT
cana-510	221	18	1	1	NUM
cana-510	221	19	−	−	NOUN
cana-510	221	20	𝜔	𝜔	NOUN
cana-510	221	21	)	)	PUNCT
cana-510	221	22	)	)	PUNCT
cana-510	222	1	∞	∞	NUM
cana-510	222	2	𝑛=2	𝑛=2	PART
cana-510	222	3	𝑒𝑛	𝑒𝑛	NOUN
cana-510	222	4	≤	≤	NUM
cana-510	222	5	1	1	NUM
cana-510	222	6	,	,	PUNCT
cana-510	222	7	that	that	ADV
cana-510	222	8	is	is	ADV
cana-510	222	9	,	,	PUNCT
cana-510	222	10	if	if	SCONJ
cana-510	222	11	∑	∑	VERB
cana-510	222	12	∑	∑	PROPN
cana-510	222	13	[	[	X
cana-510	222	14	𝜔(𝑛	𝜔(𝑛	X
cana-510	222	15	−	−	PROPN
cana-510	222	16	1	1	NUM
cana-510	222	17	)	)	PUNCT
cana-510	222	18	+	+	NOUN
cana-510	222	19	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	222	20	+	+	CCONJ
cana-510	222	21	1	1	NUM
cana-510	222	22	−	−	NOUN
cana-510	222	23	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	222	24	,	,	PUNCT
cana-510	222	25	𝜗	𝜗	NOUN
cana-510	222	26	)	)	PUNCT
cana-510	222	27	ς(𝜎	ς(𝜎	NOUN
cana-510	222	28	+	+	CCONJ
cana-510	222	29	(	(	PUNCT
cana-510	222	30	1	1	NUM
cana-510	222	31	−	−	NOUN
cana-510	222	32	𝜔	𝜔	NOUN
cana-510	222	33	)	)	PUNCT
cana-510	222	34	)	)	PUNCT
cana-510	223	1	𝑠	𝑠	PROPN
cana-510	223	2	𝑗=1	𝑗=1	PROPN
cana-510	223	3	∞	∞	PROPN
cana-510	223	4	𝑛=2	𝑛=2	X
cana-510	223	5	𝑐𝑗𝑎𝑛,𝑗	𝑐𝑗𝑎𝑛,𝑗	NOUN
cana-510	223	6	=	=	PUNCT
cana-510	224	1	∑	∑	PUNCT
cana-510	224	2	𝑐𝑗	𝑐𝑗	INTJ
cana-510	224	3	𝑠	𝑠	INTJ
cana-510	224	4	𝑗=1	𝑗=1	PUNCT
cana-510	224	5	∑	∑	PUNCT
cana-510	225	1	[	[	X
cana-510	225	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	225	3	−	−	PROPN
cana-510	225	4	1	1	NUM
cana-510	225	5	)	)	PUNCT
cana-510	225	6	+	+	NOUN
cana-510	225	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	225	8	+	+	CCONJ
cana-510	225	9	1	1	NUM
cana-510	225	10	−	−	NOUN
cana-510	225	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	225	12	,	,	PUNCT
cana-510	225	13	𝜗	𝜗	NOUN
cana-510	225	14	)	)	PUNCT
cana-510	225	15	ς(𝜎	ς(𝜎	NOUN
cana-510	225	16	+	+	CCONJ
cana-510	225	17	(	(	PUNCT
cana-510	225	18	1	1	NUM
cana-510	225	19	−	−	NOUN
cana-510	225	20	𝜔	𝜔	NOUN
cana-510	225	21	)	)	PUNCT
cana-510	225	22	)	)	PUNCT
cana-510	226	1	∞	∞	NUM
cana-510	226	2	𝑛=2	𝑛=2	NOUN
cana-510	226	3	𝑎𝑛,𝑗	𝑎𝑛,𝑗	ADV
cana-510	226	4	≤	≤	NOUN
cana-510	226	5	∑	∑	PUNCT
cana-510	226	6	𝑐𝑗	𝑐𝑗	NOUN
cana-510	226	7	𝑠	𝑠	NOUN
cana-510	226	8	𝑗=1	𝑗=1	PUNCT
cana-510	226	9	=	=	SYM
cana-510	226	10	1	1	X
cana-510	226	11	.	.	PUNCT
cana-510	226	12	theorem	theorem	VERB
cana-510	226	13	14	14	NUM
cana-510	226	14	.	.	PUNCT
cana-510	227	1	let	let	VERB
cana-510	227	2	𝑢	𝑢	NOUN
cana-510	227	3	,	,	PUNCT
cana-510	227	4	𝑔	𝑔	PROPN
cana-510	227	5	∈	∈	PROPN
cana-510	227	6	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	227	7	,	,	PUNCT
cana-510	227	8	𝜎	𝜎	PROPN
cana-510	227	9	,	,	PUNCT
cana-510	227	10	ς	ς	PROPN
cana-510	227	11	,	,	PUNCT
cana-510	227	12	𝜗	𝜗	NOUN
cana-510	227	13	)	)	PUNCT
cana-510	227	14	.	.	PUNCT
cana-510	228	1	then	then	ADV
cana-510	228	2	ℎ(𝑧	ℎ(𝑧	VERB
cana-510	228	3	)	)	PUNCT
cana-510	229	1	=	=	SYM
cana-510	229	2	𝑧	𝑧	PRON
cana-510	229	3	−	−	NOUN
cana-510	229	4	∑	∑	INTJ
cana-510	229	5	(	(	PUNCT
cana-510	229	6	𝑎𝑛	𝑎𝑛	PROPN
cana-510	229	7	2	2	NUM
cana-510	229	8	+	+	CCONJ
cana-510	229	9	𝑏𝑛	𝑏𝑛	ADP
cana-510	229	10	2)∞	2)∞	NUM
cana-510	229	11	𝑛=2	𝑛=2	NOUN
cana-510	229	12	𝑧𝑛	𝑧𝑛	ADP
cana-510	229	13	∈	∈	PROPN
cana-510	229	14	𝑇𝑆(𝜔	𝑇𝑆(𝜔	PROPN
cana-510	229	15	,	,	PUNCT
cana-510	229	16	𝜎	𝜎	PROPN
cana-510	229	17	,	,	PUNCT
cana-510	229	18	ς	ς	PROPN
cana-510	229	19	,	,	PUNCT
cana-510	229	20	𝜗	𝜗	NOUN
cana-510	229	21	)	)	PUNCT
cana-510	229	22	,	,	PUNCT
cana-510	229	23	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	PROPN
cana-510	229	24	𝜁	𝜁	PROPN
cana-510	229	25	≥	≥	NOUN
cana-510	229	26	2𝜔(𝑛	2𝜔(𝑛	NUM
cana-510	229	27	−	−	NOUN
cana-510	229	28	1)ς2(𝜎	1)ς2(𝜎	NUM
cana-510	229	29	+	+	CCONJ
cana-510	229	30	(	(	PUNCT
cana-510	229	31	1	1	NUM
cana-510	229	32	−	−	NOUN
cana-510	229	33	𝜔	𝜔	NOUN
cana-510	229	34	)	)	PUNCT
cana-510	229	35	)	)	PUNCT
cana-510	230	1	[	[	X
cana-510	230	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	230	3	−	−	PROPN
cana-510	230	4	1	1	NUM
cana-510	230	5	)	)	PUNCT
cana-510	230	6	+	+	NOUN
cana-510	230	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	230	8	+	+	CCONJ
cana-510	230	9	1	1	NUM
cana-510	230	10	−	−	NOUN
cana-510	230	11	𝜔)]2𝛩(𝑛	𝜔)]2𝛩(𝑛	ADJ
cana-510	230	12	,	,	PUNCT
cana-510	230	13	𝜗	𝜗	NOUN
cana-510	230	14	)	)	PUNCT
cana-510	230	15	−	−	PROPN
cana-510	230	16	2ς2(𝜎	2ς2(𝜎	NOUN
cana-510	230	17	+	+	CCONJ
cana-510	230	18	(	(	PUNCT
cana-510	230	19	1	1	NUM
cana-510	230	20	−	−	PROPN
cana-510	230	21	𝜔))(𝑛𝜎	𝜔))(𝑛𝜎	NOUN
cana-510	230	22	+	+	NUM
cana-510	230	23	1	1	NUM
cana-510	230	24	−	−	NOUN
cana-510	230	25	𝜔	𝜔	NOUN
cana-510	230	26	)	)	PUNCT
cana-510	230	27	.	.	PUNCT
cana-510	231	1	proof	proof	NOUN
cana-510	231	2	.	.	PUNCT
cana-510	232	1	since	since	SCONJ
cana-510	232	2	𝑢	𝑢	X
cana-510	232	3	,	,	PUNCT
cana-510	232	4	𝑔	𝑔	PROPN
cana-510	232	5	∈	∈	PROPN
cana-510	232	6	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	232	7	,	,	PUNCT
cana-510	232	8	𝜎	𝜎	PROPN
cana-510	232	9	,	,	PUNCT
cana-510	232	10	ς	ς	PROPN
cana-510	232	11	,	,	PUNCT
cana-510	232	12	𝜗	𝜗	NOUN
cana-510	232	13	)	)	PUNCT
cana-510	232	14	,	,	PUNCT
cana-510	232	15	so	so	ADV
cana-510	232	16	theorem	theorem	VERB
cana-510	232	17	2.1	2.1	NUM
cana-510	232	18	,	,	PUNCT
cana-510	232	19	yields	yield	VERB
cana-510	232	20	∑	∑	PUNCT
cana-510	232	21	[	[	PUNCT
cana-510	232	22	(	(	PUNCT
cana-510	232	23	𝜔(𝑛	𝜔(𝑛	PROPN
cana-510	232	24	−	−	PROPN
cana-510	232	25	1	1	NUM
cana-510	232	26	)	)	PUNCT
cana-510	232	27	+	+	NOUN
cana-510	232	28	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	232	29	+	+	CCONJ
cana-510	232	30	1	1	NUM
cana-510	232	31	−	−	NOUN
cana-510	232	32	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	232	33	,	,	PUNCT
cana-510	232	34	𝜗	𝜗	NOUN
cana-510	232	35	)	)	PUNCT
cana-510	232	36	ς(𝜎	ς(𝜎	NOUN
cana-510	232	37	+	+	CCONJ
cana-510	232	38	(	(	PUNCT
cana-510	232	39	1	1	NUM
cana-510	232	40	−	−	NOUN
cana-510	232	41	𝜔	𝜔	NOUN
cana-510	232	42	)	)	PUNCT
cana-510	232	43	)	)	PUNCT
cana-510	233	1	𝑎𝑛	𝑎𝑛	PROPN
cana-510	233	2	]	]	X
cana-510	233	3	2∞	2∞	NUM
cana-510	233	4	𝑛=2	𝑛=2	NOUN
cana-510	233	5	≤	≤	NUM
cana-510	233	6	1	1	NUM
cana-510	233	7	and	and	CCONJ
cana-510	233	8	∑	∑	ADP
cana-510	233	9	[	[	PUNCT
cana-510	233	10	(	(	PUNCT
cana-510	233	11	𝜔(𝑛	𝜔(𝑛	PROPN
cana-510	233	12	−	−	PROPN
cana-510	233	13	1	1	NUM
cana-510	233	14	)	)	PUNCT
cana-510	233	15	+	+	NOUN
cana-510	233	16	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	233	17	+	+	CCONJ
cana-510	233	18	1	1	NUM
cana-510	233	19	−	−	NOUN
cana-510	233	20	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	233	21	,	,	PUNCT
cana-510	233	22	𝜗	𝜗	NOUN
cana-510	233	23	)	)	PUNCT
cana-510	233	24	ς(𝜎	ς(𝜎	NOUN
cana-510	233	25	+	+	CCONJ
cana-510	233	26	(	(	PUNCT
cana-510	233	27	1	1	NUM
cana-510	233	28	−	−	NOUN
cana-510	233	29	𝜔	𝜔	NOUN
cana-510	233	30	)	)	PUNCT
cana-510	233	31	)	)	PUNCT
cana-510	234	1	𝑏𝑛	𝑏𝑛	ADP
cana-510	234	2	]	]	X
cana-510	234	3	2∞	2∞	NUM
cana-510	234	4	𝑛=2	𝑛=2	NOUN
cana-510	234	5	≤	≤	NUM
cana-510	234	6	1	1	NUM
cana-510	234	7	.	.	PUNCT
cana-510	235	1	we	we	PRON
cana-510	235	2	obtain	obtain	VERB
cana-510	235	3	from	from	ADP
cana-510	235	4	the	the	DET
cana-510	235	5	last	last	ADJ
cana-510	235	6	two	two	NUM
cana-510	235	7	inequalities	inequality	NOUN
cana-510	235	8	∑	∑	PROPN
cana-510	235	9	1	1	NUM
cana-510	235	10	2	2	NUM
cana-510	235	11	∞	∞	NUM
cana-510	235	12	𝑛=2	𝑛=2	NOUN
cana-510	235	13	[	[	PUNCT
cana-510	235	14	(	(	PUNCT
cana-510	235	15	𝜔(𝑛	𝜔(𝑛	PROPN
cana-510	235	16	−	−	PROPN
cana-510	235	17	1	1	NUM
cana-510	235	18	)	)	PUNCT
cana-510	235	19	+	+	NOUN
cana-510	235	20	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	235	21	+	+	CCONJ
cana-510	235	22	1	1	NUM
cana-510	235	23	−	−	NOUN
cana-510	235	24	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	235	25	,	,	PUNCT
cana-510	235	26	𝜗	𝜗	NOUN
cana-510	235	27	)	)	PUNCT
cana-510	235	28	ς(𝜎	ς(𝜎	NOUN
cana-510	235	29	+	+	CCONJ
cana-510	235	30	(	(	PUNCT
cana-510	235	31	1	1	NUM
cana-510	235	32	−	−	NOUN
cana-510	235	33	𝜔	𝜔	NOUN
cana-510	235	34	)	)	PUNCT
cana-510	235	35	)	)	PUNCT
cana-510	235	36	]	]	PUNCT
cana-510	236	1	2	2	X
cana-510	236	2	(	(	PUNCT
cana-510	236	3	𝑎𝑛	𝑎𝑛	PROPN
cana-510	236	4	2	2	NUM
cana-510	236	5	+	+	CCONJ
cana-510	236	6	𝑏𝑛	𝑏𝑛	ADP
cana-510	236	7	2	2	NUM
cana-510	236	8	)	)	PUNCT
cana-510	236	9	≤	≤	NOUN
cana-510	236	10	1	1	NUM
cana-510	236	11	.	.	PUNCT
cana-510	237	1	(	(	PUNCT
cana-510	237	2	7.1	7.1	NUM
cana-510	237	3	)	)	PUNCT
cana-510	237	4	but	but	CCONJ
cana-510	237	5	ℎ(𝑧	ℎ(𝑧	PROPN
cana-510	237	6	)	)	PUNCT
cana-510	237	7	∈	∈	PROPN
cana-510	237	8	𝑇𝑆(𝜔	𝑇𝑆(𝜔	NOUN
cana-510	237	9	,	,	PUNCT
cana-510	237	10	𝜎	𝜎	PROPN
cana-510	237	11	,	,	PUNCT
cana-510	237	12	𝜁	𝜁	PROPN
cana-510	237	13	,	,	PUNCT
cana-510	237	14	𝑞	𝑞	NOUN
cana-510	237	15	,	,	PUNCT
cana-510	237	16	𝑚	𝑚	NOUN
cana-510	237	17	)	)	PUNCT
cana-510	237	18	,	,	PUNCT
cana-510	237	19	if	if	SCONJ
cana-510	237	20	and	and	CCONJ
cana-510	238	1	only	only	ADV
cana-510	238	2	if	if	SCONJ
cana-510	238	3	∑	∑	PROPN
cana-510	238	4	[	[	X
cana-510	238	5	𝜔(𝑛	𝜔(𝑛	X
cana-510	238	6	−	−	PROPN
cana-510	238	7	1	1	NUM
cana-510	238	8	)	)	PUNCT
cana-510	238	9	+	+	CCONJ
cana-510	238	10	𝜁(𝑛𝜎	𝜁(𝑛𝜎	VERB
cana-510	238	11	+	+	CCONJ
cana-510	238	12	1	1	NUM
cana-510	238	13	−	−	NOUN
cana-510	238	14	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	238	15	,	,	PUNCT
cana-510	238	16	𝜗	𝜗	NOUN
cana-510	238	17	)	)	PUNCT
cana-510	238	18	𝜁(𝜎	𝜁(𝜎	PROPN
cana-510	238	19	+	+	CCONJ
cana-510	238	20	(	(	PUNCT
cana-510	238	21	1	1	NUM
cana-510	238	22	−	−	NOUN
cana-510	238	23	𝜔	𝜔	NOUN
cana-510	238	24	)	)	PUNCT
cana-510	238	25	)	)	PUNCT
cana-510	239	1	∞	∞	NUM
cana-510	239	2	𝑛=2	𝑛=2	PROPN
cana-510	239	3	(	(	PUNCT
cana-510	239	4	𝑎𝑛	𝑎𝑛	NOUN
cana-510	239	5	2	2	NUM
cana-510	239	6	+	+	CCONJ
cana-510	239	7	𝑏𝑛	𝑏𝑛	ADP
cana-510	239	8	2	2	NUM
cana-510	239	9	)	)	PUNCT
cana-510	239	10	≤	≤	NUM
cana-510	239	11	1	1	NUM
cana-510	239	12	,	,	PUNCT
cana-510	239	13	(	(	PUNCT
cana-510	239	14	7.2	7.2	NUM
cana-510	239	15	)	)	PUNCT
cana-510	239	16	where	where	SCONJ
cana-510	239	17	0	0	PUNCT
cana-510	239	18	<	<	X
cana-510	239	19	𝜁	𝜁	X
cana-510	239	20	<	<	X
cana-510	239	21	1	1	NUM
cana-510	239	22	,	,	PUNCT
cana-510	239	23	however	however	ADV
cana-510	239	24	(	(	PUNCT
cana-510	239	25	7.1	7.1	NUM
cana-510	239	26	)	)	PUNCT
cana-510	239	27	implies	imply	VERB
cana-510	239	28	(	(	PUNCT
cana-510	239	29	7.2	7.2	NUM
cana-510	239	30	)	)	PUNCT
cana-510	239	31	if	if	SCONJ
cana-510	239	32	communications	communication	NOUN
cana-510	239	33	on	on	ADP
cana-510	239	34	applied	apply	VERB
cana-510	239	35	nonlinear	nonlinear	ADJ
cana-510	239	36	analysis	analysis	NOUN
cana-510	239	37	issn	issn	NOUN
cana-510	239	38	:	:	PUNCT
cana-510	239	39	1074	1074	NUM
cana-510	239	40	-	-	PUNCT
cana-510	239	41	133x	133x	NUM
cana-510	239	42	vol	vol	NOUN
cana-510	239	43	31	31	NUM
cana-510	239	44	no	no	NOUN
cana-510	239	45	.	.	NOUN
cana-510	239	46	2	2	NUM
cana-510	239	47	(	(	PUNCT
cana-510	239	48	2024	2024	NUM
cana-510	239	49	)	)	PUNCT
cana-510	239	50	33	33	NUM
cana-510	239	51	https://internationalpubls.com	https://internationalpubls.com	X
cana-510	240	1	[	[	X
cana-510	240	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	240	3	−	−	PROPN
cana-510	240	4	1	1	NUM
cana-510	240	5	)	)	PUNCT
cana-510	240	6	+	+	CCONJ
cana-510	240	7	𝜁(𝑛𝜎	𝜁(𝑛𝜎	VERB
cana-510	240	8	+	+	CCONJ
cana-510	240	9	1	1	NUM
cana-510	240	10	−	−	NOUN
cana-510	240	11	𝜔)]𝛩(𝑛	𝜔)]𝛩(𝑛	INTJ
cana-510	240	12	,	,	PUNCT
cana-510	240	13	𝜗	𝜗	NOUN
cana-510	240	14	)	)	PUNCT
cana-510	240	15	𝜁(𝜎	𝜁(𝜎	PROPN
cana-510	240	16	+	+	CCONJ
cana-510	240	17	(	(	PUNCT
cana-510	240	18	1	1	NUM
cana-510	240	19	−	−	NOUN
cana-510	240	20	𝜔	𝜔	NOUN
cana-510	240	21	)	)	PUNCT
cana-510	240	22	)	)	PUNCT
cana-510	240	23	≤	≤	NUM
cana-510	240	24	1	1	NUM
cana-510	240	25	2	2	NUM
cana-510	240	26	[	[	PUNCT
cana-510	240	27	(	(	PUNCT
cana-510	240	28	𝜔(𝑛	𝜔(𝑛	PROPN
cana-510	240	29	−	−	PROPN
cana-510	240	30	1	1	NUM
cana-510	240	31	)	)	PUNCT
cana-510	240	32	+	+	NOUN
cana-510	240	33	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	240	34	+	+	CCONJ
cana-510	240	35	1	1	NUM
cana-510	240	36	−	−	NOUN
cana-510	240	37	𝜔))𝛩(𝑛	𝜔))𝛩(𝑛	PROPN
cana-510	240	38	,	,	PUNCT
cana-510	240	39	𝜗	𝜗	NOUN
cana-510	240	40	)	)	PUNCT
cana-510	240	41	ς(𝜎	ς(𝜎	NOUN
cana-510	240	42	+	+	CCONJ
cana-510	240	43	(	(	PUNCT
cana-510	240	44	1	1	NUM
cana-510	240	45	−	−	NOUN
cana-510	240	46	𝜔	𝜔	NOUN
cana-510	240	47	)	)	PUNCT
cana-510	240	48	)	)	PUNCT
cana-510	240	49	]	]	PUNCT
cana-510	241	1	2	2	X
cana-510	241	2	.	.	X
cana-510	241	3	simplifying	simplify	VERB
cana-510	241	4	,	,	PUNCT
cana-510	241	5	we	we	PRON
cana-510	241	6	get	get	VERB
cana-510	241	7	𝜁	𝜁	DET
cana-510	241	8	≥	≥	NOUN
cana-510	241	9	2𝜔(𝑛	2𝜔(𝑛	NUM
cana-510	241	10	−	−	NOUN
cana-510	241	11	1)ς2(𝜎	1)ς2(𝜎	NUM
cana-510	241	12	+	+	CCONJ
cana-510	241	13	(	(	PUNCT
cana-510	241	14	1	1	NUM
cana-510	241	15	−	−	NOUN
cana-510	241	16	𝜔	𝜔	NOUN
cana-510	241	17	)	)	PUNCT
cana-510	241	18	)	)	PUNCT
cana-510	242	1	[	[	X
cana-510	242	2	𝜔(𝑛	𝜔(𝑛	X
cana-510	242	3	−	−	PROPN
cana-510	242	4	1	1	NUM
cana-510	242	5	)	)	PUNCT
cana-510	242	6	+	+	NOUN
cana-510	242	7	ς(𝑛𝜎	ς(𝑛𝜎	NUM
cana-510	242	8	+	+	CCONJ
cana-510	242	9	1	1	NUM
cana-510	242	10	−	−	NOUN
cana-510	242	11	𝜔)]2𝛩(𝑛	𝜔)]2𝛩(𝑛	ADJ
cana-510	242	12	,	,	PUNCT
cana-510	242	13	𝜗	𝜗	NOUN
cana-510	242	14	)	)	PUNCT
cana-510	242	15	−	−	PROPN
cana-510	242	16	2ς2(𝜎	2ς2(𝜎	NOUN
cana-510	242	17	+	+	CCONJ
cana-510	242	18	(	(	PUNCT
cana-510	242	19	1	1	NUM
cana-510	242	20	−	−	PROPN
cana-510	242	21	𝜔))(𝑛𝜎	𝜔))(𝑛𝜎	NOUN
cana-510	242	22	+	+	NUM
cana-510	242	23	1	1	NUM
cana-510	242	24	−	−	NOUN
cana-510	242	25	𝜔	𝜔	NOUN
cana-510	242	26	)	)	PUNCT
cana-510	242	27	.	.	PUNCT
cana-510	243	1	refrences	refrence	VERB
cana-510	244	1	[	[	X
cana-510	244	2	1	1	X
cana-510	244	3	]	]	X
cana-510	244	4	cho	cho	PROPN
cana-510	244	5	,	,	PUNCT
cana-510	244	6	n.	n.	PROPN
cana-510	244	7	e.	e.	PROPN
cana-510	244	8	,	,	PUNCT
cana-510	244	9	woo	woo	PROPN
cana-510	244	10	,	,	PUNCT
cana-510	244	11	s.	s.	PROPN
cana-510	244	12	y.	y.	PROPN
cana-510	244	13	and	and	CCONJ
cana-510	244	14	owa	owa	PROPN
cana-510	244	15	,	,	PUNCT
cana-510	244	16	s.	s.	PROPN
cana-510	244	17	,	,	PUNCT
cana-510	244	18	uniform	uniform	ADJ
cana-510	244	19	convexity	convexity	NOUN
cana-510	244	20	properties	property	NOUN
cana-510	244	21	for	for	ADP
cana-510	244	22	hypergeometric	hypergeometric	ADJ
cana-510	244	23	functions	function	NOUN
cana-510	244	24	,	,	PUNCT
cana-510	244	25	fract	fract	NOUN
cana-510	244	26	.	.	PUNCT
cana-510	245	1	calc	calc	PROPN
cana-510	245	2	.	.	PUNCT
cana-510	246	1	appl	appl	PROPN
cana-510	246	2	.	.	PUNCT
cana-510	247	1	anal	anal	PROPN
cana-510	247	2	.	.	PROPN
cana-510	247	3	,	,	PUNCT
cana-510	247	4	5(3	5(3	NUM
cana-510	247	5	)	)	PUNCT
cana-510	247	6	(	(	PUNCT
cana-510	247	7	2002	2002	NUM
cana-510	247	8	)	)	PUNCT
cana-510	247	9	,	,	PUNCT
cana-510	247	10	303	303	NUM
cana-510	247	11	313	313	NUM
cana-510	247	12	.	.	PUNCT
cana-510	248	1	[	[	X
cana-510	248	2	2	2	NUM
cana-510	248	3	]	]	X
cana-510	248	4	de	de	X
cana-510	248	5	branges	brange	NOUN
cana-510	248	6	,	,	PUNCT
cana-510	248	7	l.	l.	PROPN
cana-510	248	8	,	,	PUNCT
cana-510	248	9	a	a	DET
cana-510	248	10	proof	proof	NOUN
cana-510	248	11	of	of	ADP
cana-510	248	12	the	the	DET
cana-510	248	13	bieberbach	bieberbach	NOUN
cana-510	248	14	conjecture	conjecture	NOUN
cana-510	248	15	,	,	PUNCT
cana-510	248	16	acta	acta	PROPN
cana-510	248	17	math	math	PROPN
cana-510	248	18	.	.	PUNCT
cana-510	248	19	,	,	PUNCT
cana-510	248	20	154(1	154(1	NUM
cana-510	248	21	-	-	SYM
cana-510	248	22	2	2	NUM
cana-510	248	23	)	)	PUNCT
cana-510	248	24	(	(	PUNCT
cana-510	248	25	1985	1985	NUM
cana-510	248	26	)	)	PUNCT
cana-510	248	27	,	,	PUNCT
cana-510	248	28	137	137	NUM
cana-510	248	29	152	152	NUM
cana-510	248	30	.	.	PUNCT
cana-510	249	1	[	[	X
cana-510	249	2	3	3	X
cana-510	249	3	]	]	X
cana-510	249	4	el	el	PROPN
cana-510	249	5	-	-	PUNCT
cana-510	249	6	deeb	deeb	PROPN
cana-510	249	7	,	,	PUNCT
cana-510	249	8	s.	s.	PROPN
cana-510	249	9	m.	m.	PROPN
cana-510	249	10	,	,	PUNCT
cana-510	249	11	bulboaca	bulboaca	NOUN
cana-510	249	12	,	,	PUNCT
cana-510	249	13	t.	t.	NOUN
cana-510	249	14	and	and	CCONJ
cana-510	249	15	dziok	dziok	NOUN
cana-510	249	16	,	,	PUNCT
cana-510	249	17	j.	j.	PROPN
cana-510	249	18	,	,	PUNCT
cana-510	249	19	pascal	pascal	ADJ
cana-510	249	20	distribution	distribution	NOUN
cana-510	249	21	series	series	NOUN
cana-510	249	22	connected	connect	VERB
cana-510	249	23	with	with	ADP
cana-510	249	24	certain	certain	ADJ
cana-510	249	25	subclasses	subclass	NOUN
cana-510	249	26	of	of	ADP
cana-510	249	27	univalent	univalent	ADJ
cana-510	249	28	functions	function	NOUN
cana-510	249	29	,	,	PUNCT
cana-510	249	30	kyungpook	kyungpook	NOUN
cana-510	249	31	math	math	NOUN
cana-510	249	32	.	.	PUNCT
cana-510	250	1	j.	j.	PROPN
cana-510	250	2	,	,	PUNCT
cana-510	250	3	59	59	NUM
cana-510	250	4	(	(	PUNCT
cana-510	250	5	2	2	NUM
cana-510	250	6	)	)	PUNCT
cana-510	250	7	(	(	PUNCT
cana-510	250	8	2019	2019	NUM
cana-510	250	9	)	)	PUNCT
cana-510	250	10	,	,	PUNCT
cana-510	250	11	301	301	NUM
cana-510	250	12	314	314	NUM
cana-510	250	13	.	.	PUNCT
cana-510	251	1	[	[	X
cana-510	251	2	4	4	NUM
cana-510	251	3	]	]	X
cana-510	251	4	kim	kim	PROPN
cana-510	251	5	,	,	PUNCT
cana-510	251	6	y.c	y.c	PROPN
cana-510	251	7	.	.	PROPN
cana-510	251	8	and	and	CCONJ
cana-510	251	9	srivastava	srivastava	PROPN
cana-510	251	10	,	,	PUNCT
cana-510	251	11	h.m	h.m	PROPN
cana-510	251	12	.	.	PROPN
cana-510	251	13	,	,	PUNCT
cana-510	251	14	fractional	fractional	ADJ
cana-510	251	15	integral	integral	ADJ
cana-510	251	16	and	and	CCONJ
cana-510	251	17	other	other	ADJ
cana-510	251	18	linear	linear	PROPN
cana-510	251	19	operators	operator	NOUN
cana-510	251	20	asso	asso	PROPN
cana-510	251	21	ciated	ciate	VERB
cana-510	251	22	with	with	ADP
cana-510	251	23	the	the	DET
cana-510	251	24	gaussian	gaussian	ADJ
cana-510	251	25	hypergeometric	hypergeometric	ADJ
cana-510	251	26	function	function	NOUN
cana-510	251	27	,	,	PUNCT
cana-510	251	28	complex	complex	ADJ
cana-510	251	29	variables	variable	NOUN
cana-510	251	30	theory	theory	NOUN
cana-510	251	31	appl	appl	PROPN
cana-510	251	32	.	.	PUNCT
cana-510	252	1	34	34	NUM
cana-510	252	2	,	,	PUNCT
cana-510	252	3	(	(	PUNCT
cana-510	252	4	1997	1997	NUM
cana-510	252	5	)	)	PUNCT
cana-510	252	6	,	,	PUNCT
cana-510	252	7	293	293	NUM
cana-510	252	8	–	–	PUNCT
cana-510	252	9	312	312	NUM
cana-510	252	10	.	.	PUNCT
cana-510	253	1	[	[	X
cana-510	253	2	5	5	NUM
cana-510	253	3	]	]	SYM
cana-510	253	4	merkes	merke	NOUN
cana-510	253	5	,	,	PUNCT
cana-510	253	6	e.	e.	PROPN
cana-510	253	7	p.	p.	PROPN
cana-510	253	8	and	and	CCONJ
cana-510	253	9	scott	scott	PROPN
cana-510	253	10	,	,	PUNCT
cana-510	253	11	w.	w.	PROPN
cana-510	253	12	t.	t.	PROPN
cana-510	253	13	,	,	PUNCT
cana-510	253	14	starlike	starlike	ADJ
cana-510	253	15	hypergeometric	hypergeometric	ADJ
cana-510	253	16	functions	function	NOUN
cana-510	253	17	,	,	PUNCT
cana-510	253	18	proc	proc	NOUN
cana-510	253	19	.	.	PUNCT
cana-510	254	1	amer	amer	PROPN
cana-510	254	2	.	.	PUNCT
cana-510	254	3	math	math	PROPN
cana-510	254	4	.	.	PUNCT
cana-510	255	1	soc	soc	PROPN
cana-510	255	2	.	.	PROPN
cana-510	255	3	,	,	PUNCT
cana-510	255	4	12	12	NUM
cana-510	255	5	(	(	PUNCT
cana-510	255	6	1961	1961	NUM
cana-510	255	7	)	)	PUNCT
cana-510	255	8	,	,	PUNCT
cana-510	255	9	885	885	NUM
cana-510	255	10	888	888	NUM
cana-510	255	11	.	.	PUNCT
cana-510	256	1	[	[	X
cana-510	256	2	6	6	NUM
cana-510	256	3	]	]	X
cana-510	256	4	mostafa	mostafa	PROPN
cana-510	256	5	,	,	PUNCT
cana-510	256	6	a.	a.	PROPN
cana-510	256	7	o.	o.	PROPN
cana-510	256	8	,	,	PUNCT
cana-510	256	9	a	a	DET
cana-510	256	10	study	study	NOUN
cana-510	256	11	on	on	ADP
cana-510	256	12	starlike	starlike	NOUN
cana-510	256	13	and	and	CCONJ
cana-510	256	14	convex	convex	NOUN
cana-510	256	15	properties	property	NOUN
cana-510	256	16	for	for	ADP
cana-510	256	17	hypergeometric	hypergeometric	ADJ
cana-510	256	18	functions	function	NOUN
cana-510	256	19	,	,	PUNCT
cana-510	256	20	jipam	jipam	NOUN
cana-510	256	21	.	.	PUNCT
cana-510	257	1	j.	j.	PROPN
cana-510	257	2	inequal	inequal	PROPN
cana-510	257	3	.	.	PUNCT
cana-510	258	1	pure	pure	ADJ
cana-510	258	2	appl	appl	PROPN
cana-510	258	3	.	.	PUNCT
cana-510	258	4	math	math	PROPN
cana-510	258	5	.	.	PUNCT
cana-510	258	6	,	,	PUNCT
cana-510	258	7	10(3	10(3	NUM
cana-510	258	8	)	)	PUNCT
cana-510	258	9	(	(	PUNCT
cana-510	258	10	2009	2009	NUM
cana-510	258	11	)	)	PUNCT
cana-510	258	12	,	,	PUNCT
cana-510	258	13	article	article	NOUN
cana-510	258	14	87	87	NUM
cana-510	258	15	,	,	PUNCT
cana-510	258	16	8	8	NUM
cana-510	258	17	pp	pp	NOUN
cana-510	258	18	.	.	PUNCT
cana-510	259	1	[	[	X
cana-510	259	2	7	7	NUM
cana-510	259	3	]	]	X
cana-510	259	4	murugusundaramoorthy	murugusundaramoorthy	ADJ
cana-510	259	5	,	,	PUNCT
cana-510	259	6	g.	g.	NOUN
cana-510	259	7	,	,	PUNCT
cana-510	259	8	subclasses	subclass	NOUN
cana-510	259	9	of	of	ADP
cana-510	259	10	starlike	starlike	NOUN
cana-510	259	11	and	and	CCONJ
cana-510	259	12	convex	convex	NOUN
cana-510	259	13	functions	function	NOUN
cana-510	259	14	involving	involve	VERB
cana-510	259	15	poisson	poisson	NOUN
cana-510	259	16	distribution	distribution	NOUN
cana-510	259	17	series	series	NOUN
cana-510	259	18	,	,	PUNCT
cana-510	259	19	afr	afr	PROPN
cana-510	259	20	.	.	PUNCT
cana-510	260	1	mat	mat	PROPN
cana-510	260	2	.	.	PROPN
cana-510	260	3	28	28	NUM
cana-510	261	1	no	no	NOUN
cana-510	261	2	.	.	NOUN
cana-510	262	1	7	7	NUM
cana-510	262	2	-	-	SYM
cana-510	262	3	8	8	NUM
cana-510	262	4	,	,	PUNCT
cana-510	262	5	(	(	PUNCT
cana-510	262	6	2017	2017	NUM
cana-510	262	7	)	)	PUNCT
cana-510	262	8	,	,	PUNCT
cana-510	262	9	1357	1357	NUM
cana-510	262	10	1366	1366	NUM
cana-510	262	11	.	.	PUNCT
cana-510	263	1	[	[	X
cana-510	263	2	8	8	NUM
cana-510	263	3	]	]	X
cana-510	263	4	murugusundaramoorthy	murugusundaramoorthy	ADJ
cana-510	263	5	,	,	PUNCT
cana-510	263	6	g.	g.	PROPN
cana-510	263	7	,	,	PUNCT
cana-510	263	8	vijaya	vijaya	PROPN
cana-510	263	9	,	,	PUNCT
cana-510	263	10	k.	k.	PROPN
cana-510	263	11	,	,	PUNCT
cana-510	263	12	and	and	CCONJ
cana-510	263	13	porwal	porwal	NOUN
cana-510	263	14	,	,	PUNCT
cana-510	263	15	s.	s.	PROPN
cana-510	263	16	,	,	PUNCT
cana-510	263	17	some	some	DET
cana-510	263	18	inclusion	inclusion	NOUN
cana-510	263	19	results	result	NOUN
cana-510	263	20	of	of	ADP
cana-510	263	21	certain	certain	ADJ
cana-510	263	22	subclass	subclass	NOUN
cana-510	263	23	of	of	ADP
cana-510	263	24	analytic	analytic	ADJ
cana-510	263	25	functions	function	NOUN
cana-510	263	26	associated	associate	VERB
cana-510	263	27	with	with	ADP
cana-510	263	28	poisson	poisson	NOUN
cana-510	263	29	distribution	distribution	NOUN
cana-510	263	30	series	series	NOUN
cana-510	263	31	,	,	PUNCT
cana-510	263	32	hacet	hacet	PROPN
cana-510	263	33	.	.	PUNCT
cana-510	264	1	j.	j.	PROPN
cana-510	264	2	math	math	PROPN
cana-510	264	3	.	.	PUNCT
cana-510	265	1	stat	stat	PROPN
cana-510	265	2	.	.	PUNCT
cana-510	266	1	45	45	NUM
cana-510	266	2	no	no	NOUN
cana-510	266	3	.	.	NOUN
cana-510	266	4	4	4	NUM
cana-510	266	5	,	,	PUNCT
cana-510	266	6	(	(	PUNCT
cana-510	266	7	2016	2016	NUM
cana-510	266	8	)	)	PUNCT
cana-510	266	9	,	,	PUNCT
cana-510	266	10	1101–1107	1101–1107	NUM
cana-510	266	11	.	.	PUNCT
cana-510	267	1	[	[	X
cana-510	267	2	9	9	NUM
cana-510	267	3	]	]	SYM
cana-510	267	4	owa	owa	PROPN
cana-510	267	5	,	,	PUNCT
cana-510	267	6	s.	s.	PROPN
cana-510	267	7	and	and	CCONJ
cana-510	267	8	srivastava	srivastava	PROPN
cana-510	267	9	,	,	PUNCT
cana-510	267	10	h.m	h.m	PROPN
cana-510	267	11	.	.	PROPN
cana-510	267	12	,	,	PUNCT
cana-510	267	13	univalent	univalent	ADJ
cana-510	267	14	and	and	CCONJ
cana-510	267	15	starlike	starlike	ADJ
cana-510	267	16	generalized	generalize	VERB
cana-510	267	17	hypergeometric	hypergeometric	ADJ
cana-510	267	18	functions	function	NOUN
cana-510	267	19	,	,	PUNCT
cana-510	267	20	can	can	AUX
cana-510	267	21	.	.	PUNCT
cana-510	268	1	j.	j.	PROPN
cana-510	268	2	math	math	PROPN
cana-510	268	3	.	.	PROPN
cana-510	269	1	39	39	NUM
cana-510	269	2	,	,	PUNCT
cana-510	269	3	,	,	PUNCT
cana-510	269	4	(	(	PUNCT
cana-510	269	5	1987	1987	NUM
cana-510	269	6	)	)	PUNCT
cana-510	269	7	,	,	PUNCT
cana-510	269	8	1057	1057	NUM
cana-510	269	9	-	-	SYM
cana-510	269	10	1077	1077	NUM
cana-510	269	11	.	.	PUNCT
cana-510	270	1	[	[	X
cana-510	270	2	10	10	NUM
cana-510	270	3	]	]	X
cana-510	270	4	silverman	silverman	NOUN
cana-510	270	5	,	,	PUNCT
cana-510	270	6	h.	h.	PROPN
cana-510	270	7	,	,	PUNCT
cana-510	270	8	starlike	starlike	NOUN
cana-510	270	9	and	and	CCONJ
cana-510	270	10	convexity	convexity	NOUN
cana-510	270	11	properties	property	NOUN
cana-510	270	12	for	for	ADP
cana-510	270	13	hypergeometric	hypergeometric	ADJ
cana-510	270	14	functions	function	NOUN
cana-510	270	15	,	,	PUNCT
cana-510	270	16	j.	j.	PROPN
cana-510	270	17	math	math	PROPN
cana-510	270	18	.	.	PUNCT
cana-510	271	1	anal	anal	PROPN
cana-510	271	2	.	.	PUNCT
cana-510	271	3	appl	appl	PROPN
cana-510	271	4	.	.	PUNCT
cana-510	272	1	172	172	NUM
cana-510	272	2	(	(	PUNCT
cana-510	272	3	2)(1993	2)(1993	NUM
cana-510	272	4	)	)	PUNCT
cana-510	272	5	,	,	PUNCT
cana-510	272	6	574	574	NUM
cana-510	272	7	581	581	NUM
cana-510	272	8	.	.	PUNCT
cana-510	273	1	[	[	X
cana-510	273	2	11	11	NUM
cana-510	273	3	]	]	X
cana-510	273	4	srivastava	srivastava	PROPN
cana-510	273	5	,	,	PUNCT
cana-510	273	6	h.	h.	PROPN
cana-510	273	7	m.	m.	PROPN
cana-510	273	8	,	,	PUNCT
cana-510	273	9	murugusundaramoorthy	murugusundaramoorthy	ADJ
cana-510	273	10	,	,	PUNCT
cana-510	273	11	g.	g.	PROPN
cana-510	273	12	and	and	CCONJ
cana-510	273	13	sivasubramanian	sivasubramanian	ADJ
cana-510	273	14	,	,	PUNCT
cana-510	273	15	s.	s.	PROPN
cana-510	273	16	,	,	PUNCT
cana-510	273	17	hypergeometric	hypergeometric	ADJ
cana-510	273	18	functions	function	NOUN
cana-510	273	19	in	in	ADP
cana-510	273	20	the	the	DET
cana-510	273	21	parabolic	parabolic	ADJ
cana-510	273	22	starlike	starlike	NOUN
cana-510	273	23	and	and	CCONJ
cana-510	273	24	uniformly	uniformly	ADV
cana-510	273	25	convex	convex	ADJ
cana-510	273	26	domains	domain	NOUN
cana-510	273	27	,	,	PUNCT
cana-510	273	28	integral	integral	ADJ
cana-510	273	29	transforms	transform	VERB
cana-510	273	30	spec	spec	NOUN
cana-510	273	31	.	.	PUNCT
cana-510	274	1	funct	funct	PROPN
cana-510	274	2	.	.	PUNCT
cana-510	275	1	18(7	18(7	NUM
cana-510	275	2	-	-	SYM
cana-510	275	3	8)	8)	NUM
cana-510	275	4	(	(	PUNCT
cana-510	275	5	2007	2007	NUM
cana-510	275	6	)	)	PUNCT
cana-510	275	7	,	,	PUNCT
cana-510	275	8	511	511	NUM
cana-510	275	9	520	520	NUM
cana-510	275	10	.	.	PUNCT
cana-510	276	1	[	[	X
cana-510	276	2	12	12	NUM
cana-510	276	3	]	]	PUNCT
cana-510	276	4	pommerenke	pommerenke	NOUN
cana-510	276	5	,	,	PUNCT
cana-510	276	6	c.	c.	PROPN
cana-510	276	7	,	,	PUNCT
cana-510	276	8	univalent	univalent	ADJ
cana-510	276	9	functions	function	NOUN
cana-510	276	10	,	,	PUNCT
cana-510	276	11	vandenhoeck	vandenhoeck	NOUN
cana-510	276	12	and	and	CCONJ
cana-510	276	13	ruprecht	ruprecht	NOUN
cana-510	276	14	,	,	PUNCT
cana-510	276	15	gottingen	gottingen	NOUN
cana-510	276	16	,	,	PUNCT
cana-510	276	17	(	(	PUNCT
cana-510	276	18	1975	1975	NUM
cana-510	276	19	)	)	PUNCT
cana-510	276	20	.	.	PUNCT
cana-510	277	1	[	[	X
cana-510	277	2	13	13	NUM
cana-510	277	3	]	]	X
cana-510	277	4	samko	samko	NOUN
cana-510	277	5	,	,	PUNCT
cana-510	277	6	s.g	s.g	PROPN
cana-510	277	7	.	.	PROPN
cana-510	277	8	,	,	PUNCT
cana-510	277	9	kilbas	kilbas	PROPN
cana-510	277	10	,	,	PUNCT
cana-510	277	11	a.	a.	NOUN
cana-510	277	12	a.	a.	PROPN
cana-510	277	13	and	and	CCONJ
cana-510	277	14	marichev	marichev	PROPN
cana-510	277	15	,	,	PUNCT
cana-510	277	16	o.i	o.i	PROPN
cana-510	277	17	.	.	PROPN
cana-510	277	18	,	,	PUNCT
cana-510	277	19	fractional	fractional	ADJ
cana-510	277	20	integral	integral	ADJ
cana-510	277	21	and	and	CCONJ
cana-510	277	22	derivatives	derivative	NOUN
cana-510	277	23	,	,	PUNCT
cana-510	277	24	theory	theory	NOUN
cana-510	277	25	and	and	CCONJ
cana-510	277	26	applications	application	NOUN
cana-510	277	27	,	,	PUNCT
cana-510	277	28	gordon	gordon	PROPN
cana-510	277	29	and	and	CCONJ
cana-510	277	30	breach	breach	PROPN
cana-510	277	31	,	,	PUNCT
cana-510	277	32	new	new	PROPN
cana-510	277	33	york	york	PROPN
cana-510	277	34	,	,	PUNCT
cana-510	277	35	philadelphia	philadelphia	PROPN
cana-510	277	36	,	,	PUNCT
cana-510	277	37	london	london	PROPN
cana-510	277	38	,	,	PUNCT
cana-510	277	39	paris	paris	PROPN
cana-510	277	40	,	,	PUNCT
cana-510	277	41	montreux	montreux	PROPN
cana-510	277	42	,	,	PUNCT
cana-510	277	43	toronto	toronto	PROPN
cana-510	277	44	and	and	CCONJ
cana-510	277	45	melbourne	melbourne	PROPN
cana-510	277	46	,	,	PUNCT
cana-510	277	47	1993	1993	NUM
cana-510	277	48	.	.	PUNCT
cana-510	278	1	[	[	X
cana-510	278	2	14	14	NUM
cana-510	278	3	]	]	PUNCT
cana-510	278	4	silverman	silverman	NOUN
cana-510	278	5	,	,	PUNCT
cana-510	278	6	h.	h.	PROPN
cana-510	278	7	,	,	PUNCT
cana-510	278	8	univalent	univalent	ADJ
cana-510	278	9	functions	function	NOUN
cana-510	278	10	with	with	ADP
cana-510	278	11	negative	negative	ADJ
cana-510	278	12	coefficients	coefficient	NOUN
cana-510	278	13	,	,	PUNCT
cana-510	278	14	proc	proc	NOUN
cana-510	278	15	.	.	PUNCT
cana-510	279	1	amer	amer	PROPN
cana-510	279	2	.	.	PUNCT
cana-510	279	3	math	math	PROPN
cana-510	279	4	.	.	PUNCT
cana-510	280	1	soc	soc	PROPN
cana-510	280	2	.	.	PUNCT
cana-510	280	3	,	,	PUNCT
cana-510	280	4	51,(1975	51,(1975	NUM
cana-510	280	5	)	)	PUNCT
cana-510	280	6	,	,	PUNCT
cana-510	280	7	109	109	NUM
cana-510	280	8	-	-	SYM
cana-510	280	9	116	116	NUM
cana-510	280	10	.	.	PUNCT
cana-510	281	1	[	[	X
cana-510	281	2	15	15	NUM
cana-510	281	3	]	]	X
cana-510	281	4	srivastava	srivastava	PROPN
cana-510	281	5	,	,	PUNCT
cana-510	281	6	h.m	h.m	PROPN
cana-510	281	7	.	.	PROPN
cana-510	281	8	and	and	CCONJ
cana-510	281	9	buschman	buschman	NOUN
cana-510	281	10	,	,	PUNCT
cana-510	281	11	r.g	r.g	PROPN
cana-510	281	12	.	.	PROPN
cana-510	281	13	,	,	PUNCT
cana-510	281	14	theory	theory	NOUN
cana-510	281	15	and	and	CCONJ
cana-510	281	16	applications	application	NOUN
cana-510	281	17	of	of	ADP
cana-510	281	18	convolution	convolution	NOUN
cana-510	281	19	integral	integral	ADJ
cana-510	281	20	equations	equation	NOUN
cana-510	281	21	,	,	PUNCT
cana-510	281	22	kluwer	kluwer	NOUN
cana-510	281	23	academic	academic	ADJ
cana-510	281	24	publishers	publisher	NOUN
cana-510	281	25	,	,	PUNCT
cana-510	281	26	dordrecht	dordrecht	PROPN
cana-510	281	27	,	,	PUNCT
cana-510	281	28	boston	boston	PROPN
cana-510	281	29	and	and	CCONJ
cana-510	281	30	london	london	PROPN
cana-510	281	31	,	,	PUNCT
cana-510	281	32	1992	1992	NUM
cana-510	281	33	.	.	PUNCT