id	sid	tid	token	lemma	pos
cana-1089	1	1	communications	communication	NOUN
cana-1089	1	2	on	on	ADP
cana-1089	1	3	applied	apply	VERB
cana-1089	1	4	nonlinear	nonlinear	ADJ
cana-1089	1	5	analysis	analysis	NOUN
cana-1089	1	6	issn	issn	NOUN
cana-1089	1	7	:	:	PUNCT
cana-1089	1	8	1074	1074	NUM
cana-1089	1	9	-	-	PUNCT
cana-1089	1	10	133x	133x	NUM
cana-1089	1	11	vol	vol	NOUN
cana-1089	1	12	31	31	NUM
cana-1089	1	13	no	no	NOUN
cana-1089	1	14	.	.	PUNCT
cana-1089	2	1	5s	5s	NUM
cana-1089	2	2	(	(	PUNCT
cana-1089	2	3	2024	2024	NUM
cana-1089	2	4	)	)	PUNCT
cana-1089	2	5	540	540	NUM
cana-1089	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	2	7	a	a	DET
cana-1089	2	8	class	class	NOUN
cana-1089	2	9	of	of	ADP
cana-1089	2	10	analytic	analytic	ADJ
cana-1089	2	11	functions	function	NOUN
cana-1089	2	12	with	with	ADP
cana-1089	2	13	respect	respect	NOUN
cana-1089	2	14	to	to	ADP
cana-1089	2	15	symmetric	symmetric	ADJ
cana-1089	2	16	points	point	NOUN
cana-1089	2	17	involving	involve	VERB
cana-1089	2	18	multiplicative	multiplicative	ADJ
cana-1089	2	19	derivative	derivative	ADJ
cana-1089	2	20	kadhavoor	kadhavoor	PROPN
cana-1089	2	21	r.	r.	PROPN
cana-1089	2	22	karthikeyan1	karthikeyan1	PROPN
cana-1089	2	23	,	,	PUNCT
cana-1089	2	24	seetharam	seetharam	NOUN
cana-1089	2	25	varadharajan2	varadharajan2	PROPN
cana-1089	2	26	1department	1department	NUM
cana-1089	2	27	of	of	ADP
cana-1089	2	28	applied	apply	VERB
cana-1089	2	29	mathematics	mathematic	NOUN
cana-1089	2	30	and	and	CCONJ
cana-1089	2	31	science	science	NOUN
cana-1089	2	32	,	,	PUNCT
cana-1089	2	33	college	college	NOUN
cana-1089	2	34	of	of	ADP
cana-1089	2	35	engineering	engineering	PROPN
cana-1089	2	36	,	,	PUNCT
cana-1089	2	37	national	national	ADJ
cana-1089	2	38	university	university	PROPN
cana-1089	2	39	of	of	ADP
cana-1089	2	40	science	science	PROPN
cana-1089	2	41	&	&	CCONJ
cana-1089	2	42	technology	technology	PROPN
cana-1089	2	43	,	,	PUNCT
cana-1089	2	44	cpo	cpo	PROPN
cana-1089	2	45	seeb	seeb	PROPN
cana-1089	2	46	111	111	NUM
cana-1089	2	47	,	,	PUNCT
cana-1089	2	48	al	al	PROPN
cana-1089	2	49	hail	hail	PROPN
cana-1089	2	50	,	,	PUNCT
cana-1089	2	51	muscat	muscat	PROPN
cana-1089	2	52	,	,	PUNCT
cana-1089	2	53	oman	oman	NOUN
cana-1089	2	54	.	.	PUNCT
cana-1089	3	1	email	email	NOUN
cana-1089	3	2	:	:	PUNCT
cana-1089	3	3	karthikeyan1979@nu.edu.om	karthikeyan1979@nu.edu.om	PROPN
cana-1089	3	4	2mathematics	2mathematics	NUM
cana-1089	3	5	section	section	NOUN
cana-1089	3	6	,	,	PUNCT
cana-1089	3	7	department	department	NOUN
cana-1089	3	8	of	of	ADP
cana-1089	3	9	information	information	NOUN
cana-1089	3	10	technology	technology	NOUN
cana-1089	3	11	,	,	PUNCT
cana-1089	3	12	university	university	NOUN
cana-1089	3	13	of	of	ADP
cana-1089	3	14	technology	technology	NOUN
cana-1089	3	15	and	and	CCONJ
cana-1089	3	16	applied	apply	VERB
cana-1089	3	17	sciences	sciences	PROPN
cana-1089	3	18	al	al	PROPN
cana-1089	3	19	mussanah	mussanah	PROPN
cana-1089	3	20	,	,	PUNCT
cana-1089	3	21	oman	oman	PROPN
cana-1089	3	22	.	.	PUNCT
cana-1089	4	1	email	email	NOUN
cana-1089	4	2	:	:	PUNCT
cana-1089	4	3	svrajanram@gmail.com	svrajanram@gmail.com	X
cana-1089	4	4	article	article	NOUN
cana-1089	4	5	history	history	NOUN
cana-1089	4	6	:	:	PUNCT
cana-1089	4	7	received	receive	VERB
cana-1089	4	8	:	:	PUNCT
cana-1089	4	9	18	18	NUM
cana-1089	4	10	-	-	SYM
cana-1089	4	11	05	05	NUM
cana-1089	4	12	-	-	PUNCT
cana-1089	4	13	2024	2024	NUM
cana-1089	4	14	revised	revise	VERB
cana-1089	4	15	:	:	PUNCT
cana-1089	4	16	20	20	NUM
cana-1089	4	17	-	-	SYM
cana-1089	4	18	06	06	NUM
cana-1089	4	19	-	-	PUNCT
cana-1089	4	20	2024	2024	NUM
cana-1089	4	21	accepted	accept	VERB
cana-1089	4	22	:	:	PUNCT
cana-1089	4	23	11	11	NUM
cana-1089	4	24	-	-	SYM
cana-1089	4	25	07	07	NUM
cana-1089	4	26	-	-	PUNCT
cana-1089	4	27	2024	2024	NUM
cana-1089	4	28	abstract	abstract	NOUN
cana-1089	4	29	:	:	PUNCT
cana-1089	4	30	here	here	ADV
cana-1089	4	31	we	we	PRON
cana-1089	4	32	explore	explore	VERB
cana-1089	4	33	the	the	DET
cana-1089	4	34	behaviour	behaviour	NOUN
cana-1089	4	35	and	and	CCONJ
cana-1089	4	36	deviations	deviation	NOUN
cana-1089	4	37	of	of	ADP
cana-1089	4	38	the	the	DET
cana-1089	4	39	geometric	geometric	ADJ
cana-1089	4	40	properties	property	NOUN
cana-1089	4	41	of	of	ADP
cana-1089	4	42	a	a	DET
cana-1089	4	43	class	class	NOUN
cana-1089	4	44	of	of	ADP
cana-1089	4	45	univalent	univalent	ADJ
cana-1089	4	46	functions	function	NOUN
cana-1089	4	47	when	when	SCONJ
cana-1089	4	48	the	the	DET
cana-1089	4	49	classical	classical	ADJ
cana-1089	4	50	derivative	derivative	NOUN
cana-1089	4	51	is	be	AUX
cana-1089	4	52	replaced	replace	VERB
cana-1089	4	53	with	with	ADP
cana-1089	4	54	a	a	DET
cana-1089	4	55	multiplicative	multiplicative	ADJ
cana-1089	4	56	derivative	derivative	NOUN
cana-1089	4	57	.	.	PUNCT
cana-1089	5	1	the	the	DET
cana-1089	5	2	primary	primary	ADJ
cana-1089	5	3	question	question	NOUN
cana-1089	5	4	that	that	SCONJ
cana-1089	5	5	we	we	PRON
cana-1089	5	6	will	will	AUX
cana-1089	5	7	be	be	AUX
cana-1089	5	8	addressing	address	VERB
cana-1089	5	9	here	here	ADV
cana-1089	5	10	is	be	AUX
cana-1089	5	11	that	that	SCONJ
cana-1089	5	12	given	give	VERB
cana-1089	5	13	a	a	DET
cana-1089	5	14	more	more	ADV
cana-1089	5	15	versatile	versatile	ADJ
cana-1089	5	16	calculus	calculus	NOUN
cana-1089	5	17	of	of	ADP
cana-1089	5	18	newton	newton	PROPN
cana-1089	5	19	and	and	CCONJ
cana-1089	5	20	euler	euler	PROPN
cana-1089	5	21	,	,	PUNCT
cana-1089	5	22	why	why	SCONJ
cana-1089	5	23	we	we	PRON
cana-1089	5	24	need	need	VERB
cana-1089	5	25	a	a	DET
cana-1089	5	26	study	study	NOUN
cana-1089	5	27	involving	involve	VERB
cana-1089	5	28	such	such	DET
cana-1089	5	29	a	a	DET
cana-1089	5	30	restrictive	restrictive	ADJ
cana-1089	5	31	calculus	calculus	NOUN
cana-1089	5	32	so	so	ADV
cana-1089	5	33	called	call	VERB
cana-1089	5	34	as	as	ADP
cana-1089	5	35	multiplicative	multiplicative	ADJ
cana-1089	5	36	calculus	calculus	NOUN
cana-1089	5	37	.	.	PUNCT
cana-1089	6	1	precisely	precisely	ADV
cana-1089	6	2	,	,	PUNCT
cana-1089	6	3	we	we	PRON
cana-1089	6	4	introduce	introduce	VERB
cana-1089	6	5	and	and	CCONJ
cana-1089	6	6	study	study	VERB
cana-1089	6	7	a	a	DET
cana-1089	6	8	new	new	ADJ
cana-1089	6	9	subclass	subclass	NOUN
cana-1089	6	10	of	of	ADP
cana-1089	6	11	analytic	analytic	ADJ
cana-1089	6	12	function	function	NOUN
cana-1089	6	13	with	with	ADP
cana-1089	6	14	respect	respect	NOUN
cana-1089	6	15	to	to	ADP
cana-1089	6	16	symmetric	symmetric	ADJ
cana-1089	6	17	points	point	NOUN
cana-1089	6	18	using	use	VERB
cana-1089	6	19	multiplicative	multiplicative	ADJ
cana-1089	6	20	derivative	derivative	NOUN
cana-1089	6	21	.	.	PUNCT
cana-1089	7	1	we	we	PRON
cana-1089	7	2	obtain	obtain	VERB
cana-1089	7	3	the	the	DET
cana-1089	7	4	estimates	estimate	NOUN
cana-1089	7	5	for	for	ADP
cana-1089	7	6	the	the	DET
cana-1089	7	7	initial	initial	ADJ
cana-1089	7	8	coefficients	coefficient	NOUN
cana-1089	7	9	and	and	CCONJ
cana-1089	7	10	fekete	fekete	NOUN
cana-1089	7	11	-	-	PUNCT
cana-1089	7	12	szegő	szegő	ADJ
cana-1089	7	13	inequalities	inequality	NOUN
cana-1089	7	14	of	of	ADP
cana-1089	7	15	the	the	DET
cana-1089	7	16	same	same	ADJ
cana-1089	7	17	.	.	PUNCT
cana-1089	8	1	we	we	PRON
cana-1089	8	2	have	have	AUX
cana-1089	8	3	included	include	VERB
cana-1089	8	4	some	some	DET
cana-1089	8	5	examples	example	NOUN
cana-1089	8	6	to	to	PART
cana-1089	8	7	establish	establish	VERB
cana-1089	8	8	the	the	DET
cana-1089	8	9	inclusion	inclusion	NOUN
cana-1089	8	10	and	and	CCONJ
cana-1089	8	11	closure	closure	NOUN
cana-1089	8	12	properties	property	NOUN
cana-1089	8	13	of	of	ADP
cana-1089	8	14	our	our	PRON
cana-1089	8	15	defined	define	VERB
cana-1089	8	16	class	class	NOUN
cana-1089	8	17	.	.	PUNCT
cana-1089	9	1	further	far	ADV
cana-1089	9	2	,	,	PUNCT
cana-1089	9	3	we	we	PRON
cana-1089	9	4	obtain	obtain	VERB
cana-1089	9	5	the	the	DET
cana-1089	9	6	logarithmic	logarithmic	ADJ
cana-1089	9	7	and	and	CCONJ
cana-1089	9	8	inverse	inverse	ADJ
cana-1089	9	9	coefficients	coefficient	NOUN
cana-1089	9	10	for	for	ADP
cana-1089	9	11	the	the	DET
cana-1089	9	12	defined	define	VERB
cana-1089	9	13	function	function	NOUN
cana-1089	9	14	class	class	NOUN
cana-1089	9	15	.	.	PUNCT
cana-1089	10	1	keywords	keyword	NOUN
cana-1089	10	2	:	:	PUNCT
cana-1089	10	3	multiplicative	multiplicative	ADJ
cana-1089	10	4	calculus	calculus	NOUN
cana-1089	10	5	,	,	PUNCT
cana-1089	10	6	starlike	starlike	NOUN
cana-1089	10	7	function	function	NOUN
cana-1089	10	8	,	,	PUNCT
cana-1089	10	9	convex	convex	NOUN
cana-1089	10	10	function	function	NOUN
cana-1089	10	11	,	,	PUNCT
cana-1089	10	12	close	close	NOUN
cana-1089	10	13	-	-	PUNCT
cana-1089	10	14	to	to	ADP
cana-1089	10	15	-	-	PUNCT
cana-1089	10	16	convex	convex	NOUN
cana-1089	10	17	function	function	NOUN
cana-1089	10	18	,	,	PUNCT
cana-1089	10	19	subordination	subordination	NOUN
cana-1089	10	20	.	.	PUNCT
cana-1089	11	1	1	1	X
cana-1089	11	2	.	.	X
cana-1089	11	3	introduction	introduction	NOUN
cana-1089	11	4	for	for	ADP
cana-1089	11	5	𝒰	𝒰	PROPN
cana-1089	11	6	=	=	PUNCT
cana-1089	11	7	{	{	PUNCT
cana-1089	11	8	𝜔	𝜔	PART
cana-1089	11	9	∈	∈	PROPN
cana-1089	11	10	ℂ	ℂ	PROPN
cana-1089	11	11	;	;	PUNCT
cana-1089	11	12	|𝜔|	|𝜔|	NOUN
cana-1089	11	13	<	<	X
cana-1089	11	14	1	1	NUM
cana-1089	11	15	}	}	PUNCT
cana-1089	11	16	,	,	PUNCT
cana-1089	11	17	we	we	PRON
cana-1089	11	18	let	let	VERB
cana-1089	11	19	𝒜	𝒜	NOUN
cana-1089	11	20	to	to	PART
cana-1089	11	21	denote	denote	VERB
cana-1089	11	22	the	the	DET
cana-1089	11	23	class	class	NOUN
cana-1089	11	24	of	of	ADP
cana-1089	11	25	functions	function	NOUN
cana-1089	11	26	analytic	analytic	ADJ
cana-1089	11	27	with	with	ADP
cana-1089	11	28	normalization	normalization	NOUN
cana-1089	11	29	𝜑(0	𝜑(0	NOUN
cana-1089	11	30	)	)	PUNCT
cana-1089	11	31	=	=	SYM
cana-1089	11	32	0	0	PUNCT
cana-1089	12	1	=	=	SYM
cana-1089	12	2	𝜑′(0	𝜑′(0	PROPN
cana-1089	12	3	)	)	PUNCT
cana-1089	12	4	−	−	PROPN
cana-1089	13	1	1	1	X
cana-1089	13	2	.	.	PUNCT
cana-1089	14	1	we	we	PRON
cana-1089	14	2	denote	denote	VERB
cana-1089	14	3	the	the	DET
cana-1089	14	4	classes	class	NOUN
cana-1089	14	5	of	of	ADP
cana-1089	14	6	starlike	starlike	NOUN
cana-1089	14	7	and	and	CCONJ
cana-1089	14	8	convex	convex	NOUN
cana-1089	14	9	function	function	NOUN
cana-1089	14	10	by	by	ADP
cana-1089	14	11	𝒮∗(𝛾	𝒮∗(𝛾	NOUN
cana-1089	14	12	)	)	PUNCT
cana-1089	14	13	and	and	CCONJ
cana-1089	14	14	𝒞(𝛾	𝒞(𝛾	X
cana-1089	14	15	)	)	PUNCT
cana-1089	14	16	respectively	respectively	ADV
cana-1089	14	17	.	.	PUNCT
cana-1089	15	1	it	it	PRON
cana-1089	15	2	is	be	AUX
cana-1089	15	3	well	well	ADV
cana-1089	15	4	-	-	PUNCT
cana-1089	15	5	known	know	VERB
cana-1089	15	6	that	that	SCONJ
cana-1089	15	7	𝒮∗(𝛾	𝒮∗(𝛾	ADJ
cana-1089	15	8	)	)	PUNCT
cana-1089	15	9	and	and	CCONJ
cana-1089	15	10	c(γ	c(γ	NOUN
cana-1089	15	11	)	)	PUNCT
cana-1089	15	12	satisfies	satisfy	VERB
cana-1089	15	13	the	the	DET
cana-1089	15	14	condition	condition	NOUN
cana-1089	15	15	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	15	16	(	(	PUNCT
cana-1089	15	17	𝜔𝜑′(𝜔	𝜔𝜑′(𝜔	PROPN
cana-1089	15	18	)	)	PUNCT
cana-1089	15	19	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	15	20	)	)	PUNCT
cana-1089	15	21	)	)	PUNCT
cana-1089	15	22	>	>	X
cana-1089	16	1	𝛾	𝛾	X
cana-1089	16	2	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1089	16	3	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	16	4	(	(	PUNCT
cana-1089	16	5	1	1	NUM
cana-1089	16	6	+	+	NUM
cana-1089	16	7	𝜔𝜑′′(𝜔	𝜔𝜑′′(𝜔	NOUN
cana-1089	16	8	)	)	PUNCT
cana-1089	16	9	𝜑′(𝜔	𝜑′(𝜔	NOUN
cana-1089	16	10	)	)	PUNCT
cana-1089	16	11	)	)	PUNCT
cana-1089	16	12	>	>	PUNCT
cana-1089	17	1	𝛾	𝛾	ADP
cana-1089	17	2	,	,	PUNCT
cana-1089	17	3	(	(	PUNCT
cana-1089	17	4	𝜔	𝜔	PROPN
cana-1089	17	5	∈	∈	PROPN
cana-1089	17	6	𝒰	𝒰	NOUN
cana-1089	17	7	;	;	PUNCT
cana-1089	17	8	0	0	NUM
cana-1089	17	9	≤	≤	NOUN
cana-1089	17	10	𝛾	𝛾	ADP
cana-1089	17	11	<	<	X
cana-1089	17	12	1	1	NUM
cana-1089	17	13	)	)	PUNCT
cana-1089	17	14	,	,	PUNCT
cana-1089	17	15	respectively	respectively	ADV
cana-1089	17	16	.	.	PUNCT
cana-1089	18	1	let	let	VERB
cana-1089	18	2	𝒫	𝒫	NOUN
cana-1089	18	3	signify	signify	VERB
cana-1089	18	4	the	the	DET
cana-1089	18	5	category	category	NOUN
cana-1089	18	6	of	of	ADP
cana-1089	18	7	functions	function	NOUN
cana-1089	18	8	that	that	PRON
cana-1089	18	9	are	be	AUX
cana-1089	18	10	analytic	analytic	ADJ
cana-1089	18	11	in	in	ADP
cana-1089	18	12	𝒰	𝒰	PROPN
cana-1089	18	13	with	with	ADP
cana-1089	18	14	𝑝(0	𝑝(0	PROPN
cana-1089	18	15	)	)	PUNCT
cana-1089	18	16	=	=	NOUN
cana-1089	18	17	1	1	NUM
cana-1089	18	18	and	and	CCONJ
cana-1089	18	19	𝑅𝑒{𝑝(𝜔	𝑅𝑒{𝑝(𝜔	NUM
cana-1089	18	20	)	)	PUNCT
cana-1089	18	21	}	}	PUNCT
cana-1089	18	22	>	>	X
cana-1089	18	23	0	0	PUNCT
cana-1089	19	1	for	for	ADP
cana-1089	19	2	all	all	PRON
cana-1089	19	3	𝜔	𝜔	PRON
cana-1089	19	4	∈	∈	NOUN
cana-1089	19	5	𝒰.	𝒰.	NOUN
cana-1089	19	6	let	let	VERB
cana-1089	19	7	𝒮	𝒮	PRON
cana-1089	19	8	denote	denote	VERB
cana-1089	19	9	the	the	DET
cana-1089	19	10	class	class	NOUN
cana-1089	19	11	of	of	ADP
cana-1089	19	12	functions	function	NOUN
cana-1089	19	13	𝜑	𝜑	X
cana-1089	19	14	∈	∈	PROPN
cana-1089	19	15	𝒜	𝒜	NOUN
cana-1089	19	16	which	which	PRON
cana-1089	19	17	are	be	AUX
cana-1089	19	18	univalent	univalent	ADJ
cana-1089	19	19	in	in	ADP
cana-1089	19	20	𝒰.	𝒰.	PROPN
cana-1089	19	21	the	the	DET
cana-1089	19	22	class	class	NOUN
cana-1089	19	23	𝒮	𝒮	PROPN
cana-1089	19	24	is	be	AUX
cana-1089	19	25	not	not	PART
cana-1089	19	26	preserved	preserve	VERB
cana-1089	19	27	under	under	ADP
cana-1089	19	28	even	even	ADV
cana-1089	19	29	the	the	DET
cana-1089	19	30	most	most	ADV
cana-1089	19	31	basic	basic	ADJ
cana-1089	19	32	operations	operation	NOUN
cana-1089	19	33	like	like	ADP
cana-1089	19	34	addition	addition	NOUN
cana-1089	19	35	or	or	CCONJ
cana-1089	19	36	subtraction	subtraction	NOUN
cana-1089	19	37	.	.	PUNCT
cana-1089	20	1	however	however	ADV
cana-1089	20	2	,	,	PUNCT
cana-1089	20	3	the	the	DET
cana-1089	20	4	class	class	NOUN
cana-1089	20	5	is	be	AUX
cana-1089	20	6	preserved	preserve	VERB
cana-1089	20	7	under	under	ADP
cana-1089	20	8	𝑘	𝑘	DET
cana-1089	20	9	−root	−root	NOUN
cana-1089	20	10	transformation	transformation	NOUN
cana-1089	20	11	.	.	PUNCT
cana-1089	21	1	it	it	PRON
cana-1089	21	2	is	be	AUX
cana-1089	21	3	well	well	ADV
cana-1089	21	4	known	know	VERB
cana-1089	21	5	that	that	SCONJ
cana-1089	21	6	if	if	SCONJ
cana-1089	21	7	𝜑	𝜑	PROPN
cana-1089	21	8	∈	∈	PROPN
cana-1089	21	9	𝒜	𝒜	NOUN
cana-1089	21	10	is	be	AUX
cana-1089	21	11	in	in	ADP
cana-1089	21	12	𝒮	𝒮	PROPN
cana-1089	21	13	,	,	PUNCT
cana-1089	21	14	then	then	ADV
cana-1089	21	15	$	$	SYM
cana-1089	21	16	[	[	X
cana-1089	21	17	𝜑(𝜔𝑘	𝜑(𝜔𝑘	NUM
cana-1089	21	18	)	)	PUNCT
cana-1089	21	19	]	]	PUNCT
cana-1089	22	1	1	1	NUM
cana-1089	22	2	𝑘	𝑘	INTJ
cana-1089	22	3	,	,	PUNCT
cana-1089	22	4	(	(	PUNCT
cana-1089	22	5	𝑘	𝑘	X
cana-1089	22	6	is	be	AUX
cana-1089	22	7	a	a	DET
cana-1089	22	8	positive	positive	ADJ
cana-1089	22	9	integer	integer	NOUN
cana-1089	22	10	)	)	PUNCT
cana-1089	22	11	is	be	AUX
cana-1089	22	12	also	also	ADV
cana-1089	22	13	in	in	ADP
cana-1089	22	14	𝒮.	𝒮.	PROPN
cana-1089	22	15	refer	refer	VERB
cana-1089	22	16	to	to	ADP
cana-1089	22	17	[	[	X
cana-1089	22	18	9	9	NUM
cana-1089	22	19	,	,	PUNCT
cana-1089	22	20	pg	pg	INTJ
cana-1089	22	21	.	.	PROPN
cana-1089	22	22	18	18	NUM
cana-1089	22	23	]	]	PUNCT
cana-1089	22	24	for	for	ADP
cana-1089	22	25	the	the	DET
cana-1089	22	26	formal	formal	ADJ
cana-1089	22	27	definition	definition	NOUN
cana-1089	22	28	of	of	ADP
cana-1089	22	29	𝑘symmetric	𝑘symmetric	ADJ
cana-1089	22	30	function	function	NOUN
cana-1089	22	31	.	.	PUNCT
cana-1089	23	1	for	for	SCONJ
cana-1089	23	2	every	every	DET
cana-1089	23	3	integer	integer	NOUN
cana-1089	23	4	𝑘	𝑘	NOUN
cana-1089	23	5	,	,	PUNCT
cana-1089	23	6	let	let	VERB
cana-1089	23	7	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	23	8	)	)	PUNCT
cana-1089	23	9	be	be	AUX
cana-1089	23	10	defined	define	VERB
cana-1089	23	11	by	by	ADP
cana-1089	23	12	the	the	DET
cana-1089	23	13	following	follow	VERB
cana-1089	23	14	equality	equality	NOUN
cana-1089	23	15	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	23	16	)	)	PUNCT
cana-1089	23	17	=	=	SYM
cana-1089	24	1	1	1	NUM
cana-1089	24	2	𝑘	𝑘	X
cana-1089	24	3	∑	∑	PROPN
cana-1089	24	4	𝜑(𝜀𝜈	𝜑(𝜀𝜈	PROPN
cana-1089	24	5	𝜔	𝜔	PROPN
cana-1089	24	6	)	)	PUNCT
cana-1089	24	7	𝜀𝜈	𝜀𝜈	ADP
cana-1089	24	8	𝑘−1	𝑘−1	PROPN
cana-1089	24	9	𝜈=0	𝜈=0	PUNCT
cana-1089	24	10	,	,	PUNCT
cana-1089	24	11	(	(	PUNCT
cana-1089	24	12	𝜑	𝜑	PROPN
cana-1089	24	13	∈	∈	PROPN
cana-1089	24	14	𝒜	𝒜	PROPN
cana-1089	24	15	)	)	PUNCT
cana-1089	24	16	.	.	PUNCT
cana-1089	25	1	(	(	PUNCT
cana-1089	25	2	1.1	1.1	NUM
cana-1089	25	3	)	)	PUNCT
cana-1089	25	4	from	from	ADP
cana-1089	25	5	(	(	PUNCT
cana-1089	25	6	1.1	1.1	NUM
cana-1089	25	7	)	)	PUNCT
cana-1089	25	8	,	,	PUNCT
cana-1089	25	9	we	we	PRON
cana-1089	25	10	see	see	VERB
cana-1089	25	11	that	that	PRON
cana-1089	25	12	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	25	13	)	)	PUNCT
cana-1089	25	14	satisfies	satisfy	VERB
cana-1089	25	15	the	the	DET
cana-1089	25	16	linearity	linearity	NOUN
cana-1089	25	17	conditions	condition	NOUN
cana-1089	25	18	.	.	PUNCT
cana-1089	26	1	sakaguchi	sakaguchi	ADJ
cana-1089	26	2	[	[	X
cana-1089	26	3	22	22	NUM
cana-1089	26	4	]	]	PUNCT
cana-1089	26	5	defined	define	VERB
cana-1089	26	6	the	the	DET
cana-1089	26	7	class	class	NOUN
cana-1089	26	8	𝒮𝑠	𝒮𝑠	PROPN
cana-1089	26	9	∗(𝛾	∗(𝛾	PROPN
cana-1089	26	10	)	)	PUNCT
cana-1089	26	11	,	,	PUNCT
cana-1089	26	12	the	the	DET
cana-1089	26	13	class	class	NOUN
cana-1089	26	14	of	of	ADP
cana-1089	26	15	function	function	NOUN
cana-1089	26	16	starlike	starlike	NOUN
cana-1089	26	17	with	with	ADP
cana-1089	26	18	respect	respect	NOUN
cana-1089	26	19	to	to	ADP
cana-1089	26	20	symmetric	symmetric	ADJ
cana-1089	26	21	points	point	NOUN
cana-1089	26	22	as	as	SCONJ
cana-1089	26	23	follows	follow	VERB
cana-1089	26	24	communications	communication	NOUN
cana-1089	26	25	on	on	ADP
cana-1089	26	26	applied	apply	VERB
cana-1089	26	27	nonlinear	nonlinear	ADJ
cana-1089	26	28	analysis	analysis	NOUN
cana-1089	26	29	issn	issn	NOUN
cana-1089	26	30	:	:	PUNCT
cana-1089	26	31	1074	1074	NUM
cana-1089	26	32	-	-	PUNCT
cana-1089	26	33	133x	133x	NUM
cana-1089	26	34	vol	vol	NOUN
cana-1089	26	35	31	31	NUM
cana-1089	26	36	no	no	NOUN
cana-1089	26	37	.	.	PUNCT
cana-1089	27	1	5s	5s	NUM
cana-1089	27	2	(	(	PUNCT
cana-1089	27	3	2024	2024	NUM
cana-1089	27	4	)	)	PUNCT
cana-1089	27	5	541	541	NUM
cana-1089	27	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	28	1	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	28	2	(	(	PUNCT
cana-1089	28	3	2𝜔𝜑′(𝜔	2𝜔𝜑′(𝜔	NUM
cana-1089	28	4	)	)	PUNCT
cana-1089	28	5	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	28	6	)	)	PUNCT
cana-1089	28	7	−	−	NOUN
cana-1089	28	8	𝜑(−𝜔	𝜑(−𝜔	NOUN
cana-1089	28	9	)	)	PUNCT
cana-1089	28	10	)	)	PUNCT
cana-1089	28	11	>	>	PUNCT
cana-1089	29	1	𝛾	𝛾	ADP
cana-1089	29	2	,	,	PUNCT
cana-1089	29	3	(	(	PUNCT
cana-1089	29	4	𝜔	𝜔	PROPN
cana-1089	29	5	∈	∈	PROPN
cana-1089	29	6	𝒰	𝒰	NOUN
cana-1089	29	7	;	;	PUNCT
cana-1089	29	8	0	0	NUM
cana-1089	29	9	≤	≤	NOUN
cana-1089	29	10	𝛾	𝛾	ADP
cana-1089	29	11	<	<	X
cana-1089	29	12	1	1	NUM
cana-1089	29	13	)	)	PUNCT
cana-1089	29	14	.	.	PUNCT
cana-1089	30	1	the	the	DET
cana-1089	30	2	functions	function	NOUN
cana-1089	30	3	belonging	belong	VERB
cana-1089	30	4	to	to	ADP
cana-1089	30	5	the	the	DET
cana-1089	30	6	class	class	NOUN
cana-1089	30	7	𝒮𝑠	𝒮𝑠	PROPN
cana-1089	30	8	∗(𝛾	∗(𝛾	PROPN
cana-1089	30	9	)	)	PUNCT
cana-1089	30	10	are	be	AUX
cana-1089	30	11	univalent	univalent	ADJ
cana-1089	30	12	(	(	PUNCT
cana-1089	30	13	see	see	VERB
cana-1089	30	14	[	[	X
cana-1089	30	15	22	22	NUM
cana-1089	30	16	]	]	PUNCT
cana-1089	30	17	)	)	PUNCT
cana-1089	30	18	.	.	PUNCT
cana-1089	31	1	extending	extend	VERB
cana-1089	31	2	the	the	DET
cana-1089	31	3	sakaguchi	sakaguchi	ADJ
cana-1089	31	4	class	class	NOUN
cana-1089	31	5	of	of	ADP
cana-1089	31	6	starlike	starlike	NOUN
cana-1089	31	7	function	function	NOUN
cana-1089	31	8	,	,	PUNCT
cana-1089	31	9	the	the	DET
cana-1089	31	10	class	class	NOUN
cana-1089	31	11	of	of	ADP
cana-1089	31	12	starlike	starlike	NOUN
cana-1089	31	13	functions	function	NOUN
cana-1089	31	14	with	with	ADP
cana-1089	31	15	respect	respect	NOUN
cana-1089	31	16	to	to	ADP
cana-1089	31	17	𝑘	𝑘	DET
cana-1089	31	18	−symmetric	−symmetric	ADJ
cana-1089	31	19	points	point	NOUN
cana-1089	31	20	denoted	denote	VERB
cana-1089	31	21	by	by	ADP
cana-1089	31	22	𝒮𝑠	𝒮𝑠	PROPN
cana-1089	31	23	𝑘(𝛾	𝑘(𝛾	PROPN
cana-1089	31	24	)	)	PUNCT
cana-1089	31	25	was	be	AUX
cana-1089	31	26	introduced	introduce	VERB
cana-1089	31	27	and	and	CCONJ
cana-1089	31	28	is	be	AUX
cana-1089	31	29	known	know	VERB
cana-1089	31	30	to	to	PART
cana-1089	31	31	satisfy	satisfy	VERB
cana-1089	31	32	the	the	DET
cana-1089	31	33	analytic	analytic	ADJ
cana-1089	31	34	characterization	characterization	NOUN
cana-1089	31	35	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	31	36	(	(	PUNCT
cana-1089	31	37	2𝜔𝜑′(𝜔	2𝜔𝜑′(𝜔	NUM
cana-1089	31	38	)	)	PUNCT
cana-1089	31	39	𝜑𝑘(𝜔	𝜑𝑘(𝜔	NOUN
cana-1089	31	40	)	)	PUNCT
cana-1089	31	41	)	)	PUNCT
cana-1089	31	42	>	>	PUNCT
cana-1089	32	1	𝛾	𝛾	ADP
cana-1089	32	2	,	,	PUNCT
cana-1089	32	3	(	(	PUNCT
cana-1089	32	4	𝑘	𝑘	X
cana-1089	32	5	=	=	SYM
cana-1089	32	6	1	1	NUM
cana-1089	32	7	,	,	PUNCT
cana-1089	32	8	2	2	NUM
cana-1089	32	9	,	,	PUNCT
cana-1089	32	10	3	3	NUM
cana-1089	32	11	,	,	PUNCT
cana-1089	32	12	…	…	PUNCT
cana-1089	32	13	)	)	PUNCT
cana-1089	32	14	,	,	PUNCT
cana-1089	32	15	(	(	PUNCT
cana-1089	32	16	1.2	1.2	NUM
cana-1089	32	17	)	)	PUNCT
cana-1089	32	18	where	where	SCONJ
cana-1089	32	19	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	32	20	)	)	PUNCT
cana-1089	32	21	=	=	SYM
cana-1089	33	1	1	1	NUM
cana-1089	33	2	𝑘	𝑘	X
cana-1089	33	3	∑	∑	PROPN
cana-1089	33	4	𝜑(𝜀𝜈	𝜑(𝜀𝜈	PROPN
cana-1089	33	5	𝜔	𝜔	PROPN
cana-1089	33	6	)	)	PUNCT
cana-1089	33	7	𝜀𝜈	𝜀𝜈	ADP
cana-1089	33	8	𝑘−1	𝑘−1	PROPN
cana-1089	33	9	𝜈=0	𝜈=0	PUNCT
cana-1089	33	10	,	,	PUNCT
cana-1089	33	11	(	(	PUNCT
cana-1089	33	12	𝜑	𝜑	PROPN
cana-1089	33	13	∈	∈	PROPN
cana-1089	33	14	𝒜	𝒜	PROPN
cana-1089	33	15	)	)	PUNCT
cana-1089	33	16	.	.	PUNCT
cana-1089	34	1	for	for	ADP
cana-1089	34	2	developments	development	NOUN
cana-1089	34	3	and	and	CCONJ
cana-1089	34	4	study	study	NOUN
cana-1089	34	5	of	of	ADP
cana-1089	34	6	various	various	ADJ
cana-1089	34	7	subclasses	subclass	NOUN
cana-1089	34	8	of	of	ADP
cana-1089	34	9	analytic	analytic	ADJ
cana-1089	34	10	functions	function	NOUN
cana-1089	34	11	with	with	ADP
cana-1089	34	12	respect	respect	NOUN
cana-1089	34	13	to	to	ADP
cana-1089	34	14	symmetric	symmetric	ADJ
cana-1089	34	15	points	point	NOUN
cana-1089	34	16	,	,	PUNCT
cana-1089	34	17	refer	refer	VERB
cana-1089	34	18	to	to	ADP
cana-1089	34	19	[	[	X
cana-1089	34	20	12	12	NUM
cana-1089	34	21	,	,	PUNCT
cana-1089	34	22	13	13	NUM
cana-1089	34	23	,	,	PUNCT
cana-1089	34	24	23	23	NUM
cana-1089	34	25	,	,	PUNCT
cana-1089	34	26	24	24	NUM
cana-1089	34	27	,	,	PUNCT
cana-1089	34	28	25	25	NUM
cana-1089	34	29	,	,	PUNCT
cana-1089	34	30	26	26	NUM
cana-1089	34	31	,	,	PUNCT
cana-1089	34	32	27	27	NUM
cana-1089	34	33	,	,	PUNCT
cana-1089	34	34	28	28	NUM
cana-1089	34	35	]	]	PUNCT
cana-1089	34	36	.	.	PUNCT
cana-1089	35	1	bashirov	bashirov	PROPN
cana-1089	35	2	,	,	PUNCT
cana-1089	35	3	kurpinar	kurpinar	ADJ
cana-1089	35	4	and	and	CCONJ
cana-1089	35	5	őzyapĭ	őzyapĭ	ADJ
cana-1089	35	6	in	in	ADP
cana-1089	35	7	[	[	X
cana-1089	35	8	5	5	NUM
cana-1089	35	9	]	]	PUNCT
cana-1089	35	10	(	(	PUNCT
cana-1089	35	11	also	also	ADV
cana-1089	35	12	see	see	VERB
cana-1089	35	13	[	[	X
cana-1089	35	14	6	6	NUM
cana-1089	35	15	,	,	PUNCT
cana-1089	35	16	7	7	NUM
cana-1089	35	17	,	,	PUNCT
cana-1089	35	18	21	21	NUM
cana-1089	35	19	]	]	PUNCT
cana-1089	35	20	)	)	PUNCT
cana-1089	35	21	studied	study	VERB
cana-1089	35	22	the	the	DET
cana-1089	35	23	properties	property	NOUN
cana-1089	35	24	of	of	ADP
cana-1089	35	25	a	a	DET
cana-1089	35	26	calculus	calculus	NOUN
cana-1089	35	27	titled	title	VERB
cana-1089	35	28	multiplicative	multiplicative	ADJ
cana-1089	35	29	calculus	calculus	NOUN
cana-1089	35	30	which	which	PRON
cana-1089	35	31	has	have	AUX
cana-1089	35	32	been	be	AUX
cana-1089	35	33	a	a	DET
cana-1089	35	34	useful	useful	ADJ
cana-1089	35	35	mathematical	mathematical	ADJ
cana-1089	35	36	tool	tool	NOUN
cana-1089	35	37	in	in	ADP
cana-1089	35	38	economics	economic	NOUN
cana-1089	35	39	and	and	CCONJ
cana-1089	35	40	finance	finance	NOUN
cana-1089	35	41	.	.	PUNCT
cana-1089	36	1	for	for	ADP
cana-1089	36	2	a	a	DET
cana-1089	36	3	positive	positive	ADJ
cana-1089	36	4	real	real	ADV
cana-1089	36	5	valued	value	VERB
cana-1089	36	6	function	function	NOUN
cana-1089	36	7	𝜑:ℛ	𝜑:ℛ	PROPN
cana-1089	36	8	→	→	SYM
cana-1089	36	9	ℛ	ℛ	PROPN
cana-1089	36	10	,	,	PUNCT
cana-1089	36	11	the	the	DET
cana-1089	36	12	multiplicative	multiplicative	ADJ
cana-1089	36	13	derivative	derivative	ADJ
cana-1089	36	14	𝜑∗	𝜑∗	NOUN
cana-1089	36	15	is	be	AUX
cana-1089	36	16	defined	define	VERB
cana-1089	36	17	as	as	SCONJ
cana-1089	36	18	follows	follow	VERB
cana-1089	36	19	𝜑∗(𝑥	𝜑∗(𝑥	PROPN
cana-1089	36	20	)	)	PUNCT
cana-1089	37	1	=	=	VERB
cana-1089	37	2	lim	lim	PROPN
cana-1089	37	3	ℎ→0	ℎ→0	NOUN
cana-1089	37	4	(	(	PUNCT
cana-1089	37	5	𝜑(𝑥	𝜑(𝑥	X
cana-1089	37	6	+	+	NUM
cana-1089	37	7	ℎ	ℎ	NOUN
cana-1089	37	8	)	)	PUNCT
cana-1089	37	9	𝜑(𝑥	𝜑(𝑥	NOUN
cana-1089	37	10	)	)	PUNCT
cana-1089	37	11	)	)	PUNCT
cana-1089	37	12	1	1	NUM
cana-1089	37	13	ℎ	ℎ	X
cana-1089	37	14	=	=	SYM
cana-1089	37	15	𝑒	𝑒	PROPN
cana-1089	37	16	𝜑′(𝑥	𝜑′(𝑥	PROPN
cana-1089	37	17	)	)	PUNCT
cana-1089	37	18	𝜑(𝑥	𝜑(𝑥	NOUN
cana-1089	37	19	)	)	PUNCT
cana-1089	37	20	=	=	SYM
cana-1089	37	21	𝑒[ln𝜑(𝑥)]′	𝑒[ln𝜑(𝑥)]′	PROPN
cana-1089	37	22	where	where	SCONJ
cana-1089	37	23	𝜑′(𝑥	𝜑′(𝑥	X
cana-1089	37	24	)	)	PUNCT
cana-1089	37	25	is	be	AUX
cana-1089	37	26	the	the	DET
cana-1089	37	27	ordinary	ordinary	ADJ
cana-1089	37	28	derivative	derivative	NOUN
cana-1089	37	29	.	.	PUNCT
cana-1089	38	1	the	the	DET
cana-1089	38	2	∗-derivative	∗-derivative	NOUN
cana-1089	38	3	of	of	ADP
cana-1089	38	4	𝜑	𝜑	NOUN
cana-1089	38	5	at	at	ADP
cana-1089	38	6	𝜔	𝜔	ADP
cana-1089	38	7	belonging	belong	VERB
cana-1089	38	8	to	to	ADP
cana-1089	38	9	a	a	DET
cana-1089	38	10	small	small	ADJ
cana-1089	38	11	neighbourhood	neighbourhood	NOUN
cana-1089	38	12	of	of	ADP
cana-1089	38	13	a	a	DET
cana-1089	38	14	domain	domain	NOUN
cana-1089	38	15	in	in	ADP
cana-1089	38	16	a	a	DET
cana-1089	38	17	complex	complex	ADJ
cana-1089	38	18	plane	plane	NOUN
cana-1089	38	19	where	where	SCONJ
cana-1089	38	20	𝜑	𝜑	PROPN
cana-1089	38	21	is	be	AUX
cana-1089	38	22	non	non	ADJ
cana-1089	38	23	-	-	ADJ
cana-1089	38	24	vanishing	vanishing	ADJ
cana-1089	38	25	differentiable	differentiable	NOUN
cana-1089	38	26	,	,	PUNCT
cana-1089	38	27	is	be	AUX
cana-1089	38	28	given	give	VERB
cana-1089	38	29	by	by	ADP
cana-1089	38	30	𝜑∗(𝜔	𝜑∗(𝜔	NOUN
cana-1089	38	31	)	)	PUNCT
cana-1089	38	32	=	=	SYM
cana-1089	38	33	𝑒	𝑒	PROPN
cana-1089	38	34	𝜑′(𝜔	𝜑′(𝜔	PROPN
cana-1089	38	35	)	)	PUNCT
cana-1089	38	36	𝜑(𝜔	𝜑(𝜔	PROPN
cana-1089	38	37	)	)	PUNCT
cana-1089	38	38	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1089	38	39	𝜑∗(𝑛)(𝜔	𝜑∗(𝑛)(𝜔	NOUN
cana-1089	38	40	)	)	PUNCT
cana-1089	38	41	=	=	SYM
cana-1089	38	42	𝑒	𝑒	PROPN
cana-1089	38	43	[	[	PUNCT
cana-1089	38	44	𝜑′(𝜔	𝜑′(𝜔	NOUN
cana-1089	38	45	)	)	PUNCT
cana-1089	38	46	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	38	47	)	)	PUNCT
cana-1089	38	48	]	]	PUNCT
cana-1089	38	49	(	(	PUNCT
cana-1089	38	50	𝑛	𝑛	NOUN
cana-1089	38	51	)	)	PUNCT
cana-1089	38	52	,	,	PUNCT
cana-1089	38	53	𝑛	𝑛	PROPN
cana-1089	38	54	=	=	SYM
cana-1089	38	55	1	1	NUM
cana-1089	38	56	,	,	PUNCT
cana-1089	38	57	2	2	NUM
cana-1089	38	58	,	,	PUNCT
cana-1089	38	59	…	…	PUNCT
cana-1089	38	60	.	.	PUNCT
cana-1089	39	1	influenced	influence	VERB
cana-1089	39	2	by	by	ADP
cana-1089	39	3	the	the	DET
cana-1089	39	4	definition	definition	NOUN
cana-1089	39	5	of	of	ADP
cana-1089	39	6	multiplicative	multiplicative	ADJ
cana-1089	39	7	derivative	derivative	NOUN
cana-1089	39	8	,	,	PUNCT
cana-1089	39	9	recently	recently	ADV
cana-1089	39	10	karthikeyan	karthikeyan	ADJ
cana-1089	39	11	and	and	CCONJ
cana-1089	39	12	murugusundaramoorthy	murugusundaramoorthy	ADJ
cana-1089	39	13	in	in	ADP
cana-1089	39	14	[	[	X
cana-1089	39	15	10	10	NUM
cana-1089	39	16	]	]	PUNCT
cana-1089	39	17	introduced	introduce	VERB
cana-1089	39	18	and	and	CCONJ
cana-1089	39	19	studied	study	VERB
cana-1089	39	20	a	a	DET
cana-1089	39	21	class	class	NOUN
cana-1089	39	22	of	of	ADP
cana-1089	39	23	analytic	analytic	ADJ
cana-1089	39	24	functions	function	NOUN
cana-1089	39	25	ℛ(𝜒	ℛ(𝜒	NOUN
cana-1089	39	26	)	)	PUNCT
cana-1089	39	27	satisfying	satisfy	VERB
cana-1089	39	28	the	the	DET
cana-1089	39	29	subordination	subordination	NOUN
cana-1089	39	30	condition	condition	NOUN
cana-1089	39	31	𝜔	𝜔	X
cana-1089	39	32	𝑒	𝑒	PROPN
cana-1089	39	33	ω2φ′(ω	ω2φ′(ω	ADJ
cana-1089	39	34	)	)	PUNCT
cana-1089	39	35	φ(ω	φ(ω	NOUN
cana-1089	39	36	)	)	PUNCT
cana-1089	40	1	φ(ω	φ(ω	NOUN
cana-1089	40	2	)	)	PUNCT
cana-1089	40	3	≺	≺	NOUN
cana-1089	40	4	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	40	5	)	)	PUNCT
cana-1089	40	6	(	(	PUNCT
cana-1089	40	7	1.3	1.3	NUM
cana-1089	40	8	)	)	PUNCT
cana-1089	40	9	where	where	SCONJ
cana-1089	40	10	𝜒	𝜒	X
cana-1089	40	11	∈	∈	PROPN
cana-1089	40	12	𝒫	𝒫	NOUN
cana-1089	40	13	and	and	CCONJ
cana-1089	40	14	𝜒(𝒰	𝜒(𝒰	NOUN
cana-1089	40	15	)	)	PUNCT
cana-1089	40	16	is	be	AUX
cana-1089	40	17	symmetric	symmetric	ADJ
cana-1089	40	18	with	with	ADP
cana-1089	40	19	respect	respect	NOUN
cana-1089	40	20	to	to	ADP
cana-1089	40	21	the	the	DET
cana-1089	40	22	real	real	ADJ
cana-1089	40	23	axis	axis	NOUN
cana-1089	40	24	which	which	PRON
cana-1089	40	25	has	have	VERB
cana-1089	40	26	a	a	DET
cana-1089	40	27	series	series	NOUN
cana-1089	40	28	expansion	expansion	NOUN
cana-1089	40	29	of	of	ADP
cana-1089	40	30	the	the	DET
cana-1089	40	31	form	form	NOUN
cana-1089	40	32	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	40	33	)	)	PUNCT
cana-1089	40	34	=	=	SYM
cana-1089	41	1	1	1	NUM
cana-1089	41	2	+	+	CCONJ
cana-1089	41	3	𝐿1𝜔	𝐿1𝜔	PROPN
cana-1089	41	4	+	+	CCONJ
cana-1089	41	5	𝐿2𝜔	𝐿2𝜔	PROPN
cana-1089	41	6	2	2	NUM
cana-1089	41	7	+	+	NUM
cana-1089	41	8	𝐿3𝜔	𝐿3𝜔	PROPN
cana-1089	41	9	3	3	NUM
cana-1089	41	10	+	+	SYM
cana-1089	41	11	⋯	⋯	PROPN
cana-1089	41	12	,	,	PUNCT
cana-1089	41	13	(	(	PUNCT
cana-1089	41	14	𝐿1	𝐿1	X
cana-1089	41	15	>	>	X
cana-1089	41	16	0	0	NUM
cana-1089	41	17	;	;	PUNCT
cana-1089	41	18	𝜔	𝜔	PROPN
cana-1089	41	19	∈	∈	PROPN
cana-1089	41	20	𝒰	𝒰	PROPN
cana-1089	41	21	)	)	PUNCT
cana-1089	41	22	.	.	PUNCT
cana-1089	42	1	(	(	PUNCT
cana-1089	42	2	1.4	1.4	NUM
cana-1089	42	3	)	)	PUNCT
cana-1089	42	4	the	the	DET
cana-1089	42	5	class	class	NOUN
cana-1089	42	6	is	be	AUX
cana-1089	42	7	non	non	ADJ
cana-1089	42	8	-	-	ADJ
cana-1089	42	9	empty	empty	ADJ
cana-1089	42	10	and	and	CCONJ
cana-1089	42	11	possess	possess	VERB
cana-1089	42	12	good	good	ADJ
cana-1089	42	13	geometrical	geometrical	ADJ
cana-1089	42	14	implications	implication	NOUN
cana-1089	42	15	but	but	CCONJ
cana-1089	42	16	it	it	PRON
cana-1089	42	17	does	do	AUX
cana-1089	42	18	not	not	PART
cana-1089	42	19	reduce	reduce	VERB
cana-1089	42	20	to	to	ADP
cana-1089	42	21	wellknown	wellknown	ADJ
cana-1089	42	22	subclasses	subclass	NOUN
cana-1089	42	23	of	of	ADP
cana-1089	42	24	𝒮.	𝒮.	PROPN
cana-1089	42	25	for	for	ADP
cana-1089	42	26	the	the	DET
cana-1089	42	27	detailed	detailed	ADJ
cana-1089	42	28	analysis	analysis	NOUN
cana-1089	42	29	and	and	CCONJ
cana-1089	42	30	closure	closure	NOUN
cana-1089	42	31	properties	property	NOUN
cana-1089	42	32	of	of	ADP
cana-1089	42	33	the	the	DET
cana-1089	42	34	class	class	NOUN
cana-1089	42	35	ℛ(𝜒	ℛ(𝜒	PROPN
cana-1089	42	36	)	)	PUNCT
cana-1089	42	37	,	,	PUNCT
cana-1089	42	38	refer	refer	VERB
cana-1089	42	39	to	to	ADP
cana-1089	42	40	[	[	X
cana-1089	42	41	10	10	NUM
cana-1089	42	42	,	,	PUNCT
cana-1089	42	43	11	11	NUM
cana-1089	42	44	]	]	PUNCT
cana-1089	42	45	.	.	PUNCT
cana-1089	43	1	throughout	throughout	ADP
cana-1089	43	2	this	this	DET
cana-1089	43	3	paper	paper	NOUN
cana-1089	43	4	,	,	PUNCT
cana-1089	43	5	we	we	PRON
cana-1089	43	6	let	let	VERB
cana-1089	43	7	γ𝑛,𝑘	γ𝑛,𝑘	PUNCT
cana-1089	43	8	=	=	SYM
cana-1089	43	9	1	1	NUM
cana-1089	43	10	𝑘	𝑘	NOUN
cana-1089	43	11	∑	∑	PROPN
cana-1089	44	1	[	[	X
cana-1089	44	2	exp	exp	X
cana-1089	44	3	2𝜋	2𝜋	NUM
cana-1089	44	4	𝑖	𝑖	SYM
cana-1089	44	5	𝑘	𝑘	X
cana-1089	44	6	]	]	X
cana-1089	44	7	(	(	PUNCT
cana-1089	44	8	𝑛−1)𝜈𝑘−1	𝑛−1)𝜈𝑘−1	PROPN
cana-1089	44	9	𝜈=0	𝜈=0	PROPN
cana-1089	44	10	and	and	CCONJ
cana-1089	44	11	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	44	12	)	)	PUNCT
cana-1089	44	13	=	=	PUNCT
cana-1089	44	14	𝜔	𝜔	X
cana-1089	44	15	+	+	NOUN
cana-1089	44	16	∑	∑	PROPN
cana-1089	44	17	γ𝑛,𝑘𝑎𝑛𝜔𝑛	γ𝑛,𝑘𝑎𝑛𝜔𝑛	PROPN
cana-1089	44	18	∞	∞	NUM
cana-1089	44	19	𝑛=2	𝑛=2	PROPN
cana-1089	44	20	.	.	PUNCT
cana-1089	45	1	communications	communication	NOUN
cana-1089	45	2	on	on	ADP
cana-1089	45	3	applied	apply	VERB
cana-1089	45	4	nonlinear	nonlinear	ADJ
cana-1089	45	5	analysis	analysis	NOUN
cana-1089	45	6	issn	issn	NOUN
cana-1089	45	7	:	:	PUNCT
cana-1089	45	8	1074	1074	NUM
cana-1089	45	9	-	-	PUNCT
cana-1089	45	10	133x	133x	NUM
cana-1089	45	11	vol	vol	NOUN
cana-1089	45	12	31	31	NUM
cana-1089	45	13	no	no	NOUN
cana-1089	45	14	.	.	PUNCT
cana-1089	46	1	5s	5s	NUM
cana-1089	46	2	(	(	PUNCT
cana-1089	46	3	2024	2024	NUM
cana-1089	46	4	)	)	PUNCT
cana-1089	46	5	542	542	NUM
cana-1089	46	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1089	46	7	motivated	motivate	VERB
cana-1089	46	8	by	by	ADP
cana-1089	46	9	ℛ(𝜒	ℛ(𝜒	PROPN
cana-1089	46	10	)	)	PUNCT
cana-1089	46	11	,	,	PUNCT
cana-1089	46	12	here	here	ADV
cana-1089	46	13	we	we	PRON
cana-1089	46	14	will	will	AUX
cana-1089	46	15	introduce	introduce	VERB
cana-1089	46	16	and	and	CCONJ
cana-1089	46	17	study	study	VERB
cana-1089	46	18	a	a	DET
cana-1089	46	19	new	new	ADJ
cana-1089	46	20	subclass	subclass	NOUN
cana-1089	46	21	of	of	ADP
cana-1089	46	22	starlike	starlike	NOUN
cana-1089	46	23	functions	function	NOUN
cana-1089	46	24	with	with	ADP
cana-1089	46	25	respect	respect	NOUN
cana-1089	46	26	to	to	ADP
cana-1089	46	27	symmetric	symmetric	ADJ
cana-1089	46	28	points	point	NOUN
cana-1089	46	29	.	.	PUNCT
cana-1089	47	1	the	the	DET
cana-1089	47	2	definition	definition	NOUN
cana-1089	47	3	of	of	ADP
cana-1089	47	4	the	the	DET
cana-1089	47	5	new	new	ADJ
cana-1089	47	6	function	function	NOUN
cana-1089	47	7	class	class	NOUN
cana-1089	47	8	are	be	AUX
cana-1089	47	9	as	as	SCONJ
cana-1089	47	10	follows	follow	VERB
cana-1089	47	11	.	.	PUNCT
cana-1089	48	1	definition	definition	NOUN
cana-1089	48	2	1.1	1.1	NUM
cana-1089	48	3	.	.	PUNCT
cana-1089	49	1	let	let	VERB
cana-1089	49	2	𝑘	𝑘	PART
cana-1089	49	3	be	be	AUX
cana-1089	49	4	chosen	choose	VERB
cana-1089	49	5	such	such	ADJ
cana-1089	49	6	that	that	SCONJ
cana-1089	49	7	γ𝑛,𝑘	γ𝑛,𝑘	PUNCT
cana-1089	49	8	=	=	SYM
cana-1089	49	9	(	(	PUNCT
cana-1089	49	10	𝑛	𝑛	PRON
cana-1089	49	11	−	−	PROPN
cana-1089	49	12	1	1	NUM
cana-1089	49	13	)	)	PUNCT
cana-1089	49	14	.	.	PUNCT
cana-1089	50	1	a	a	DET
cana-1089	50	2	function	function	NOUN
cana-1089	50	3	𝜑	𝜑	X
cana-1089	50	4	∈	∈	PROPN
cana-1089	50	5	𝒜	𝒜	NOUN
cana-1089	50	6	is	be	AUX
cana-1089	50	7	said	say	VERB
cana-1089	50	8	to	to	PART
cana-1089	50	9	be	be	AUX
cana-1089	50	10	in	in	ADP
cana-1089	50	11	ℳ𝑘(𝜒	ℳ𝑘(𝜒	NUM
cana-1089	50	12	)	)	PUNCT
cana-1089	50	13	,	,	PUNCT
cana-1089	50	14	if	if	SCONJ
cana-1089	50	15	it	it	PRON
cana-1089	50	16	satisfies	satisfy	VERB
cana-1089	50	17	the	the	DET
cana-1089	50	18	following	follow	VERB
cana-1089	50	19	condition	condition	NOUN
cana-1089	50	20	ω	ω	PROPN
cana-1089	50	21	f∗(ω	f∗(ω	PROPN
cana-1089	50	22	)	)	PUNCT
cana-1089	50	23	𝑒	𝑒	ADP
cana-1089	50	24	φk(ω	φk(ω	NUM
cana-1089	50	25	)	)	PUNCT
cana-1089	50	26	≺	≺	NOUN
cana-1089	50	27	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	50	28	)	)	PUNCT
cana-1089	50	29	,	,	PUNCT
cana-1089	50	30	(	(	PUNCT
cana-1089	50	31	ω	ω	PROPN
cana-1089	50	32	∈	∈	PROPN
cana-1089	50	33	𝒰	𝒰	PROPN
cana-1089	50	34	;	;	PUNCT
cana-1089	50	35	e	e	X
cana-1089	50	36	=	=	SYM
cana-1089	50	37	\exp(1	\exp(1	PROPN
cana-1089	50	38	)	)	PUNCT
cana-1089	50	39	)	)	PUNCT
cana-1089	50	40	(	(	PUNCT
cana-1089	50	41	1.5	1.5	NUM
cana-1089	50	42	)	)	PUNCT
cana-1089	50	43	where	where	SCONJ
cana-1089	50	44	𝐹∗(𝜔	𝐹∗(𝜔	NOUN
cana-1089	50	45	)	)	PUNCT
cana-1089	50	46	=	=	SYM
cana-1089	50	47	𝑒	𝑒	PROPN
cana-1089	50	48	ωφ′(ω	ωφ′(ω	NOUN
cana-1089	50	49	)	)	PUNCT
cana-1089	50	50	φ(ω	φ(ω	PROPN
cana-1089	50	51	)	)	PUNCT
cana-1089	50	52	,	,	PUNCT
cana-1089	50	53	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	50	54	)	)	PUNCT
cana-1089	50	55	=	=	SYM
cana-1089	51	1	1	1	NUM
cana-1089	51	2	𝑘	𝑘	X
cana-1089	51	3	∑	∑	PROPN
cana-1089	51	4	𝜑(𝜀𝜈	𝜑(𝜀𝜈	PROPN
cana-1089	51	5	𝜔	𝜔	PROPN
cana-1089	51	6	)	)	PUNCT
cana-1089	51	7	𝜀𝜈	𝜀𝜈	ADP
cana-1089	51	8	𝑘−1	𝑘−1	PROPN
cana-1089	51	9	𝜈=0	𝜈=0	PUNCT
cana-1089	51	10	,	,	PUNCT
cana-1089	51	11	𝜒	𝜒	X
cana-1089	51	12	∈	∈	PROPN
cana-1089	51	13	𝒫	𝒫	NOUN
cana-1089	51	14	and	and	CCONJ
cana-1089	51	15	𝜒(𝒰	𝜒(𝒰	NOUN
cana-1089	51	16	)	)	PUNCT
cana-1089	51	17	is	be	AUX
cana-1089	51	18	defined	define	VERB
cana-1089	51	19	as	as	ADP
cana-1089	51	20	in	in	ADP
cana-1089	51	21	(	(	PUNCT
cana-1089	51	22	1.4	1.4	NUM
cana-1089	51	23	)	)	PUNCT
cana-1089	51	24	.	.	PUNCT
cana-1089	52	1	the	the	DET
cana-1089	52	2	class	class	NOUN
cana-1089	52	3	ℳ𝑘(𝜒	ℳ𝑘(𝜒	NOUN
cana-1089	52	4	)	)	PUNCT
cana-1089	52	5	has	have	AUX
cana-1089	52	6	been	be	AUX
cana-1089	52	7	defined	define	VERB
cana-1089	52	8	by	by	ADP
cana-1089	52	9	replacing	replace	VERB
cana-1089	52	10	the	the	DET
cana-1089	52	11	classical	classical	ADJ
cana-1089	52	12	derivative	derivative	NOUN
cana-1089	52	13	with	with	ADP
cana-1089	52	14	a	a	DET
cana-1089	52	15	multiplicative	multiplicative	ADJ
cana-1089	52	16	derivative	derivative	NOUN
cana-1089	52	17	in	in	ADP
cana-1089	52	18	(	(	PUNCT
cana-1089	52	19	1.2	1.2	NUM
cana-1089	52	20	)	)	PUNCT
cana-1089	52	21	.	.	PUNCT
cana-1089	53	1	note	note	VERB
cana-1089	53	2	that	that	SCONJ
cana-1089	53	3	𝑘	𝑘	NOUN
cana-1089	53	4	=	=	SYM
cana-1089	53	5	1	1	NUM
cana-1089	53	6	is	be	AUX
cana-1089	53	7	not	not	PART
cana-1089	53	8	admissible	admissible	ADJ
cana-1089	53	9	in	in	ADP
cana-1089	53	10	ℳ𝑘(𝜒	ℳ𝑘(𝜒	NUM
cana-1089	53	11	)	)	PUNCT
cana-1089	53	12	,	,	PUNCT
cana-1089	53	13	so	so	CCONJ
cana-1089	53	14	the	the	DET
cana-1089	53	15	class	class	NOUN
cana-1089	53	16	of	of	ADP
cana-1089	53	17	function	function	NOUN
cana-1089	53	18	𝜑	𝜑	PROPN
cana-1089	53	19	∈	∈	PROPN
cana-1089	53	20	𝒜	𝒜	NOUN
cana-1089	53	21	satisfying	satisfy	VERB
cana-1089	53	22	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	53	23	(	(	PUNCT
cana-1089	53	24	𝜔	𝜔	PART
cana-1089	53	25	𝐹∗(𝜔	𝐹∗(𝜔	NOUN
cana-1089	53	26	)	)	PUNCT
cana-1089	53	27	𝑒	𝑒	ADP
cana-1089	53	28	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	53	29	)	)	PUNCT
cana-1089	53	30	)	)	PUNCT
cana-1089	53	31	>	>	X
cana-1089	54	1	0	0	NUM
cana-1089	54	2	,	,	PUNCT
cana-1089	54	3	(	(	PUNCT
cana-1089	54	4	𝐹∗(𝜔	𝐹∗(𝜔	NOUN
cana-1089	54	5	)	)	PUNCT
cana-1089	54	6	=	=	SYM
cana-1089	54	7	𝑒	𝑒	PROPN
cana-1089	54	8	ωφ′(ω	ωφ′(ω	NOUN
cana-1089	54	9	)	)	PUNCT
cana-1089	54	10	φ(ω	φ(ω	PROPN
cana-1089	54	11	)	)	PUNCT
cana-1089	54	12	)	)	PUNCT
cana-1089	54	13	fails	fail	VERB
cana-1089	54	14	to	to	PART
cana-1089	54	15	exist	exist	VERB
cana-1089	54	16	.	.	PUNCT
cana-1089	55	1	the	the	DET
cana-1089	55	2	reason	reason	NOUN
cana-1089	55	3	for	for	ADP
cana-1089	55	4	imposing	impose	VERB
cana-1089	55	5	𝑘	𝑘	DET
cana-1089	55	6	≠	≠	PROPN
cana-1089	55	7	1	1	NUM
cana-1089	55	8	in	in	ADP
cana-1089	55	9	ℳ𝑘(𝜒	ℳ𝑘(𝜒	NUM
cana-1089	55	10	)	)	PUNCT
cana-1089	55	11	is	be	AUX
cana-1089	55	12	that	that	SCONJ
cana-1089	55	13	we	we	PRON
cana-1089	55	14	would	would	AUX
cana-1089	55	15	be	be	AUX
cana-1089	55	16	unable	unable	ADJ
cana-1089	55	17	to	to	PART
cana-1089	55	18	work	work	VERB
cana-1089	55	19	within	within	ADP
cana-1089	55	20	the	the	DET
cana-1089	55	21	existing	exist	VERB
cana-1089	55	22	framework	framework	NOUN
cana-1089	55	23	,	,	PUNCT
cana-1089	55	24	since	since	SCONJ
cana-1089	55	25	the	the	DET
cana-1089	55	26	requirement	requirement	NOUN
cana-1089	55	27	of	of	ADP
cana-1089	55	28	the	the	DET
cana-1089	55	29	condition	condition	NOUN
cana-1089	55	30	of	of	ADP
cana-1089	55	31	𝐿1	𝐿1	PROPN
cana-1089	55	32	to	to	PART
cana-1089	55	33	be	be	AUX
cana-1089	55	34	non	non	ADJ
cana-1089	55	35	-	-	ADJ
cana-1089	55	36	zero	zero	NUM
cana-1089	55	37	would	would	AUX
cana-1089	55	38	be	be	AUX
cana-1089	55	39	violated	violate	VERB
cana-1089	55	40	.	.	PUNCT
cana-1089	56	1	alternatively	alternatively	ADV
cana-1089	56	2	,	,	PUNCT
cana-1089	56	3	we	we	PRON
cana-1089	56	4	will	will	AUX
cana-1089	56	5	now	now	ADV
cana-1089	56	6	define	define	VERB
cana-1089	56	7	a	a	DET
cana-1089	56	8	class	class	NOUN
cana-1089	56	9	which	which	PRON
cana-1089	56	10	would	would	AUX
cana-1089	56	11	be	be	AUX
cana-1089	56	12	defined	define	VERB
cana-1089	56	13	for	for	ADP
cana-1089	56	14	𝑘	𝑘	NOUN
cana-1089	56	15	=	=	SYM
cana-1089	56	16	1	1	NUM
cana-1089	56	17	.	.	PUNCT
cana-1089	56	18	definition	definition	NOUN
cana-1089	56	19	1.2	1.2	NUM
cana-1089	56	20	.	.	PUNCT
cana-1089	57	1	let	let	VERB
cana-1089	57	2	𝑘	𝑘	PART
cana-1089	57	3	be	be	AUX
cana-1089	57	4	chosen	choose	VERB
cana-1089	57	5	such	such	ADJ
cana-1089	57	6	that	that	SCONJ
cana-1089	57	7	γ𝑛,𝑘	γ𝑛,𝑘	ADP
cana-1089	57	8	≠	≠	PROPN
cana-1089	57	9	0	0	NUM
cana-1089	57	10	.	.	PUNCT
cana-1089	58	1	a	a	DET
cana-1089	58	2	function	function	NOUN
cana-1089	58	3	𝜑	𝜑	X
cana-1089	58	4	∈	∈	PROPN
cana-1089	58	5	𝒜	𝒜	NOUN
cana-1089	58	6	is	be	AUX
cana-1089	58	7	said	say	VERB
cana-1089	58	8	to	to	PART
cana-1089	58	9	be	be	AUX
cana-1089	58	10	in	in	ADP
cana-1089	58	11	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	58	12	)	)	PUNCT
cana-1089	58	13	,	,	PUNCT
cana-1089	58	14	if	if	SCONJ
cana-1089	58	15	it	it	PRON
cana-1089	58	16	satisfies	satisfy	VERB
cana-1089	58	17	the	the	DET
cana-1089	58	18	following	follow	VERB
cana-1089	58	19	condition	condition	NOUN
cana-1089	58	20	ω	ω	PROPN
cana-1089	58	21	𝑒	𝑒	PROPN
cana-1089	58	22	𝜔2φ′(ω	𝜔2φ′(ω	PROPN
cana-1089	58	23	)	)	PUNCT
cana-1089	58	24	φ(ω	φ(ω	NOUN
cana-1089	58	25	)	)	PUNCT
cana-1089	58	26	φk(ω	φk(ω	NUM
cana-1089	58	27	)	)	PUNCT
cana-1089	58	28	≺	≺	NOUN
cana-1089	58	29	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	58	30	)	)	PUNCT
cana-1089	58	31	,	,	PUNCT
cana-1089	58	32	(	(	PUNCT
cana-1089	58	33	ω	ω	PROPN
cana-1089	58	34	∈	∈	PROPN
cana-1089	58	35	𝒰	𝒰	PROPN
cana-1089	58	36	)	)	PUNCT
cana-1089	58	37	,	,	PUNCT
cana-1089	58	38	(	(	PUNCT
cana-1089	58	39	1.6	1.6	NUM
cana-1089	58	40	)	)	PUNCT
cana-1089	58	41	where	where	SCONJ
cana-1089	58	42	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	58	43	)	)	PUNCT
cana-1089	58	44	is	be	AUX
cana-1089	58	45	defined	define	VERB
cana-1089	58	46	as	as	ADP
cana-1089	58	47	in	in	ADP
cana-1089	58	48	(	(	PUNCT
cana-1089	58	49	1.1	1.1	NUM
cana-1089	58	50	)	)	PUNCT
cana-1089	58	51	,	,	PUNCT
cana-1089	58	52	𝜒	𝜒	X
cana-1089	58	53	∈	∈	PROPN
cana-1089	58	54	𝒫	𝒫	NOUN
cana-1089	58	55	and	and	CCONJ
cana-1089	58	56	𝜒(𝒰)$	𝜒(𝒰)$	NOUN
cana-1089	58	57	is	be	AUX
cana-1089	58	58	defined	define	VERB
cana-1089	58	59	as	as	ADP
cana-1089	58	60	in	in	ADP
cana-1089	58	61	(	(	PUNCT
cana-1089	58	62	1.4	1.4	NUM
cana-1089	58	63	)	)	PUNCT
cana-1089	58	64	.	.	PUNCT
cana-1089	59	1	from	from	ADP
cana-1089	59	2	the	the	DET
cana-1089	59	3	study	study	NOUN
cana-1089	59	4	of	of	ADP
cana-1089	59	5	[	[	X
cana-1089	59	6	10	10	NUM
cana-1089	59	7	]	]	PUNCT
cana-1089	59	8	,	,	PUNCT
cana-1089	59	9	we	we	PRON
cana-1089	59	10	find	find	VERB
cana-1089	59	11	that	that	SCONJ
cana-1089	59	12	classes	class	NOUN
cana-1089	59	13	involving	involve	VERB
cana-1089	59	14	the	the	DET
cana-1089	59	15	multiplicative	multiplicative	ADJ
cana-1089	59	16	derivative	derivative	NOUN
cana-1089	59	17	does	do	AUX
cana-1089	59	18	not	not	PART
cana-1089	59	19	have	have	VERB
cana-1089	59	20	any	any	DET
cana-1089	59	21	well	well	ADV
cana-1089	59	22	-	-	PUNCT
cana-1089	59	23	known	know	VERB
cana-1089	59	24	classes	class	NOUN
cana-1089	59	25	as	as	ADP
cana-1089	59	26	its	its	PRON
cana-1089	59	27	special	special	ADJ
cana-1089	59	28	cases	case	NOUN
cana-1089	59	29	.	.	PUNCT
cana-1089	60	1	but	but	CCONJ
cana-1089	60	2	these	these	DET
cana-1089	60	3	classes	class	NOUN
cana-1089	60	4	had	have	VERB
cana-1089	60	5	very	very	ADV
cana-1089	60	6	good	good	ADJ
cana-1089	60	7	geometric	geometric	ADJ
cana-1089	60	8	behaviour	behaviour	NOUN
cana-1089	60	9	when	when	SCONJ
cana-1089	60	10	compared	compare	VERB
cana-1089	60	11	to	to	ADP
cana-1089	60	12	various	various	ADJ
cana-1089	60	13	other	other	ADJ
cana-1089	60	14	subclasses	subclass	NOUN
cana-1089	60	15	of	of	ADP
cana-1089	60	16	analytic	analytic	ADJ
cana-1089	60	17	functions	function	NOUN
cana-1089	60	18	.	.	PUNCT
cana-1089	61	1	letting	let	VERB
cana-1089	61	2	𝑘	𝑘	X
cana-1089	61	3	=	=	SYM
cana-1089	61	4	2	2	NUM
cana-1089	61	5	and	and	CCONJ
cana-1089	61	6	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	61	7	)	)	PUNCT
cana-1089	62	1	=	=	PUNCT
cana-1089	63	1	1+𝜔	1+𝜔	NUM
cana-1089	63	2	1−𝜔	1−𝜔	NUM
cana-1089	63	3	in	in	ADP
cana-1089	63	4	(	(	PUNCT
cana-1089	63	5	1.5	1.5	NUM
cana-1089	63	6	)	)	PUNCT
cana-1089	63	7	,	,	PUNCT
cana-1089	63	8	we	we	PRON
cana-1089	63	9	get	get	VERB
cana-1089	64	1	the	the	DET
cana-1089	64	2	following	follow	VERB
cana-1089	64	3	familiar	familiar	ADJ
cana-1089	64	4	analytic	analytic	ADJ
cana-1089	64	5	characterizations	characterization	NOUN
cana-1089	64	6	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	64	7	(	(	PUNCT
cana-1089	64	8	2𝜔	2𝜔	NOUN
cana-1089	64	9	𝑒	𝑒	X
cana-1089	64	10	𝜔𝜑′(𝑤	𝜔𝜑′(𝑤	PROPN
cana-1089	64	11	)	)	PUNCT
cana-1089	64	12	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	64	13	)	)	PUNCT
cana-1089	64	14	−1	−1	NOUN
cana-1089	64	15	𝜑(𝜔)−𝜑(−𝜔	𝜑(𝜔)−𝜑(−𝜔	PROPN
cana-1089	64	16	)	)	PUNCT
cana-1089	64	17	)	)	PUNCT
cana-1089	64	18	>	>	X
cana-1089	65	1	0	0	X
cana-1089	65	2	.	.	PUNCT
cana-1089	66	1	(	(	PUNCT
cana-1089	66	2	1.7	1.7	NUM
cana-1089	66	3	)	)	PUNCT
cana-1089	66	4	notice	notice	VERB
cana-1089	66	5	that	that	SCONJ
cana-1089	66	6	the	the	DET
cana-1089	66	7	expression	expression	NOUN
cana-1089	66	8	in	in	ADP
cana-1089	66	9	(	(	PUNCT
cana-1089	66	10	1.7	1.7	NUM
cana-1089	66	11	)	)	PUNCT
cana-1089	66	12	is	be	AUX
cana-1089	66	13	similar	similar	ADJ
cana-1089	66	14	to	to	ADP
cana-1089	66	15	the	the	DET
cana-1089	66	16	analytic	analytic	ADJ
cana-1089	66	17	characterization	characterization	NOUN
cana-1089	66	18	of	of	ADP
cana-1089	66	19	𝒮𝑠	𝒮𝑠	PROPN
cana-1089	66	20	∗(0	∗(0	PROPN
cana-1089	66	21	)	)	PUNCT
cana-1089	66	22	.	.	PUNCT
cana-1089	67	1	also	also	ADV
cana-1089	67	2	letting	let	VERB
cana-1089	67	3	𝑘	𝑘	X
cana-1089	67	4	=	=	NOUN
cana-1089	67	5	1	1	NUM
cana-1089	67	6	in	in	ADP
cana-1089	67	7	definition	definition	NOUN
cana-1089	67	8	1.2	1.2	NUM
cana-1089	67	9	,	,	PUNCT
cana-1089	67	10	the	the	DET
cana-1089	67	11	class	class	NOUN
cana-1089	67	12	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	67	13	)	)	PUNCT
cana-1089	67	14	reduces	reduce	VERB
cana-1089	67	15	to	to	ADP
cana-1089	67	16	the	the	DET
cana-1089	67	17	class	class	NOUN
cana-1089	67	18	ℛ(𝜒	ℛ(𝜒	NOUN
cana-1089	67	19	)	)	PUNCT
cana-1089	67	20	studied	study	VERB
cana-1089	67	21	by	by	ADP
cana-1089	67	22	karthikeyan	karthikeyan	NOUN
cana-1089	67	23	and	and	CCONJ
cana-1089	67	24	murugusundaramoorthy	murugusundaramoorthy	ADJ
cana-1089	67	25	in	in	ADP
cana-1089	67	26	[	[	X
cana-1089	67	27	10	10	NUM
cana-1089	67	28	]	]	PUNCT
cana-1089	67	29	.	.	PUNCT
cana-1089	68	1	2	2	X
cana-1089	68	2	.	.	NOUN
cana-1089	68	3	coefficients	coefficient	NOUN
cana-1089	68	4	inequalities	inequality	NOUN
cana-1089	68	5	of	of	ADP
cana-1089	68	6	functions	function	NOUN
cana-1089	68	7	in	in	ADP
cana-1089	68	8	𝓜𝒌(𝝌	𝓜𝒌(𝝌	NOUN
cana-1089	68	9	)	)	PUNCT
cana-1089	68	10	and	and	CCONJ
cana-1089	68	11	𝓛𝒌(𝝌	𝓛𝒌(𝝌	NUM
cana-1089	68	12	)	)	PUNCT
cana-1089	68	13	now	now	ADV
cana-1089	68	14	we	we	PRON
cana-1089	68	15	will	will	AUX
cana-1089	68	16	find	find	VERB
cana-1089	68	17	the	the	DET
cana-1089	68	18	solution	solution	NOUN
cana-1089	68	19	to	to	ADP
cana-1089	68	20	the	the	DET
cana-1089	68	21	fekete	fekete	NOUN
cana-1089	68	22	-	-	PUNCT
cana-1089	68	23	szegő	szegő	ADJ
cana-1089	68	24	problem	problem	NOUN
cana-1089	68	25	for	for	ADP
cana-1089	68	26	𝜑	𝜑	PROPN
cana-1089	68	27	∈	∈	PROPN
cana-1089	68	28	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	68	29	)	)	PUNCT
cana-1089	68	30	.	.	PUNCT
cana-1089	69	1	lemma	lemma	PROPN
cana-1089	69	2	2.1	2.1	NUM
cana-1089	70	1	[	[	X
cana-1089	70	2	15	15	NUM
cana-1089	70	3	]	]	X
cana-1089	70	4	if	if	SCONJ
cana-1089	70	5	𝑑(𝜔	𝑑(𝜔	X
cana-1089	70	6	)	)	PUNCT
cana-1089	70	7	=	=	SYM
cana-1089	70	8	1	1	NUM
cana-1089	70	9	+	+	CCONJ
cana-1089	70	10	∑	∑	ADP
cana-1089	70	11	𝑑𝑘	𝑑𝑘	ADV
cana-1089	70	12	∞	∞	NUM
cana-1089	70	13	𝑘=1	𝑘=1	ADJ
cana-1089	70	14	𝜔𝑘	𝜔𝑘	ADP
cana-1089	70	15	∈	∈	PROPN
cana-1089	70	16	𝒫	𝒫	PROPN
cana-1089	70	17	,	,	PUNCT
cana-1089	70	18	and	and	CCONJ
cana-1089	70	19	𝜌	𝜌	NOUN
cana-1089	70	20	is	be	AUX
cana-1089	70	21	complex	complex	ADJ
cana-1089	70	22	number	number	NOUN
cana-1089	70	23	,	,	PUNCT
cana-1089	70	24	then	then	ADV
cana-1089	70	25	|𝑑2	|𝑑2	NOUN
cana-1089	71	1	−	−	NOUN
cana-1089	71	2	𝜌	𝜌	ADP
cana-1089	71	3	𝑑1	𝑑1	NOUN
cana-1089	71	4	2|	2|	NUM
cana-1089	71	5	≤	≤	PROPN
cana-1089	72	1	2max{1	2max{1	NUM
cana-1089	72	2	;	;	PUNCT
cana-1089	72	3	|2𝜌	|2𝜌	VERB
cana-1089	72	4	−	−	PROPN
cana-1089	72	5	1|	1|	NUM
cana-1089	72	6	}	}	PUNCT
cana-1089	72	7	,	,	PUNCT
cana-1089	72	8	communications	communication	NOUN
cana-1089	72	9	on	on	ADP
cana-1089	72	10	applied	apply	VERB
cana-1089	72	11	nonlinear	nonlinear	ADJ
cana-1089	72	12	analysis	analysis	NOUN
cana-1089	72	13	issn	issn	NOUN
cana-1089	72	14	:	:	PUNCT
cana-1089	72	15	1074	1074	NUM
cana-1089	72	16	-	-	PUNCT
cana-1089	72	17	133x	133x	NUM
cana-1089	72	18	vol	vol	NOUN
cana-1089	72	19	31	31	NUM
cana-1089	72	20	no	no	NOUN
cana-1089	72	21	.	.	PUNCT
cana-1089	73	1	5s	5s	NUM
cana-1089	73	2	(	(	PUNCT
cana-1089	73	3	2024	2024	NUM
cana-1089	73	4	)	)	PUNCT
cana-1089	73	5	543	543	NUM
cana-1089	73	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	73	7	and	and	CCONJ
cana-1089	73	8	the	the	DET
cana-1089	73	9	result	result	NOUN
cana-1089	73	10	is	be	AUX
cana-1089	73	11	sharp	sharp	ADJ
cana-1089	73	12	.	.	PUNCT
cana-1089	74	1	theorem	theorem	VERB
cana-1089	74	2	2.1	2.1	NUM
cana-1089	74	3	.	.	PUNCT
cana-1089	75	1	if	if	SCONJ
cana-1089	75	2	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	75	3	)	)	PUNCT
cana-1089	75	4	∈	∈	NOUN
cana-1089	75	5	ℳ𝑘(𝜒	ℳ𝑘(𝜒	NOUN
cana-1089	75	6	)	)	PUNCT
cana-1089	75	7	,	,	PUNCT
cana-1089	75	8	then	then	ADV
cana-1089	75	9	we	we	PRON
cana-1089	75	10	have	have	VERB
cana-1089	75	11	|𝑎2|	|𝑎2|	NOUN
cana-1089	75	12	≤	≤	PROPN
cana-1089	75	13	𝐿1	𝐿1	PROPN
cana-1089	75	14	|1	|1	PRON
cana-1089	76	1	−	−	PROPN
cana-1089	77	1	γ2,𝑘|	γ2,𝑘|	PROPN
cana-1089	77	2	(	(	PUNCT
cana-1089	77	3	2.1	2.1	NUM
cana-1089	77	4	)	)	PUNCT
cana-1089	77	5	|	|	ADV
cana-1089	77	6	𝑎3|	𝑎3|	NOUN
cana-1089	77	7	≤	≤	NUM
cana-1089	77	8	𝐿1	𝐿1	VERB
cana-1089	77	9	|2	|2	NUM
cana-1089	77	10	−	−	PRON
cana-1089	77	11	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	77	12	max	max	PROPN
cana-1089	77	13	{	{	PUNCT
cana-1089	77	14	1	1	NUM
cana-1089	77	15	;	;	PUNCT
cana-1089	77	16	|	|	ADV
cana-1089	77	17	𝐿2	𝐿2	NOUN
cana-1089	77	18	𝐿1	𝐿1	PROPN
cana-1089	77	19	−	−	PROPN
cana-1089	77	20	𝐿1	𝐿1	PROPN
cana-1089	78	1	(	(	PUNCT
cana-1089	78	2	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	78	3	2	2	NUM
cana-1089	78	4	−	−	PROPN
cana-1089	78	5	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	78	6	−	−	PROPN
cana-1089	78	7	1	1	NUM
cana-1089	78	8	2	2	NUM
cana-1089	78	9	)	)	PUNCT
cana-1089	78	10	(	(	PUNCT
cana-1089	78	11	1	1	NUM
cana-1089	78	12	−	−	PROPN
cana-1089	78	13	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	78	14	)	)	PUNCT
cana-1089	78	15	2	2	NUM
cana-1089	78	16	|	|	NOUN
cana-1089	78	17	}	}	PUNCT
cana-1089	78	18	(	(	PUNCT
cana-1089	78	19	2.2	2.2	NUM
cana-1089	78	20	)	)	PUNCT
cana-1089	78	21	and	and	CCONJ
cana-1089	78	22	for	for	ADP
cana-1089	78	23	all	all	DET
cana-1089	78	24	𝜌	𝜌	ADP
cana-1089	78	25	∈	∈	PROPN
cana-1089	78	26	ℂ	ℂ	PROPN
cana-1089	78	27	|𝑎3	|𝑎3	VERB
cana-1089	78	28	−	−	PROPN
cana-1089	78	29	𝜌	𝜌	ADP
cana-1089	78	30	𝑎2	𝑎2	NOUN
cana-1089	78	31	2|	2|	NUM
cana-1089	78	32	≤	≤	NOUN
cana-1089	78	33	𝐿1	𝐿1	VERB
cana-1089	78	34	|2	|2	NUM
cana-1089	78	35	−	−	PRON
cana-1089	78	36	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	78	37	max	max	PROPN
cana-1089	78	38	{	{	PUNCT
cana-1089	78	39	1	1	NUM
cana-1089	78	40	;	;	PUNCT
cana-1089	78	41	|	|	ADV
cana-1089	78	42	𝐿2	𝐿2	NOUN
cana-1089	78	43	𝐿1	𝐿1	PROPN
cana-1089	78	44	−	−	PROPN
cana-1089	78	45	𝐿1	𝐿1	PROPN
cana-1089	79	1	(	(	PUNCT
cana-1089	79	2	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	79	3	2	2	NUM
cana-1089	79	4	−	−	PROPN
cana-1089	79	5	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	79	6	−	−	PROPN
cana-1089	79	7	1	1	NUM
cana-1089	79	8	2	2	NUM
cana-1089	79	9	)	)	PUNCT
cana-1089	79	10	(	(	PUNCT
cana-1089	79	11	1	1	NUM
cana-1089	79	12	−	−	PROPN
cana-1089	79	13	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	79	14	)	)	PUNCT
cana-1089	79	15	2	2	NUM
cana-1089	79	16	−	−	NOUN
cana-1089	79	17	𝐿1𝜌(2	𝐿1𝜌(2	PROPN
cana-1089	79	18	−	−	PROPN
cana-1089	79	19	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	79	20	)	)	PUNCT
cana-1089	79	21	(	(	PUNCT
cana-1089	79	22	1	1	NUM
cana-1089	79	23	−	−	PROPN
cana-1089	79	24	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	79	25	)	)	PUNCT
cana-1089	79	26	2	2	NUM
cana-1089	79	27	|	|	NOUN
cana-1089	79	28	}	}	PUNCT
cana-1089	79	29	.	.	PUNCT
cana-1089	80	1	(	(	PUNCT
cana-1089	80	2	2.3	2.3	NUM
cana-1089	80	3	)	)	PUNCT
cana-1089	80	4	the	the	DET
cana-1089	80	5	inequality	inequality	NOUN
cana-1089	80	6	is	be	AUX
cana-1089	80	7	sharp	sharp	ADJ
cana-1089	80	8	for	for	SCONJ
cana-1089	80	9	each	each	DET
cana-1089	80	10	𝜌	𝜌	X
cana-1089	80	11	∈	∈	NOUN
cana-1089	80	12	ℂ.	ℂ.	ADJ
cana-1089	80	13	proof	proof	NOUN
cana-1089	80	14	.	.	PUNCT
cana-1089	81	1	as	as	ADP
cana-1089	81	2	𝜑	𝜑	PROPN
cana-1089	81	3	∈	∈	PROPN
cana-1089	81	4	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	81	5	)	)	PUNCT
cana-1089	81	6	,	,	PUNCT
cana-1089	81	7	by	by	ADP
cana-1089	81	8	(	(	PUNCT
cana-1089	81	9	1.5	1.5	NUM
cana-1089	81	10	)	)	PUNCT
cana-1089	81	11	we	we	PRON
cana-1089	81	12	have	have	VERB
cana-1089	81	13	𝜔	𝜔	PART
cana-1089	81	14	𝐹∗(𝜔	𝐹∗(𝜔	NOUN
cana-1089	81	15	)	)	PUNCT
cana-1089	81	16	𝑒	𝑒	ADP
cana-1089	81	17	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PRON
cana-1089	81	18	)	)	PUNCT
cana-1089	81	19	=	=	SYM
cana-1089	81	20	𝜒[𝑤(𝜔	𝜒[𝑤(𝜔	NOUN
cana-1089	81	21	)	)	PUNCT
cana-1089	81	22	]	]	PUNCT
cana-1089	81	23	.	.	PUNCT
cana-1089	82	1	(	(	PUNCT
cana-1089	82	2	2.4	2.4	NUM
cana-1089	82	3	)	)	PUNCT
cana-1089	82	4	thus	thus	ADV
cana-1089	82	5	,	,	PUNCT
cana-1089	82	6	let	let	VERB
cana-1089	82	7	𝜗	𝜗	PRON
cana-1089	82	8	∈	∈	NOUN
cana-1089	82	9	𝒫	𝒫	NOUN
cana-1089	82	10	be	be	NOUN
cana-1089	82	11	of	of	ADP
cana-1089	82	12	the	the	DET
cana-1089	82	13	form	form	NOUN
cana-1089	82	14	𝜗(𝜔	𝜗(𝜔	NOUN
cana-1089	82	15	)	)	PUNCT
cana-1089	82	16	=	=	SYM
cana-1089	83	1	1	1	NUM
cana-1089	83	2	+	+	CCONJ
cana-1089	83	3	∑	∑	ADP
cana-1089	83	4	𝜗𝑘𝜔	𝜗𝑘𝜔	NOUN
cana-1089	83	5	𝑘	𝑘	DET
cana-1089	83	6	∞	∞	PROPN
cana-1089	83	7	𝑘=1	𝑘=1	PROPN
cana-1089	83	8	and	and	CCONJ
cana-1089	83	9	defined	define	VERB
cana-1089	83	10	by	by	ADP
cana-1089	83	11	𝜗(𝜔	𝜗(𝜔	NOUN
cana-1089	83	12	)	)	PUNCT
cana-1089	83	13	=	=	SYM
cana-1089	83	14	1	1	NUM
cana-1089	83	15	+	+	NUM
cana-1089	83	16	𝑤(𝜔	𝑤(𝜔	NOUN
cana-1089	83	17	)	)	PUNCT
cana-1089	83	18	1	1	NUM
cana-1089	83	19	−	−	PROPN
cana-1089	83	20	𝑤(𝜔	𝑤(𝜔	NOUN
cana-1089	83	21	)	)	PUNCT
cana-1089	83	22	,	,	PUNCT
cana-1089	83	23	𝜔	𝜔	PROPN
cana-1089	83	24	∈	∈	ADJ
cana-1089	83	25	𝒰	𝒰	NOUN
cana-1089	83	26	.	.	PUNCT
cana-1089	84	1	on	on	ADP
cana-1089	84	2	computation	computation	NOUN
cana-1089	84	3	,	,	PUNCT
cana-1089	84	4	the	the	DET
cana-1089	84	5	right	right	ADJ
cana-1089	84	6	hand	hand	NOUN
cana-1089	84	7	side	side	NOUN
cana-1089	84	8	of	of	ADP
cana-1089	84	9	(	(	PUNCT
cana-1089	84	10	2.4	2.4	NUM
cana-1089	84	11	)	)	PUNCT
cana-1089	84	12	𝜒[𝑤(𝜔	𝜒[𝑤(𝜔	NOUN
cana-1089	84	13	)	)	PUNCT
cana-1089	84	14	]	]	PUNCT
cana-1089	84	15	=	=	SYM
cana-1089	84	16	1	1	NUM
cana-1089	84	17	+	+	CCONJ
cana-1089	84	18	𝜗1𝐿1	𝜗1𝐿1	NUM
cana-1089	84	19	2	2	NUM
cana-1089	84	20	𝜔	𝜔	NOUN
cana-1089	84	21	+	+	NOUN
cana-1089	84	22	𝐿1	𝐿1	X
cana-1089	84	23	2	2	NUM
cana-1089	84	24	[	[	NOUN
cana-1089	84	25	𝜗2	𝜗2	X
cana-1089	84	26	−	−	NOUN
cana-1089	84	27	𝜗1	𝜗1	X
cana-1089	84	28	2	2	NUM
cana-1089	84	29	2	2	NUM
cana-1089	84	30	(	(	PUNCT
cana-1089	84	31	1	1	NUM
cana-1089	84	32	−	−	NOUN
cana-1089	84	33	𝐿2	𝐿2	PROPN
cana-1089	84	34	𝐿1	𝐿1	PROPN
cana-1089	84	35	)	)	PUNCT
cana-1089	84	36	]	]	PUNCT
cana-1089	85	1	𝜔2	𝜔2	PROPN
cana-1089	85	2	+	+	CCONJ
cana-1089	85	3	⋯.	⋯.	PROPN
cana-1089	85	4	(	(	PUNCT
cana-1089	85	5	2.5	2.5	NUM
cana-1089	85	6	)	)	PUNCT
cana-1089	85	7	the	the	DET
cana-1089	85	8	left	left	ADJ
cana-1089	85	9	hand	hand	NOUN
cana-1089	85	10	side	side	NOUN
cana-1089	85	11	of	of	ADP
cana-1089	85	12	(	(	PUNCT
cana-1089	85	13	2.4	2.4	NUM
cana-1089	85	14	)	)	PUNCT
cana-1089	85	15	will	will	AUX
cana-1089	85	16	be	be	AUX
cana-1089	85	17	of	of	ADP
cana-1089	85	18	the	the	DET
cana-1089	85	19	form	form	NOUN
cana-1089	85	20	𝜔	𝜔	PRON
cana-1089	85	21	𝐹∗(𝜔	𝐹∗(𝜔	NOUN
cana-1089	85	22	)	)	PUNCT
cana-1089	85	23	𝑒	𝑒	ADP
cana-1089	85	24	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PRON
cana-1089	85	25	)	)	PUNCT
cana-1089	85	26	=	=	SYM
cana-1089	86	1	1	1	NUM
cana-1089	86	2	+	+	CCONJ
cana-1089	86	3	𝑎2[1	𝑎2[1	PROPN
cana-1089	86	4	−	−	NOUN
cana-1089	86	5	γ2,𝑘]𝜔	γ2,𝑘]𝜔	NOUN
cana-1089	86	6	+	+	PUNCT
cana-1089	87	1	[	[	X
cana-1089	87	2	(	(	PUNCT
cana-1089	87	3	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	87	4	2	2	NUM
cana-1089	87	5	−	−	PROPN
cana-1089	87	6	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	87	7	−	−	PROPN
cana-1089	87	8	1	1	NUM
cana-1089	87	9	2	2	NUM
cana-1089	87	10	)	)	PUNCT
cana-1089	87	11	𝑎2	𝑎2	NOUN
cana-1089	87	12	2	2	NUM
cana-1089	87	13	+	+	CCONJ
cana-1089	87	14	(	(	PUNCT
cana-1089	87	15	2	2	NUM
cana-1089	87	16	−	−	NOUN
cana-1089	87	17	γ3,𝑘)𝑎3]𝜔2	γ3,𝑘)𝑎3]𝜔2	NOUN
cana-1089	87	18	+	+	CCONJ
cana-1089	87	19	⋯.	⋯.	PROPN
cana-1089	87	20	(	(	PUNCT
cana-1089	87	21	2.6	2.6	NUM
cana-1089	87	22	)	)	PUNCT
cana-1089	87	23	from	from	ADP
cana-1089	87	24	(	(	PUNCT
cana-1089	87	25	2.5	2.5	NUM
cana-1089	87	26	)	)	PUNCT
cana-1089	87	27	and	and	CCONJ
cana-1089	87	28	(	(	PUNCT
cana-1089	87	29	2.6	2.6	NUM
cana-1089	87	30	)	)	PUNCT
cana-1089	87	31	,	,	PUNCT
cana-1089	87	32	we	we	PRON
cana-1089	87	33	obtain	obtain	VERB
cana-1089	87	34	𝑎2	𝑎2	NOUN
cana-1089	87	35	=	=	SYM
cana-1089	87	36	1	1	NUM
cana-1089	87	37	(	(	PUNCT
cana-1089	87	38	1	1	NUM
cana-1089	87	39	−	−	PROPN
cana-1089	87	40	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	87	41	)	)	PUNCT
cana-1089	87	42	[	[	PUNCT
cana-1089	87	43	𝜗1𝐿1	𝜗1𝐿1	NUM
cana-1089	87	44	2	2	NUM
cana-1089	87	45	]	]	PUNCT
cana-1089	87	46	(	(	PUNCT
cana-1089	87	47	2.7	2.7	NUM
cana-1089	87	48	)	)	PUNCT
cana-1089	87	49	and	and	CCONJ
cana-1089	87	50	𝑎3	𝑎3	PROPN
cana-1089	87	51	=	=	SYM
cana-1089	87	52	𝐿1	𝐿1	PROPN
cana-1089	87	53	2(2	2(2	NUM
cana-1089	87	54	−	−	NOUN
cana-1089	88	1	γ3,𝑘	γ3,𝑘	NUM
cana-1089	88	2	)	)	PUNCT
cana-1089	89	1	[	[	X
cana-1089	89	2	𝜗2	𝜗2	NOUN
cana-1089	89	3	−	−	NOUN
cana-1089	89	4	𝜗1	𝜗1	X
cana-1089	89	5	2	2	NUM
cana-1089	89	6	2	2	NUM
cana-1089	89	7	(	(	PUNCT
cana-1089	89	8	1	1	NUM
cana-1089	89	9	−	−	NOUN
cana-1089	89	10	𝐿2	𝐿2	NOUN
cana-1089	89	11	𝐿1	𝐿1	PROPN
cana-1089	89	12	−	−	PROPN
cana-1089	89	13	𝐿1	𝐿1	PROPN
cana-1089	89	14	(	(	PUNCT
cana-1089	89	15	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	89	16	2	2	NUM
cana-1089	89	17	−	−	PROPN
cana-1089	89	18	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	89	19	−	−	PROPN
cana-1089	89	20	1	1	NUM
cana-1089	89	21	2	2	NUM
cana-1089	89	22	)	)	PUNCT
cana-1089	89	23	(	(	PUNCT
cana-1089	89	24	1	1	NUM
cana-1089	89	25	−	−	PROPN
cana-1089	89	26	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	89	27	)	)	PUNCT
cana-1089	89	28	2	2	NUM
cana-1089	89	29	)	)	PUNCT
cana-1089	89	30	]	]	PUNCT
cana-1089	89	31	(	(	PUNCT
cana-1089	89	32	2.8	2.8	NUM
cana-1089	89	33	)	)	PUNCT
cana-1089	89	34	equations	equation	NOUN
cana-1089	89	35	(	(	PUNCT
cana-1089	89	36	2.1	2.1	NUM
cana-1089	89	37	)	)	PUNCT
cana-1089	89	38	can	can	AUX
cana-1089	89	39	be	be	AUX
cana-1089	89	40	obtained	obtain	VERB
cana-1089	89	41	by	by	ADP
cana-1089	89	42	applying	apply	VERB
cana-1089	89	43	the	the	DET
cana-1089	89	44	well	well	ADV
cana-1089	89	45	-	-	PUNCT
cana-1089	89	46	known	know	VERB
cana-1089	89	47	result	result	NOUN
cana-1089	89	48	of	of	ADP
cana-1089	89	49	|𝜗1|	|𝜗1|	NOUN
cana-1089	89	50	≤	≤	ADJ
cana-1089	89	51	2	2	NUM
cana-1089	89	52	in	in	ADP
cana-1089	89	53	(	(	PUNCT
cana-1089	89	54	2.7	2.7	NUM
cana-1089	89	55	)	)	PUNCT
cana-1089	89	56	.	.	PUNCT
cana-1089	90	1	applying	apply	VERB
cana-1089	90	2	lemma	lemma	PROPN
cana-1089	90	3	2.1	2.1	NUM
cana-1089	90	4	in	in	ADP
cana-1089	90	5	(	(	PUNCT
cana-1089	90	6	2.8	2.8	NUM
cana-1089	90	7	)	)	PUNCT
cana-1089	90	8	,	,	PUNCT
cana-1089	90	9	we	we	PRON
cana-1089	90	10	get	get	VERB
cana-1089	90	11	(	(	PUNCT
cana-1089	90	12	2.2	2.2	NUM
cana-1089	90	13	)	)	PUNCT
cana-1089	90	14	.	.	PUNCT
cana-1089	91	1	communications	communication	NOUN
cana-1089	91	2	on	on	ADP
cana-1089	91	3	applied	apply	VERB
cana-1089	91	4	nonlinear	nonlinear	ADJ
cana-1089	91	5	analysis	analysis	NOUN
cana-1089	91	6	issn	issn	NOUN
cana-1089	91	7	:	:	PUNCT
cana-1089	91	8	1074	1074	NUM
cana-1089	91	9	-	-	PUNCT
cana-1089	91	10	133x	133x	NUM
cana-1089	91	11	vol	vol	NOUN
cana-1089	91	12	31	31	NUM
cana-1089	91	13	no	no	NOUN
cana-1089	91	14	.	.	PUNCT
cana-1089	92	1	5s	5s	NUM
cana-1089	92	2	(	(	PUNCT
cana-1089	92	3	2024	2024	NUM
cana-1089	92	4	)	)	PUNCT
cana-1089	92	5	544	544	NUM
cana-1089	92	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	92	7	now	now	ADV
cana-1089	92	8	to	to	PART
cana-1089	92	9	prove	prove	VERB
cana-1089	92	10	the	the	DET
cana-1089	92	11	fekete	fekete	NOUN
cana-1089	92	12	-	-	PUNCT
cana-1089	92	13	szegő	szegő	ADJ
cana-1089	92	14	inequality	inequality	NOUN
cana-1089	92	15	for	for	ADP
cana-1089	92	16	the	the	DET
cana-1089	92	17	class	class	NOUN
cana-1089	92	18	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	92	19	)	)	PUNCT
cana-1089	92	20	,	,	PUNCT
cana-1089	92	21	we	we	PRON
cana-1089	92	22	consider	consider	VERB
cana-1089	92	23	|𝑎3	|𝑎3	VERB
cana-1089	92	24	−	−	PROPN
cana-1089	92	25	𝜌	𝜌	ADP
cana-1089	92	26	𝑎2	𝑎2	VERB
cana-1089	92	27	2|	2|	NUM
cana-1089	93	1	=	=	PUNCT
cana-1089	93	2	|	|	ADV
cana-1089	93	3	𝐿1	𝐿1	VERB
cana-1089	93	4	2(2	2(2	NUM
cana-1089	93	5	−	−	PROPN
cana-1089	94	1	γ3,𝑘	γ3,𝑘	NUM
cana-1089	94	2	)	)	PUNCT
cana-1089	95	1	[	[	X
cana-1089	95	2	𝜗2	𝜗2	NOUN
cana-1089	95	3	−	−	NOUN
cana-1089	95	4	𝜗1	𝜗1	X
cana-1089	95	5	2	2	NUM
cana-1089	95	6	2	2	NUM
cana-1089	95	7	(	(	PUNCT
cana-1089	95	8	1	1	NUM
cana-1089	95	9	−	−	NOUN
cana-1089	95	10	𝐿2	𝐿2	NOUN
cana-1089	95	11	𝐿1	𝐿1	PROPN
cana-1089	95	12	−	−	PROPN
cana-1089	95	13	𝐿1	𝐿1	PROPN
cana-1089	95	14	(	(	PUNCT
cana-1089	95	15	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	95	16	2	2	NUM
cana-1089	95	17	−	−	PROPN
cana-1089	95	18	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	95	19	−	−	PROPN
cana-1089	95	20	1	1	NUM
cana-1089	95	21	2	2	NUM
cana-1089	95	22	)	)	PUNCT
cana-1089	95	23	(	(	PUNCT
cana-1089	95	24	1	1	NUM
cana-1089	95	25	−	−	PROPN
cana-1089	95	26	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	95	27	)	)	PUNCT
cana-1089	95	28	2	2	NUM
cana-1089	95	29	)	)	PUNCT
cana-1089	95	30	]	]	PUNCT
cana-1089	96	1	−	−	PUNCT
cana-1089	96	2	𝜌	𝜌	X
cana-1089	96	3	𝜗1	𝜗1	X
cana-1089	96	4	2	2	NUM
cana-1089	96	5	𝐿1	𝐿1	NOUN
cana-1089	96	6	2	2	NUM
cana-1089	96	7	4(1	4(1	NOUN
cana-1089	96	8	−	−	PROPN
cana-1089	96	9	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	96	10	)	)	PUNCT
cana-1089	96	11	2|	2|	NUM
cana-1089	97	1	=	=	PUNCT
cana-1089	97	2	|	|	ADV
cana-1089	97	3	𝐿1	𝐿1	VERB
cana-1089	97	4	2(2	2(2	NUM
cana-1089	97	5	−	−	PROPN
cana-1089	98	1	γ3,𝑘	γ3,𝑘	NUM
cana-1089	98	2	)	)	PUNCT
cana-1089	99	1	[	[	X
cana-1089	99	2	𝜗2	𝜗2	NOUN
cana-1089	99	3	−	−	NOUN
cana-1089	99	4	𝜗1	𝜗1	X
cana-1089	99	5	2	2	NUM
cana-1089	99	6	2	2	NUM
cana-1089	99	7	(	(	PUNCT
cana-1089	99	8	1	1	NUM
cana-1089	99	9	−	−	NOUN
cana-1089	99	10	𝐿2	𝐿2	NOUN
cana-1089	99	11	𝐿1	𝐿1	PROPN
cana-1089	99	12	−	−	PROPN
cana-1089	99	13	𝐿1	𝐿1	PROPN
cana-1089	99	14	(	(	PUNCT
cana-1089	99	15	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	99	16	2	2	NUM
cana-1089	99	17	−	−	PROPN
cana-1089	99	18	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	99	19	−	−	PROPN
cana-1089	99	20	1	1	NUM
cana-1089	99	21	2	2	NUM
cana-1089	99	22	)	)	PUNCT
cana-1089	99	23	(	(	PUNCT
cana-1089	99	24	1	1	NUM
cana-1089	99	25	−	−	PROPN
cana-1089	99	26	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	99	27	)	)	PUNCT
cana-1089	99	28	2	2	NUM
cana-1089	100	1	+	+	CCONJ
cana-1089	100	2	𝐿1𝜌(2	𝐿1𝜌(2	PROPN
cana-1089	100	3	−	−	PROPN
cana-1089	100	4	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	100	5	)	)	PUNCT
cana-1089	100	6	(	(	PUNCT
cana-1089	100	7	1	1	NUM
cana-1089	100	8	−	−	PROPN
cana-1089	100	9	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	100	10	)	)	PUNCT
cana-1089	100	11	2	2	NUM
cana-1089	100	12	)	)	PUNCT
cana-1089	100	13	]	]	PUNCT
cana-1089	100	14	|	|	ADV
cana-1089	100	15	.	.	PUNCT
cana-1089	101	1	using	use	VERB
cana-1089	101	2	the	the	DET
cana-1089	101	3	triangle	triangle	NOUN
cana-1089	101	4	inequality	inequality	NOUN
cana-1089	101	5	and	and	CCONJ
cana-1089	101	6	lemma	lemma	PROPN
cana-1089	101	7	2.1	2.1	NUM
cana-1089	101	8	in	in	ADP
cana-1089	101	9	the	the	DET
cana-1089	101	10	above	above	ADJ
cana-1089	101	11	equality	equality	NOUN
cana-1089	101	12	,	,	PUNCT
cana-1089	101	13	we	we	PRON
cana-1089	101	14	can	can	AUX
cana-1089	101	15	obtain	obtain	VERB
cana-1089	101	16	(	(	PUNCT
cana-1089	101	17	2.3	2.3	NUM
cana-1089	101	18	)	)	PUNCT
cana-1089	101	19	.	.	PUNCT
cana-1089	102	1	let	let	VERB
cana-1089	102	2	𝑘	𝑘	X
cana-1089	102	3	=	=	SYM
cana-1089	102	4	2	2	NUM
cana-1089	102	5	in	in	ADP
cana-1089	102	6	theorem	theorem	NOUN
cana-1089	102	7	2.1	2.1	NUM
cana-1089	102	8	,	,	PUNCT
cana-1089	102	9	we	we	PRON
cana-1089	102	10	have	have	VERB
cana-1089	102	11	the	the	DET
cana-1089	102	12	following	following	NOUN
cana-1089	102	13	.	.	PUNCT
cana-1089	103	1	corollary	corollary	ADJ
cana-1089	103	2	2.1	2.1	NUM
cana-1089	103	3	.	.	PUNCT
cana-1089	104	1	let	let	VERB
cana-1089	104	2	𝜑	𝜑	PRON
cana-1089	104	3	∈	∈	PROPN
cana-1089	104	4	ℳ2(𝜒	ℳ2(𝜒	PROPN
cana-1089	104	5	)	)	PUNCT
cana-1089	104	6	.	.	PUNCT
cana-1089	105	1	then	then	ADV
cana-1089	105	2	,	,	PUNCT
cana-1089	105	3	|𝑎2|	|𝑎2|	NOUN
cana-1089	105	4	≤	≤	NUM
cana-1089	105	5	𝐿1	𝐿1	PROPN
cana-1089	105	6	,	,	PUNCT
cana-1089	105	7	|𝑎3	|𝑎3	VERB
cana-1089	105	8	|	|	ADV
cana-1089	105	9	≤	≤	NUM
cana-1089	105	10	𝐿1	𝐿1	ADP
cana-1089	105	11	max	max	PROPN
cana-1089	105	12	{	{	PUNCT
cana-1089	105	13	1	1	NUM
cana-1089	105	14	;	;	PUNCT
cana-1089	105	15	|	|	ADV
cana-1089	105	16	𝐿2	𝐿2	NOUN
cana-1089	105	17	𝐿1	𝐿1	PROPN
cana-1089	105	18	+	+	CCONJ
cana-1089	105	19	𝐿1	𝐿1	PROPN
cana-1089	105	20	2	2	NUM
cana-1089	105	21	|	|	NOUN
cana-1089	105	22	}	}	PUNCT
cana-1089	105	23	and	and	CCONJ
cana-1089	105	24	for	for	ADP
cana-1089	105	25	a	a	DET
cana-1089	105	26	complex	complex	ADJ
cana-1089	105	27	number	number	NOUN
cana-1089	105	28	𝜌	𝜌	NOUN
cana-1089	105	29	,	,	PUNCT
cana-1089	105	30	|𝑎3	|𝑎3	VERB
cana-1089	105	31	−	−	PROPN
cana-1089	105	32	𝜌	𝜌	ADP
cana-1089	105	33	𝑎2	𝑎2	NOUN
cana-1089	105	34	2|	2|	NUM
cana-1089	105	35	≤	≤	PROPN
cana-1089	105	36	𝐿1	𝐿1	ADP
cana-1089	105	37	max	max	PROPN
cana-1089	105	38	{	{	PUNCT
cana-1089	105	39	1	1	NUM
cana-1089	105	40	;	;	PUNCT
cana-1089	105	41	|	|	ADV
cana-1089	105	42	𝐿2	𝐿2	NOUN
cana-1089	105	43	𝐿1	𝐿1	PROPN
cana-1089	105	44	+	+	CCONJ
cana-1089	105	45	𝐿1	𝐿1	PROPN
cana-1089	105	46	2	2	NUM
cana-1089	105	47	(	(	PUNCT
cana-1089	105	48	1	1	NUM
cana-1089	105	49	−	−	PROPN
cana-1089	105	50	2𝜌)|	2𝜌)|	NUM
cana-1089	105	51	}	}	PUNCT
cana-1089	105	52	the	the	DET
cana-1089	105	53	inequality	inequality	NOUN
cana-1089	105	54	is	be	AUX
cana-1089	105	55	sharp	sharp	ADJ
cana-1089	105	56	for	for	SCONJ
cana-1089	105	57	each	each	DET
cana-1089	105	58	𝜌	𝜌	X
cana-1089	105	59	∈	∈	NOUN
cana-1089	105	60	ℂ.	ℂ.	ADJ
cana-1089	105	61	proof	proof	NOUN
cana-1089	105	62	.	.	PUNCT
cana-1089	106	1	by	by	ADP
cana-1089	106	2	the	the	DET
cana-1089	106	3	definition	definition	NOUN
cana-1089	106	4	of	of	ADP
cana-1089	106	5	φk(ω	φk(ω	NUM
cana-1089	106	6	)	)	PUNCT
cana-1089	106	7	,	,	PUNCT
cana-1089	106	8	we	we	PRON
cana-1089	106	9	have	have	VERB
cana-1089	106	10	𝜑𝑘(𝜔	𝜑𝑘(𝜔	NOUN
cana-1089	106	11	)	)	PUNCT
cana-1089	106	12	=	=	SYM
cana-1089	107	1	1	1	NUM
cana-1089	107	2	𝑘	𝑘	X
cana-1089	107	3	∑	∑	PROPN
cana-1089	107	4	𝜑(𝜀𝜈	𝜑(𝜀𝜈	PROPN
cana-1089	107	5	𝜔	𝜔	PROPN
cana-1089	107	6	)	)	PUNCT
cana-1089	107	7	𝜀𝜈	𝜀𝜈	ADP
cana-1089	107	8	𝑘−1	𝑘−1	PROPN
cana-1089	107	9	𝜈=0	𝜈=0	NOUN
cana-1089	107	10	=	=	PUNCT
cana-1089	107	11	𝜔	𝜔	X
cana-1089	107	12	+	+	NOUN
cana-1089	107	13	∑	∑	PUNCT
cana-1089	107	14	γ𝑛,𝑘	γ𝑛,𝑘	ADP
cana-1089	107	15	𝑎𝑛	𝑎𝑛	NUM
cana-1089	107	16	∞	∞	NUM
cana-1089	107	17	𝑛=2	𝑛=2	NOUN
cana-1089	107	18	𝜔𝑛	𝜔𝑛	NOUN
cana-1089	107	19	,	,	PUNCT
cana-1089	107	20	where	where	SCONJ
cana-1089	107	21	γ𝑛,𝑘	γ𝑛,𝑘	PUNCT
cana-1089	107	22	=	=	SYM
cana-1089	108	1	1	1	NUM
cana-1089	108	2	𝑘	𝑘	NOUN
cana-1089	108	3	∑	∑	PROPN
cana-1089	109	1	[	[	X
cana-1089	109	2	exp	exp	X
cana-1089	109	3	(	(	PUNCT
cana-1089	109	4	2𝜋𝑖	2𝜋𝑖	NOUN
cana-1089	109	5	𝑘	𝑘	NOUN
cana-1089	109	6	)	)	PUNCT
cana-1089	109	7	]	]	PUNCT
cana-1089	109	8	(	(	PUNCT
cana-1089	109	9	𝑛−1)𝜈	𝑛−1)𝜈	NUM
cana-1089	109	10	𝑘−1	𝑘−1	PROPN
cana-1089	110	1	𝜈=0	𝜈=0	PROPN
cana-1089	110	2	.	.	PUNCT
cana-1089	111	1	it	it	PRON
cana-1089	111	2	can	can	AUX
cana-1089	111	3	be	be	AUX
cana-1089	111	4	easily	easily	ADV
cana-1089	111	5	seen	see	VERB
cana-1089	111	6	that	that	SCONJ
cana-1089	111	7	γ2,2	γ2,2	PROPN
cana-1089	111	8	=	=	NOUN
cana-1089	111	9	1	1	NUM
cana-1089	111	10	2	2	NUM
cana-1089	111	11	∑[exp(𝜋𝑖)]𝜈	∑[exp(𝜋𝑖)]𝜈	NOUN
cana-1089	111	12	1	1	NUM
cana-1089	111	13	𝜈=0	𝜈=0	NOUN
cana-1089	111	14	=	=	SYM
cana-1089	111	15	0	0	NUM
cana-1089	111	16	,	,	PUNCT
cana-1089	111	17	γ3,2	γ3,2	NOUN
cana-1089	111	18	=	=	NOUN
cana-1089	111	19	1	1	NUM
cana-1089	111	20	2	2	NUM
cana-1089	111	21	∑[exp(𝜋𝑖)]2𝜈	∑[exp(𝜋𝑖)]2𝜈	NOUN
cana-1089	111	22	1	1	NUM
cana-1089	112	1	𝜈=0	𝜈=0	NOUN
cana-1089	112	2	=	=	SYM
cana-1089	112	3	1	1	X
cana-1089	112	4	.	.	X
cana-1089	113	1	substituting	substitute	VERB
cana-1089	113	2	the	the	DET
cana-1089	113	3	above	above	ADJ
cana-1089	113	4	expression	expression	NOUN
cana-1089	113	5	in	in	ADP
cana-1089	113	6	(	(	PUNCT
cana-1089	113	7	2.1	2.1	NUM
cana-1089	113	8	)	)	PUNCT
cana-1089	113	9	,	,	PUNCT
cana-1089	113	10	(	(	PUNCT
cana-1089	113	11	2.2	2.2	NUM
cana-1089	113	12	)	)	PUNCT
cana-1089	113	13	and	and	CCONJ
cana-1089	113	14	(	(	PUNCT
cana-1089	113	15	2.3	2.3	NUM
cana-1089	113	16	)	)	PUNCT
cana-1089	113	17	,	,	PUNCT
cana-1089	113	18	we	we	PRON
cana-1089	113	19	obtain	obtain	VERB
cana-1089	113	20	the	the	DET
cana-1089	113	21	assertion	assertion	NOUN
cana-1089	113	22	of	of	ADP
cana-1089	113	23	the	the	DET
cana-1089	113	24	corollary	corollary	NOUN
cana-1089	113	25	.	.	PUNCT
cana-1089	114	1	fixing	fix	VERB
cana-1089	114	2	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	114	3	)	)	PUNCT
cana-1089	114	4	to	to	PART
cana-1089	114	5	be	be	AUX
cana-1089	114	6	well	well	ADV
cana-1089	114	7	-	-	PUNCT
cana-1089	114	8	known	know	VERB
cana-1089	114	9	conic	conic	ADJ
cana-1089	114	10	regions	region	NOUN
cana-1089	114	11	,	,	PUNCT
cana-1089	114	12	we	we	PRON
cana-1089	114	13	can	can	AUX
cana-1089	114	14	obtain	obtain	VERB
cana-1089	114	15	several	several	ADJ
cana-1089	114	16	applications	application	NOUN
cana-1089	114	17	of	of	ADP
cana-1089	114	18	our	our	PRON
cana-1089	114	19	result	result	NOUN
cana-1089	114	20	.	.	PUNCT
cana-1089	115	1	but	but	CCONJ
cana-1089	115	2	here	here	ADV
cana-1089	115	3	we	we	PRON
cana-1089	115	4	will	will	AUX
cana-1089	115	5	restrict	restrict	VERB
cana-1089	115	6	to	to	ADP
cana-1089	115	7	pointing	point	VERB
cana-1089	115	8	out	out	ADP
cana-1089	115	9	the	the	DET
cana-1089	115	10	case	case	NOUN
cana-1089	115	11	when	when	SCONJ
cana-1089	115	12	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	115	13	)	)	PUNCT
cana-1089	115	14	is	be	AUX
cana-1089	115	15	known	know	VERB
cana-1089	115	16	to	to	PART
cana-1089	115	17	be	be	AUX
cana-1089	115	18	extremal	extremal	ADJ
cana-1089	115	19	.	.	PUNCT
cana-1089	116	1	letting	let	VERB
cana-1089	116	2	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	116	3	)	)	PUNCT
cana-1089	117	1	=	=	PUNCT
cana-1089	118	1	1+𝜔	1+𝜔	NUM
cana-1089	118	2	1−𝜔	1−𝜔	NUM
cana-1089	118	3	in	in	ADP
cana-1089	118	4	corollary	corollary	ADJ
cana-1089	118	5	2.1	2.1	NUM
cana-1089	118	6	,	,	PUNCT
cana-1089	118	7	we	we	PRON
cana-1089	118	8	get	get	VERB
cana-1089	118	9	corollary	corollary	ADJ
cana-1089	118	10	2.2	2.2	NUM
cana-1089	118	11	.	.	PUNCT
cana-1089	119	1	let	let	VERB
cana-1089	119	2	𝜑	𝜑	PRON
cana-1089	119	3	∈	∈	PROPN
cana-1089	119	4	𝒜	𝒜	NOUN
cana-1089	119	5	satisfy	satisfy	VERB
cana-1089	119	6	the	the	DET
cana-1089	119	7	condition	condition	NOUN
cana-1089	120	1	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	120	2	(	(	PUNCT
cana-1089	120	3	2𝜔	2𝜔	NOUN
cana-1089	120	4	𝑒	𝑒	ADP
cana-1089	120	5	𝜔𝜑′(𝜔	𝜔𝜑′(𝜔	PROPN
cana-1089	120	6	)	)	PUNCT
cana-1089	120	7	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	120	8	)	)	PUNCT
cana-1089	120	9	−1	−1	NOUN
cana-1089	120	10	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	120	11	)	)	PUNCT
cana-1089	120	12	−	−	NOUN
cana-1089	120	13	𝜑(−𝜔	𝜑(−𝜔	NOUN
cana-1089	120	14	)	)	PUNCT
cana-1089	120	15	)	)	PUNCT
cana-1089	121	1	>	>	X
cana-1089	121	2	0	0	X
cana-1089	121	3	.	.	PUNCT
cana-1089	122	1	then	then	ADV
cana-1089	122	2	,	,	PUNCT
cana-1089	122	3	|𝑎2|	|𝑎2|	NOUN
cana-1089	122	4	≤	≤	NOUN
cana-1089	122	5	2	2	NUM
cana-1089	122	6	,	,	PUNCT
cana-1089	122	7	|𝑎3|	|𝑎3|	NOUN
cana-1089	122	8	≤	≤	NUM
cana-1089	122	9	4	4	NUM
cana-1089	122	10	communications	communication	NOUN
cana-1089	122	11	on	on	ADP
cana-1089	122	12	applied	apply	VERB
cana-1089	122	13	nonlinear	nonlinear	ADJ
cana-1089	122	14	analysis	analysis	NOUN
cana-1089	122	15	issn	issn	NOUN
cana-1089	122	16	:	:	PUNCT
cana-1089	122	17	1074	1074	NUM
cana-1089	122	18	-	-	PUNCT
cana-1089	122	19	133x	133x	NUM
cana-1089	122	20	vol	vol	NOUN
cana-1089	122	21	31	31	NUM
cana-1089	122	22	no	no	NOUN
cana-1089	122	23	.	.	PUNCT
cana-1089	123	1	5s	5s	NUM
cana-1089	123	2	(	(	PUNCT
cana-1089	123	3	2024	2024	NUM
cana-1089	123	4	)	)	PUNCT
cana-1089	123	5	545	545	NUM
cana-1089	123	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	123	7	and	and	CCONJ
cana-1089	123	8	for	for	ADP
cana-1089	123	9	a	a	DET
cana-1089	123	10	complex	complex	ADJ
cana-1089	123	11	number	number	NOUN
cana-1089	123	12	𝜌	𝜌	NOUN
cana-1089	123	13	,	,	PUNCT
cana-1089	123	14	|𝑎3	|𝑎3	VERB
cana-1089	123	15	−	−	PROPN
cana-1089	123	16	𝜌	𝜌	ADP
cana-1089	123	17	𝑎2	𝑎2	NOUN
cana-1089	123	18	2|	2|	NUM
cana-1089	123	19	≤	≤	PROPN
cana-1089	124	1	2max{1	2max{1	NUM
cana-1089	124	2	;	;	PUNCT
cana-1089	124	3	2|1	2|1	NUM
cana-1089	124	4	−	−	PROPN
cana-1089	124	5	𝜌|	𝜌|	PROPN
cana-1089	124	6	}	}	PUNCT
cana-1089	124	7	theorem	theorem	VERB
cana-1089	124	8	2.2	2.2	NUM
cana-1089	124	9	.	.	PUNCT
cana-1089	125	1	if	if	SCONJ
cana-1089	125	2	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	125	3	)	)	PUNCT
cana-1089	125	4	∈	∈	PROPN
cana-1089	125	5	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	125	6	)	)	PUNCT
cana-1089	125	7	,	,	PUNCT
cana-1089	125	8	then	then	ADV
cana-1089	125	9	we	we	PRON
cana-1089	125	10	have	have	VERB
cana-1089	125	11	|𝑎2|	|𝑎2|	NOUN
cana-1089	125	12	≤	≤	NOUN
cana-1089	125	13	1	1	NUM
cana-1089	125	14	|γ2,𝑘|	|γ2,𝑘|	NOUN
cana-1089	125	15	|𝐿1	|𝐿1	PUNCT
cana-1089	126	1	+	+	X
cana-1089	126	2	1|	1|	NUM
cana-1089	126	3	.	.	PUNCT
cana-1089	127	1	(	(	PUNCT
cana-1089	127	2	2.9	2.9	NUM
cana-1089	127	3	)	)	PUNCT
cana-1089	127	4	|𝑎3|	|𝑎3|	NOUN
cana-1089	127	5	≤	≤	X
cana-1089	127	6	𝐿1	𝐿1	PROPN
cana-1089	127	7	|γ3,𝑘|	|γ3,𝑘|	PRON
cana-1089	127	8	{	{	PUNCT
cana-1089	127	9	max	max	PROPN
cana-1089	127	10	{	{	PUNCT
cana-1089	127	11	1	1	NUM
cana-1089	127	12	,	,	PUNCT
cana-1089	127	13	|	|	ADV
cana-1089	127	14	𝐿2	𝐿2	NOUN
cana-1089	127	15	𝐿1	𝐿1	VERB
cana-1089	127	16	−	−	PROPN
cana-1089	127	17	𝐿1|	𝐿1|	PROPN
cana-1089	127	18	}	}	PUNCT
cana-1089	127	19	+	+	CCONJ
cana-1089	128	1	|	|	ADV
cana-1089	128	2	1	1	X
cana-1089	128	3	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	129	1	+	+	NUM
cana-1089	129	2	1|	1|	NUM
cana-1089	130	1	+	+	CCONJ
cana-1089	130	2	3	3	NUM
cana-1089	130	3	2|𝐿1|	2|𝐿1|	NUM
cana-1089	130	4	}	}	PUNCT
cana-1089	130	5	(	(	PUNCT
cana-1089	130	6	2.10	2.10	NUM
cana-1089	130	7	)	)	PUNCT
cana-1089	130	8	and	and	CCONJ
cana-1089	130	9	for	for	ADP
cana-1089	130	10	all	all	PRON
cana-1089	130	11	𝜌	𝜌	ADP
cana-1089	130	12	∈	∈	PROPN
cana-1089	130	13	ℂ	ℂ	PROPN
cana-1089	130	14	|𝑎3	|𝑎3	VERB
cana-1089	130	15	−	−	PROPN
cana-1089	130	16	𝜌	𝜌	PART
cana-1089	130	17	𝑎2	𝑎2	NOUN
cana-1089	130	18	2	2	NUM
cana-1089	130	19	|	|	NOUN
cana-1089	130	20	≤	≤	X
cana-1089	130	21	𝐿1	𝐿1	PROPN
cana-1089	131	1	|γ3,𝑘|	|γ3,𝑘|	PRON
cana-1089	131	2	[	[	X
cana-1089	131	3	max	max	X
cana-1089	131	4	{	{	PUNCT
cana-1089	131	5	1	1	NUM
cana-1089	131	6	,	,	PUNCT
cana-1089	131	7	|	|	ADV
cana-1089	131	8	𝐿2	𝐿2	NOUN
cana-1089	131	9	𝐿1	𝐿1	VERB
cana-1089	131	10	−	−	PROPN
cana-1089	131	11	𝐿1	𝐿1	PROPN
cana-1089	132	1	+	+	CCONJ
cana-1089	132	2	𝜌𝐿1γ3,𝑘	𝜌𝐿1γ3,𝑘	NOUN
cana-1089	132	3	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	132	4	2	2	NUM
cana-1089	132	5	|	|	NOUN
cana-1089	132	6	}	}	PUNCT
cana-1089	133	1	+	+	CCONJ
cana-1089	133	2	|	|	ADV
cana-1089	133	3	1	1	X
cana-1089	133	4	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	133	5	+	+	CCONJ
cana-1089	133	6	1	1	NUM
cana-1089	133	7	−	−	PROPN
cana-1089	133	8	2𝜌γ3,𝑘	2𝜌γ3,𝑘	PROPN
cana-1089	133	9	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	133	10	2	2	NUM
cana-1089	134	1	|	|	ADV
cana-1089	134	2	+	+	CCONJ
cana-1089	134	3	1	1	NUM
cana-1089	134	4	2|𝐿1|	2|𝐿1|	NUM
cana-1089	134	5	|3	|3	NUM
cana-1089	134	6	−	−	PROPN
cana-1089	134	7	−	−	PROPN
cana-1089	134	8	2𝜌γ3,𝑘	2𝜌γ3,𝑘	PROPN
cana-1089	134	9	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	134	10	2	2	NUM
cana-1089	134	11	|	|	NOUN
cana-1089	134	12	]	]	X
cana-1089	134	13	(	(	PUNCT
cana-1089	134	14	2.11	2.11	NUM
cana-1089	134	15	)	)	PUNCT
cana-1089	134	16	the	the	DET
cana-1089	134	17	inequality	inequality	NOUN
cana-1089	134	18	is	be	AUX
cana-1089	134	19	sharp	sharp	ADJ
cana-1089	134	20	for	for	SCONJ
cana-1089	134	21	each	each	DET
cana-1089	134	22	𝜌	𝜌	X
cana-1089	134	23	∈	∈	NOUN
cana-1089	134	24	ℂ.	ℂ.	ADJ
cana-1089	134	25	proof	proof	NOUN
cana-1089	134	26	.	.	PUNCT
cana-1089	135	1	expanding	expand	VERB
cana-1089	135	2	the	the	DET
cana-1089	135	3	left	left	ADJ
cana-1089	135	4	hand	hand	NOUN
cana-1089	135	5	side	side	NOUN
cana-1089	135	6	of	of	ADP
cana-1089	135	7	(	(	PUNCT
cana-1089	135	8	1.6	1.6	NUM
cana-1089	135	9	)	)	PUNCT
cana-1089	135	10	and	and	CCONJ
cana-1089	135	11	simplifying	simplify	VERB
cana-1089	135	12	the	the	DET
cana-1089	135	13	expansion	expansion	NOUN
cana-1089	135	14	we	we	PRON
cana-1089	135	15	get	get	VERB
cana-1089	135	16	𝜔	𝜔	PRON
cana-1089	135	17	𝑒	𝑒	NOUN
cana-1089	135	18	𝜔2𝜑′(𝜔	𝜔2𝜑′(𝜔	NOUN
cana-1089	135	19	)	)	PUNCT
cana-1089	135	20	𝜑(𝜔	𝜑(𝜔	PROPN
cana-1089	135	21	)	)	PUNCT
cana-1089	135	22	𝜑𝑘(𝜔	𝜑𝑘(𝜔	PUNCT
cana-1089	135	23	)	)	PUNCT
cana-1089	135	24	=	=	SYM
cana-1089	135	25	1	1	NUM
cana-1089	135	26	+	+	CCONJ
cana-1089	135	27	(	(	PUNCT
cana-1089	135	28	1	1	NUM
cana-1089	135	29	−	−	NOUN
cana-1089	135	30	𝑎2γ2,𝑘)𝜔	𝑎2γ2,𝑘)𝜔	NOUN
cana-1089	135	31	+	+	CCONJ
cana-1089	135	32	(	(	PUNCT
cana-1089	135	33	1	1	NUM
cana-1089	135	34	2	2	NUM
cana-1089	135	35	+	+	NOUN
cana-1089	135	36	𝑎2	𝑎2	NOUN
cana-1089	135	37	−	−	PROPN
cana-1089	135	38	𝑎2γ2,𝑘	𝑎2γ2,𝑘	PROPN
cana-1089	135	39	+	+	CCONJ
cana-1089	135	40	𝑎2	𝑎2	PROPN
cana-1089	135	41	2γ2,𝑘	2γ2,𝑘	NUM
cana-1089	135	42	2	2	NUM
cana-1089	135	43	−	−	PROPN
cana-1089	135	44	𝑎3γ3,𝑘)𝜔2	𝑎3γ3,𝑘)𝜔2	PROPN
cana-1089	135	45	+	+	NUM
cana-1089	135	46	⋯.	⋯.	PROPN
cana-1089	135	47	(	(	PUNCT
cana-1089	135	48	2.12	2.12	NUM
cana-1089	135	49	)	)	PUNCT
cana-1089	135	50	given	give	VERB
cana-1089	135	51	𝜑	𝜑	PRON
cana-1089	135	52	∈	∈	PROPN
cana-1089	135	53	ℒ𝑘	ℒ𝑘	PROPN
cana-1089	135	54	(	(	PUNCT
cana-1089	135	55	𝜒	𝜒	NOUN
cana-1089	135	56	)	)	PUNCT
cana-1089	135	57	,	,	PUNCT
cana-1089	135	58	so	so	CCONJ
cana-1089	135	59	the	the	DET
cana-1089	135	60	right	right	ADJ
cana-1089	135	61	hand	hand	NOUN
cana-1089	135	62	side	side	NOUN
cana-1089	135	63	of	of	ADP
cana-1089	135	64	the	the	DET
cana-1089	135	65	expansion	expansion	NOUN
cana-1089	135	66	of	of	ADP
cana-1089	135	67	(	(	PUNCT
cana-1089	135	68	1.6	1.6	NUM
cana-1089	135	69	)	)	PUNCT
cana-1089	135	70	is	be	AUX
cana-1089	135	71	the	the	DET
cana-1089	135	72	same	same	ADJ
cana-1089	135	73	as	as	ADP
cana-1089	135	74	(	(	PUNCT
cana-1089	135	75	2.5	2.5	NUM
cana-1089	135	76	)	)	PUNCT
cana-1089	135	77	.	.	PUNCT
cana-1089	136	1	from	from	ADP
cana-1089	136	2	(	(	PUNCT
cana-1089	136	3	2.12	2.12	NUM
cana-1089	136	4	)	)	PUNCT
cana-1089	136	5	and	and	CCONJ
cana-1089	136	6	(	(	PUNCT
cana-1089	136	7	2.5	2.5	NUM
cana-1089	136	8	)	)	PUNCT
cana-1089	136	9	,	,	PUNCT
cana-1089	136	10	we	we	PRON
cana-1089	136	11	obtain	obtain	VERB
cana-1089	136	12	𝑎2	𝑎2	NOUN
cana-1089	136	13	=	=	PUNCT
cana-1089	137	1	−	−	PROPN
cana-1089	137	2	1	1	NUM
cana-1089	137	3	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	137	4	[	[	PUNCT
cana-1089	137	5	𝜗1𝐿1	𝜗1𝐿1	NUM
cana-1089	137	6	2	2	NUM
cana-1089	137	7	−	−	NOUN
cana-1089	137	8	1	1	NUM
cana-1089	137	9	]	]	PUNCT
cana-1089	137	10	(	(	PUNCT
cana-1089	137	11	2.13	2.13	NUM
cana-1089	137	12	)	)	PUNCT
cana-1089	137	13	and	and	CCONJ
cana-1089	137	14	𝑎3	𝑎3	PROPN
cana-1089	137	15	=	=	PUNCT
cana-1089	137	16	−	−	PROPN
cana-1089	137	17	1	1	NUM
cana-1089	137	18	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	137	19	{	{	PUNCT
cana-1089	137	20	𝐿1	𝐿1	PROPN
cana-1089	137	21	2	2	NUM
cana-1089	137	22	[	[	NOUN
cana-1089	137	23	𝜗2	𝜗2	X
cana-1089	137	24	−	−	NOUN
cana-1089	137	25	𝜗1	𝜗1	X
cana-1089	137	26	2	2	NUM
cana-1089	137	27	2	2	NUM
cana-1089	137	28	(	(	PUNCT
cana-1089	137	29	1	1	NUM
cana-1089	137	30	−	−	NOUN
cana-1089	137	31	𝐿2	𝐿2	PROPN
cana-1089	137	32	𝐿1	𝐿1	PROPN
cana-1089	137	33	+	+	CCONJ
cana-1089	137	34	𝐿1	𝐿1	PROPN
cana-1089	137	35	)	)	PUNCT
cana-1089	137	36	]	]	PUNCT
cana-1089	138	1	+	+	CCONJ
cana-1089	138	2	𝜗1𝐿1	𝜗1𝐿1	NUM
cana-1089	138	3	2	2	NUM
cana-1089	138	4	[	[	PUNCT
cana-1089	138	5	1	1	NUM
cana-1089	138	6	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	138	7	+	+	CCONJ
cana-1089	138	8	1	1	X
cana-1089	138	9	]	]	SYM
cana-1089	138	10	−	−	PROPN
cana-1089	138	11	3	3	NUM
cana-1089	138	12	2	2	NUM
cana-1089	138	13	}	}	PUNCT
cana-1089	138	14	.	.	PUNCT
cana-1089	139	1	(	(	PUNCT
cana-1089	139	2	2.14	2.14	NUM
cana-1089	139	3	)	)	PUNCT
cana-1089	139	4	equations	equation	NOUN
cana-1089	139	5	(	(	PUNCT
cana-1089	139	6	2.9	2.9	NUM
cana-1089	139	7	)	)	PUNCT
cana-1089	139	8	can	can	AUX
cana-1089	139	9	be	be	AUX
cana-1089	139	10	obtained	obtain	VERB
cana-1089	139	11	by	by	ADP
cana-1089	139	12	applying	apply	VERB
cana-1089	139	13	the	the	DET
cana-1089	139	14	well	well	ADV
cana-1089	139	15	-	-	PUNCT
cana-1089	139	16	known	know	VERB
cana-1089	139	17	result	result	NOUN
cana-1089	139	18	of	of	ADP
cana-1089	139	19	|𝜗1|	|𝜗1|	NOUN
cana-1089	139	20	≤	≤	ADJ
cana-1089	139	21	2	2	NUM
cana-1089	139	22	in	in	ADP
cana-1089	139	23	(	(	PUNCT
cana-1089	139	24	2.13	2.13	NUM
cana-1089	139	25	)	)	PUNCT
cana-1089	139	26	applying	apply	VERB
cana-1089	139	27	lemma	lemma	PROPN
cana-1089	139	28	2.1	2.1	NUM
cana-1089	139	29	in	in	ADP
cana-1089	139	30	(	(	PUNCT
cana-1089	139	31	2.14	2.14	NUM
cana-1089	139	32	)	)	PUNCT
cana-1089	139	33	,	,	PUNCT
cana-1089	139	34	we	we	PRON
cana-1089	139	35	get	get	VERB
cana-1089	139	36	(	(	PUNCT
cana-1089	139	37	2.10	2.10	NUM
cana-1089	139	38	)	)	PUNCT
cana-1089	139	39	.	.	PUNCT
cana-1089	140	1	now	now	ADV
cana-1089	140	2	to	to	PART
cana-1089	140	3	prove	prove	VERB
cana-1089	140	4	the	the	DET
cana-1089	140	5	fekete	fekete	NOUN
cana-1089	140	6	-	-	PUNCT
cana-1089	140	7	szegő	szegő	ADJ
cana-1089	140	8	inequality	inequality	NOUN
cana-1089	140	9	for	for	ADP
cana-1089	140	10	the	the	DET
cana-1089	140	11	class	class	NOUN
cana-1089	140	12	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	140	13	)	)	PUNCT
cana-1089	140	14	,	,	PUNCT
cana-1089	140	15	we	we	PRON
cana-1089	140	16	consider	consider	VERB
cana-1089	140	17	|𝑎3	|𝑎3	VERB
cana-1089	140	18	−	−	PROPN
cana-1089	140	19	𝜌	𝜌	ADP
cana-1089	140	20	𝑎2	𝑎2	NOUN
cana-1089	140	21	2|	2|	NUM
cana-1089	141	1	=	=	PUNCT
cana-1089	142	1	|	|	ADV
cana-1089	142	2	1	1	NUM
cana-1089	142	3	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	142	4	{	{	PUNCT
cana-1089	142	5	𝐿1	𝐿1	PROPN
cana-1089	142	6	2	2	NUM
cana-1089	142	7	[	[	NOUN
cana-1089	142	8	𝜗2	𝜗2	X
cana-1089	142	9	−	−	NOUN
cana-1089	142	10	𝜗1	𝜗1	X
cana-1089	142	11	2	2	NUM
cana-1089	142	12	2	2	NUM
cana-1089	142	13	(	(	PUNCT
cana-1089	142	14	1	1	NUM
cana-1089	142	15	−	−	NOUN
cana-1089	142	16	𝐿2	𝐿2	PROPN
cana-1089	142	17	𝐿1	𝐿1	PROPN
cana-1089	142	18	+	+	CCONJ
cana-1089	142	19	𝐿1	𝐿1	PROPN
cana-1089	142	20	)	)	PUNCT
cana-1089	142	21	]	]	PUNCT
cana-1089	143	1	+	+	CCONJ
cana-1089	143	2	𝜗1𝐿1	𝜗1𝐿1	NUM
cana-1089	143	3	2	2	NUM
cana-1089	143	4	[	[	PUNCT
cana-1089	143	5	1	1	NUM
cana-1089	143	6	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	143	7	+	+	CCONJ
cana-1089	143	8	1	1	X
cana-1089	143	9	]	]	SYM
cana-1089	143	10	−	−	PROPN
cana-1089	143	11	3	3	NUM
cana-1089	143	12	2	2	NUM
cana-1089	143	13	}	}	PUNCT
cana-1089	143	14	+	+	CCONJ
cana-1089	143	15	𝜌	𝜌	X
cana-1089	143	16	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	143	17	2	2	NUM
cana-1089	143	18	[	[	PUNCT
cana-1089	143	19	𝜗1	𝜗1	X
cana-1089	143	20	2	2	NUM
cana-1089	143	21	𝐿1	𝐿1	NOUN
cana-1089	143	22	2	2	NUM
cana-1089	143	23	4	4	NUM
cana-1089	143	24	−	−	NOUN
cana-1089	143	25	𝜗1𝐿1	𝜗1𝐿1	NUM
cana-1089	143	26	+	+	NUM
cana-1089	143	27	1]|	1]|	NUM
cana-1089	143	28	.	.	PUNCT
cana-1089	144	1	using	use	VERB
cana-1089	144	2	the	the	DET
cana-1089	144	3	triangle	triangle	NOUN
cana-1089	144	4	inequality	inequality	NOUN
cana-1089	144	5	and	and	CCONJ
cana-1089	144	6	lemma	lemma	PROPN
cana-1089	144	7	2.1	2.1	NUM
cana-1089	144	8	in	in	ADP
cana-1089	144	9	the	the	DET
cana-1089	144	10	above	above	ADJ
cana-1089	144	11	equality	equality	NOUN
cana-1089	144	12	,	,	PUNCT
cana-1089	144	13	we	we	PRON
cana-1089	144	14	can	can	AUX
cana-1089	144	15	obtain	obtain	VERB
cana-1089	144	16	(	(	PUNCT
cana-1089	144	17	2.11	2.11	NUM
cana-1089	144	18	)	)	PUNCT
cana-1089	144	19	.	.	PUNCT
cana-1089	145	1	communications	communication	NOUN
cana-1089	145	2	on	on	ADP
cana-1089	145	3	applied	apply	VERB
cana-1089	145	4	nonlinear	nonlinear	ADJ
cana-1089	145	5	analysis	analysis	NOUN
cana-1089	145	6	issn	issn	NOUN
cana-1089	145	7	:	:	PUNCT
cana-1089	145	8	1074	1074	NUM
cana-1089	145	9	-	-	PUNCT
cana-1089	145	10	133x	133x	NUM
cana-1089	145	11	vol	vol	NOUN
cana-1089	145	12	31	31	NUM
cana-1089	145	13	no	no	NOUN
cana-1089	145	14	.	.	PUNCT
cana-1089	146	1	5s	5s	NUM
cana-1089	146	2	(	(	PUNCT
cana-1089	146	3	2024	2024	NUM
cana-1089	146	4	)	)	PUNCT
cana-1089	146	5	546	546	NUM
cana-1089	146	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	146	7	corollary	corollary	ADJ
cana-1089	146	8	2.3	2.3	NUM
cana-1089	146	9	.	.	PUNCT
cana-1089	147	1	[	[	X
cana-1089	147	2	10	10	NUM
cana-1089	147	3	]	]	X
cana-1089	147	4	if	if	SCONJ
cana-1089	147	5	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	147	6	)	)	PUNCT
cana-1089	147	7	∈	∈	PROPN
cana-1089	147	8	ℒ1(𝜒	ℒ1(𝜒	PROPN
cana-1089	147	9	)	)	PUNCT
cana-1089	147	10	,	,	PUNCT
cana-1089	147	11	then	then	ADV
cana-1089	147	12	we	we	PRON
cana-1089	147	13	have	have	VERB
cana-1089	147	14	|𝑎2|	|𝑎2|	NOUN
cana-1089	147	15	≤	≤	NOUN
cana-1089	147	16	1	1	NUM
cana-1089	147	17	+	+	CCONJ
cana-1089	147	18	𝐿1	𝐿1	PROPN
cana-1089	147	19	.	.	PUNCT
cana-1089	148	1	|𝑎3|	|𝑎3|	PROPN
cana-1089	148	2	≤	≤	PUNCT
cana-1089	148	3	𝐿1	𝐿1	VERB
cana-1089	149	1	[	[	X
cana-1089	149	2	max	max	X
cana-1089	149	3	{	{	PUNCT
cana-1089	149	4	1	1	NUM
cana-1089	149	5	,	,	PUNCT
cana-1089	149	6	|	|	ADV
cana-1089	149	7	𝐿2	𝐿2	NOUN
cana-1089	149	8	𝐿1	𝐿1	VERB
cana-1089	149	9	−	−	PROPN
cana-1089	149	10	𝐿1|	𝐿1|	PROPN
cana-1089	149	11	}	}	PUNCT
cana-1089	149	12	+	+	CCONJ
cana-1089	150	1	3	3	NUM
cana-1089	150	2	2|𝐿1|	2|𝐿1|	NUM
cana-1089	150	3	+	+	CCONJ
cana-1089	150	4	2	2	NUM
cana-1089	150	5	]	]	PUNCT
cana-1089	150	6	and	and	CCONJ
cana-1089	150	7	for	for	ADP
cana-1089	150	8	all	all	DET
cana-1089	150	9	𝜌	𝜌	ADP
cana-1089	150	10	∈	∈	PROPN
cana-1089	150	11	ℂ	ℂ	PROPN
cana-1089	150	12	|𝑎3	|𝑎3	VERB
cana-1089	150	13	−	−	PROPN
cana-1089	150	14	𝜌	𝜌	ADP
cana-1089	150	15	𝑎2	𝑎2	NOUN
cana-1089	150	16	2|	2|	NUM
cana-1089	150	17	≤	≤	NOUN
cana-1089	150	18	𝐿1	𝐿1	VERB
cana-1089	151	1	[	[	X
cana-1089	151	2	max	max	X
cana-1089	151	3	{	{	PUNCT
cana-1089	151	4	1	1	NUM
cana-1089	151	5	,	,	PUNCT
cana-1089	151	6	|	|	ADV
cana-1089	151	7	𝐿2	𝐿2	NOUN
cana-1089	151	8	𝐿1	𝐿1	VERB
cana-1089	151	9	−	−	ADP
cana-1089	151	10	𝐿1(1	𝐿1(1	ADJ
cana-1089	151	11	−	−	PROPN
cana-1089	151	12	𝜌)|	𝜌)|	NOUN
cana-1089	151	13	}	}	PUNCT
cana-1089	151	14	+	+	CCONJ
cana-1089	151	15	1	1	NUM
cana-1089	151	16	2|𝐿1|	2|𝐿1|	NUM
cana-1089	151	17	|3	|3	NUM
cana-1089	152	1	−	−	PROPN
cana-1089	153	1	2𝜌|	2𝜌|	NUM
cana-1089	154	1	+	+	NUM
cana-1089	154	2	2|1	2|1	NUM
cana-1089	154	3	−	−	NOUN
cana-1089	154	4	𝜌|	𝜌|	PROPN
cana-1089	154	5	]	]	PUNCT
cana-1089	154	6	.	.	PUNCT
cana-1089	155	1	the	the	DET
cana-1089	155	2	inequality	inequality	NOUN
cana-1089	155	3	is	be	AUX
cana-1089	155	4	sharp	sharp	ADJ
cana-1089	155	5	for	for	SCONJ
cana-1089	155	6	each	each	PRON
cana-1089	155	7	𝜌	𝜌	X
cana-1089	155	8	∈	∈	PROPN
cana-1089	155	9	ℂ.	ℂ.	NOUN
cana-1089	155	10	letting	let	VERB
cana-1089	155	11	𝜒(𝜔	𝜒(𝜔	NOUN
cana-1089	155	12	)	)	PUNCT
cana-1089	156	1	=	=	PUNCT
cana-1089	157	1	1+𝜔	1+𝜔	NUM
cana-1089	157	2	1−𝜔	1−𝜔	NUM
cana-1089	157	3	in	in	ADP
cana-1089	157	4	corollary	corollary	ADJ
cana-1089	157	5	2.3	2.3	NUM
cana-1089	157	6	,	,	PUNCT
cana-1089	157	7	we	we	PRON
cana-1089	157	8	get	get	VERB
cana-1089	157	9	corollary	corollary	ADJ
cana-1089	157	10	2.4	2.4	NUM
cana-1089	157	11	let	let	VERB
cana-1089	157	12	𝜑	𝜑	PRON
cana-1089	157	13	∈	∈	PROPN
cana-1089	157	14	𝒜	𝒜	NOUN
cana-1089	157	15	satisfy	satisfy	VERB
cana-1089	157	16	the	the	DET
cana-1089	157	17	condition	condition	NOUN
cana-1089	157	18	𝑅𝑒	𝑅𝑒	PROPN
cana-1089	157	19	(	(	PUNCT
cana-1089	157	20	𝜔𝑒	𝜔𝑒	X
cana-1089	157	21	𝜔2𝜑′(𝜔	𝜔2𝜑′(𝜔	NOUN
cana-1089	157	22	)	)	PUNCT
cana-1089	157	23	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	157	24	)	)	PUNCT
cana-1089	157	25	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	157	26	)	)	PUNCT
cana-1089	157	27	)	)	PUNCT
cana-1089	157	28	>	>	X
cana-1089	158	1	0	0	X
cana-1089	158	2	.	.	PUNCT
cana-1089	159	1	then	then	ADV
cana-1089	159	2	,	,	PUNCT
cana-1089	159	3	|𝑎2|	|𝑎2|	NOUN
cana-1089	159	4	≤	≤	NOUN
cana-1089	159	5	3	3	NUM
cana-1089	159	6	,	,	PUNCT
cana-1089	159	7	|𝑎3|	|𝑎3|	NOUN
cana-1089	159	8	≤	≤	ADV
cana-1089	159	9	15	15	NUM
cana-1089	159	10	2	2	NUM
cana-1089	159	11	and	and	CCONJ
cana-1089	159	12	for	for	ADP
cana-1089	159	13	a	a	DET
cana-1089	159	14	complex	complex	ADJ
cana-1089	159	15	number	number	NOUN
cana-1089	159	16	𝜌	𝜌	NOUN
cana-1089	159	17	,	,	PUNCT
cana-1089	159	18	|𝑎3	|𝑎3	VERB
cana-1089	159	19	−	−	PROPN
cana-1089	159	20	𝜌	𝜌	ADP
cana-1089	159	21	𝑎2	𝑎2	NOUN
cana-1089	159	22	2|	2|	NUM
cana-1089	159	23	≤	≤	ADV
cana-1089	159	24	2	2	NUM
cana-1089	159	25	[	[	X
cana-1089	159	26	max{1	max{1	NOUN
cana-1089	159	27	,	,	PUNCT
cana-1089	159	28	|2𝜌	|2𝜌	VERB
cana-1089	159	29	−	−	PROPN
cana-1089	159	30	1|	1|	NUM
cana-1089	159	31	}	}	PUNCT
cana-1089	159	32	+	+	NUM
cana-1089	159	33	2|1	2|1	NUM
cana-1089	159	34	−	−	NOUN
cana-1089	159	35	𝜌|	𝜌|	NOUN
cana-1089	159	36	+	+	CCONJ
cana-1089	159	37	1	1	NUM
cana-1089	159	38	4	4	NUM
cana-1089	159	39	|3	|3	NUM
cana-1089	159	40	−	−	PROPN
cana-1089	159	41	2𝜌|	2𝜌|	NUM
cana-1089	159	42	]	]	PUNCT
cana-1089	159	43	.	.	PUNCT
cana-1089	160	1	the	the	DET
cana-1089	160	2	inequality	inequality	NOUN
cana-1089	160	3	is	be	AUX
cana-1089	160	4	sharp	sharp	ADJ
cana-1089	160	5	for	for	SCONJ
cana-1089	160	6	each	each	PRON
cana-1089	160	7	𝜌	𝜌	X
cana-1089	160	8	∈	∈	PROPN
cana-1089	160	9	ℂ.	ℂ.	NOUN
cana-1089	160	10	3	3	NUM
cana-1089	160	11	.	.	PUNCT
cana-1089	161	1	coefficient	coefficient	NOUN
cana-1089	161	2	estimates	estimate	NOUN
cana-1089	161	3	for	for	ADP
cana-1089	161	4	the	the	DET
cana-1089	161	5	inverse	inverse	NOUN
cana-1089	161	6	functions	function	NOUN
cana-1089	161	7	.	.	PUNCT
cana-1089	162	1	in	in	ADP
cana-1089	162	2	this	this	DET
cana-1089	162	3	section	section	NOUN
cana-1089	162	4	,	,	PUNCT
cana-1089	162	5	we	we	PRON
cana-1089	162	6	will	will	AUX
cana-1089	162	7	find	find	VERB
cana-1089	162	8	the	the	DET
cana-1089	162	9	coefficient	coefficient	NOUN
cana-1089	162	10	estimates	estimate	NOUN
cana-1089	162	11	for	for	ADP
cana-1089	162	12	the	the	DET
cana-1089	162	13	inverse	inverse	NOUN
cana-1089	162	14	functions	function	NOUN
cana-1089	162	15	of	of	ADP
cana-1089	162	16	𝜑	𝜑	NOUN
cana-1089	162	17	belonging	belong	VERB
cana-1089	162	18	to	to	ADP
cana-1089	162	19	the	the	DET
cana-1089	162	20	classes	class	NOUN
cana-1089	162	21	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	162	22	)	)	PUNCT
cana-1089	162	23	and	and	CCONJ
cana-1089	162	24	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	162	25	)	)	PUNCT
cana-1089	162	26	.	.	PUNCT
cana-1089	163	1	refer	refer	VERB
cana-1089	163	2	to	to	ADP
cana-1089	163	3	[	[	PUNCT
cana-1089	163	4	14	14	NUM
cana-1089	163	5	18	18	NUM
cana-1089	163	6	]	]	PUNCT
cana-1089	163	7	for	for	ADP
cana-1089	163	8	its	its	PRON
cana-1089	163	9	relevance	relevance	NOUN
cana-1089	163	10	and	and	CCONJ
cana-1089	163	11	application	application	NOUN
cana-1089	163	12	in	in	ADP
cana-1089	163	13	the	the	DET
cana-1089	163	14	field	field	NOUN
cana-1089	163	15	of	of	ADP
cana-1089	163	16	univalent	univalent	ADJ
cana-1089	163	17	function	function	NOUN
cana-1089	163	18	theory	theory	NOUN
cana-1089	163	19	.	.	PUNCT
cana-1089	164	1	the	the	DET
cana-1089	164	2	following	following	ADJ
cana-1089	164	3	result	result	NOUN
cana-1089	164	4	would	would	AUX
cana-1089	164	5	help	help	VERB
cana-1089	164	6	us	we	PRON
cana-1089	164	7	to	to	PART
cana-1089	164	8	obtain	obtain	VERB
cana-1089	164	9	the	the	DET
cana-1089	164	10	coefficient	coefficient	NOUN
cana-1089	164	11	estimates	estimate	NOUN
cana-1089	164	12	for	for	ADP
cana-1089	164	13	𝜑−1	𝜑−1	PROPN
cana-1089	164	14	(	(	PUNCT
cana-1089	164	15	provided	provide	VERB
cana-1089	164	16	it	it	PRON
cana-1089	164	17	exists	exist	VERB
cana-1089	164	18	)	)	PUNCT
cana-1089	164	19	,	,	PUNCT
cana-1089	164	20	form	form	VERB
cana-1089	164	21	the	the	DET
cana-1089	164	22	coefficient	coefficient	NOUN
cana-1089	164	23	estimates	estimate	NOUN
cana-1089	164	24	of	of	ADP
cana-1089	164	25	𝜑.	𝜑.	PROPN
cana-1089	164	26	lemma	lemma	PROPN
cana-1089	164	27	3.1	3.1	NUM
cana-1089	164	28	.	.	PUNCT
cana-1089	165	1	[	[	X
cana-1089	165	2	9	9	NUM
cana-1089	165	3	,	,	PUNCT
cana-1089	165	4	p.	p.	NOUN
cana-1089	165	5	56	56	NUM
cana-1089	165	6	]	]	X
cana-1089	165	7	if	if	SCONJ
cana-1089	165	8	the	the	DET
cana-1089	165	9	function	function	NOUN
cana-1089	165	10	𝜑	𝜑	X
cana-1089	165	11	∈	∈	PROPN
cana-1089	165	12	𝒜	𝒜	PROPN
cana-1089	165	13	and	and	CCONJ
cana-1089	165	14	𝜑−1	𝜑−1	PROPN
cana-1089	165	15	=	=	PROPN
cana-1089	165	16	𝑔(𝑤	𝑔(𝑤	PROPN
cana-1089	165	17	)	)	PUNCT
cana-1089	165	18	given	give	VERB
cana-1089	165	19	by	by	ADP
cana-1089	165	20	𝑔(𝑤	𝑔(𝑤	NOUN
cana-1089	165	21	)	)	PUNCT
cana-1089	165	22	=	=	PUNCT
cana-1089	166	1	𝑤	𝑤	ADP
cana-1089	166	2	+	+	NOUN
cana-1089	166	3	∑	∑	PROPN
cana-1089	166	4	𝑏𝑘	𝑏𝑘	NUM
cana-1089	166	5	∞	∞	NUM
cana-1089	166	6	𝑘=2	𝑘=2	INTJ
cana-1089	166	7	𝑤𝑘	𝑤𝑘	INTJ
cana-1089	166	8	(	(	PUNCT
cana-1089	166	9	3.1	3.1	NUM
cana-1089	166	10	)	)	PUNCT
cana-1089	166	11	are	be	AUX
cana-1089	166	12	inverse	inverse	NOUN
cana-1089	166	13	functions	function	NOUN
cana-1089	166	14	,	,	PUNCT
cana-1089	166	15	then	then	ADV
cana-1089	166	16	for	for	ADP
cana-1089	166	17	𝑘	𝑘	PRON
cana-1089	166	18	≥	≥	NUM
cana-1089	166	19	2	2	NUM
cana-1089	166	20	communications	communication	NOUN
cana-1089	166	21	on	on	ADP
cana-1089	166	22	applied	apply	VERB
cana-1089	166	23	nonlinear	nonlinear	ADJ
cana-1089	166	24	analysis	analysis	NOUN
cana-1089	166	25	issn	issn	NOUN
cana-1089	166	26	:	:	PUNCT
cana-1089	166	27	1074	1074	NUM
cana-1089	166	28	-	-	PUNCT
cana-1089	166	29	133x	133x	NUM
cana-1089	166	30	vol	vol	NOUN
cana-1089	166	31	31	31	NUM
cana-1089	166	32	no	no	NOUN
cana-1089	166	33	.	.	PUNCT
cana-1089	167	1	5s	5s	NUM
cana-1089	167	2	(	(	PUNCT
cana-1089	167	3	2024	2024	NUM
cana-1089	167	4	)	)	PUNCT
cana-1089	167	5	547	547	NUM
cana-1089	167	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	167	7	𝑏𝑘	𝑏𝑘	NOUN
cana-1089	167	8	=	=	SYM
cana-1089	167	9	(	(	PUNCT
cana-1089	167	10	−1)𝑘+1	−1)𝑘+1	NOUN
cana-1089	167	11	𝑘	𝑘	X
cana-1089	167	12	!	!	PUNCT
cana-1089	168	1	[	[	PUNCT
cana-1089	168	2	𝑘𝑎2	𝑘𝑎2	NOUN
cana-1089	168	3	1	1	NUM
cana-1089	168	4	0	0	NUM
cana-1089	168	5	⋯	⋯	ADP
cana-1089	168	6	0	0	NUM
cana-1089	168	7	2𝑘𝑎3	2𝑘𝑎3	NUM
cana-1089	168	8	(	(	PUNCT
cana-1089	168	9	𝑘	𝑘	X
cana-1089	168	10	+	+	X
cana-1089	168	11	1)𝑎2	1)𝑎2	NUM
cana-1089	168	12	2	2	NUM
cana-1089	168	13	⋯	⋯	ADP
cana-1089	168	14	0	0	NUM
cana-1089	168	15	3𝑘𝑎4	3𝑘𝑎4	NUM
cana-1089	168	16	⋮	⋮	NOUN
cana-1089	168	17	(	(	PUNCT
cana-1089	168	18	𝑘	𝑘	PRON
cana-1089	168	19	−	−	PROPN
cana-1089	168	20	1)𝑘𝑎𝑘	1)𝑘𝑎𝑘	NUM
cana-1089	168	21	(	(	PUNCT
cana-1089	168	22	2𝑘	2𝑘	NUM
cana-1089	168	23	+	+	PROPN
cana-1089	168	24	1)𝑎3	1)𝑎3	NUM
cana-1089	168	25	⋮	⋮	NOUN
cana-1089	168	26	[	[	X
cana-1089	168	27	𝑘(𝑘	𝑘(𝑘	PROPN
cana-1089	168	28	−	−	NOUN
cana-1089	168	29	2	2	NUM
cana-1089	168	30	)	)	PUNCT
cana-1089	168	31	+	+	CCONJ
cana-1089	169	1	1]𝑎𝑘−1	1]𝑎𝑘−1	NUM
cana-1089	170	1	(	(	PUNCT
cana-1089	170	2	𝑘	𝑘	X
cana-1089	170	3	+	+	X
cana-1089	170	4	1)𝑎2	1)𝑎2	NUM
cana-1089	170	5	⋯	⋯	NOUN
cana-1089	170	6	0	0	NUM
cana-1089	170	7	⋮	⋮	NOUN
cana-1089	170	8	⋮	⋮	NOUN
cana-1089	170	9	(	(	PUNCT
cana-1089	170	10	𝑘	𝑘	PRON
cana-1089	170	11	−	−	NOUN
cana-1089	170	12	2	2	NUM
cana-1089	170	13	)	)	PUNCT
cana-1089	171	1	[	[	X
cana-1089	171	2	𝑘(𝑘	𝑘(𝑘	PROPN
cana-1089	171	3	−	−	NOUN
cana-1089	171	4	3	3	NUM
cana-1089	171	5	)	)	PUNCT
cana-1089	171	6	+	+	CCONJ
cana-1089	171	7	2]𝑎𝑘−2	2]𝑎𝑘−2	NUM
cana-1089	171	8	⋯	⋯	NOUN
cana-1089	171	9	(	(	PUNCT
cana-1089	171	10	2𝑘	2𝑘	NUM
cana-1089	171	11	−	−	PROPN
cana-1089	171	12	2)𝑎2	2)𝑎2	NUM
cana-1089	171	13	]	]	PUNCT
cana-1089	171	14	(	(	PUNCT
cana-1089	171	15	3.2	3.2	NUM
cana-1089	171	16	)	)	PUNCT
cana-1089	171	17	the	the	DET
cana-1089	171	18	elements	element	NOUN
cana-1089	171	19	of	of	ADP
cana-1089	171	20	the	the	DET
cana-1089	171	21	determinant	determinant	ADJ
cana-1089	171	22	in	in	ADP
cana-1089	171	23	(	(	PUNCT
cana-1089	171	24	3.2	3.2	NUM
cana-1089	171	25	)	)	PUNCT
cana-1089	171	26	are	be	AUX
cana-1089	171	27	given	give	VERB
cana-1089	171	28	by	by	ADP
cana-1089	171	29	θij	θij	NOUN
cana-1089	171	30	=	=	X
cana-1089	171	31	{	{	PUNCT
cana-1089	172	1	[	[	X
cana-1089	172	2	(	(	PUNCT
cana-1089	172	3	i	i	PRON
cana-1089	172	4	−	−	PROPN
cana-1089	172	5	j	j	PROPN
cana-1089	172	6	+	+	CCONJ
cana-1089	172	7	1)k	1)k	NUM
cana-1089	172	8	+	+	CCONJ
cana-1089	172	9	j	j	PROPN
cana-1089	173	1	−	−	PROPN
cana-1089	173	2	1]ai−j+2	1]ai−j+2	NUM
cana-1089	173	3	,	,	PUNCT
cana-1089	173	4	if	if	SCONJ
cana-1089	173	5	𝑖	𝑖	ADP
cana-1089	173	6	+	+	NOUN
cana-1089	173	7	1	1	NUM
cana-1089	173	8	≥	≥	NOUN
cana-1089	173	9	𝑗	𝑗	PROPN
cana-1089	173	10	0	0	NUM
cana-1089	173	11	,	,	PUNCT
cana-1089	173	12	if	if	SCONJ
cana-1089	173	13	𝑖	𝑖	ADP
cana-1089	173	14	+	+	ADP
cana-1089	173	15	1	1	NUM
cana-1089	173	16	<	<	X
cana-1089	173	17	𝑗	𝑗	NOUN
cana-1089	173	18	.	.	PUNCT
cana-1089	174	1	the	the	DET
cana-1089	174	2	functions	function	NOUN
cana-1089	174	3	in	in	ADP
cana-1089	174	4	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	174	5	)	)	PUNCT
cana-1089	174	6	need	need	AUX
cana-1089	174	7	not	not	PART
cana-1089	174	8	be	be	AUX
cana-1089	174	9	univalent	univalent	ADJ
cana-1089	174	10	,	,	PUNCT
cana-1089	174	11	but	but	CCONJ
cana-1089	174	12	since	since	SCONJ
cana-1089	174	13	𝜑′(0	𝜑′(0	NOUN
cana-1089	174	14	)	)	PUNCT
cana-1089	174	15	=	=	SYM
cana-1089	174	16	1	1	NUM
cana-1089	174	17	≠	≠	PROPN
cana-1089	174	18	0	0	NUM
cana-1089	174	19	for	for	ADP
cana-1089	174	20	all	all	PRON
cana-1089	174	21	𝜑	𝜑	PRON
cana-1089	174	22	∈	∈	NOUN
cana-1089	174	23	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	174	24	)	)	PUNCT
cana-1089	174	25	and	and	CCONJ
cana-1089	174	26	𝜑(0	𝜑(0	NOUN
cana-1089	174	27	)	)	PUNCT
cana-1089	174	28	=	=	SYM
cana-1089	175	1	0	0	NUM
cana-1089	175	2	,	,	PUNCT
cana-1089	175	3	there	there	PRON
cana-1089	175	4	exist	exist	VERB
cana-1089	175	5	an	an	DET
cana-1089	175	6	inverse	inverse	NOUN
cana-1089	175	7	function	function	NOUN
cana-1089	175	8	in	in	ADP
cana-1089	175	9	some	some	DET
cana-1089	175	10	small	small	ADJ
cana-1089	175	11	disk	disk	NOUN
cana-1089	175	12	with	with	ADP
cana-1089	175	13	center	center	NOUN
cana-1089	175	14	at	at	ADP
cana-1089	175	15	𝑤	𝑤	ADP
cana-1089	175	16	=	=	SYM
cana-1089	175	17	0	0	PROPN
cana-1089	175	18	.	.	PUNCT
cana-1089	175	19	theorem	theorem	VERB
cana-1089	175	20	3.1	3.1	NUM
cana-1089	175	21	.	.	PUNCT
cana-1089	176	1	let	let	VERB
cana-1089	176	2	𝜑	𝜑	PRON
cana-1089	176	3	∈	∈	VERB
cana-1089	176	4	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	176	5	)	)	PUNCT
cana-1089	176	6	and	and	CCONJ
cana-1089	176	7	let	let	VERB
cana-1089	176	8	𝜑−1	𝜑−1	PROPN
cana-1089	176	9	be	be	AUX
cana-1089	176	10	the	the	DET
cana-1089	176	11	inverse	inverse	NOUN
cana-1089	176	12	of	of	ADP
cana-1089	176	13	𝜑	𝜑	NOUN
cana-1089	176	14	defined	define	VERB
cana-1089	176	15	by	by	ADP
cana-1089	176	16	𝜑−1(𝑤	𝜑−1(𝑤	PROPN
cana-1089	176	17	)	)	PUNCT
cana-1089	176	18	=	=	PUNCT
cana-1089	177	1	𝑤	𝑤	ADP
cana-1089	177	2	+	+	NOUN
cana-1089	177	3	∑	∑	PROPN
cana-1089	177	4	𝑏𝑘	𝑏𝑘	NUM
cana-1089	177	5	∞	∞	NUM
cana-1089	177	6	𝑘=2	𝑘=2	NOUN
cana-1089	177	7	𝑤𝑘	𝑤𝑘	INTJ
cana-1089	177	8	,	,	PUNCT
cana-1089	177	9	(	(	PUNCT
cana-1089	177	10	|	|	ADV
cana-1089	177	11	𝑤|	𝑤|	VERB
cana-1089	177	12	<	<	X
cana-1089	177	13	𝑟	𝑟	NOUN
cana-1089	177	14	;	;	PUNCT
cana-1089	177	15	𝑟	𝑟	PRON
cana-1089	177	16	≥	≥	NUM
cana-1089	177	17	1	1	NUM
cana-1089	177	18	4	4	NUM
cana-1089	177	19	)	)	PUNCT
cana-1089	177	20	,	,	PUNCT
cana-1089	177	21	then	then	ADV
cana-1089	177	22	|	|	ADV
cana-1089	177	23	𝑏2|	𝑏2|	ADJ
cana-1089	177	24	≤	≤	NOUN
cana-1089	177	25	𝐿1	𝐿1	VERB
cana-1089	177	26	|	|	ADV
cana-1089	177	27	1	1	NUM
cana-1089	177	28	−	−	NOUN
cana-1089	177	29	γ2,𝑘|	γ2,𝑘|	PROPN
cana-1089	177	30	|𝑏3|	|𝑏3|	NOUN
cana-1089	177	31	≤	≤	NUM
cana-1089	177	32	𝐿1	𝐿1	NOUN
cana-1089	178	1	|	|	ADV
cana-1089	178	2	2	2	NUM
cana-1089	178	3	−	−	NOUN
cana-1089	178	4	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	178	5	max	max	PROPN
cana-1089	178	6	{	{	PUNCT
cana-1089	178	7	1	1	NUM
cana-1089	178	8	;	;	PUNCT
cana-1089	178	9	|	|	ADV
cana-1089	178	10	𝐿2	𝐿2	NOUN
cana-1089	178	11	𝐿1	𝐿1	PROPN
cana-1089	179	1	−	−	PROPN
cana-1089	179	2	𝐿1	𝐿1	PROPN
cana-1089	180	1	(	(	PUNCT
cana-1089	180	2	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	180	3	2	2	NUM
cana-1089	180	4	−	−	PROPN
cana-1089	180	5	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	180	6	−	−	PROPN
cana-1089	180	7	1	1	NUM
cana-1089	180	8	2	2	NUM
cana-1089	180	9	)	)	PUNCT
cana-1089	180	10	(	(	PUNCT
cana-1089	180	11	1	1	NUM
cana-1089	180	12	−	−	PROPN
cana-1089	180	13	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	180	14	)	)	PUNCT
cana-1089	180	15	2	2	NUM
cana-1089	180	16	−	−	PROPN
cana-1089	180	17	2𝐿1(2	2𝐿1(2	NUM
cana-1089	180	18	−	−	NOUN
cana-1089	180	19	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	180	20	)	)	PUNCT
cana-1089	180	21	(	(	PUNCT
cana-1089	180	22	1	1	NUM
cana-1089	180	23	−	−	PROPN
cana-1089	180	24	γ2,k	γ2,k	PROPN
cana-1089	180	25	)	)	PUNCT
cana-1089	180	26	2	2	NUM
cana-1089	180	27	|	|	NOUN
cana-1089	180	28	}	}	PUNCT
cana-1089	180	29	and	and	CCONJ
cana-1089	180	30	for	for	ADP
cana-1089	180	31	a	a	DET
cana-1089	180	32	complex	complex	ADJ
cana-1089	180	33	number	number	NOUN
cana-1089	180	34	𝜏	𝜏	NOUN
cana-1089	180	35	,	,	PUNCT
cana-1089	180	36	|b3	|b3	NOUN
cana-1089	180	37	−	−	PROPN
cana-1089	180	38	τ	τ	PROPN
cana-1089	180	39	a2	a2	PROPN
cana-1089	180	40	2	2	NUM
cana-1089	180	41	|	|	ADV
cana-1089	180	42	≤	≤	X
cana-1089	180	43	𝐿1	𝐿1	NOUN
cana-1089	181	1	|	|	ADV
cana-1089	181	2	2	2	NUM
cana-1089	181	3	−	−	NOUN
cana-1089	181	4	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	181	5	max	max	PROPN
cana-1089	181	6	{	{	PUNCT
cana-1089	181	7	1	1	NUM
cana-1089	181	8	;	;	PUNCT
cana-1089	181	9	|	|	ADV
cana-1089	181	10	𝐿2	𝐿2	NOUN
cana-1089	181	11	𝐿1	𝐿1	PROPN
cana-1089	181	12	−	−	PROPN
cana-1089	181	13	𝐿1	𝐿1	PROPN
cana-1089	182	1	(	(	PUNCT
cana-1089	182	2	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	182	3	2	2	NUM
cana-1089	182	4	−	−	PROPN
cana-1089	182	5	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	182	6	−	−	PROPN
cana-1089	182	7	1	1	NUM
cana-1089	182	8	2	2	NUM
cana-1089	182	9	)	)	PUNCT
cana-1089	182	10	(	(	PUNCT
cana-1089	182	11	1	1	NUM
cana-1089	182	12	−	−	PROPN
cana-1089	182	13	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	182	14	)	)	PUNCT
cana-1089	182	15	2	2	NUM
cana-1089	182	16	−	−	NOUN
cana-1089	182	17	(	(	PUNCT
cana-1089	182	18	𝜏	𝜏	PROPN
cana-1089	182	19	−	−	PROPN
cana-1089	182	20	2)𝐿1(2	2)𝐿1(2	NUM
cana-1089	182	21	−	−	PROPN
cana-1089	182	22	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	182	23	)	)	PUNCT
cana-1089	182	24	(	(	PUNCT
cana-1089	182	25	1	1	NUM
cana-1089	182	26	−	−	PROPN
cana-1089	182	27	γ2,k	γ2,k	PROPN
cana-1089	182	28	)	)	PUNCT
cana-1089	182	29	2	2	NUM
cana-1089	182	30	|	|	NOUN
cana-1089	182	31	}	}	PUNCT
cana-1089	182	32	proof	proof	NOUN
cana-1089	182	33	.	.	PUNCT
cana-1089	183	1	from	from	ADP
cana-1089	183	2	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	183	3	)	)	PUNCT
cana-1089	183	4	=	=	PUNCT
cana-1089	183	5	𝜔	𝜔	PRON
cana-1089	183	6	+	+	ADJ
cana-1089	183	7	∑	∑	PART
cana-1089	183	8	𝑎𝑛𝜔𝑛∞	𝑎𝑛𝜔𝑛∞	NOUN
cana-1089	183	9	𝑛=2	𝑛=2	PUNCT
cana-1089	183	10	and	and	CCONJ
cana-1089	183	11	(	(	PUNCT
cana-1089	183	12	3.1	3.1	NUM
cana-1089	183	13	)	)	PUNCT
cana-1089	183	14	,	,	PUNCT
cana-1089	183	15	we	we	PRON
cana-1089	183	16	have	have	VERB
cana-1089	183	17	𝑏2	𝑏2	NOUN
cana-1089	183	18	=	=	PUNCT
cana-1089	184	1	−𝑎2	−𝑎2	ADJ
cana-1089	184	2	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1089	184	3	𝑏3	𝑏3	NOUN
cana-1089	184	4	=	=	SYM
cana-1089	184	5	2𝑎2	2𝑎2	NUM
cana-1089	184	6	2	2	NUM
cana-1089	184	7	−	−	NOUN
cana-1089	184	8	𝑎3	𝑎3	PROPN
cana-1089	184	9	.	.	PUNCT
cana-1089	185	1	the	the	DET
cana-1089	185	2	estimate	estimate	NOUN
cana-1089	185	3	for	for	ADP
cana-1089	185	4	|	|	ADV
cana-1089	185	5	𝑏2|	𝑏2|	VERB
cana-1089	185	6	=	=	PUNCT
cana-1089	185	7	|	|	ADV
cana-1089	185	8	𝑎2|	𝑎2|	PRON
cana-1089	185	9	follows	follow	VERB
cana-1089	185	10	immediately	immediately	ADV
cana-1089	185	11	from	from	ADP
cana-1089	185	12	(	(	PUNCT
cana-1089	185	13	2.7	2.7	NUM
cana-1089	185	14	)	)	PUNCT
cana-1089	185	15	.	.	PUNCT
cana-1089	186	1	letting	let	VERB
cana-1089	186	2	𝜌	𝜌	X
cana-1089	186	3	=	=	SYM
cana-1089	186	4	2	2	NUM
cana-1089	186	5	in	in	ADP
cana-1089	186	6	(	(	PUNCT
cana-1089	186	7	2.3	2.3	NUM
cana-1089	186	8	)	)	PUNCT
cana-1089	186	9	,	,	PUNCT
cana-1089	186	10	we	we	PRON
cana-1089	186	11	get	get	VERB
cana-1089	186	12	the	the	DET
cana-1089	186	13	estimate	estimate	NOUN
cana-1089	186	14	|𝑏3|	|𝑏3|	NOUN
cana-1089	186	15	.	.	PUNCT
cana-1089	187	1	to	to	PART
cana-1089	187	2	find	find	VERB
cana-1089	187	3	the	the	DET
cana-1089	187	4	fekete	fekete	NOUN
cana-1089	187	5	-	-	PUNCT
cana-1089	187	6	szegő	szegő	ADJ
cana-1089	187	7	inequality	inequality	NOUN
cana-1089	187	8	for	for	ADP
cana-1089	187	9	𝜑−1	𝜑−1	PROPN
cana-1089	187	10	,	,	PUNCT
cana-1089	187	11	consider	consider	VERB
cana-1089	187	12	|𝑏3	|𝑏3	PRON
cana-1089	187	13	−	−	PROPN
cana-1089	187	14	𝜏	𝜏	PROPN
cana-1089	187	15	𝑏2	𝑏2	NOUN
cana-1089	187	16	2|	2|	NUM
cana-1089	187	17	=	=	SYM
cana-1089	187	18	|2𝑎2	|2𝑎2	SYM
cana-1089	187	19	2	2	NUM
cana-1089	187	20	−	−	NOUN
cana-1089	187	21	𝑎3	𝑎3	PROPN
cana-1089	187	22	−	−	PROPN
cana-1089	187	23	𝜏	𝜏	PROPN
cana-1089	187	24	𝑎2	𝑎2	NOUN
cana-1089	187	25	2|	2|	NUM
cana-1089	187	26	=	=	PUNCT
cana-1089	187	27	|𝑎3	|𝑎3	VERB
cana-1089	187	28	−	−	PROPN
cana-1089	187	29	(	(	PUNCT
cana-1089	187	30	𝜏	𝜏	NOUN
cana-1089	187	31	−	−	PROPN
cana-1089	187	32	2)𝑎2	2)𝑎2	NUM
cana-1089	187	33	2|	2|	NUM
cana-1089	187	34	.	.	PUNCT
cana-1089	188	1	changing	change	VERB
cana-1089	188	2	𝜌	𝜌	X
cana-1089	188	3	=	=	SYM
cana-1089	188	4	(	(	PUNCT
cana-1089	188	5	𝜏	𝜏	NOUN
cana-1089	188	6	−	−	NOUN
cana-1089	188	7	2	2	NUM
cana-1089	188	8	)	)	PUNCT
cana-1089	188	9	in	in	ADP
cana-1089	188	10	the	the	DET
cana-1089	188	11	(	(	PUNCT
cana-1089	188	12	2.3	2.3	NUM
cana-1089	188	13	)	)	PUNCT
cana-1089	188	14	,	,	PUNCT
cana-1089	188	15	we	we	PRON
cana-1089	188	16	get	get	VERB
cana-1089	188	17	the	the	DET
cana-1089	188	18	desired	desire	VERB
cana-1089	188	19	result	result	NOUN
cana-1089	188	20	.	.	PUNCT
cana-1089	189	1	analogous	analogous	ADJ
cana-1089	189	2	to	to	ADP
cana-1089	189	3	the	the	DET
cana-1089	189	4	results	result	NOUN
cana-1089	189	5	obtained	obtain	VERB
cana-1089	189	6	in	in	ADP
cana-1089	189	7	theorem	theorem	ADJ
cana-1089	189	8	3.2	3.2	NUM
cana-1089	189	9	,	,	PUNCT
cana-1089	189	10	we	we	PRON
cana-1089	189	11	can	can	AUX
cana-1089	189	12	easily	easily	ADV
cana-1089	189	13	get	get	VERB
cana-1089	189	14	the	the	DET
cana-1089	189	15	following	follow	VERB
cana-1089	189	16	result	result	NOUN
cana-1089	189	17	.	.	PUNCT
cana-1089	190	1	theorem	theorem	ADJ
cana-1089	190	2	3.2	3.2	NUM
cana-1089	190	3	let	let	VERB
cana-1089	190	4	𝜑	𝜑	NOUN
cana-1089	190	5	∈	∈	PROPN
cana-1089	190	6	ℒ𝑘(𝜒)$	ℒ𝑘(𝜒)$	PROPN
cana-1089	190	7	and	and	CCONJ
cana-1089	190	8	let	let	VERB
cana-1089	190	9	𝜑−1	𝜑−1	PROPN
cana-1089	190	10	be	be	AUX
cana-1089	190	11	the	the	DET
cana-1089	190	12	inverse	inverse	NOUN
cana-1089	190	13	of	of	ADP
cana-1089	190	14	𝜑	𝜑	NOUN
cana-1089	190	15	defined	define	VERB
cana-1089	190	16	by	by	ADP
cana-1089	190	17	𝜑−1(𝑤	𝜑−1(𝑤	PROPN
cana-1089	190	18	)	)	PUNCT
cana-1089	190	19	=	=	PUNCT
cana-1089	191	1	𝑤	𝑤	ADP
cana-1089	191	2	+	+	NOUN
cana-1089	191	3	∑	∑	PROPN
cana-1089	191	4	𝑏𝑘	𝑏𝑘	NUM
cana-1089	191	5	∞	∞	NUM
cana-1089	191	6	𝑘=2	𝑘=2	NOUN
cana-1089	191	7	𝑤𝑘	𝑤𝑘	INTJ
cana-1089	191	8	,	,	PUNCT
cana-1089	191	9	(	(	PUNCT
cana-1089	191	10	|	|	ADV
cana-1089	191	11	𝑤|	𝑤|	VERB
cana-1089	191	12	<	<	X
cana-1089	191	13	𝑟	𝑟	NOUN
cana-1089	191	14	;	;	PUNCT
cana-1089	191	15	𝑟	𝑟	PRON
cana-1089	191	16	≥	≥	NUM
cana-1089	191	17	1	1	NUM
cana-1089	191	18	4	4	NUM
cana-1089	191	19	)	)	PUNCT
cana-1089	191	20	,	,	PUNCT
cana-1089	191	21	communications	communication	NOUN
cana-1089	191	22	on	on	ADP
cana-1089	191	23	applied	apply	VERB
cana-1089	191	24	nonlinear	nonlinear	ADJ
cana-1089	191	25	analysis	analysis	NOUN
cana-1089	191	26	issn	issn	NOUN
cana-1089	191	27	:	:	PUNCT
cana-1089	191	28	1074	1074	NUM
cana-1089	191	29	-	-	PUNCT
cana-1089	191	30	133x	133x	NUM
cana-1089	191	31	vol	vol	NOUN
cana-1089	191	32	31	31	NUM
cana-1089	191	33	no	no	NOUN
cana-1089	191	34	.	.	PUNCT
cana-1089	192	1	5s	5s	NUM
cana-1089	192	2	(	(	PUNCT
cana-1089	192	3	2024	2024	NUM
cana-1089	192	4	)	)	PUNCT
cana-1089	192	5	548	548	NUM
cana-1089	192	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	192	7	then$$	then$$	X
cana-1089	193	1	|	|	ADV
cana-1089	193	2	𝑏2|	𝑏2|	ADJ
cana-1089	193	3	≤	≤	NOUN
cana-1089	193	4	1	1	NUM
cana-1089	193	5	|	|	ADV
cana-1089	193	6	γ2,𝑘|	γ2,𝑘|	PUNCT
cana-1089	194	1	[	[	X
cana-1089	194	2	𝐿1	𝐿1	NOUN
cana-1089	194	3	+	+	X
cana-1089	194	4	1	1	NUM
cana-1089	194	5	]	]	PUNCT
cana-1089	194	6	,	,	PUNCT
cana-1089	194	7	|𝑏3|	|𝑏3|	VERB
cana-1089	194	8	≤	≤	NUM
cana-1089	194	9	𝐿1	𝐿1	NOUN
cana-1089	194	10	|	|	ADV
cana-1089	194	11	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	195	1	[	[	X
cana-1089	195	2	max	max	X
cana-1089	195	3	{	{	PUNCT
cana-1089	195	4	1	1	NUM
cana-1089	195	5	;	;	PUNCT
cana-1089	195	6	|	|	ADV
cana-1089	195	7	𝐿2	𝐿2	NOUN
cana-1089	195	8	𝐿1	𝐿1	PROPN
cana-1089	195	9	−	−	PROPN
cana-1089	195	10	𝐿1	𝐿1	PROPN
cana-1089	196	1	+	+	CCONJ
cana-1089	196	2	2𝐿1γ3,𝑘	2𝐿1γ3,𝑘	NUM
cana-1089	196	3	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	196	4	2	2	NUM
cana-1089	196	5	|	|	NOUN
cana-1089	196	6	}	}	PUNCT
cana-1089	196	7	+	+	CCONJ
cana-1089	196	8	|1	|1	NUM
cana-1089	197	1	+	+	NUM
cana-1089	197	2	1	1	NUM
cana-1089	197	3	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	197	4	−	−	ADP
cana-1089	197	5	4γ3,𝑘	4γ3,𝑘	NUM
cana-1089	198	1	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	198	2	2	2	NUM
cana-1089	199	1	|	|	ADV
cana-1089	200	1	+	+	CCONJ
cana-1089	200	2	1	1	NUM
cana-1089	201	1	2|	2|	NUM
cana-1089	201	2	𝐿1|	𝐿1|	PROPN
cana-1089	201	3	|3	|3	PUNCT
cana-1089	202	1	−	−	PROPN
cana-1089	203	1	4γ3,𝑘	4γ3,𝑘	NUM
cana-1089	204	1	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	204	2	2	2	NUM
cana-1089	204	3	|	|	NOUN
cana-1089	204	4	]	]	PUNCT
cana-1089	204	5	and	and	CCONJ
cana-1089	204	6	for	for	ADP
cana-1089	204	7	a	a	DET
cana-1089	204	8	complex	complex	ADJ
cana-1089	204	9	number	number	NOUN
cana-1089	204	10	𝜏	𝜏	NUM
cana-1089	204	11	,	,	PUNCT
cana-1089	204	12	|𝑏3	|𝑏3	PRON
cana-1089	204	13	−	−	X
cana-1089	204	14	𝜏	𝜏	PROPN
cana-1089	204	15	𝑎2	𝑎2	NOUN
cana-1089	204	16	2|	2|	PROPN
cana-1089	204	17	≤	≤	PROPN
cana-1089	204	18	𝐿1	𝐿1	PROPN
cana-1089	204	19	|	|	ADV
cana-1089	204	20	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	205	1	[	[	X
cana-1089	205	2	max	max	X
cana-1089	205	3	{	{	PUNCT
cana-1089	205	4	1	1	NUM
cana-1089	205	5	;	;	PUNCT
cana-1089	205	6	|	|	ADV
cana-1089	205	7	𝐿2	𝐿2	NOUN
cana-1089	205	8	𝐿1	𝐿1	PROPN
cana-1089	205	9	−	−	PROPN
cana-1089	205	10	𝐿1	𝐿1	PROPN
cana-1089	206	1	+	+	CCONJ
cana-1089	206	2	(	(	PUNCT
cana-1089	206	3	𝜏	𝜏	PRON
cana-1089	206	4	−	−	PROPN
cana-1089	206	5	2)𝐿1γ3,𝑘	2)𝐿1γ3,𝑘	NUM
cana-1089	206	6	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	206	7	2	2	NUM
cana-1089	206	8	|	|	NOUN
cana-1089	206	9	}	}	PUNCT
cana-1089	206	10	+	+	CCONJ
cana-1089	206	11	|1	|1	NUM
cana-1089	207	1	+	+	NUM
cana-1089	207	2	1	1	NUM
cana-1089	207	3	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	207	4	−	−	PROPN
cana-1089	207	5	2(𝜏	2(𝜏	NUM
cana-1089	207	6	−	−	PROPN
cana-1089	207	7	2)γ3,𝑘	2)γ3,𝑘	PUNCT
cana-1089	208	1	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	208	2	2	2	NUM
cana-1089	209	1	|	|	ADV
cana-1089	210	1	+	+	CCONJ
cana-1089	210	2	1	1	NUM
cana-1089	211	1	2|	2|	NUM
cana-1089	211	2	𝐿1|	𝐿1|	PROPN
cana-1089	211	3	|3	|3	NUM
cana-1089	212	1	−	−	PROPN
cana-1089	212	2	2(𝜏	2(𝜏	NUM
cana-1089	213	1	−	−	NUM
cana-1089	213	2	2)γ3,𝑘	2)γ3,𝑘	PUNCT
cana-1089	214	1	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	214	2	2	2	NUM
cana-1089	214	3	|	|	NOUN
cana-1089	214	4	]	]	X
cana-1089	214	5	.	.	PUNCT
cana-1089	215	1	4	4	X
cana-1089	215	2	.	.	X
cana-1089	215	3	logarithmic	logarithmic	ADJ
cana-1089	215	4	coefficients	coefficient	NOUN
cana-1089	215	5	milin	milin	PROPN
cana-1089	215	6	in	in	ADP
cana-1089	215	7	[	[	X
cana-1089	215	8	16	16	NUM
cana-1089	215	9	]	]	PUNCT
cana-1089	215	10	studied	study	VERB
cana-1089	215	11	the	the	DET
cana-1089	215	12	properties	property	NOUN
cana-1089	215	13	of	of	ADP
cana-1089	215	14	the	the	DET
cana-1089	215	15	logarithmic	logarithmic	ADJ
cana-1089	215	16	coefficients	coefficient	NOUN
cana-1089	215	17	to	to	PART
cana-1089	215	18	obtain	obtain	VERB
cana-1089	215	19	the	the	DET
cana-1089	215	20	bounds	bound	NOUN
cana-1089	215	21	of	of	ADP
cana-1089	215	22	the	the	DET
cana-1089	215	23	taylor	taylor	PROPN
cana-1089	215	24	coefficients	coefficient	NOUN
cana-1089	215	25	of	of	ADP
cana-1089	215	26	univalent	univalent	ADJ
cana-1089	215	27	functions	function	NOUN
cana-1089	215	28	.	.	PUNCT
cana-1089	216	1	the	the	DET
cana-1089	216	2	milin	milin	PROPN
cana-1089	216	3	conjuncture	conjuncture	NOUN
cana-1089	216	4	about	about	ADP
cana-1089	216	5	the	the	DET
cana-1089	216	6	inequalities	inequality	NOUN
cana-1089	216	7	of	of	ADP
cana-1089	216	8	the	the	DET
cana-1089	216	9	logarithmic	logarithmic	ADJ
cana-1089	216	10	coefficients	coefficient	NOUN
cana-1089	216	11	garnered	garner	VERB
cana-1089	216	12	the	the	DET
cana-1089	216	13	attention	attention	NOUN
cana-1089	216	14	several	several	ADJ
cana-1089	216	15	researchers	researcher	NOUN
cana-1089	216	16	in	in	ADP
cana-1089	216	17	those	those	DET
cana-1089	216	18	period	period	NOUN
cana-1089	216	19	of	of	ADP
cana-1089	216	20	time	time	NOUN
cana-1089	216	21	,	,	PUNCT
cana-1089	216	22	because	because	SCONJ
cana-1089	216	23	proving	prove	VERB
cana-1089	216	24	milin	milin	NOUN
cana-1089	216	25	conjuncture	conjuncture	NOUN
cana-1089	216	26	would	would	AUX
cana-1089	216	27	imply	imply	AUX
cana-1089	216	28	proving	prove	VERB
cana-1089	216	29	robertson	robertson	PROPN
cana-1089	216	30	conjecture	conjecture	NOUN
cana-1089	216	31	and	and	CCONJ
cana-1089	216	32	the	the	DET
cana-1089	216	33	bieberbach	bieberbach	NOUN
cana-1089	216	34	conjecture	conjecture	VERB
cana-1089	216	35	.	.	PUNCT
cana-1089	217	1	refer	refer	VERB
cana-1089	217	2	to	to	ADP
cana-1089	217	3	ponnusamy	ponnusamy	NOUN
cana-1089	217	4	et	et	NOUN
cana-1089	217	5	al	al	PROPN
cana-1089	217	6	.	.	PUNCT
cana-1089	218	1	[	[	X
cana-1089	218	2	18	18	NUM
cana-1089	218	3	,	,	PUNCT
cana-1089	218	4	19	19	NUM
cana-1089	218	5	20	20	NUM
cana-1089	218	6	]	]	PUNCT
cana-1089	218	7	and	and	CCONJ
cana-1089	218	8	[	[	X
cana-1089	218	9	1	1	NUM
cana-1089	218	10	,	,	PUNCT
cana-1089	218	11	2	2	NUM
cana-1089	218	12	,	,	PUNCT
cana-1089	218	13	3	3	NUM
cana-1089	218	14	,	,	PUNCT
cana-1089	218	15	4	4	NUM
cana-1089	218	16	,	,	PUNCT
cana-1089	218	17	17	17	NUM
cana-1089	218	18	]	]	PUNCT
cana-1089	218	19	for	for	ADP
cana-1089	218	20	the	the	DET
cana-1089	218	21	detailed	detailed	ADJ
cana-1089	218	22	study	study	NOUN
cana-1089	218	23	on	on	ADP
cana-1089	218	24	properties	property	NOUN
cana-1089	218	25	and	and	CCONJ
cana-1089	218	26	significance	significance	NOUN
cana-1089	218	27	of	of	ADP
cana-1089	218	28	the	the	DET
cana-1089	218	29	logarithmic	logarithmic	ADJ
cana-1089	218	30	coefficients	coefficient	NOUN
cana-1089	218	31	.	.	PUNCT
cana-1089	219	1	the	the	DET
cana-1089	219	2	logarithmic	logarithmic	ADJ
cana-1089	219	3	coefficients	coefficient	NOUN
cana-1089	219	4	𝑑𝑛	𝑑𝑛	NOUN
cana-1089	219	5	of	of	ADP
cana-1089	219	6	a	a	DET
cana-1089	219	7	function	function	NOUN
cana-1089	219	8	𝜑	𝜑	X
cana-1089	219	9	∈	∈	PROPN
cana-1089	219	10	𝒜	𝒜	NOUN
cana-1089	219	11	such	such	ADJ
cana-1089	219	12	that	that	DET
cana-1089	219	13	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	219	14	)	)	PUNCT
cana-1089	219	15	𝜔	𝜔	ADP
cana-1089	219	16	≠	≠	PROPN
cana-1089	219	17	0	0	NUM
cana-1089	219	18	for	for	SCONJ
cana-1089	219	19	all	all	DET
cana-1089	219	20	𝜔	𝜔	DET
cana-1089	219	21	∈	∈	NOUN
cana-1089	219	22	𝒰	𝒰	NOUN
cana-1089	219	23	is	be	AUX
cana-1089	219	24	defined	define	VERB
cana-1089	219	25	by	by	ADP
cana-1089	219	26	log	log	NOUN
cana-1089	219	27	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	219	28	)	)	PUNCT
cana-1089	219	29	=	=	SYM
cana-1089	219	30	2	2	NUM
cana-1089	219	31	∑	∑	ADP
cana-1089	219	32	𝑑𝑛𝜔𝑛	𝑑𝑛𝜔𝑛	PROPN
cana-1089	219	33	∞	∞	PROPN
cana-1089	219	34	𝑛=1	𝑛=1	PROPN
cana-1089	219	35	.	.	PUNCT
cana-1089	220	1	note	note	VERB
cana-1089	220	2	that	that	SCONJ
cana-1089	220	3	for	for	ADP
cana-1089	220	4	all	all	DET
cana-1089	220	5	functions	function	NOUN
cana-1089	220	6	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	220	7	)	)	PUNCT
cana-1089	220	8	∈	∈	NOUN
cana-1089	220	9	ℳ𝑘(𝜒	ℳ𝑘(𝜒	NOUN
cana-1089	220	10	)	)	PUNCT
cana-1089	220	11	and	and	CCONJ
cana-1089	220	12	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	220	13	)	)	PUNCT
cana-1089	220	14	,	,	PUNCT
cana-1089	220	15	the	the	DET
cana-1089	220	16	relation	relation	NOUN
cana-1089	220	17	(	(	PUNCT
cana-1089	220	18	4.1	4.1	NUM
cana-1089	220	19	)	)	PUNCT
cana-1089	220	20	is	be	AUX
cana-1089	220	21	well	well	ADV
cana-1089	220	22	-	-	PUNCT
cana-1089	220	23	defined	define	VERB
cana-1089	220	24	.	.	PUNCT
cana-1089	221	1	theorem	theorem	VERB
cana-1089	221	2	4.1	4.1	NUM
cana-1089	221	3	.	.	PUNCT
cana-1089	222	1	if	if	SCONJ
cana-1089	222	2	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	222	3	)	)	PUNCT
cana-1089	222	4	∈	∈	NOUN
cana-1089	222	5	ℳ𝑘(𝜒	ℳ𝑘(𝜒	NOUN
cana-1089	222	6	)	)	PUNCT
cana-1089	222	7	,	,	PUNCT
cana-1089	222	8	with	with	ADP
cana-1089	222	9	the	the	DET
cana-1089	222	10	logarithmic	logarithmic	ADJ
cana-1089	222	11	coefficients	coefficient	NOUN
cana-1089	222	12	given	give	VERB
cana-1089	222	13	by	by	ADP
cana-1089	222	14	(	(	PUNCT
cana-1089	222	15	4.1	4.1	NUM
cana-1089	222	16	)	)	PUNCT
cana-1089	222	17	,	,	PUNCT
cana-1089	222	18	then	then	ADV
cana-1089	222	19	,	,	PUNCT
cana-1089	222	20	|𝑑1|	|𝑑1|	VERB
cana-1089	222	21	≤	≤	NUM
cana-1089	222	22	𝐿1	𝐿1	PROPN
cana-1089	223	1	2|	2|	NUM
cana-1089	223	2	1	1	NUM
cana-1089	223	3	−	−	PROPN
cana-1089	223	4	γ2,𝑘|	γ2,𝑘|	PROPN
cana-1089	223	5	|𝑑2|	|𝑑2|	NOUN
cana-1089	223	6	≤	≤	NOUN
cana-1089	223	7	𝐿1	𝐿1	PROPN
cana-1089	223	8	2|	2|	NUM
cana-1089	223	9	2	2	NUM
cana-1089	223	10	−	−	PRON
cana-1089	224	1	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	225	1	max	max	PROPN
cana-1089	226	1	{	{	PUNCT
cana-1089	227	1	1	1	NUM
cana-1089	227	2	;	;	PUNCT
cana-1089	227	3	|	|	ADV
cana-1089	227	4	𝐿2	𝐿2	NOUN
cana-1089	227	5	𝐿1	𝐿1	PROPN
cana-1089	227	6	−	−	PROPN
cana-1089	227	7	𝐿1	𝐿1	PROPN
cana-1089	228	1	(	(	PUNCT
cana-1089	228	2	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	228	3	2	2	NUM
cana-1089	228	4	−	−	PROPN
cana-1089	228	5	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	228	6	−	−	PROPN
cana-1089	228	7	1	1	NUM
cana-1089	228	8	2	2	NUM
cana-1089	228	9	)	)	PUNCT
cana-1089	228	10	(	(	PUNCT
cana-1089	228	11	1	1	NUM
cana-1089	228	12	−	−	PROPN
cana-1089	228	13	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	228	14	)	)	PUNCT
cana-1089	228	15	2	2	NUM
cana-1089	228	16	−	−	NOUN
cana-1089	228	17	𝐿1(2	𝐿1(2	NOUN
cana-1089	228	18	−	−	PROPN
cana-1089	228	19	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	228	20	)	)	PUNCT
cana-1089	228	21	2(1	2(1	NUM
cana-1089	228	22	−	−	PROPN
cana-1089	228	23	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	228	24	)	)	PUNCT
cana-1089	228	25	2|	2|	NUM
cana-1089	228	26	}	}	PUNCT
cana-1089	228	27	.	.	PUNCT
cana-1089	229	1	for	for	ADP
cana-1089	229	2	𝜇	𝜇	ADP
cana-1089	229	3	∈	∈	PROPN
cana-1089	229	4	ℂ	ℂ	PROPN
cana-1089	229	5	,	,	PUNCT
cana-1089	229	6	we	we	PRON
cana-1089	229	7	have	have	VERB
cana-1089	229	8	|𝑑2	|𝑑2	NOUN
cana-1089	229	9	−	−	NOUN
cana-1089	229	10	𝜇𝑑1	𝜇𝑑1	NOUN
cana-1089	229	11	2|	2|	PROPN
cana-1089	229	12	≤	≤	NOUN
cana-1089	229	13	𝐿1	𝐿1	PROPN
cana-1089	229	14	2|	2|	NUM
cana-1089	230	1	2	2	NUM
cana-1089	230	2	−	−	PRON
cana-1089	230	3	γ3,𝑘|	γ3,𝑘|	PROPN
cana-1089	230	4	max	max	PROPN
cana-1089	230	5	{	{	PUNCT
cana-1089	230	6	1	1	NUM
cana-1089	230	7	;	;	PUNCT
cana-1089	230	8	|	|	ADV
cana-1089	230	9	𝐿2	𝐿2	NOUN
cana-1089	230	10	𝐿1	𝐿1	PROPN
cana-1089	230	11	−	−	PROPN
cana-1089	230	12	𝐿1	𝐿1	PROPN
cana-1089	231	1	(	(	PUNCT
cana-1089	231	2	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	231	3	2	2	NUM
cana-1089	231	4	−	−	PROPN
cana-1089	231	5	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	231	6	−	−	PROPN
cana-1089	231	7	1	1	NUM
cana-1089	231	8	2	2	NUM
cana-1089	231	9	)	)	PUNCT
cana-1089	231	10	(	(	PUNCT
cana-1089	231	11	1	1	NUM
cana-1089	231	12	−	−	PROPN
cana-1089	231	13	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	231	14	)	)	PUNCT
cana-1089	231	15	2	2	NUM
cana-1089	231	16	−	−	NOUN
cana-1089	231	17	𝐿1(1	𝐿1(1	ADJ
cana-1089	231	18	+	+	NUM
cana-1089	231	19	𝜇)(2	𝜇)(2	NUM
cana-1089	231	20	−	−	PROPN
cana-1089	231	21	γ3,𝑘	γ3,𝑘	PROPN
cana-1089	231	22	)	)	PUNCT
cana-1089	231	23	2(1	2(1	NUM
cana-1089	231	24	−	−	PROPN
cana-1089	231	25	γ2,𝑘	γ2,𝑘	NOUN
cana-1089	231	26	)	)	PUNCT
cana-1089	231	27	2	2	NUM
cana-1089	231	28	|	|	NOUN
cana-1089	231	29	}	}	PUNCT
cana-1089	231	30	.	.	PUNCT
cana-1089	232	1	(	(	PUNCT
cana-1089	232	2	4.2	4.2	NUM
cana-1089	232	3	)	)	PUNCT
cana-1089	232	4	communications	communication	NOUN
cana-1089	232	5	on	on	ADP
cana-1089	232	6	applied	apply	VERB
cana-1089	232	7	nonlinear	nonlinear	ADJ
cana-1089	232	8	analysis	analysis	NOUN
cana-1089	232	9	issn	issn	NOUN
cana-1089	232	10	:	:	PUNCT
cana-1089	232	11	1074	1074	NUM
cana-1089	232	12	-	-	PUNCT
cana-1089	232	13	133x	133x	NUM
cana-1089	232	14	vol	vol	NOUN
cana-1089	232	15	31	31	NUM
cana-1089	232	16	no	no	NOUN
cana-1089	232	17	.	.	PUNCT
cana-1089	233	1	5s	5s	NUM
cana-1089	233	2	(	(	PUNCT
cana-1089	233	3	2024	2024	NUM
cana-1089	233	4	)	)	PUNCT
cana-1089	233	5	549	549	NUM
cana-1089	233	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	233	7	proof	proof	NOUN
cana-1089	233	8	.	.	PUNCT
cana-1089	234	1	from	from	ADP
cana-1089	234	2	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	234	3	)	)	PUNCT
cana-1089	234	4	=	=	PUNCT
cana-1089	234	5	𝜔	𝜔	PRON
cana-1089	234	6	+	+	ADJ
cana-1089	234	7	∑	∑	PART
cana-1089	234	8	𝑎𝑛𝜔𝑛∞	𝑎𝑛𝜔𝑛∞	NOUN
cana-1089	234	9	𝑛=2	𝑛=2	PUNCT
cana-1089	234	10	and	and	CCONJ
cana-1089	234	11	equating	equate	VERB
cana-1089	234	12	the	the	DET
cana-1089	234	13	first	first	ADJ
cana-1089	234	14	two	two	NUM
cana-1089	234	15	coefficients	coefficient	NOUN
cana-1089	234	16	of	of	ADP
cana-1089	234	17	the	the	DET
cana-1089	234	18	relation	relation	NOUN
cana-1089	234	19	(	(	PUNCT
cana-1089	234	20	4.1	4.1	NUM
cana-1089	234	21	)	)	PUNCT
cana-1089	234	22	,	,	PUNCT
cana-1089	234	23	we	we	PRON
cana-1089	234	24	get	get	VERB
cana-1089	234	25	𝑑1	𝑑1	NOUN
cana-1089	234	26	=	=	SYM
cana-1089	234	27	𝑎2	𝑎2	NOUN
cana-1089	234	28	2	2	NUM
cana-1089	234	29	,	,	PUNCT
cana-1089	234	30	𝑑2	𝑑2	NOUN
cana-1089	234	31	=	=	NOUN
cana-1089	234	32	1	1	NUM
cana-1089	234	33	2	2	NUM
cana-1089	234	34	(	(	PUNCT
cana-1089	234	35	𝑎3	𝑎3	PROPN
cana-1089	234	36	−	−	PROPN
cana-1089	234	37	𝑎2	𝑎2	NOUN
cana-1089	234	38	2	2	NUM
cana-1089	234	39	2	2	NUM
cana-1089	234	40	)	)	PUNCT
cana-1089	234	41	.	.	PUNCT
cana-1089	235	1	using	use	VERB
cana-1089	235	2	(	(	PUNCT
cana-1089	235	3	2.1	2.1	NUM
cana-1089	235	4	)	)	PUNCT
cana-1089	235	5	and	and	CCONJ
cana-1089	235	6	(	(	PUNCT
cana-1089	235	7	2.3	2.3	NUM
cana-1089	235	8	)	)	PUNCT
cana-1089	235	9	in	in	ADP
cana-1089	235	10	the	the	DET
cana-1089	235	11	above	above	ADJ
cana-1089	235	12	expression	expression	NOUN
cana-1089	235	13	,	,	PUNCT
cana-1089	235	14	we	we	PRON
cana-1089	235	15	can	can	AUX
cana-1089	235	16	find	find	VERB
cana-1089	235	17	the	the	DET
cana-1089	235	18	estimates	estimate	NOUN
cana-1089	235	19	for	for	ADP
cana-1089	235	20	𝑑1	𝑑1	NOUN
cana-1089	235	21	and	and	CCONJ
cana-1089	235	22	𝑑2	𝑑2	NOUN
cana-1089	235	23	.	.	PUNCT
cana-1089	236	1	to	to	PART
cana-1089	236	2	find	find	VERB
cana-1089	236	3	the	the	DET
cana-1089	236	4	estimate	estimate	NOUN
cana-1089	236	5	(	(	PUNCT
cana-1089	236	6	4.2	4.2	NUM
cana-1089	236	7	)	)	PUNCT
cana-1089	236	8	,	,	PUNCT
cana-1089	236	9	consider	consider	VERB
cana-1089	236	10	|𝑑2	|𝑑2	NOUN
cana-1089	236	11	−	−	NOUN
cana-1089	236	12	𝜇	𝜇	ADP
cana-1089	236	13	𝑑1	𝑑1	NOUN
cana-1089	236	14	2|	2|	NUM
cana-1089	236	15	=	=	SYM
cana-1089	237	1	1	1	NUM
cana-1089	237	2	2	2	NUM
cana-1089	238	1	[	[	X
cana-1089	238	2	𝑎3	𝑎3	ADJ
cana-1089	238	3	−	−	NOUN
cana-1089	238	4	1	1	NUM
cana-1089	238	5	+	+	CCONJ
cana-1089	238	6	𝜇	𝜇	DET
cana-1089	238	7	2	2	NUM
cana-1089	238	8	𝑎2	𝑎2	NOUN
cana-1089	238	9	2	2	NUM
cana-1089	238	10	]	]	PUNCT
cana-1089	238	11	changing	change	VERB
cana-1089	238	12	𝜌	𝜌	X
cana-1089	238	13	=	=	SYM
cana-1089	238	14	(	(	PUNCT
cana-1089	238	15	1+\𝑚𝑢	1+\𝑚𝑢	PROPN
cana-1089	238	16	)	)	PUNCT
cana-1089	238	17	2	2	NUM
cana-1089	238	18	in	in	ADP
cana-1089	238	19	(	(	PUNCT
cana-1089	238	20	2.3	2.3	NUM
cana-1089	238	21	)	)	PUNCT
cana-1089	238	22	and	and	CCONJ
cana-1089	238	23	simplifying	simplify	VERB
cana-1089	238	24	,	,	PUNCT
cana-1089	238	25	we	we	PRON
cana-1089	238	26	get	get	VERB
cana-1089	238	27	the	the	DET
cana-1089	238	28	desired	desire	VERB
cana-1089	238	29	result	result	NOUN
cana-1089	238	30	.	.	PUNCT
cana-1089	239	1	for	for	ADP
cana-1089	239	2	completeness	completeness	NOUN
cana-1089	239	3	,	,	PUNCT
cana-1089	239	4	we	we	PRON
cana-1089	239	5	just	just	ADV
cana-1089	239	6	state	state	VERB
cana-1089	239	7	the	the	DET
cana-1089	239	8	following	following	NOUN
cana-1089	239	9	.	.	PUNCT
cana-1089	240	1	theorem	theorem	VERB
cana-1089	240	2	4.2	4.2	NUM
cana-1089	240	3	.	.	PUNCT
cana-1089	241	1	if	if	SCONJ
cana-1089	241	2	𝜑(𝜔	𝜑(𝜔	NOUN
cana-1089	241	3	)	)	PUNCT
cana-1089	241	4	∈	∈	PROPN
cana-1089	241	5	𝐿𝑘(𝜒	𝐿𝑘(𝜒	NOUN
cana-1089	241	6	)	)	PUNCT
cana-1089	241	7	,	,	PUNCT
cana-1089	241	8	with	with	ADP
cana-1089	241	9	the	the	DET
cana-1089	241	10	logarithmic	logarithmic	ADJ
cana-1089	241	11	coefficients	coefficient	NOUN
cana-1089	241	12	given	give	VERB
cana-1089	241	13	by	by	ADP
cana-1089	241	14	(	(	PUNCT
cana-1089	241	15	4.1	4.1	NUM
cana-1089	241	16	)	)	PUNCT
cana-1089	241	17	,	,	PUNCT
cana-1089	241	18	then	then	ADV
cana-1089	241	19	,	,	PUNCT
cana-1089	241	20	|𝑑1|	|𝑑1|	VERB
cana-1089	241	21	≤	≤	NUM
cana-1089	241	22	1	1	NUM
cana-1089	241	23	2|	2|	NUM
cana-1089	241	24	γ2,𝑘|	γ2,𝑘|	PUNCT
cana-1089	242	1	[	[	X
cana-1089	242	2	𝐿1	𝐿1	X
cana-1089	242	3	+	+	X
cana-1089	242	4	1	1	X
cana-1089	242	5	]	]	PUNCT
cana-1089	242	6	|𝑑2|	|𝑑2|	NOUN
cana-1089	242	7	≤	≤	X
cana-1089	242	8	𝐿1	𝐿1	VERB
cana-1089	242	9	2|	2|	X
cana-1089	242	10	γ3,𝑘|	γ3,𝑘|	PUNCT
cana-1089	243	1	[	[	X
cana-1089	243	2	max	max	X
cana-1089	243	3	{	{	PUNCT
cana-1089	243	4	1	1	NUM
cana-1089	243	5	;	;	PUNCT
cana-1089	243	6	|	|	ADV
cana-1089	243	7	𝐿2	𝐿2	NOUN
cana-1089	243	8	𝐿1	𝐿1	PROPN
cana-1089	243	9	−	−	PROPN
cana-1089	243	10	𝐿1	𝐿1	PROPN
cana-1089	244	1	+	+	CCONJ
cana-1089	244	2	2	2	NUM
cana-1089	244	3	𝐿1γ3,𝑘	𝐿1γ3,𝑘	NOUN
cana-1089	244	4	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	244	5	2	2	NUM
cana-1089	244	6	|	|	NOUN
cana-1089	244	7	}	}	PUNCT
cana-1089	244	8	+	+	CCONJ
cana-1089	244	9	|1	|1	NUM
cana-1089	245	1	+	+	NUM
cana-1089	245	2	1	1	NUM
cana-1089	245	3	γ2,k	γ2,k	PROPN
cana-1089	245	4	−	−	PROPN
cana-1089	245	5	4γ3,k	4γ3,k	PRON
cana-1089	245	6	γ2,k	γ2,k	ADJ
cana-1089	245	7	2	2	NUM
cana-1089	246	1	|	|	ADV
cana-1089	247	1	+	+	CCONJ
cana-1089	247	2	1	1	NUM
cana-1089	247	3	2|	2|	NUM
cana-1089	247	4	l1|	l1|	PROPN
cana-1089	247	5	|3	|3	PUNCT
cana-1089	248	1	−	−	PRON
cana-1089	248	2	4γ3,k	4γ3,k	PRON
cana-1089	248	3	γ2,k	γ2,k	ADJ
cana-1089	248	4	2	2	NUM
cana-1089	248	5	|	|	NOUN
cana-1089	248	6	]	]	X
cana-1089	248	7	.	.	PUNCT
cana-1089	249	1	for	for	ADP
cana-1089	249	2	𝜇	𝜇	ADP
cana-1089	249	3	∈	∈	PROPN
cana-1089	249	4	ℂ	ℂ	PROPN
cana-1089	249	5	,	,	PUNCT
cana-1089	249	6	we	we	PRON
cana-1089	249	7	have	have	VERB
cana-1089	249	8	|𝑑2	|𝑑2	NOUN
cana-1089	249	9	−	−	NOUN
cana-1089	249	10	𝜇𝑑1	𝜇𝑑1	NOUN
cana-1089	249	11	2|	2|	PROPN
cana-1089	249	12	≤	≤	NOUN
cana-1089	249	13	𝐿1	𝐿1	VERB
cana-1089	250	1	2|	2|	X
cana-1089	250	2	γ3,𝑘|	γ3,𝑘|	PUNCT
cana-1089	251	1	[	[	X
cana-1089	251	2	max	max	X
cana-1089	251	3	{	{	PUNCT
cana-1089	251	4	1	1	NUM
cana-1089	251	5	;	;	PUNCT
cana-1089	251	6	|	|	ADV
cana-1089	251	7	𝐿2	𝐿2	NOUN
cana-1089	251	8	𝐿1	𝐿1	PROPN
cana-1089	251	9	−	−	PROPN
cana-1089	251	10	𝐿1	𝐿1	PROPN
cana-1089	252	1	+	+	CCONJ
cana-1089	252	2	(	(	PUNCT
cana-1089	252	3	1	1	NUM
cana-1089	252	4	+	+	NUM
cana-1089	252	5	𝜇)𝐿1γ3,𝑘	𝜇)𝐿1γ3,𝑘	NOUN
cana-1089	252	6	γ2,𝑘	γ2,𝑘	PROPN
cana-1089	252	7	2	2	NUM
cana-1089	252	8	|	|	NOUN
cana-1089	252	9	}	}	PUNCT
cana-1089	252	10	+	+	CCONJ
cana-1089	252	11	|1	|1	NUM
cana-1089	252	12	+	+	NUM
cana-1089	252	13	1	1	NUM
cana-1089	252	14	γ2,k	γ2,k	PROPN
cana-1089	252	15	−	−	NOUN
cana-1089	252	16	(	(	PUNCT
cana-1089	252	17	1	1	NUM
cana-1089	252	18	+	+	CCONJ
cana-1089	252	19	𝜇)γ3,k	𝜇)γ3,k	X
cana-1089	252	20	γ2,k	γ2,k	PROPN
cana-1089	252	21	2	2	NUM
cana-1089	252	22	|	|	ADV
cana-1089	253	1	+	+	CCONJ
cana-1089	253	2	1	1	NUM
cana-1089	253	3	2|	2|	NUM
cana-1089	253	4	l1|	l1|	PROPN
cana-1089	253	5	|3	|3	PUNCT
cana-1089	254	1	−	−	PROPN
cana-1089	254	2	(	(	PUNCT
cana-1089	254	3	1	1	NUM
cana-1089	254	4	+	+	CCONJ
cana-1089	254	5	𝜇)γ3,k	𝜇)γ3,k	X
cana-1089	254	6	γ2,k	γ2,k	PROPN
cana-1089	254	7	2	2	NUM
cana-1089	254	8	|	|	NOUN
cana-1089	254	9	]	]	PUNCT
cana-1089	254	10	.	.	PUNCT
cana-1089	255	1	conclusions	conclusion	NOUN
cana-1089	255	2	:	:	PUNCT
cana-1089	255	3	the	the	DET
cana-1089	255	4	function	function	NOUN
cana-1089	255	5	class	class	NOUN
cana-1089	255	6	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	255	7	)	)	PUNCT
cana-1089	255	8	was	be	AUX
cana-1089	255	9	defined	define	VERB
cana-1089	255	10	by	by	ADP
cana-1089	255	11	replacing	replace	VERB
cana-1089	255	12	the	the	DET
cana-1089	255	13	classical	classical	ADJ
cana-1089	255	14	derivative	derivative	NOUN
cana-1089	255	15	with	with	ADP
cana-1089	255	16	a	a	DET
cana-1089	255	17	multiplicative	multiplicative	ADJ
cana-1089	255	18	derivative	derivative	NOUN
cana-1089	255	19	in	in	ADP
cana-1089	255	20	the	the	DET
cana-1089	255	21	well	well	ADV
cana-1089	255	22	-	-	PUNCT
cana-1089	255	23	known	know	VERB
cana-1089	255	24	class	class	NOUN
cana-1089	255	25	of	of	ADP
cana-1089	255	26	starlike	starlike	NOUN
cana-1089	255	27	functions	function	NOUN
cana-1089	255	28	with	with	ADP
cana-1089	255	29	respect	respect	NOUN
cana-1089	255	30	to	to	ADP
cana-1089	255	31	symmetric	symmetric	ADJ
cana-1089	255	32	points	point	NOUN
cana-1089	255	33	.	.	PUNCT
cana-1089	256	1	the	the	DET
cana-1089	256	2	definition	definition	NOUN
cana-1089	256	3	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	256	4	)	)	PUNCT
cana-1089	256	5	is	be	AUX
cana-1089	256	6	not	not	PART
cana-1089	256	7	defined	define	VERB
cana-1089	256	8	for	for	SCONJ
cana-1089	256	9	all	all	DET
cana-1089	256	10	integers	integer	NOUN
cana-1089	256	11	𝑘.	𝑘.	VERB
cana-1089	256	12	in	in	ADP
cana-1089	256	13	fact	fact	NOUN
cana-1089	256	14	the	the	DET
cana-1089	256	15	class	class	NOUN
cana-1089	256	16	exist	exist	VERB
cana-1089	256	17	only	only	ADV
cana-1089	256	18	for	for	ADP
cana-1089	256	19	the	the	DET
cana-1089	256	20	integers	integer	NOUN
cana-1089	256	21	values	value	NOUN
cana-1089	256	22	of	of	ADP
cana-1089	256	23	𝑘	𝑘	NOUN
cana-1089	256	24	,	,	PUNCT
cana-1089	256	25	for	for	ADP
cana-1089	256	26	which	which	PRON
cana-1089	256	27	γ𝑛,𝑘	γ𝑛,𝑘	ADP
cana-1089	256	28	≠	≠	PROPN
cana-1089	256	29	(	(	PUNCT
cana-1089	256	30	𝑛	𝑛	PRON
cana-1089	256	31	−	−	PROPN
cana-1089	256	32	1	1	NUM
cana-1089	256	33	)	)	PUNCT
cana-1089	256	34	.	.	PUNCT
cana-1089	257	1	hence	hence	ADV
cana-1089	257	2	we	we	PRON
cana-1089	257	3	defined	define	VERB
cana-1089	257	4	a	a	DET
cana-1089	257	5	class	class	NOUN
cana-1089	257	6	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	257	7	)	)	PUNCT
cana-1089	257	8	,	,	PUNCT
cana-1089	257	9	influenced	influence	VERB
cana-1089	257	10	by	by	ADP
cana-1089	257	11	the	the	DET
cana-1089	257	12	multiplicative	multiplicative	ADJ
cana-1089	257	13	derivative	derivative	NOUN
cana-1089	257	14	which	which	PRON
cana-1089	257	15	will	will	AUX
cana-1089	257	16	be	be	AUX
cana-1089	257	17	defined	define	VERB
cana-1089	257	18	for	for	ADP
cana-1089	257	19	γ𝑛,𝑘	γ𝑛,𝑘	PUNCT
cana-1089	257	20	=	=	SYM
cana-1089	257	21	(	(	PUNCT
cana-1089	257	22	𝑛	𝑛	PRON
cana-1089	257	23	−	−	PROPN
cana-1089	257	24	1	1	NUM
cana-1089	257	25	)	)	PUNCT
cana-1089	257	26	.	.	PUNCT
cana-1089	258	1	letting	let	VERB
cana-1089	258	2	𝜒	𝜒	PRON
cana-1089	258	3	to	to	PART
cana-1089	258	4	be	be	AUX
cana-1089	258	5	a	a	DET
cana-1089	258	6	specific	specific	ADJ
cana-1089	258	7	conic	conic	ADJ
cana-1089	258	8	region	region	NOUN
cana-1089	258	9	and	and	CCONJ
cana-1089	258	10	varying	vary	VERB
cana-1089	258	11	parameters	parameter	NOUN
cana-1089	258	12	involved	involve	VERB
cana-1089	258	13	in	in	ADP
cana-1089	258	14	the	the	DET
cana-1089	258	15	definitions	definition	NOUN
cana-1089	258	16	1.1	1.1	NUM
cana-1089	258	17	and	and	CCONJ
cana-1089	258	18	1.2	1.2	NUM
cana-1089	258	19	,	,	PUNCT
cana-1089	258	20	the	the	DET
cana-1089	258	21	function	function	NOUN
cana-1089	258	22	classes	class	NOUN
cana-1089	258	23	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	258	24	)	)	PUNCT
cana-1089	258	25	and	and	CCONJ
cana-1089	258	26	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	258	27	)	)	PUNCT
cana-1089	258	28	will	will	AUX
cana-1089	258	29	reduce	reduce	VERB
cana-1089	258	30	to	to	ADP
cana-1089	258	31	classes	class	NOUN
cana-1089	258	32	having	have	VERB
cana-1089	258	33	good	good	ADJ
cana-1089	258	34	geometry	geometry	NOUN
cana-1089	258	35	.	.	PUNCT
cana-1089	259	1	our	our	PRON
cana-1089	259	2	main	main	ADJ
cana-1089	259	3	results	result	NOUN
cana-1089	259	4	have	have	VERB
cana-1089	259	5	wide	wide	ADJ
cana-1089	259	6	applications	application	NOUN
cana-1089	259	7	.	.	PUNCT
cana-1089	260	1	the	the	DET
cana-1089	260	2	classical	classical	ADJ
cana-1089	260	3	starlike	starlike	NOUN
cana-1089	260	4	functions	function	NOUN
cana-1089	260	5	with	with	ADP
cana-1089	260	6	respect	respect	NOUN
cana-1089	260	7	to	to	ADP
cana-1089	260	8	𝑘-symmetric	𝑘-symmetric	ADJ
cana-1089	260	9	points	point	NOUN
cana-1089	260	10	are	be	AUX
cana-1089	260	11	known	know	VERB
cana-1089	260	12	to	to	PART
cana-1089	260	13	be	be	AUX
cana-1089	260	14	univalent	univalent	ADJ
cana-1089	260	15	.	.	PUNCT
cana-1089	261	1	but	but	CCONJ
cana-1089	261	2	the	the	DET
cana-1089	261	3	classes	class	NOUN
cana-1089	261	4	ℳ𝑘(𝜒	ℳ𝑘(𝜒	ADV
cana-1089	261	5	)	)	PUNCT
cana-1089	261	6	and	and	CCONJ
cana-1089	261	7	ℒ𝑘(𝜒	ℒ𝑘(𝜒	NOUN
cana-1089	261	8	)	)	PUNCT
cana-1089	261	9	are	be	AUX
cana-1089	261	10	neither	neither	CCONJ
cana-1089	261	11	a	a	DET
cana-1089	261	12	subclass	subclass	NOUN
cana-1089	261	13	nor	nor	CCONJ
cana-1089	261	14	a	a	DET
cana-1089	261	15	generalized	generalized	ADJ
cana-1089	261	16	class	class	NOUN
cana-1089	261	17	of	of	ADP
cana-1089	261	18	univalent	univalent	ADJ
cana-1089	261	19	functions	function	NOUN
cana-1089	261	20	.	.	PUNCT
cana-1089	262	1	so	so	ADV
cana-1089	262	2	,	,	PUNCT
cana-1089	262	3	the	the	DET
cana-1089	262	4	scope	scope	NOUN
cana-1089	262	5	of	of	ADP
cana-1089	262	6	further	further	ADJ
cana-1089	262	7	research	research	NOUN
cana-1089	262	8	of	of	ADP
cana-1089	262	9	this	this	DET
cana-1089	262	10	paper	paper	NOUN
cana-1089	262	11	are	be	AUX
cana-1089	262	12	to	to	PART
cana-1089	262	13	explore	explore	VERB
cana-1089	262	14	the	the	DET
cana-1089	262	15	relationship	relationship	NOUN
cana-1089	262	16	and	and	CCONJ
cana-1089	262	17	closure	closure	NOUN
cana-1089	262	18	properties	property	NOUN
cana-1089	262	19	with	with	ADP
cana-1089	262	20	the	the	DET
cana-1089	262	21	known	know	VERB
cana-1089	262	22	classes	class	NOUN
cana-1089	262	23	like	like	ADP
cana-1089	262	24	spirallike	spirallike	NOUN
cana-1089	262	25	,	,	PUNCT
cana-1089	262	26	starlike	starlike	NOUN
cana-1089	262	27	,	,	PUNCT
cana-1089	262	28	and	and	CCONJ
cana-1089	262	29	convex	convex	PROPN
cana-1089	262	30	.	.	PUNCT
cana-1089	263	1	conflicts	conflict	NOUN
cana-1089	263	2	of	of	ADP
cana-1089	263	3	interest	interest	NOUN
cana-1089	263	4	:	:	PUNCT
cana-1089	263	5	both	both	DET
cana-1089	263	6	authors	author	NOUN
cana-1089	263	7	declare	declare	VERB
cana-1089	263	8	that	that	SCONJ
cana-1089	263	9	they	they	PRON
cana-1089	263	10	have	have	VERB
cana-1089	263	11	no	no	DET
cana-1089	263	12	conflict	conflict	NOUN
cana-1089	263	13	of	of	ADP
cana-1089	263	14	interest	interest	NOUN
cana-1089	263	15	.	.	PUNCT
cana-1089	264	1	funding	funding	NOUN
cana-1089	264	2	:	:	PUNCT
cana-1089	264	3	this	this	DET
cana-1089	264	4	research	research	NOUN
cana-1089	264	5	study	study	NOUN
cana-1089	264	6	received	receive	VERB
cana-1089	264	7	no	no	DET
cana-1089	264	8	external	external	ADJ
cana-1089	264	9	funding	funding	NOUN
cana-1089	264	10	.	.	PUNCT
cana-1089	265	1	institutional	institutional	ADJ
cana-1089	265	2	review	review	PROPN
cana-1089	265	3	board	board	NOUN
cana-1089	265	4	statement	statement	NOUN
cana-1089	265	5	:	:	PUNCT
cana-1089	265	6	not	not	PART
cana-1089	265	7	applicable	applicable	ADJ
cana-1089	265	8	.	.	PUNCT
cana-1089	266	1	data	datum	NOUN
cana-1089	266	2	availability	availability	NOUN
cana-1089	266	3	:	:	PUNCT
cana-1089	266	4	no	no	DET
cana-1089	266	5	data	datum	NOUN
cana-1089	266	6	was	be	AUX
cana-1089	266	7	used	use	VERB
cana-1089	266	8	to	to	PART
cana-1089	266	9	support	support	VERB
cana-1089	266	10	this	this	DET
cana-1089	266	11	study	study	NOUN
cana-1089	266	12	.	.	PUNCT
cana-1089	267	1	communications	communication	NOUN
cana-1089	267	2	on	on	ADP
cana-1089	267	3	applied	apply	VERB
cana-1089	267	4	nonlinear	nonlinear	ADJ
cana-1089	267	5	analysis	analysis	NOUN
cana-1089	267	6	issn	issn	NOUN
cana-1089	267	7	:	:	PUNCT
cana-1089	267	8	1074	1074	NUM
cana-1089	267	9	-	-	PUNCT
cana-1089	267	10	133x	133x	NUM
cana-1089	267	11	vol	vol	NOUN
cana-1089	267	12	31	31	NUM
cana-1089	267	13	no	no	NOUN
cana-1089	267	14	.	.	PUNCT
cana-1089	268	1	5s	5s	NUM
cana-1089	268	2	(	(	PUNCT
cana-1089	268	3	2024	2024	NUM
cana-1089	268	4	)	)	PUNCT
cana-1089	268	5	550	550	NUM
cana-1089	268	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	268	7	references	reference	NOUN
cana-1089	268	8	:	:	PUNCT
cana-1089	269	1	[	[	X
cana-1089	269	2	1	1	X
cana-1089	269	3	]	]	X
cana-1089	269	4	d.	d.	PROPN
cana-1089	269	5	alimohammadi	alimohammadi	PROPN
cana-1089	269	6	,	,	PUNCT
cana-1089	269	7	n.	n.	PROPN
cana-1089	269	8	e.	e.	PROPN
cana-1089	269	9	cho	cho	PROPN
cana-1089	269	10	,	,	PUNCT
cana-1089	269	11	e.	e.	PROPN
cana-1089	269	12	a.	a.	PROPN
cana-1089	269	13	adegani	adegani	PROPN
cana-1089	269	14	and	and	CCONJ
cana-1089	269	15	a.	a.	NOUN
cana-1089	269	16	motamednezhad	motamednezhad	PROPN
cana-1089	269	17	,	,	PUNCT
cana-1089	269	18	argument	argument	NOUN
cana-1089	269	19	and	and	CCONJ
cana-1089	269	20	coefficient	coefficient	NOUN
cana-1089	269	21	estimates	estimate	NOUN
cana-1089	269	22	for	for	ADP
cana-1089	269	23	certain	certain	ADJ
cana-1089	269	24	analytic	analytic	ADJ
cana-1089	269	25	functions	function	NOUN
cana-1089	269	26	,	,	PUNCT
cana-1089	269	27	mathematics	mathematic	NOUN
cana-1089	269	28	,	,	PUNCT
cana-1089	269	29	8	8	NUM
cana-1089	269	30	(	(	PUNCT
cana-1089	269	31	2020	2020	NUM
cana-1089	269	32	)	)	PUNCT
cana-1089	269	33	,	,	PUNCT
cana-1089	269	34	no.1	no.1	NUM
cana-1089	269	35	,	,	PUNCT
cana-1089	269	36	88	88	NUM
cana-1089	269	37	;	;	PUNCT
cana-1089	269	38	doi	doi	NOUN
cana-1089	269	39	.	.	PUNCT
cana-1089	269	40	:	:	PUNCT
cana-1089	270	1	https://doi.org/10.3390/math8010088	https://doi.org/10.3390/math8010088	PROPN
cana-1089	271	1	[	[	X
cana-1089	271	2	2	2	X
cana-1089	271	3	]	]	PUNCT
cana-1089	271	4	d.	d.	PROPN
cana-1089	271	5	alimohammadi	alimohammadi	PROPN
cana-1089	271	6	,	,	PUNCT
cana-1089	271	7	e.	e.	PROPN
cana-1089	271	8	a.	a.	PROPN
cana-1089	271	9	adegani	adegani	PROPN
cana-1089	271	10	and	and	CCONJ
cana-1089	271	11	t.	t.	NOUN
cana-1089	271	12	bulboacă	bulboacă	NOUN
cana-1089	271	13	and	and	CCONJ
cana-1089	271	14	n.	n.	PROPN
cana-1089	271	15	e.	e.	PROPN
cana-1089	271	16	cho	cho	PROPN
cana-1089	271	17	,	,	PUNCT
cana-1089	271	18	logarithmic	logarithmic	ADJ
cana-1089	271	19	coefficients	coefficient	NOUN
cana-1089	271	20	for	for	ADP
cana-1089	271	21	classes	class	NOUN
cana-1089	271	22	related	relate	VERB
cana-1089	271	23	to	to	ADP
cana-1089	271	24	convex	convex	NOUN
cana-1089	271	25	functions	function	NOUN
cana-1089	271	26	,	,	PUNCT
cana-1089	271	27	bull	bull	NOUN
cana-1089	271	28	.	.	PUNCT
cana-1089	272	1	malays	malays	PROPN
cana-1089	272	2	.	.	PUNCT
cana-1089	273	1	math	math	NOUN
cana-1089	273	2	.	.	PUNCT
cana-1089	274	1	sci	sci	PROPN
cana-1089	274	2	.	.	PROPN
cana-1089	274	3	soc	soc	PROPN
cana-1089	274	4	.	.	PUNCT
cana-1089	274	5	,	,	PUNCT
cana-1089	274	6	44	44	NUM
cana-1089	274	7	(	(	PUNCT
cana-1089	274	8	2021	2021	NUM
cana-1089	274	9	)	)	PUNCT
cana-1089	274	10	,	,	PUNCT
cana-1089	274	11	2659–2673	2659–2673	NUM
cana-1089	274	12	.	.	PUNCT
cana-1089	275	1	doi	doi	PROPN
cana-1089	275	2	.	.	PUNCT
cana-1089	275	3	:	:	PUNCT
cana-1089	276	1	https://doi.org/10.1007/s40840-02101085-z	https://doi.org/10.1007/s40840-02101085-z	PROPN
cana-1089	277	1	[	[	X
cana-1089	277	2	3	3	X
cana-1089	277	3	]	]	X
cana-1089	277	4	e.	e.	PROPN
cana-1089	277	5	a.	a.	PROPN
cana-1089	277	6	adegani	adegani	PROPN
cana-1089	277	7	,	,	PUNCT
cana-1089	277	8	n.	n.	PROPN
cana-1089	277	9	e.	e.	PROPN
cana-1089	277	10	cho	cho	PROPN
cana-1089	277	11	and	and	CCONJ
cana-1089	277	12	m.	m.	NOUN
cana-1089	277	13	jafari	jafari	PROPN
cana-1089	277	14	,	,	PUNCT
cana-1089	277	15	logarithmic	logarithmic	ADJ
cana-1089	277	16	coefficients	coefficient	NOUN
cana-1089	277	17	for	for	ADP
cana-1089	277	18	univalent	univalent	ADJ
cana-1089	277	19	functions	function	NOUN
cana-1089	277	20	defined	define	VERB
cana-1089	277	21	by	by	ADP
cana-1089	277	22	subordination	subordination	NOUN
cana-1089	277	23	,	,	PUNCT
cana-1089	277	24	mathematics	mathematic	NOUN
cana-1089	277	25	,	,	PUNCT
cana-1089	277	26	7	7	NUM
cana-1089	277	27	(	(	PUNCT
cana-1089	277	28	2019	2019	NUM
cana-1089	277	29	)	)	PUNCT
cana-1089	277	30	,	,	PUNCT
cana-1089	277	31	no.5	no.5	PROPN
cana-1089	277	32	,	,	PUNCT
cana-1089	277	33	408	408	NUM
cana-1089	277	34	;	;	PUNCT
cana-1089	277	35	doi	doi	NOUN
cana-1089	277	36	.	.	PUNCT
cana-1089	277	37	:	:	PUNCT
cana-1089	277	38	https://doi.org/10.3390/math7050408	https://doi.org/10.3390/math7050408	X
cana-1089	277	39	.	.	PUNCT
cana-1089	278	1	[	[	X
cana-1089	278	2	4	4	X
cana-1089	278	3	]	]	PUNCT
cana-1089	278	4	e.	e.	PROPN
cana-1089	278	5	a.	a.	PROPN
cana-1089	278	6	adegani	adegani	PROPN
cana-1089	278	7	,	,	PUNCT
cana-1089	278	8	t.	t.	NOUN
cana-1089	278	9	bulboacă	bulboacă	NOUN
cana-1089	278	10	,	,	PUNCT
cana-1089	278	11	n.	n.	PROPN
cana-1089	278	12	h.	h.	PROPN
cana-1089	278	13	mohammed	mohammed	PROPN
cana-1089	278	14	and	and	CCONJ
cana-1089	278	15	p.	p.	PROPN
cana-1089	278	16	zaprawa	zaprawa	PROPN
cana-1089	278	17	,	,	PUNCT
cana-1089	278	18	solution	solution	NOUN
cana-1089	278	19	of	of	ADP
cana-1089	278	20	logarithmic	logarithmic	ADJ
cana-1089	278	21	coefficients	coefficient	NOUN
cana-1089	278	22	conjectures	conjecture	VERB
cana-1089	278	23	for	for	ADP
cana-1089	278	24	some	some	DET
cana-1089	278	25	classes	class	NOUN
cana-1089	278	26	of	of	ADP
cana-1089	278	27	convex	convex	NOUN
cana-1089	278	28	functions	function	NOUN
cana-1089	278	29	,	,	PUNCT
cana-1089	278	30	mathematica	mathematica	PROPN
cana-1089	278	31	slovaca	slovaca	PROPN
cana-1089	278	32	,	,	PUNCT
cana-1089	278	33	73	73	NUM
cana-1089	278	34	(	(	PUNCT
cana-1089	278	35	2023	2023	NUM
cana-1089	278	36	)	)	PUNCT
cana-1089	278	37	,	,	PUNCT
cana-1089	278	38	no.1	no.1	NUM
cana-1089	278	39	,	,	PUNCT
cana-1089	278	40	pp	pp	ADJ
cana-1089	278	41	.	.	PUNCT
cana-1089	279	1	79	79	NUM
cana-1089	279	2	-	-	SYM
cana-1089	279	3	88	88	NUM
cana-1089	279	4	.	.	PUNCT
cana-1089	279	5	doi	doi	PROPN
cana-1089	279	6	.	.	PUNCT
cana-1089	279	7	:	:	PUNCT
cana-1089	279	8	https://doi.org/10.1515/ms-2023-0009	https://doi.org/10.1515/ms-2023-0009	X
cana-1089	280	1	[	[	X
cana-1089	280	2	5	5	NUM
cana-1089	280	3	]	]	PUNCT
cana-1089	280	4	a.	a.	PROPN
cana-1089	280	5	e.	e.	PROPN
cana-1089	280	6	bashirov	bashirov	PROPN
cana-1089	280	7	,	,	PUNCT
cana-1089	280	8	e.	e.	PROPN
cana-1089	280	9	m.	m.	PROPN
cana-1089	280	10	kurpinar	kurpinar	PROPN
cana-1089	280	11	and	and	CCONJ
cana-1089	280	12	a.	a.	NOUN
cana-1089	280	13	őzyapıcı	őzyapıcı	PROPN
cana-1089	280	14	,	,	PUNCT
cana-1089	280	15	multiplicative	multiplicative	ADJ
cana-1089	280	16	calculus	calculus	NOUN
cana-1089	280	17	and	and	CCONJ
cana-1089	280	18	its	its	PRON
cana-1089	280	19	applications	application	NOUN
cana-1089	280	20	,	,	PUNCT
cana-1089	280	21	j.	j.	PROPN
cana-1089	280	22	math	math	PROPN
cana-1089	280	23	.	.	PUNCT
cana-1089	281	1	anal	anal	PROPN
cana-1089	281	2	.	.	PUNCT
cana-1089	281	3	appl	appl	PROPN
cana-1089	281	4	.	.	PUNCT
cana-1089	282	1	337	337	NUM
cana-1089	282	2	(	(	PUNCT
cana-1089	282	3	2008	2008	NUM
cana-1089	282	4	)	)	PUNCT
cana-1089	282	5	,	,	PUNCT
cana-1089	282	6	no	no	INTJ
cana-1089	282	7	.	.	NOUN
cana-1089	282	8	1	1	NUM
cana-1089	282	9	,	,	PUNCT
cana-1089	282	10	36–48	36–48	NUM
cana-1089	282	11	.	.	PUNCT
cana-1089	283	1	doi	doi	PROPN
cana-1089	283	2	:	:	PUNCT
cana-1089	283	3	https://doi.org/10.1016/j.jmaa.2007.03.081	https://doi.org/10.1016/j.jmaa.2007.03.081	PROPN
cana-1089	283	4	.	.	PUNCT
cana-1089	284	1	[	[	X
cana-1089	284	2	6	6	NUM
cana-1089	284	3	]	]	PUNCT
cana-1089	284	4	a.	a.	PROPN
cana-1089	284	5	e.	e.	PROPN
cana-1089	284	6	bashirov	bashirov	PROPN
cana-1089	284	7	,	,	PUNCT
cana-1089	284	8	e.mısırlı	e.mısırlı	PROPN
cana-1089	284	9	,	,	PUNCT
cana-1089	284	10	y.	y.	NOUN
cana-1089	284	11	tandoğdu	tandoğdu	NOUN
cana-1089	284	12	and	and	CCONJ
cana-1089	284	13	a.	a.	NOUN
cana-1089	284	14	őzyapıcı	őzyapıcı	PROPN
cana-1089	284	15	,	,	PUNCT
cana-1089	284	16	on	on	ADP
cana-1089	284	17	modeling	modeling	NOUN
cana-1089	284	18	with	with	ADP
cana-1089	284	19	multiplicative	multiplicative	ADJ
cana-1089	284	20	differential	differential	ADJ
cana-1089	284	21	equations	equation	NOUN
cana-1089	284	22	,	,	PUNCT
cana-1089	284	23	appl	appl	PROPN
cana-1089	284	24	.	.	PROPN
cana-1089	284	25	math	math	PROPN
cana-1089	284	26	.	.	PUNCT
cana-1089	285	1	j.	j.	PROPN
cana-1089	285	2	chin	chin	PROPN
cana-1089	285	3	.	.	PROPN
cana-1089	286	1	univ	univ	PROPN
cana-1089	286	2	.	.	PROPN
cana-1089	287	1	26	26	NUM
cana-1089	287	2	,	,	PUNCT
cana-1089	287	3	425–438	425–438	NUM
cana-1089	287	4	(	(	PUNCT
cana-1089	287	5	2011	2011	NUM
cana-1089	287	6	)	)	PUNCT
cana-1089	287	7	.	.	PUNCT
cana-1089	288	1	doi	doi	PROPN
cana-1089	288	2	.	.	PUNCT
cana-1089	288	3	:	:	PUNCT
cana-1089	288	4	https://doi.org/10.1007/s11766-011-2767-6	https://doi.org/10.1007/s11766-011-2767-6	NUM
cana-1089	289	1	[	[	X
cana-1089	289	2	7	7	NUM
cana-1089	289	3	]	]	PUNCT
cana-1089	289	4	a.	a.	PROPN
cana-1089	289	5	e.	e.	PROPN
cana-1089	289	6	bashirov	bashirov	PROPN
cana-1089	289	7	and	and	CCONJ
cana-1089	289	8	m.	m.	PROPN
cana-1089	289	9	riza	riza	PROPN
cana-1089	289	10	,	,	PUNCT
cana-1089	289	11	on	on	ADP
cana-1089	289	12	complex	complex	ADJ
cana-1089	289	13	multiplicative	multiplicative	ADJ
cana-1089	289	14	differentiation	differentiation	NOUN
cana-1089	289	15	.	.	PUNCT
cana-1089	290	1	twms	twms	PROPN
cana-1089	290	2	j.	j.	PROPN
cana-1089	290	3	appl	appl	PROPN
cana-1089	290	4	.	.	PUNCT
cana-1089	291	1	eng	eng	PROPN
cana-1089	291	2	.	.	PROPN
cana-1089	291	3	math	math	PROPN
cana-1089	291	4	.	.	PUNCT
cana-1089	292	1	1	1	NUM
cana-1089	292	2	(	(	PUNCT
cana-1089	292	3	2011	2011	NUM
cana-1089	292	4	)	)	PUNCT
cana-1089	292	5	,	,	PUNCT
cana-1089	292	6	no.1	no.1	NUM
cana-1089	292	7	,	,	PUNCT
cana-1089	292	8	75–85	75–85	NOUN
cana-1089	292	9	.	.	PUNCT
cana-1089	293	1	[	[	X
cana-1089	293	2	8	8	NUM
cana-1089	293	3	]	]	X
cana-1089	293	4	d.	d.	PROPN
cana-1089	293	5	breaz	breaz	PROPN
cana-1089	293	6	,	,	PUNCT
cana-1089	293	7	k.	k.	PROPN
cana-1089	293	8	r.	r.	PROPN
cana-1089	293	9	karthikeyan	karthikeyan	PROPN
cana-1089	293	10	,	,	PUNCT
cana-1089	293	11	e.	e.	PROPN
cana-1089	293	12	umadevi	umadevi	PROPN
cana-1089	293	13	and	and	CCONJ
cana-1089	293	14	a.	a.	NOUN
cana-1089	293	15	senguttuvan	senguttuvan	PROPN
cana-1089	293	16	,	,	PUNCT
cana-1089	293	17	some	some	DET
cana-1089	293	18	properties	property	NOUN
cana-1089	293	19	of	of	ADP
cana-1089	293	20	bazilevič	bazilevič	NOUN
cana-1089	293	21	functions	function	NOUN
cana-1089	293	22	involving	involve	VERB
cana-1089	293	23	srivastava	srivastava	PROPN
cana-1089	293	24	–	–	PUNCT
cana-1089	293	25	tomovski	tomovski	ADJ
cana-1089	293	26	operator	operator	NOUN
cana-1089	293	27	.	.	PUNCT
cana-1089	294	1	axioms	axiom	NOUN
cana-1089	294	2	,	,	PUNCT
cana-1089	294	3	2022	2022	NUM
cana-1089	294	4	,	,	PUNCT
cana-1089	294	5	11	11	NUM
cana-1089	294	6	,	,	PUNCT
cana-1089	294	7	687	687	NUM
cana-1089	294	8	.	.	PUNCT
cana-1089	295	1	doi	doi	NOUN
cana-1089	295	2	.	.	PUNCT
cana-1089	295	3	:	:	PUNCT
cana-1089	296	1	https://doi.org/10.3390/axioms11120687	https://doi.org/10.3390/axioms11120687	NOUN
cana-1089	296	2	[	[	X
cana-1089	296	3	9	9	NUM
cana-1089	296	4	]	]	PUNCT
cana-1089	296	5	a.	a.	PROPN
cana-1089	296	6	w.	w.	PROPN
cana-1089	296	7	goodman	goodman	PROPN
cana-1089	296	8	,	,	PUNCT
cana-1089	296	9	univalent	univalent	ADJ
cana-1089	296	10	functions	function	NOUN
cana-1089	296	11	.	.	PUNCT
cana-1089	297	1	vol	vol	NOUN
cana-1089	297	2	.	.	PUNCT
cana-1089	297	3	ii	ii	PROPN
cana-1089	297	4	,	,	PUNCT
cana-1089	297	5	mariner	mariner	PROPN
cana-1089	297	6	publishing	publishing	PROPN
cana-1089	297	7	co.	co.	PROPN
cana-1089	297	8	,	,	PUNCT
cana-1089	297	9	inc	inc	PROPN
cana-1089	297	10	.	.	PROPN
cana-1089	297	11	,	,	PUNCT
cana-1089	297	12	tampa	tampa	PROPN
cana-1089	297	13	,	,	PUNCT
cana-1089	297	14	fl	fl	PROPN
cana-1089	297	15	,	,	PUNCT
cana-1089	297	16	1983	1983	NUM
cana-1089	297	17	.	.	PUNCT
cana-1089	298	1	[	[	X
cana-1089	298	2	10	10	NUM
cana-1089	298	3	]	]	PUNCT
cana-1089	298	4	k.	k.	PROPN
cana-1089	298	5	r.	r.	PROPN
cana-1089	298	6	karthikeyan	karthikeyan	PROPN
cana-1089	298	7	and	and	CCONJ
cana-1089	298	8	g.	g.	PROPN
cana-1089	298	9	murugusundaramoorthy	murugusundaramoorthy	PROPN
cana-1089	298	10	,	,	PUNCT
cana-1089	298	11	properties	property	NOUN
cana-1089	298	12	of	of	ADP
cana-1089	298	13	a	a	DET
cana-1089	298	14	class	class	NOUN
cana-1089	298	15	of	of	ADP
cana-1089	298	16	analytic	analytic	ADJ
cana-1089	298	17	functions	function	NOUN
cana-1089	298	18	influenced	influence	VERB
cana-1089	298	19	by	by	ADP
cana-1089	298	20	multiplicative	multiplicative	ADJ
cana-1089	298	21	calculus	calculus	NOUN
cana-1089	298	22	,	,	PUNCT
cana-1089	298	23	fractal	fractal	ADJ
cana-1089	298	24	fract	fract	NOUN
cana-1089	298	25	.	.	PUNCT
cana-1089	298	26	,	,	PUNCT
cana-1089	298	27	8	8	NUM
cana-1089	298	28	(	(	PUNCT
cana-1089	298	29	2024	2024	NUM
cana-1089	298	30	)	)	PUNCT
cana-1089	298	31	,	,	PUNCT
cana-1089	298	32	no.3	no.3	VERB
cana-1089	298	33	,	,	PUNCT
cana-1089	298	34	131	131	NUM
cana-1089	298	35	.	.	PUNCT
cana-1089	298	36	doi	doi	PROPN
cana-1089	298	37	.	.	PUNCT
cana-1089	298	38	:	:	PUNCT
cana-1089	299	1	https://doi.org/10.3390/fractalfract8030131	https://doi.org/10.3390/fractalfract8030131	VERB
cana-1089	300	1	[	[	X
cana-1089	300	2	11	11	NUM
cana-1089	300	3	]	]	PUNCT
cana-1089	300	4	k.	k.	PROPN
cana-1089	300	5	r.	r.	PROPN
cana-1089	300	6	karthikeyan	karthikeyan	PROPN
cana-1089	300	7	,	,	PUNCT
cana-1089	300	8	e.	e.	PROPN
cana-1089	300	9	umadevi	umadevi	PROPN
cana-1089	300	10	and	and	CCONJ
cana-1089	300	11	g.	g.	PROPN
cana-1089	300	12	murugusundaramoorthy	murugusundaramoorthy	ADJ
cana-1089	300	13	,	,	PUNCT
cana-1089	300	14	subclasses	subclass	NOUN
cana-1089	300	15	of	of	ADP
cana-1089	300	16	close	close	NOUN
cana-1089	300	17	-	-	PUNCT
cana-1089	300	18	to	to	ADP
cana-1089	300	19	-	-	PUNCT
cana-1089	300	20	convex	convex	NOUN
cana-1089	300	21	functions	function	NOUN
cana-1089	300	22	involving	involve	VERB
cana-1089	300	23	the	the	DET
cana-1089	300	24	multiplicative	multiplicative	ADJ
cana-1089	300	25	derivative	derivative	NOUN
cana-1089	300	26	,	,	PUNCT
cana-1089	300	27	preprint	preprint	NOUN
cana-1089	300	28	.	.	PUNCT
cana-1089	301	1	[	[	X
cana-1089	301	2	12	12	NUM
cana-1089	301	3	]	]	PUNCT
cana-1089	301	4	k.	k.	PROPN
cana-1089	301	5	r.	r.	PROPN
cana-1089	301	6	karthikeyan	karthikeyan	PROPN
cana-1089	301	7	and	and	CCONJ
cana-1089	301	8	a.	a.	PROPN
cana-1089	301	9	senguttuvan	senguttuvan	PROPN
cana-1089	301	10	,	,	PUNCT
cana-1089	301	11	on	on	ADP
cana-1089	301	12	a	a	DET
cana-1089	301	13	characterization	characterization	NOUN
cana-1089	301	14	of	of	ADP
cana-1089	301	15	starlike	starlike	NOUN
cana-1089	301	16	functions	function	NOUN
cana-1089	301	17	with	with	ADP
cana-1089	301	18	respect	respect	NOUN
cana-1089	301	19	to	to	ADP
cana-1089	301	20	(	(	PUNCT
cana-1089	301	21	2𝑗	2𝑗	NOUN
cana-1089	301	22	,	,	PUNCT
cana-1089	301	23	ℓ	ℓ	NOUN
cana-1089	301	24	)	)	PUNCT
cana-1089	301	25	−symmetric	−symmetric	ADJ
cana-1089	301	26	conjugate	conjugate	ADJ
cana-1089	301	27	points	point	NOUN
cana-1089	301	28	,	,	PUNCT
cana-1089	301	29	asian	asian	ADJ
cana-1089	301	30	-	-	PUNCT
cana-1089	301	31	european	european	PROPN
cana-1089	301	32	j.	j.	PROPN
cana-1089	301	33	math	math	PROPN
cana-1089	301	34	.	.	PUNCT
cana-1089	302	1	,	,	PUNCT
cana-1089	302	2	16	16	NUM
cana-1089	302	3	(	(	PUNCT
cana-1089	302	4	2023	2023	NUM
cana-1089	302	5	)	)	PUNCT
cana-1089	302	6	,	,	PUNCT
cana-1089	302	7	no.10	no.10	NOUN
cana-1089	302	8	,	,	PUNCT
cana-1089	302	9	2350180	2350180	NUM
cana-1089	302	10	.	.	PUNCT
cana-1089	303	1	doi	doi	PROPN
cana-1089	303	2	.	.	PUNCT
cana-1089	303	3	:	:	PUNCT
cana-1089	304	1	https://doi.org/10.1142/s1793557123501802	https://doi.org/10.1142/s1793557123501802	PROPN
cana-1089	304	2	[	[	X
cana-1089	304	3	13	13	NUM
cana-1089	304	4	]	]	PUNCT
cana-1089	304	5	k.	k.	PROPN
cana-1089	304	6	r.	r.	PROPN
cana-1089	304	7	karthikeyan	karthikeyan	PROPN
cana-1089	304	8	,	,	PUNCT
cana-1089	304	9	g.	g.	PROPN
cana-1089	304	10	murugusundaramoorthy	murugusundaramoorthy	PROPN
cana-1089	304	11	,	,	PUNCT
cana-1089	304	12	s.	s.	PROPN
cana-1089	304	13	d.	d.	PROPN
cana-1089	304	14	purohit	purohit	PROPN
cana-1089	304	15	and	and	CCONJ
cana-1089	304	16	d.	d.	PROPN
cana-1089	304	17	l.	l.	PROPN
cana-1089	304	18	suthar	suthar	PROPN
cana-1089	304	19	,	,	PUNCT
cana-1089	304	20	certain	certain	ADJ
cana-1089	304	21	class	class	NOUN
cana-1089	304	22	of	of	ADP
cana-1089	304	23	analytic	analytic	ADJ
cana-1089	304	24	functions	function	NOUN
cana-1089	304	25	with	with	ADP
cana-1089	304	26	respect	respect	NOUN
cana-1089	304	27	to	to	ADP
cana-1089	304	28	symmetric	symmetric	ADJ
cana-1089	304	29	points	point	NOUN
cana-1089	304	30	defined	define	VERB
cana-1089	304	31	by	by	ADP
cana-1089	304	32	𝑄	𝑄	PROPN
cana-1089	304	33	−calculus	−calculus	PROPN
cana-1089	304	34	,	,	PUNCT
cana-1089	304	35	j.	j.	PROPN
cana-1089	304	36	math	math	PROPN
cana-1089	304	37	.	.	PUNCT
cana-1089	304	38	2021	2021	NUM
cana-1089	304	39	,	,	PUNCT
cana-1089	304	40	art	art	NOUN
cana-1089	304	41	.	.	PUNCT
cana-1089	305	1	i	i	PRON
cana-1089	305	2	d	d	PROPN
cana-1089	305	3	8298848	8298848	NUM
cana-1089	305	4	,	,	PUNCT
cana-1089	305	5	9	9	NUM
cana-1089	305	6	pp	pp	NOUN
cana-1089	305	7	.	.	PUNCT
cana-1089	306	1	[	[	X
cana-1089	306	2	14	14	NUM
cana-1089	306	3	]	]	PUNCT
cana-1089	306	4	k.	k.	PROPN
cana-1089	306	5	r.	r.	PROPN
cana-1089	306	6	karthikeyan	karthikeyan	PROPN
cana-1089	306	7	,	,	PUNCT
cana-1089	306	8	g.	g.	PROPN
cana-1089	306	9	murugusundaramoorthy	murugusundaramoorthy	PROPN
cana-1089	306	10	and	and	CCONJ
cana-1089	306	11	n.	n.	PROPN
cana-1089	306	12	e.	e.	PROPN
cana-1089	306	13	cho	cho	PROPN
cana-1089	306	14	,	,	PUNCT
cana-1089	306	15	some	some	DET
cana-1089	306	16	inequalities	inequality	NOUN
cana-1089	306	17	on	on	ADP
cana-1089	306	18	bazilevič	bazilevič	PROPN
cana-1089	306	19	class	class	PROPN
cana-1089	306	20	of	of	ADP
cana-1089	306	21	functions	function	NOUN
cana-1089	306	22	involving	involve	VERB
cana-1089	306	23	quasi	quasi	NOUN
cana-1089	306	24	-	-	NOUN
cana-1089	306	25	subordination	subordination	NOUN
cana-1089	306	26	,	,	PUNCT
cana-1089	306	27	aims	aim	VERB
cana-1089	306	28	math	math	NOUN
cana-1089	306	29	.	.	PUNCT
cana-1089	307	1	6	6	NUM
cana-1089	307	2	(	(	PUNCT
cana-1089	307	3	2021	2021	NUM
cana-1089	307	4	)	)	PUNCT
cana-1089	307	5	,	,	PUNCT
cana-1089	307	6	no.7	no.7	PROPN
cana-1089	307	7	,	,	PUNCT
cana-1089	307	8	7111	7111	NUM
cana-1089	307	9	-	-	SYM
cana-1089	307	10	-7124	-7124	NOUN
cana-1089	307	11	.	.	PUNCT
cana-1089	308	1	[	[	X
cana-1089	308	2	15	15	NUM
cana-1089	308	3	]	]	X
cana-1089	308	4	w.	w.	PROPN
cana-1089	308	5	c.	c.	PROPN
cana-1089	308	6	ma	ma	PROPN
cana-1089	308	7	and	and	CCONJ
cana-1089	308	8	d.	d.	PROPN
cana-1089	308	9	minda	minda	PROPN
cana-1089	308	10	,	,	PUNCT
cana-1089	308	11	a	a	DET
cana-1089	308	12	unified	unified	ADJ
cana-1089	308	13	treatment	treatment	NOUN
cana-1089	308	14	of	of	ADP
cana-1089	308	15	some	some	DET
cana-1089	308	16	special	special	ADJ
cana-1089	308	17	classes	class	NOUN
cana-1089	308	18	of	of	ADP
cana-1089	308	19	univalent	univalent	ADJ
cana-1089	308	20	functions	function	NOUN
cana-1089	308	21	,	,	PUNCT
cana-1089	308	22	in	in	ADP
cana-1089	308	23	proceedings	proceeding	NOUN
cana-1089	308	24	of	of	ADP
cana-1089	308	25	the	the	DET
cana-1089	308	26	conference	conference	NOUN
cana-1089	308	27	on	on	ADP
cana-1089	308	28	complex	complex	ADJ
cana-1089	308	29	analysis	analysis	NOUN
cana-1089	308	30	(	(	PUNCT
cana-1089	308	31	tianjin	tianjin	NOUN
cana-1089	308	32	,	,	PUNCT
cana-1089	308	33	1992	1992	NUM
cana-1089	308	34	)	)	PUNCT
cana-1089	308	35	,	,	PUNCT
cana-1089	308	36	157	157	NUM
cana-1089	308	37	-	-	PUNCT
cana-1089	308	38	-169	-169	PROPN
cana-1089	308	39	,	,	PUNCT
cana-1089	308	40	conf	conf	NOUN
cana-1089	308	41	.	.	PUNCT
cana-1089	309	1	proc	proc	PROPN
cana-1089	309	2	.	.	PUNCT
cana-1089	310	1	lecture	lecture	NOUN
cana-1089	310	2	notes	note	VERB
cana-1089	310	3	anal	anal	ADJ
cana-1089	310	4	.	.	PUNCT
cana-1089	311	1	,	,	PUNCT
cana-1089	311	2	i	i	PRON
cana-1089	311	3	,	,	PUNCT
cana-1089	311	4	int	int	PROPN
cana-1089	311	5	.	.	PUNCT
cana-1089	312	1	press	press	PROPN
cana-1089	312	2	,	,	PUNCT
cana-1089	312	3	cambridge	cambridge	PROPN
cana-1089	312	4	,	,	PUNCT
cana-1089	312	5	ma	ma	PROPN
cana-1089	312	6	.	.	PUNCT
cana-1089	313	1	[	[	X
cana-1089	313	2	16	16	NUM
cana-1089	313	3	]	]	X
cana-1089	313	4	i.	i.	PROPN
cana-1089	313	5	m.	m.	PROPN
cana-1089	313	6	milin	milin	PROPN
cana-1089	313	7	,	,	PUNCT
cana-1089	313	8	univalent	univalent	ADJ
cana-1089	313	9	functions	function	NOUN
cana-1089	313	10	and	and	CCONJ
cana-1089	313	11	orthonormal	orthonormal	ADJ
cana-1089	313	12	systems	system	NOUN
cana-1089	313	13	.	.	PUNCT
cana-1089	314	1	nauka	nauka	PROPN
cana-1089	314	2	,	,	PUNCT
cana-1089	314	3	moscow	moscow	PROPN
cana-1089	314	4	(	(	PUNCT
cana-1089	314	5	1971	1971	NUM
cana-1089	314	6	)	)	PUNCT
cana-1089	314	7	.	.	PUNCT
cana-1089	315	1	(	(	PUNCT
cana-1089	315	2	in	in	ADP
cana-1089	315	3	russian	russian	PROPN
cana-1089	315	4	)	)	PUNCT
cana-1089	315	5	;	;	PUNCT
cana-1089	315	6	english	english	ADJ
cana-1089	315	7	translation	translation	NOUN
cana-1089	315	8	:	:	PUNCT
cana-1089	315	9	american	american	PROPN
cana-1089	315	10	mathematical	mathematical	PROPN
cana-1089	315	11	society	society	NOUN
cana-1089	315	12	,	,	PUNCT
cana-1089	315	13	providence	providence	NOUN
cana-1089	315	14	(	(	PUNCT
cana-1089	315	15	1977	1977	NUM
cana-1089	315	16	)	)	PUNCT
cana-1089	315	17	.	.	PUNCT
cana-1089	316	1	[	[	X
cana-1089	316	2	17	17	NUM
cana-1089	316	3	]	]	PUNCT
cana-1089	316	4	m.	m.	NOUN
cana-1089	316	5	obradovič	obradovič	PROPN
cana-1089	316	6	,	,	PUNCT
cana-1089	316	7	s.	s.	PROPN
cana-1089	316	8	ponnusamy	ponnusamy	PROPN
cana-1089	316	9	and	and	CCONJ
cana-1089	316	10	k.	k.	PROPN
cana-1089	316	11	j.	j.	PROPN
cana-1089	316	12	wirths	wirths	PROPN
cana-1089	316	13	,	,	PUNCT
cana-1089	316	14	logarithmic	logarithmic	ADJ
cana-1089	316	15	coefficients	coefficient	NOUN
cana-1089	316	16	and	and	CCONJ
cana-1089	316	17	a	a	DET
cana-1089	316	18	coefficient	coefficient	NOUN
cana-1089	316	19	conjecture	conjecture	NOUN
cana-1089	316	20	for	for	ADP
cana-1089	316	21	univalent	univalent	ADJ
cana-1089	316	22	functions	function	NOUN
cana-1089	316	23	,	,	PUNCT
cana-1089	316	24	monatsh	monatsh	ADJ
cana-1089	316	25	math	math	NOUN
cana-1089	316	26	,	,	PUNCT
cana-1089	316	27	185	185	NUM
cana-1089	316	28	(	(	PUNCT
cana-1089	316	29	2018	2018	NUM
cana-1089	316	30	)	)	PUNCT
cana-1089	316	31	,	,	PUNCT
cana-1089	316	32	489–501	489–501	NUM
cana-1089	316	33	.	.	PUNCT
cana-1089	317	1	doi	doi	PROPN
cana-1089	317	2	.	.	PUNCT
cana-1089	317	3	:	:	PUNCT
cana-1089	317	4	https://doi.org/10.1007/s00605-017-1024-3	https://doi.org/10.1007/s00605-017-1024-3	NUM
cana-1089	317	5	.	.	PUNCT
cana-1089	318	1	[	[	X
cana-1089	318	2	18	18	NUM
cana-1089	318	3	]	]	X
cana-1089	318	4	s.	s.	PROPN
cana-1089	318	5	ponnusamy	ponnusamy	PROPN
cana-1089	318	6	,	,	PUNCT
cana-1089	318	7	n.	n.	PROPN
cana-1089	318	8	l.	l.	PROPN
cana-1089	318	9	sharma	sharma	PROPN
cana-1089	318	10	and	and	CCONJ
cana-1089	318	11	k.	k.	PROPN
cana-1089	318	12	j.	j.	PROPN
cana-1089	318	13	wirths	wirths	PROPN
cana-1089	318	14	,	,	PUNCT
cana-1089	318	15	logarithmic	logarithmic	ADJ
cana-1089	318	16	coefficients	coefficient	NOUN
cana-1089	318	17	of	of	ADP
cana-1089	318	18	the	the	DET
cana-1089	318	19	inverse	inverse	NOUN
cana-1089	318	20	of	of	ADP
cana-1089	318	21	univalent	univalent	ADJ
cana-1089	318	22	functions	function	NOUN
cana-1089	318	23	,	,	PUNCT
cana-1089	318	24	results	result	VERB
cana-1089	318	25	math	math	NOUN
cana-1089	318	26	.	.	PUNCT
cana-1089	319	1	73	73	NUM
cana-1089	319	2	(	(	PUNCT
cana-1089	319	3	2018	2018	NUM
cana-1089	319	4	)	)	PUNCT
cana-1089	319	5	,	,	PUNCT
cana-1089	319	6	160	160	NUM
cana-1089	319	7	.	.	PUNCT
cana-1089	319	8	doi	doi	PROPN
cana-1089	319	9	.	.	PUNCT
cana-1089	319	10	;	;	PUNCT
cana-1089	319	11	https://doi.org/10.1007/s00025-018-0921-7	https://doi.org/10.1007/s00025-018-0921-7	PROPN
cana-1089	319	12	[	[	X
cana-1089	319	13	19	19	NUM
cana-1089	319	14	]	]	X
cana-1089	319	15	s.	s.	PROPN
cana-1089	319	16	ponnusamy	ponnusamy	PROPN
cana-1089	319	17	,	,	PUNCT
cana-1089	319	18	n.	n.	PROPN
cana-1089	319	19	l.	l.	PROPN
cana-1089	319	20	sharma	sharma	PROPN
cana-1089	319	21	and	and	CCONJ
cana-1089	319	22	k.	k.	PROPN
cana-1089	319	23	j.	j.	PROPN
cana-1089	319	24	wirths	wirths	PROPN
cana-1089	319	25	,	,	PUNCT
cana-1089	319	26	logarithmic	logarithmic	ADJ
cana-1089	319	27	coefficients	coefficient	NOUN
cana-1089	319	28	problems	problem	NOUN
cana-1089	319	29	in	in	ADP
cana-1089	319	30	families	family	NOUN
cana-1089	319	31	related	relate	VERB
cana-1089	319	32	to	to	ADP
cana-1089	319	33	starlike	starlike	NOUN
cana-1089	319	34	and	and	CCONJ
cana-1089	319	35	convex	convex	NOUN
cana-1089	319	36	functions	function	NOUN
cana-1089	319	37	,	,	PUNCT
cana-1089	319	38	journal	journal	NOUN
cana-1089	319	39	of	of	ADP
cana-1089	319	40	the	the	DET
cana-1089	319	41	australian	australian	ADJ
cana-1089	319	42	mathematical	mathematical	ADJ
cana-1089	319	43	society	society	NOUN
cana-1089	319	44	,	,	PUNCT
cana-1089	319	45	109	109	NUM
cana-1089	319	46	(	(	PUNCT
cana-1089	319	47	2020	2020	NUM
cana-1089	319	48	)	)	PUNCT
cana-1089	319	49	,	,	PUNCT
cana-1089	319	50	no.2	no.2	PROPN
cana-1089	319	51	,	,	PUNCT
cana-1089	319	52	230–49	230–49	NUM
cana-1089	319	53	.	.	PUNCT
cana-1089	320	1	doi	doi	NOUN
cana-1089	320	2	:	:	PUNCT
cana-1089	320	3	10.1017	10.1017	NUM
cana-1089	320	4	/	/	SYM
cana-1089	320	5	s1446788719000065	s1446788719000065	NOUN
cana-1089	320	6	.	.	PUNCT
cana-1089	321	1	[	[	X
cana-1089	321	2	20	20	NUM
cana-1089	321	3	]	]	PUNCT
cana-1089	321	4	s.	s.	PROPN
cana-1089	321	5	ponnusamy	ponnusamy	NOUN
cana-1089	321	6	and	and	CCONJ
cana-1089	321	7	t.	t.	PROPN
cana-1089	321	8	sugawa	sugawa	PROPN
cana-1089	321	9	,	,	PUNCT
cana-1089	321	10	sharp	sharp	ADJ
cana-1089	321	11	inequalities	inequality	NOUN
cana-1089	321	12	for	for	ADP
cana-1089	321	13	logarithmic	logarithmic	ADJ
cana-1089	321	14	coefficients	coefficient	NOUN
cana-1089	321	15	and	and	CCONJ
cana-1089	321	16	their	their	PRON
cana-1089	321	17	applications	application	NOUN
cana-1089	321	18	,	,	PUNCT
cana-1089	321	19	bulletin	bulletin	PROPN
cana-1089	321	20	des	des	PROPN
cana-1089	321	21	sciences	sciences	PROPN
cana-1089	321	22	mathématiques	mathématique	NOUN
cana-1089	321	23	,	,	PUNCT
cana-1089	321	24	166	166	NUM
cana-1089	321	25	(	(	PUNCT
cana-1089	321	26	2021	2021	NUM
cana-1089	321	27	)	)	PUNCT
cana-1089	321	28	,	,	PUNCT
cana-1089	321	29	23	23	NUM
cana-1089	321	30	pages	page	NOUN
cana-1089	321	31	;	;	PUNCT
cana-1089	321	32	article	article	NOUN
cana-1089	321	33	102931	102931	NUM
cana-1089	321	34	;	;	PUNCT
cana-1089	321	35	doi	doi	NOUN
cana-1089	321	36	:	:	PUNCT
cana-1089	321	37	https://doi.org/10.1016	https://doi.org/10.1016	NOUN
cana-1089	321	38	/	/	SYM
cana-1089	321	39	j.bulsci.2020.102931	j.bulsci.2020.102931	NOUN
cana-1089	321	40	[	[	X
cana-1089	321	41	21	21	NUM
cana-1089	321	42	]	]	X
cana-1089	321	43	m.	m.	NOUN
cana-1089	321	44	riza	riza	PROPN
cana-1089	321	45	,	,	PUNCT
cana-1089	321	46	a.	a.	NOUN
cana-1089	321	47	őzyapıcı	őzyapıcı	PROPN
cana-1089	321	48	and	and	CCONJ
cana-1089	321	49	e.	e.	PROPN
cana-1089	321	50	mısırlı	mısırlı	PROPN
cana-1089	321	51	,	,	PUNCT
cana-1089	321	52	multiplicative	multiplicative	ADJ
cana-1089	321	53	finite	finite	ADJ
cana-1089	321	54	difference	difference	NOUN
cana-1089	321	55	methods	method	NOUN
cana-1089	321	56	,	,	PUNCT
cana-1089	321	57	quart	quart	NOUN
cana-1089	321	58	.	.	PUNCT
cana-1089	322	1	appl	appl	PROPN
cana-1089	322	2	.	.	PROPN
cana-1089	322	3	math	math	PROPN
cana-1089	322	4	.	.	PUNCT
cana-1089	322	5	,	,	PUNCT
cana-1089	322	6	67(2009	67(2009	NUM
cana-1089	322	7	)	)	PUNCT
cana-1089	322	8	,	,	PUNCT
cana-1089	322	9	no.4	no.4	PROPN
cana-1089	322	10	,	,	PUNCT
cana-1089	322	11	745–754	745–754	NUM
cana-1089	322	12	.	.	PUNCT
cana-1089	323	1	communications	communication	NOUN
cana-1089	323	2	on	on	ADP
cana-1089	323	3	applied	apply	VERB
cana-1089	323	4	nonlinear	nonlinear	ADJ
cana-1089	323	5	analysis	analysis	NOUN
cana-1089	323	6	issn	issn	NOUN
cana-1089	323	7	:	:	PUNCT
cana-1089	323	8	1074	1074	NUM
cana-1089	323	9	-	-	PUNCT
cana-1089	323	10	133x	133x	NUM
cana-1089	323	11	vol	vol	NOUN
cana-1089	323	12	31	31	NUM
cana-1089	323	13	no	no	NOUN
cana-1089	323	14	.	.	PUNCT
cana-1089	324	1	5s	5s	NUM
cana-1089	324	2	(	(	PUNCT
cana-1089	324	3	2024	2024	NUM
cana-1089	324	4	)	)	PUNCT
cana-1089	324	5	551	551	NUM
cana-1089	324	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1089	325	1	[	[	X
cana-1089	325	2	22	22	NUM
cana-1089	325	3	]	]	PUNCT
cana-1089	325	4	k.	k.	PROPN
cana-1089	325	5	sakaguchi	sakaguchi	PROPN
cana-1089	325	6	,	,	PUNCT
cana-1089	325	7	on	on	ADP
cana-1089	325	8	a	a	DET
cana-1089	325	9	certain	certain	ADJ
cana-1089	325	10	univalent	univalent	ADJ
cana-1089	325	11	mapping	mapping	NOUN
cana-1089	325	12	,	,	PUNCT
cana-1089	325	13	j.	j.	PROPN
cana-1089	325	14	math	math	PROPN
cana-1089	325	15	.	.	PUNCT
cana-1089	326	1	soc	soc	PROPN
cana-1089	326	2	.	.	PUNCT
cana-1089	327	1	japan	japan	PROPN
cana-1089	327	2	,	,	PUNCT
cana-1089	327	3	11	11	NUM
cana-1089	327	4	(	(	PUNCT
cana-1089	327	5	1959	1959	NUM
cana-1089	327	6	)	)	PUNCT
cana-1089	327	7	,	,	PUNCT
cana-1089	327	8	72	72	NUM
cana-1089	327	9	-	-	SYM
cana-1089	327	10	-75	-75	NOUN
cana-1089	327	11	.	.	PUNCT
cana-1089	328	1	[	[	X
cana-1089	328	2	23	23	NUM
cana-1089	328	3	]	]	X
cana-1089	328	4	h.	h.	PROPN
cana-1089	328	5	m.	m.	PROPN
cana-1089	328	6	srivastava	srivastava	PROPN
cana-1089	328	7	,	,	PUNCT
cana-1089	328	8	s.	s.	PROPN
cana-1089	328	9	z.	z.	PROPN
cana-1089	328	10	h.	h.	PROPN
cana-1089	328	11	bukhari	bukhari	PROPN
cana-1089	328	12	and	and	CCONJ
cana-1089	328	13	m.	m.	PROPN
cana-1089	328	14	nazir	nazir	PROPN
cana-1089	328	15	,	,	PUNCT
cana-1089	328	16	a	a	DET
cana-1089	328	17	subclass	subclass	NOUN
cana-1089	328	18	of	of	ADP
cana-1089	328	19	𝛼	𝛼	PRON
cana-1089	328	20	−convex	−convex	NOUN
cana-1089	328	21	functions	function	NOUN
cana-1089	328	22	with	with	ADP
cana-1089	328	23	respect	respect	NOUN
cana-1089	328	24	to	to	ADP
cana-1089	328	25	(	(	PUNCT
cana-1089	328	26	2𝑗	2𝑗	NOUN
cana-1089	328	27	,	,	PUNCT
cana-1089	328	28	𝑘	𝑘	NOUN
cana-1089	328	29	)	)	PUNCT
cana-1089	328	30	−symmetric	−symmetric	ADJ
cana-1089	328	31	conjugate	conjugate	ADJ
cana-1089	328	32	points	point	NOUN
cana-1089	328	33	,	,	PUNCT
cana-1089	328	34	bull	bull	NOUN
cana-1089	328	35	.	.	PUNCT
cana-1089	329	1	iranian	iranian	ADJ
cana-1089	329	2	math	math	PROPN
cana-1089	329	3	.	.	PUNCT
cana-1089	330	1	soc	soc	PROPN
cana-1089	330	2	.	.	PUNCT
cana-1089	331	1	44	44	NUM
cana-1089	331	2	(	(	PUNCT
cana-1089	331	3	2018	2018	NUM
cana-1089	331	4	)	)	PUNCT
cana-1089	331	5	,	,	PUNCT
cana-1089	331	6	no.5	no.5	PROPN
cana-1089	331	7	,	,	PUNCT
cana-1089	331	8	1227	1227	NUM
cana-1089	331	9	-	-	SYM
cana-1089	331	10	1242	1242	NUM
cana-1089	331	11	.	.	PUNCT
cana-1089	332	1	[	[	X
cana-1089	332	2	24	24	NUM
cana-1089	332	3	]	]	PUNCT
cana-1089	332	4	z.-g	z.-g	PROPN
cana-1089	332	5	.	.	PUNCT
cana-1089	333	1	wang	wang	PROPN
cana-1089	333	2	and	and	CCONJ
cana-1089	333	3	y.-p	y.-p	PROPN
cana-1089	333	4	.	.	PUNCT
cana-1089	334	1	jiang	jiang	PROPN
cana-1089	334	2	,	,	PUNCT
cana-1089	334	3	some	some	DET
cana-1089	334	4	properties	property	NOUN
cana-1089	334	5	of	of	ADP
cana-1089	334	6	certain	certain	ADJ
cana-1089	334	7	subclasses	subclass	NOUN
cana-1089	334	8	of	of	ADP
cana-1089	334	9	close	close	NOUN
cana-1089	334	10	-	-	PUNCT
cana-1089	334	11	to	to	ADP
cana-1089	334	12	-	-	PUNCT
cana-1089	334	13	convex	convex	ADJ
cana-1089	334	14	and	and	CCONJ
cana-1089	334	15	quasi	quasi	ADJ
cana-1089	334	16	-	-	ADJ
cana-1089	334	17	convex	convex	ADJ
cana-1089	334	18	functions	function	NOUN
cana-1089	334	19	with	with	ADP
cana-1089	334	20	respect	respect	NOUN
cana-1089	334	21	to	to	ADP
cana-1089	334	22	2𝑘-symmetric	2𝑘-symmetric	NUM
cana-1089	334	23	conjugate	conjugate	ADJ
cana-1089	334	24	points	point	NOUN
cana-1089	334	25	,	,	PUNCT
cana-1089	334	26	bull	bull	NOUN
cana-1089	334	27	.	.	PUNCT
cana-1089	335	1	iranian	iranian	ADJ
cana-1089	335	2	math	math	PROPN
cana-1089	335	3	.	.	PUNCT
cana-1089	336	1	soc	soc	PROPN
cana-1089	336	2	.	.	PUNCT
cana-1089	337	1	36	36	NUM
cana-1089	337	2	(	(	PUNCT
cana-1089	337	3	2010	2010	NUM
cana-1089	337	4	)	)	PUNCT
cana-1089	337	5	,	,	PUNCT
cana-1089	337	6	no.~2	no.~2	VERB
cana-1089	337	7	,	,	PUNCT
cana-1089	337	8	217	217	NUM
cana-1089	337	9	-	-	SYM
cana-1089	337	10	-238	-238	NUM
cana-1089	337	11	.	.	PUNCT
cana-1089	338	1	[	[	X
cana-1089	338	2	25	25	NUM
cana-1089	338	3	]	]	PUNCT
cana-1089	338	4	z.-g	z.-g	PROPN
cana-1089	338	5	.	.	PUNCT
cana-1089	339	1	wang	wang	PROPN
cana-1089	339	2	and	and	CCONJ
cana-1089	339	3	c.-y	c.-y	NOUN
cana-1089	339	4	.	.	PUNCT
cana-1089	340	1	gao	gao	PROPN
cana-1089	340	2	,	,	PUNCT
cana-1089	340	3	on	on	ADP
cana-1089	340	4	starlike	starlike	NOUN
cana-1089	340	5	and	and	CCONJ
cana-1089	340	6	convex	convex	NOUN
cana-1089	340	7	functions	function	NOUN
cana-1089	340	8	with	with	ADP
cana-1089	340	9	respect	respect	NOUN
cana-1089	340	10	to	to	ADP
cana-1089	340	11	2𝑘-symmetric	2𝑘-symmetric	NUM
cana-1089	340	12	conjugate	conjugate	ADJ
cana-1089	340	13	points	point	NOUN
cana-1089	340	14	,	,	PUNCT
cana-1089	340	15	tamsui	tamsui	PROPN
cana-1089	340	16	oxf	oxf	PROPN
cana-1089	340	17	.	.	PUNCT
cana-1089	341	1	j.	j.	PROPN
cana-1089	341	2	math	math	PROPN
cana-1089	341	3	.	.	PUNCT
cana-1089	342	1	sci	sci	PROPN
cana-1089	342	2	.	.	PROPN
cana-1089	343	1	24	24	NUM
cana-1089	343	2	(	(	PUNCT
cana-1089	343	3	2008	2008	NUM
cana-1089	343	4	)	)	PUNCT
cana-1089	343	5	,	,	PUNCT
cana-1089	343	6	no.~3	no.~3	ADP
cana-1089	343	7	,	,	PUNCT
cana-1089	343	8	277	277	NUM
cana-1089	343	9	-	-	PUNCT
cana-1089	343	10	-287	-287	NOUN
cana-1089	343	11	.	.	PUNCT
cana-1089	344	1	[	[	X
cana-1089	344	2	26	26	NUM
cana-1089	344	3	]	]	PUNCT
cana-1089	344	4	z.-g	z.-g	PROPN
cana-1089	344	5	.	.	PUNCT
cana-1089	345	1	wang	wang	PROPN
cana-1089	345	2	,	,	PUNCT
cana-1089	345	3	some	some	DET
cana-1089	345	4	subclasses	subclass	NOUN
cana-1089	345	5	of	of	ADP
cana-1089	345	6	close	close	NOUN
cana-1089	345	7	-	-	PUNCT
cana-1089	345	8	to	to	ADP
cana-1089	345	9	-	-	PUNCT
cana-1089	345	10	convex	convex	ADJ
cana-1089	345	11	and	and	CCONJ
cana-1089	345	12	quasi	quasi	ADJ
cana-1089	345	13	-	-	ADJ
cana-1089	345	14	convex	convex	ADJ
cana-1089	345	15	functions	function	NOUN
cana-1089	345	16	,	,	PUNCT
cana-1089	345	17	mat	mat	PROPN
cana-1089	345	18	.	.	PROPN
cana-1089	345	19	vesnik	vesnik	PROPN
cana-1089	345	20	,	,	PUNCT
cana-1089	345	21	59	59	NUM
cana-1089	345	22	(	(	PUNCT
cana-1089	345	23	2007	2007	NUM
cana-1089	345	24	)	)	PUNCT
cana-1089	345	25	,	,	PUNCT
cana-1089	345	26	no	no	INTJ
cana-1089	345	27	.	.	NOUN
cana-1089	345	28	1	1	NUM
cana-1089	345	29	-	-	SYM
cana-1089	345	30	2	2	NUM
cana-1089	345	31	,	,	PUNCT
cana-1089	345	32	65	65	NUM
cana-1089	345	33	-	-	SYM
cana-1089	345	34	73	73	NUM
cana-1089	345	35	.	.	PUNCT
cana-1089	346	1	[	[	X
cana-1089	346	2	27	27	NUM
cana-1089	346	3	]	]	PUNCT
cana-1089	346	4	z.-g	z.-g	PROPN
cana-1089	346	5	.	.	PUNCT
cana-1089	347	1	wang	wang	PROPN
cana-1089	347	2	and	and	CCONJ
cana-1089	347	3	c.-y	c.-y	NOUN
cana-1089	347	4	.	.	PUNCT
cana-1089	348	1	gao	gao	PROPN
cana-1089	348	2	,	,	PUNCT
cana-1089	348	3	some	some	DET
cana-1089	348	4	subclasses	subclass	NOUN
cana-1089	348	5	of	of	ADP
cana-1089	348	6	close	close	NOUN
cana-1089	348	7	-	-	PUNCT
cana-1089	348	8	to	to	ADP
cana-1089	348	9	-	-	PUNCT
cana-1089	348	10	convex	convex	ADJ
cana-1089	348	11	and	and	CCONJ
cana-1089	348	12	quasi	quasi	ADJ
cana-1089	348	13	-	-	ADJ
cana-1089	348	14	convex	convex	ADJ
cana-1089	348	15	functions	function	NOUN
cana-1089	348	16	with	with	ADP
cana-1089	348	17	respect	respect	NOUN
cana-1089	348	18	to	to	ADP
cana-1089	348	19	𝑘symmetric	𝑘symmetric	ADJ
cana-1089	348	20	points	point	NOUN
cana-1089	348	21	,	,	PUNCT
cana-1089	348	22	gen	gen	PROPN
cana-1089	348	23	.	.	PROPN
cana-1089	348	24	math	math	PROPN
cana-1089	348	25	.	.	PUNCT
cana-1089	348	26	,	,	PUNCT
cana-1089	348	27	15	15	NUM
cana-1089	348	28	(	(	PUNCT
cana-1089	348	29	2007	2007	NUM
cana-1089	348	30	)	)	PUNCT
cana-1089	348	31	,	,	PUNCT
cana-1089	348	32	no.4	no.4	PROPN
cana-1089	348	33	,	,	PUNCT
cana-1089	348	34	107	107	PROPN
cana-1089	348	35	-	-	SYM
cana-1089	348	36	-119	-119	NOUN
cana-1089	348	37	.	.	PUNCT
cana-1089	349	1	[	[	X
cana-1089	349	2	28	28	NUM
cana-1089	349	3	]	]	PUNCT
cana-1089	349	4	z.-g	z.-g	PROPN
cana-1089	349	5	.	.	PUNCT
cana-1089	350	1	wang	wang	PROPN
cana-1089	350	2	,	,	PUNCT
cana-1089	350	3	c.-y	c.-y	NOUN
cana-1089	350	4	.	.	PUNCT
cana-1089	351	1	gao	gao	PROPN
cana-1089	351	2	and	and	CCONJ
cana-1089	351	3	s.-m	s.-m	PROPN
cana-1089	351	4	.	.	PUNCT
cana-1089	352	1	yuan	yuan	NOUN
cana-1089	352	2	,	,	PUNCT
cana-1089	352	3	on	on	ADP
cana-1089	352	4	certain	certain	ADJ
cana-1089	352	5	subclasses	subclass	NOUN
cana-1089	352	6	of	of	ADP
cana-1089	352	7	close	close	NOUN
cana-1089	352	8	-	-	PUNCT
cana-1089	352	9	to	to	ADP
cana-1089	352	10	-	-	PUNCT
cana-1089	352	11	convex	convex	ADJ
cana-1089	352	12	and	and	CCONJ
cana-1089	352	13	quasi	quasi	ADJ
cana-1089	352	14	-	-	ADJ
cana-1089	352	15	convex	convex	ADJ
cana-1089	352	16	functions	function	NOUN
cana-1089	352	17	with	with	ADP
cana-1089	352	18	respect	respect	NOUN
cana-1089	352	19	to	to	ADP
cana-1089	352	20	𝑘-symmetric	𝑘-symmetric	ADJ
cana-1089	352	21	points	point	NOUN
cana-1089	352	22	,	,	PUNCT
cana-1089	352	23	j.	j.	PROPN
cana-1089	352	24	math	math	PROPN
cana-1089	352	25	.	.	PUNCT
cana-1089	353	1	anal	anal	PROPN
cana-1089	353	2	.	.	PUNCT
cana-1089	353	3	appl	appl	PROPN
cana-1089	353	4	.	.	PUNCT
cana-1089	354	1	322	322	NUM
cana-1089	354	2	(	(	PUNCT
cana-1089	354	3	2006	2006	NUM
cana-1089	354	4	)	)	PUNCT
cana-1089	354	5	,	,	PUNCT
cana-1089	354	6	no.~1	no.~1	NOUN
cana-1089	354	7	,	,	PUNCT
cana-1089	354	8	97	97	NUM
cana-1089	354	9	-	-	SYM
cana-1089	354	10	-106	-106	NOUN
cana-1089	354	11	.	.	PUNCT