id	sid	tid	token	lemma	pos
cana-758	1	1	communications	communication	NOUN
cana-758	1	2	on	on	ADP
cana-758	1	3	applied	apply	VERB
cana-758	1	4	nonlinear	nonlinear	ADJ
cana-758	1	5	analysis	analysis	NOUN
cana-758	1	6	issn	issn	NOUN
cana-758	1	7	:	:	PUNCT
cana-758	1	8	1074	1074	NUM
cana-758	1	9	-	-	PUNCT
cana-758	1	10	133x	133x	NUM
cana-758	1	11	vol	vol	NOUN
cana-758	1	12	31	31	NUM
cana-758	1	13	no	no	NOUN
cana-758	1	14	.	.	PUNCT
cana-758	2	1	3s	3s	NUM
cana-758	2	2	(	(	PUNCT
cana-758	2	3	2024	2024	NUM
cana-758	2	4	)	)	PUNCT
cana-758	2	5	186	186	NUM
cana-758	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	2	7	certain	certain	ADJ
cana-758	2	8	properties	property	NOUN
cana-758	2	9	on	on	ADP
cana-758	2	10	univalent	univalent	ADJ
cana-758	2	11	functions	function	NOUN
cana-758	2	12	related	relate	VERB
cana-758	2	13	to	to	ADP
cana-758	2	14	new	new	ADJ
cana-758	2	15	linear	linear	ADJ
cana-758	2	16	and	and	CCONJ
cana-758	2	17	integral	integral	ADJ
cana-758	2	18	operators	operator	NOUN
cana-758	2	19	lect	lect	VERB
cana-758	2	20	.	.	PUNCT
cana-758	3	1	dr	dr	PROPN
cana-758	3	2	.	.	PUNCT
cana-758	4	1	bassim	bassim	PROPN
cana-758	4	2	kareem	kareem	PROPN
cana-758	4	3	mihsin	mihsin	PROPN
cana-758	4	4	general	general	ADJ
cana-758	4	5	directorate	directorate	NOUN
cana-758	4	6	of	of	ADP
cana-758	4	7	education	education	NOUN
cana-758	4	8	in	in	ADP
cana-758	4	9	karbala	karbala	PROPN
cana-758	4	10	,	,	PUNCT
cana-758	4	11	iraq	iraq	PROPN
cana-758	4	12	email	email	NOUN
cana-758	4	13	:	:	PUNCT
cana-758	4	14	bassim_kareem@karbala.edu.iq	bassim_kareem@karbala.edu.iq	VERB
cana-758	4	15	basmk3756@gmail.com	basmk3756@gmail.com	NOUN
cana-758	4	16	article	article	NOUN
cana-758	4	17	history	history	NOUN
cana-758	4	18	:	:	PUNCT
cana-758	4	19	received	receive	VERB
cana-758	4	20	:	:	PUNCT
cana-758	4	21	06	06	NUM
cana-758	4	22	-	-	PUNCT
cana-758	4	23	04	04	NUM
cana-758	4	24	-	-	PUNCT
cana-758	4	25	2024	2024	NUM
cana-758	4	26	revised	revise	VERB
cana-758	4	27	:	:	PUNCT
cana-758	4	28	28	28	NUM
cana-758	4	29	-	-	SYM
cana-758	4	30	05	05	NUM
cana-758	4	31	-	-	PUNCT
cana-758	4	32	2024	2024	NUM
cana-758	4	33	accepted	accept	VERB
cana-758	4	34	:	:	PUNCT
cana-758	4	35	12	12	NUM
cana-758	4	36	-	-	PUNCT
cana-758	4	37	06	06	NUM
cana-758	4	38	-	-	PUNCT
cana-758	4	39	2024	2024	NUM
cana-758	4	40	abstract	abstract	NOUN
cana-758	4	41	:	:	PUNCT
cana-758	4	42	we	we	PRON
cana-758	4	43	are	be	AUX
cana-758	4	44	implementing	implement	VERB
cana-758	4	45	the	the	DET
cana-758	4	46	two	two	NUM
cana-758	4	47	new	new	ADJ
cana-758	4	48	operators	operator	NOUN
cana-758	4	49	,	,	PUNCT
cana-758	4	50	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	4	51	𝕞	𝕞	PRON
cana-758	4	52	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	4	53	)	)	PUNCT
cana-758	4	54	and	and	CCONJ
cana-758	4	55	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	4	56	𝕞	𝕞	DET
cana-758	4	57	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	4	58	)	)	PUNCT
cana-758	4	59	say	say	VERB
cana-758	4	60	linear	linear	ADJ
cana-758	4	61	and	and	CCONJ
cana-758	4	62	integral	integral	ADJ
cana-758	4	63	operators	operator	NOUN
cana-758	4	64	respectively	respectively	ADV
cana-758	4	65	,	,	PUNCT
cana-758	4	66	of	of	ADP
cana-758	4	67	analaytic	analaytic	ADJ
cana-758	4	68	functions	function	NOUN
cana-758	4	69	in	in	ADP
cana-758	4	70	open	open	ADJ
cana-758	4	71	unit	unit	NOUN
cana-758	4	72	disk	disk	NOUN
cana-758	4	73	ʋ	ʋ	PROPN
cana-758	4	74	,	,	PUNCT
cana-758	4	75	to	to	ADP
cana-758	4	76	pedimenting	pedimente	VERB
cana-758	4	77	new	new	ADJ
cana-758	4	78	results	result	NOUN
cana-758	4	79	for	for	ADP
cana-758	4	80	superordination	superordination	NOUN
cana-758	4	81	and	and	CCONJ
cana-758	4	82	subordination	subordination	NOUN
cana-758	4	83	.	.	PUNCT
cana-758	5	1	we	we	PRON
cana-758	5	2	conclude	conclude	VERB
cana-758	5	3	several	several	ADJ
cana-758	5	4	sandwich	sandwich	NOUN
cana-758	5	5	-	-	PUNCT
cana-758	5	6	type	type	NOUN
cana-758	5	7	results	result	NOUN
cana-758	5	8	are	be	AUX
cana-758	5	9	the	the	DET
cana-758	5	10	master	master	NOUN
cana-758	5	11	goal	goal	NOUN
cana-758	5	12	for	for	ADP
cana-758	5	13	this	this	DET
cana-758	5	14	paper	paper	NOUN
cana-758	5	15	.	.	PUNCT
cana-758	6	1	keywords	keyword	NOUN
cana-758	6	2	:	:	PUNCT
cana-758	6	3	analytic	analytic	ADJ
cana-758	6	4	functions	function	NOUN
cana-758	6	5	,	,	PUNCT
cana-758	6	6	multivalent	multivalent	NOUN
cana-758	6	7	functions	function	NOUN
cana-758	6	8	,	,	PUNCT
cana-758	6	9	hadamart	hadamart	NOUN
cana-758	6	10	product	product	NOUN
cana-758	6	11	,	,	PUNCT
cana-758	6	12	differential	differential	ADJ
cana-758	6	13	subordination	subordination	NOUN
cana-758	6	14	,	,	PUNCT
cana-758	6	15	superordination	superordination	NOUN
cana-758	6	16	,	,	PUNCT
cana-758	6	17	sandwich	sandwich	NOUN
cana-758	6	18	theorems	theorem	NOUN
cana-758	6	19	,	,	PUNCT
cana-758	6	20	dominant	dominant	ADJ
cana-758	6	21	,	,	PUNCT
cana-758	6	22	subordinant	subordinant	NOUN
cana-758	6	23	.	.	PUNCT
cana-758	7	1	1	1	X
cana-758	7	2	.	.	X
cana-758	7	3	introduction	introduction	NOUN
cana-758	7	4	suppose	suppose	VERB
cana-758	7	5	that	that	SCONJ
cana-758	7	6	ℬ	ℬ	NOUN
cana-758	7	7	to	to	PART
cana-758	7	8	be	be	AUX
cana-758	7	9	class	class	NOUN
cana-758	7	10	functions	function	NOUN
cana-758	7	11	intailing	intaile	VERB
cana-758	7	12	the	the	DET
cana-758	7	13	following	follow	VERB
cana-758	7	14	function	function	NOUN
cana-758	7	15	:	:	PUNCT
cana-758	7	16	𝑓(𝓏	𝑓(𝓏	X
cana-758	7	17	)	)	PUNCT
cana-758	8	1	=	=	SYM
cana-758	8	2	𝒶	𝒶	X
cana-758	9	1	+	+	CCONJ
cana-758	9	2	𝒶𝔫	𝒶𝔫	VERB
cana-758	9	3	𝓏𝔫	𝓏𝔫	ADV
cana-758	9	4	+	+	NUM
cana-758	9	5	𝒶𝔫+1𝓏𝔫+1	𝒶𝔫+1𝓏𝔫+1	X
cana-758	9	6	+	+	PROPN
cana-758	9	7	⋯	⋯	PROPN
cana-758	9	8	(	(	PUNCT
cana-758	9	9	𝒶	𝒶	PROPN
cana-758	9	10	∈	∈	PROPN
cana-758	9	11	₵	₵	PROPN
cana-758	9	12	,	,	PUNCT
cana-758	9	13	(	(	PUNCT
cana-758	9	14	𝔫	𝔫	NOUN
cana-758	9	15	∈	∈	PROPN
cana-758	9	16	ℕ	ℕ	PROPN
cana-758	9	17	=	=	SYM
cana-758	9	18	{	{	PUNCT
cana-758	9	19	1,2	1,2	NUM
cana-758	9	20	,	,	PUNCT
cana-758	9	21	…	…	PUNCT
cana-758	9	22	}	}	PUNCT
cana-758	9	23	;	;	PUNCT
cana-758	9	24	𝓏	𝓏	PROPN
cana-758	9	25	∈	∈	PROPN
cana-758	9	26	ʋ	ʋ	NOUN
cana-758	9	27	)	)	PUNCT
cana-758	9	28	,	,	PUNCT
cana-758	9	29	(	(	PUNCT
cana-758	9	30	1,1	1,1	X
cana-758	9	31	)	)	PUNCT
cana-758	9	32	where	where	SCONJ
cana-758	9	33	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	9	34	)	)	PUNCT
cana-758	9	35	be	be	AUX
cana-758	9	36	analytic	analytic	ADJ
cana-758	9	37	to	to	PART
cana-758	9	38	open	open	VERB
cana-758	9	39	unit	unit	NOUN
cana-758	9	40	disk	disk	NOUN
cana-758	9	41	ʋ	ʋ	NOUN
cana-758	9	42	=	=	PUNCT
cana-758	9	43	{	{	PUNCT
cana-758	9	44	𝓏	𝓏	PROPN
cana-758	9	45	∈	∈	PROPN
cana-758	9	46	₵	₵	PROPN
cana-758	9	47	∶	∶	NOUN
cana-758	9	48	|𝓏|	|𝓏|	NOUN
cana-758	9	49	<	<	X
cana-758	9	50	1	1	NUM
cana-758	9	51	}	}	PUNCT
cana-758	9	52	.	.	PUNCT
cana-758	10	1	assume	assume	VERB
cana-758	10	2	𝑀[𝒶	𝑀[𝒶	PROPN
cana-758	10	3	,	,	PUNCT
cana-758	10	4	𝔫	𝔫	X
cana-758	10	5	]	]	PUNCT
cana-758	10	6	be	be	AUX
cana-758	10	7	subclass	subclass	NOUN
cana-758	10	8	of	of	ADP
cana-758	10	9	the	the	DET
cana-758	10	10	function	function	NOUN
cana-758	10	11	𝑓	𝑓	DET
cana-758	10	12	∈	∈	PROPN
cana-758	10	13	𝒢	𝒢	PROPN
cana-758	10	14	.	.	PUNCT
cana-758	11	1	for	for	SCONJ
cana-758	11	2	𝒶	𝒶	PROPN
cana-758	11	3	∈	∈	PROPN
cana-758	11	4	₵	₵	NOUN
cana-758	11	5	and	and	CCONJ
cana-758	11	6	𝔫	𝔫	NOUN
cana-758	11	7	to	to	PART
cana-758	11	8	be	be	AUX
cana-758	11	9	positive	positive	ADJ
cana-758	11	10	integer	integer	NOUN
cana-758	11	11	number	number	NOUN
cana-758	11	12	,	,	PUNCT
cana-758	11	13	if	if	SCONJ
cana-758	11	14	𝑓	𝑓	DET
cana-758	11	15	∈	∈	PROPN
cana-758	11	16	𝒢	𝒢	NOUN
cana-758	11	17	defined	define	VERB
cana-758	11	18	by	by	ADP
cana-758	11	19	(	(	PUNCT
cana-758	11	20	1.1	1.1	NUM
cana-758	11	21	)	)	PUNCT
cana-758	11	22	and	and	CCONJ
cana-758	11	23	𝑔	𝑔	PROPN
cana-758	11	24	∈	∈	PROPN
cana-758	11	25	𝒢	𝒢	PROPN
cana-758	11	26	is	be	AUX
cana-758	11	27	given	give	VERB
cana-758	11	28	by	by	ADP
cana-758	11	29	formula	formula	NOUN
cana-758	11	30	:	:	PUNCT
cana-758	11	31	𝑓(𝓏	𝑓(𝓏	X
cana-758	11	32	)	)	PUNCT
cana-758	12	1	=	=	SYM
cana-758	12	2	𝓏	𝓏	PROPN
cana-758	12	3	+	+	CCONJ
cana-758	12	4	∑	∑	PUNCT
cana-758	12	5	𝒶𝔫𝓏𝔫∞	𝒶𝔫𝓏𝔫∞	NUM
cana-758	12	6	𝔫=2	𝔫=2	PROPN
cana-758	12	7	,	,	PUNCT
cana-758	12	8	𝑔(𝓏	𝑔(𝓏	PROPN
cana-758	12	9	)	)	PUNCT
cana-758	12	10	=	=	PUNCT
cana-758	12	11	𝓏	𝓏	PROPN
cana-758	13	1	+	+	CCONJ
cana-758	13	2	∑	∑	PROPN
cana-758	13	3	𝑏𝔫𝓏𝔫∞	𝑏𝔫𝓏𝔫∞	PROPN
cana-758	13	4	𝔫=2	𝔫=2	PROPN
cana-758	13	5	by	by	ADP
cana-758	13	6	ussing	usse	VERB
cana-758	13	7	convolution	convolution	NOUN
cana-758	13	8	of	of	ADP
cana-758	13	9	𝑓	𝑓	DET
cana-758	13	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-758	13	11	𝑔	𝑔	PROPN
cana-758	13	12	to	to	PART
cana-758	13	13	get	get	VERB
cana-758	13	14	(	(	PUNCT
cana-758	13	15	𝑓	𝑓	DET
cana-758	13	16	∗	∗	NOUN
cana-758	13	17	𝑔)(𝓏	𝑔)(𝓏	NOUN
cana-758	13	18	)	)	PUNCT
cana-758	13	19	=	=	SYM
cana-758	13	20	𝓏	𝓏	PROPN
cana-758	14	1	+	+	CCONJ
cana-758	14	2	∑	∑	ADV
cana-758	14	3	𝒶𝔫𝑏𝔫𝓏𝔫∞	𝒶𝔫𝑏𝔫𝓏𝔫∞	X
cana-758	14	4	𝔫=2	𝔫=2	NOUN
cana-758	14	5	=	=	PUNCT
cana-758	14	6	(	(	PUNCT
cana-758	14	7	𝑔	𝑔	NOUN
cana-758	14	8	∗	∗	NOUN
cana-758	14	9	𝑓)(𝓏	𝑓)(𝓏	NOUN
cana-758	14	10	)	)	PUNCT
cana-758	14	11	.	.	PUNCT
cana-758	15	1	if	if	SCONJ
cana-758	15	2	the	the	DET
cana-758	15	3	functions	function	NOUN
cana-758	15	4	𝑓	𝑓	PRON
cana-758	15	5	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-758	15	6	𝑔	𝑔	AUX
cana-758	15	7	be	be	AUX
cana-758	15	8	analytic	analytic	ADJ
cana-758	15	9	functions	function	NOUN
cana-758	15	10	in	in	ADP
cana-758	15	11	,	,	PUNCT
cana-758	15	12	so	so	ADV
cana-758	15	13	𝑓	𝑓	PRON
cana-758	15	14	be	be	AUX
cana-758	15	15	subordinate	subordinate	ADJ
cana-758	15	16	to	to	ADP
cana-758	15	17	𝑔	𝑔	PROPN
cana-758	15	18	in	in	ADP
cana-758	15	19	ʋ	ʋ	PROPN
cana-758	15	20	,	,	PUNCT
cana-758	15	21	for	for	SCONJ
cana-758	15	22	that	that	SCONJ
cana-758	15	23	we	we	PRON
cana-758	15	24	can	can	AUX
cana-758	15	25	say	say	VERB
cana-758	15	26	𝑓	𝑓	DET
cana-758	15	27	(	(	PUNCT
cana-758	15	28	𝓏	𝓏	NOUN
cana-758	15	29	)	)	PUNCT
cana-758	15	30	≺	≺	NOUN
cana-758	15	31	𝑔	𝑔	PROPN
cana-758	15	32	(	(	PUNCT
cana-758	15	33	𝓏	𝓏	PROPN
cana-758	15	34	)	)	PUNCT
cana-758	15	35	,	,	PUNCT
cana-758	15	36	if	if	SCONJ
cana-758	15	37	existing	exist	VERB
cana-758	15	38	a	a	DET
cana-758	15	39	schwarz	schwarz	NOUN
cana-758	15	40	function	function	NOUN
cana-758	15	41	ѡ(𝓏	ѡ(𝓏	NOUN
cana-758	15	42	)	)	PUNCT
cana-758	15	43	to	to	PART
cana-758	15	44	be	be	AUX
cana-758	15	45	analytic	analytic	ADJ
cana-758	15	46	within	within	ADP
cana-758	15	47	ʋ	ʋ	NOUN
cana-758	15	48	to	to	PART
cana-758	15	49	satisfy	satisfy	VERB
cana-758	15	50	the	the	DET
cana-758	15	51	conditions	condition	NOUN
cana-758	15	52	that	that	PRON
cana-758	15	53	|ѡ(𝓏)|	|ѡ(𝓏)|	PROPN
cana-758	15	54	<	<	X
cana-758	15	55	1	1	NUM
cana-758	15	56	(	(	PUNCT
cana-758	15	57	𝓏	𝓏	PROPN
cana-758	15	58	∈	∈	PROPN
cana-758	15	59	ʋ	ʋ	NOUN
cana-758	15	60	)	)	PUNCT
cana-758	15	61	and	and	CCONJ
cana-758	15	62	ѡ(0	ѡ(0	PROPN
cana-758	15	63	)	)	PUNCT
cana-758	15	64	=	=	SYM
cana-758	15	65	0	0	PUNCT
cana-758	16	1	and	and	CCONJ
cana-758	16	2	where	where	SCONJ
cana-758	16	3	𝑓	𝑓	DET
cana-758	16	4	(	(	PUNCT
cana-758	16	5	𝓏	𝓏	NOUN
cana-758	16	6	)	)	PUNCT
cana-758	16	7	=	=	SYM
cana-758	16	8	𝑔	𝑔	PROPN
cana-758	16	9	(	(	PUNCT
cana-758	16	10	ѡ(𝓏	ѡ(𝓏	NOUN
cana-758	16	11	)	)	PUNCT
cana-758	16	12	)	)	PUNCT
cana-758	16	13	,	,	PUNCT
cana-758	16	14	(	(	PUNCT
cana-758	16	15	𝓏	𝓏	PROPN
cana-758	16	16	∈	∈	PROPN
cana-758	16	17	ʋ	ʋ	PROPN
cana-758	16	18	)	)	PUNCT
cana-758	16	19	.	.	PUNCT
cana-758	17	1	as	as	ADP
cana-758	17	2	additional	additional	ADJ
cana-758	17	3	,	,	PUNCT
cana-758	17	4	to	to	ADP
cana-758	17	5	that	that	PRON
cana-758	17	6	if	if	SCONJ
cana-758	17	7	𝑔	𝑔	PRON
cana-758	17	8	be	be	AUX
cana-758	17	9	univalent	univalent	ADJ
cana-758	17	10	function	function	NOUN
cana-758	17	11	in	in	ADP
cana-758	17	12	ʋ	ʋ	PROPN
cana-758	17	13	,	,	PUNCT
cana-758	17	14	so	so	SCONJ
cana-758	17	15	we	we	PRON
cana-758	17	16	satisfy	satisfy	VERB
cana-758	17	17	the	the	DET
cana-758	17	18	equivalence	equivalence	NOUN
cana-758	17	19	relation	relation	NOUN
cana-758	17	20	link	link	NOUN
cana-758	17	21	(	(	PUNCT
cana-758	17	22	see	see	VERB
cana-758	18	1	[	[	X
cana-758	18	2	15],[16]and[18	15],[16]and[18	NUM
cana-758	18	3	]	]	PUNCT
cana-758	18	4	)	)	PUNCT
cana-758	19	1	𝑓	𝑓	PROPN
cana-758	19	2	(	(	PUNCT
cana-758	19	3	ʋ	ʋ	NOUN
cana-758	19	4	)	)	PUNCT
cana-758	19	5	⊂	⊂	PROPN
cana-758	19	6	𝑔(ʋ	𝑔(ʋ	PROPN
cana-758	19	7	)	)	PUNCT
cana-758	19	8	,	,	PUNCT
cana-758	19	9	(	(	PUNCT
cana-758	19	10	𝓏	𝓏	PROPN
cana-758	19	11	∈	∈	PROPN
cana-758	19	12	ʋ	ʋ	NOUN
cana-758	19	13	)	)	PUNCT
cana-758	19	14	and	and	CCONJ
cana-758	19	15	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	19	16	)	)	PUNCT
cana-758	19	17	≺	≺	NOUN
cana-758	19	18	𝑔(𝓏	𝑔(𝓏	NOUN
cana-758	19	19	)	)	PUNCT
cana-758	19	20	⇔	⇔	X
cana-758	19	21	𝑓	𝑓	PROPN
cana-758	19	22	(	(	PUNCT
cana-758	19	23	0	0	NUM
cana-758	19	24	)	)	PUNCT
cana-758	19	25	=	=	SYM
cana-758	19	26	𝑔(0	𝑔(0	NOUN
cana-758	19	27	)	)	PUNCT
cana-758	19	28	.	.	PUNCT
cana-758	20	1	definition	definition	NOUN
cana-758	20	2	(	(	PUNCT
cana-758	20	3	1.1)[15	1.1)[15	PROPN
cana-758	20	4	]	]	X
cana-758	20	5	:	:	PUNCT
cana-758	20	6	assume	assume	VERB
cana-758	20	7	that	that	SCONJ
cana-758	20	8	ȴ(𝑧	ȴ(𝑧	NOUN
cana-758	20	9	)	)	PUNCT
cana-758	20	10	to	to	PART
cana-758	20	11	be	be	AUX
cana-758	20	12	univalent	univalent	ADJ
cana-758	20	13	in	in	ADP
cana-758	20	14	ʋ	ʋ	X
cana-758	20	15	and	and	CCONJ
cana-758	20	16	ə:₵3	ə:₵3	ADJ
cana-758	20	17	×	×	NOUN
cana-758	20	18	ʋ	ʋ	PROPN
cana-758	20	19	⇢	⇢	NOUN
cana-758	20	20	₵	₵	PROPN
cana-758	20	21	.	.	PUNCT
cana-758	21	1	let	let	VERB
cana-758	21	2	𝒫(𝓏	𝒫(𝓏	PRON
cana-758	21	3	)	)	PUNCT
cana-758	21	4	be	be	AUX
cana-758	21	5	analytic	analytic	ADJ
cana-758	21	6	in	in	ADP
cana-758	21	7	ʋ	ʋ	X
cana-758	21	8	to	to	PART
cana-758	21	9	satisfy	satisfy	VERB
cana-758	21	10	the	the	DET
cana-758	21	11	following	follow	VERB
cana-758	21	12	differential	differential	ADJ
cana-758	21	13	subordination	subordination	NOUN
cana-758	21	14	of	of	ADP
cana-758	21	15	second	second	ADJ
cana-758	21	16	–	–	PUNCT
cana-758	21	17	order	order	NOUN
cana-758	21	18	:	:	PUNCT
cana-758	21	19	ə	ə	PROPN
cana-758	21	20	(	(	PUNCT
cana-758	21	21	𝒫(𝓏	𝒫(𝓏	PROPN
cana-758	21	22	)	)	PUNCT
cana-758	21	23	,	,	PUNCT
cana-758	21	24	𝓏(𝒫(𝓏	𝓏(𝒫(𝓏	PROPN
cana-758	21	25	)	)	PUNCT
cana-758	21	26	)	)	PUNCT
cana-758	22	1	′	′	NUM
cana-758	22	2	,	,	PUNCT
cana-758	22	3	𝓏2(𝒫(𝓏	𝓏2(𝒫(𝓏	NUM
cana-758	22	4	)	)	PUNCT
cana-758	22	5	)	)	PUNCT
cana-758	23	1	′′	′′	PROPN
cana-758	23	2	;	;	PUNCT
cana-758	23	3	𝓏	𝓏	X
cana-758	23	4	)	)	PUNCT
cana-758	23	5	≺	≺	NOUN
cana-758	23	6	ȴ(𝓏	ȴ(𝓏	NOUN
cana-758	23	7	)	)	PUNCT
cana-758	23	8	,	,	PUNCT
cana-758	23	9	(	(	PUNCT
cana-758	23	10	1.2	1.2	NUM
cana-758	23	11	)	)	PUNCT
cana-758	23	12	communications	communication	NOUN
cana-758	23	13	on	on	ADP
cana-758	23	14	applied	apply	VERB
cana-758	23	15	nonlinear	nonlinear	ADJ
cana-758	23	16	analysis	analysis	NOUN
cana-758	23	17	issn	issn	NOUN
cana-758	23	18	:	:	PUNCT
cana-758	23	19	1074	1074	NUM
cana-758	23	20	-	-	PUNCT
cana-758	23	21	133x	133x	NUM
cana-758	23	22	vol	vol	NOUN
cana-758	23	23	31	31	NUM
cana-758	23	24	no	no	NOUN
cana-758	23	25	.	.	PUNCT
cana-758	24	1	3s	3s	NUM
cana-758	24	2	(	(	PUNCT
cana-758	24	3	2024	2024	NUM
cana-758	24	4	)	)	PUNCT
cana-758	24	5	187	187	NUM
cana-758	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	24	7	therefore	therefore	ADV
cana-758	24	8	the	the	DET
cana-758	24	9	equation	equation	NOUN
cana-758	24	10	(	(	PUNCT
cana-758	24	11	1.2	1.2	NUM
cana-758	24	12	)	)	PUNCT
cana-758	24	13	of	of	ADP
cana-758	24	14	differential	differential	ADJ
cana-758	24	15	subordination	subordination	NOUN
cana-758	24	16	have	have	VERB
cana-758	24	17	the	the	DET
cana-758	24	18	solution	solution	NOUN
cana-758	24	19	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	24	20	)	)	PUNCT
cana-758	24	21	.	.	PUNCT
cana-758	25	1	the	the	DET
cana-758	25	2	formula	formula	NOUN
cana-758	25	3	(	(	PUNCT
cana-758	25	4	1.2	1.2	NUM
cana-758	25	5	)	)	PUNCT
cana-758	25	6	represent	represent	VERB
cana-758	25	7	the	the	DET
cana-758	25	8	solution	solution	NOUN
cana-758	25	9	of	of	ADP
cana-758	25	10	differential	differential	ADJ
cana-758	25	11	subordination	subordination	NOUN
cana-758	25	12	which	which	PRON
cana-758	25	13	has	have	VERB
cana-758	25	14	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	25	15	)	)	PUNCT
cana-758	25	16	as	as	ADP
cana-758	25	17	a	a	DET
cana-758	25	18	dominant	dominant	ADJ
cana-758	25	19	,	,	PUNCT
cana-758	25	20	or	or	CCONJ
cana-758	25	21	more	more	ADJ
cana-758	25	22	,	,	PUNCT
cana-758	25	23	additional	additional	ADJ
cana-758	25	24	to	to	ADP
cana-758	25	25	that	that	PRON
cana-758	25	26	to	to	PART
cana-758	25	27	be	be	AUX
cana-758	25	28	simply	simply	ADV
cana-758	25	29	dominant	dominant	ADJ
cana-758	25	30	,	,	PUNCT
cana-758	25	31	if	if	SCONJ
cana-758	25	32	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	25	33	)	)	PUNCT
cana-758	25	34	≺	≺	NOUN
cana-758	25	35	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	25	36	)	)	PUNCT
cana-758	25	37	to	to	ADP
cana-758	25	38	all	all	DET
cana-758	25	39	𝒫(𝓏	𝒫(𝓏	NOUN
cana-758	25	40	)	)	PUNCT
cana-758	25	41	for	for	ADP
cana-758	25	42	that	that	PRON
cana-758	25	43	will	will	AUX
cana-758	25	44	satisfy	satisfy	VERB
cana-758	25	45	(	(	PUNCT
cana-758	25	46	1.2).if	1.2).if	NUM
cana-758	25	47	ϥ̃(𝓏	ϥ̃(𝓏	PROPN
cana-758	25	48	)	)	PUNCT
cana-758	25	49	is	be	AUX
cana-758	25	50	univalent	univalent	ADJ
cana-758	25	51	dominant	dominant	ADJ
cana-758	25	52	which	which	DET
cana-758	25	53	satisfys	satisfys	ADP
cana-758	25	54	ϥ̃(𝓏	ϥ̃(𝓏	PROPN
cana-758	25	55	)	)	PUNCT
cana-758	25	56	≺	≺	NOUN
cana-758	25	57	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	25	58	)	)	PUNCT
cana-758	25	59	for	for	ADP
cana-758	25	60	every	every	DET
cana-758	25	61	dominant	dominant	ADJ
cana-758	25	62	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	25	63	)	)	PUNCT
cana-758	25	64	for	for	ADP
cana-758	25	65	(	(	PUNCT
cana-758	25	66	1.2	1.2	NUM
cana-758	25	67	)	)	PUNCT
cana-758	25	68	,	,	PUNCT
cana-758	25	69	so	so	CCONJ
cana-758	25	70	the	the	DET
cana-758	25	71	formula	formula	NOUN
cana-758	25	72	(	(	PUNCT
cana-758	25	73	1.2	1.2	NUM
cana-758	25	74	)	)	PUNCT
cana-758	25	75	will	will	AUX
cana-758	25	76	be	be	AUX
cana-758	25	77	satisfied	satisfy	VERB
cana-758	25	78	by	by	ADP
cana-758	25	79	the	the	DET
cana-758	25	80	best	good	ADJ
cana-758	25	81	dominant	dominant	NOUN
cana-758	25	82	.	.	PUNCT
cana-758	26	1	definition	definition	NOUN
cana-758	26	2	(	(	PUNCT
cana-758	26	3	1.2)([15]𝑎𝑙𝑠𝑜	1.2)([15]𝑎𝑙𝑠𝑜	NOUN
cana-758	26	4	𝑠𝑒𝑒[16	𝑠𝑒𝑒[16	NOUN
cana-758	26	5	]	]	PUNCT
cana-758	26	6	)	)	PUNCT
cana-758	27	1	∶	∶	NOUN
cana-758	27	2	assume	assume	VERB
cana-758	27	3	that	that	SCONJ
cana-758	27	4	ə:₵3	ə:₵3	VERB
cana-758	27	5	×	×	NOUN
cana-758	27	6	ʋ	ʋ	SYM
cana-758	27	7	⇢	⇢	NOUN
cana-758	27	8	₵	₵	NOUN
cana-758	27	9	and	and	CCONJ
cana-758	27	10	assume	assume	VERB
cana-758	27	11	the	the	DET
cana-758	27	12	function	function	NOUN
cana-758	27	13	ȴ(𝓏	ȴ(𝓏	NOUN
cana-758	27	14	)	)	PUNCT
cana-758	27	15	which	which	PRON
cana-758	27	16	be	be	VERB
cana-758	27	17	analytic	analytic	ADJ
cana-758	27	18	in	in	ADP
cana-758	27	19	ʋ	ʋ	PROPN
cana-758	27	20	.	.	PUNCT
cana-758	28	1	if	if	SCONJ
cana-758	28	2	ə	ə	PROPN
cana-758	28	3	(	(	PUNCT
cana-758	28	4	𝒫(𝓏	𝒫(𝓏	PROPN
cana-758	28	5	)	)	PUNCT
cana-758	28	6	,	,	PUNCT
cana-758	28	7	𝓏(𝒫(𝓏	𝓏(𝒫(𝓏	PROPN
cana-758	28	8	)	)	PUNCT
cana-758	28	9	)	)	PUNCT
cana-758	29	1	′	′	NUM
cana-758	29	2	,	,	PUNCT
cana-758	29	3	𝓏2(𝒫(𝓏	𝓏2(𝒫(𝓏	NUM
cana-758	29	4	)	)	PUNCT
cana-758	29	5	)	)	PUNCT
cana-758	30	1	′′	′′	PROPN
cana-758	30	2	;	;	PUNCT
cana-758	30	3	𝓏	𝓏	X
cana-758	30	4	)	)	PUNCT
cana-758	30	5	and	and	CCONJ
cana-758	30	6	the	the	DET
cana-758	30	7	univalent	univalent	ADJ
cana-758	30	8	function	function	NOUN
cana-758	30	9	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	30	10	)	)	PUNCT
cana-758	30	11	within	within	ADP
cana-758	30	12	ʋ	ʋ	PRON
cana-758	30	13	where	where	SCONJ
cana-758	30	14	𝒫(𝓏	𝒫(𝓏	X
cana-758	30	15	)	)	PUNCT
cana-758	30	16	satisfy	satisfy	VERB
cana-758	30	17	the	the	DET
cana-758	30	18	following	follow	VERB
cana-758	30	19	differential	differential	ADJ
cana-758	30	20	superordination	superordination	NOUN
cana-758	30	21	of	of	ADP
cana-758	30	22	second	second	ADJ
cana-758	30	23	–	–	PUNCT
cana-758	30	24	order	order	NOUN
cana-758	30	25	:	:	PUNCT
cana-758	30	26	ȴ(𝓏	ȴ(𝓏	NOUN
cana-758	30	27	)	)	PUNCT
cana-758	30	28	≺	≺	NOUN
cana-758	30	29	ə	ə	PROPN
cana-758	30	30	(	(	PUNCT
cana-758	30	31	𝒫(𝓏	𝒫(𝓏	PROPN
cana-758	30	32	)	)	PUNCT
cana-758	30	33	,	,	PUNCT
cana-758	30	34	𝓏(𝒫(𝓏	𝓏(𝒫(𝓏	PROPN
cana-758	30	35	)	)	PUNCT
cana-758	30	36	)	)	PUNCT
cana-758	30	37	′	′	NUM
cana-758	30	38	,	,	PUNCT
cana-758	30	39	𝓏2(𝒫(𝓏	𝓏2(𝒫(𝓏	NUM
cana-758	30	40	)	)	PUNCT
cana-758	30	41	)	)	PUNCT
cana-758	31	1	′′	′′	PROPN
cana-758	31	2	;	;	PUNCT
cana-758	31	3	𝓏	𝓏	X
cana-758	31	4	)	)	PUNCT
cana-758	31	5	,	,	PUNCT
cana-758	31	6	(	(	PUNCT
cana-758	31	7	1.3	1.3	NUM
cana-758	31	8	)	)	PUNCT
cana-758	31	9	then	then	ADV
cana-758	31	10	the	the	DET
cana-758	31	11	equation	equation	NOUN
cana-758	31	12	(	(	PUNCT
cana-758	31	13	1.3	1.3	NUM
cana-758	31	14	)	)	PUNCT
cana-758	31	15	have	have	VERB
cana-758	31	16	the	the	DET
cana-758	31	17	differential	differential	ADJ
cana-758	31	18	superordination	superordination	NOUN
cana-758	31	19	solution	solution	NOUN
cana-758	31	20	of	of	ADP
cana-758	31	21	(	(	PUNCT
cana-758	31	22	1.3	1.3	NUM
cana-758	31	23	)	)	PUNCT
cana-758	31	24	say	say	VERB
cana-758	31	25	𝒫(𝓏	𝒫(𝓏	NOUN
cana-758	31	26	)	)	PUNCT
cana-758	31	27	.	.	PUNCT
cana-758	32	1	equation	equation	NOUN
cana-758	32	2	(	(	PUNCT
cana-758	32	3	1.3	1.3	NUM
cana-758	32	4	)	)	PUNCT
cana-758	32	5	leads	lead	VERB
cana-758	32	6	to	to	ADP
cana-758	32	7	the	the	DET
cana-758	32	8	solution	solution	NOUN
cana-758	32	9	of	of	ADP
cana-758	32	10	subordinant	subordinant	NOUN
cana-758	32	11	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	32	12	)	)	PUNCT
cana-758	32	13	,	,	PUNCT
cana-758	32	14	where	where	SCONJ
cana-758	32	15	must	must	AUX
cana-758	32	16	be	be	AUX
cana-758	32	17	analytic	analytic	ADJ
cana-758	32	18	function	function	NOUN
cana-758	32	19	or	or	CCONJ
cana-758	32	20	we	we	PRON
cana-758	32	21	can	can	AUX
cana-758	32	22	say	say	VERB
cana-758	32	23	that	that	DET
cana-758	32	24	subordinant	subordinant	NOUN
cana-758	32	25	will	will	AUX
cana-758	32	26	be	be	AUX
cana-758	32	27	more	more	ADV
cana-758	32	28	simple	simple	ADJ
cana-758	32	29	when	when	SCONJ
cana-758	32	30	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	32	31	)	)	PUNCT
cana-758	32	32	≺	≺	NOUN
cana-758	32	33	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	32	34	)	)	PUNCT
cana-758	32	35	for	for	ADP
cana-758	32	36	every	every	DET
cana-758	32	37	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	32	38	)	)	PUNCT
cana-758	32	39	halds	hald	NOUN
cana-758	32	40	(	(	PUNCT
cana-758	32	41	1.3	1.3	NUM
cana-758	32	42	)	)	PUNCT
cana-758	32	43	.	.	PUNCT
cana-758	33	1	the	the	DET
cana-758	33	2	function	function	NOUN
cana-758	33	3	ϥ̃(𝓏	ϥ̃(𝓏	PROPN
cana-758	33	4	)	)	PUNCT
cana-758	33	5	be	be	AUX
cana-758	33	6	univalent	univalent	ADJ
cana-758	33	7	subordinant	subordinant	NOUN
cana-758	33	8	which	which	PRON
cana-758	33	9	satisfy	satisfy	VERB
cana-758	33	10	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	33	11	)	)	PUNCT
cana-758	33	12	≺	≺	NOUN
cana-758	33	13	ϥ̃(𝓏	ϥ̃(𝓏	PROPN
cana-758	33	14	)	)	PUNCT
cana-758	33	15	to	to	ADP
cana-758	33	16	all	all	DET
cana-758	33	17	subordinants	subordinant	NOUN
cana-758	33	18	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	33	19	)	)	PUNCT
cana-758	33	20	of	of	ADP
cana-758	33	21	(	(	PUNCT
cana-758	33	22	1.3	1.3	NUM
cana-758	33	23	)	)	PUNCT
cana-758	33	24	,	,	PUNCT
cana-758	33	25	will	will	AUX
cana-758	33	26	be	be	AUX
cana-758	33	27	best	well	ADV
cana-758	33	28	subordinant	subordinant	NOUN
cana-758	33	29	.	.	PUNCT
cana-758	34	1	in	in	ADP
cana-758	34	2	[	[	X
cana-758	34	3	16	16	NUM
cana-758	34	4	]	]	PUNCT
cana-758	34	5	,	,	PUNCT
cana-758	34	6	miller	miller	PROPN
cana-758	34	7	and	and	CCONJ
cana-758	34	8	macanu	macanu	NOUN
cana-758	34	9	have	have	AUX
cana-758	34	10	obtained	obtain	VERB
cana-758	34	11	sufficient	sufficient	ADJ
cana-758	34	12	conditions	condition	NOUN
cana-758	34	13	to	to	PART
cana-758	34	14	functions	function	VERB
cana-758	34	15	ȴ	ȴ	PRON
cana-758	34	16	,	,	PUNCT
cana-758	34	17	ϥ	ϥ	PROPN
cana-758	34	18	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-758	34	19	ə	ə	PROPN
cana-758	34	20	for	for	ADP
cana-758	34	21	where	where	SCONJ
cana-758	34	22	the	the	DET
cana-758	34	23	implicationts	implicationt	NOUN
cana-758	34	24	given	give	VERB
cana-758	34	25	by	by	ADP
cana-758	34	26	:	:	PUNCT
cana-758	34	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	34	28	)	)	PUNCT
cana-758	34	29	≺	≺	NOUN
cana-758	34	30	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	34	31	)	)	PUNCT
cana-758	34	32	⇒	⇒	NOUN
cana-758	34	33	ȴ(𝓏	ȴ(𝓏	NOUN
cana-758	34	34	)	)	PUNCT
cana-758	34	35	≺	≺	NOUN
cana-758	34	36	ə(𝒫(𝓏	ə(𝒫(𝓏	NUM
cana-758	34	37	)	)	PUNCT
cana-758	34	38	,	,	PUNCT
cana-758	34	39	𝓏𝒫′(𝓏	𝓏𝒫′(𝓏	PROPN
cana-758	34	40	)	)	PUNCT
cana-758	34	41	,	,	PUNCT
cana-758	34	42	𝓏2𝒫′′(𝓏	𝓏2𝒫′′(𝓏	PROPN
cana-758	34	43	)	)	PUNCT
cana-758	34	44	;	;	PUNCT
cana-758	34	45	𝓏	𝓏	X
cana-758	34	46	)	)	PUNCT
cana-758	34	47	(	(	PUNCT
cana-758	34	48	1.4	1.4	NUM
cana-758	34	49	)	)	PUNCT
cana-758	34	50	by	by	ADP
cana-758	34	51	taking	take	VERB
cana-758	34	52	the	the	DET
cana-758	34	53	results	result	NOUN
cana-758	34	54	(	(	PUNCT
cana-758	34	55	see	see	VERB
cana-758	34	56	[	[	X
cana-758	34	57	1,2,4,5,6,7,,8,919	1,2,4,5,6,7,,8,919	NOUN
cana-758	34	58	]	]	PUNCT
cana-758	34	59	)	)	PUNCT
cana-758	34	60	for	for	ADP
cana-758	34	61	obtaining	obtain	VERB
cana-758	34	62	sufficient	sufficient	ADJ
cana-758	34	63	conditions	condition	NOUN
cana-758	34	64	to	to	PART
cana-758	34	65	normalized	normalize	VERB
cana-758	34	66	analytic	analytic	ADJ
cana-758	34	67	functions	function	NOUN
cana-758	34	68	for	for	ADP
cana-758	34	69	satisfing	satisfe	VERB
cana-758	34	70	:	:	PUNCT
cana-758	34	71	ϥ1(𝓏	ϥ1(𝓏	NOUN
cana-758	34	72	)	)	PUNCT
cana-758	34	73	≺	≺	NOUN
cana-758	34	74	𝓏𝑓′(𝓏	𝓏𝑓′(𝓏	NOUN
cana-758	34	75	)	)	PUNCT
cana-758	34	76	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	34	77	)	)	PUNCT
cana-758	34	78	≺	≺	VERB
cana-758	34	79	ϥ2(𝓏	ϥ2(𝓏	SYM
cana-758	34	80	)	)	PUNCT
cana-758	34	81	,	,	PUNCT
cana-758	34	82	such	such	ADJ
cana-758	34	83	that	that	SCONJ
cana-758	34	84	the	the	DET
cana-758	34	85	two	two	NUM
cana-758	34	86	tbe	tbe	VERB
cana-758	34	87	univalent	univalent	ADJ
cana-758	34	88	function	function	NOUN
cana-758	34	89	in	in	ADP
cana-758	34	90	ʋ	ʋ	X
cana-758	34	91	sayϥ1(𝓏)𝑎𝑛𝑑	sayϥ1(𝓏)𝑎𝑛𝑑	X
cana-758	34	92	ϥ2(𝓏	ϥ2(𝓏	NOUN
cana-758	34	93	)	)	PUNCT
cana-758	35	1	where	where	SCONJ
cana-758	35	2	ϥ1(0	ϥ1(0	NOUN
cana-758	35	3	)	)	PUNCT
cana-758	35	4	=	=	PUNCT
cana-758	35	5	ϥ2(0	ϥ2(0	NOUN
cana-758	35	6	)	)	PUNCT
cana-758	35	7	=	=	SYM
cana-758	35	8	1	1	X
cana-758	35	9	.	.	PUNCT
cana-758	36	1	addition	addition	NOUN
cana-758	36	2	to	to	ADP
cana-758	36	3	that	that	PRON
cana-758	36	4	,	,	PUNCT
cana-758	36	5	el	el	PROPN
cana-758	36	6	-	-	NOUN
cana-758	36	7	ashwah	ashwah	NOUN
cana-758	36	8	and	and	CCONJ
cana-758	36	9	aouf	aouf	PROPN
cana-758	37	1	[	[	X
cana-758	37	2	23	23	NUM
cana-758	37	3	]	]	PUNCT
cana-758	37	4	,	,	PUNCT
cana-758	37	5	ali	ali	PROPN
cana-758	37	6	et	et	PROPN
cana-758	37	7	al	al	PROPN
cana-758	37	8	.	.	PUNCT
cana-758	38	1	[	[	X
cana-758	38	2	3	3	NUM
cana-758	38	3	]	]	PUNCT
cana-758	38	4	,	,	PUNCT
cana-758	38	5	atshan	atshan	NOUN
cana-758	38	6	and	and	CCONJ
cana-758	38	7	hadi	hadi	NOUN
cana-758	39	1	[	[	X
cana-758	39	2	6	6	NUM
cana-758	39	3	]	]	PUNCT
cana-758	39	4	,	,	PUNCT
cana-758	39	5	atshan	atshan	PROPN
cana-758	39	6	and	and	CCONJ
cana-758	39	7	ali	ali	PROPN
cana-758	40	1	[	[	X
cana-758	40	2	4	4	NUM
cana-758	40	3	]	]	PUNCT
cana-758	40	4	,	,	PUNCT
cana-758	40	5	and	and	CCONJ
cana-758	40	6	gochhayat	gochhayat	PROPN
cana-758	41	1	[	[	X
cana-758	41	2	25	25	NUM
cana-758	41	3	]	]	PUNCT
cana-758	41	4	derived	derive	VERB
cana-758	41	5	some	some	DET
cana-758	41	6	superordination	superordination	NOUN
cana-758	41	7	and	and	CCONJ
cana-758	41	8	subordination	subordination	NOUN
cana-758	41	9	results	result	NOUN
cana-758	41	10	to	to	ADP
cana-758	41	11	analytic	analytic	ADJ
cana-758	41	12	functions	function	NOUN
cana-758	41	13	in	in	ADP
cana-758	41	14	ʋ	ʋ	X
cana-758	41	15	.	.	PUNCT
cana-758	42	1	recently	recently	ADV
cana-758	42	2	,	,	PUNCT
cana-758	42	3	al	al	PROPN
cana-758	42	4	-	-	PUNCT
cana-758	42	5	ameedee	ameedee	PROPN
cana-758	42	6	et	et	PROPN
cana-758	42	7	al	al	PROPN
cana-758	42	8	.	.	PUNCT
cana-758	43	1	[	[	X
cana-758	43	2	1	1	X
cana-758	43	3	]	]	PUNCT
cana-758	43	4	,	,	PUNCT
cana-758	43	5	atshan	atshan	ADP
cana-758	43	6	et	et	PROPN
cana-758	43	7	al	al	PROPN
cana-758	43	8	.	.	PUNCT
cana-758	44	1	[	[	X
cana-758	44	2	4,5	4,5	X
cana-758	44	3	]	]	PUNCT
cana-758	44	4	and	and	CCONJ
cana-758	44	5	gochhayat	gochhayat	PROPN
cana-758	44	6	[	[	X
cana-758	44	7	24	24	NUM
cana-758	44	8	]	]	PUNCT
cana-758	44	9	got	get	VERB
cana-758	44	10	sandwich	sandwich	NOUN
cana-758	44	11	results	result	NOUN
cana-758	44	12	to	to	ADP
cana-758	44	13	some	some	DET
cana-758	44	14	classes	class	NOUN
cana-758	44	15	of	of	ADP
cana-758	44	16	analytic	analytic	ADJ
cana-758	44	17	functions	function	NOUN
cana-758	44	18	the	the	DET
cana-758	44	19	function	function	NOUN
cana-758	44	20	𝜙𝕥(𝓏	𝜙𝕥(𝓏	NOUN
cana-758	44	21	,	,	PUNCT
cana-758	44	22	𝕥	𝕥	NOUN
cana-758	44	23	,	,	PUNCT
cana-758	44	24	𝕤	𝕤	NOUN
cana-758	44	25	)	)	PUNCT
cana-758	44	26	define	define	VERB
cana-758	44	27	the	the	DET
cana-758	44	28	following	follow	VERB
cana-758	44	29	series	series	NOUN
cana-758	44	30	:	:	PUNCT
cana-758	44	31	𝜙𝕥(𝓏	𝜙𝕥(𝓏	NOUN
cana-758	44	32	,	,	PUNCT
cana-758	44	33	𝕥	𝕥	NOUN
cana-758	44	34	,	,	PUNCT
cana-758	44	35	𝕤	𝕤	NOUN
cana-758	44	36	)	)	PUNCT
cana-758	44	37	=	=	PUNCT
cana-758	45	1	∑	∑	PROPN
cana-758	45	2	𝓏𝑛	𝓏𝑛	INTJ
cana-758	45	3	(	(	PUNCT
cana-758	45	4	𝕣+𝑛𝕤	𝕣+𝑛𝕤	ADJ
cana-758	45	5	𝕤	𝕤	NOUN
cana-758	45	6	)	)	PUNCT
cana-758	45	7	𝕥	𝕥	NOUN
cana-758	45	8	∞	∞	NUM
cana-758	45	9	𝑛=0	𝑛=0	PROPN
cana-758	45	10	.	.	PUNCT
cana-758	46	1	where	where	SCONJ
cana-758	46	2	𝓏	𝓏	PROPN
cana-758	46	3	∈	∈	PROPN
cana-758	46	4	ʋ	ʋ	PROPN
cana-758	46	5	,	,	PUNCT
cana-758	46	6	𝕣	𝕣	DET
cana-758	46	7	∈	∈	PROPN
cana-758	46	8	₵	₵	NOUN
cana-758	46	9	∖	∖	X
cana-758	46	10	𝑧0	𝑧0	VERB
cana-758	46	11	−	−	PROPN
cana-758	46	12	=	=	PUNCT
cana-758	46	13	{	{	PUNCT
cana-758	46	14	0	0	NUM
cana-758	46	15	,	,	PUNCT
cana-758	46	16	−1	−1	NOUN
cana-758	46	17	,	,	PUNCT
cana-758	46	18	−2	−2	NOUN
cana-758	46	19	,	,	PUNCT
cana-758	46	20	…	…	PUNCT
cana-758	46	21	}	}	PUNCT
cana-758	46	22	,	,	PUNCT
cana-758	46	23	𝕥	𝕥	PROPN
cana-758	46	24	∈	∈	PROPN
cana-758	46	25	₵	₵	NOUN
cana-758	46	26	,	,	PUNCT
cana-758	46	27	𝑅𝑒(𝕥	𝑅𝑒(𝕥	ADV
cana-758	46	28	)	)	PUNCT
cana-758	46	29	>	>	X
cana-758	46	30	1	1	NUM
cana-758	46	31	,	,	PUNCT
cana-758	46	32	𝓏	𝓏	PROPN
cana-758	46	33	∈	∈	PROPN
cana-758	46	34	𝜕ʋ	𝜕ʋ	NOUN
cana-758	46	35	,	,	PUNCT
cana-758	46	36	𝕤	𝕤	PRON
cana-758	46	37	∈	∈	NOUN
cana-758	47	1	𝑁	𝑁	ADJ
cana-758	47	2	∖	∖	NOUN
cana-758	47	3	{	{	PUNCT
cana-758	47	4	1	1	NUM
cana-758	47	5	}	}	PUNCT
cana-758	47	6	.	.	PUNCT
cana-758	48	1	the	the	DET
cana-758	48	2	following	following	NOUN
cana-758	48	3	normalized	normalize	VERB
cana-758	48	4	function	function	NOUN
cana-758	48	5	𝒥𝕣,𝕤	𝒥𝕣,𝕤	PROPN
cana-758	48	6	𝕥	𝕥	PROPN
cana-758	48	7	(	(	PUNCT
cana-758	48	8	𝓏	𝓏	NOUN
cana-758	48	9	)	)	PUNCT
cana-758	48	10	defined	define	VERB
cana-758	48	11	as	as	ADP
cana-758	48	12	:	:	PUNCT
cana-758	48	13	𝒥𝕣,𝕤	𝒥𝕣,𝕤	PROPN
cana-758	48	14	𝕥	𝕥	PROPN
cana-758	48	15	(	(	PUNCT
cana-758	48	16	𝓏	𝓏	NOUN
cana-758	48	17	)	)	PUNCT
cana-758	48	18	=	=	SYM
cana-758	48	19	(	(	PUNCT
cana-758	48	20	𝕣	𝕣	X
cana-758	48	21	𝕤	𝕤	NOUN
cana-758	48	22	+	+	NOUN
cana-758	48	23	1	1	X
cana-758	48	24	)	)	PUNCT
cana-758	48	25	𝕥	𝕥	NOUN
cana-758	48	26	[	[	X
cana-758	48	27	𝜙𝕥(𝓏	𝜙𝕥(𝓏	X
cana-758	48	28	,	,	PUNCT
cana-758	48	29	𝕥	𝕥	NOUN
cana-758	48	30	,	,	PUNCT
cana-758	48	31	𝕤	𝕤	NOUN
cana-758	48	32	)	)	PUNCT
cana-758	48	33	−	−	PROPN
cana-758	49	1	(	(	PUNCT
cana-758	49	2	𝕣	𝕣	NOUN
cana-758	49	3	𝕤	𝕤	NOUN
cana-758	49	4	)	)	PUNCT
cana-758	49	5	−𝕥	−𝕥	NOUN
cana-758	49	6	]	]	PUNCT
cana-758	49	7	=	=	PUNCT
cana-758	49	8	𝓏	𝓏	PROPN
cana-758	49	9	+	+	CCONJ
cana-758	49	10	∑	∑	PROPN
cana-758	49	11	(	(	PUNCT
cana-758	49	12	𝕣+𝕤	𝕣+𝕤	X
cana-758	49	13	𝕣+𝓃𝕤	𝕣+𝓃𝕤	ADV
cana-758	49	14	)	)	PUNCT
cana-758	49	15	𝕥	𝕥	PROPN
cana-758	49	16	∞	∞	PROPN
cana-758	49	17	𝑛=0	𝑛=0	PROPN
cana-758	49	18	𝒶𝑛𝓏𝑛	𝒶𝑛𝓏𝑛	PROPN
cana-758	49	19	,	,	PUNCT
cana-758	49	20	𝓏	𝓏	PROPN
cana-758	49	21	∈	∈	PROPN
cana-758	49	22	ʋ	ʋ	X
cana-758	49	23	,	,	PUNCT
cana-758	49	24	(	(	PUNCT
cana-758	49	25	1	1	NUM
cana-758	49	26	.	.	NOUN
cana-758	49	27	5	5	NUM
cana-758	49	28	)	)	PUNCT
cana-758	49	29	see	see	VERB
cana-758	49	30	more	more	ADJ
cana-758	49	31	[	[	X
cana-758	49	32	22	22	NUM
cana-758	49	33	]	]	PUNCT
cana-758	49	34	.	.	PUNCT
cana-758	50	1	communications	communication	NOUN
cana-758	50	2	on	on	ADP
cana-758	50	3	applied	apply	VERB
cana-758	50	4	nonlinear	nonlinear	ADJ
cana-758	50	5	analysis	analysis	NOUN
cana-758	50	6	issn	issn	NOUN
cana-758	50	7	:	:	PUNCT
cana-758	50	8	1074	1074	NUM
cana-758	50	9	-	-	PUNCT
cana-758	50	10	133x	133x	NUM
cana-758	50	11	vol	vol	NOUN
cana-758	50	12	31	31	NUM
cana-758	50	13	no	no	NOUN
cana-758	50	14	.	.	PUNCT
cana-758	51	1	3s	3s	NUM
cana-758	51	2	(	(	PUNCT
cana-758	51	3	2024	2024	NUM
cana-758	51	4	)	)	PUNCT
cana-758	51	5	188	188	NUM
cana-758	51	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	51	7	swamy	swamy	NOUN
cana-758	52	1	[	[	X
cana-758	52	2	21	21	NUM
cana-758	52	3	]	]	X
cana-758	52	4	defined	define	VERB
cana-758	52	5	linear	linear	NOUN
cana-758	52	6	operator	operator	NOUN
cana-758	52	7	ℱ𝕦,𝕧	ℱ𝕦,𝕧	PROPN
cana-758	52	8	𝕞	𝕞	PRON
cana-758	52	9	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	52	10	)	)	PUNCT
cana-758	52	11	for	for	ADP
cana-758	52	12	𝕞	𝕞	PROPN
cana-758	52	13	∈	∈	PROPN
cana-758	52	14	ℕ0	ℕ0	NOUN
cana-758	52	15	=	=	SYM
cana-758	52	16	ℕ	ℕ	PROPN
cana-758	52	17	∪	∪	X
cana-758	52	18	{	{	PUNCT
cana-758	52	19	0	0	NUM
cana-758	52	20	}	}	PUNCT
cana-758	52	21	,	,	PUNCT
cana-758	52	22	𝕦	𝕦	PROPN
cana-758	52	23	∈	∈	PROPN
cana-758	52	24	𝑅	𝑅	PROPN
cana-758	52	25	,	,	PUNCT
cana-758	52	26	𝕧	𝕧	PROPN
cana-758	52	27	≥	≥	NOUN
cana-758	52	28	0	0	NUM
cana-758	52	29	,	,	PUNCT
cana-758	52	30	𝕦	𝕦	PROPN
cana-758	52	31	+	+	CCONJ
cana-758	52	32	𝕧	𝕧	X
cana-758	52	33	>	>	X
cana-758	52	34	0	0	PUNCT
cana-758	52	35	and	and	CCONJ
cana-758	52	36	define	define	VERB
cana-758	52	37	by	by	ADP
cana-758	52	38	:	:	PUNCT
cana-758	52	39	ℱ𝕦,𝕧	ℱ𝕦,𝕧	PROPN
cana-758	52	40	𝕞	𝕞	X
cana-758	52	41	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	52	42	)	)	PUNCT
cana-758	52	43	=	=	SYM
cana-758	53	1	𝓏	𝓏	PROPN
cana-758	53	2	+	+	CCONJ
cana-758	53	3	∑	∑	PROPN
cana-758	53	4	(	(	PUNCT
cana-758	53	5	𝕦+𝔫𝕧	𝕦+𝔫𝕧	NOUN
cana-758	53	6	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	53	7	)	)	PUNCT
cana-758	53	8	𝕞	𝕞	PRON
cana-758	53	9	∞	∞	PROPN
cana-758	53	10	𝒏=𝟐	𝒏=𝟐	PRON
cana-758	53	11	𝒶𝑛𝓏𝑛	𝒶𝑛𝓏𝑛	NOUN
cana-758	53	12	.	.	PUNCT
cana-758	54	1	(	(	PUNCT
cana-758	54	2	1.6	1.6	NUM
cana-758	54	3	)	)	PUNCT
cana-758	54	4	see	see	VERB
cana-758	54	5	[	[	X
cana-758	54	6	10,11,12,13,14,17,20,21	10,11,12,13,14,17,20,21	NUM
cana-758	54	7	]	]	PUNCT
cana-758	54	8	.	.	PUNCT
cana-758	55	1	definition	definition	NOUN
cana-758	55	2	(	(	PUNCT
cana-758	55	3	1.3	1.3	NUM
cana-758	55	4	):	):	PUNCT
cana-758	55	5	suppose	suppose	VERB
cana-758	55	6	𝑓	𝑓	DET
cana-758	55	7	∈	∈	PROPN
cana-758	55	8	𝒢	𝒢	PROPN
cana-758	55	9	,	,	PUNCT
cana-758	55	10	𝓏	𝓏	PROPN
cana-758	55	11	∈	∈	PROPN
cana-758	55	12	ʋ	ʋ	PROPN
cana-758	55	13	,	,	PUNCT
cana-758	55	14	𝕣	𝕣	DET
cana-758	55	15	∈	∈	PROPN
cana-758	55	16	₵	₵	NOUN
cana-758	55	17	∖	∖	X
cana-758	55	18	𝑧0	𝑧0	VERB
cana-758	55	19	−	−	PROPN
cana-758	56	1	=	=	PUNCT
cana-758	56	2	{	{	PUNCT
cana-758	56	3	0	0	NUM
cana-758	56	4	,	,	PUNCT
cana-758	56	5	−1	−1	NOUN
cana-758	56	6	,	,	PUNCT
cana-758	56	7	−2	−2	NOUN
cana-758	56	8	,	,	PUNCT
cana-758	56	9	…	…	PUNCT
cana-758	56	10	}	}	PUNCT
cana-758	56	11	,	,	PUNCT
cana-758	56	12	𝕥	𝕥	PROPN
cana-758	56	13	∈	∈	PROPN
cana-758	56	14	₵	₵	NOUN
cana-758	56	15	,	,	PUNCT
cana-758	56	16	,	,	PUNCT
cana-758	56	17	𝕞	𝕞	PROPN
cana-758	56	18	∈	∈	NOUN
cana-758	56	19	ℕ0	ℕ0	NOUN
cana-758	56	20	=	=	SYM
cana-758	56	21	ℕ	ℕ	PROPN
cana-758	56	22	∪	∪	X
cana-758	56	23	{	{	PUNCT
cana-758	56	24	0	0	NUM
cana-758	56	25	}	}	PUNCT
cana-758	56	26	,	,	PUNCT
cana-758	56	27	𝕦	𝕦	PROPN
cana-758	56	28	∈	∈	PROPN
cana-758	56	29	𝑅	𝑅	PROPN
cana-758	56	30	,	,	PUNCT
cana-758	56	31	𝕤	𝕤	PRON
cana-758	56	32	∈	∈	NOUN
cana-758	56	33	𝑁	𝑁	ADJ
cana-758	56	34	∖	∖	NOUN
cana-758	56	35	{	{	PUNCT
cana-758	56	36	1	1	NUM
cana-758	56	37	}	}	PUNCT
cana-758	56	38	,	,	PUNCT
cana-758	56	39	𝕧	𝕧	PROPN
cana-758	56	40	≥	≥	NOUN
cana-758	56	41	0	0	NUM
cana-758	56	42	,	,	PUNCT
cana-758	56	43	𝕦	𝕦	PROPN
cana-758	56	44	+	+	CCONJ
cana-758	56	45	𝕧	𝕧	X
cana-758	56	46	>	>	X
cana-758	56	47	0	0	NUM
cana-758	56	48	,	,	PUNCT
cana-758	56	49	𝑅𝑒(𝕥	𝑅𝑒(𝕥	ADV
cana-758	56	50	)	)	PUNCT
cana-758	56	51	>	>	X
cana-758	56	52	1	1	NUM
cana-758	56	53	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	PROPN
cana-758	56	54	𝓏	𝓏	PROPN
cana-758	56	55	∈	∈	PROPN
cana-758	56	56	𝜕ʋ	𝜕ʋ	NOUN
cana-758	56	57	and	and	CCONJ
cana-758	56	58	we	we	PRON
cana-758	56	59	define	define	VERB
cana-758	56	60	new	new	ADJ
cana-758	56	61	operator	operator	NOUN
cana-758	56	62	:	:	PUNCT
cana-758	56	63	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	57	1	𝕞	𝕞	ADP
cana-758	57	2	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	57	3	):	):	PUNCT
cana-758	57	4	𝒢	𝒢	PROPN
cana-758	57	5	→	→	SYM
cana-758	57	6	𝒢	𝒢	PROPN
cana-758	57	7	,	,	PUNCT
cana-758	57	8	where	where	SCONJ
cana-758	57	9	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	57	10	𝕞	𝕞	PRON
cana-758	57	11	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	57	12	)	)	PUNCT
cana-758	57	13	=	=	SYM
cana-758	57	14	𝒥𝕣,𝕤	𝒥𝕣,𝕤	PROPN
cana-758	57	15	𝕥	𝕥	PROPN
cana-758	57	16	(	(	PUNCT
cana-758	57	17	𝓏	𝓏	NOUN
cana-758	57	18	)	)	PUNCT
cana-758	57	19	∗	∗	VERB
cana-758	57	20	ℱ𝕦,𝕧	ℱ𝕦,𝕧	PROPN
cana-758	57	21	𝕞	𝕞	DET
cana-758	57	22	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	57	23	)	)	PUNCT
cana-758	57	24	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	57	25	𝕞	𝕞	DET
cana-758	57	26	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	57	27	)	)	PUNCT
cana-758	57	28	=	=	PUNCT
cana-758	58	1	𝔃	𝔃	X
cana-758	58	2	+	+	ADV
cana-758	58	3	∑	∑	PROPN
cana-758	58	4	(	(	PUNCT
cana-758	58	5	𝕣+𝕤	𝕣+𝕤	PROPN
cana-758	58	6	𝕣+𝕤𝔫	𝕣+𝕤𝔫	NOUN
cana-758	58	7	)	)	PUNCT
cana-758	58	8	𝕥	𝕥	PROPN
cana-758	58	9	(	(	PUNCT
cana-758	58	10	𝕦+𝖓𝕧	𝕦+𝖓𝕧	PROPN
cana-758	58	11	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	58	12	)	)	PUNCT
cana-758	58	13	𝕞	𝕞	PRON
cana-758	58	14	∞	∞	PROPN
cana-758	58	15	𝒏=𝟐	𝒏=𝟐	PRON
cana-758	58	16	𝒶𝑛𝓏𝑛.	𝒶𝑛𝓏𝑛.	NOUN
cana-758	58	17	(	(	PUNCT
cana-758	58	18	1.7	1.7	NUM
cana-758	58	19	)	)	PUNCT
cana-758	58	20	we	we	PRON
cana-758	58	21	have	have	VERB
cana-758	58	22	from	from	ADP
cana-758	58	23	(	(	PUNCT
cana-758	58	24	1.7	1.7	NUM
cana-758	58	25	)	)	PUNCT
cana-758	58	26	that	that	SCONJ
cana-758	58	27	:	:	PUNCT
cana-758	58	28	(	(	PUNCT
cana-758	58	29	𝕦	𝕦	PROPN
cana-758	58	30	+	+	NUM
cana-758	58	31	𝕧)𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝕧)𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	58	32	𝕞+1	𝕞+1	NUM
cana-758	58	33	𝑓(𝓏	𝑓(𝓏	X
cana-758	58	34	)	)	PUNCT
cana-758	58	35	=	=	PUNCT
cana-758	58	36	𝕦𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝕦𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	58	37	𝕞	𝕞	DET
cana-758	58	38	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	58	39	)	)	PUNCT
cana-758	59	1	−	−	PROPN
cana-758	59	2	𝕧𝓏	𝕧𝓏	PROPN
cana-758	59	3	(	(	PUNCT
cana-758	59	4	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	59	5	𝕞	𝕞	X
cana-758	59	6	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	59	7	)	)	PUNCT
cana-758	59	8	)	)	PUNCT
cana-758	59	9	′	′	NOUN
cana-758	59	10	,	,	PUNCT
cana-758	59	11	𝕧	𝕧	PROPN
cana-758	59	12	>	>	X
cana-758	59	13	0	0	PUNCT
cana-758	59	14	.	.	PUNCT
cana-758	60	1	(	(	PUNCT
cana-758	60	2	1.8	1.8	NUM
cana-758	60	3	)	)	PUNCT
cana-758	60	4	we	we	PRON
cana-758	60	5	observe	observe	VERB
cana-758	60	6	that	that	SCONJ
cana-758	60	7	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	60	8	𝕞	𝕞	PRON
cana-758	61	1	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	61	2	):	):	PUNCT
cana-758	61	3	𝒢	𝒢	PROPN
cana-758	61	4	→	→	SYM
cana-758	61	5	𝒢	𝒢	PROPN
cana-758	61	6	is	be	AUX
cana-758	61	7	an	an	DET
cana-758	61	8	integral	integral	ADJ
cana-758	61	9	operator	operator	NOUN
cana-758	61	10	and	and	CCONJ
cana-758	61	11	for	for	ADP
cana-758	61	12	f	f	PROPN
cana-758	61	13	given	give	VERB
cana-758	61	14	by	by	ADP
cana-758	61	15	(	(	PUNCT
cana-758	61	16	1.2	1.2	NUM
cana-758	61	17	)	)	PUNCT
cana-758	61	18	we	we	PRON
cana-758	61	19	have	have	VERB
cana-758	61	20	:	:	PUNCT
cana-758	61	21	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	61	22	𝕞	𝕞	PRON
cana-758	61	23	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	61	24	)	)	PUNCT
cana-758	61	25	=	=	PUNCT
cana-758	62	1	𝔃	𝔃	X
cana-758	62	2	+	+	ADV
cana-758	62	3	∑	∑	PROPN
cana-758	62	4	(	(	PUNCT
cana-758	62	5	𝕣+𝕤	𝕣+𝕤	PROPN
cana-758	62	6	𝕣+𝕤𝔫	𝕣+𝕤𝔫	NOUN
cana-758	62	7	)	)	PUNCT
cana-758	62	8	𝕥	𝕥	PROPN
cana-758	62	9	(	(	PUNCT
cana-758	62	10	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	62	11	𝕦+𝔫𝕧	𝕦+𝔫𝕧	NOUN
cana-758	62	12	)	)	PUNCT
cana-758	62	13	𝕞	𝕞	DET
cana-758	62	14	∞	∞	PROPN
cana-758	62	15	𝒏=𝟐	𝒏=𝟐	PRON
cana-758	62	16	𝒶𝑛𝓏𝑛	𝒶𝑛𝓏𝑛	NOUN
cana-758	62	17	,	,	PUNCT
cana-758	62	18	𝓏	𝓏	PROPN
cana-758	62	19	∈	∈	PROPN
cana-758	62	20	ʋ	ʋ	X
cana-758	62	21	.	.	PUNCT
cana-758	63	1	(	(	PUNCT
cana-758	63	2	1.9	1.9	NUM
cana-758	63	3	)	)	PUNCT
cana-758	63	4	it	it	PRON
cana-758	63	5	follows	follow	VERB
cana-758	63	6	form	form	NOUN
cana-758	63	7	(	(	PUNCT
cana-758	63	8	1.9	1.9	NUM
cana-758	63	9	)	)	PUNCT
cana-758	63	10	that	that	PRON
cana-758	63	11	:	:	PUNCT
cana-758	63	12	(	(	PUNCT
cana-758	63	13	𝕦	𝕦	PROPN
cana-758	63	14	+	+	X
cana-758	63	15	𝕧)𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝕧)𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	X
cana-758	63	16	𝕞	𝕞	DET
cana-758	63	17	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	63	18	)	)	PUNCT
cana-758	63	19	=	=	PUNCT
cana-758	63	20	𝕦𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝕦𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	X
cana-758	63	21	𝕞+1	𝕞+1	NUM
cana-758	63	22	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	63	23	)	)	PUNCT
cana-758	64	1	+	+	CCONJ
cana-758	64	2	𝕧𝓏	𝕧𝓏	X
cana-758	64	3	(	(	PUNCT
cana-758	64	4	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	64	5	𝕞+1	𝕞+1	NUM
cana-758	64	6	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	64	7	)	)	PUNCT
cana-758	64	8	)	)	PUNCT
cana-758	64	9	′	′	X
cana-758	64	10	.	.	PUNCT
cana-758	65	1	(	(	PUNCT
cana-758	65	2	1.10	1.10	NUM
cana-758	65	3	)	)	PUNCT
cana-758	65	4	now	now	ADV
cana-758	65	5	,	,	PUNCT
cana-758	65	6	in	in	ADP
cana-758	65	7	this	this	DET
cana-758	65	8	work	work	NOUN
cana-758	65	9	has	have	AUX
cana-758	65	10	been	be	AUX
cana-758	65	11	dedicated	dedicate	VERB
cana-758	65	12	to	to	PART
cana-758	65	13	derive	derive	VERB
cana-758	65	14	several	several	ADJ
cana-758	65	15	superordination	superordination	NOUN
cana-758	65	16	,	,	PUNCT
cana-758	65	17	subordination	subordination	NOUN
cana-758	65	18	and	and	CCONJ
cana-758	65	19	sandwich	sandwich	NOUN
cana-758	65	20	results	result	NOUN
cana-758	65	21	of	of	ADP
cana-758	65	22	differential	differential	NOUN
cana-758	65	23	containing	contain	VERB
cana-758	65	24	new	new	ADJ
cana-758	65	25	operators	operator	NOUN
cana-758	65	26	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	VERB
cana-758	65	27	𝕞	𝕞	DET
cana-758	65	28	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	65	29	)	)	PUNCT
cana-758	65	30	and	and	CCONJ
cana-758	65	31	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	65	32	𝕞	𝕞	PRON
cana-758	65	33	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	65	34	)	)	PUNCT
cana-758	65	35	.	.	PUNCT
cana-758	66	1	2	2	X
cana-758	66	2	.	.	X
cana-758	66	3	preliminaries	preliminary	NOUN
cana-758	66	4	constructing	construct	VERB
cana-758	66	5	our	our	PRON
cana-758	66	6	major	major	ADJ
cana-758	66	7	results	result	NOUN
cana-758	66	8	,	,	PUNCT
cana-758	66	9	some	some	DET
cana-758	66	10	following	follow	VERB
cana-758	66	11	lemmas	lemmas	PROPN
cana-758	66	12	will	will	AUX
cana-758	66	13	be	be	AUX
cana-758	66	14	needed	need	VERB
cana-758	66	15	with	with	ADP
cana-758	66	16	its	its	PRON
cana-758	66	17	references	reference	NOUN
cana-758	66	18	,	,	PUNCT
cana-758	66	19	(	(	PUNCT
cana-758	66	20	see	see	VERB
cana-758	66	21	also	also	ADV
cana-758	66	22	[	[	X
cana-758	66	23	23	23	NUM
cana-758	66	24	]	]	PUNCT
cana-758	66	25	)	)	PUNCT
cana-758	66	26	.	.	PUNCT
cana-758	67	1	definition	definition	NOUN
cana-758	67	2	(	(	PUNCT
cana-758	67	3	2.1)[15	2.1)[15	NOUN
cana-758	67	4	]	]	PUNCT
cana-758	67	5	:	:	PUNCT
cana-758	67	6	called	call	VERB
cana-758	67	7	by	by	ADP
cana-758	67	8	𝚀	𝚀	PROPN
cana-758	67	9	which	which	PRON
cana-758	67	10	represent	represent	VERB
cana-758	67	11	all	all	PRON
cana-758	67	12	𝑓	𝑓	DET
cana-758	67	13	functions	function	NOUN
cana-758	67	14	,	,	PUNCT
cana-758	67	15	they	they	PRON
cana-758	67	16	must	must	AUX
cana-758	67	17	be	be	AUX
cana-758	67	18	analytic	analytic	ADJ
cana-758	67	19	and	and	CCONJ
cana-758	67	20	one	one	NUM
cana-758	67	21	–	–	PUNCT
cana-758	67	22	to	to	ADP
cana-758	67	23	–	–	PUNCT
cana-758	67	24	one	one	NUM
cana-758	67	25	on	on	ADP
cana-758	67	26	ʋ	ʋ	X
cana-758	67	27	∖	∖	PRON
cana-758	67	28	𝐸(𝑓	𝐸(𝑓	NUM
cana-758	67	29	)	)	PUNCT
cana-758	67	30	,	,	PUNCT
cana-758	67	31	where	where	SCONJ
cana-758	67	32	𝐸(𝑓	𝐸(𝑓	VERB
cana-758	67	33	)	)	PUNCT
cana-758	67	34	=	=	NOUN
cana-758	67	35	{	{	PUNCT
cana-758	67	36	𝜁	𝜁	PROPN
cana-758	67	37	∈	∈	PROPN
cana-758	67	38	𝜕ʋ	𝜕ʋ	NOUN
cana-758	67	39	:	:	PUNCT
cana-758	67	40	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-758	67	41	𝓏→𝜁	𝓏→𝜁	PROPN
cana-758	67	42	𝑓(𝓏	𝑓(𝓏	X
cana-758	67	43	)	)	PUNCT
cana-758	67	44	=	=	SYM
cana-758	68	1	∞	∞	PROPN
cana-758	68	2	}	}	PUNCT
cana-758	68	3	and	and	CCONJ
cana-758	68	4	ʋ̅	ʋ̅	PUNCT
cana-758	68	5	=	=	NOUN
cana-758	68	6	ʋ	ʋ	X
cana-758	68	7	∪	∪	X
cana-758	68	8	{	{	PUNCT
cana-758	68	9	𝓏	𝓏	PROPN
cana-758	68	10	∈	∈	PROPN
cana-758	68	11	𝜕ʋ	𝜕ʋ	NOUN
cana-758	68	12	}	}	PUNCT
cana-758	68	13	,	,	PUNCT
cana-758	68	14	in	in	ADP
cana-758	68	15	addition	addition	NOUN
cana-758	68	16	to	to	ADP
cana-758	68	17	that	that	PRON
cana-758	68	18	𝑓′(𝜁	𝑓′(𝜁	ADJ
cana-758	68	19	)	)	PUNCT
cana-758	68	20	≠	≠	PROPN
cana-758	68	21	0	0	NUM
cana-758	68	22	of	of	ADP
cana-758	68	23	𝜁	𝜁	PROPN
cana-758	68	24	∈	∈	PROPN
cana-758	68	25	𝜕ʋ\𝐸(𝑓	𝜕ʋ\𝐸(𝑓	NUM
cana-758	68	26	)	)	PUNCT
cana-758	68	27	.	.	PUNCT
cana-758	69	1	more	more	ADJ
cana-758	69	2	that	that	SCONJ
cana-758	69	3	,	,	PUNCT
cana-758	69	4	we	we	PRON
cana-758	69	5	consider	consider	VERB
cana-758	69	6	a	a	DET
cana-758	69	7	subclass	subclass	NOUN
cana-758	69	8	of	of	ADP
cana-758	69	9	𝚀	𝚀	NOUN
cana-758	69	10	to	to	ADP
cana-758	69	11	𝑓(0	𝑓(0	PROPN
cana-758	69	12	)	)	PUNCT
cana-758	70	1	=	=	SYM
cana-758	70	2	𝒶	𝒶	PROPN
cana-758	70	3	is	be	AUX
cana-758	70	4	called	call	VERB
cana-758	70	5	𝚀(𝒶	𝚀(𝒶	ADP
cana-758	70	6	)	)	PUNCT
cana-758	70	7	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-758	70	8	𝚀(1	𝚀(1	NOUN
cana-758	70	9	)	)	PUNCT
cana-758	70	10	=	=	NOUN
cana-758	70	11	𝚀1𝑎𝑛𝑑	𝚀1𝑎𝑛𝑑	X
cana-758	70	12	𝚀(0	𝚀(0	NUM
cana-758	70	13	)	)	PUNCT
cana-758	70	14	=	=	SYM
cana-758	70	15	𝚀0	𝚀0	PROPN
cana-758	70	16	.	.	PUNCT
cana-758	71	1	lemma	lemma	PROPN
cana-758	71	2	(	(	PUNCT
cana-758	71	3	2.2	2.2	NUM
cana-758	71	4	)	)	PUNCT
cana-758	72	1	[	[	X
cana-758	72	2	15	15	NUM
cana-758	72	3	]	]	PRON
cana-758	72	4	:	:	PUNCT
cana-758	72	5	assume	assume	VERB
cana-758	72	6	the	the	DET
cana-758	72	7	function	function	NOUN
cana-758	72	8	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	72	9	)	)	PUNCT
cana-758	72	10	be	be	AUX
cana-758	72	11	univalent	univalent	ADJ
cana-758	72	12	and	and	CCONJ
cana-758	72	13	convex	convex	VERB
cana-758	72	14	in	in	ADP
cana-758	72	15	ʋ	ʋ	PRON
cana-758	72	16	assume	assume	VERB
cana-758	72	17	𝛽	𝛽	PROPN
cana-758	72	18	∈	∈	PROPN
cana-758	72	19	₵	₵	NOUN
cana-758	72	20	∕	∕	NOUN
cana-758	72	21	{	{	PUNCT
cana-758	72	22	0	0	NUM
cana-758	72	23	}	}	PUNCT
cana-758	72	24	𝛼	𝛼	PROPN
cana-758	72	25	∈	∈	PROPN
cana-758	72	26	₵	₵	NOUN
cana-758	72	27	,	,	PUNCT
cana-758	72	28	,	,	PUNCT
cana-758	72	29	𝛽	𝛽	PROPN
cana-758	72	30	≠	≠	PROPN
cana-758	72	31	0	0	NUM
cana-758	72	32	,	,	PUNCT
cana-758	72	33	and	and	CCONJ
cana-758	72	34	suppose	suppose	VERB
cana-758	72	35	𝑅𝑒	𝑅𝑒	VERB
cana-758	72	36	{	{	PUNCT
cana-758	72	37	1	1	NUM
cana-758	72	38	+	+	PROPN
cana-758	72	39	𝓏ϥ	𝓏ϥ	PROPN
cana-758	72	40	"	"	PUNCT
cana-758	72	41	(	(	PUNCT
cana-758	72	42	𝓏	𝓏	PROPN
cana-758	72	43	)	)	PUNCT
cana-758	72	44	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	72	45	)	)	PUNCT
cana-758	72	46	}	}	PUNCT
cana-758	72	47	>	>	X
cana-758	72	48	max	max	PROPN
cana-758	72	49	{	{	PUNCT
cana-758	72	50	0	0	NUM
cana-758	72	51	,	,	PUNCT
cana-758	72	52	−𝑅𝑒	−𝑅𝑒	NOUN
cana-758	72	53	(	(	PUNCT
cana-758	72	54	α	α	NOUN
cana-758	72	55	β	β	NOUN
cana-758	72	56	)	)	PUNCT
cana-758	72	57	}	}	PUNCT
cana-758	72	58	.	.	PUNCT
cana-758	73	1	if	if	SCONJ
cana-758	73	2	𝑃(𝓏	𝑃(𝓏	NUM
cana-758	73	3	)	)	PUNCT
cana-758	73	4	is	be	AUX
cana-758	73	5	analytic	analytic	ADJ
cana-758	73	6	function	function	NOUN
cana-758	73	7	in	in	ADP
cana-758	73	8	ʋ	ʋ	PROPN
cana-758	73	9	,	,	PUNCT
cana-758	73	10	and	and	CCONJ
cana-758	73	11	communications	communication	NOUN
cana-758	73	12	on	on	ADP
cana-758	73	13	applied	apply	VERB
cana-758	73	14	nonlinear	nonlinear	ADJ
cana-758	73	15	analysis	analysis	NOUN
cana-758	73	16	issn	issn	NOUN
cana-758	73	17	:	:	PUNCT
cana-758	73	18	1074	1074	NUM
cana-758	73	19	-	-	PUNCT
cana-758	73	20	133x	133x	NUM
cana-758	73	21	vol	vol	NOUN
cana-758	73	22	31	31	NUM
cana-758	73	23	no	no	NOUN
cana-758	73	24	.	.	PUNCT
cana-758	74	1	3s	3s	NUM
cana-758	74	2	(	(	PUNCT
cana-758	74	3	2024	2024	NUM
cana-758	74	4	)	)	PUNCT
cana-758	74	5	189	189	NUM
cana-758	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	74	7	𝛼𝒫	𝛼𝒫	PROPN
cana-758	74	8	(	(	PUNCT
cana-758	74	9	𝓏	𝓏	NOUN
cana-758	74	10	)	)	PUNCT
cana-758	74	11	+	+	CCONJ
cana-758	74	12	𝛽𝓏(𝒫(𝓏	𝛽𝓏(𝒫(𝓏	NOUN
cana-758	74	13	)	)	PUNCT
cana-758	74	14	)	)	PUNCT
cana-758	75	1	′	′	NUM
cana-758	75	2	≺	≺	NOUN
cana-758	75	3	𝛼ϥ	𝛼ϥ	CCONJ
cana-758	75	4	(	(	PUNCT
cana-758	75	5	𝓏	𝓏	NOUN
cana-758	75	6	)	)	PUNCT
cana-758	75	7	+	+	CCONJ
cana-758	75	8	𝛽𝓏(ϥ(𝓏	𝛽𝓏(ϥ(𝓏	NOUN
cana-758	75	9	)	)	PUNCT
cana-758	75	10	)	)	PUNCT
cana-758	75	11	′	′	NOUN
cana-758	75	12	,	,	PUNCT
cana-758	75	13	(	(	PUNCT
cana-758	75	14	2.1	2.1	NUM
cana-758	75	15	)	)	PUNCT
cana-758	75	16	hence	hence	ADV
cana-758	75	17	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	75	18	)	)	PUNCT
cana-758	75	19	will	will	AUX
cana-758	75	20	be	be	AUX
cana-758	75	21	best	well	ADV
cana-758	75	22	dominant	dominant	ADJ
cana-758	75	23	and	and	CCONJ
cana-758	75	24	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	75	25	)	)	PUNCT
cana-758	75	26	≺	≺	NOUN
cana-758	75	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	75	28	)	)	PUNCT
cana-758	75	29	.	.	PUNCT
cana-758	76	1	lemma	lemma	PROPN
cana-758	76	2	(	(	PUNCT
cana-758	76	3	2.3	2.3	NUM
cana-758	76	4	)	)	PUNCT
cana-758	77	1	[	[	X
cana-758	77	2	15	15	NUM
cana-758	77	3	]	]	PUNCT
cana-758	77	4	:	:	PUNCT
cana-758	77	5	let	let	VERB
cana-758	77	6	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	77	7	)	)	PUNCT
cana-758	77	8	is	be	AUX
cana-758	77	9	belongs	belong	VERB
cana-758	77	10	in	in	ADP
cana-758	77	11	ʋ	ʋ	PROPN
cana-758	77	12	with	with	ADP
cana-758	77	13	𝑞(0	𝑞(0	PROPN
cana-758	77	14	)	)	PUNCT
cana-758	77	15	=	=	SYM
cana-758	78	1	1,where	1,where	X
cana-758	78	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	78	3	)	)	PUNCT
cana-758	78	4	is	be	AUX
cana-758	78	5	convex	convex	ADJ
cana-758	78	6	and	and	CCONJ
cana-758	78	7	univalent	univalent	ADJ
cana-758	78	8	.	.	PUNCT
cana-758	79	1	let	let	VERB
cana-758	80	1	𝑅𝑒	𝑅𝑒	PROPN
cana-758	80	2	(	(	PUNCT
cana-758	80	3	𝛽	𝛽	NOUN
cana-758	80	4	)	)	PUNCT
cana-758	80	5	>	>	X
cana-758	80	6	0	0	PUNCT
cana-758	80	7	and	and	CCONJ
cana-758	80	8	𝛽	𝛽	PROPN
cana-758	80	9	∈	∈	PROPN
cana-758	80	10	₵	₵	NOUN
cana-758	80	11	.	.	PUNCT
cana-758	81	1	if	if	SCONJ
cana-758	81	2	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	81	3	)	)	PUNCT
cana-758	81	4	∈	∈	PROPN
cana-758	81	5	𝑀[ϥ(0	𝑀[ϥ(0	PROPN
cana-758	81	6	)	)	PUNCT
cana-758	81	7	,	,	PUNCT
cana-758	81	8	1	1	X
cana-758	81	9	]	]	PUNCT
cana-758	81	10	∩	∩	ADJ
cana-758	81	11	𝚀	𝚀	PROPN
cana-758	81	12	and	and	CCONJ
cana-758	81	13	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	81	14	)	)	PUNCT
cana-758	81	15	+	+	CCONJ
cana-758	81	16	𝛽𝓏(𝒫(𝓏	𝛽𝓏(𝒫(𝓏	NOUN
cana-758	81	17	)	)	PUNCT
cana-758	81	18	)	)	PUNCT
cana-758	81	19	′	′	NUM
cana-758	81	20	is	be	AUX
cana-758	81	21	univalent	univalent	ADJ
cana-758	81	22	in	in	ADP
cana-758	81	23	ʋ	ʋ	PROPN
cana-758	81	24	,	,	PUNCT
cana-758	81	25	then	then	ADV
cana-758	81	26	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	81	27	)	)	PUNCT
cana-758	82	1	+	+	CCONJ
cana-758	82	2	𝛽𝓏(ϥ(𝓏))′	𝛽𝓏(ϥ(𝓏))′	PROPN
cana-758	82	3	≺	≺	NOUN
cana-758	82	4	𝒫(𝓏	𝒫(𝓏	NUM
cana-758	82	5	)	)	PUNCT
cana-758	82	6	+	+	CCONJ
cana-758	82	7	𝛽𝓏(𝒫(𝓏))′	𝛽𝓏(𝒫(𝓏))′	PROPN
cana-758	82	8	,	,	PUNCT
cana-758	82	9	which	which	PRON
cana-758	82	10	implies	imply	VERB
cana-758	82	11	that	that	SCONJ
cana-758	82	12	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	82	13	)	)	PUNCT
cana-758	82	14	≺	≺	NOUN
cana-758	82	15	𝒫(𝓏	𝒫(𝓏	SYM
cana-758	82	16	)	)	PUNCT
cana-758	82	17	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-758	82	18	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	82	19	)	)	PUNCT
cana-758	82	20	will	will	AUX
cana-758	82	21	be	be	AUX
cana-758	82	22	best	well	ADV
cana-758	82	23	subordinant	subordinant	NOUN
cana-758	82	24	.	.	PUNCT
cana-758	83	1	3	3	X
cana-758	83	2	.	.	X
cana-758	83	3	subordination	subordination	NOUN
cana-758	83	4	results	result	NOUN
cana-758	83	5	:	:	PUNCT
cana-758	83	6	theorem	theorem	NOUN
cana-758	83	7	(	(	PUNCT
cana-758	83	8	3.1	3.1	NUM
cana-758	83	9	):	):	PUNCT
cana-758	83	10	let	let	VERB
cana-758	83	11	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	83	12	)	)	PUNCT
cana-758	83	13	is	be	AUX
cana-758	83	14	belongs	belong	VERB
cana-758	83	15	in	in	ADP
cana-758	83	16	ʋ	ʋ	PROPN
cana-758	83	17	with	with	ADP
cana-758	83	18	ϥ(0	ϥ(0	PROPN
cana-758	83	19	)	)	PUNCT
cana-758	83	20	=	=	SYM
cana-758	84	1	1,where	1,where	X
cana-758	84	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	84	3	)	)	PUNCT
cana-758	84	4	is	be	AUX
cana-758	84	5	convex	convex	ADJ
cana-758	84	6	and	and	CCONJ
cana-758	84	7	univalent	univalent	ADJ
cana-758	84	8	.	.	PUNCT
cana-758	85	1	let	let	VERB
cana-758	85	2	𝜉	𝜉	PRON
cana-758	85	3	∈	∈	PROPN
cana-758	85	4	₵	₵	PROPN
cana-758	85	5	∗	∗	NOUN
cana-758	85	6	,	,	PUNCT
cana-758	85	7	𝜇	𝜇	ADP
cana-758	85	8	,	,	PUNCT
cana-758	85	9	𝕧	𝕧	PROPN
cana-758	85	10	>	>	X
cana-758	85	11	0	0	NUM
cana-758	85	12	,	,	PUNCT
cana-758	85	13	𝕦	𝕦	ADJ
cana-758	85	14	real	real	ADJ
cana-758	85	15	number	number	NOUN
cana-758	85	16	such	such	ADJ
cana-758	85	17	that	that	SCONJ
cana-758	85	18	𝕦	𝕦	PROPN
cana-758	85	19	+	+	X
cana-758	85	20	𝕧	𝕧	X
cana-758	85	21	>	>	X
cana-758	85	22	0	0	PUNCT
cana-758	85	23	andsuppose	andsuppose	ADJ
cana-758	85	24	that	that	SCONJ
cana-758	85	25	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	85	26	)	)	PUNCT
cana-758	85	27	satisfies	satisfie	NOUN
cana-758	85	28	:	:	PUNCT
cana-758	85	29	𝑅𝑒	𝑅𝑒	VERB
cana-758	85	30	{	{	PUNCT
cana-758	85	31	1	1	NUM
cana-758	85	32	+	+	NUM
cana-758	85	33	𝑧ϥ	𝑧ϥ	PRON
cana-758	85	34	"	"	PUNCT
cana-758	85	35	(	(	PUNCT
cana-758	85	36	𝓏	𝓏	NOUN
cana-758	85	37	)	)	PUNCT
cana-758	85	38	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	85	39	)	)	PUNCT
cana-758	85	40	}	}	PUNCT
cana-758	85	41	>	>	X
cana-758	85	42	max	max	PROPN
cana-758	85	43	{	{	PUNCT
cana-758	85	44	0	0	NUM
cana-758	85	45	,	,	PUNCT
cana-758	85	46	𝑅𝑒	𝑅𝑒	PROPN
cana-758	85	47	(	(	PUNCT
cana-758	85	48	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADJ
cana-758	85	49	)	)	PUNCT
cana-758	85	50	𝜉𝕧	𝜉𝕧	NOUN
cana-758	85	51	)	)	PUNCT
cana-758	85	52	}	}	PUNCT
cana-758	85	53	.	.	PUNCT
cana-758	86	1	(	(	PUNCT
cana-758	86	2	3.1	3.1	NUM
cana-758	86	3	)	)	PUNCT
cana-758	86	4	let	let	VERB
cana-758	86	5	𝑓	𝑓	DET
cana-758	86	6	∈	∈	NOUN
cana-758	86	7	𝒢	𝒢	PROPN
cana-758	86	8	holds	hold	VERB
cana-758	86	9	the	the	DET
cana-758	86	10	subordination	subordination	NOUN
cana-758	86	11	𝔇(𝕞	𝔇(𝕞	NUM
cana-758	86	12	,	,	PUNCT
cana-758	86	13	𝜉	𝜉	PROPN
cana-758	86	14	,	,	PUNCT
cana-758	86	15	𝜇	𝜇	ADP
cana-758	86	16	,	,	PUNCT
cana-758	86	17	𝕦	𝕦	ADJ
cana-758	86	18	,	,	PUNCT
cana-758	86	19	𝕧	𝕧	NOUN
cana-758	86	20	)	)	PUNCT
cana-758	86	21	≺	≺	NOUN
cana-758	86	22	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	86	23	)	)	PUNCT
cana-758	87	1	+	+	CCONJ
cana-758	87	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	87	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADJ
cana-758	87	4	)	)	PUNCT
cana-758	87	5	𝓏ϥ′(𝓏	𝓏ϥ′(𝓏	NOUN
cana-758	87	6	)	)	PUNCT
cana-758	87	7	,	,	PUNCT
cana-758	87	8	(	(	PUNCT
cana-758	87	9	3.2	3.2	NUM
cana-758	87	10	)	)	PUNCT
cana-758	87	11	where	where	SCONJ
cana-758	87	12	𝔇(𝕞	𝔇(𝕞	NOUN
cana-758	87	13	,	,	PUNCT
cana-758	87	14	𝜉	𝜉	X
cana-758	87	15	,	,	PUNCT
cana-758	87	16	𝜇	𝜇	AUX
cana-758	87	17	,	,	PUNCT
cana-758	87	18	𝕦	𝕦	ADJ
cana-758	87	19	,	,	PUNCT
cana-758	87	20	𝕧	𝕧	NOUN
cana-758	87	21	)	)	PUNCT
cana-758	87	22	=	=	PUNCT
cana-758	87	23	(	(	PUNCT
cana-758	87	24	1	1	NUM
cana-758	87	25	−	−	NOUN
cana-758	87	26	𝜉	𝜉	NOUN
cana-758	87	27	)	)	PUNCT
cana-758	87	28	(	(	PUNCT
cana-758	87	29	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	87	30	𝕞	𝕞	X
cana-758	87	31	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	87	32	)	)	PUNCT
cana-758	87	33	𝓏	𝓏	PROPN
cana-758	87	34	)	)	PUNCT
cana-758	87	35	𝜇	𝜇	ADP
cana-758	87	36	+	+	X
cana-758	87	37	𝜉	𝜉	X
cana-758	87	38	(	(	PUNCT
cana-758	87	39	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	87	40	𝕞	𝕞	X
cana-758	87	41	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	87	42	)	)	PUNCT
cana-758	87	43	𝓏	𝓏	PROPN
cana-758	87	44	)	)	PUNCT
cana-758	87	45	𝜇	𝜇	ADP
cana-758	87	46	+	+	X
cana-758	87	47	(	(	PUNCT
cana-758	87	48	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	87	49	𝕞+1	𝕞+1	NUM
cana-758	87	50	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	87	51	)	)	PUNCT
cana-758	87	52	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	87	53	𝕞	𝕞	DET
cana-758	87	54	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	87	55	)	)	PUNCT
cana-758	87	56	)	)	PUNCT
cana-758	87	57	,	,	PUNCT
cana-758	87	58	(	(	PUNCT
cana-758	87	59	3.3	3.3	NUM
cana-758	87	60	)	)	PUNCT
cana-758	87	61	then	then	ADV
cana-758	87	62	(	(	PUNCT
cana-758	87	63	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	87	64	𝕞	𝕞	X
cana-758	87	65	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	87	66	)	)	PUNCT
cana-758	87	67	𝓏	𝓏	PROPN
cana-758	87	68	)	)	PUNCT
cana-758	87	69	𝜇	𝜇	ADP
cana-758	87	70	≺	≺	NOUN
cana-758	87	71	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	87	72	)	)	PUNCT
cana-758	87	73	,	,	PUNCT
cana-758	87	74	(	(	PUNCT
cana-758	87	75	3.4	3.4	NUM
cana-758	87	76	)	)	PUNCT
cana-758	87	77	and	and	CCONJ
cana-758	87	78	the	the	DET
cana-758	87	79	equation	equation	NOUN
cana-758	87	80	(	(	PUNCT
cana-758	87	81	3.2	3.2	NUM
cana-758	87	82	)	)	PUNCT
cana-758	87	83	have	have	VERB
cana-758	87	84	the	the	DET
cana-758	87	85	best	good	ADJ
cana-758	87	86	dominant	dominant	NOUN
cana-758	87	87	say	say	VERB
cana-758	87	88	ϥ	ϥ	NOUN
cana-758	87	89	.	.	PUNCT
cana-758	88	1	proof	proof	NOUN
cana-758	88	2	:	:	PUNCT
cana-758	88	3	put	put	VERB
cana-758	88	4	𝒮(𝓏	𝒮(𝓏	NOUN
cana-758	88	5	)	)	PUNCT
cana-758	88	6	=	=	PRON
cana-758	88	7	(	(	PUNCT
cana-758	89	1	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	89	2	𝕞	𝕞	X
cana-758	89	3	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	89	4	)	)	PUNCT
cana-758	89	5	𝓏	𝓏	PROPN
cana-758	89	6	)	)	PUNCT
cana-758	89	7	𝜇	𝜇	ADP
cana-758	89	8	,	,	PUNCT
cana-758	89	9	𝓏	𝓏	PROPN
cana-758	89	10	∈	∈	PROPN
cana-758	89	11	ʋ	ʋ	NOUN
cana-758	89	12	.	.	PUNCT
cana-758	90	1	(	(	PUNCT
cana-758	90	2	3.5	3.5	NUM
cana-758	90	3	)	)	PUNCT
cana-758	90	4	hence	hence	ADV
cana-758	90	5	differentiate	differentiate	VERB
cana-758	90	6	(	(	PUNCT
cana-758	90	7	3.5	3.5	NUM
cana-758	90	8	)	)	PUNCT
cana-758	90	9	logarithmically	logarithmically	ADV
cana-758	90	10	according	accord	VERB
cana-758	90	11	to	to	ADP
cana-758	90	12	𝓏	𝓏	NUM
cana-758	90	13	,	,	PUNCT
cana-758	90	14	and	and	CCONJ
cana-758	90	15	taking	take	VERB
cana-758	90	16	identity	identity	NOUN
cana-758	90	17	(	(	PUNCT
cana-758	90	18	1.8	1.8	NUM
cana-758	90	19	)	)	PUNCT
cana-758	90	20	in	in	ADP
cana-758	90	21	resultant	resultant	NOUN
cana-758	90	22	equation	equation	NOUN
cana-758	90	23	,	,	PUNCT
cana-758	90	24	to	to	PART
cana-758	90	25	get	get	VERB
cana-758	90	26	𝓏	𝓏	PROPN
cana-758	90	27	𝒮′(𝓏	𝒮′(𝓏	NOUN
cana-758	90	28	)	)	PUNCT
cana-758	90	29	𝒮(𝓏	𝒮(𝓏	NUM
cana-758	90	30	)	)	PUNCT
cana-758	90	31	=	=	SYM
cana-758	90	32	𝜇	𝜇	X
cana-758	90	33	(	(	PUNCT
cana-758	90	34	(	(	PUNCT
cana-758	90	35	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	90	36	)	)	PUNCT
cana-758	90	37	𝕧	𝕧	NOUN
cana-758	90	38	)	)	PUNCT
cana-758	90	39	(	(	PUNCT
cana-758	90	40	𝓏𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝓏𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	VERB
cana-758	90	41	𝕞+1	𝕞+1	NUM
cana-758	90	42	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	90	43	)	)	PUNCT
cana-758	90	44	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	90	45	𝕞	𝕞	DET
cana-758	90	46	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	90	47	)	)	PUNCT
cana-758	91	1	−	−	PROPN
cana-758	91	2	1),and	1),and	NUM
cana-758	91	3	which	which	PRON
cana-758	91	4	can	can	AUX
cana-758	91	5	be	be	AUX
cana-758	91	6	written	write	VERB
cana-758	91	7	as	as	ADP
cana-758	91	8	𝕧	𝕧	PROPN
cana-758	91	9	𝜇(𝕦	𝜇(𝕦	NOUN
cana-758	91	10	+	+	CCONJ
cana-758	91	11	𝕧	𝕧	NOUN
cana-758	91	12	)	)	PUNCT
cana-758	91	13	𝓏𝒮′(𝓏	𝓏𝒮′(𝓏	PROPN
cana-758	91	14	)	)	PUNCT
cana-758	92	1	=	=	SYM
cana-758	92	2	(	(	PUNCT
cana-758	92	3	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	92	4	𝕞	𝕞	X
cana-758	92	5	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	92	6	)	)	PUNCT
cana-758	92	7	𝓏	𝓏	PROPN
cana-758	92	8	)	)	PUNCT
cana-758	92	9	𝜇	𝜇	X
cana-758	92	10	(	(	PUNCT
cana-758	92	11	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	92	12	𝕞+1	𝕞+1	NUM
cana-758	92	13	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	92	14	)	)	PUNCT
cana-758	92	15	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	92	16	𝕞	𝕞	DET
cana-758	92	17	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	92	18	)	)	PUNCT
cana-758	92	19	−	−	PROPN
cana-758	93	1	1	1	X
cana-758	93	2	)	)	PUNCT
cana-758	93	3	thus	thus	ADV
cana-758	93	4	the	the	DET
cana-758	93	5	equation	equation	NOUN
cana-758	93	6	of	of	ADP
cana-758	93	7	subordination	subordination	NOUN
cana-758	93	8	which	which	PRON
cana-758	93	9	is	be	AUX
cana-758	93	10	represented	represent	VERB
cana-758	93	11	by	by	ADP
cana-758	93	12	(	(	PUNCT
cana-758	93	13	3.2	3.2	NUM
cana-758	93	14	)	)	PUNCT
cana-758	93	15	be	be	AUX
cana-758	93	16	equivalent	equivalent	ADJ
cana-758	93	17	by	by	ADP
cana-758	93	18	𝒮(𝓏	𝒮(𝓏	NUM
cana-758	93	19	)	)	PUNCT
cana-758	93	20	+	+	CCONJ
cana-758	93	21	𝜉𝕧	𝜉𝕧	ADP
cana-758	93	22	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADV
cana-758	93	23	)	)	PUNCT
cana-758	93	24	𝓏𝒮′(𝓏	𝓏𝒮′(𝓏	PROPN
cana-758	93	25	)	)	PUNCT
cana-758	93	26	≺	≺	NOUN
cana-758	93	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	93	28	)	)	PUNCT
cana-758	94	1	+	+	CCONJ
cana-758	94	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	94	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADJ
cana-758	94	4	)	)	PUNCT
cana-758	94	5	𝓏ϥ′(𝓏	𝓏ϥ′(𝓏	NOUN
cana-758	94	6	)	)	PUNCT
cana-758	94	7	.	.	PUNCT
cana-758	95	1	by	by	ADP
cana-758	95	2	apply	apply	VERB
cana-758	95	3	lemma	lemma	PROPN
cana-758	95	4	(	(	PUNCT
cana-758	95	5	2.2)2.1	2.2)2.1	NUM
cana-758	95	6	when	when	SCONJ
cana-758	95	7	𝜎	𝜎	PROPN
cana-758	95	8	=	=	X
cana-758	95	9	𝜉𝕧	𝜉𝕧	ADP
cana-758	95	10	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	NUM
cana-758	95	11	)	)	PUNCT
cana-758	95	12	,	,	PUNCT
cana-758	95	13	the	the	DET
cana-758	95	14	proof	proof	NOUN
cana-758	95	15	of	of	ADP
cana-758	95	16	theorem(3.1	theorem(3.1	ADJ
cana-758	95	17	)	)	PUNCT
cana-758	95	18	is	be	AUX
cana-758	95	19	complete	complete	ADJ
cana-758	95	20	.	.	PUNCT
cana-758	96	1	communications	communication	NOUN
cana-758	96	2	on	on	ADP
cana-758	96	3	applied	apply	VERB
cana-758	96	4	nonlinear	nonlinear	ADJ
cana-758	96	5	analysis	analysis	NOUN
cana-758	96	6	issn	issn	NOUN
cana-758	96	7	:	:	PUNCT
cana-758	96	8	1074	1074	NUM
cana-758	96	9	-	-	PUNCT
cana-758	96	10	133x	133x	NUM
cana-758	96	11	vol	vol	NOUN
cana-758	96	12	31	31	NUM
cana-758	96	13	no	no	NOUN
cana-758	96	14	.	.	PUNCT
cana-758	97	1	3s	3s	NUM
cana-758	97	2	(	(	PUNCT
cana-758	97	3	2024	2024	NUM
cana-758	97	4	)	)	PUNCT
cana-758	97	5	190	190	NUM
cana-758	97	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	97	7	now	now	ADV
cana-758	97	8	,	,	PUNCT
cana-758	97	9	in	in	ADP
cana-758	97	10	theorem	theorem	NOUN
cana-758	97	11	above	above	ADV
cana-758	97	12	,	,	PUNCT
cana-758	97	13	we	we	PRON
cana-758	97	14	put	put	VERB
cana-758	97	15	,	,	PUNCT
cana-758	97	16	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	97	17	)	)	PUNCT
cana-758	97	18	=	=	PUNCT
cana-758	98	1	1+𝐴𝓏	1+𝐴𝓏	NUM
cana-758	98	2	1+𝐵𝓏	1+𝐵𝓏	NUM
cana-758	98	3	,	,	PUNCT
cana-758	98	4	so	so	ADV
cana-758	98	5	we	we	PRON
cana-758	98	6	get	get	VERB
cana-758	98	7	the	the	DET
cana-758	98	8	following	follow	VERB
cana-758	98	9	corollary	corollary	NOUN
cana-758	98	10	.	.	PUNCT
cana-758	99	1	corollary	corollary	ADJ
cana-758	99	2	(	(	PUNCT
cana-758	99	3	3.1	3.1	NUM
cana-758	99	4	):	):	PUNCT
cana-758	99	5	assume𝜉	assume𝜉	VERB
cana-758	99	6	,	,	PUNCT
cana-758	99	7	𝐴	𝐴	PROPN
cana-758	99	8	,	,	PUNCT
cana-758	99	9	𝐵	𝐵	PROPN
cana-758	99	10	∈	∈	PROPN
cana-758	99	11	₵	₵	NOUN
cana-758	99	12	,	,	PUNCT
cana-758	99	13	𝐴	𝐴	PROPN
cana-758	99	14	≠	≠	PROPN
cana-758	99	15	𝐵	𝐵	PROPN
cana-758	99	16	,	,	PUNCT
cana-758	99	17	|𝐵|	|𝐵|	VERB
cana-758	99	18	<	<	X
cana-758	99	19	1	1	NUM
cana-758	99	20	,	,	PUNCT
cana-758	99	21	𝜇	𝜇	X
cana-758	99	22	>	>	X
cana-758	99	23	0	0	PROPN
cana-758	99	24	,	,	PUNCT
cana-758	99	25	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	99	26	)	)	PUNCT
cana-758	99	27	>	>	SYM
cana-758	99	28	0	0	NUM
cana-758	100	1	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-758	100	2	𝕦	𝕦	ADJ
cana-758	100	3	real	real	ADJ
cana-758	100	4	number	number	NOUN
cana-758	100	5	such	such	ADJ
cana-758	100	6	that	that	SCONJ
cana-758	100	7	𝕦	𝕦	PROPN
cana-758	100	8	+	+	X
cana-758	100	9	𝕧	𝕧	X
cana-758	100	10	>	>	X
cana-758	100	11	0	0	NUM
cana-758	100	12	,	,	PUNCT
cana-758	100	13	if	if	SCONJ
cana-758	100	14	𝑓	𝑓	DET
cana-758	100	15	∈	∈	NOUN
cana-758	100	16	𝒢	𝒢	NOUN
cana-758	100	17	satisfy	satisfy	VERB
cana-758	100	18	the	the	DET
cana-758	100	19	recent	recent	ADJ
cana-758	100	20	subordination	subordination	NOUN
cana-758	100	21	case	case	NOUN
cana-758	100	22	:	:	PUNCT
cana-758	100	23	𝔇(𝕞	𝔇(𝕞	NUM
cana-758	100	24	,	,	PUNCT
cana-758	100	25	𝜉	𝜉	PROPN
cana-758	100	26	,	,	PUNCT
cana-758	100	27	𝜇	𝜇	ADP
cana-758	100	28	,	,	PUNCT
cana-758	100	29	𝕦	𝕦	ADJ
cana-758	100	30	,	,	PUNCT
cana-758	100	31	𝕧	𝕧	NOUN
cana-758	100	32	)	)	PUNCT
cana-758	100	33	≺	≺	NOUN
cana-758	100	34	(	(	PUNCT
cana-758	100	35	1+𝐴𝓏	1+𝐴𝓏	NUM
cana-758	100	36	1+𝐵𝓏	1+𝐵𝓏	NOUN
cana-758	100	37	)	)	PUNCT
cana-758	101	1	+	+	CCONJ
cana-758	101	2	𝜉𝕧	𝜉𝕧	X
cana-758	101	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADV
cana-758	101	4	)	)	PUNCT
cana-758	101	5	(	(	PUNCT
cana-758	101	6	(	(	PUNCT
cana-758	101	7	𝐴+𝐵)𝓏	𝐴+𝐵)𝓏	PROPN
cana-758	101	8	(	(	PUNCT
cana-758	101	9	1+𝐵𝓏)2	1+𝐵𝓏)2	NUM
cana-758	101	10	)	)	PUNCT
cana-758	101	11	,	,	PUNCT
cana-758	101	12	such	such	ADJ
cana-758	101	13	that	that	SCONJ
cana-758	101	14	𝔇(𝕞	𝔇(𝕞	NUM
cana-758	101	15	,	,	PUNCT
cana-758	101	16	𝜉	𝜉	PROPN
cana-758	101	17	,	,	PUNCT
cana-758	101	18	𝜇	𝜇	ADP
cana-758	101	19	,	,	PUNCT
cana-758	101	20	𝕦	𝕦	ADJ
cana-758	101	21	,	,	PUNCT
cana-758	101	22	𝕧	𝕧	NOUN
cana-758	101	23	)	)	PUNCT
cana-758	101	24	known	know	VERB
cana-758	101	25	by	by	ADP
cana-758	101	26	(	(	PUNCT
cana-758	101	27	3.3	3.3	NUM
cana-758	101	28	)	)	PUNCT
cana-758	101	29	,	,	PUNCT
cana-758	101	30	then	then	ADV
cana-758	101	31	(	(	PUNCT
cana-758	101	32	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	101	33	𝕞	𝕞	X
cana-758	101	34	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	101	35	)	)	PUNCT
cana-758	101	36	𝓏	𝓏	PROPN
cana-758	101	37	)	)	PUNCT
cana-758	101	38	𝜇	𝜇	ADP
cana-758	101	39	≺	≺	NOUN
cana-758	101	40	1	1	NUM
cana-758	101	41	+	+	CCONJ
cana-758	101	42	𝐴𝓏	𝐴𝓏	PROPN
cana-758	101	43	1	1	NUM
cana-758	101	44	+	+	CCONJ
cana-758	101	45	𝐵𝓏	𝐵𝓏	PROPN
cana-758	101	46	,	,	PUNCT
cana-758	101	47	and	and	CCONJ
cana-758	101	48	will	will	AUX
cana-758	101	49	be	be	AUX
cana-758	101	50	best	well	ADV
cana-758	101	51	dominant	dominant	ADJ
cana-758	101	52	say	say	VERB
cana-758	102	1	1+𝐴𝓏	1+𝐴𝓏	NUM
cana-758	102	2	1+𝐵𝓏	1+𝐵𝓏	NUM
cana-758	102	3	.	.	PUNCT
cana-758	103	1	now	now	ADV
cana-758	103	2	,	,	PUNCT
cana-758	103	3	put	put	VERB
cana-758	103	4	𝕞	𝕞	PRON
cana-758	103	5	=	=	NOUN
cana-758	103	6	0	0	NUM
cana-758	103	7	in	in	ADP
cana-758	103	8	the	the	DET
cana-758	103	9	theorem	theorem	NOUN
cana-758	103	10	above	above	ADV
cana-758	103	11	,	,	PUNCT
cana-758	103	12	to	to	PART
cana-758	103	13	get	get	VERB
cana-758	103	14	a	a	DET
cana-758	103	15	new	new	ADJ
cana-758	103	16	result	result	NOUN
cana-758	103	17	.	.	PUNCT
cana-758	104	1	corollary	corollary	ADJ
cana-758	104	2	(	(	PUNCT
cana-758	104	3	3.2	3.2	NUM
cana-758	104	4	):	):	PUNCT
cana-758	104	5	let	let	VERB
cana-758	104	6	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	104	7	)	)	PUNCT
cana-758	104	8	is	be	AUX
cana-758	104	9	belongs	belong	VERB
cana-758	104	10	in	in	ADP
cana-758	104	11	ʋ	ʋ	PROPN
cana-758	104	12	with	with	ADP
cana-758	104	13	ϥ(0	ϥ(0	PROPN
cana-758	104	14	)	)	PUNCT
cana-758	104	15	=	=	SYM
cana-758	105	1	1,where	1,where	X
cana-758	105	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	105	3	)	)	PUNCT
cana-758	105	4	is	be	AUX
cana-758	105	5	univalent	univalent	ADJ
cana-758	105	6	.	.	PUNCT
cana-758	106	1	let	let	VERB
cana-758	106	2	𝜉	𝜉	PRON
cana-758	106	3	∈	∈	PROPN
cana-758	106	4	₵	₵	PROPN
cana-758	106	5	∗	∗	NOUN
cana-758	106	6	,	,	PUNCT
cana-758	106	7	𝜇	𝜇	ADP
cana-758	106	8	,	,	PUNCT
cana-758	106	9	𝕧	𝕧	PROPN
cana-758	106	10	>	>	X
cana-758	106	11	0	0	NUM
cana-758	106	12	,	,	PUNCT
cana-758	106	13	𝕦	𝕦	ADJ
cana-758	106	14	real	real	ADJ
cana-758	106	15	number	number	NOUN
cana-758	106	16	such	such	ADJ
cana-758	106	17	that	that	SCONJ
cana-758	106	18	𝕦	𝕦	PROPN
cana-758	106	19	+	+	X
cana-758	106	20	𝕧	𝕧	X
cana-758	106	21	>	>	X
cana-758	106	22	0	0	PUNCT
cana-758	106	23	andsuppose	andsuppose	ADJ
cana-758	106	24	that	that	SCONJ
cana-758	106	25	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	106	26	)	)	PUNCT
cana-758	106	27	satisfies	satisfie	NOUN
cana-758	106	28	:	:	PUNCT
cana-758	106	29	let	let	VERB
cana-758	106	30	𝑓	𝑓	DET
cana-758	106	31	∈	∈	PROPN
cana-758	106	32	𝒢	𝒢	PROPN
cana-758	106	33	holds	hold	VERB
cana-758	106	34	the	the	DET
cana-758	106	35	subordination	subordination	NOUN
cana-758	106	36	𝔇1(0	𝔇1(0	NOUN
cana-758	106	37	,	,	PUNCT
cana-758	106	38	𝜉	𝜉	PROPN
cana-758	106	39	,	,	PUNCT
cana-758	106	40	𝜇	𝜇	ADP
cana-758	106	41	,	,	PUNCT
cana-758	106	42	𝕦	𝕦	ADJ
cana-758	106	43	,	,	PUNCT
cana-758	106	44	𝕧	𝕧	NOUN
cana-758	106	45	)	)	PUNCT
cana-758	106	46	≺	≺	NOUN
cana-758	106	47	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	106	48	)	)	PUNCT
cana-758	107	1	+	+	CCONJ
cana-758	107	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	107	3	𝜇(𝕦	𝜇(𝕦	NOUN
cana-758	107	4	+	+	CCONJ
cana-758	107	5	𝕧	𝕧	NOUN
cana-758	107	6	)	)	PUNCT
cana-758	107	7	𝓏ϥ′(𝓏	𝓏ϥ′(𝓏	PROPN
cana-758	107	8	)	)	PUNCT
cana-758	107	9	,	,	PUNCT
cana-758	107	10	where	where	SCONJ
cana-758	107	11	𝔇1(0	𝔇1(0	NOUN
cana-758	107	12	,	,	PUNCT
cana-758	107	13	𝜉	𝜉	PROPN
cana-758	107	14	,	,	PUNCT
cana-758	107	15	𝜇	𝜇	ADP
cana-758	107	16	,	,	PUNCT
cana-758	107	17	𝕦	𝕦	ADJ
cana-758	107	18	,	,	PUNCT
cana-758	107	19	𝕧	𝕧	NOUN
cana-758	107	20	)	)	PUNCT
cana-758	107	21	=	=	PUNCT
cana-758	107	22	(	(	PUNCT
cana-758	107	23	1	1	NUM
cana-758	107	24	−	−	NOUN
cana-758	107	25	𝜉	𝜉	NOUN
cana-758	107	26	)	)	PUNCT
cana-758	107	27	(	(	PUNCT
cana-758	107	28	𝒯𝕣,𝕤,𝕥	𝒯𝕣,𝕤,𝕥	PROPN
cana-758	107	29	,	,	PUNCT
cana-758	107	30	0	0	PUNCT
cana-758	107	31	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	107	32	)	)	PUNCT
cana-758	107	33	𝓏	𝓏	PROPN
cana-758	107	34	)	)	PUNCT
cana-758	107	35	𝜇	𝜇	ADP
cana-758	107	36	+	+	X
cana-758	107	37	𝜉	𝜉	X
cana-758	107	38	(	(	PUNCT
cana-758	107	39	𝒯𝕣,𝕤,𝕥	𝒯𝕣,𝕤,𝕥	PROPN
cana-758	107	40	,	,	PUNCT
cana-758	107	41	,	,	PUNCT
cana-758	107	42	0	0	NUM
cana-758	108	1	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	108	2	)	)	PUNCT
cana-758	108	3	𝓏	𝓏	PROPN
cana-758	108	4	)	)	PUNCT
cana-758	108	5	𝜇	𝜇	ADP
cana-758	108	6	+	+	X
cana-758	108	7	(	(	PUNCT
cana-758	108	8	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	108	9	1	1	NUM
cana-758	108	10	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	108	11	)	)	PUNCT
cana-758	108	12	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	108	13	)	)	PUNCT
cana-758	108	14	)	)	PUNCT
cana-758	108	15	,	,	PUNCT
cana-758	108	16	then	then	ADV
cana-758	108	17	(	(	PUNCT
cana-758	108	18	𝒯𝕣,𝕤,𝕥	𝒯𝕣,𝕤,𝕥	PROPN
cana-758	108	19	,	,	PUNCT
cana-758	108	20	0	0	PUNCT
cana-758	108	21	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	108	22	)	)	PUNCT
cana-758	108	23	𝓏	𝓏	PROPN
cana-758	108	24	)	)	PUNCT
cana-758	108	25	𝜇	𝜇	ADP
cana-758	108	26	≺	≺	NOUN
cana-758	108	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	108	28	)	)	PUNCT
cana-758	108	29	,	,	PUNCT
cana-758	108	30	and	and	CCONJ
cana-758	108	31	the	the	DET
cana-758	108	32	equation	equation	NOUN
cana-758	108	33	(	(	PUNCT
cana-758	108	34	3.2	3.2	NUM
cana-758	108	35	)	)	PUNCT
cana-758	108	36	have	have	VERB
cana-758	108	37	the	the	DET
cana-758	108	38	best	good	ADJ
cana-758	108	39	dominant	dominant	NOUN
cana-758	108	40	say	say	VERB
cana-758	108	41	ϥ	ϥ	PROPN
cana-758	108	42	.	.	PUNCT
cana-758	109	1	now	now	ADV
cana-758	109	2	,	,	PUNCT
cana-758	109	3	put	put	VERB
cana-758	109	4	𝕦	𝕦	ADJ
cana-758	109	5	=	=	PUNCT
cana-758	109	6	𝕧	𝕧	NOUN
cana-758	109	7	=	=	SYM
cana-758	109	8	1	1	NUM
cana-758	109	9	in	in	ADP
cana-758	109	10	the	the	DET
cana-758	109	11	theorem	theorem	NOUN
cana-758	109	12	above	above	ADV
cana-758	109	13	,	,	PUNCT
cana-758	109	14	to	to	PART
cana-758	109	15	get	get	VERB
cana-758	109	16	a	a	DET
cana-758	109	17	new	new	ADJ
cana-758	109	18	result	result	NOUN
cana-758	109	19	.	.	PUNCT
cana-758	110	1	corollary	corollary	ADJ
cana-758	110	2	(	(	PUNCT
cana-758	110	3	3.3	3.3	NUM
cana-758	110	4	):	):	PUNCT
cana-758	110	5	let	let	VERB
cana-758	110	6	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	110	7	)	)	PUNCT
cana-758	110	8	is	be	AUX
cana-758	110	9	belongs	belong	VERB
cana-758	110	10	in	in	ADP
cana-758	110	11	ʋ	ʋ	PROPN
cana-758	110	12	with	with	ADP
cana-758	110	13	ϥ(0	ϥ(0	PROPN
cana-758	110	14	)	)	PUNCT
cana-758	110	15	=	=	SYM
cana-758	111	1	1,where	1,where	X
cana-758	111	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	111	3	)	)	PUNCT
cana-758	111	4	is	be	AUX
cana-758	111	5	univalent	univalent	ADJ
cana-758	111	6	.	.	PUNCT
cana-758	112	1	let	let	VERB
cana-758	112	2	𝜉	𝜉	PRON
cana-758	112	3	∈	∈	PROPN
cana-758	112	4	₵	₵	PROPN
cana-758	112	5	∗	∗	NOUN
cana-758	112	6	,	,	PUNCT
cana-758	112	7	𝜇	𝜇	X
cana-758	112	8	>	>	X
cana-758	112	9	0	0	NUM
cana-758	112	10	,	,	PUNCT
cana-758	112	11	and	and	CCONJ
cana-758	112	12	suppose	suppose	VERB
cana-758	112	13	that	that	SCONJ
cana-758	112	14	(	(	PUNCT
cana-758	112	15	3.1	3.1	NUM
cana-758	112	16	)	)	PUNCT
cana-758	112	17	holds	hold	VERB
cana-758	112	18	.	.	PUNCT
cana-758	113	1	let	let	VERB
cana-758	113	2	𝑓	𝑓	DET
cana-758	113	3	∈	∈	NOUN
cana-758	113	4	𝒢	𝒢	PROPN
cana-758	113	5	holds	hold	VERB
cana-758	113	6	the	the	DET
cana-758	113	7	subordination	subordination	NOUN
cana-758	113	8	𝔇2(𝕞	𝔇2(𝕞	NOUN
cana-758	113	9	,	,	PUNCT
cana-758	113	10	𝜉	𝜉	X
cana-758	113	11	,	,	PUNCT
cana-758	113	12	𝜇	𝜇	ADP
cana-758	113	13	,	,	PUNCT
cana-758	113	14	1,1	1,1	NUM
cana-758	113	15	)	)	PUNCT
cana-758	113	16	≺	≺	NOUN
cana-758	113	17	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	113	18	)	)	PUNCT
cana-758	114	1	+	+	CCONJ
cana-758	114	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	114	3	𝜇(𝕦	𝜇(𝕦	NOUN
cana-758	114	4	+	+	CCONJ
cana-758	114	5	𝕧	𝕧	NOUN
cana-758	114	6	)	)	PUNCT
cana-758	114	7	𝓏ϥ′(𝓏	𝓏ϥ′(𝓏	PROPN
cana-758	114	8	)	)	PUNCT
cana-758	114	9	,	,	PUNCT
cana-758	114	10	where	where	SCONJ
cana-758	114	11	𝔇2(𝕞	𝔇2(𝕞	PROPN
cana-758	114	12	,	,	PUNCT
cana-758	114	13	𝜉	𝜉	X
cana-758	114	14	,	,	PUNCT
cana-758	114	15	𝜇	𝜇	ADP
cana-758	114	16	,	,	PUNCT
cana-758	114	17	1,1	1,1	NUM
cana-758	114	18	)	)	PUNCT
cana-758	114	19	=	=	PUNCT
cana-758	114	20	(	(	PUNCT
cana-758	114	21	1	1	NUM
cana-758	114	22	−	−	NOUN
cana-758	114	23	𝜉	𝜉	NOUN
cana-758	114	24	)	)	PUNCT
cana-758	114	25	(	(	PUNCT
cana-758	114	26	𝒯𝕣,𝕤,𝕥,,1,1	𝒯𝕣,𝕤,𝕥,,1,1	NOUN
cana-758	114	27	𝕞	𝕞	PRON
cana-758	114	28	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	114	29	)	)	PUNCT
cana-758	114	30	𝓏	𝓏	PROPN
cana-758	114	31	)	)	PUNCT
cana-758	114	32	𝜇	𝜇	ADP
cana-758	114	33	+	+	X
cana-758	114	34	𝜉	𝜉	X
cana-758	114	35	(	(	PUNCT
cana-758	114	36	𝒯𝕣,𝕤,𝕥,,1,1	𝒯𝕣,𝕤,𝕥,,1,1	NOUN
cana-758	114	37	𝕞	𝕞	PRON
cana-758	114	38	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	114	39	)	)	PUNCT
cana-758	114	40	𝓏	𝓏	PROPN
cana-758	114	41	)	)	PUNCT
cana-758	114	42	𝜇	𝜇	ADP
cana-758	114	43	+	+	X
cana-758	114	44	(	(	PUNCT
cana-758	114	45	𝒯𝕣,𝕤,𝕥,,1,1	𝒯𝕣,𝕤,𝕥,,1,1	NOUN
cana-758	114	46	𝕞+1	𝕞+1	NUM
cana-758	114	47	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	114	48	)	)	PUNCT
cana-758	114	49	𝒯𝕣,𝕤,𝕥,,1,1	𝒯𝕣,𝕤,𝕥,,1,1	NOUN
cana-758	114	50	𝕞	𝕞	DET
cana-758	114	51	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	114	52	)	)	PUNCT
cana-758	114	53	)	)	PUNCT
cana-758	115	1	,	,	PUNCT
cana-758	115	2	then	then	ADV
cana-758	115	3	(	(	PUNCT
cana-758	115	4	𝒯𝕣,𝕤,𝕥,,1,1	𝒯𝕣,𝕤,𝕥,,1,1	NOUN
cana-758	115	5	𝕞	𝕞	PRON
cana-758	115	6	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	115	7	)	)	PUNCT
cana-758	115	8	𝓏	𝓏	PROPN
cana-758	115	9	)	)	PUNCT
cana-758	115	10	𝜇	𝜇	ADP
cana-758	115	11	≺	≺	NOUN
cana-758	115	12	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	115	13	)	)	PUNCT
cana-758	115	14	,	,	PUNCT
cana-758	115	15	and	and	CCONJ
cana-758	115	16	the	the	DET
cana-758	115	17	equation	equation	NOUN
cana-758	115	18	(	(	PUNCT
cana-758	115	19	3.2	3.2	NUM
cana-758	115	20	)	)	PUNCT
cana-758	115	21	have	have	VERB
cana-758	115	22	the	the	DET
cana-758	115	23	best	good	ADJ
cana-758	115	24	dominant	dominant	NOUN
cana-758	115	25	say	say	VERB
cana-758	115	26	ϥ	ϥ	PROPN
cana-758	115	27	.	.	PUNCT
cana-758	116	1	communications	communication	NOUN
cana-758	116	2	on	on	ADP
cana-758	116	3	applied	apply	VERB
cana-758	116	4	nonlinear	nonlinear	ADJ
cana-758	116	5	analysis	analysis	NOUN
cana-758	116	6	issn	issn	NOUN
cana-758	116	7	:	:	PUNCT
cana-758	116	8	1074	1074	NUM
cana-758	116	9	-	-	PUNCT
cana-758	116	10	133x	133x	NUM
cana-758	116	11	vol	vol	NOUN
cana-758	116	12	31	31	NUM
cana-758	116	13	no	no	NOUN
cana-758	116	14	.	.	PUNCT
cana-758	117	1	3s	3s	NUM
cana-758	117	2	(	(	PUNCT
cana-758	117	3	2024	2024	NUM
cana-758	117	4	)	)	PUNCT
cana-758	117	5	191	191	NUM
cana-758	117	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	117	7	by	by	ADP
cana-758	117	8	the	the	DET
cana-758	117	9	same	same	ADJ
cana-758	117	10	way	way	NOUN
cana-758	117	11	we	we	PRON
cana-758	117	12	can	can	AUX
cana-758	117	13	take	take	VERB
cana-758	117	14	theorem	theorem	NOUN
cana-758	117	15	(	(	PUNCT
cana-758	117	16	3.1	3.1	NUM
cana-758	117	17	)	)	PUNCT
cana-758	117	18	,	,	PUNCT
cana-758	117	19	to	to	PART
cana-758	117	20	prove	prove	VERB
cana-758	117	21	the	the	DET
cana-758	117	22	following	follow	VERB
cana-758	117	23	theorems	theorem	NOUN
cana-758	117	24	by	by	ADP
cana-758	117	25	using	use	VERB
cana-758	117	26	the	the	DET
cana-758	117	27	identity(1.10	identity(1.10	NUM
cana-758	117	28	)	)	PUNCT
cana-758	117	29	.	.	PUNCT
cana-758	118	1	theorem	theorem	NOUN
cana-758	118	2	(	(	PUNCT
cana-758	118	3	3.2	3.2	NUM
cana-758	118	4	):	):	PUNCT
cana-758	118	5	let	let	VERB
cana-758	118	6	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	118	7	)	)	PUNCT
cana-758	118	8	is	be	AUX
cana-758	118	9	belongs	belong	VERB
cana-758	118	10	in	in	ADP
cana-758	118	11	ʋ	ʋ	PROPN
cana-758	118	12	with	with	ADP
cana-758	118	13	ϥ(0	ϥ(0	PROPN
cana-758	118	14	)	)	PUNCT
cana-758	118	15	=	=	SYM
cana-758	119	1	1,where	1,where	X
cana-758	119	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	119	3	)	)	PUNCT
cana-758	119	4	is	be	AUX
cana-758	119	5	convex	convex	ADJ
cana-758	119	6	and	and	CCONJ
cana-758	119	7	univalent	univalent	ADJ
cana-758	119	8	.	.	PUNCT
cana-758	120	1	let	let	VERB
cana-758	120	2	𝜉	𝜉	PRON
cana-758	120	3	∈	∈	PROPN
cana-758	120	4	₵	₵	PROPN
cana-758	120	5	∗	∗	NOUN
cana-758	120	6	,	,	PUNCT
cana-758	120	7	𝕧	𝕧	X
cana-758	120	8	>	>	X
cana-758	120	9	0	0	NUM
cana-758	120	10	,	,	PUNCT
cana-758	120	11	𝕦	𝕦	ADJ
cana-758	120	12	real	real	ADJ
cana-758	120	13	number	number	NOUN
cana-758	120	14	such	such	ADJ
cana-758	120	15	that	that	SCONJ
cana-758	120	16	𝕦	𝕦	PROPN
cana-758	120	17	+	+	X
cana-758	120	18	𝕧	𝕧	X
cana-758	120	19	>	>	X
cana-758	120	20	0	0	PUNCT
cana-758	120	21	andsuppose	andsuppose	ADJ
cana-758	120	22	that	that	SCONJ
cana-758	120	23	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	120	24	)	)	PUNCT
cana-758	120	25	satisfies	satisfie	NOUN
cana-758	120	26	:	:	PUNCT
cana-758	120	27	𝑅𝑒	𝑅𝑒	VERB
cana-758	120	28	{	{	PUNCT
cana-758	120	29	1	1	NUM
cana-758	120	30	+	+	NUM
cana-758	120	31	𝑧ϥ	𝑧ϥ	PRON
cana-758	120	32	"	"	PUNCT
cana-758	120	33	(	(	PUNCT
cana-758	120	34	𝓏	𝓏	NOUN
cana-758	120	35	)	)	PUNCT
cana-758	120	36	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	120	37	)	)	PUNCT
cana-758	120	38	}	}	PUNCT
cana-758	120	39	>	>	X
cana-758	120	40	max	max	PROPN
cana-758	120	41	{	{	PUNCT
cana-758	120	42	0	0	NUM
cana-758	120	43	,	,	PUNCT
cana-758	120	44	−𝑅𝑒	−𝑅𝑒	NOUN
cana-758	120	45	(	(	PUNCT
cana-758	120	46	𝜇	𝜇	X
cana-758	120	47	𝜉	𝜉	NOUN
cana-758	120	48	)	)	PUNCT
cana-758	120	49	}	}	PUNCT
cana-758	120	50	.	.	PUNCT
cana-758	121	1	(	(	PUNCT
cana-758	121	2	3.6	3.6	NUM
cana-758	121	3	)	)	PUNCT
cana-758	121	4	let	let	VERB
cana-758	121	5	𝑓	𝑓	DET
cana-758	121	6	∈	∈	NOUN
cana-758	121	7	𝒢	𝒢	PROPN
cana-758	121	8	holds	hold	VERB
cana-758	121	9	the	the	DET
cana-758	121	10	subordination	subordination	NOUN
cana-758	121	11	𝔏(𝜇	𝔏(𝜇	ADP
cana-758	121	12	,	,	PUNCT
cana-758	121	13	𝕞	𝕞	NOUN
cana-758	121	14	,	,	PUNCT
cana-758	121	15	𝕦	𝕦	ADJ
cana-758	121	16	,	,	PUNCT
cana-758	121	17	𝕧	𝕧	PROPN
cana-758	121	18	,	,	PUNCT
cana-758	121	19	𝜉	𝜉	NOUN
cana-758	121	20	)	)	PUNCT
cana-758	121	21	≺	≺	NOUN
cana-758	121	22	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	121	23	)	)	PUNCT
cana-758	122	1	+	+	CCONJ
cana-758	122	2	𝜉	𝜉	X
cana-758	122	3	𝜇	𝜇	X
cana-758	122	4	𝓏ϥ′(𝓏	𝓏ϥ′(𝓏	NOUN
cana-758	122	5	)	)	PUNCT
cana-758	122	6	,	,	PUNCT
cana-758	122	7	(	(	PUNCT
cana-758	122	8	3.7	3.7	NUM
cana-758	122	9	)	)	PUNCT
cana-758	123	1	where	where	SCONJ
cana-758	123	2	𝔏(𝜇	𝔏(𝜇	NOUN
cana-758	123	3	,	,	PUNCT
cana-758	123	4	𝕞	𝕞	ADJ
cana-758	123	5	,	,	PUNCT
cana-758	123	6	𝕦	𝕦	AUX
cana-758	123	7	,	,	PUNCT
cana-758	123	8	𝕧	𝕧	PROPN
cana-758	123	9	,	,	PUNCT
cana-758	123	10	𝜉	𝜉	NOUN
cana-758	123	11	)	)	PUNCT
cana-758	123	12	=	=	SYM
cana-758	123	13	(	(	PUNCT
cana-758	123	14	1	1	NUM
cana-758	123	15	−	−	NOUN
cana-758	123	16	𝜉	𝜉	NOUN
cana-758	123	17	)	)	PUNCT
cana-758	123	18	(	(	PUNCT
cana-758	123	19	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	123	20	𝕧	𝕧	NOUN
cana-758	123	21	)	)	PUNCT
cana-758	123	22	(	(	PUNCT
cana-758	123	23	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	SYM
cana-758	123	24	𝕞+1	𝕞+1	NUM
cana-758	123	25	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	123	26	)	)	PUNCT
cana-758	123	27	𝓏	𝓏	AUX
cana-758	123	28	)	)	PUNCT
cana-758	123	29	𝜇	𝜇	ADP
cana-758	123	30	+	+	X
cana-758	123	31	𝜉	𝜉	X
cana-758	123	32	(	(	PUNCT
cana-758	123	33	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	123	34	𝕧	𝕧	PROPN
cana-758	123	35	)	)	PUNCT
cana-758	123	36	(	(	PUNCT
cana-758	123	37	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	SYM
cana-758	123	38	𝕞+1	𝕞+1	NUM
cana-758	123	39	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	123	40	)	)	PUNCT
cana-758	123	41	𝓏	𝓏	PROPN
cana-758	123	42	)	)	PUNCT
cana-758	123	43	𝜇	𝜇	ADP
cana-758	123	44	+	+	X
cana-758	123	45	(	(	PUNCT
cana-758	123	46	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	123	47	𝕞	𝕞	DET
cana-758	123	48	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	123	49	)	)	PUNCT
cana-758	123	50	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	123	51	𝕞+1	𝕞+1	NUM
cana-758	123	52	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	123	53	)	)	PUNCT
cana-758	123	54	)	)	PUNCT
cana-758	123	55	,	,	PUNCT
cana-758	123	56	(	(	PUNCT
cana-758	123	57	3.8	3.8	NUM
cana-758	123	58	)	)	PUNCT
cana-758	123	59	then	then	ADV
cana-758	123	60	(	(	PUNCT
cana-758	123	61	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	123	62	𝕞+1	𝕞+1	NUM
cana-758	123	63	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	123	64	)	)	PUNCT
cana-758	123	65	𝓏	𝓏	PROPN
cana-758	123	66	)	)	PUNCT
cana-758	123	67	𝜇	𝜇	ADP
cana-758	123	68	≺	≺	NOUN
cana-758	123	69	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	123	70	)	)	PUNCT
cana-758	123	71	,	,	PUNCT
cana-758	123	72	(	(	PUNCT
cana-758	123	73	3.9	3.9	NUM
cana-758	123	74	)	)	PUNCT
cana-758	123	75	and	and	CCONJ
cana-758	123	76	the	the	DET
cana-758	123	77	equation	equation	NOUN
cana-758	123	78	(	(	PUNCT
cana-758	123	79	3.7	3.7	NUM
cana-758	123	80	)	)	PUNCT
cana-758	123	81	have	have	VERB
cana-758	123	82	the	the	DET
cana-758	123	83	best	good	ADJ
cana-758	123	84	dominant	dominant	NOUN
cana-758	123	85	say	say	VERB
cana-758	123	86	ϥ	ϥ	NOUN
cana-758	123	87	.	.	PUNCT
cana-758	124	1	proof	proof	NOUN
cana-758	124	2	:	:	PUNCT
cana-758	124	3	put	put	VERB
cana-758	124	4	𝒮(𝓏	𝒮(𝓏	NOUN
cana-758	124	5	)	)	PUNCT
cana-758	124	6	=	=	PUNCT
cana-758	124	7	(	(	PUNCT
cana-758	124	8	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	124	9	𝕞	𝕞	PRON
cana-758	124	10	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	124	11	)	)	PUNCT
cana-758	124	12	𝓏	𝓏	PROPN
cana-758	124	13	)	)	PUNCT
cana-758	124	14	𝜇	𝜇	ADP
cana-758	124	15	,	,	PUNCT
cana-758	124	16	𝓏	𝓏	PROPN
cana-758	124	17	∈	∈	PROPN
cana-758	124	18	ʋ	ʋ	PROPN
cana-758	124	19	.	.	PUNCT
cana-758	125	1	(	(	PUNCT
cana-758	125	2	3.10	3.10	NUM
cana-758	125	3	)	)	PUNCT
cana-758	125	4	hence	hence	ADV
cana-758	125	5	differentiate	differentiate	VERB
cana-758	125	6	(	(	PUNCT
cana-758	125	7	3.10	3.10	NUM
cana-758	125	8	)	)	PUNCT
cana-758	125	9	logarithmically	logarithmically	ADV
cana-758	125	10	according	accord	VERB
cana-758	125	11	to	to	ADP
cana-758	125	12	𝓏	𝓏	NUM
cana-758	125	13	,	,	PUNCT
cana-758	125	14	and	and	CCONJ
cana-758	125	15	taking	take	VERB
cana-758	125	16	identity	identity	NOUN
cana-758	125	17	(	(	PUNCT
cana-758	125	18	1.10	1.10	NUM
cana-758	125	19	)	)	PUNCT
cana-758	125	20	in	in	ADP
cana-758	125	21	resultant	resultant	NOUN
cana-758	125	22	equation	equation	NOUN
cana-758	125	23	,	,	PUNCT
cana-758	125	24	to	to	PART
cana-758	125	25	get	get	VERB
cana-758	125	26	𝓏	𝓏	PROPN
cana-758	125	27	𝒮′(𝓏	𝒮′(𝓏	NOUN
cana-758	125	28	)	)	PUNCT
cana-758	125	29	𝒮(𝓏	𝒮(𝓏	NUM
cana-758	125	30	)	)	PUNCT
cana-758	125	31	=	=	SYM
cana-758	125	32	𝜇	𝜇	X
cana-758	125	33	(	(	PUNCT
cana-758	125	34	(	(	PUNCT
cana-758	125	35	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	125	36	)	)	PUNCT
cana-758	125	37	𝕧	𝕧	NOUN
cana-758	125	38	)	)	PUNCT
cana-758	125	39	(	(	PUNCT
cana-758	125	40	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	125	41	𝕞	𝕞	PRON
cana-758	125	42	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	125	43	)	)	PUNCT
cana-758	125	44	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	125	45	𝕞+1	𝕞+1	NUM
cana-758	125	46	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	125	47	)	)	PUNCT
cana-758	126	1	−	−	PROPN
cana-758	126	2	1),and	1),and	NUM
cana-758	126	3	which	which	PRON
cana-758	126	4	can	can	AUX
cana-758	126	5	be	be	AUX
cana-758	126	6	written	write	VERB
cana-758	126	7	as	as	ADP
cana-758	126	8	𝕧	𝕧	PROPN
cana-758	126	9	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADV
cana-758	126	10	)	)	PUNCT
cana-758	126	11	𝓏𝒮′(𝓏	𝓏𝒮′(𝓏	NOUN
cana-758	126	12	)	)	PUNCT
cana-758	127	1	=	=	PUNCT
cana-758	127	2	(	(	PUNCT
cana-758	127	3	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	127	4	𝕞+1	𝕞+1	NUM
cana-758	127	5	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	127	6	)	)	PUNCT
cana-758	127	7	𝓏	𝓏	PROPN
cana-758	127	8	)	)	PUNCT
cana-758	127	9	𝜇	𝜇	X
cana-758	127	10	(	(	PUNCT
cana-758	127	11	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	127	12	𝕞	𝕞	DET
cana-758	127	13	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	127	14	)	)	PUNCT
cana-758	127	15	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	127	16	𝕞+1	𝕞+1	NUM
cana-758	127	17	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	127	18	)	)	PUNCT
cana-758	127	19	−	−	PROPN
cana-758	127	20	1	1	NUM
cana-758	127	21	)	)	PUNCT
cana-758	127	22	.	.	PUNCT
cana-758	128	1	thus	thus	ADV
cana-758	128	2	the	the	DET
cana-758	128	3	equation	equation	NOUN
cana-758	128	4	of	of	ADP
cana-758	128	5	subordination	subordination	NOUN
cana-758	128	6	which	which	PRON
cana-758	128	7	is	be	AUX
cana-758	128	8	represented	represent	VERB
cana-758	128	9	by	by	ADP
cana-758	128	10	(	(	PUNCT
cana-758	128	11	3.7	3.7	NUM
cana-758	128	12	)	)	PUNCT
cana-758	128	13	be	be	AUX
cana-758	128	14	equivalent	equivalent	ADJ
cana-758	128	15	by	by	ADP
cana-758	128	16	𝒮(𝓏	𝒮(𝓏	NUM
cana-758	128	17	)	)	PUNCT
cana-758	128	18	+	+	CCONJ
cana-758	128	19	𝜉	𝜉	X
cana-758	128	20	𝜇	𝜇	ADP
cana-758	128	21	𝓏𝒮′(𝓏	𝓏𝒮′(𝓏	NOUN
cana-758	128	22	)	)	PUNCT
cana-758	128	23	≺	≺	NOUN
cana-758	128	24	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	128	25	)	)	PUNCT
cana-758	129	1	+	+	CCONJ
cana-758	129	2	𝜉	𝜉	X
cana-758	129	3	𝜇	𝜇	X
cana-758	129	4	𝓏ϥ′(𝓏	𝓏ϥ′(𝓏	NOUN
cana-758	129	5	)	)	PUNCT
cana-758	129	6	.	.	PUNCT
cana-758	130	1	by	by	ADP
cana-758	130	2	apply	apply	VERB
cana-758	130	3	lemma	lemma	PROPN
cana-758	130	4	(	(	PUNCT
cana-758	130	5	2.2	2.2	NUM
cana-758	130	6	)	)	PUNCT
cana-758	130	7	when	when	SCONJ
cana-758	130	8	𝜎	𝜎	NOUN
cana-758	130	9	=	=	X
cana-758	130	10	𝜉	𝜉	X
cana-758	130	11	𝜇	𝜇	X
cana-758	130	12	,	,	PUNCT
cana-758	130	13	the	the	DET
cana-758	130	14	proof	proof	NOUN
cana-758	130	15	of	of	ADP
cana-758	130	16	theorem(3.2	theorem(3.2	NOUN
cana-758	130	17	)	)	PUNCT
cana-758	130	18	is	be	AUX
cana-758	130	19	complete	complete	ADJ
cana-758	130	20	therefore	therefore	ADV
cana-758	130	21	,	,	PUNCT
cana-758	130	22	we	we	PRON
cana-758	130	23	put	put	VERB
cana-758	130	24	𝕧	𝕧	NOUN
cana-758	130	25	=	=	NOUN
cana-758	130	26	1	1	NUM
cana-758	130	27	in	in	ADP
cana-758	130	28	above	above	ADP
cana-758	130	29	theorem	theorem	VERB
cana-758	130	30	to	to	PART
cana-758	130	31	obtain	obtain	VERB
cana-758	130	32	the	the	DET
cana-758	130	33	results	result	NOUN
cana-758	130	34	below	below	ADV
cana-758	130	35	.	.	PUNCT
cana-758	131	1	corollary(3.4	corollary(3.4	NOUN
cana-758	131	2	):	):	PUNCT
cana-758	131	3	let	let	VERB
cana-758	131	4	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	131	5	)	)	PUNCT
cana-758	131	6	is	be	AUX
cana-758	131	7	belongs	belong	VERB
cana-758	131	8	in	in	ADP
cana-758	131	9	ʋ	ʋ	PROPN
cana-758	131	10	with	with	ADP
cana-758	131	11	ϥ(0	ϥ(0	PROPN
cana-758	131	12	)	)	PUNCT
cana-758	131	13	=	=	SYM
cana-758	132	1	1,where	1,where	X
cana-758	132	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	132	3	)	)	PUNCT
cana-758	132	4	is	be	AUX
cana-758	132	5	univalent	univalent	ADJ
cana-758	132	6	.	.	PUNCT
cana-758	133	1	let	let	VERB
cana-758	133	2	𝜉	𝜉	PRON
cana-758	133	3	∈	∈	PROPN
cana-758	133	4	₵	₵	PROPN
cana-758	133	5	∗	∗	NOUN
cana-758	133	6	,	,	PUNCT
cana-758	133	7	𝜇	𝜇	X
cana-758	133	8	>	>	X
cana-758	133	9	0	0	PROPN
cana-758	133	10	,	,	PUNCT
cana-758	133	11	𝕦	𝕦	ADJ
cana-758	133	12	real	real	ADJ
cana-758	133	13	number	number	NOUN
cana-758	133	14	such	such	ADJ
cana-758	133	15	that	that	SCONJ
cana-758	133	16	𝕦	𝕦	PROPN
cana-758	133	17	+	+	X
cana-758	133	18	𝕧	𝕧	X
cana-758	133	19	>	>	X
cana-758	133	20	0	0	PUNCT
cana-758	133	21	andsuppose	andsuppose	ADJ
cana-758	133	22	that	that	SCONJ
cana-758	133	23	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	133	24	)	)	PUNCT
cana-758	133	25	satisfies	satisfie	NOUN
cana-758	133	26	:	:	PUNCT
cana-758	133	27	𝑅𝑒	𝑅𝑒	VERB
cana-758	133	28	{	{	PUNCT
cana-758	133	29	1	1	NUM
cana-758	133	30	+	+	NUM
cana-758	133	31	𝑧ϥ	𝑧ϥ	PRON
cana-758	133	32	"	"	PUNCT
cana-758	133	33	(	(	PUNCT
cana-758	133	34	𝓏	𝓏	NOUN
cana-758	133	35	)	)	PUNCT
cana-758	133	36	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	133	37	)	)	PUNCT
cana-758	133	38	}	}	PUNCT
cana-758	133	39	>	>	X
cana-758	133	40	max	max	PROPN
cana-758	133	41	{	{	PUNCT
cana-758	133	42	0	0	NUM
cana-758	133	43	,	,	PUNCT
cana-758	133	44	−𝑅𝑒	−𝑅𝑒	NOUN
cana-758	133	45	(	(	PUNCT
cana-758	133	46	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADJ
cana-758	133	47	)	)	PUNCT
cana-758	133	48	𝜉𝕧	𝜉𝕧	NOUN
cana-758	133	49	)	)	PUNCT
cana-758	133	50	}	}	PUNCT
cana-758	133	51	.	.	PUNCT
cana-758	134	1	(	(	PUNCT
cana-758	134	2	3.11	3.11	NUM
cana-758	134	3	)	)	PUNCT
cana-758	134	4	let	let	VERB
cana-758	134	5	𝑓	𝑓	DET
cana-758	134	6	∈	∈	NOUN
cana-758	134	7	𝒢	𝒢	PROPN
cana-758	134	8	holds	hold	VERB
cana-758	134	9	the	the	DET
cana-758	134	10	subordination	subordination	NOUN
cana-758	134	11	communications	communication	NOUN
cana-758	134	12	on	on	ADP
cana-758	134	13	applied	apply	VERB
cana-758	134	14	nonlinear	nonlinear	ADJ
cana-758	134	15	analysis	analysis	NOUN
cana-758	134	16	issn	issn	NOUN
cana-758	134	17	:	:	PUNCT
cana-758	134	18	1074	1074	NUM
cana-758	134	19	-	-	PUNCT
cana-758	134	20	133x	133x	NUM
cana-758	134	21	vol	vol	NOUN
cana-758	134	22	31	31	NUM
cana-758	134	23	no	no	NOUN
cana-758	134	24	.	.	PUNCT
cana-758	135	1	3s	3s	NUM
cana-758	135	2	(	(	PUNCT
cana-758	135	3	2024	2024	NUM
cana-758	135	4	)	)	PUNCT
cana-758	135	5	192	192	NUM
cana-758	135	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	135	7	𝔏1(𝜇	𝔏1(𝜇	PROPN
cana-758	135	8	,	,	PUNCT
cana-758	135	9	𝕞	𝕞	X
cana-758	135	10	,	,	PUNCT
cana-758	135	11	𝕦	𝕦	ADJ
cana-758	135	12	,	,	PUNCT
cana-758	135	13	𝕧	𝕧	PROPN
cana-758	135	14	,	,	PUNCT
cana-758	135	15	𝜉	𝜉	NOUN
cana-758	135	16	)	)	PUNCT
cana-758	135	17	≺	≺	NOUN
cana-758	135	18	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	135	19	)	)	PUNCT
cana-758	136	1	+	+	CCONJ
cana-758	136	2	𝜉	𝜉	X
cana-758	136	3	𝜇	𝜇	X
cana-758	136	4	𝓏ϥ′(𝓏	𝓏ϥ′(𝓏	NOUN
cana-758	136	5	)	)	PUNCT
cana-758	136	6	,	,	PUNCT
cana-758	136	7	(	(	PUNCT
cana-758	136	8	3.12	3.12	NUM
cana-758	136	9	)	)	PUNCT
cana-758	137	1	where	where	SCONJ
cana-758	137	2	𝔏1(𝜇	𝔏1(𝜇	PROPN
cana-758	137	3	,	,	PUNCT
cana-758	137	4	𝕞	𝕞	PROPN
cana-758	137	5	,	,	PUNCT
cana-758	137	6	𝕦	𝕦	ADJ
cana-758	137	7	,	,	PUNCT
cana-758	137	8	𝕧	𝕧	PROPN
cana-758	137	9	,	,	PUNCT
cana-758	137	10	𝜉	𝜉	NOUN
cana-758	137	11	)	)	PUNCT
cana-758	137	12	=	=	SYM
cana-758	137	13	(	(	PUNCT
cana-758	137	14	1	1	NUM
cana-758	137	15	−	−	NOUN
cana-758	137	16	𝜉	𝜉	NOUN
cana-758	137	17	)	)	PUNCT
cana-758	137	18	(	(	PUNCT
cana-758	137	19	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	137	20	𝕧	𝕧	NOUN
cana-758	137	21	)	)	PUNCT
cana-758	137	22	(	(	PUNCT
cana-758	137	23	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	SYM
cana-758	137	24	𝕞+1	𝕞+1	NUM
cana-758	137	25	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	137	26	)	)	PUNCT
cana-758	137	27	𝓏	𝓏	AUX
cana-758	137	28	)	)	PUNCT
cana-758	137	29	𝜇	𝜇	ADP
cana-758	137	30	+	+	X
cana-758	137	31	𝜉	𝜉	X
cana-758	137	32	(	(	PUNCT
cana-758	137	33	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	137	34	𝕧	𝕧	PROPN
cana-758	137	35	)	)	PUNCT
cana-758	137	36	(	(	PUNCT
cana-758	137	37	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	SYM
cana-758	137	38	𝕞+1	𝕞+1	NUM
cana-758	137	39	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	137	40	)	)	PUNCT
cana-758	137	41	𝓏	𝓏	PROPN
cana-758	137	42	)	)	PUNCT
cana-758	137	43	𝜇	𝜇	ADP
cana-758	137	44	+	+	X
cana-758	137	45	(	(	PUNCT
cana-758	137	46	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	137	47	𝕞	𝕞	PRON
cana-758	137	48	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	137	49	)	)	PUNCT
cana-758	137	50	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	137	51	𝕞+1	𝕞+1	NUM
cana-758	137	52	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	137	53	)	)	PUNCT
cana-758	137	54	)	)	PUNCT
cana-758	138	1	(	(	PUNCT
cana-758	138	2	3.13	3.13	NUM
cana-758	138	3	)	)	PUNCT
cana-758	138	4	then	then	ADV
cana-758	138	5	(	(	PUNCT
cana-758	138	6	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	138	7	𝕞+1	𝕞+1	NUM
cana-758	138	8	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	138	9	)	)	PUNCT
cana-758	138	10	𝓏	𝓏	PROPN
cana-758	138	11	)	)	PUNCT
cana-758	138	12	𝜇	𝜇	ADP
cana-758	138	13	≺	≺	NOUN
cana-758	138	14	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	138	15	)	)	PUNCT
cana-758	138	16	,	,	PUNCT
cana-758	138	17	(	(	PUNCT
cana-758	138	18	3.14	3.14	NUM
cana-758	138	19	)	)	PUNCT
cana-758	138	20	and	and	CCONJ
cana-758	138	21	the	the	DET
cana-758	138	22	equation	equation	NOUN
cana-758	138	23	(	(	PUNCT
cana-758	138	24	3.7	3.7	NUM
cana-758	138	25	)	)	PUNCT
cana-758	138	26	have	have	VERB
cana-758	138	27	the	the	DET
cana-758	138	28	best	good	ADJ
cana-758	138	29	dominant	dominant	NOUN
cana-758	138	30	say	say	VERB
cana-758	138	31	ϥ	ϥ	PROPN
cana-758	138	32	.	.	PUNCT
cana-758	139	1	4	4	X
cana-758	139	2	.	.	X
cana-758	139	3	superordination	superordination	NOUN
cana-758	139	4	results	result	NOUN
cana-758	139	5	:	:	PUNCT
cana-758	139	6	theorem	theorem	NOUN
cana-758	139	7	(	(	PUNCT
cana-758	139	8	4.1	4.1	NUM
cana-758	139	9	)	)	PUNCT
cana-758	139	10	:	:	PUNCT
cana-758	139	11	assume	assume	VERB
cana-758	139	12	that	that	SCONJ
cana-758	139	13	the	the	DET
cana-758	139	14	function	function	NOUN
cana-758	139	15	ϥ	ϥ	PROPN
cana-758	139	16	are	be	AUX
cana-758	139	17	univalent	univalent	ADJ
cana-758	139	18	and	and	CCONJ
cana-758	139	19	convex	convex	VERB
cana-758	139	20	in	in	ADP
cana-758	139	21	ʋ	ʋ	X
cana-758	139	22	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	VERB
cana-758	139	23	ϥ(0	ϥ(0	PROPN
cana-758	139	24	)	)	PUNCT
cana-758	140	1	=	=	SYM
cana-758	140	2	1	1	NUM
cana-758	140	3	,	,	PUNCT
cana-758	140	4	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	140	5	)	)	PUNCT
cana-758	140	6	>	>	X
cana-758	140	7	0	0	NUM
cana-758	140	8	,	,	PUNCT
cana-758	140	9	𝜉	𝜉	PROPN
cana-758	140	10	∈	∈	PROPN
cana-758	140	11	₵	₵	PROPN
cana-758	140	12	,	,	PUNCT
cana-758	140	13	𝜇	𝜇	ADV
cana-758	140	14	,	,	PUNCT
cana-758	140	15	𝕧	𝕧	PROPN
cana-758	140	16	>	>	X
cana-758	140	17	0	0	NUM
cana-758	140	18	,	,	PUNCT
cana-758	140	19	𝕦	𝕦	PROPN
cana-758	140	20	∈	∈	PROPN
cana-758	140	21	𝑅	𝑅	PROPN
cana-758	140	22	such	such	ADJ
cana-758	140	23	that	that	SCONJ
cana-758	140	24	𝕦	𝕦	PROPN
cana-758	140	25	+	+	X
cana-758	140	26	𝕧	𝕧	X
cana-758	140	27	>	>	X
cana-758	140	28	0	0	NUM
cana-758	140	29	.	.	PUNCT
cana-758	141	1	if	if	SCONJ
cana-758	141	2	𝑓	𝑓	DET
cana-758	141	3	∈	∈	PROPN
cana-758	141	4	𝒢	𝒢	NOUN
cana-758	141	5	,	,	PUNCT
cana-758	141	6	where	where	SCONJ
cana-758	141	7	(	(	PUNCT
cana-758	141	8	𝒯𝕣,𝕤,𝕥,,1,1	𝒯𝕣,𝕤,𝕥,,1,1	NOUN
cana-758	141	9	𝕞	𝕞	PRON
cana-758	141	10	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	141	11	)	)	PUNCT
cana-758	141	12	𝓏	𝓏	PROPN
cana-758	141	13	)	)	PUNCT
cana-758	141	14	𝜇	𝜇	ADP
cana-758	141	15	∈	∈	PROPN
cana-758	141	16	𝑀[ϥ(0	𝑀[ϥ(0	PROPN
cana-758	141	17	)	)	PUNCT
cana-758	141	18	,	,	PUNCT
cana-758	141	19	1	1	X
cana-758	141	20	]	]	PUNCT
cana-758	141	21	∩	∩	ADJ
cana-758	141	22	𝚀	𝚀	NOUN
cana-758	141	23	.	.	PUNCT
cana-758	142	1	(	(	PUNCT
cana-758	142	2	4.1	4.1	NUM
cana-758	142	3	)	)	PUNCT
cana-758	142	4	if	if	SCONJ
cana-758	142	5	𝔇(𝕞	𝔇(𝕞	NOUN
cana-758	142	6	,	,	PUNCT
cana-758	142	7	𝜉	𝜉	PROPN
cana-758	142	8	,	,	PUNCT
cana-758	142	9	𝜇	𝜇	ADP
cana-758	142	10	,	,	PUNCT
cana-758	142	11	𝕦	𝕦	ADJ
cana-758	142	12	,	,	PUNCT
cana-758	142	13	𝕧)is	𝕧)is	ADJ
cana-758	142	14	univalent	univalent	ADJ
cana-758	142	15	function	function	NOUN
cana-758	142	16	in	in	ADP
cana-758	142	17	ʋ	ʋ	PROPN
cana-758	142	18	as	as	SCONJ
cana-758	142	19	defined	define	VERB
cana-758	142	20	by	by	ADP
cana-758	142	21	(	(	PUNCT
cana-758	142	22	3.3),and	3.3),and	NUM
cana-758	142	23	satisfies	satisfy	VERB
cana-758	142	24	the	the	DET
cana-758	142	25	superordination	superordination	NOUN
cana-758	142	26	case	case	NOUN
cana-758	142	27	below	below	ADV
cana-758	142	28	;	;	PUNCT
cana-758	142	29	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	142	30	)	)	PUNCT
cana-758	143	1	+	+	CCONJ
cana-758	143	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	143	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	NUM
cana-758	143	4	)	)	PUNCT
cana-758	143	5	𝓏	𝓏	PROPN
cana-758	143	6	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	143	7	)	)	PUNCT
cana-758	143	8	≺	≺	NOUN
cana-758	143	9	𝔇(𝕞	𝔇(𝕞	NOUN
cana-758	143	10	,	,	PUNCT
cana-758	143	11	𝜉	𝜉	PROPN
cana-758	143	12	,	,	PUNCT
cana-758	143	13	𝜇	𝜇	ADP
cana-758	143	14	,	,	PUNCT
cana-758	143	15	𝕦	𝕦	ADJ
cana-758	143	16	,	,	PUNCT
cana-758	143	17	𝕧	𝕧	NOUN
cana-758	143	18	)	)	PUNCT
cana-758	143	19	,	,	PUNCT
cana-758	143	20	(	(	PUNCT
cana-758	143	21	4.2	4.2	NUM
cana-758	143	22	)	)	PUNCT
cana-758	143	23	then	then	ADV
cana-758	143	24	ϥ(z	ϥ(z	PROPN
cana-758	143	25	)	)	PUNCT
cana-758	143	26	≺	≺	NOUN
cana-758	143	27	(	(	PUNCT
cana-758	143	28	𝒯𝕣,𝕤,𝕥,,1,1	𝒯𝕣,𝕤,𝕥,,1,1	NOUN
cana-758	143	29	𝕞	𝕞	PRON
cana-758	143	30	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	143	31	)	)	PUNCT
cana-758	143	32	𝓏	𝓏	PROPN
cana-758	143	33	)	)	PUNCT
cana-758	143	34	𝜇	𝜇	ADP
cana-758	143	35	,	,	PUNCT
cana-758	143	36	(	(	PUNCT
cana-758	143	37	4.3	4.3	NUM
cana-758	143	38	)	)	PUNCT
cana-758	143	39	and	and	CCONJ
cana-758	143	40	ϥ(z	ϥ(z	PROPN
cana-758	143	41	)	)	PUNCT
cana-758	143	42	will	will	AUX
cana-758	143	43	be	be	AUX
cana-758	143	44	best	good	ADJ
cana-758	143	45	subordination	subordination	NOUN
cana-758	143	46	.	.	PUNCT
cana-758	144	1	proof	proof	NOUN
cana-758	144	2	:	:	PUNCT
cana-758	144	3	put	put	VERB
cana-758	144	4	𝒮(𝓏	𝒮(𝓏	NOUN
cana-758	144	5	)	)	PUNCT
cana-758	144	6	=	=	PRON
cana-758	144	7	(	(	PUNCT
cana-758	145	1	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	145	2	𝕞	𝕞	X
cana-758	145	3	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	145	4	)	)	PUNCT
cana-758	145	5	𝓏	𝓏	PROPN
cana-758	145	6	)	)	PUNCT
cana-758	145	7	𝜇	𝜇	ADP
cana-758	145	8	,	,	PUNCT
cana-758	145	9	𝓏	𝓏	PROPN
cana-758	145	10	∈	∈	PROPN
cana-758	145	11	ʋ	ʋ	NOUN
cana-758	145	12	.	.	PUNCT
cana-758	146	1	(	(	PUNCT
cana-758	146	2	3.4	3.4	NUM
cana-758	146	3	)	)	PUNCT
cana-758	146	4	hence	hence	ADV
cana-758	146	5	differentiate	differentiate	VERB
cana-758	146	6	(	(	PUNCT
cana-758	146	7	3.4	3.4	NUM
cana-758	146	8	)	)	PUNCT
cana-758	146	9	logarithmically	logarithmically	ADV
cana-758	146	10	according	accord	VERB
cana-758	146	11	to	to	ADP
cana-758	146	12	𝓏	𝓏	NUM
cana-758	146	13	,	,	PUNCT
cana-758	146	14	and	and	CCONJ
cana-758	146	15	taking	take	VERB
cana-758	146	16	identity	identity	NOUN
cana-758	146	17	(	(	PUNCT
cana-758	146	18	1.8	1.8	NUM
cana-758	146	19	)	)	PUNCT
cana-758	146	20	in	in	ADP
cana-758	146	21	resultant	resultant	NOUN
cana-758	146	22	equation	equation	NOUN
cana-758	146	23	,	,	PUNCT
cana-758	146	24	to	to	PART
cana-758	146	25	get	get	VERB
cana-758	146	26	𝓏	𝓏	PROPN
cana-758	146	27	𝒮′(𝓏	𝒮′(𝓏	NOUN
cana-758	146	28	)	)	PUNCT
cana-758	146	29	𝒮(𝓏	𝒮(𝓏	NUM
cana-758	146	30	)	)	PUNCT
cana-758	146	31	=	=	SYM
cana-758	146	32	𝜇	𝜇	X
cana-758	146	33	(	(	PUNCT
cana-758	146	34	(	(	PUNCT
cana-758	146	35	𝕦+𝕧	𝕦+𝕧	NUM
cana-758	146	36	)	)	PUNCT
cana-758	146	37	𝕧	𝕧	NOUN
cana-758	146	38	)	)	PUNCT
cana-758	146	39	(	(	PUNCT
cana-758	146	40	𝓏𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝓏𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	VERB
cana-758	146	41	𝕞+1	𝕞+1	NUM
cana-758	146	42	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	146	43	)	)	PUNCT
cana-758	146	44	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	146	45	𝕞	𝕞	DET
cana-758	146	46	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	146	47	)	)	PUNCT
cana-758	147	1	−	−	PROPN
cana-758	147	2	1),and	1),and	NUM
cana-758	147	3	which	which	PRON
cana-758	147	4	can	can	AUX
cana-758	147	5	be	be	AUX
cana-758	147	6	written	write	VERB
cana-758	147	7	as	as	ADP
cana-758	147	8	𝕧	𝕧	PROPN
cana-758	147	9	𝜇(𝕦	𝜇(𝕦	NOUN
cana-758	147	10	+	+	CCONJ
cana-758	147	11	𝕧	𝕧	NOUN
cana-758	147	12	)	)	PUNCT
cana-758	147	13	𝓏𝒮′(𝓏	𝓏𝒮′(𝓏	PROPN
cana-758	147	14	)	)	PUNCT
cana-758	148	1	=	=	SYM
cana-758	148	2	(	(	PUNCT
cana-758	148	3	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	148	4	𝕞	𝕞	X
cana-758	148	5	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	148	6	)	)	PUNCT
cana-758	148	7	𝓏	𝓏	PROPN
cana-758	148	8	)	)	PUNCT
cana-758	148	9	𝜇	𝜇	X
cana-758	148	10	(	(	PUNCT
cana-758	148	11	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	148	12	𝕞+1	𝕞+1	NUM
cana-758	148	13	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	148	14	)	)	PUNCT
cana-758	148	15	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	148	16	𝕞	𝕞	DET
cana-758	148	17	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	148	18	)	)	PUNCT
cana-758	148	19	−	−	PROPN
cana-758	149	1	1	1	X
cana-758	149	2	)	)	PUNCT
cana-758	149	3	thus	thus	ADV
cana-758	149	4	the	the	DET
cana-758	149	5	equation	equation	NOUN
cana-758	149	6	of	of	ADP
cana-758	149	7	subordination	subordination	NOUN
cana-758	149	8	which	which	PRON
cana-758	149	9	is	be	AUX
cana-758	149	10	represented	represent	VERB
cana-758	149	11	by	by	ADP
cana-758	149	12	(	(	PUNCT
cana-758	149	13	4.2	4.2	NUM
cana-758	149	14	)	)	PUNCT
cana-758	149	15	be	be	AUX
cana-758	149	16	equivalent	equivalent	ADJ
cana-758	149	17	by	by	ADP
cana-758	149	18	𝒮(𝓏	𝒮(𝓏	NUM
cana-758	149	19	)	)	PUNCT
cana-758	149	20	+	+	CCONJ
cana-758	149	21	𝜉𝕧	𝜉𝕧	ADP
cana-758	149	22	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADV
cana-758	149	23	)	)	PUNCT
cana-758	149	24	𝓏𝒮′(𝓏	𝓏𝒮′(𝓏	NOUN
cana-758	149	25	)	)	PUNCT
cana-758	149	26	≺	≺	NOUN
cana-758	149	27	𝑝(𝓏	𝑝(𝓏	NOUN
cana-758	149	28	)	)	PUNCT
cana-758	150	1	+	+	CCONJ
cana-758	150	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	150	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	NOUN
cana-758	150	4	)	)	PUNCT
cana-758	150	5	𝓏𝑝′(𝓏	𝓏𝑝′(𝓏	NOUN
cana-758	150	6	)	)	PUNCT
cana-758	150	7	.	.	PUNCT
cana-758	151	1	by	by	ADP
cana-758	151	2	apply	apply	VERB
cana-758	151	3	lemma	lemma	PROPN
cana-758	151	4	(	(	PUNCT
cana-758	151	5	2.2	2.2	NUM
cana-758	151	6	)	)	PUNCT
cana-758	151	7	,	,	PUNCT
cana-758	151	8	the	the	DET
cana-758	151	9	proof	proof	NOUN
cana-758	151	10	of	of	ADP
cana-758	151	11	theorem(4.1	theorem(4.1	NOUN
cana-758	151	12	)	)	PUNCT
cana-758	151	13	is	be	AUX
cana-758	151	14	complete	complete	ADJ
cana-758	151	15	.	.	PUNCT
cana-758	152	1	now	now	ADV
cana-758	152	2	,	,	PUNCT
cana-758	152	3	in	in	ADP
cana-758	152	4	theorem	theorem	NOUN
cana-758	152	5	above	above	ADV
cana-758	152	6	,	,	PUNCT
cana-758	152	7	we	we	PRON
cana-758	152	8	put	put	VERB
cana-758	152	9	,	,	PUNCT
cana-758	152	10	𝕞	𝕞	X
cana-758	152	11	=	=	NOUN
cana-758	152	12	0	0	NUM
cana-758	152	13	,	,	PUNCT
cana-758	152	14	so	so	ADV
cana-758	152	15	we	we	PRON
cana-758	152	16	get	get	VERB
cana-758	152	17	the	the	DET
cana-758	152	18	following	follow	VERB
cana-758	152	19	corollary	corollary	NOUN
cana-758	152	20	.	.	PUNCT
cana-758	153	1	communications	communication	NOUN
cana-758	153	2	on	on	ADP
cana-758	153	3	applied	apply	VERB
cana-758	153	4	nonlinear	nonlinear	ADJ
cana-758	153	5	analysis	analysis	NOUN
cana-758	153	6	issn	issn	NOUN
cana-758	153	7	:	:	PUNCT
cana-758	153	8	1074	1074	NUM
cana-758	153	9	-	-	PUNCT
cana-758	153	10	133x	133x	NUM
cana-758	153	11	vol	vol	NOUN
cana-758	153	12	31	31	NUM
cana-758	153	13	no	no	NOUN
cana-758	153	14	.	.	PUNCT
cana-758	154	1	3s	3s	NUM
cana-758	154	2	(	(	PUNCT
cana-758	154	3	2024	2024	NUM
cana-758	154	4	)	)	PUNCT
cana-758	154	5	193	193	NUM
cana-758	154	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	154	7	corollary	corollary	NOUN
cana-758	154	8	(	(	PUNCT
cana-758	154	9	4.1	4.1	NUM
cana-758	154	10	)	)	PUNCT
cana-758	154	11	:	:	PUNCT
cana-758	154	12	let	let	VERB
cana-758	154	13	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	154	14	)	)	PUNCT
cana-758	154	15	is	be	AUX
cana-758	154	16	belongs	belong	VERB
cana-758	154	17	in	in	ADP
cana-758	154	18	ʋ	ʋ	PROPN
cana-758	154	19	with	with	ADP
cana-758	154	20	ϥ(0	ϥ(0	PROPN
cana-758	154	21	)	)	PUNCT
cana-758	154	22	=	=	SYM
cana-758	155	1	1,where	1,where	X
cana-758	155	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	155	3	)	)	PUNCT
cana-758	155	4	is	be	AUX
cana-758	155	5	convex	convex	ADJ
cana-758	155	6	and	and	CCONJ
cana-758	155	7	univalent	univalent	ADJ
cana-758	155	8	,	,	PUNCT
cana-758	155	9	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	155	10	)	)	PUNCT
cana-758	155	11	>	>	X
cana-758	155	12	0	0	NUM
cana-758	155	13	,	,	PUNCT
cana-758	155	14	𝜉	𝜉	PROPN
cana-758	155	15	∈	∈	PROPN
cana-758	155	16	₵	₵	PROPN
cana-758	155	17	,	,	PUNCT
cana-758	155	18	𝜇	𝜇	ADV
cana-758	155	19	,	,	PUNCT
cana-758	155	20	𝕧	𝕧	PROPN
cana-758	155	21	>	>	X
cana-758	155	22	0	0	NUM
cana-758	155	23	,	,	PUNCT
cana-758	155	24	𝕦	𝕦	PROPN
cana-758	155	25	∈	∈	NOUN
cana-758	155	26	𝑅.	𝑅.	PUNCT
cana-758	155	27	if	if	SCONJ
cana-758	155	28	𝑓	𝑓	DET
cana-758	155	29	∈	∈	PROPN
cana-758	155	30	𝒢	𝒢	NOUN
cana-758	155	31	,	,	PUNCT
cana-758	155	32	where	where	SCONJ
cana-758	155	33	(	(	PUNCT
cana-758	155	34	𝒯𝕣,𝕤,𝕥	𝒯𝕣,𝕤,𝕥	PROPN
cana-758	155	35	𝕞	𝕞	PRON
cana-758	155	36	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	155	37	)	)	PUNCT
cana-758	155	38	𝓏	𝓏	PROPN
cana-758	155	39	)	)	PUNCT
cana-758	155	40	𝜇	𝜇	ADP
cana-758	155	41	∈	∈	PROPN
cana-758	155	42	𝑀[ϥ(0	𝑀[ϥ(0	PROPN
cana-758	155	43	)	)	PUNCT
cana-758	155	44	,	,	PUNCT
cana-758	155	45	1	1	X
cana-758	155	46	]	]	PUNCT
cana-758	155	47	∩	∩	ADJ
cana-758	155	48	𝚀	𝚀	NOUN
cana-758	155	49	.	.	PUNCT
cana-758	156	1	(	(	PUNCT
cana-758	156	2	4.5	4.5	NUM
cana-758	156	3	)	)	PUNCT
cana-758	156	4	if	if	SCONJ
cana-758	156	5	𝔇1(0	𝔇1(0	NOUN
cana-758	156	6	,	,	PUNCT
cana-758	156	7	𝜉	𝜉	X
cana-758	156	8	,	,	PUNCT
cana-758	156	9	𝜇	𝜇	ADP
cana-758	156	10	,	,	PUNCT
cana-758	156	11	𝕦	𝕦	ADJ
cana-758	156	12	,	,	PUNCT
cana-758	156	13	𝕧)is	𝕧)is	ADJ
cana-758	156	14	univalent	univalent	ADJ
cana-758	156	15	function	function	NOUN
cana-758	156	16	as	as	SCONJ
cana-758	156	17	defined	define	VERB
cana-758	156	18	by	by	ADP
cana-758	156	19	(	(	PUNCT
cana-758	156	20	3.3),and	3.3),and	NUM
cana-758	156	21	satisfies	satisfy	VERB
cana-758	156	22	the	the	DET
cana-758	156	23	superordination	superordination	NOUN
cana-758	156	24	case	case	NOUN
cana-758	156	25	below	below	ADV
cana-758	156	26	;	;	PUNCT
cana-758	156	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	156	28	)	)	PUNCT
cana-758	157	1	+	+	CCONJ
cana-758	157	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	157	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	NUM
cana-758	157	4	)	)	PUNCT
cana-758	157	5	𝓏	𝓏	PROPN
cana-758	157	6	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	157	7	)	)	PUNCT
cana-758	157	8	≺	≺	NOUN
cana-758	157	9	𝔇1(0	𝔇1(0	NOUN
cana-758	157	10	,	,	PUNCT
cana-758	157	11	𝜉	𝜉	PROPN
cana-758	157	12	,	,	PUNCT
cana-758	157	13	𝜇	𝜇	ADP
cana-758	157	14	,	,	PUNCT
cana-758	157	15	𝕦	𝕦	ADJ
cana-758	157	16	,	,	PUNCT
cana-758	157	17	𝕧	𝕧	NOUN
cana-758	157	18	)	)	PUNCT
cana-758	157	19	,	,	PUNCT
cana-758	157	20	(	(	PUNCT
cana-758	157	21	4.6	4.6	NUM
cana-758	157	22	)	)	PUNCT
cana-758	157	23	then	then	ADV
cana-758	157	24	ϥ(z	ϥ(z	PROPN
cana-758	157	25	)	)	PUNCT
cana-758	157	26	≺	≺	NOUN
cana-758	157	27	(	(	PUNCT
cana-758	157	28	𝒯𝕣,𝕤,𝕥	𝒯𝕣,𝕤,𝕥	PROPN
cana-758	157	29	𝕞	𝕞	PRON
cana-758	157	30	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	157	31	)	)	PUNCT
cana-758	157	32	𝓏	𝓏	PROPN
cana-758	157	33	)	)	PUNCT
cana-758	157	34	𝜇	𝜇	ADP
cana-758	157	35	,	,	PUNCT
cana-758	157	36	(	(	PUNCT
cana-758	157	37	4.7	4.7	NUM
cana-758	157	38	)	)	PUNCT
cana-758	157	39	and	and	CCONJ
cana-758	157	40	ϥ(z	ϥ(z	PROPN
cana-758	157	41	)	)	PUNCT
cana-758	157	42	will	will	AUX
cana-758	157	43	be	be	AUX
cana-758	157	44	best	good	ADJ
cana-758	157	45	subordination	subordination	NOUN
cana-758	157	46	.	.	PUNCT
cana-758	158	1	now	now	ADV
cana-758	158	2	,	,	PUNCT
cana-758	158	3	in	in	ADP
cana-758	158	4	theorem	theorem	NOUN
cana-758	158	5	above	above	ADV
cana-758	158	6	,	,	PUNCT
cana-758	158	7	we	we	PRON
cana-758	158	8	put	put	VERB
cana-758	158	9	,	,	PUNCT
cana-758	158	10	𝕧	𝕧	PROPN
cana-758	158	11	=	=	SYM
cana-758	158	12	1	1	NUM
cana-758	158	13	,	,	PUNCT
cana-758	158	14	so	so	ADV
cana-758	158	15	to	to	PART
cana-758	158	16	obtain	obtain	VERB
cana-758	158	17	corollary	corollary	ADJ
cana-758	158	18	below	below	ADV
cana-758	158	19	.	.	PUNCT
cana-758	159	1	corollary	corollary	ADJ
cana-758	159	2	(	(	PUNCT
cana-758	159	3	4.2	4.2	NUM
cana-758	159	4	)	)	PUNCT
cana-758	159	5	:	:	PUNCT
cana-758	159	6	let	let	VERB
cana-758	159	7	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	159	8	)	)	PUNCT
cana-758	159	9	is	be	AUX
cana-758	159	10	belongs	belong	VERB
cana-758	159	11	in	in	ADP
cana-758	159	12	ʋ	ʋ	PROPN
cana-758	159	13	with	with	ADP
cana-758	159	14	ϥ(0	ϥ(0	PROPN
cana-758	159	15	)	)	PUNCT
cana-758	159	16	=	=	SYM
cana-758	160	1	1,where	1,where	X
cana-758	160	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	160	3	)	)	PUNCT
cana-758	160	4	is	be	AUX
cana-758	160	5	convex	convex	ADJ
cana-758	160	6	and	and	CCONJ
cana-758	160	7	univalent	univalent	ADJ
cana-758	160	8	,	,	PUNCT
cana-758	160	9	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	160	10	)	)	PUNCT
cana-758	160	11	>	>	X
cana-758	160	12	0	0	NUM
cana-758	160	13	,	,	PUNCT
cana-758	160	14	𝜉	𝜉	PROPN
cana-758	160	15	∈	∈	PROPN
cana-758	160	16	₵	₵	NOUN
cana-758	160	17	,	,	PUNCT
cana-758	160	18	𝜇	𝜇	ADP
cana-758	160	19	>	>	X
cana-758	160	20	0,such	0,such	PROPN
cana-758	160	21	that	that	SCONJ
cana-758	160	22	𝕦	𝕦	ADJ
cana-758	160	23	+	+	CCONJ
cana-758	160	24	𝕧	𝕧	X
cana-758	160	25	>	>	X
cana-758	160	26	0	0	NUM
cana-758	160	27	.	.	PUNCT
cana-758	161	1	if	if	SCONJ
cana-758	161	2	𝑓	𝑓	DET
cana-758	161	3	∈	∈	PROPN
cana-758	161	4	𝒢	𝒢	NOUN
cana-758	161	5	,	,	PUNCT
cana-758	161	6	where	where	SCONJ
cana-758	161	7	(	(	PUNCT
cana-758	161	8	𝒯𝕣,𝕤,𝕥,,𝕦,1	𝒯𝕣,𝕤,𝕥,,𝕦,1	NOUN
cana-758	161	9	𝕞	𝕞	X
cana-758	161	10	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	161	11	)	)	PUNCT
cana-758	161	12	𝓏	𝓏	PROPN
cana-758	161	13	)	)	PUNCT
cana-758	161	14	𝜇	𝜇	ADP
cana-758	161	15	∈	∈	PROPN
cana-758	161	16	𝑀[ϥ(0	𝑀[ϥ(0	PROPN
cana-758	161	17	)	)	PUNCT
cana-758	161	18	,	,	PUNCT
cana-758	161	19	1	1	X
cana-758	161	20	]	]	PUNCT
cana-758	161	21	∩	∩	ADJ
cana-758	161	22	𝚀	𝚀	NOUN
cana-758	161	23	.	.	PUNCT
cana-758	162	1	(	(	PUNCT
cana-758	162	2	4.8	4.8	NUM
cana-758	162	3	)	)	PUNCT
cana-758	162	4	if	if	SCONJ
cana-758	162	5	𝔇(𝕞	𝔇(𝕞	NOUN
cana-758	162	6	,	,	PUNCT
cana-758	162	7	𝜉	𝜉	PROPN
cana-758	162	8	,	,	PUNCT
cana-758	162	9	𝜇	𝜇	ADP
cana-758	162	10	,	,	PUNCT
cana-758	162	11	𝕦	𝕦	ADJ
cana-758	162	12	,	,	PUNCT
cana-758	162	13	1)is	1)is	NUM
cana-758	162	14	univalent	univalent	ADJ
cana-758	162	15	function	function	NOUN
cana-758	162	16	as	as	SCONJ
cana-758	162	17	defined	define	VERB
cana-758	162	18	by	by	ADP
cana-758	162	19	(	(	PUNCT
cana-758	162	20	3.3),and	3.3),and	NUM
cana-758	162	21	satisfies	satisfy	VERB
cana-758	162	22	the	the	DET
cana-758	162	23	superordination	superordination	NOUN
cana-758	162	24	case	case	NOUN
cana-758	162	25	below	below	ADV
cana-758	162	26	;	;	PUNCT
cana-758	162	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	162	28	)	)	PUNCT
cana-758	163	1	+	+	CCONJ
cana-758	163	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	163	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	NUM
cana-758	163	4	)	)	PUNCT
cana-758	163	5	𝓏	𝓏	PROPN
cana-758	163	6	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	163	7	)	)	PUNCT
cana-758	163	8	≺	≺	NOUN
cana-758	163	9	𝔇3(𝕞	𝔇3(𝕞	NUM
cana-758	163	10	,	,	PUNCT
cana-758	163	11	𝜉	𝜉	X
cana-758	163	12	,	,	PUNCT
cana-758	163	13	𝜇	𝜇	ADP
cana-758	163	14	,	,	PUNCT
cana-758	163	15	𝕦	𝕦	ADJ
cana-758	163	16	,	,	PUNCT
cana-758	163	17	1	1	NUM
cana-758	163	18	)	)	PUNCT
cana-758	163	19	,	,	PUNCT
cana-758	163	20	(	(	PUNCT
cana-758	163	21	4.9	4.9	NUM
cana-758	163	22	)	)	PUNCT
cana-758	163	23	where	where	SCONJ
cana-758	163	24	𝔇3(𝕞	𝔇3(𝕞	ADJ
cana-758	163	25	,	,	PUNCT
cana-758	163	26	𝜉	𝜉	X
cana-758	163	27	,	,	PUNCT
cana-758	163	28	𝜇	𝜇	ADP
cana-758	163	29	,	,	PUNCT
cana-758	163	30	𝕦	𝕦	ADJ
cana-758	163	31	,	,	PUNCT
cana-758	163	32	1	1	NUM
cana-758	163	33	)	)	PUNCT
cana-758	163	34	=	=	SYM
cana-758	163	35	(	(	PUNCT
cana-758	163	36	1	1	NUM
cana-758	163	37	−	−	NOUN
cana-758	163	38	𝜉	𝜉	NOUN
cana-758	163	39	)	)	PUNCT
cana-758	163	40	(	(	PUNCT
cana-758	163	41	𝒯𝕣,𝕤,𝕥,,𝕦1	𝒯𝕣,𝕤,𝕥,,𝕦1	PROPN
cana-758	163	42	𝕞	𝕞	DET
cana-758	163	43	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	163	44	)	)	PUNCT
cana-758	163	45	𝓏	𝓏	PROPN
cana-758	163	46	)	)	PUNCT
cana-758	163	47	𝜇	𝜇	ADP
cana-758	163	48	+	+	X
cana-758	163	49	𝜉	𝜉	X
cana-758	163	50	(	(	PUNCT
cana-758	163	51	𝒯𝕣,𝕤,𝕥,,𝕦,1	𝒯𝕣,𝕤,𝕥,,𝕦,1	NOUN
cana-758	163	52	𝕞	𝕞	X
cana-758	163	53	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	163	54	)	)	PUNCT
cana-758	163	55	𝓏	𝓏	PROPN
cana-758	163	56	)	)	PUNCT
cana-758	163	57	𝜇	𝜇	ADP
cana-758	163	58	+	+	X
cana-758	163	59	(	(	PUNCT
cana-758	163	60	𝒯𝕣,𝕤,𝕥,,𝕦,1	𝒯𝕣,𝕤,𝕥,,𝕦,1	NOUN
cana-758	163	61	𝕞+1	𝕞+1	NUM
cana-758	163	62	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	163	63	)	)	PUNCT
cana-758	163	64	𝒯𝕣,𝕤,𝕥,,𝕦,1	𝒯𝕣,𝕤,𝕥,,𝕦,1	NOUN
cana-758	163	65	𝕞	𝕞	X
cana-758	163	66	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	163	67	)	)	PUNCT
cana-758	163	68	)	)	PUNCT
cana-758	163	69	then	then	ADV
cana-758	163	70	ϥ(z	ϥ(z	PROPN
cana-758	163	71	)	)	PUNCT
cana-758	163	72	≺	≺	NOUN
cana-758	163	73	(	(	PUNCT
cana-758	163	74	𝒯𝕣,𝕤,𝕥,,𝕦,1	𝒯𝕣,𝕤,𝕥,,𝕦,1	NOUN
cana-758	163	75	𝕞	𝕞	X
cana-758	163	76	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	163	77	)	)	PUNCT
cana-758	163	78	𝓏	𝓏	PROPN
cana-758	163	79	)	)	PUNCT
cana-758	163	80	𝜇	𝜇	ADP
cana-758	163	81	,	,	PUNCT
cana-758	163	82	(	(	PUNCT
cana-758	163	83	4.10	4.10	NUM
cana-758	163	84	)	)	PUNCT
cana-758	163	85	and	and	CCONJ
cana-758	163	86	ϥ(z	ϥ(z	PROPN
cana-758	163	87	)	)	PUNCT
cana-758	163	88	will	will	AUX
cana-758	163	89	be	be	AUX
cana-758	163	90	best	good	ADJ
cana-758	163	91	subordination	subordination	NOUN
cana-758	163	92	.	.	PUNCT
cana-758	164	1	theorem	theorem	NOUN
cana-758	164	2	(	(	PUNCT
cana-758	164	3	4.2	4.2	NUM
cana-758	164	4	)	)	PUNCT
cana-758	164	5	:	:	PUNCT
cana-758	164	6	assume	assume	VERB
cana-758	164	7	that	that	SCONJ
cana-758	164	8	the	the	DET
cana-758	164	9	function	function	NOUN
cana-758	164	10	ϥ	ϥ	PROPN
cana-758	164	11	are	be	AUX
cana-758	164	12	univalent	univalent	ADJ
cana-758	164	13	and	and	CCONJ
cana-758	164	14	convex	convex	VERB
cana-758	164	15	in	in	ADP
cana-758	164	16	ʋ	ʋ	X
cana-758	164	17	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	VERB
cana-758	164	18	ϥ(0	ϥ(0	PROPN
cana-758	164	19	)	)	PUNCT
cana-758	165	1	=	=	SYM
cana-758	165	2	1	1	NUM
cana-758	165	3	,	,	PUNCT
cana-758	165	4	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	165	5	)	)	PUNCT
cana-758	165	6	>	>	X
cana-758	165	7	0	0	NUM
cana-758	165	8	,	,	PUNCT
cana-758	165	9	𝜉	𝜉	PROPN
cana-758	165	10	∈	∈	PROPN
cana-758	165	11	₵	₵	PROPN
cana-758	165	12	,	,	PUNCT
cana-758	165	13	𝜇	𝜇	ADV
cana-758	165	14	,	,	PUNCT
cana-758	165	15	𝕧	𝕧	PROPN
cana-758	165	16	>	>	X
cana-758	165	17	0,such	0,such	NOUN
cana-758	165	18	that	that	SCONJ
cana-758	165	19	𝕦	𝕦	ADJ
cana-758	165	20	+	+	CCONJ
cana-758	165	21	𝕧	𝕧	X
cana-758	165	22	>	>	X
cana-758	165	23	0	0	NUM
cana-758	165	24	.	.	PUNCT
cana-758	166	1	if	if	SCONJ
cana-758	166	2	𝑓	𝑓	DET
cana-758	166	3	∈	∈	PROPN
cana-758	166	4	𝒢	𝒢	NOUN
cana-758	166	5	,	,	PUNCT
cana-758	166	6	where	where	SCONJ
cana-758	166	7	(	(	PUNCT
cana-758	166	8	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	166	9	𝕞+1	𝕞+1	NUM
cana-758	166	10	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	166	11	)	)	PUNCT
cana-758	166	12	𝓏	𝓏	PROPN
cana-758	166	13	)	)	PUNCT
cana-758	166	14	𝜇	𝜇	ADP
cana-758	166	15	∈	∈	PROPN
cana-758	166	16	𝑀[ϥ(0	𝑀[ϥ(0	PROPN
cana-758	166	17	)	)	PUNCT
cana-758	166	18	,	,	PUNCT
cana-758	166	19	1	1	X
cana-758	166	20	]	]	PUNCT
cana-758	166	21	∩	∩	ADJ
cana-758	166	22	𝚀	𝚀	NOUN
cana-758	166	23	.	.	PUNCT
cana-758	167	1	(	(	PUNCT
cana-758	167	2	4.11	4.11	NUM
cana-758	167	3	)	)	PUNCT
cana-758	167	4	if	if	SCONJ
cana-758	167	5	𝔏(𝜇	𝔏(𝜇	NOUN
cana-758	167	6	,	,	PUNCT
cana-758	167	7	𝕞	𝕞	ADJ
cana-758	167	8	,	,	PUNCT
cana-758	167	9	𝕦	𝕦	ADJ
cana-758	167	10	,	,	PUNCT
cana-758	167	11	𝕧	𝕧	PROPN
cana-758	167	12	,	,	PUNCT
cana-758	167	13	𝜉)is	𝜉)is	ADJ
cana-758	167	14	univalent	univalent	ADJ
cana-758	167	15	function	function	NOUN
cana-758	167	16	as	as	SCONJ
cana-758	167	17	defined	define	VERB
cana-758	167	18	by	by	ADP
cana-758	167	19	(	(	PUNCT
cana-758	167	20	3.8),and	3.8),and	NUM
cana-758	167	21	satisfies	satisfy	VERB
cana-758	167	22	the	the	DET
cana-758	167	23	superordination	superordination	NOUN
cana-758	167	24	case	case	NOUN
cana-758	167	25	below	below	ADV
cana-758	167	26	;	;	PUNCT
cana-758	167	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	167	28	)	)	PUNCT
cana-758	168	1	+	+	CCONJ
cana-758	168	2	𝜉	𝜉	X
cana-758	168	3	𝜇	𝜇	X
cana-758	168	4	𝓏	𝓏	PROPN
cana-758	168	5	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	168	6	)	)	PUNCT
cana-758	168	7	≺	≺	NOUN
cana-758	168	8	𝔏(𝜇	𝔏(𝜇	NOUN
cana-758	168	9	,	,	PUNCT
cana-758	168	10	𝕞	𝕞	NOUN
cana-758	168	11	,	,	PUNCT
cana-758	168	12	𝕦	𝕦	ADJ
cana-758	168	13	,	,	PUNCT
cana-758	168	14	𝕧	𝕧	PROPN
cana-758	168	15	,	,	PUNCT
cana-758	168	16	𝜉	𝜉	NOUN
cana-758	168	17	)	)	PUNCT
cana-758	168	18	,	,	PUNCT
cana-758	168	19	(	(	PUNCT
cana-758	168	20	4.12	4.12	NUM
cana-758	168	21	)	)	PUNCT
cana-758	168	22	then	then	ADV
cana-758	168	23	ϥ(z	ϥ(z	PROPN
cana-758	168	24	)	)	PUNCT
cana-758	168	25	≺	≺	NOUN
cana-758	168	26	(	(	PUNCT
cana-758	168	27	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	168	28	𝕞+1	𝕞+1	NUM
cana-758	168	29	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	168	30	)	)	PUNCT
cana-758	168	31	𝓏	𝓏	PROPN
cana-758	168	32	)	)	PUNCT
cana-758	168	33	𝜇	𝜇	ADP
cana-758	168	34	,	,	PUNCT
cana-758	168	35	(	(	PUNCT
cana-758	168	36	4.13	4.13	NUM
cana-758	168	37	)	)	PUNCT
cana-758	168	38	and	and	CCONJ
cana-758	168	39	ϥ(z	ϥ(z	PROPN
cana-758	168	40	)	)	PUNCT
cana-758	168	41	will	will	AUX
cana-758	168	42	be	be	AUX
cana-758	168	43	best	good	ADJ
cana-758	168	44	subordination	subordination	NOUN
cana-758	168	45	.	.	PUNCT
cana-758	169	1	communications	communication	NOUN
cana-758	169	2	on	on	ADP
cana-758	169	3	applied	apply	VERB
cana-758	169	4	nonlinear	nonlinear	ADJ
cana-758	169	5	analysis	analysis	NOUN
cana-758	169	6	issn	issn	NOUN
cana-758	169	7	:	:	PUNCT
cana-758	169	8	1074	1074	NUM
cana-758	169	9	-	-	PUNCT
cana-758	169	10	133x	133x	NUM
cana-758	169	11	vol	vol	NOUN
cana-758	169	12	31	31	NUM
cana-758	169	13	no	no	NOUN
cana-758	169	14	.	.	PUNCT
cana-758	170	1	3s	3s	NUM
cana-758	170	2	(	(	PUNCT
cana-758	170	3	2024	2024	NUM
cana-758	170	4	)	)	PUNCT
cana-758	170	5	194	194	NUM
cana-758	170	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	170	7	thus	thus	ADV
cana-758	170	8	by	by	ADP
cana-758	170	9	taking	take	VERB
cana-758	170	10	lemma	lemma	PROPN
cana-758	170	11	(	(	PUNCT
cana-758	170	12	2.3	2.3	NUM
cana-758	170	13	)	)	PUNCT
cana-758	170	14	,	,	PUNCT
cana-758	170	15	we	we	PRON
cana-758	170	16	obtain	obtain	VERB
cana-758	170	17	result	result	NOUN
cana-758	170	18	that	that	SCONJ
cana-758	170	19	required	require	VERB
cana-758	170	20	.	.	PUNCT
cana-758	171	1	now	now	ADV
cana-758	171	2	,	,	PUNCT
cana-758	171	3	in	in	ADP
cana-758	171	4	theorem	theorem	NOUN
cana-758	171	5	above	above	ADV
cana-758	171	6	,	,	PUNCT
cana-758	171	7	we	we	PRON
cana-758	171	8	put	put	VERB
cana-758	171	9	,	,	PUNCT
cana-758	171	10	𝕧	𝕧	PROPN
cana-758	171	11	=	=	SYM
cana-758	171	12	1	1	NUM
cana-758	171	13	,	,	PUNCT
cana-758	171	14	so	so	ADV
cana-758	171	15	to	to	PART
cana-758	171	16	obtain	obtain	VERB
cana-758	171	17	corollary	corollary	ADJ
cana-758	171	18	below	below	ADV
cana-758	171	19	.	.	PUNCT
cana-758	172	1	corollary	corollary	ADJ
cana-758	172	2	(	(	PUNCT
cana-758	172	3	4.3	4.3	NUM
cana-758	172	4	)	)	PUNCT
cana-758	172	5	:	:	PUNCT
cana-758	172	6	let	let	VERB
cana-758	172	7	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	172	8	)	)	PUNCT
cana-758	172	9	is	be	AUX
cana-758	172	10	belongs	belong	VERB
cana-758	172	11	in	in	ADP
cana-758	172	12	ʋ	ʋ	PROPN
cana-758	172	13	with	with	ADP
cana-758	172	14	ϥ(0	ϥ(0	PROPN
cana-758	172	15	)	)	PUNCT
cana-758	172	16	=	=	SYM
cana-758	173	1	1,where	1,where	X
cana-758	173	2	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	173	3	)	)	PUNCT
cana-758	173	4	is	be	AUX
cana-758	173	5	convex	convex	ADJ
cana-758	173	6	and	and	CCONJ
cana-758	173	7	univalent	univalent	ADJ
cana-758	173	8	,	,	PUNCT
cana-758	173	9	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	173	10	)	)	PUNCT
cana-758	173	11	>	>	X
cana-758	173	12	0	0	NUM
cana-758	173	13	,	,	PUNCT
cana-758	173	14	𝜉	𝜉	PROPN
cana-758	173	15	∈	∈	PROPN
cana-758	173	16	₵	₵	NOUN
cana-758	173	17	,	,	PUNCT
cana-758	173	18	𝜇	𝜇	ADP
cana-758	173	19	>	>	X
cana-758	173	20	0	0	PROPN
cana-758	173	21	,	,	PUNCT
cana-758	173	22	𝕦	𝕦	ADJ
cana-758	173	23	𝑏𝑒	𝑏𝑒	ADP
cana-758	173	24	real	real	ADJ
cana-758	173	25	number	number	NOUN
cana-758	173	26	.	.	PUNCT
cana-758	174	1	if	if	SCONJ
cana-758	174	2	𝑓	𝑓	DET
cana-758	174	3	∈	∈	PROPN
cana-758	174	4	𝒢	𝒢	NOUN
cana-758	174	5	,	,	PUNCT
cana-758	174	6	where	where	SCONJ
cana-758	174	7	(	(	PUNCT
cana-758	174	8	𝔩𝕣,𝕤,𝕥,,𝕦,1	𝔩𝕣,𝕤,𝕥,,𝕦,1	NUM
cana-758	174	9	𝕞+1	𝕞+1	NUM
cana-758	174	10	𝑓(𝓏	𝑓(𝓏	X
cana-758	174	11	)	)	PUNCT
cana-758	174	12	𝓏	𝓏	PROPN
cana-758	174	13	)	)	PUNCT
cana-758	174	14	𝜇	𝜇	ADP
cana-758	174	15	∈	∈	PROPN
cana-758	174	16	𝑀[ϥ(0	𝑀[ϥ(0	PROPN
cana-758	174	17	)	)	PUNCT
cana-758	174	18	,	,	PUNCT
cana-758	174	19	1	1	X
cana-758	174	20	]	]	PUNCT
cana-758	174	21	∩	∩	ADJ
cana-758	174	22	𝚀	𝚀	NOUN
cana-758	174	23	.	.	PUNCT
cana-758	175	1	(	(	PUNCT
cana-758	175	2	4.14	4.14	NUM
cana-758	175	3	)	)	PUNCT
cana-758	175	4	if	if	SCONJ
cana-758	175	5	𝔏1(𝜇	𝔏1(𝜇	PROPN
cana-758	175	6	,	,	PUNCT
cana-758	175	7	𝕞	𝕞	PROPN
cana-758	175	8	,	,	PUNCT
cana-758	175	9	𝕦	𝕦	ADJ
cana-758	175	10	,	,	PUNCT
cana-758	175	11	1	1	NUM
cana-758	175	12	,	,	PUNCT
cana-758	175	13	𝜉)is	𝜉)is	ADJ
cana-758	175	14	univalent	univalent	ADJ
cana-758	175	15	function	function	NOUN
cana-758	175	16	as	as	SCONJ
cana-758	175	17	defined	define	VERB
cana-758	175	18	by	by	ADP
cana-758	175	19	(	(	PUNCT
cana-758	175	20	3.8),and	3.8),and	NUM
cana-758	175	21	satisfies	satisfy	VERB
cana-758	175	22	the	the	DET
cana-758	175	23	superordination	superordination	NOUN
cana-758	175	24	case	case	NOUN
cana-758	175	25	below	below	ADV
cana-758	175	26	;	;	PUNCT
cana-758	175	27	ϥ(𝓏	ϥ(𝓏	PROPN
cana-758	175	28	)	)	PUNCT
cana-758	176	1	+	+	CCONJ
cana-758	176	2	𝜉	𝜉	X
cana-758	176	3	𝜇	𝜇	X
cana-758	176	4	𝓏	𝓏	PROPN
cana-758	176	5	ϥ′(𝓏	ϥ′(𝓏	NOUN
cana-758	176	6	)	)	PUNCT
cana-758	176	7	≺	≺	NOUN
cana-758	176	8	𝔏1(𝜇	𝔏1(𝜇	PROPN
cana-758	176	9	,	,	PUNCT
cana-758	176	10	𝕞	𝕞	PROPN
cana-758	176	11	,	,	PUNCT
cana-758	176	12	𝕦	𝕦	ADJ
cana-758	176	13	,	,	PUNCT
cana-758	176	14	1	1	NUM
cana-758	176	15	,	,	PUNCT
cana-758	176	16	𝜉	𝜉	NOUN
cana-758	176	17	)	)	PUNCT
cana-758	176	18	,	,	PUNCT
cana-758	176	19	(	(	PUNCT
cana-758	176	20	4.15	4.15	NUM
cana-758	176	21	)	)	PUNCT
cana-758	176	22	then	then	ADV
cana-758	176	23	ϥ(z	ϥ(z	PROPN
cana-758	176	24	)	)	PUNCT
cana-758	176	25	≺	≺	NOUN
cana-758	176	26	(	(	PUNCT
cana-758	176	27	𝔩𝕣,𝕤,𝕥,,𝕦,1	𝔩𝕣,𝕤,𝕥,,𝕦,1	X
cana-758	176	28	𝕞+1	𝕞+1	NUM
cana-758	176	29	𝑓(𝓏	𝑓(𝓏	X
cana-758	176	30	)	)	PUNCT
cana-758	176	31	𝓏	𝓏	PROPN
cana-758	176	32	)	)	PUNCT
cana-758	176	33	𝜇	𝜇	ADP
cana-758	176	34	,	,	PUNCT
cana-758	176	35	(	(	PUNCT
cana-758	176	36	4.16	4.16	NUM
cana-758	176	37	)	)	PUNCT
cana-758	176	38	and	and	CCONJ
cana-758	176	39	ϥ(z	ϥ(z	PROPN
cana-758	176	40	)	)	PUNCT
cana-758	176	41	will	will	AUX
cana-758	176	42	be	be	AUX
cana-758	176	43	best	good	ADJ
cana-758	176	44	subordination	subordination	NOUN
cana-758	176	45	.	.	PUNCT
cana-758	177	1	thus	thus	ADV
cana-758	177	2	by	by	ADP
cana-758	177	3	taking	take	VERB
cana-758	177	4	lemma	lemma	PROPN
cana-758	177	5	(	(	PUNCT
cana-758	177	6	2.3	2.3	NUM
cana-758	177	7	)	)	PUNCT
cana-758	177	8	,	,	PUNCT
cana-758	177	9	we	we	PRON
cana-758	177	10	obtain	obtain	VERB
cana-758	177	11	result	result	NOUN
cana-758	177	12	that	that	SCONJ
cana-758	177	13	required	require	VERB
cana-758	177	14	.	.	PUNCT
cana-758	178	1	5	5	X
cana-758	178	2	.	.	X
cana-758	178	3	sandwich	sandwich	NOUN
cana-758	178	4	results	result	NOUN
cana-758	178	5	joining	join	VERB
cana-758	178	6	theorems	theorem	NOUN
cana-758	178	7	(	(	PUNCT
cana-758	178	8	3.1	3.1	NUM
cana-758	178	9	)	)	PUNCT
cana-758	178	10	and	and	CCONJ
cana-758	178	11	(	(	PUNCT
cana-758	178	12	4.1	4.1	NUM
cana-758	178	13	)	)	PUNCT
cana-758	178	14	,	,	PUNCT
cana-758	178	15	in	in	ADP
cana-758	178	16	order	order	NOUN
cana-758	178	17	to	to	PART
cana-758	178	18	get	get	VERB
cana-758	178	19	sandwich	sandwich	NOUN
cana-758	178	20	theorem	theorem	NOUN
cana-758	178	21	(	(	PUNCT
cana-758	178	22	5.1	5.1	NUM
cana-758	178	23	)	)	PUNCT
cana-758	178	24	.	.	PUNCT
cana-758	179	1	theorem	theorem	NOUN
cana-758	179	2	(	(	PUNCT
cana-758	179	3	5.1	5.1	NUM
cana-758	179	4	)	)	PUNCT
cana-758	179	5	:	:	PUNCT
cana-758	179	6	assume	assume	VERB
cana-758	179	7	the	the	DET
cana-758	179	8	two	two	NUM
cana-758	179	9	convex	convex	NOUN
cana-758	179	10	functions	function	NOUN
cana-758	179	11	in	in	ADP
cana-758	179	12	ʋ	ʋ	PRON
cana-758	179	13	sayϥ1(𝓏	sayϥ1(𝓏	NOUN
cana-758	179	14	)	)	PUNCT
cana-758	179	15	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-758	179	16	ϥ2(𝓏	ϥ2(𝓏	PROPN
cana-758	179	17	)	)	PUNCT
cana-758	179	18	together	together	ADV
cana-758	179	19	with	with	ADP
cana-758	179	20	ϥ1(0	ϥ1(0	PROPN
cana-758	179	21	)	)	PUNCT
cana-758	179	22	=	=	PUNCT
cana-758	179	23	ϥ2(0	ϥ2(0	NOUN
cana-758	179	24	)	)	PUNCT
cana-758	179	25	=	=	PUNCT
cana-758	179	26	1.suppose	1.suppose	NUM
cana-758	179	27	that	that	PRON
cana-758	179	28	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	179	29	)	)	PUNCT
cana-758	179	30	>	>	X
cana-758	179	31	0	0	NUM
cana-758	179	32	,	,	PUNCT
cana-758	179	33	𝜉	𝜉	PROPN
cana-758	179	34	∈	∈	PROPN
cana-758	179	35	₵	₵	PROPN
cana-758	179	36	,	,	PUNCT
cana-758	179	37	𝜇	𝜇	ADV
cana-758	179	38	,	,	PUNCT
cana-758	179	39	𝕧	𝕧	PROPN
cana-758	179	40	>	>	X
cana-758	179	41	0	0	NUM
cana-758	179	42	,	,	PUNCT
cana-758	179	43	𝕦	𝕦	ADV
cana-758	179	44	be	be	VERB
cana-758	179	45	real	real	ADJ
cana-758	179	46	number	number	NOUN
cana-758	179	47	such	such	ADJ
cana-758	179	48	that	that	SCONJ
cana-758	179	49	𝕦	𝕦	PROPN
cana-758	179	50	+	+	X
cana-758	179	51	𝕧	𝕧	X
cana-758	179	52	>	>	X
cana-758	179	53	0	0	NUM
cana-758	179	54	.	.	PUNCT
cana-758	179	55	.i𝑓	.i𝑓	PUNCT
cana-758	180	1	𝑓	𝑓	DET
cana-758	180	2	∈	∈	PROPN
cana-758	180	3	𝒢	𝒢	PROPN
cana-758	180	4	,	,	PUNCT
cana-758	180	5	where	where	SCONJ
cana-758	180	6	(	(	PUNCT
cana-758	180	7	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	180	8	𝕞	𝕞	X
cana-758	180	9	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	180	10	)	)	PUNCT
cana-758	180	11	𝓏	𝓏	PROPN
cana-758	180	12	)	)	PUNCT
cana-758	180	13	𝜇	𝜇	ADP
cana-758	180	14	∈	∈	PROPN
cana-758	180	15	𝑀[ϥ(0	𝑀[ϥ(0	PROPN
cana-758	180	16	)	)	PUNCT
cana-758	180	17	,	,	PUNCT
cana-758	180	18	1	1	X
cana-758	180	19	]	]	PUNCT
cana-758	180	20	∩	∩	ADJ
cana-758	180	21	𝚀	𝚀	NOUN
cana-758	180	22	,	,	PUNCT
cana-758	180	23	and	and	CCONJ
cana-758	180	24	𝔇(𝕞	𝔇(𝕞	NUM
cana-758	180	25	,	,	PUNCT
cana-758	180	26	𝜉	𝜉	PROPN
cana-758	180	27	,	,	PUNCT
cana-758	180	28	𝜇	𝜇	ADP
cana-758	180	29	,	,	PUNCT
cana-758	180	30	𝕦	𝕦	ADJ
cana-758	180	31	,	,	PUNCT
cana-758	180	32	𝕧	𝕧	NOUN
cana-758	180	33	)	)	PUNCT
cana-758	180	34	which	which	PRON
cana-758	180	35	is	be	AUX
cana-758	180	36	given	give	VERB
cana-758	180	37	by	by	ADP
cana-758	180	38	(	(	PUNCT
cana-758	180	39	3.3	3.3	NUM
cana-758	180	40	)	)	PUNCT
cana-758	180	41	be	be	AUX
cana-758	180	42	univalent	univalent	ADJ
cana-758	180	43	function	function	NOUN
cana-758	180	44	and	and	CCONJ
cana-758	180	45	holds	hold	VERB
cana-758	180	46	ϥ1(𝓏	ϥ1(𝓏	NOUN
cana-758	180	47	)	)	PUNCT
cana-758	180	48	+	+	CCONJ
cana-758	180	49	𝜉𝕧	𝜉𝕧	ADP
cana-758	180	50	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADV
cana-758	180	51	)	)	PUNCT
cana-758	180	52	𝓏ϥ′	𝓏ϥ′	NOUN
cana-758	180	53	1	1	NUM
cana-758	180	54	(	(	PUNCT
cana-758	180	55	𝓏	𝓏	NOUN
cana-758	180	56	)	)	PUNCT
cana-758	180	57	≺	≺	NOUN
cana-758	180	58	𝔇(𝕞	𝔇(𝕞	NOUN
cana-758	180	59	,	,	PUNCT
cana-758	180	60	𝜉	𝜉	PROPN
cana-758	180	61	,	,	PUNCT
cana-758	180	62	𝜇	𝜇	ADP
cana-758	180	63	,	,	PUNCT
cana-758	180	64	𝕦	𝕦	ADJ
cana-758	180	65	,	,	PUNCT
cana-758	180	66	𝕧	𝕧	NOUN
cana-758	180	67	)	)	PUNCT
cana-758	180	68	≺	≺	NOUN
cana-758	180	69	ϥ2(𝓏	ϥ2(𝓏	SYM
cana-758	180	70	)	)	PUNCT
cana-758	181	1	+	+	CCONJ
cana-758	181	2	𝜉𝕧	𝜉𝕧	ADP
cana-758	181	3	𝜇(𝕦+𝕧	𝜇(𝕦+𝕧	ADV
cana-758	181	4	)	)	PUNCT
cana-758	181	5	𝓏ϥ′	𝓏ϥ′	NOUN
cana-758	181	6	2	2	NUM
cana-758	181	7	(	(	PUNCT
cana-758	181	8	𝓏	𝓏	NOUN
cana-758	181	9	)	)	PUNCT
cana-758	181	10	,	,	PUNCT
cana-758	181	11	(	(	PUNCT
cana-758	181	12	5.1	5.1	NUM
cana-758	181	13	)	)	PUNCT
cana-758	181	14	implies	imply	VERB
cana-758	181	15	ϥ1(𝓏	ϥ1(𝓏	NOUN
cana-758	181	16	)	)	PUNCT
cana-758	181	17	≺	≺	NOUN
cana-758	181	18	(	(	PUNCT
cana-758	181	19	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	181	20	𝕞	𝕞	X
cana-758	181	21	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	181	22	)	)	PUNCT
cana-758	181	23	𝓏	𝓏	PROPN
cana-758	181	24	)	)	PUNCT
cana-758	181	25	𝜇	𝜇	ADP
cana-758	181	26	≺	≺	NOUN
cana-758	181	27	ϥ2(𝓏),with	ϥ2(𝓏),with	ADP
cana-758	181	28	ϥ1(𝓏	ϥ1(𝓏	NOUN
cana-758	181	29	)	)	PUNCT
cana-758	181	30	best	good	ADJ
cana-758	181	31	subordinant	subordinant	NOUN
cana-758	181	32	and	and	CCONJ
cana-758	181	33	ϥ2(𝓏	ϥ2(𝓏	NOUN
cana-758	181	34	)	)	PUNCT
cana-758	181	35	best	well	ADV
cana-758	181	36	dominant	dominant	ADJ
cana-758	181	37	(	(	PUNCT
cana-758	181	38	5.1	5.1	NUM
cana-758	181	39	)	)	PUNCT
cana-758	181	40	respectively	respectively	ADV
cana-758	181	41	.	.	PUNCT
cana-758	182	1	joining	join	VERB
cana-758	182	2	theorems	theorem	NOUN
cana-758	182	3	(	(	PUNCT
cana-758	182	4	3.2	3.2	NUM
cana-758	182	5	)	)	PUNCT
cana-758	182	6	and	and	CCONJ
cana-758	182	7	(	(	PUNCT
cana-758	182	8	4.2	4.2	NUM
cana-758	182	9	)	)	PUNCT
cana-758	182	10	,	,	PUNCT
cana-758	182	11	in	in	ADP
cana-758	182	12	order	order	NOUN
cana-758	182	13	to	to	PART
cana-758	182	14	get	get	VERB
cana-758	182	15	sandwich	sandwich	NOUN
cana-758	182	16	theorem	theorem	NOUN
cana-758	182	17	(	(	PUNCT
cana-758	182	18	5.2	5.2	NUM
cana-758	182	19	)	)	PUNCT
cana-758	182	20	.	.	PUNCT
cana-758	183	1	theorem	theorem	NOUN
cana-758	183	2	(	(	PUNCT
cana-758	183	3	5.2	5.2	NUM
cana-758	183	4	)	)	PUNCT
cana-758	183	5	:	:	PUNCT
cana-758	183	6	assume	assume	VERB
cana-758	183	7	the	the	DET
cana-758	183	8	two	two	NUM
cana-758	183	9	univalent	univalent	ADJ
cana-758	183	10	convex	convex	NOUN
cana-758	183	11	functions	function	NOUN
cana-758	183	12	in	in	ADP
cana-758	183	13	ʋ	ʋ	PRON
cana-758	183	14	sayϥ1(𝓏	sayϥ1(𝓏	NOUN
cana-758	183	15	)	)	PUNCT
cana-758	183	16	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-758	183	17	ϥ2(𝓏	ϥ2(𝓏	PROPN
cana-758	183	18	)	)	PUNCT
cana-758	183	19	together	together	ADV
cana-758	183	20	with	with	ADP
cana-758	183	21	ϥ1(0	ϥ1(0	PROPN
cana-758	183	22	)	)	PUNCT
cana-758	183	23	=	=	PUNCT
cana-758	183	24	ϥ2(0	ϥ2(0	NOUN
cana-758	183	25	)	)	PUNCT
cana-758	183	26	=	=	SYM
cana-758	184	1	1	1	X
cana-758	184	2	.	.	PUNCT
cana-758	184	3	suppose	suppose	VERB
cana-758	184	4	that	that	SCONJ
cana-758	184	5	𝑅𝑒(𝜉	𝑅𝑒(𝜉	PROPN
cana-758	184	6	)	)	PUNCT
cana-758	184	7	>	>	X
cana-758	184	8	0	0	NUM
cana-758	184	9	,	,	PUNCT
cana-758	184	10	𝜉	𝜉	PROPN
cana-758	184	11	∈	∈	PROPN
cana-758	184	12	₵	₵	PROPN
cana-758	184	13	,	,	PUNCT
cana-758	184	14	𝜇	𝜇	ADV
cana-758	184	15	,	,	PUNCT
cana-758	184	16	𝕧	𝕧	PROPN
cana-758	184	17	>	>	X
cana-758	184	18	0	0	NUM
cana-758	184	19	,	,	PUNCT
cana-758	184	20	𝕦	𝕦	ADJ
cana-758	184	21	real	real	ADJ
cana-758	184	22	number	number	NOUN
cana-758	184	23	such	such	ADJ
cana-758	184	24	that	that	SCONJ
cana-758	184	25	𝕦	𝕦	PROPN
cana-758	184	26	+	+	X
cana-758	184	27	𝕧	𝕧	X
cana-758	184	28	>	>	X
cana-758	184	29	0	0	NUM
cana-758	184	30	.	.	PUNCT
cana-758	184	31	.i𝑓	.i𝑓	PUNCT
cana-758	185	1	𝑓	𝑓	DET
cana-758	185	2	∈	∈	PROPN
cana-758	185	3	𝒢	𝒢	PROPN
cana-758	185	4	,	,	PUNCT
cana-758	185	5	where	where	SCONJ
cana-758	185	6	(	(	PUNCT
cana-758	185	7	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	185	8	𝕞	𝕞	PRON
cana-758	185	9	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	185	10	)	)	PUNCT
cana-758	185	11	𝓏	𝓏	PROPN
cana-758	185	12	)	)	PUNCT
cana-758	185	13	𝜇	𝜇	ADP
cana-758	185	14	∈	∈	PROPN
cana-758	185	15	𝑀[ϥ(0),1	𝑀[ϥ(0),1	NOUN
cana-758	185	16	]	]	PUNCT
cana-758	185	17	∩	∩	PROPN
cana-758	185	18	𝚀	𝚀	PROPN
cana-758	185	19	,	,	PUNCT
cana-758	185	20	assume	assume	VERB
cana-758	185	21	univalent	univalent	ADJ
cana-758	185	22	function	function	NOUN
cana-758	185	23	in	in	ADP
cana-758	185	24	ʋ	ʋ	PRON
cana-758	185	25	say	say	VERB
cana-758	185	26	𝔏(𝜇	𝔏(𝜇	ADP
cana-758	185	27	,	,	PUNCT
cana-758	185	28	𝕞	𝕞	NOUN
cana-758	185	29	,	,	PUNCT
cana-758	185	30	𝕦	𝕦	ADJ
cana-758	185	31	,	,	PUNCT
cana-758	185	32	𝕧	𝕧	PROPN
cana-758	185	33	,	,	PUNCT
cana-758	185	34	𝜉	𝜉	NOUN
cana-758	185	35	)	)	PUNCT
cana-758	185	36	,	,	PUNCT
cana-758	185	37	then	then	ADV
cana-758	185	38	ϥ1(𝓏	ϥ1(𝓏	NOUN
cana-758	185	39	)	)	PUNCT
cana-758	186	1	+	+	CCONJ
cana-758	186	2	𝜉	𝜉	X
cana-758	186	3	𝜇	𝜇	X
cana-758	186	4	ϥ2	ϥ2	PROPN
cana-758	186	5	1	1	NUM
cana-758	186	6	(	(	PUNCT
cana-758	186	7	𝓏	𝓏	NOUN
cana-758	186	8	)	)	PUNCT
cana-758	186	9	≺	≺	NOUN
cana-758	186	10	𝔏(𝜇	𝔏(𝜇	NOUN
cana-758	186	11	,	,	PUNCT
cana-758	186	12	𝕞	𝕞	NOUN
cana-758	186	13	,	,	PUNCT
cana-758	186	14	𝕦	𝕦	ADJ
cana-758	186	15	,	,	PUNCT
cana-758	186	16	𝕧	𝕧	PROPN
cana-758	186	17	,	,	PUNCT
cana-758	186	18	𝜉	𝜉	NOUN
cana-758	186	19	)	)	PUNCT
cana-758	186	20	≺	≺	NOUN
cana-758	186	21	ϥ2(𝓏	ϥ2(𝓏	SYM
cana-758	186	22	)	)	PUNCT
cana-758	187	1	+	+	CCONJ
cana-758	187	2	𝜉	𝜉	SYM
cana-758	187	3	𝜇	𝜇	ADP
cana-758	187	4	𝓏ϥ2	𝓏ϥ2	NOUN
cana-758	187	5	′	′	NUM
cana-758	187	6	(	(	PUNCT
cana-758	187	7	𝓏	𝓏	NOUN
cana-758	187	8	)	)	PUNCT
cana-758	187	9	.	.	PUNCT
cana-758	188	1	(	(	PUNCT
cana-758	188	2	5.2	5.2	NUM
cana-758	188	3	)	)	PUNCT
cana-758	188	4	communications	communication	NOUN
cana-758	188	5	on	on	ADP
cana-758	188	6	applied	apply	VERB
cana-758	188	7	nonlinear	nonlinear	ADJ
cana-758	188	8	analysis	analysis	NOUN
cana-758	188	9	issn	issn	NOUN
cana-758	188	10	:	:	PUNCT
cana-758	188	11	1074	1074	NUM
cana-758	188	12	-	-	PUNCT
cana-758	188	13	133x	133x	NUM
cana-758	188	14	vol	vol	NOUN
cana-758	188	15	31	31	NUM
cana-758	188	16	no	no	NOUN
cana-758	188	17	.	.	PUNCT
cana-758	189	1	3s	3s	NUM
cana-758	189	2	(	(	PUNCT
cana-758	189	3	2024	2024	NUM
cana-758	189	4	)	)	PUNCT
cana-758	189	5	195	195	NUM
cana-758	189	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	189	7	implies	imply	VERB
cana-758	189	8	ϥ1(𝓏	ϥ1(𝓏	NOUN
cana-758	189	9	)	)	PUNCT
cana-758	189	10	≺	≺	NOUN
cana-758	189	11	(	(	PUNCT
cana-758	189	12	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	189	13	𝕞	𝕞	PRON
cana-758	189	14	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	189	15	)	)	PUNCT
cana-758	189	16	𝓏	𝓏	PROPN
cana-758	189	17	)	)	PUNCT
cana-758	189	18	𝜇	𝜇	ADP
cana-758	189	19	≺	≺	NOUN
cana-758	189	20	ϥ2(𝓏	ϥ2(𝓏	SYM
cana-758	189	21	)	)	PUNCT
cana-758	189	22	,	,	PUNCT
cana-758	189	23	and	and	CCONJ
cana-758	189	24	with	with	ADP
cana-758	189	25	ϥ1(𝓏	ϥ1(𝓏	NOUN
cana-758	189	26	)	)	PUNCT
cana-758	189	27	best	good	ADJ
cana-758	189	28	subordinant	subordinant	NOUN
cana-758	189	29	and	and	CCONJ
cana-758	189	30	ϥ2(𝓏	ϥ2(𝓏	NOUN
cana-758	189	31	)	)	PUNCT
cana-758	189	32	best	well	ADV
cana-758	189	33	dominant	dominant	ADJ
cana-758	189	34	(	(	PUNCT
cana-758	189	35	5.2	5.2	NUM
cana-758	189	36	)	)	PUNCT
cana-758	189	37	respectively	respectively	ADV
cana-758	189	38	.	.	PUNCT
cana-758	190	1	6	6	X
cana-758	190	2	.	.	X
cana-758	190	3	conclusions	conclusion	NOUN
cana-758	190	4	the	the	DET
cana-758	190	5	main	main	ADJ
cana-758	190	6	aim	aim	NOUN
cana-758	190	7	of	of	ADP
cana-758	190	8	our	our	PRON
cana-758	190	9	present	present	ADJ
cana-758	190	10	work	work	NOUN
cana-758	190	11	is	be	AUX
cana-758	190	12	dedecated	dedecate	VERB
cana-758	190	13	to	to	PART
cana-758	190	14	give	give	VERB
cana-758	190	15	a	a	DET
cana-758	190	16	new	new	ADJ
cana-758	190	17	results	result	NOUN
cana-758	190	18	connected	connect	VERB
cana-758	190	19	by	by	ADP
cana-758	190	20	new	new	ADJ
cana-758	190	21	operators	operator	NOUN
cana-758	190	22	,	,	PUNCT
cana-758	190	23	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	𝒯𝕣,𝕤,𝕥,,𝕦,𝕧	PROPN
cana-758	190	24	𝕞	𝕞	PRON
cana-758	190	25	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	190	26	)	)	PUNCT
cana-758	190	27	and	and	CCONJ
cana-758	190	28	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	𝔩𝕣,𝕤,𝕥,,𝕦,𝕧	NUM
cana-758	190	29	𝕞	𝕞	DET
cana-758	190	30	𝑓(𝓏	𝑓(𝓏	PROPN
cana-758	190	31	)	)	PUNCT
cana-758	190	32	say	say	VERB
cana-758	190	33	linear	linear	ADJ
cana-758	190	34	and	and	CCONJ
cana-758	190	35	integral	integral	ADJ
cana-758	190	36	operators	operator	NOUN
cana-758	190	37	respectively	respectively	ADV
cana-758	190	38	for	for	ADP
cana-758	190	39	univalent	univalent	ADJ
cana-758	190	40	function	function	NOUN
cana-758	190	41	in	in	ADP
cana-758	190	42	open	open	ADJ
cana-758	190	43	unit	unit	NOUN
cana-758	190	44	disc	disc	NOUN
cana-758	190	45	ʋ	ʋ	PROPN
cana-758	190	46	,	,	PUNCT
cana-758	190	47	by	by	ADP
cana-758	190	48	using	use	VERB
cana-758	190	49	differential	differential	ADJ
cana-758	190	50	superordinations	superordination	NOUN
cana-758	190	51	and	and	CCONJ
cana-758	190	52	subordinations	subordination	NOUN
cana-758	190	53	.	.	PUNCT
cana-758	191	1	the	the	DET
cana-758	191	2	introduced	introduce	VERB
cana-758	191	3	results	result	NOUN
cana-758	191	4	have	have	VERB
cana-758	191	5	properties	property	NOUN
cana-758	191	6	of	of	ADP
cana-758	191	7	differential	differential	ADJ
cana-758	191	8	subordinations	subordination	NOUN
cana-758	191	9	which	which	PRON
cana-758	191	10	are	be	AUX
cana-758	191	11	analogous	analogous	ADJ
cana-758	191	12	to	to	PART
cana-758	191	13	differential	differential	VERB
cana-758	191	14	superordination	superordination	NOUN
cana-758	191	15	properties	property	NOUN
cana-758	191	16	in	in	ADP
cana-758	191	17	sandwich	sandwich	NOUN
cana-758	191	18	theorm	theorm	NOUN
cana-758	191	19	,	,	PUNCT
cana-758	191	20	in	in	ADP
cana-758	191	21	supplement	supplement	NOUN
cana-758	191	22	to	to	ADP
cana-758	191	23	that	that	PRON
cana-758	191	24	the	the	DET
cana-758	191	25	results	result	NOUN
cana-758	191	26	of	of	ADP
cana-758	191	27	this	this	DET
cana-758	191	28	work	work	NOUN
cana-758	191	29	include	include	VERB
cana-758	191	30	with	with	ADP
cana-758	191	31	a	a	DET
cana-758	191	32	new	new	ADJ
cana-758	191	33	ideas	idea	NOUN
cana-758	191	34	,	,	PUNCT
cana-758	191	35	which	which	PRON
cana-758	191	36	can	can	AUX
cana-758	191	37	be	be	AUX
cana-758	191	38	applied	apply	VERB
cana-758	191	39	on	on	ADP
cana-758	191	40	analytic	analytic	ADJ
cana-758	191	41	and	and	CCONJ
cana-758	191	42	multivalent	multivalent	NOUN
cana-758	191	43	functions	function	NOUN
cana-758	191	44	theory	theory	NOUN
cana-758	191	45	.	.	PUNCT
cana-758	192	1	refrences	refrence	VERB
cana-758	193	1	[	[	X
cana-758	193	2	1	1	X
cana-758	193	3	]	]	PUNCT
cana-758	193	4	s.	s.	PROPN
cana-758	193	5	a.	a.	PROPN
cana-758	193	6	alameedee	alameedee	PROPN
cana-758	193	7	,	,	PUNCT
cana-758	193	8	w.	w.	PROPN
cana-758	193	9	g.	g.	PROPN
cana-758	193	10	atshan	atshan	PROPN
cana-758	193	11	and	and	CCONJ
cana-758	193	12	f.	f.	PROPN
cana-758	193	13	a.	a.	PROPN
cana-758	193	14	al	al	PROPN
cana-758	193	15	-	-	PUNCT
cana-758	193	16	maamori	maamori	PROPN
cana-758	193	17	,	,	PUNCT
cana-758	193	18	on	on	ADP
cana-758	193	19	sandwich	sandwich	NOUN
cana-758	193	20	results	result	NOUN
cana-758	193	21	of	of	ADP
cana-758	193	22	univalent	univalent	ADJ
cana-758	193	23	functions	function	NOUN
cana-758	193	24	defined	define	VERB
cana-758	193	25	by	by	ADP
cana-758	193	26	a	a	DET
cana-758	193	27	linear	linear	ADJ
cana-758	193	28	operator	operator	NOUN
cana-758	193	29	,	,	PUNCT
cana-758	193	30	journal	journal	NOUN
cana-758	193	31	of	of	ADP
cana-758	193	32	interdisciplinary	interdisciplinary	ADJ
cana-758	193	33	mathematics	mathematic	NOUN
cana-758	193	34	,	,	PUNCT
cana-758	193	35	23(4)(2020	23(4)(2020	NUM
cana-758	193	36	)	)	PUNCT
cana-758	193	37	,	,	PUNCT
cana-758	193	38	803	803	NUM
cana-758	193	39	-	-	SYM
cana-758	193	40	809	809	NUM
cana-758	193	41	.	.	PUNCT
cana-758	194	1	[	[	X
cana-758	194	2	2	2	X
cana-758	194	3	]	]	PUNCT
cana-758	194	4	s.	s.	PROPN
cana-758	194	5	a.	a.	PROPN
cana-758	194	6	al	al	PROPN
cana-758	194	7	-	-	PUNCT
cana-758	194	8	ameedee	ameedee	PROPN
cana-758	194	9	,	,	PUNCT
cana-758	194	10	w.	w.	PROPN
cana-758	194	11	g.	g.	PROPN
cana-758	194	12	atshan	atshan	PROPN
cana-758	194	13	and	and	CCONJ
cana-758	194	14	f.	f.	PROPN
cana-758	194	15	a.	a.	PROPN
cana-758	194	16	al	al	PROPN
cana-758	194	17	-	-	PUNCT
cana-758	194	18	maamori	maamori	PROPN
cana-758	194	19	,	,	PUNCT
cana-758	194	20	some	some	DET
cana-758	194	21	new	new	ADJ
cana-758	194	22	results	result	NOUN
cana-758	194	23	of	of	ADP
cana-758	194	24	differential	differential	ADJ
cana-758	194	25	subordiantions	subordiantion	NOUN
cana-758	194	26	for	for	ADP
cana-758	194	27	higher	high	ADJ
cana-758	194	28	-	-	PUNCT
cana-758	194	29	order	order	NOUN
cana-758	194	30	derivatives	derivative	NOUN
cana-758	194	31	of	of	ADP
cana-758	194	32	multivalent	multivalent	NOUN
cana-758	194	33	functions	function	NOUN
cana-758	194	34	,	,	PUNCT
cana-758	194	35	journal	journal	NOUN
cana-758	194	36	of	of	ADP
cana-758	194	37	physics	physics	NOUN
cana-758	194	38	:	:	PUNCT
cana-758	194	39	conference	conference	NOUN
cana-758	194	40	series	series	NOUN
cana-758	194	41	,	,	PUNCT
cana-758	194	42	1804	1804	NUM
cana-758	194	43	(	(	PUNCT
cana-758	194	44	2021	2021	NUM
cana-758	194	45	)	)	PUNCT
cana-758	194	46	012111	012111	NUM
cana-758	194	47	,	,	PUNCT
cana-758	194	48	1	1	NUM
cana-758	194	49	-	-	SYM
cana-758	194	50	11	11	NUM
cana-758	194	51	.	.	PUNCT
cana-758	195	1	[	[	X
cana-758	195	2	3	3	NUM
cana-758	195	3	]	]	X
cana-758	195	4	r.	r.	PROPN
cana-758	195	5	m.	m.	PROPN
cana-758	195	6	ali	ali	PROPN
cana-758	195	7	,	,	PUNCT
cana-758	195	8	v.	v.	ADP
cana-758	195	9	ravichandran	ravichandran	NOUN
cana-758	195	10	and	and	CCONJ
cana-758	195	11	n.	n.	PROPN
cana-758	195	12	seenivasagan	seenivasagan	PROPN
cana-758	195	13	,	,	PUNCT
cana-758	195	14	differential	differential	ADJ
cana-758	195	15	subordination	subordination	NOUN
cana-758	195	16	and	and	CCONJ
cana-758	195	17	superordination	superordination	NOUN
cana-758	195	18	of	of	ADP
cana-758	195	19	analytic	analytic	ADJ
cana-758	195	20	functions	function	NOUN
cana-758	195	21	defined	define	VERB
cana-758	195	22	by	by	ADP
cana-758	195	23	the	the	DET
cana-758	195	24	multiplier	multipli	ADJ
cana-758	195	25	,	,	PUNCT
cana-758	195	26	math	math	NOUN
cana-758	195	27	.	.	PUNCT
cana-758	196	1	transformation	transformation	NOUN
cana-758	196	2	inequal	inequal	PROPN
cana-758	196	3	.	.	PUNCT
cana-758	197	1	appl	appl	PROPN
cana-758	197	2	.	.	PROPN
cana-758	197	3	,	,	PUNCT
cana-758	197	4	12	12	NUM
cana-758	197	5	(	(	PUNCT
cana-758	197	6	2009	2009	NUM
cana-758	197	7	)	)	PUNCT
cana-758	197	8	,	,	PUNCT
cana-758	197	9	123	123	NUM
cana-758	197	10	-	-	SYM
cana-758	197	11	139	139	NUM
cana-758	197	12	.	.	PUNCT
cana-758	198	1	[	[	X
cana-758	198	2	4	4	X
cana-758	198	3	]	]	PUNCT
cana-758	198	4	w.	w.	PROPN
cana-758	198	5	g.	g.	PROPN
cana-758	198	6	atshan	atshan	PROPN
cana-758	198	7	and	and	CCONJ
cana-758	198	8	a.	a.	PROPN
cana-758	198	9	a.	a.	PROPN
cana-758	198	10	r.	r.	PROPN
cana-758	198	11	ali	ali	PROPN
cana-758	198	12	,	,	PUNCT
cana-758	198	13	on	on	ADP
cana-758	198	14	some	some	DET
cana-758	198	15	sandwich	sandwich	NOUN
cana-758	198	16	theorems	theorem	NOUN
cana-758	198	17	of	of	ADP
cana-758	198	18	analytic	analytic	ADJ
cana-758	198	19	functions	function	NOUN
cana-758	198	20	involving	involve	VERB
cana-758	198	21	noor	noor	PROPN
cana-758	198	22	–	–	PUNCT
cana-758	198	23	sǎlǎgean	sǎlǎgean	ADJ
cana-758	198	24	operator	operator	NOUN
cana-758	198	25	,	,	PUNCT
cana-758	198	26	advances	advance	NOUN
cana-758	198	27	in	in	ADP
cana-758	198	28	mathematics	mathematic	NOUN
cana-758	198	29	:	:	PUNCT
cana-758	198	30	scientific	scientific	ADJ
cana-758	198	31	journal	journal	NOUN
cana-758	198	32	,	,	PUNCT
cana-758	198	33	9(10)(2020	9(10)(2020	PROPN
cana-758	198	34	)	)	PUNCT
cana-758	198	35	,	,	PUNCT
cana-758	198	36	8455	8455	NUM
cana-758	198	37	-	-	SYM
cana-758	198	38	8467	8467	NUM
cana-758	198	39	.	.	PUNCT
cana-758	199	1	[	[	X
cana-758	199	2	5	5	X
cana-758	199	3	]	]	PUNCT
cana-758	199	4	w.	w.	PROPN
cana-758	199	5	g.	g.	PROPN
cana-758	199	6	atshan	atshan	PROPN
cana-758	199	7	,	,	PUNCT
cana-758	199	8	a.	a.	PROPN
cana-758	199	9	h.	h.	PROPN
cana-758	199	10	battor	battor	PROPN
cana-758	199	11	and	and	CCONJ
cana-758	199	12	a.	a.	PROPN
cana-758	199	13	f.	f.	PROPN
cana-758	199	14	abaas	abaas	PROPN
cana-758	199	15	,	,	PUNCT
cana-758	199	16	some	some	DET
cana-758	199	17	sandwich	sandwich	NOUN
cana-758	199	18	theorems	theorem	NOUN
cana-758	199	19	for	for	ADP
cana-758	199	20	meromorphic	meromorphic	ADJ
cana-758	199	21	univalent	univalent	ADJ
cana-758	199	22	functions	function	NOUN
cana-758	199	23	defined	define	VERB
cana-758	199	24	by	by	ADP
cana-758	199	25	new	new	ADJ
cana-758	199	26	integral	integral	ADJ
cana-758	199	27	operator	operator	NOUN
cana-758	199	28	,	,	PUNCT
cana-758	199	29	journal	journal	NOUN
cana-758	199	30	of	of	ADP
cana-758	199	31	interdisciplinary	interdisciplinary	ADJ
cana-758	199	32	mathematics	mathematic	NOUN
cana-758	199	33	,	,	PUNCT
cana-758	199	34	24(3)(2021	24(3)(2021	NUM
cana-758	199	35	)	)	PUNCT
cana-758	199	36	,	,	PUNCT
cana-758	199	37	579	579	NUM
cana-758	199	38	-	-	SYM
cana-758	199	39	591	591	NUM
cana-758	199	40	.	.	PUNCT
cana-758	200	1	[	[	X
cana-758	200	2	6	6	NUM
cana-758	200	3	]	]	PUNCT
cana-758	200	4	w.	w.	PROPN
cana-758	200	5	g.	g.	PROPN
cana-758	200	6	atshan	atshan	PROPN
cana-758	200	7	and	and	CCONJ
cana-758	200	8	r.	r.	PROPN
cana-758	200	9	a.	a.	PROPN
cana-758	200	10	hadi	hadi	PROPN
cana-758	200	11	,	,	PUNCT
cana-758	200	12	some	some	DET
cana-758	200	13	differential	differential	ADJ
cana-758	200	14	subordination	subordination	NOUN
cana-758	200	15	and	and	CCONJ
cana-758	200	16	superordination	superordination	NOUN
cana-758	200	17	results	result	NOUN
cana-758	200	18	of	of	ADP
cana-758	200	19	p	p	NOUN
cana-758	200	20	-	-	PUNCT
cana-758	200	21	valent	valent	NOUN
cana-758	200	22	functions	function	NOUN
cana-758	200	23	defined	define	VERB
cana-758	200	24	by	by	ADP
cana-758	200	25	differential	differential	ADJ
cana-758	200	26	operator	operator	NOUN
cana-758	200	27	,	,	PUNCT
cana-758	200	28	journal	journal	NOUN
cana-758	200	29	of	of	ADP
cana-758	200	30	physics	physics	PROPN
cana-758	200	31	:	:	PUNCT
cana-758	200	32	conference	conference	NOUN
cana-758	200	33	series	series	NOUN
cana-758	200	34	,	,	PUNCT
cana-758	200	35	1664	1664	NUM
cana-758	200	36	(	(	PUNCT
cana-758	200	37	2020	2020	NUM
cana-758	200	38	)	)	PUNCT
cana-758	200	39	012043	012043	NUM
cana-758	200	40	,	,	PUNCT
cana-758	200	41	1	1	NUM
cana-758	200	42	-	-	SYM
cana-758	200	43	15	15	NUM
cana-758	200	44	.	.	PUNCT
cana-758	201	1	[	[	X
cana-758	201	2	7	7	X
cana-758	201	3	]	]	X
cana-758	201	4	w.	w.	PROPN
cana-758	201	5	g.	g.	PROPN
cana-758	201	6	atshan	atshan	PROPN
cana-758	201	7	and	and	CCONJ
cana-758	201	8	s.	s.	PROPN
cana-758	201	9	r.	r.	PROPN
cana-758	201	10	kulkarni	kulkarni	PROPN
cana-758	201	11	,	,	PUNCT
cana-758	201	12	on	on	ADP
cana-758	201	13	application	application	NOUN
cana-758	201	14	of	of	ADP
cana-758	201	15	differential	differential	ADJ
cana-758	201	16	subordination	subordination	NOUN
cana-758	201	17	for	for	ADP
cana-758	201	18	certain	certain	ADJ
cana-758	201	19	subclass	subclass	NOUN
cana-758	201	20	of	of	ADP
cana-758	201	21	meromorphically	meromorphically	ADV
cana-758	201	22	p	p	ADJ
cana-758	201	23	-	-	PUNCT
cana-758	201	24	valent	valent	NOUN
cana-758	201	25	functions	function	NOUN
cana-758	201	26	with	with	ADP
cana-758	201	27	positive	positive	ADJ
cana-758	201	28	coefficients	coefficient	NOUN
cana-758	201	29	defined	define	VERB
cana-758	201	30	by	by	ADP
cana-758	201	31	linear	linear	ADJ
cana-758	201	32	operator	operator	NOUN
cana-758	201	33	,	,	PUNCT
cana-758	201	34	journal	journal	NOUN
cana-758	201	35	of	of	ADP
cana-758	201	36	inequalities	inequality	NOUN
cana-758	201	37	in	in	ADP
cana-758	201	38	pure	pure	ADJ
cana-758	201	39	and	and	CCONJ
cana-758	201	40	applied	applied	ADJ
cana-758	201	41	mathematics	mathematic	NOUN
cana-758	201	42	,	,	PUNCT
cana-758	201	43	10(2)(2009	10(2)(2009	NUM
cana-758	201	44	)	)	PUNCT
cana-758	201	45	,	,	PUNCT
cana-758	201	46	article	article	NOUN
cana-758	201	47	53	53	NUM
cana-758	201	48	,	,	PUNCT
cana-758	201	49	11	11	NUM
cana-758	201	50	pp	pp	NOUN
cana-758	201	51	.	.	PUNCT
cana-758	202	1	[	[	X
cana-758	202	2	8	8	X
cana-758	202	3	]	]	X
cana-758	202	4	t.	t.	PROPN
cana-758	202	5	bullboacă	bullboacă	PROPN
cana-758	202	6	,	,	PUNCT
cana-758	202	7	classes	class	NOUN
cana-758	202	8	of	of	ADP
cana-758	202	9	first	first	ADJ
cana-758	202	10	order	order	NOUN
cana-758	202	11	differential	differential	ADJ
cana-758	202	12	superordinations	superordination	NOUN
cana-758	202	13	,	,	PUNCT
cana-758	202	14	demonstration	demonstration	NOUN
cana-758	202	15	math	math	NOUN
cana-758	202	16	.	.	PUNCT
cana-758	203	1	,	,	PUNCT
cana-758	203	2	35(2	35(2	NUM
cana-758	203	3	)	)	PUNCT
cana-758	203	4	(	(	PUNCT
cana-758	203	5	2002	2002	NUM
cana-758	203	6	)	)	PUNCT
cana-758	203	7	,	,	PUNCT
cana-758	203	8	287	287	NUM
cana-758	203	9	287	287	NUM
cana-758	203	10	.	.	PUNCT
cana-758	204	1	[	[	X
cana-758	204	2	9	9	NUM
cana-758	204	3	]	]	X
cana-758	204	4	t.	t.	PROPN
cana-758	204	5	bullboacă	bullboacă	PROPN
cana-758	204	6	,	,	PUNCT
cana-758	204	7	differential	differential	ADJ
cana-758	204	8	subordinations	subordination	NOUN
cana-758	204	9	and	and	CCONJ
cana-758	204	10	superordinations	superordination	NOUN
cana-758	204	11	,	,	PUNCT
cana-758	204	12	recent	recent	ADJ
cana-758	204	13	results	result	NOUN
cana-758	204	14	,	,	PUNCT
cana-758	204	15	house	house	NOUN
cana-758	204	16	of	of	ADP
cana-758	204	17	scientific	scientific	ADJ
cana-758	204	18	book	book	NOUN
cana-758	204	19	publ.cluj	publ.cluj	NOUN
cana-758	204	20	-	-	NOUN
cana-758	204	21	napoca	napoca	NOUN
cana-758	204	22	(	(	PUNCT
cana-758	204	23	2005	2005	NUM
cana-758	204	24	)	)	PUNCT
cana-758	204	25	.	.	PUNCT
cana-758	205	1	[	[	X
cana-758	205	2	10	10	NUM
cana-758	205	3	]	]	X
cana-758	205	4	j.	j.	PROPN
cana-758	205	5	choi	choi	PROPN
cana-758	205	6	and	and	CCONJ
cana-758	205	7	h.	h.	PROPN
cana-758	205	8	m	m	PROPN
cana-758	205	9	.	.	PUNCT
cana-758	206	1	srivastava	srivastava	PROPN
cana-758	206	2	,	,	PUNCT
cana-758	206	3	certain	certain	ADJ
cana-758	206	4	families	family	NOUN
cana-758	206	5	of	of	ADP
cana-758	206	6	series	series	NOUN
cana-758	206	7	associated	associate	VERB
cana-758	206	8	with	with	ADP
cana-758	206	9	the	the	DET
cana-758	206	10	hurwitz	hurwitz	PROPN
cana-758	206	11	–	–	PUNCT
cana-758	206	12	lerch	lerch	PROPN
cana-758	206	13	zeta	zeta	PROPN
cana-758	206	14	function	function	PROPN
cana-758	206	15	,	,	PUNCT
cana-758	206	16	appl	appl	PROPN
cana-758	206	17	.	.	PROPN
cana-758	206	18	math	math	PROPN
cana-758	206	19	.	.	PUNCT
cana-758	207	1	comput	comput	NOUN
cana-758	207	2	.	.	PUNCT
cana-758	207	3	,	,	PUNCT
cana-758	207	4	170(2005	170(2005	NUM
cana-758	207	5	)	)	PUNCT
cana-758	207	6	,	,	PUNCT
cana-758	207	7	399	399	NUM
cana-758	207	8	-	-	SYM
cana-758	207	9	409	409	NUM
cana-758	207	10	.	.	PUNCT
cana-758	208	1	[	[	X
cana-758	208	2	11	11	NUM
cana-758	208	3	]	]	X
cana-758	208	4	c.	c.	PROPN
cana-758	208	5	ferreira	ferreira	PROPN
cana-758	208	6	and	and	CCONJ
cana-758	208	7	j.	j.	PROPN
cana-758	208	8	l.	l.	PROPN
cana-758	208	9	lopez	lopez	PROPN
cana-758	208	10	,	,	PUNCT
cana-758	208	11	asymptotic	asymptotic	ADJ
cana-758	208	12	expansions	expansion	NOUN
cana-758	208	13	of	of	ADP
cana-758	208	14	the	the	DET
cana-758	208	15	hurwitz	hurwitz	PROPN
cana-758	208	16	–	–	PUNCT
cana-758	208	17	lerch	lerch	PROPN
cana-758	208	18	zeta	zeta	PROPN
cana-758	208	19	function	function	PROPN
cana-758	208	20	,	,	PUNCT
cana-758	208	21	j.	j.	PROPN
cana-758	208	22	math	math	PROPN
cana-758	208	23	.	.	PUNCT
cana-758	209	1	anal	anal	PROPN
cana-758	209	2	.	.	PUNCT
cana-758	209	3	,	,	PUNCT
cana-758	209	4	298(2004	298(2004	NUM
cana-758	209	5	)	)	PUNCT
cana-758	209	6	,	,	PUNCT
cana-758	209	7	210	210	NUM
cana-758	209	8	-	-	SYM
cana-758	209	9	224	224	NUM
cana-758	209	10	.	.	PUNCT
cana-758	210	1	[	[	X
cana-758	210	2	12	12	NUM
cana-758	210	3	]	]	PUNCT
cana-758	210	4	s.	s.	PROPN
cana-758	210	5	d.	d.	PROPN
cana-758	210	6	lin	lin	PROPN
cana-758	210	7	and	and	CCONJ
cana-758	210	8	h.m	h.m	PROPN
cana-758	210	9	.	.	PROPN
cana-758	210	10	srivastava	srivastava	PROPN
cana-758	210	11	,	,	PUNCT
cana-758	210	12	some	some	DET
cana-758	210	13	families	family	NOUN
cana-758	210	14	of	of	ADP
cana-758	210	15	the	the	DET
cana-758	210	16	hurwitz	hurwitz	PROPN
cana-758	210	17	-	-	PUNCT
cana-758	210	18	lerch	lerch	PROPN
cana-758	210	19	zeta	zeta	PROPN
cana-758	210	20	function	function	PROPN
cana-758	210	21	and	and	CCONJ
cana-758	210	22	associated	associate	VERB
cana-758	210	23	fractional	fractional	ADJ
cana-758	210	24	derivative	derivative	ADJ
cana-758	210	25	and	and	CCONJ
cana-758	210	26	other	other	ADJ
cana-758	210	27	integral	integral	ADJ
cana-758	210	28	representation	representation	NOUN
cana-758	210	29	,	,	PUNCT
cana-758	210	30	appl.math	appl.math	PROPN
cana-758	210	31	.	.	PUNCT
cana-758	210	32	comp	comp	NOUN
cana-758	210	33	.	.	PUNCT
cana-758	211	1	,	,	PUNCT
cana-758	211	2	154(2004	154(2004	NUM
cana-758	211	3	)	)	PUNCT
cana-758	211	4	,	,	PUNCT
cana-758	211	5	725	725	PROPN
cana-758	211	6	-733	-733	ADJ
cana-758	211	7	.	.	PUNCT
cana-758	212	1	[	[	X
cana-758	212	2	13	13	NUM
cana-758	212	3	]	]	PUNCT
cana-758	212	4	s.	s.	PROPN
cana-758	212	5	d.	d.	PROPN
cana-758	212	6	lin	lin	PROPN
cana-758	212	7	and	and	CCONJ
cana-758	212	8	h.m	h.m	PROPN
cana-758	212	9	.	.	PROPN
cana-758	212	10	srivastava	srivastava	PROPN
cana-758	212	11	and	and	CCONJ
cana-758	212	12	p.	p.	PROPN
cana-758	212	13	y.	y.	PROPN
cana-758	212	14	wang	wang	PROPN
cana-758	212	15	,	,	PUNCT
cana-758	212	16	some	some	DET
cana-758	212	17	expansion	expansion	NOUN
cana-758	212	18	formulas	formula	VERB
cana-758	212	19	for	for	ADP
cana-758	212	20	a	a	DET
cana-758	212	21	class	class	NOUN
cana-758	212	22	of	of	ADP
cana-758	212	23	generalized	generalized	ADJ
cana-758	212	24	hurwitzlerch	hurwitzlerch	NOUN
cana-758	212	25	zeta	zeta	NOUN
cana-758	212	26	function	function	NOUN
cana-758	212	27	,	,	PUNCT
cana-758	212	28	integral	integral	ADJ
cana-758	212	29	transforms	transform	VERB
cana-758	212	30	spec.funct	spec.funct	NOUN
cana-758	212	31	.	.	NOUN
cana-758	212	32	17(2006	17(2006	NUM
cana-758	212	33	)	)	PUNCT
cana-758	212	34	,	,	PUNCT
cana-758	212	35	817	817	NUM
cana-758	212	36	-827	-827	PROPN
cana-758	212	37	.	.	PUNCT
cana-758	213	1	[	[	X
cana-758	213	2	14	14	NUM
cana-758	213	3	]	]	X
cana-758	213	4	q.	q.	PROPN
cana-758	213	5	m.	m.	PROPN
cana-758	213	6	luo	luo	PROPN
cana-758	213	7	and	and	CCONJ
cana-758	213	8	h.	h.	PROPN
cana-758	213	9	m.	m.	PROPN
cana-758	213	10	srivastava	srivastava	PROPN
cana-758	213	11	,	,	PUNCT
cana-758	213	12	some	some	DET
cana-758	213	13	generalization	generalization	NOUN
cana-758	213	14	of	of	ADP
cana-758	213	15	the	the	DET
cana-758	213	16	apodtol	apodtol	NOUN
cana-758	213	17	–	–	PUNCT
cana-758	213	18	bernoulli	bernoulli	NOUN
cana-758	213	19	and	and	CCONJ
cana-758	213	20	apostol	apostol	NOUN
cana-758	213	21	-	-	PUNCT
cana-758	213	22	polnomials	polnomial	NOUN
cana-758	213	23	,	,	PUNCT
cana-758	213	24	j.	j.	PROPN
cana-758	213	25	math	math	PROPN
cana-758	213	26	.	.	PUNCT
cana-758	214	1	anal	anal	PROPN
cana-758	214	2	.	.	PUNCT
cana-758	215	1	,	,	PUNCT
cana-758	215	2	308(2005	308(2005	NUM
cana-758	215	3	)	)	PUNCT
cana-758	215	4	,	,	PUNCT
cana-758	215	5	290	290	NUM
cana-758	215	6	-	-	SYM
cana-758	215	7	302	302	NUM
cana-758	215	8	.	.	PUNCT
cana-758	216	1	[	[	X
cana-758	216	2	15	15	NUM
cana-758	216	3	]	]	X
cana-758	216	4	s.	s.	PROPN
cana-758	216	5	s.	s.	PROPN
cana-758	216	6	miller	miller	PROPN
cana-758	216	7	and	and	CCONJ
cana-758	216	8	p.	p.	PROPN
cana-758	216	9	t.	t.	PROPN
cana-758	216	10	mocanu	mocanu	PROPN
cana-758	216	11	,	,	PUNCT
cana-758	216	12	differential	differential	ADJ
cana-758	216	13	subordinations	subordination	NOUN
cana-758	216	14	:	:	PUNCT
cana-758	216	15	theory	theory	NOUN
cana-758	216	16	and	and	CCONJ
cana-758	216	17	applications	application	NOUN
cana-758	216	18	mathematics	mathematic	NOUN
cana-758	216	19	,	,	PUNCT
cana-758	216	20	225	225	NUM
cana-758	216	21	,	,	PUNCT
cana-758	216	22	marcel	marcel	PROPN
cana-758	216	23	dekker	dekker	PROPN
cana-758	216	24	,	,	PUNCT
cana-758	216	25	new	new	PROPN
cana-758	216	26	york	york	PROPN
cana-758	216	27	and	and	CCONJ
cana-758	216	28	basel	basel	PROPN
cana-758	216	29	,	,	PUNCT
cana-758	216	30	(	(	PUNCT
cana-758	216	31	2000	2000	NUM
cana-758	216	32	)	)	PUNCT
cana-758	216	33	.	.	PUNCT
cana-758	217	1	[	[	X
cana-758	217	2	16	16	NUM
cana-758	217	3	]	]	PUNCT
cana-758	217	4	s.	s.	PROPN
cana-758	217	5	s.	s.	PROPN
cana-758	217	6	miller	miller	PROPN
cana-758	217	7	and	and	CCONJ
cana-758	217	8	p.	p.	PROPN
cana-758	217	9	t.	t.	PROPN
cana-758	217	10	mocanu	mocanu	PROPN
cana-758	217	11	,	,	PUNCT
cana-758	217	12	subordinations	subordination	NOUN
cana-758	217	13	of	of	ADP
cana-758	217	14	differential	differential	ADJ
cana-758	217	15	superodinations	superodination	NOUN
cana-758	217	16	,	,	PUNCT
cana-758	217	17	complex	complex	ADJ
cana-758	217	18	variables	variable	NOUN
cana-758	217	19	,	,	PUNCT
cana-758	217	20	48(10	48(10	NUM
cana-758	217	21	)	)	PUNCT
cana-758	217	22	(	(	PUNCT
cana-758	217	23	2003	2003	NUM
cana-758	217	24	)	)	PUNCT
cana-758	217	25	,	,	PUNCT
cana-758	217	26	815	815	NUM
cana-758	217	27	-	-	SYM
cana-758	217	28	826	826	NUM
cana-758	217	29	.	.	PUNCT
cana-758	218	1	[	[	X
cana-758	218	2	17	17	NUM
cana-758	218	3	]	]	X
cana-758	218	4	g.	g.	PROPN
cana-758	218	5	s.	s.	PROPN
cana-758	218	6	sălăgean	sălăgean	PROPN
cana-758	218	7	,	,	PUNCT
cana-758	218	8	subclasses	subclass	NOUN
cana-758	218	9	of	of	ADP
cana-758	218	10	univalent	univalent	ADJ
cana-758	218	11	functions	function	NOUN
cana-758	218	12	,	,	PUNCT
cana-758	218	13	lecture	lecture	NOUN
cana-758	218	14	notes	note	NOUN
cana-758	218	15	in	in	ADP
cana-758	218	16	math	math	NOUN
cana-758	218	17	.	.	PUNCT
cana-758	218	18	,	,	PUNCT
cana-758	218	19	1013	1013	NUM
cana-758	218	20	,	,	PUNCT
cana-758	218	21	springerverlag	springerverlag	NOUN
cana-758	218	22	,	,	PUNCT
cana-758	218	23	(	(	PUNCT
cana-758	218	24	1983	1983	NUM
cana-758	218	25	)	)	PUNCT
cana-758	218	26	,	,	PUNCT
cana-758	218	27	362	362	NUM
cana-758	218	28	-	-	SYM
cana-758	218	29	372	372	NUM
cana-758	218	30	.	.	PUNCT
cana-758	219	1	communications	communication	NOUN
cana-758	219	2	on	on	ADP
cana-758	219	3	applied	apply	VERB
cana-758	219	4	nonlinear	nonlinear	ADJ
cana-758	219	5	analysis	analysis	NOUN
cana-758	219	6	issn	issn	NOUN
cana-758	219	7	:	:	PUNCT
cana-758	219	8	1074	1074	NUM
cana-758	219	9	-	-	PUNCT
cana-758	219	10	133x	133x	NUM
cana-758	219	11	vol	vol	NOUN
cana-758	219	12	31	31	NUM
cana-758	219	13	no	no	NOUN
cana-758	219	14	.	.	PUNCT
cana-758	220	1	3s	3s	NUM
cana-758	220	2	(	(	PUNCT
cana-758	220	3	2024	2024	NUM
cana-758	220	4	)	)	PUNCT
cana-758	220	5	196	196	NUM
cana-758	220	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-758	221	1	[	[	X
cana-758	221	2	18	18	NUM
cana-758	221	3	]	]	PUNCT
cana-758	221	4	t.	t.	PROPN
cana-758	221	5	n.	n.	PROPN
cana-758	221	6	shanmugam	shanmugam	PROPN
cana-758	221	7	,	,	PUNCT
cana-758	221	8	v.	v.	ADP
cana-758	221	9	ravichandran	ravichandran	NOUN
cana-758	221	10	and	and	CCONJ
cana-758	221	11	s.	s.	PROPN
cana-758	221	12	sivasubramanian	sivasubramanian	PROPN
cana-758	221	13	,	,	PUNCT
cana-758	221	14	differential	differential	ADJ
cana-758	221	15	sandwich	sandwich	NOUN
cana-758	221	16	theorems	theorem	NOUN
cana-758	221	17	for	for	ADP
cana-758	221	18	some	some	DET
cana-758	221	19	subclasses	subclass	NOUN
cana-758	221	20	of	of	ADP
cana-758	221	21	analytic	analytic	ADJ
cana-758	221	22	functions	function	NOUN
cana-758	221	23	,	,	PUNCT
cana-758	221	24	aust	aust	PROPN
cana-758	221	25	.	.	PUNCT
cana-758	222	1	j.	j.	PROPN
cana-758	222	2	math	math	PROPN
cana-758	222	3	.	.	PUNCT
cana-758	223	1	anal	anal	PROPN
cana-758	223	2	.	.	PUNCT
cana-758	224	1	appl	appl	PROPN
cana-758	224	2	.	.	PROPN
cana-758	224	3	,	,	PUNCT
cana-758	224	4	3	3	NUM
cana-758	224	5	(	(	PUNCT
cana-758	224	6	1	1	NUM
cana-758	224	7	)	)	PUNCT
cana-758	224	8	(	(	PUNCT
cana-758	224	9	2006),1	2006),1	NUM
cana-758	224	10	-	-	SYM
cana-758	224	11	11	11	NUM
cana-758	224	12	.	.	PUNCT
cana-758	225	1	[	[	X
cana-758	225	2	19	19	NUM
cana-758	225	3	]	]	PUNCT
cana-758	225	4	t.	t.	PROPN
cana-758	225	5	n.	n.	PROPN
cana-758	225	6	shanmugam	shanmugam	PROPN
cana-758	225	7	,	,	PUNCT
cana-758	225	8	v.	v.	CCONJ
cana-758	225	9	rvichandran	rvichandran	PROPN
cana-758	225	10	and	and	CCONJ
cana-758	225	11	s.	s.	PROPN
cana-758	225	12	sivasubramanian	sivasubramanian	PROPN
cana-758	225	13	,	,	PUNCT
cana-758	225	14	differential	differential	ADJ
cana-758	225	15	sandwich	sandwich	NOUN
cana-758	225	16	theorems	theorem	NOUN
cana-758	225	17	for	for	ADP
cana-758	225	18	subclasses	subclass	NOUN
cana-758	225	19	of	of	ADP
cana-758	225	20	analytic	analytic	ADJ
cana-758	225	21	functions	function	NOUN
cana-758	225	22	,	,	PUNCT
cana-758	225	23	aust	aust	PROPN
cana-758	225	24	.j	.j	PROPN
cana-758	225	25	.	.	PUNCT
cana-758	226	1	math	math	PROPN
cana-758	226	2	.anal	.anal	PROPN
cana-758	226	3	.	.	PUNCT
cana-758	227	1	,	,	PUNCT
cana-758	227	2	3	3	NUM
cana-758	227	3	,	,	PUNCT
cana-758	227	4	article	article	NOUN
cana-758	227	5	8(2006	8(2006	NUM
cana-758	227	6	)	)	PUNCT
cana-758	227	7	.,1	.,1	NOUN
cana-758	227	8	-11	-11	PUNCT
cana-758	227	9	.	.	PUNCT
cana-758	228	1	[	[	X
cana-758	228	2	20	20	NUM
cana-758	228	3	]	]	X
cana-758	228	4	h.	h.	PROPN
cana-758	228	5	m.	m.	PROPN
cana-758	228	6	srivastava	srivastava	PROPN
cana-758	228	7	and	and	CCONJ
cana-758	228	8	a.	a.	NOUN
cana-758	228	9	a.	a.	PROPN
cana-758	228	10	attiya	attiya	PROPN
cana-758	228	11	,	,	PUNCT
cana-758	228	12	an	an	DET
cana-758	228	13	integral	integral	ADJ
cana-758	228	14	operator	operator	NOUN
cana-758	228	15	associated	associate	VERB
cana-758	228	16	with	with	ADP
cana-758	228	17	the	the	DET
cana-758	228	18	hurwitz	hurwitz	PROPN
cana-758	228	19	-	-	PUNCT
cana-758	228	20	lerch	lerch	PROPN
cana-758	228	21	zeta	zeta	PROPN
cana-758	228	22	function	function	PROPN
cana-758	228	23	and	and	CCONJ
cana-758	228	24	differential	differential	ADJ
cana-758	228	25	subordination	subordination	NOUN
cana-758	228	26	,	,	PUNCT
cana-758	228	27	integral	integral	ADJ
cana-758	228	28	transforms	transform	VERB
cana-758	228	29	spec.funct	spec.funct	PROPN
cana-758	228	30	.	.	PUNCT
cana-758	228	31	,18,(2007),207	,18,(2007),207	NOUN
cana-758	228	32	-	-	PUNCT
cana-758	228	33	216	216	NUM
cana-758	228	34	.	.	PUNCT
cana-758	229	1	[	[	X
cana-758	229	2	21	21	NUM
cana-758	229	3	]	]	X
cana-758	229	4	s.	s.	PROPN
cana-758	229	5	r.	r.	PROPN
cana-758	229	6	swamy	swamy	PROPN
cana-758	229	7	,	,	PUNCT
cana-758	229	8	inclusion	inclusion	NOUN
cana-758	229	9	properties	property	NOUN
cana-758	229	10	of	of	ADP
cana-758	229	11	certion	certion	NOUN
cana-758	229	12	subclasses	subclass	NOUN
cana-758	229	13	of	of	ADP
cana-758	229	14	analytic	analytic	ADJ
cana-758	229	15	functions	function	NOUN
cana-758	229	16	,	,	PUNCT
cana-758	229	17	international	international	PROPN
cana-758	229	18	mathematical	mathematical	ADJ
cana-758	229	19	forum	forum	PROPN
cana-758	229	20	,	,	PUNCT
cana-758	229	21	vol	vol	NOUN
cana-758	229	22	.	.	PROPN
cana-758	230	1	7	7	NUM
cana-758	230	2	,	,	PUNCT
cana-758	230	3	no	no	DET
cana-758	230	4	.36	.36	NUM
cana-758	230	5	,	,	PUNCT
cana-758	230	6	(	(	PUNCT
cana-758	230	7	2012	2012	NUM
cana-758	230	8	)	)	PUNCT
cana-758	230	9	,	,	PUNCT
cana-758	230	10	1751	1751	NUM
cana-758	230	11	-1760	-1760	PROPN
cana-758	230	12	.	.	PUNCT
cana-758	231	1	[	[	X
cana-758	231	2	22	22	NUM
cana-758	231	3	]	]	PUNCT
cana-758	231	4	s.	s.	PROPN
cana-758	231	5	r.	r.	PROPN
cana-758	231	6	swamy	swamy	PROPN
cana-758	231	7	,	,	PUNCT
cana-758	231	8	sandwich	sandwich	NOUN
cana-758	231	9	theorems	theorem	NOUN
cana-758	231	10	for	for	ADP
cana-758	231	11	p	p	NOUN
cana-758	231	12	-	-	PUNCT
cana-758	231	13	valent	valent	NOUN
cana-758	231	14	functionsdefined	functionsdefine	VERB
cana-758	231	15	by	by	ADP
cana-758	231	16	certain	certain	ADJ
cana-758	231	17	integral	integral	ADJ
cana-758	231	18	operator	operator	NOUN
cana-758	231	19	,	,	PUNCT
cana-758	231	20	int	int	NOUN
cana-758	231	21	.	.	PUNCT
cana-758	232	1	j.	j.	PROPN
cana-758	232	2	math.arch	math.arch	PROPN
cana-758	232	3	.	.	PUNCT
cana-758	233	1	,4(3),101	,4(3),101	PUNCT
cana-758	233	2	-	-	PUNCT
cana-758	233	3	107,2013	107,2013	NUM
cana-758	233	4	.	.	PUNCT
cana-758	234	1	[	[	X
cana-758	234	2	23	23	NUM
cana-758	234	3	]	]	X
cana-758	234	4	r.	r.	PROPN
cana-758	234	5	m.	m.	PROPN
cana-758	234	6	el	el	PROPN
cana-758	234	7	-	-	NOUN
cana-758	234	8	ashwah	ashwah	NOUN
cana-758	234	9	and	and	CCONJ
cana-758	234	10	m.	m.	PROPN
cana-758	234	11	k.	k.	PROPN
cana-758	234	12	aouf	aouf	PROPN
cana-758	234	13	,	,	PUNCT
cana-758	234	14	differential	differential	ADJ
cana-758	234	15	subordination	subordination	NOUN
cana-758	234	16	and	and	CCONJ
cana-758	234	17	superordination	superordination	NOUN
cana-758	234	18	for	for	ADP
cana-758	234	19	certain	certain	ADJ
cana-758	234	20	subclasses	subclass	NOUN
cana-758	234	21	of	of	ADP
cana-758	234	22	pvalent	pvalent	NOUN
cana-758	234	23	functions	function	NOUN
cana-758	234	24	,	,	PUNCT
cana-758	234	25	mathematical	mathematical	ADJ
cana-758	234	26	and	and	CCONJ
cana-758	234	27	computer	computer	NOUN
cana-758	234	28	modelling	modelling	NOUN
cana-758	234	29	,	,	PUNCT
cana-758	234	30	51(5	51(5	PROPN
cana-758	234	31	-	-	PUNCT
cana-758	234	32	6)(2010	6)(2010	NOUN
cana-758	234	33	)	)	PUNCT
cana-758	234	34	,	,	PUNCT
cana-758	234	35	349	349	NUM
cana-758	234	36	-	-	SYM
cana-758	234	37	360	360	NUM
cana-758	234	38	.	.	PUNCT
cana-758	235	1	[	[	X
cana-758	235	2	24	24	NUM
cana-758	235	3	]	]	X
cana-758	235	4	p.	p.	NOUN
cana-758	235	5	gochhayat	gochhayat	PROPN
cana-758	235	6	,	,	PUNCT
cana-758	235	7	sandwich	sandwich	NOUN
cana-758	235	8	-	-	PUNCT
cana-758	235	9	type	type	NOUN
cana-758	235	10	theorems	theorem	NOUN
cana-758	235	11	of	of	ADP
cana-758	235	12	some	some	DET
cana-758	235	13	subclasses	subclass	NOUN
cana-758	235	14	of	of	ADP
cana-758	235	15	multivalent	multivalent	NOUN
cana-758	235	16	functions	function	NOUN
cana-758	235	17	involving	involve	VERB
cana-758	235	18	dziok	dziok	NOUN
cana-758	235	19	-	-	PUNCT
cana-758	235	20	srivastava	srivastava	PROPN
cana-758	235	21	operator	operator	NOUN
cana-758	235	22	,	,	PUNCT
cana-758	235	23	acta	acta	PROPN
cana-758	235	24	univ	univ	PROPN
cana-758	235	25	.	.	PUNCT
cana-758	236	1	apulensis,32(2012),31	apulensis,32(2012),31	PROPN
cana-758	236	2	-	-	PUNCT
cana-758	236	3	47	47	NUM
cana-758	236	4	.	.	PUNCT
cana-758	237	1	[	[	X
cana-758	237	2	25	25	NUM
cana-758	237	3	]	]	X
cana-758	237	4	p.	p.	NOUN
cana-758	237	5	gochhayat	gochhayat	PROPN
cana-758	237	6	,	,	PUNCT
cana-758	237	7	sandwich	sandwich	NOUN
cana-758	237	8	-	-	PUNCT
cana-758	237	9	type	type	NOUN
cana-758	237	10	results	result	NOUN
cana-758	237	11	for	for	ADP
cana-758	237	12	a	a	DET
cana-758	237	13	class	class	NOUN
cana-758	237	14	of	of	ADP
cana-758	237	15	functions	function	NOUN
cana-758	237	16	defined	define	VERB
cana-758	237	17	by	by	ADP
cana-758	237	18	a	a	DET
cana-758	237	19	generalized	generalize	VERB
cana-758	237	20	differential	differential	NOUN
cana-758	237	21	operator	operator	NOUN
cana-758	237	22	,	,	PUNCT
cana-758	237	23	math	math	NOUN
cana-758	237	24	.	.	PUNCT
cana-758	238	1	vesink,65(2)(2013),178	vesink,65(2)(2013),178	ADV
cana-758	238	2	-	-	SYM
cana-758	238	3	186	186	NUM
cana-758	238	4	.	.	PUNCT