id sid tid token lemma pos cana-4831 1 1 communications communication NOUN cana-4831 1 2 on on ADP cana-4831 1 3 applied apply VERB cana-4831 1 4 nonlinear nonlinear ADJ cana-4831 1 5 analysis analysis NOUN cana-4831 1 6 issn issn NOUN cana-4831 1 7 : : PUNCT cana-4831 1 8 1074 1074 NUM cana-4831 1 9 - - PUNCT cana-4831 1 10 133x 133x NUM cana-4831 1 11 vol vol VERB cana-4831 1 12 32 32 NUM cana-4831 1 13 no no NOUN cana-4831 1 14 . . PUNCT cana-4831 2 1 10s 10 NOUN cana-4831 2 2 ( ( PUNCT cana-4831 2 3 2025 2025 NUM cana-4831 2 4 ) ) PUNCT cana-4831 2 5 382 382 NUM cana-4831 2 6 https://internationalpubls.com https://internationalpubls.com X cana-4831 2 7 subclasses subclass NOUN cana-4831 2 8 on on ADP cana-4831 2 9 negative negative ADJ cana-4831 2 10 coefficints coefficint NOUN cana-4831 2 11 by by ADP cana-4831 2 12 linear linear PROPN cana-4831 2 13 differential differential NOUN cana-4831 2 14 operator operator NOUN cana-4831 2 15 annapoorna annapoorna VERB cana-4831 2 16 s. s. PROPN cana-4831 2 17 * * PROPN cana-4831 2 18 and and CCONJ cana-4831 2 19 dileep dileep PROPN cana-4831 2 20 l. l. PROPN cana-4831 2 21 * * PROPN cana-4831 2 22 * * PROPN cana-4831 2 23 vidyavardhaka vidyavardhaka PROPN cana-4831 2 24 college college NOUN cana-4831 2 25 of of ADP cana-4831 2 26 engineering engineering NOUN cana-4831 2 27 , , PUNCT cana-4831 2 28 mysuru mysuru PROPN cana-4831 2 29 , , PUNCT cana-4831 2 30 india-570 india-570 NOUN cana-4831 2 31 002 002 NUM cana-4831 2 32 visvesvaraya visvesvaraya ADP cana-4831 2 33 technological technological ADJ cana-4831 2 34 university university NOUN cana-4831 2 35 , , PUNCT cana-4831 2 36 belagavi belagavi VERB cana-4831 2 37 article article NOUN cana-4831 2 38 history history NOUN cana-4831 2 39 : : PUNCT cana-4831 2 40 received receive VERB cana-4831 2 41 : : PUNCT cana-4831 2 42 12 12 NUM cana-4831 2 43 - - SYM cana-4831 2 44 01 01 NUM cana-4831 2 45 - - PUNCT cana-4831 2 46 2025 2025 NUM cana-4831 2 47 revised revise VERB cana-4831 2 48 : : PUNCT cana-4831 2 49 15 15 NUM cana-4831 2 50 - - NUM cana-4831 2 51 02 02 NUM cana-4831 2 52 - - PUNCT cana-4831 2 53 2025 2025 NUM cana-4831 2 54 accepted accept VERB cana-4831 2 55 : : PUNCT cana-4831 2 56 01 01 NUM cana-4831 2 57 - - SYM cana-4831 2 58 03 03 NUM cana-4831 2 59 - - PUNCT cana-4831 2 60 2025 2025 NUM cana-4831 2 61 abstract abstract NOUN cana-4831 2 62 : : PUNCT cana-4831 2 63 in in ADP cana-4831 2 64 this this DET cana-4831 2 65 paper paper NOUN cana-4831 2 66 , , PUNCT cana-4831 2 67 we we PRON cana-4831 2 68 use use VERB cana-4831 2 69 the the DET cana-4831 2 70 linear linear ADJ cana-4831 2 71 operator operator NOUN cana-4831 2 72 π΄π‘†πœ†,π‘ž π΄π‘†πœ†,π‘ž PROPN cana-4831 2 73 𝛿,𝑛 𝛿,𝑛 PROPN cana-4831 2 74 to to PART cana-4831 2 75 define define VERB cana-4831 2 76 the the DET cana-4831 2 77 class class NOUN cana-4831 2 78 𝑇𝑛(𝛼 𝑇𝑛(𝛼 NOUN cana-4831 2 79 , , PUNCT cana-4831 2 80 𝛽 𝛽 NOUN cana-4831 2 81 , , PUNCT cana-4831 2 82 𝛿 𝛿 ADJ cana-4831 2 83 , , PUNCT cana-4831 2 84 πœ† πœ† NOUN cana-4831 2 85 ; ; PUNCT cana-4831 2 86 π‘ž π‘ž NOUN cana-4831 2 87 ) ) PUNCT cana-4831 2 88 . . PUNCT cana-4831 3 1 we we PRON cana-4831 3 2 derive derive VERB cana-4831 3 3 coefficient coefficient NOUN cana-4831 3 4 estimates estimate NOUN cana-4831 3 5 and and CCONJ cana-4831 3 6 numerous numerous ADJ cana-4831 3 7 other other ADJ cana-4831 3 8 features feature NOUN cana-4831 3 9 for for ADP cana-4831 3 10 functions function NOUN cana-4831 3 11 that that PRON cana-4831 3 12 fall fall VERB cana-4831 3 13 under under ADP cana-4831 3 14 this this DET cana-4831 3 15 class class NOUN cana-4831 3 16 . . PUNCT cana-4831 4 1 we we PRON cana-4831 4 2 identify identify VERB cana-4831 4 3 the the DET cana-4831 4 4 extreme extreme ADJ cana-4831 4 5 points point NOUN cana-4831 4 6 and and CCONJ cana-4831 4 7 integral integral ADJ cana-4831 4 8 means mean NOUN cana-4831 4 9 as as ADV cana-4831 4 10 well well ADV cana-4831 4 11 . . PUNCT cana-4831 5 1 keywords keyword NOUN cana-4831 5 2 : : PUNCT cana-4831 5 3 analytic analytic ADJ cana-4831 5 4 function function NOUN cana-4831 5 5 , , PUNCT cana-4831 5 6 linear linear ADJ cana-4831 5 7 differential differential NOUN cana-4831 5 8 operator operator NOUN cana-4831 5 9 , , PUNCT cana-4831 5 10 coefficient coefficient NOUN cana-4831 5 11 inequalities inequality NOUN cana-4831 5 12 , , PUNCT cana-4831 5 13 extreme extreme ADJ cana-4831 5 14 points point NOUN cana-4831 5 15 and and CCONJ cana-4831 5 16 integral integral ADJ cana-4831 5 17 means mean NOUN cana-4831 5 18 . . PUNCT cana-4831 6 1 ams am NOUN cana-4831 6 2 classification classification NOUN cana-4831 6 3 : : PUNCT cana-4831 6 4 primary primary ADJ cana-4831 6 5 30c45 30c45 NUM cana-4831 6 6 ; ; PUNCT cana-4831 6 7 secondary secondary ADJ cana-4831 6 8 30c50;30c80 30c50;30c80 PROPN cana-4831 6 9 1 1 NUM cana-4831 6 10 . . PUNCT cana-4831 6 11 introduction introduction NOUN cana-4831 6 12 linear linear PROPN cana-4831 6 13 differential differential NOUN cana-4831 6 14 operators operator NOUN cana-4831 6 15 are be AUX cana-4831 6 16 crucial crucial ADJ cana-4831 6 17 in in ADP cana-4831 6 18 geometric geometric ADJ cana-4831 6 19 function function NOUN cana-4831 6 20 theory theory NOUN cana-4831 6 21 , , PUNCT cana-4831 6 22 a a DET cana-4831 6 23 branch branch NOUN cana-4831 6 24 of of ADP cana-4831 6 25 mathematics mathematic NOUN cana-4831 6 26 that that PRON cana-4831 6 27 studies study VERB cana-4831 6 28 the the DET cana-4831 6 29 properties property NOUN cana-4831 6 30 of of ADP cana-4831 6 31 functions function NOUN cana-4831 6 32 and and CCONJ cana-4831 6 33 their their PRON cana-4831 6 34 translations translation NOUN cana-4831 6 35 in in ADP cana-4831 6 36 geometric geometric ADJ cana-4831 6 37 contexts contexts NOUN cana-4831 6 38 . . PUNCT cana-4831 7 1 in in ADP cana-4831 7 2 particular particular ADJ cana-4831 7 3 , , PUNCT cana-4831 7 4 linear linear ADJ cana-4831 7 5 differential differential NOUN cana-4831 7 6 operators operator NOUN cana-4831 7 7 are be AUX cana-4831 7 8 used use VERB cana-4831 7 9 to to PART cana-4831 7 10 study study VERB cana-4831 7 11 the the DET cana-4831 7 12 characteristics characteristic NOUN cana-4831 7 13 of of ADP cana-4831 7 14 conformal conformal ADJ cana-4831 7 15 mappings mapping NOUN cana-4831 7 16 , , PUNCT cana-4831 7 17 quasi quasi ADJ cana-4831 7 18 - - ADJ cana-4831 7 19 conformal conformal ADJ cana-4831 7 20 mappings mapping NOUN cana-4831 7 21 and and CCONJ cana-4831 7 22 other other ADJ cana-4831 7 23 types type NOUN cana-4831 7 24 of of ADP cana-4831 7 25 mappings mapping NOUN cana-4831 7 26 between between ADP cana-4831 7 27 riemann riemann PROPN cana-4831 7 28 surfaces surface NOUN cana-4831 7 29 and and CCONJ cana-4831 7 30 other other ADJ cana-4831 7 31 geometric geometric ADJ cana-4831 7 32 objects object NOUN cana-4831 7 33 . . PUNCT cana-4831 8 1 numerous numerous ADJ cana-4831 8 2 aspects aspect NOUN cana-4831 8 3 of of ADP cana-4831 8 4 functions function NOUN cana-4831 8 5 and and CCONJ cana-4831 8 6 mappings mapping NOUN cana-4831 8 7 , , PUNCT cana-4831 8 8 including include VERB cana-4831 8 9 their their PRON cana-4831 8 10 regularity regularity NOUN cana-4831 8 11 , , PUNCT cana-4831 8 12 smoothness smoothness ADJ cana-4831 8 13 , , PUNCT cana-4831 8 14 and and CCONJ cana-4831 8 15 geometric geometric ADJ cana-4831 8 16 features feature NOUN cana-4831 8 17 like like ADP cana-4831 8 18 curvature curvature NOUN cana-4831 8 19 and and CCONJ cana-4831 8 20 conformality conformality NOUN cana-4831 8 21 , , PUNCT cana-4831 8 22 are be AUX cana-4831 8 23 explored explore VERB cana-4831 8 24 using use VERB cana-4831 8 25 linear linear PROPN cana-4831 8 26 operators operator NOUN cana-4831 8 27 . . PUNCT cana-4831 9 1 let let VERB cana-4831 9 2 𝐴 𝐴 PROPN cana-4831 9 3 be be AUX cana-4831 9 4 the the DET cana-4831 9 5 class class NOUN cana-4831 9 6 of of ADP cana-4831 9 7 functions function NOUN cana-4831 9 8 𝑓 𝑓 PRON cana-4831 9 9 of of ADP cana-4831 9 10 the the DET cana-4831 9 11 form form NOUN cana-4831 9 12 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 9 13 ) ) PUNCT cana-4831 9 14 = = PUNCT cana-4831 10 1 𝑧 𝑧 PROPN cana-4831 10 2 + + PUNCT cana-4831 10 3 βˆ‘ βˆ‘ PROPN cana-4831 10 4 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-4831 10 5 𝑗=2 𝑗=2 PROPN cana-4831 10 6 , , PUNCT cana-4831 10 7 ( ( PUNCT cana-4831 10 8 1.1 1.1 NUM cana-4831 10 9 ) ) PUNCT cana-4831 10 10 which which PRON cana-4831 10 11 are be AUX cana-4831 10 12 analytic analytic ADJ cana-4831 10 13 in in ADP cana-4831 10 14 the the DET cana-4831 10 15 open open ADJ cana-4831 10 16 unit unit NOUN cana-4831 10 17 disc disc VERB cana-4831 10 18 π‘ˆ π‘ˆ PROPN cana-4831 10 19 = = PUNCT cana-4831 10 20 { { PUNCT cana-4831 10 21 𝑧 𝑧 PROPN cana-4831 10 22 ∈ ∈ PROPN cana-4831 10 23 𝐢 𝐢 PROPN cana-4831 10 24 ; ; PUNCT cana-4831 10 25 |𝑧| |𝑧| PROPN cana-4831 10 26 < < X cana-4831 10 27 1 1 NUM cana-4831 10 28 } } PUNCT cana-4831 10 29 . . PUNCT cana-4831 11 1 let let VERB cana-4831 11 2 𝑇 𝑇 PROPN cana-4831 11 3 denote denote VERB cana-4831 11 4 the the DET cana-4831 11 5 subclass subclass NOUN cana-4831 11 6 of of ADP cana-4831 11 7 𝐴 𝐴 PROPN cana-4831 11 8 in in ADP cana-4831 11 9 π‘ˆ π‘ˆ PROPN cana-4831 11 10 , , PUNCT cana-4831 11 11 consisting consist VERB cana-4831 11 12 of of ADP cana-4831 11 13 analytic analytic ADJ cana-4831 11 14 functions function NOUN cana-4831 11 15 whose whose DET cana-4831 11 16 non non ADJ cana-4831 11 17 - - ADJ cana-4831 11 18 zero zero NUM cana-4831 11 19 coefficients coefficient NOUN cana-4831 11 20 from from ADP cana-4831 11 21 the the DET cana-4831 11 22 second second ADJ cana-4831 11 23 terms term NOUN cana-4831 11 24 onwards onwards ADV cana-4831 11 25 are be AUX cana-4831 11 26 negative negative ADJ cana-4831 11 27 . . PUNCT cana-4831 12 1 that that ADV cana-4831 12 2 is is ADV cana-4831 12 3 , , PUNCT cana-4831 12 4 an an DET cana-4831 12 5 analytic analytic ADJ cana-4831 12 6 function function NOUN cana-4831 12 7 𝑓 𝑓 PRON cana-4831 12 8 ∈ ∈ PROPN cana-4831 12 9 𝑇 𝑇 PROPN cana-4831 12 10 if if SCONJ cana-4831 12 11 it it PRON cana-4831 12 12 has have VERB cana-4831 12 13 taylor taylor PROPN cana-4831 12 14 series series PROPN cana-4831 12 15 expansion expansion NOUN cana-4831 12 16 of of ADP cana-4831 12 17 the the DET cana-4831 12 18 form form NOUN cana-4831 12 19 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 12 20 ) ) PUNCT cana-4831 13 1 = = PUNCT cana-4831 13 2 𝑧 𝑧 PRON cana-4831 13 3 βˆ’ βˆ’ PROPN cana-4831 13 4 βˆ‘ βˆ‘ SYM cana-4831 13 5 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-4831 13 6 𝑗=2 𝑗=2 PROPN cana-4831 13 7 , , PUNCT cana-4831 13 8 ( ( PUNCT cana-4831 13 9 π‘Žπ‘— π‘Žπ‘— ADP cana-4831 13 10 β‰₯ β‰₯ NOUN cana-4831 13 11 0 0 NUM cana-4831 13 12 ) ) PUNCT cana-4831 13 13 ( ( PUNCT cana-4831 13 14 1.2 1.2 NUM cana-4831 13 15 ) ) PUNCT cana-4831 13 16 which which PRON cana-4831 13 17 are be AUX cana-4831 13 18 univalent univalent ADJ cana-4831 13 19 in in ADP cana-4831 13 20 the the DET cana-4831 13 21 open open ADJ cana-4831 13 22 unit unit NOUN cana-4831 13 23 disc disc VERB cana-4831 13 24 π‘ˆ. π‘ˆ. PROPN cana-4831 13 25 using use VERB cana-4831 13 26 the the DET cana-4831 13 27 idea idea NOUN cana-4831 13 28 of of ADP cana-4831 13 29 convolution convolution NOUN cana-4831 13 30 , , PUNCT cana-4831 13 31 annapoorna annapoorna NOUN cana-4831 13 32 s s PART cana-4831 13 33 and and CCONJ cana-4831 13 34 dileep dileep PROPN cana-4831 13 35 l l PROPN cana-4831 14 1 [ [ X cana-4831 14 2 4 4 NUM cana-4831 14 3 ] ] PUNCT cana-4831 14 4 , , PUNCT cana-4831 14 5 introduced introduce VERB cana-4831 14 6 a a DET cana-4831 14 7 linear linear ADJ cana-4831 14 8 differential differential NOUN cana-4831 14 9 operator operator NOUN cana-4831 14 10 𝑨𝑺𝝀,𝒒 𝑨𝑺𝝀,𝒒 PROPN cana-4831 14 11 𝜹,𝒏 𝜹,𝒏 NOUN cana-4831 14 12 ∢ ∢ PROPN cana-4831 14 13 𝑨 𝑨 PROPN cana-4831 14 14 β†’ β†’ SYM cana-4831 14 15 𝑨 𝑨 PROPN cana-4831 14 16 defined define VERB cana-4831 14 17 by by ADP cana-4831 14 18 𝑨𝑺𝝀,𝒒 𝑨𝑺𝝀,𝒒 PROPN cana-4831 14 19 𝜹,𝒏 𝜹,𝒏 PROPN cana-4831 14 20 𝒇(𝒛 𝒇(𝒛 NOUN cana-4831 14 21 ) ) PUNCT cana-4831 14 22 = = PUNCT cana-4831 15 1 [ [ X cana-4831 15 2 ( ( PUNCT cana-4831 15 3 𝟏 𝟏 NUM cana-4831 15 4 βˆ’ βˆ’ NOUN cana-4831 16 1 𝝀)[𝟏 𝝀)[𝟏 NOUN cana-4831 16 2 + + CCONJ cana-4831 16 3 ( ( PUNCT cana-4831 16 4 𝒋 𝒋 X cana-4831 16 5 βˆ’ βˆ’ X cana-4831 16 6 𝟏)𝜹]𝒏 𝟏)𝜹]𝒏 NOUN cana-4831 16 7 + + CCONJ cana-4831 17 1 π€πš½(𝒂 π€πš½(𝒂 ADJ cana-4831 17 2 , , PUNCT cana-4831 17 3 𝒄 𝒄 NOUN cana-4831 17 4 ) ) PUNCT cana-4831 17 5 ] ] PUNCT cana-4831 18 1 βˆ— βˆ— NOUN cana-4831 18 2 𝒇(𝒛 𝒇(𝒛 NUM cana-4831 18 3 ) ) PUNCT cana-4831 18 4 . . PUNCT cana-4831 19 1 for for ADP cana-4831 19 2 functions function NOUN cana-4831 19 3 𝑓 𝑓 DET cana-4831 19 4 ∈ ∈ PROPN cana-4831 19 5 𝐴 𝐴 PROPN cana-4831 19 6 of of ADP cana-4831 19 7 the the DET cana-4831 19 8 form form NOUN cana-4831 19 9 ( ( PUNCT cana-4831 19 10 1.1 1.1 NUM cana-4831 19 11 ) ) PUNCT cana-4831 19 12 , , PUNCT cana-4831 19 13 we we PRON cana-4831 19 14 have have VERB cana-4831 19 15 𝑨𝑺𝝀,𝒒 𝑨𝑺𝝀,𝒒 PROPN cana-4831 19 16 𝜹,𝒏 𝜹,𝒏 PROPN cana-4831 19 17 𝒇(𝒛 𝒇(𝒛 NOUN cana-4831 19 18 ) ) PUNCT cana-4831 19 19 = = PUNCT cana-4831 20 1 𝑧 𝑧 PROPN cana-4831 20 2 + + CCONJ cana-4831 20 3 βˆ‘ βˆ‘ PROPN cana-4831 20 4 π΅πœ† π΅πœ† PROPN cana-4831 20 5 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 20 6 , , PUNCT cana-4831 20 7 𝑐 𝑐 PROPN cana-4831 20 8 , , PUNCT cana-4831 20 9 𝑗 𝑗 NOUN cana-4831 20 10 , , PUNCT cana-4831 20 11 𝑛 𝑛 ADJ cana-4831 20 12 ; ; PUNCT cana-4831 20 13 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 20 14 𝑧 𝑧 X cana-4831 20 15 π‘—βˆž π‘—βˆž X cana-4831 20 16 𝑗=2 𝑗=2 PUNCT cana-4831 20 17 ( ( PUNCT cana-4831 20 18 1.3 1.3 NUM cana-4831 20 19 ) ) PUNCT cana-4831 20 20 communications communication NOUN cana-4831 20 21 on on ADP cana-4831 20 22 applied apply VERB cana-4831 20 23 nonlinear nonlinear ADJ cana-4831 20 24 analysis analysis NOUN cana-4831 20 25 issn issn NOUN cana-4831 20 26 : : PUNCT cana-4831 20 27 1074 1074 NUM cana-4831 20 28 - - PUNCT cana-4831 20 29 133x 133x NUM cana-4831 20 30 vol vol VERB cana-4831 20 31 32 32 NUM cana-4831 20 32 no no NOUN cana-4831 20 33 . . PUNCT cana-4831 21 1 10s 10 NOUN cana-4831 21 2 ( ( PUNCT cana-4831 21 3 2025 2025 NUM cana-4831 21 4 ) ) PUNCT cana-4831 21 5 383 383 NUM cana-4831 21 6 https://internationalpubls.com https://internationalpubls.com X cana-4831 21 7 where where SCONJ cana-4831 21 8 π΅πœ† π΅πœ† PROPN cana-4831 21 9 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 21 10 , , PUNCT cana-4831 21 11 𝑐 𝑐 PROPN cana-4831 21 12 , , PUNCT cana-4831 21 13 𝑗 𝑗 NOUN cana-4831 21 14 , , PUNCT cana-4831 21 15 𝑛 𝑛 PROPN cana-4831 21 16 ; ; PUNCT cana-4831 21 17 π‘ž π‘ž X cana-4831 21 18 ) ) PUNCT cana-4831 21 19 = = SYM cana-4831 22 1 [ [ X cana-4831 22 2 ( ( PUNCT cana-4831 22 3 𝟏 𝟏 NUM cana-4831 22 4 βˆ’ βˆ’ NOUN cana-4831 23 1 𝝀)[𝟏 𝝀)[𝟏 NOUN cana-4831 23 2 + + CCONJ cana-4831 23 3 ( ( PUNCT cana-4831 23 4 𝒋 𝒋 X cana-4831 23 5 βˆ’ βˆ’ X cana-4831 23 6 𝟏)𝜹]𝒏 𝟏)𝜹]𝒏 NOUN cana-4831 23 7 + + CCONJ cana-4831 23 8 𝝀 𝝀 X cana-4831 23 9 ( ( PUNCT cana-4831 23 10 𝒂)π’‹βˆ’πŸ 𝒂)π’‹βˆ’πŸ PROPN cana-4831 23 11 ( ( PUNCT cana-4831 23 12 𝒄)π’‹βˆ’πŸ 𝒄)π’‹βˆ’πŸ PROPN cana-4831 23 13 ] ] PUNCT cana-4831 23 14 π‘ž π‘ž X cana-4831 23 15 ( ( PUNCT cana-4831 23 16 1.4 1.4 NUM cana-4831 23 17 ) ) PUNCT cana-4831 23 18 𝑛 𝑛 DET cana-4831 23 19 ∈ ∈ PROPN cana-4831 23 20 β„•0 β„•0 NOUN cana-4831 23 21 , , PUNCT cana-4831 23 22 πœ† πœ† DET cana-4831 23 23 β‰₯ β‰₯ NOUN cana-4831 23 24 0 0 NUM cana-4831 23 25 , , PUNCT cana-4831 23 26 𝛿 𝛿 PRON cana-4831 23 27 β‰₯ β‰₯ NOUN cana-4831 23 28 0 0 NUM cana-4831 23 29 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-4831 23 30 π‘Ž π‘Ž PROPN cana-4831 23 31 , , PUNCT cana-4831 23 32 𝑐 𝑐 PROPN cana-4831 23 33 ∈ ∈ PROPN cana-4831 23 34 ℝ ℝ PROPN cana-4831 23 35 \β„€. \β„€. NOUN cana-4831 23 36 here here ADV cana-4831 23 37 ( ( PUNCT cana-4831 23 38 π‘Ž)𝑗 π‘Ž)𝑗 X cana-4831 23 39 is be AUX cana-4831 23 40 the the DET cana-4831 23 41 pochhammer pochhammer NOUN cana-4831 23 42 symbol symbol NOUN cana-4831 23 43 defined define VERB cana-4831 23 44 interms interm NOUN cana-4831 23 45 of of ADP cana-4831 23 46 the the DET cana-4831 23 47 gamma gamma NOUN cana-4831 23 48 function function NOUN cana-4831 23 49 by by ADP cana-4831 23 50 , , PUNCT cana-4831 23 51 ( ( PUNCT cana-4831 23 52 π‘Ž)𝑗 π‘Ž)𝑗 X cana-4831 23 53 = = PUNCT cana-4831 23 54 Ξ³(π‘Ž+𝑗 Ξ³(π‘Ž+𝑗 NOUN cana-4831 23 55 ) ) PUNCT cana-4831 23 56 Ξ³(π‘Ž Ξ³(π‘Ž PROPN cana-4831 23 57 ) ) PUNCT cana-4831 23 58 = = PRON cana-4831 23 59 { { PUNCT cana-4831 23 60 1 1 NUM cana-4831 23 61 , , PUNCT cana-4831 23 62 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-4831 23 63 𝑗 𝑗 NOUN cana-4831 23 64 = = SYM cana-4831 23 65 0 0 NUM cana-4831 23 66 π‘Ž(π‘Ž π‘Ž(π‘Ž PROPN cana-4831 23 67 + + CCONJ cana-4831 23 68 1)(π‘Ž 1)(π‘Ž NUM cana-4831 23 69 + + CCONJ cana-4831 23 70 2 2 NUM cana-4831 23 71 ) ) PUNCT cana-4831 23 72 β‹― β‹― VERB cana-4831 23 73 ( ( PUNCT cana-4831 23 74 π‘Ž π‘Ž X cana-4831 24 1 + + NOUN cana-4831 24 2 𝑗 𝑗 ADJ cana-4831 24 3 βˆ’ βˆ’ NOUN cana-4831 24 4 1 1 NUM cana-4831 24 5 ) ) PUNCT cana-4831 24 6 , , PUNCT cana-4831 24 7 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-4831 24 8 𝑗 𝑗 PRON cana-4831 24 9 ∈ ∈ PROPN cana-4831 24 10 β„• β„• PROPN cana-4831 24 11 . . PUNCT cana-4831 25 1 we we PRON cana-4831 25 2 obtain obtain VERB cana-4831 25 3 the the DET cana-4831 25 4 al al PROPN cana-4831 25 5 - - PUNCT cana-4831 25 6 oboudi oboudi ADJ cana-4831 25 7 differential differential NOUN cana-4831 25 8 operator operator NOUN cana-4831 25 9 [ [ X cana-4831 25 10 2 2 NUM cana-4831 25 11 ] ] PUNCT cana-4831 25 12 , , PUNCT cana-4831 25 13 for for ADP cana-4831 25 14 a a DET cana-4831 25 15 range range NOUN cana-4831 25 16 of of ADP cana-4831 25 17 parametric parametric ADJ cana-4831 25 18 values value NOUN cana-4831 25 19 of of ADP cana-4831 25 20 π‘ž π‘ž PROPN cana-4831 25 21 β†’ β†’ SYM cana-4831 25 22 1βˆ’ 1βˆ’ NUM cana-4831 25 23 , , PUNCT cana-4831 25 24 πœ† πœ† NOUN cana-4831 25 25 = = SYM cana-4831 25 26 0 0 PROPN cana-4831 25 27 . . PUNCT cana-4831 26 1 the the DET cana-4831 26 2 carlson carlson PROPN cana-4831 26 3 - - PUNCT cana-4831 26 4 shaffer shaffer NOUN cana-4831 26 5 operator operator NOUN cana-4831 26 6 [ [ X cana-4831 26 7 5 5 NUM cana-4831 26 8 ] ] PUNCT cana-4831 26 9 , , PUNCT cana-4831 26 10 is be AUX cana-4831 26 11 obtained obtain VERB cana-4831 26 12 for for ADP cana-4831 26 13 a a DET cana-4831 26 14 range range NOUN cana-4831 26 15 of of ADP cana-4831 26 16 parametric parametric ADJ cana-4831 26 17 values value NOUN cana-4831 26 18 of of ADP cana-4831 26 19 π‘ž π‘ž PROPN cana-4831 26 20 β†’ β†’ SYM cana-4831 26 21 1βˆ’ 1βˆ’ NUM cana-4831 26 22 , , PUNCT cana-4831 26 23 πœ† πœ† NOUN cana-4831 26 24 = = SYM cana-4831 26 25 1 1 X cana-4831 26 26 . . X cana-4831 27 1 for for ADP cana-4831 27 2 a a DET cana-4831 27 3 varied varied ADJ cana-4831 27 4 parametric parametric ADJ cana-4831 27 5 values value NOUN cana-4831 27 6 of of ADP cana-4831 27 7 πœ† πœ† NOUN cana-4831 27 8 = = SYM cana-4831 27 9 0 0 NUM cana-4831 27 10 , , PUNCT cana-4831 27 11 we we PRON cana-4831 27 12 get get VERB cana-4831 27 13 the the DET cana-4831 27 14 differential differential ADJ cana-4831 27 15 operator operator NOUN cana-4831 27 16 investigated investigate VERB cana-4831 27 17 by by ADP cana-4831 27 18 dileep dileep PROPN cana-4831 27 19 l l PROPN cana-4831 27 20 and and CCONJ cana-4831 27 21 mallige mallige PROPN cana-4831 27 22 rajeev rajeev PROPN cana-4831 28 1 [ [ X cana-4831 28 2 9 9 NUM cana-4831 28 3 ] ] PUNCT cana-4831 28 4 . . PUNCT cana-4831 29 1 we we PRON cana-4831 29 2 obtain obtain VERB cana-4831 29 3 the the DET cana-4831 29 4 operator operator NOUN cana-4831 29 5 studied study VERB cana-4831 29 6 by by ADP cana-4831 29 7 dileep dileep PROPN cana-4831 29 8 l l PROPN cana-4831 29 9 and and CCONJ cana-4831 29 10 s s X cana-4831 29 11 latha latha NOUN cana-4831 30 1 [ [ X cana-4831 30 2 8 8 NUM cana-4831 30 3 ] ] PUNCT cana-4831 30 4 , , PUNCT cana-4831 30 5 for for ADP cana-4831 30 6 π‘ž π‘ž X cana-4831 30 7 β†’ β†’ SYM cana-4831 30 8 1βˆ’ 1βˆ’ NUM cana-4831 30 9 , , PUNCT cana-4831 30 10 𝛿 𝛿 ADJ cana-4831 30 11 = = ADJ cana-4831 30 12 1 1 NUM cana-4831 30 13 . . PUNCT cana-4831 30 14 now now ADV cana-4831 30 15 using use VERB cana-4831 30 16 linear linear PROPN cana-4831 30 17 differential differential ADJ cana-4831 30 18 operator operator NOUN cana-4831 30 19 𝑨𝑺𝝀,𝒒 𝑨𝑺𝝀,𝒒 PRON cana-4831 30 20 𝜹,𝒏 𝜹,𝒏 NOUN cana-4831 30 21 , , PUNCT cana-4831 30 22 we we PRON cana-4831 30 23 define define VERB cana-4831 30 24 the the DET cana-4831 30 25 following follow VERB cana-4831 30 26 subclass subclass NOUN cana-4831 30 27 of of ADP cana-4831 30 28 𝑇. 𝑇. PROPN cana-4831 30 29 let let VERB cana-4831 30 30 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 30 31 , , PUNCT cana-4831 30 32 𝜷 𝜷 NOUN cana-4831 30 33 , , PUNCT cana-4831 30 34 𝜹 𝜹 X cana-4831 30 35 , , PUNCT cana-4831 30 36 𝝀 𝝀 NOUN cana-4831 30 37 ; ; PUNCT cana-4831 30 38 𝒒 𝒒 X cana-4831 30 39 ) ) PUNCT cana-4831 30 40 be be AUX cana-4831 30 41 the the DET cana-4831 30 42 subclass subclass NOUN cana-4831 30 43 of of ADP cana-4831 30 44 𝑇 𝑇 PROPN cana-4831 30 45 consisting consist VERB cana-4831 30 46 of of ADP cana-4831 30 47 functions function NOUN cana-4831 30 48 which which PRON cana-4831 30 49 satisfy satisfy VERB cana-4831 30 50 the the DET cana-4831 30 51 conditions condition NOUN cana-4831 30 52 𝑅 𝑅 PROPN cana-4831 30 53 { { PUNCT cana-4831 30 54 𝑧(𝑨𝑺𝝀,𝒒 𝑧(𝑨𝑺𝝀,𝒒 NOUN cana-4831 30 55 𝜹,𝒏𝑓 𝜹,𝒏𝑓 PROPN cana-4831 30 56 ) ) PUNCT cana-4831 30 57 β€² β€² NUM cana-4831 31 1 𝛽𝑧 𝛽𝑧 NOUN cana-4831 31 2 ( ( PUNCT cana-4831 31 3 𝑨𝑺𝝀,𝒒 𝑨𝑺𝝀,𝒒 X cana-4831 31 4 𝜹,𝒏𝒇 𝜹,𝒏𝒇 NUM cana-4831 31 5 ) ) PUNCT cana-4831 31 6 β€² β€² PUNCT cana-4831 32 1 + + ADJ cana-4831 32 2 ( ( PUNCT cana-4831 32 3 πŸβˆ’πœ·)𝑨𝑺𝝀,𝒒 πŸβˆ’πœ·)𝑨𝑺𝝀,𝒒 NOUN cana-4831 32 4 𝜹,𝒏𝒇 𝜹,𝒏𝒇 PROPN cana-4831 32 5 } } PUNCT cana-4831 32 6 > > PUNCT cana-4831 32 7 𝛼 𝛼 X cana-4831 32 8 , , PUNCT cana-4831 32 9 ( ( PUNCT cana-4831 32 10 1.5 1.5 NUM cana-4831 32 11 ) ) PUNCT cana-4831 32 12 for for ADP cana-4831 32 13 some some DET cana-4831 32 14 𝛼 𝛼 NOUN cana-4831 32 15 , , PUNCT cana-4831 32 16 𝛽 𝛽 PROPN cana-4831 32 17 ( ( PUNCT cana-4831 32 18 0 0 NUM cana-4831 32 19 ≀ ≀ NUM cana-4831 32 20 𝛼 𝛼 NOUN cana-4831 32 21 , , PUNCT cana-4831 32 22 𝛽 𝛽 NOUN cana-4831 32 23 < < X cana-4831 32 24 1 1 NUM cana-4831 32 25 ) ) PUNCT cana-4831 32 26 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB cana-4831 32 27 𝑛 𝑛 PROPN cana-4831 32 28 ∈ ∈ PROPN cana-4831 32 29 β„•0 β„•0 NOUN cana-4831 32 30 . . PUNCT cana-4831 33 1 for for ADP cana-4831 33 2 a a DET cana-4831 33 3 different different ADJ cana-4831 33 4 parametric parametric ADJ cana-4831 33 5 values value NOUN cana-4831 33 6 of of ADP cana-4831 33 7 π‘ž π‘ž PROPN cana-4831 33 8 β†’ β†’ SYM cana-4831 33 9 1βˆ’ 1βˆ’ NUM cana-4831 33 10 , , PUNCT cana-4831 33 11 𝛿 𝛿 ADJ cana-4831 33 12 = = SYM cana-4831 33 13 1 1 NUM cana-4831 33 14 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN cana-4831 33 15 πœ† πœ† PROPN cana-4831 34 1 = = SYM cana-4831 34 2 0 0 NUM cana-4831 35 1 the the DET cana-4831 35 2 above above ADJ cana-4831 35 3 class class NOUN cana-4831 35 4 reduces reduce VERB cana-4831 35 5 to to ADP cana-4831 35 6 the the DET cana-4831 35 7 class class NOUN cana-4831 35 8 defined define VERB cana-4831 35 9 by by ADP cana-4831 35 10 dileep dileep PROPN cana-4831 35 11 l l PROPN cana-4831 35 12 and and CCONJ cana-4831 35 13 s s X cana-4831 35 14 latha latha NOUN cana-4831 36 1 [ [ X cana-4831 36 2 8 8 NUM cana-4831 36 3 ] ] SYM cana-4831 36 4 . . PUNCT cana-4831 37 1 2 2 X cana-4831 37 2 . . X cana-4831 37 3 prime prime ADJ cana-4831 37 4 results result NOUN cana-4831 37 5 : : PUNCT cana-4831 37 6 theorem theorem VERB cana-4831 37 7 2.1 2.1 NUM cana-4831 37 8 : : PUNCT cana-4831 37 9 a a DET cana-4831 37 10 function function NOUN cana-4831 37 11 𝑓 𝑓 PRON cana-4831 37 12 defined define VERB cana-4831 37 13 by by ADP cana-4831 37 14 ( ( PUNCT cana-4831 37 15 1.2 1.2 NUM cana-4831 37 16 ) ) PUNCT cana-4831 37 17 is be AUX cana-4831 37 18 in in ADP cana-4831 37 19 the the DET cana-4831 37 20 class class NOUN cana-4831 37 21 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 37 22 , , PUNCT cana-4831 37 23 𝜷 𝜷 NOUN cana-4831 37 24 , , PUNCT cana-4831 37 25 𝜹 𝜹 X cana-4831 37 26 , , PUNCT cana-4831 37 27 𝝀 𝝀 NOUN cana-4831 37 28 ; ; PUNCT cana-4831 37 29 𝒒 𝒒 X cana-4831 37 30 ) ) PUNCT cana-4831 37 31 if if SCONJ cana-4831 38 1 and and CCONJ cana-4831 38 2 only only ADV cana-4831 38 3 if if SCONJ cana-4831 38 4 βˆ‘ βˆ‘ PUNCT cana-4831 38 5 π΅πœ† π΅πœ† PROPN cana-4831 38 6 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 38 7 , , PUNCT cana-4831 38 8 𝑐 𝑐 PROPN cana-4831 38 9 , , PUNCT cana-4831 38 10 𝑗 𝑗 NOUN cana-4831 38 11 , , PUNCT cana-4831 38 12 𝑛 𝑛 ADJ cana-4831 38 13 ; ; PUNCT cana-4831 38 14 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 39 1 [ [ X cana-4831 39 2 𝑗 𝑗 INTJ cana-4831 39 3 βˆ’ βˆ’ NOUN cana-4831 39 4 𝛼 𝛼 SYM cana-4831 39 5 + + NOUN cana-4831 39 6 𝛼𝛽 𝛼𝛽 PROPN cana-4831 39 7 βˆ’ βˆ’ PROPN cana-4831 39 8 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 39 9 ] ] X cana-4831 39 10 < < X cana-4831 39 11 1 1 NUM cana-4831 39 12 βˆ’ βˆ’ NOUN cana-4831 39 13 𝛼,∞ 𝛼,∞ PROPN cana-4831 39 14 𝑗=2 𝑗=2 X cana-4831 39 15 ( ( PUNCT cana-4831 39 16 2.1 2.1 NUM cana-4831 39 17 ) ) PUNCT cana-4831 39 18 where where SCONJ cana-4831 39 19 , , PUNCT cana-4831 39 20 𝛼 𝛼 X cana-4831 39 21 , , PUNCT cana-4831 39 22 𝛽 𝛽 PROPN cana-4831 39 23 ( ( PUNCT cana-4831 39 24 0 0 NUM cana-4831 39 25 ≀ ≀ NUM cana-4831 39 26 𝛼 𝛼 NOUN cana-4831 39 27 , , PUNCT cana-4831 39 28 𝛽 𝛽 NOUN cana-4831 39 29 < < X cana-4831 39 30 1 1 NUM cana-4831 39 31 ) ) PUNCT cana-4831 39 32 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB cana-4831 39 33 𝑛 𝑛 PROPN cana-4831 39 34 ∈ ∈ PROPN cana-4831 39 35 β„•0 β„•0 NOUN cana-4831 39 36 . . PUNCT cana-4831 40 1 proof proof NOUN cana-4831 40 2 : : PUNCT cana-4831 40 3 suppose suppose VERB cana-4831 40 4 𝑓 𝑓 DET cana-4831 40 5 ∈ ∈ NOUN cana-4831 40 6 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 40 7 , , PUNCT cana-4831 40 8 𝜷 𝜷 NOUN cana-4831 40 9 , , PUNCT cana-4831 40 10 𝜹 𝜹 X cana-4831 40 11 , , PUNCT cana-4831 40 12 𝝀 𝝀 NOUN cana-4831 40 13 ; ; PUNCT cana-4831 40 14 𝒒 𝒒 X cana-4831 40 15 ) ) PUNCT cana-4831 40 16 . . PUNCT cana-4831 41 1 then then ADV cana-4831 41 2 𝑅 𝑅 PROPN cana-4831 41 3 { { PUNCT cana-4831 41 4 𝑧(𝑨𝑺𝝀,𝒒 𝑧(𝑨𝑺𝝀,𝒒 X cana-4831 41 5 𝜹,𝒏𝑓)β€² 𝜹,𝒏𝑓)β€² X cana-4831 41 6 𝛽𝑧 𝛽𝑧 X cana-4831 41 7 ( ( PUNCT cana-4831 41 8 𝑨𝑺𝝀,𝒒 𝑨𝑺𝝀,𝒒 X cana-4831 41 9 𝜹,𝒏𝒇 𝜹,𝒏𝒇 PUNCT cana-4831 41 10 ) ) PUNCT cana-4831 41 11 β€² β€² PUNCT cana-4831 42 1 + + CCONJ cana-4831 42 2 ( ( PUNCT cana-4831 42 3 𝟏 𝟏 NUM cana-4831 42 4 βˆ’ βˆ’ PROPN cana-4831 42 5 𝜷)𝑨𝑺𝝀,𝒒 𝜷)𝑨𝑺𝝀,𝒒 NOUN cana-4831 42 6 𝜹,𝒏𝒇 𝜹,𝒏𝒇 PUNCT cana-4831 42 7 } } PUNCT cana-4831 42 8 > > X cana-4831 42 9 𝛼 𝛼 X cana-4831 42 10 𝑅 𝑅 PROPN cana-4831 42 11 { { PUNCT cana-4831 42 12 𝑧 𝑧 PRON cana-4831 42 13 βˆ’ βˆ’ PROPN cana-4831 42 14 βˆ‘ βˆ‘ PUNCT cana-4831 42 15 π‘—π΅πœ† π‘—π΅πœ† PROPN cana-4831 42 16 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 42 17 , , PUNCT cana-4831 42 18 𝑐 𝑐 PROPN cana-4831 42 19 , , PUNCT cana-4831 42 20 𝑗 𝑗 NOUN cana-4831 42 21 , , PUNCT cana-4831 42 22 𝑛 𝑛 ADJ cana-4831 42 23 ; ; PUNCT cana-4831 42 24 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 42 25 𝑧 𝑧 X cana-4831 42 26 π‘—βˆž π‘—βˆž VERB cana-4831 42 27 𝑗=2 𝑗=2 X cana-4831 43 1 𝛽 𝛽 NOUN cana-4831 43 2 [ [ X cana-4831 43 3 𝑧 𝑧 X cana-4831 43 4 βˆ’ βˆ’ PROPN cana-4831 43 5 βˆ‘ βˆ‘ PUNCT cana-4831 43 6 π‘—π΅πœ† π‘—π΅πœ† PROPN cana-4831 43 7 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 43 8 , , PUNCT cana-4831 43 9 𝑐 𝑐 PROPN cana-4831 43 10 , , PUNCT cana-4831 43 11 𝑗 𝑗 NOUN cana-4831 43 12 , , PUNCT cana-4831 43 13 𝑛 𝑛 ADJ cana-4831 43 14 ; ; PUNCT cana-4831 43 15 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 43 16 π‘§π‘—βˆž π‘§π‘—βˆž PROPN cana-4831 43 17 𝑗=2 𝑗=2 X cana-4831 43 18 ] ] PUNCT cana-4831 44 1 + + CCONJ cana-4831 44 2 ( ( PUNCT cana-4831 44 3 𝟏 𝟏 X cana-4831 44 4 βˆ’ βˆ’ NOUN cana-4831 44 5 𝜷)[𝑧 𝜷)[𝑧 INTJ cana-4831 44 6 βˆ’ βˆ’ VERB cana-4831 44 7 βˆ‘ βˆ‘ PROPN cana-4831 44 8 π΅πœ† π΅πœ† PROPN cana-4831 44 9 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 44 10 , , PUNCT cana-4831 44 11 𝑐 𝑐 PROPN cana-4831 44 12 , , PUNCT cana-4831 44 13 𝑗 𝑗 NOUN cana-4831 44 14 , , PUNCT cana-4831 44 15 𝑛 𝑛 ADJ cana-4831 44 16 ; ; PUNCT cana-4831 44 17 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 44 18 π‘§π‘—βˆž π‘§π‘—βˆž PROPN cana-4831 44 19 𝑗=2 𝑗=2 X cana-4831 44 20 ] ] PUNCT cana-4831 44 21 } } PUNCT cana-4831 44 22 > > X cana-4831 44 23 𝛼 𝛼 NOUN cana-4831 44 24 communications communication NOUN cana-4831 44 25 on on ADP cana-4831 44 26 applied apply VERB cana-4831 44 27 nonlinear nonlinear ADJ cana-4831 44 28 analysis analysis NOUN cana-4831 44 29 issn issn NOUN cana-4831 44 30 : : PUNCT cana-4831 44 31 1074 1074 NUM cana-4831 44 32 - - PUNCT cana-4831 44 33 133x 133x NUM cana-4831 44 34 vol vol VERB cana-4831 44 35 32 32 NUM cana-4831 44 36 no no NOUN cana-4831 44 37 . . PUNCT cana-4831 45 1 10s 10 NOUN cana-4831 45 2 ( ( PUNCT cana-4831 45 3 2025 2025 NUM cana-4831 45 4 ) ) PUNCT cana-4831 45 5 384 384 NUM cana-4831 45 6 https://internationalpubls.com https://internationalpubls.com X cana-4831 45 7 𝑅 𝑅 PROPN cana-4831 45 8 { { PUNCT cana-4831 45 9 π‘§βˆ’βˆ‘ π‘§βˆ’βˆ‘ PROPN cana-4831 45 10 π‘—π΅πœ† π‘—π΅πœ† NOUN cana-4831 45 11 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— NOUN cana-4831 45 12 𝑧 𝑧 PROPN cana-4831 45 13 π‘—βˆž π‘—βˆž PUNCT cana-4831 45 14 𝑗=2 𝑗=2 X cana-4831 46 1 π’›βˆ’βˆ‘ π’›βˆ’βˆ‘ VERB cana-4831 46 2 π΅πœ† π΅πœ† PRON cana-4831 46 3 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— NOUN cana-4831 46 4 𝑧 𝑧 NOUN cana-4831 46 5 π‘—βˆž π‘—βˆž PROPN cana-4831 46 6 𝑗=2 𝑗=2 PUNCT cana-4831 46 7 } } PUNCT cana-4831 46 8 > > PUNCT cana-4831 46 9 𝛼. 𝛼. PROPN cana-4831 46 10 let let VERB cana-4831 46 11 𝑧 𝑧 PRON cana-4831 46 12 β†’ β†’ X cana-4831 46 13 1 1 NUM cana-4831 46 14 , , PUNCT cana-4831 46 15 then then ADV cana-4831 46 16 we we PRON cana-4831 46 17 get get VERB cana-4831 46 18 1 1 NUM cana-4831 46 19 βˆ’ βˆ’ NOUN cana-4831 46 20 βˆ‘ βˆ‘ PUNCT cana-4831 46 21 π‘—π΅πœ† π‘—π΅πœ† PROPN cana-4831 46 22 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 46 23 , , PUNCT cana-4831 46 24 𝑐 𝑐 PROPN cana-4831 46 25 , , PUNCT cana-4831 46 26 𝑗 𝑗 NOUN cana-4831 46 27 , , PUNCT cana-4831 46 28 𝑛 𝑛 ADJ cana-4831 46 29 ; ; PUNCT cana-4831 47 1 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 47 2 ∞ ∞ PROPN cana-4831 47 3 𝑗=2 𝑗=2 X cana-4831 48 1 > > X cana-4831 48 2 𝛼 𝛼 X cana-4831 48 3 { { PUNCT cana-4831 48 4 1 1 NUM cana-4831 48 5 βˆ’ βˆ’ NOUN cana-4831 48 6 βˆ‘ βˆ‘ PUNCT cana-4831 48 7 π΅πœ† π΅πœ† PROPN cana-4831 48 8 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 48 9 , , PUNCT cana-4831 48 10 𝑐 𝑐 PROPN cana-4831 48 11 , , PUNCT cana-4831 48 12 𝑗 𝑗 NOUN cana-4831 48 13 , , PUNCT cana-4831 48 14 𝑛 𝑛 NOUN cana-4831 48 15 ; ; PUNCT cana-4831 48 16 π‘ž)[𝛽(𝑗 π‘ž)[𝛽(𝑗 ADP cana-4831 48 17 βˆ’ βˆ’ PROPN cana-4831 48 18 1 1 NUM cana-4831 48 19 ) ) PUNCT cana-4831 48 20 + + CCONJ cana-4831 48 21 1]π‘Žπ‘— 1]π‘Žπ‘— NUM cana-4831 48 22 ∞ ∞ NUM cana-4831 48 23 𝑗=2 𝑗=2 PROPN cana-4831 48 24 } } PUNCT cana-4831 48 25 βˆ‘ βˆ‘ PROPN cana-4831 48 26 π‘—π΅πœ† π‘—π΅πœ† PROPN cana-4831 48 27 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 48 28 , , PUNCT cana-4831 48 29 𝑐 𝑐 PROPN cana-4831 48 30 , , PUNCT cana-4831 48 31 𝑗 𝑗 NOUN cana-4831 48 32 , , PUNCT cana-4831 48 33 𝑛 𝑛 ADJ cana-4831 48 34 ; ; PUNCT cana-4831 48 35 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 48 36 ∞ ∞ PROPN cana-4831 48 37 𝑗=2 𝑗=2 PUNCT cana-4831 49 1 βˆ’ βˆ’ NOUN cana-4831 49 2 𝛼 𝛼 NOUN cana-4831 49 3 βˆ‘ βˆ‘ PUNCT cana-4831 49 4 π΅πœ† π΅πœ† PROPN cana-4831 49 5 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 49 6 , , PUNCT cana-4831 49 7 𝑐 𝑐 PROPN cana-4831 49 8 , , PUNCT cana-4831 49 9 𝑗 𝑗 NOUN cana-4831 49 10 , , PUNCT cana-4831 49 11 𝑛 𝑛 NOUN cana-4831 49 12 ; ; PUNCT cana-4831 49 13 π‘ž)[𝛽(𝑗 π‘ž)[𝛽(𝑗 ADP cana-4831 49 14 βˆ’ βˆ’ PROPN cana-4831 49 15 1 1 NUM cana-4831 49 16 ) ) PUNCT cana-4831 49 17 + + CCONJ cana-4831 49 18 1]π‘Žπ‘— 1]π‘Žπ‘— NUM cana-4831 49 19 ∞ ∞ NUM cana-4831 49 20 𝑗=2 𝑗=2 X cana-4831 50 1 < < X cana-4831 50 2 1 1 NUM cana-4831 50 3 βˆ’ βˆ’ NUM cana-4831 50 4 𝛼 𝛼 NOUN cana-4831 50 5 βˆ‘ βˆ‘ PUNCT cana-4831 50 6 π΅πœ† π΅πœ† PROPN cana-4831 50 7 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 50 8 , , PUNCT cana-4831 50 9 𝑐 𝑐 PROPN cana-4831 50 10 , , PUNCT cana-4831 50 11 𝑗 𝑗 NOUN cana-4831 50 12 , , PUNCT cana-4831 50 13 𝑛 𝑛 ADJ cana-4831 50 14 ; ; PUNCT cana-4831 50 15 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 51 1 [ [ X cana-4831 51 2 𝑗 𝑗 INTJ cana-4831 51 3 βˆ’ βˆ’ NOUN cana-4831 51 4 𝛼 𝛼 SYM cana-4831 51 5 + + NOUN cana-4831 51 6 𝛼𝛽 𝛼𝛽 PROPN cana-4831 51 7 βˆ’ βˆ’ PROPN cana-4831 51 8 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 51 9 ] ] X cana-4831 51 10 < < X cana-4831 51 11 1 1 NUM cana-4831 51 12 βˆ’ βˆ’ PROPN cana-4831 51 13 𝛼. 𝛼. NOUN cana-4831 51 14 ∞ ∞ PROPN cana-4831 51 15 𝑗=2 𝑗=2 PUNCT cana-4831 51 16 conversely conversely ADV cana-4831 51 17 , , PUNCT cana-4831 51 18 assume assume VERB cana-4831 51 19 that that SCONJ cana-4831 51 20 ( ( PUNCT cana-4831 51 21 2.1 2.1 NUM cana-4831 51 22 ) ) PUNCT cana-4831 51 23 be be AUX cana-4831 51 24 true true ADJ cana-4831 51 25 . . PUNCT cana-4831 52 1 we we PRON cana-4831 52 2 have have VERB cana-4831 52 3 to to PART cana-4831 52 4 show show VERB cana-4831 52 5 that that SCONJ cana-4831 52 6 ( ( PUNCT cana-4831 52 7 1.5 1.5 NUM cana-4831 52 8 ) ) PUNCT cana-4831 52 9 is be AUX cana-4831 52 10 satisfied satisfied ADJ cana-4831 52 11 or or CCONJ cana-4831 52 12 equivalently equivalently ADV cana-4831 52 13 | | ADV cana-4831 52 14 𝑧(𝑨𝑺𝝀,𝒒 𝑧(𝑨𝑺𝝀,𝒒 X cana-4831 52 15 𝜹,𝒏𝑓)β€² 𝜹,𝒏𝑓)β€² X cana-4831 52 16 𝛽𝑧 𝛽𝑧 X cana-4831 52 17 ( ( PUNCT cana-4831 52 18 𝑨𝑺𝝀,𝒒 𝑨𝑺𝝀,𝒒 X cana-4831 52 19 𝜹,𝒏𝒇 𝜹,𝒏𝒇 PUNCT cana-4831 52 20 ) ) PUNCT cana-4831 52 21 β€² β€² PUNCT cana-4831 53 1 + + CCONJ cana-4831 53 2 ( ( PUNCT cana-4831 53 3 𝟏 𝟏 NUM cana-4831 53 4 βˆ’ βˆ’ PROPN cana-4831 53 5 𝜷)𝑨𝑺𝝀,𝒒 𝜷)𝑨𝑺𝝀,𝒒 PROPN cana-4831 53 6 𝜹,𝒏𝒇 𝜹,𝒏𝒇 NUM cana-4831 53 7 βˆ’ βˆ’ PROPN cana-4831 53 8 1| 1| NUM cana-4831 53 9 < < X cana-4831 53 10 1 1 NUM cana-4831 53 11 βˆ’ βˆ’ PROPN cana-4831 53 12 𝛼. 𝛼. NOUN cana-4831 53 13 but but CCONJ cana-4831 53 14 | | ADV cana-4831 53 15 𝑧 𝑧 ADV cana-4831 53 16 βˆ’ βˆ’ PROPN cana-4831 53 17 βˆ‘ βˆ‘ PUNCT cana-4831 53 18 π‘—π΅πœ† π‘—π΅πœ† PROPN cana-4831 53 19 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 53 20 , , PUNCT cana-4831 53 21 𝑐 𝑐 PROPN cana-4831 53 22 , , PUNCT cana-4831 53 23 𝑗 𝑗 NOUN cana-4831 53 24 , , PUNCT cana-4831 53 25 𝑛 𝑛 ADJ cana-4831 53 26 ; ; PUNCT cana-4831 53 27 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 53 28 𝑧 𝑧 X cana-4831 53 29 π‘—βˆž π‘—βˆž VERB cana-4831 53 30 𝑗=2 𝑗=2 PROPN cana-4831 54 1 𝒛 𝒛 INTJ cana-4831 54 2 βˆ’ βˆ’ NOUN cana-4831 54 3 βˆ‘ βˆ‘ PUNCT cana-4831 54 4 π΅πœ† π΅πœ† PROPN cana-4831 54 5 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 54 6 , , PUNCT cana-4831 54 7 𝑐 𝑐 PROPN cana-4831 54 8 , , PUNCT cana-4831 54 9 𝑗 𝑗 NOUN cana-4831 54 10 , , PUNCT cana-4831 54 11 𝑛 𝑛 NOUN cana-4831 54 12 ; ; PUNCT cana-4831 54 13 π‘ž)[𝛽(𝑗 π‘ž)[𝛽(𝑗 ADP cana-4831 54 14 βˆ’ βˆ’ PROPN cana-4831 54 15 1 1 NUM cana-4831 54 16 ) ) PUNCT cana-4831 54 17 + + CCONJ cana-4831 55 1 1]π‘Žπ‘— 1]π‘Žπ‘— NUM cana-4831 55 2 π‘§π‘—βˆž π‘§π‘—βˆž PROPN cana-4831 55 3 𝑗=2 𝑗=2 PUNCT cana-4831 56 1 βˆ’ βˆ’ PROPN cana-4831 57 1 1| 1| NUM cana-4831 58 1 = = SYM cana-4831 59 1 | | ADV cana-4831 59 2 βˆ‘ βˆ‘ PUNCT cana-4831 59 3 π΅πœ† π΅πœ† PROPN cana-4831 59 4 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 59 5 , , PUNCT cana-4831 59 6 𝑐 𝑐 PROPN cana-4831 59 7 , , PUNCT cana-4831 59 8 𝑗 𝑗 NOUN cana-4831 59 9 , , PUNCT cana-4831 59 10 𝑛 𝑛 ADJ cana-4831 59 11 ; ; PUNCT cana-4831 59 12 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 59 13 ( ( PUNCT cana-4831 59 14 𝑗 𝑗 INTJ cana-4831 59 15 βˆ’ βˆ’ PROPN cana-4831 59 16 1)(𝛽 1)(𝛽 NUM cana-4831 59 17 βˆ’ βˆ’ NOUN cana-4831 59 18 1)π‘§π‘—βˆž 1)π‘§π‘—βˆž NOUN cana-4831 59 19 𝑗=2 𝑗=2 PROPN cana-4831 60 1 𝒛 𝒛 CCONJ cana-4831 60 2 βˆ’ βˆ’ NOUN cana-4831 60 3 βˆ‘ βˆ‘ PUNCT cana-4831 60 4 π΅πœ† π΅πœ† PROPN cana-4831 60 5 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 60 6 , , PUNCT cana-4831 60 7 𝑐 𝑐 PROPN cana-4831 60 8 , , PUNCT cana-4831 60 9 𝑗 𝑗 NOUN cana-4831 60 10 , , PUNCT cana-4831 60 11 𝑛 𝑛 NOUN cana-4831 60 12 ; ; PUNCT cana-4831 60 13 π‘ž)[𝛽(𝑗 π‘ž)[𝛽(𝑗 ADP cana-4831 60 14 βˆ’ βˆ’ PROPN cana-4831 60 15 1 1 NUM cana-4831 60 16 ) ) PUNCT cana-4831 60 17 + + CCONJ cana-4831 61 1 1]π‘Žπ‘— 1]π‘Žπ‘— NUM cana-4831 61 2 π‘§π‘—βˆž π‘§π‘—βˆž PROPN cana-4831 61 3 𝑗=2 𝑗=2 PROPN cana-4831 62 1 | | ADV cana-4831 62 2 ≀ ≀ X cana-4831 62 3 βˆ‘ βˆ‘ PUNCT cana-4831 62 4 π΅πœ† π΅πœ† PROPN cana-4831 62 5 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— PROPN cana-4831 62 6 ( ( PUNCT cana-4831 62 7 π‘—βˆ’1)(π›½βˆ’1)|𝑧𝑗|∞ π‘—βˆ’1)(π›½βˆ’1)|𝑧𝑗|∞ PROPN cana-4831 62 8 𝑗=2 𝑗=2 X cana-4831 62 9 |𝒛|βˆ’βˆ‘ |𝒛|βˆ’βˆ‘ PUNCT cana-4831 63 1 π΅πœ† π΅πœ† PRON cana-4831 63 2 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— NOUN cana-4831 63 3 |𝑧 |𝑧 NOUN cana-4831 63 4 𝑗|∞ 𝑗|∞ X cana-4831 63 5 𝑗=2 𝑗=2 PUNCT cana-4831 63 6 ≀ ≀ NUM cana-4831 63 7 βˆ‘ βˆ‘ PUNCT cana-4831 63 8 π΅πœ† π΅πœ† PROPN cana-4831 63 9 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)π‘Žπ‘— PROPN cana-4831 63 10 ( ( PUNCT cana-4831 63 11 π‘—βˆ’1)(π›½βˆ’1)∞ π‘—βˆ’1)(π›½βˆ’1)∞ SYM cana-4831 63 12 𝑗=2 𝑗=2 PROPN cana-4831 63 13 1βˆ’βˆ‘ 1βˆ’βˆ‘ NUM cana-4831 64 1 π΅πœ† π΅πœ† PRON cana-4831 64 2 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[𝛽(π‘—βˆ’1)+1]π‘Žπ‘— VERB cana-4831 64 3 ∞ ∞ PROPN cana-4831 64 4 𝑗=2 𝑗=2 PROPN cana-4831 64 5 . . PUNCT cana-4831 65 1 the the DET cana-4831 65 2 last last ADJ cana-4831 65 3 expression expression NOUN cana-4831 65 4 is be AUX cana-4831 65 5 bounded bound VERB cana-4831 65 6 by by ADP cana-4831 65 7 1 1 NUM cana-4831 65 8 βˆ’ βˆ’ NOUN cana-4831 65 9 𝛼 𝛼 NOUN cana-4831 65 10 if if SCONJ cana-4831 65 11 βˆ‘ βˆ‘ PUNCT cana-4831 65 12 π΅πœ† π΅πœ† PROPN cana-4831 65 13 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 65 14 , , PUNCT cana-4831 65 15 𝑐 𝑐 PROPN cana-4831 65 16 , , PUNCT cana-4831 65 17 𝑗 𝑗 NOUN cana-4831 65 18 , , PUNCT cana-4831 65 19 𝑛 𝑛 ADJ cana-4831 65 20 ; ; PUNCT cana-4831 65 21 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 65 22 ( ( PUNCT cana-4831 65 23 𝑗 𝑗 INTJ cana-4831 66 1 βˆ’ βˆ’ NUM cana-4831 66 2 1)(𝛽 1)(𝛽 NUM cana-4831 66 3 βˆ’ βˆ’ NOUN cana-4831 66 4 1 1 NUM cana-4831 66 5 ) ) PUNCT cana-4831 66 6 ∞ ∞ NUM cana-4831 66 7 𝑗=2 𝑗=2 PROPN cana-4831 66 8 ≀ ≀ NUM cana-4831 66 9 ( ( PUNCT cana-4831 66 10 1 1 NUM cana-4831 66 11 βˆ’ βˆ’ NOUN cana-4831 66 12 𝛼 𝛼 NOUN cana-4831 66 13 ) ) PUNCT cana-4831 66 14 ( ( PUNCT cana-4831 66 15 1 1 NUM cana-4831 66 16 βˆ’ βˆ’ NOUN cana-4831 66 17 βˆ‘ βˆ‘ PUNCT cana-4831 66 18 π΅πœ† π΅πœ† PROPN cana-4831 66 19 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 66 20 , , PUNCT cana-4831 66 21 𝑐 𝑐 PROPN cana-4831 66 22 , , PUNCT cana-4831 66 23 𝑗 𝑗 NOUN cana-4831 66 24 , , PUNCT cana-4831 66 25 𝑛 𝑛 NOUN cana-4831 66 26 ; ; PUNCT cana-4831 66 27 π‘ž)[𝛽(𝑗 π‘ž)[𝛽(𝑗 ADP cana-4831 66 28 βˆ’ βˆ’ PROPN cana-4831 66 29 1 1 NUM cana-4831 66 30 ) ) PUNCT cana-4831 66 31 + + CCONJ cana-4831 66 32 1]π‘Žπ‘— 1]π‘Žπ‘— NUM cana-4831 66 33 ∞ ∞ NUM cana-4831 66 34 𝑗=2 𝑗=2 PROPN cana-4831 66 35 ) ) PUNCT cana-4831 66 36 βˆ‘ βˆ‘ PUNCT cana-4831 67 1 π΅πœ† π΅πœ† PRON cana-4831 67 2 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 67 3 , , PUNCT cana-4831 67 4 𝑐 𝑐 PROPN cana-4831 67 5 , , PUNCT cana-4831 67 6 𝑗 𝑗 NOUN cana-4831 67 7 , , PUNCT cana-4831 67 8 𝑛 𝑛 ADJ cana-4831 67 9 ; ; PUNCT cana-4831 67 10 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 68 1 [ [ X cana-4831 68 2 𝑗 𝑗 INTJ cana-4831 68 3 βˆ’ βˆ’ NOUN cana-4831 68 4 𝛼 𝛼 SYM cana-4831 68 5 + + NOUN cana-4831 68 6 𝛼𝛽 𝛼𝛽 PROPN cana-4831 68 7 βˆ’ βˆ’ PROPN cana-4831 68 8 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 68 9 ] ] X cana-4831 68 10 < < X cana-4831 68 11 1 1 NUM cana-4831 68 12 βˆ’ βˆ’ PROPN cana-4831 68 13 𝛼 𝛼 NOUN cana-4831 68 14 , , PUNCT cana-4831 68 15 ∞ ∞ PROPN cana-4831 68 16 𝑗=2 𝑗=2 PROPN cana-4831 68 17 which which PRON cana-4831 68 18 is be AUX cana-4831 68 19 true true ADJ cana-4831 68 20 by by ADP cana-4831 68 21 hypothesis hypothesis NOUN cana-4831 68 22 . . PUNCT cana-4831 69 1 this this PRON cana-4831 69 2 completes complete VERB cana-4831 69 3 the the DET cana-4831 69 4 assertion assertion NOUN cana-4831 69 5 of of ADP cana-4831 69 6 theorem theorem ADJ cana-4831 69 7 2.1 2.1 NUM cana-4831 69 8 for for ADP cana-4831 69 9 parametric parametric ADJ cana-4831 69 10 values value NOUN cana-4831 69 11 of of ADP cana-4831 69 12 π‘ž π‘ž PROPN cana-4831 69 13 β†’ β†’ SYM cana-4831 69 14 1βˆ’ 1βˆ’ NUM cana-4831 69 15 , , PUNCT cana-4831 69 16 𝛿 𝛿 ADJ cana-4831 69 17 = = ADJ cana-4831 69 18 1 1 NUM cana-4831 69 19 , , PUNCT cana-4831 69 20 πœ† πœ† NOUN cana-4831 69 21 = = SYM cana-4831 69 22 0 0 NUM cana-4831 69 23 and and CCONJ cana-4831 69 24 different different ADJ cana-4831 69 25 values value NOUN cana-4831 69 26 of of ADP cana-4831 69 27 𝑛 𝑛 PROPN cana-4831 69 28 ( ( PUNCT cana-4831 69 29 𝑛 𝑛 PROPN cana-4831 69 30 = = SYM cana-4831 69 31 0 0 NUM cana-4831 69 32 , , PUNCT cana-4831 69 33 1 1 NUM cana-4831 69 34 ) ) PUNCT cana-4831 69 35 in in ADP cana-4831 69 36 the the DET cana-4831 69 37 above above ADJ cana-4831 69 38 theorem theorem NOUN cana-4831 69 39 , , PUNCT cana-4831 69 40 we we PRON cana-4831 69 41 have have VERB cana-4831 69 42 the the DET cana-4831 69 43 following follow VERB cana-4831 69 44 results result NOUN cana-4831 69 45 of of ADP cana-4831 69 46 a a DET cana-4831 69 47 o o NOUN cana-4831 69 48 mostafa mostafa PROPN cana-4831 70 1 [ [ X cana-4831 70 2 15 15 NUM cana-4831 70 3 ] ] PUNCT cana-4831 70 4 . . PUNCT cana-4831 71 1 corollary corollary ADJ cana-4831 71 2 2.2 2.2 NUM cana-4831 71 3 : : PUNCT cana-4831 71 4 ( ( PUNCT cana-4831 71 5 π’Š π’Š NOUN cana-4831 71 6 ) ) PUNCT cana-4831 71 7 a a DET cana-4831 71 8 function function NOUN cana-4831 71 9 𝑓 𝑓 PRON cana-4831 71 10 defined define VERB cana-4831 71 11 by by ADP cana-4831 71 12 ( ( PUNCT cana-4831 71 13 1.2 1.2 NUM cana-4831 71 14 ) ) PUNCT cana-4831 71 15 is be AUX cana-4831 71 16 in in ADP cana-4831 71 17 the the DET cana-4831 71 18 class class NOUN cana-4831 71 19 𝑻(𝜢 𝑻(𝜢 PROPN cana-4831 71 20 , , PUNCT cana-4831 71 21 𝜷 𝜷 NOUN cana-4831 71 22 ) ) PUNCT cana-4831 71 23 if if SCONJ cana-4831 72 1 and and CCONJ cana-4831 72 2 only only ADV cana-4831 72 3 if if SCONJ cana-4831 72 4 communications communication NOUN cana-4831 72 5 on on ADP cana-4831 72 6 applied apply VERB cana-4831 72 7 nonlinear nonlinear ADJ cana-4831 72 8 analysis analysis NOUN cana-4831 72 9 issn issn NOUN cana-4831 72 10 : : PUNCT cana-4831 72 11 1074 1074 NUM cana-4831 72 12 - - PUNCT cana-4831 72 13 133x 133x NUM cana-4831 72 14 vol vol VERB cana-4831 72 15 32 32 NUM cana-4831 72 16 no no NOUN cana-4831 72 17 . . PUNCT cana-4831 73 1 10s 10 NOUN cana-4831 73 2 ( ( PUNCT cana-4831 73 3 2025 2025 NUM cana-4831 73 4 ) ) PUNCT cana-4831 73 5 385 385 NUM cana-4831 73 6 https://internationalpubls.com https://internationalpubls.com X cana-4831 73 7 βˆ‘ βˆ‘ PUNCT cana-4831 74 1 [ [ X cana-4831 74 2 𝑗 𝑗 INTJ cana-4831 74 3 βˆ’ βˆ’ NOUN cana-4831 74 4 𝛼 𝛼 SYM cana-4831 74 5 + + NOUN cana-4831 74 6 𝛼𝛽 𝛼𝛽 NOUN cana-4831 74 7 βˆ’ βˆ’ NUM cana-4831 74 8 𝛼𝛽𝑗]π‘Žπ‘— 𝛼𝛽𝑗]π‘Žπ‘— NOUN cana-4831 74 9 ≀ ≀ NUM cana-4831 74 10 1 1 NUM cana-4831 74 11 βˆ’ βˆ’ PROPN cana-4831 74 12 𝛼. 𝛼. NOUN cana-4831 74 13 ∞ ∞ PROPN cana-4831 74 14 𝑗=2 𝑗=2 PROPN cana-4831 74 15 ( ( PUNCT cana-4831 74 16 𝑖𝑖 𝑖𝑖 NOUN cana-4831 74 17 ) ) PUNCT cana-4831 74 18 a a DET cana-4831 74 19 function function NOUN cana-4831 74 20 𝑓 𝑓 PRON cana-4831 74 21 defined define VERB cana-4831 74 22 by by ADP cana-4831 74 23 ( ( PUNCT cana-4831 74 24 1.2 1.2 NUM cana-4831 74 25 ) ) PUNCT cana-4831 74 26 is be AUX cana-4831 74 27 in in ADP cana-4831 74 28 the the DET cana-4831 74 29 class class NOUN cana-4831 74 30 π‘ͺ(𝜢 π‘ͺ(𝜢 PROPN cana-4831 74 31 , , PUNCT cana-4831 74 32 𝜷 𝜷 NOUN cana-4831 74 33 ) ) PUNCT cana-4831 74 34 if if SCONJ cana-4831 74 35 and and CCONJ cana-4831 74 36 only only ADV cana-4831 74 37 if if SCONJ cana-4831 74 38 βˆ‘ βˆ‘ PUNCT cana-4831 74 39 𝑗[𝑗 𝑗[𝑗 NOUN cana-4831 74 40 βˆ’ βˆ’ NOUN cana-4831 74 41 𝛼 𝛼 PROPN cana-4831 74 42 + + NOUN cana-4831 74 43 𝛼𝛽 𝛼𝛽 NOUN cana-4831 74 44 βˆ’ βˆ’ NUM cana-4831 74 45 𝛼𝛽𝑗]π‘Žπ‘— 𝛼𝛽𝑗]π‘Žπ‘— NOUN cana-4831 74 46 ≀ ≀ NUM cana-4831 74 47 1 1 NUM cana-4831 74 48 βˆ’ βˆ’ PROPN cana-4831 74 49 𝛼. 𝛼. NOUN cana-4831 74 50 ∞ ∞ PROPN cana-4831 74 51 𝑗=2 𝑗=2 PROPN cana-4831 75 1 corollary corollary NOUN cana-4831 76 1 2.3 2.3 NUM cana-4831 76 2 : : PUNCT cana-4831 76 3 if if SCONJ cana-4831 76 4 𝑓 𝑓 DET cana-4831 76 5 ∈ ∈ NOUN cana-4831 76 6 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 76 7 , , PUNCT cana-4831 76 8 𝜷 𝜷 NOUN cana-4831 76 9 , , PUNCT cana-4831 76 10 𝜹 𝜹 X cana-4831 76 11 , , PUNCT cana-4831 76 12 𝝀 𝝀 NOUN cana-4831 76 13 ; ; PUNCT cana-4831 76 14 𝒒 𝒒 X cana-4831 76 15 ) ) PUNCT cana-4831 76 16 , , PUNCT cana-4831 76 17 then then ADV cana-4831 76 18 |π‘Žπ‘—| |π‘Žπ‘—| VERB cana-4831 76 19 ≀ ≀ NOUN cana-4831 76 20 1βˆ’π›Ό 1βˆ’π›Ό NUM cana-4831 77 1 π΅πœ† π΅πœ† PRON cana-4831 77 2 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[π‘—βˆ’π›Ό+π›Όπ›½βˆ’π›Όπ›½π‘— 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[π‘—βˆ’π›Ό+π›Όπ›½βˆ’π›Όπ›½π‘— NOUN cana-4831 77 3 ] ] PUNCT cana-4831 77 4 . . PUNCT cana-4831 78 1 theorem theorem ADJ cana-4831 78 2 2.4 2.4 NUM cana-4831 78 3 : : PUNCT cana-4831 78 4 let let VERB cana-4831 78 5 0 0 NUM cana-4831 78 6 ≀ ≀ NUM cana-4831 78 7 𝛼 𝛼 X cana-4831 78 8 < < X cana-4831 78 9 1 1 NUM cana-4831 78 10 , , PUNCT cana-4831 78 11 0 0 NUM cana-4831 78 12 ≀ ≀ NUM cana-4831 78 13 𝛽1 𝛽1 NOUN cana-4831 78 14 ≀ ≀ NOUN cana-4831 78 15 𝛽2 𝛽2 NOUN cana-4831 78 16 < < X cana-4831 78 17 1 1 NUM cana-4831 78 18 , , PUNCT cana-4831 78 19 𝑛 𝑛 DET cana-4831 78 20 ∈ ∈ PROPN cana-4831 78 21 𝑁0 𝑁0 VERB cana-4831 78 22 , , PUNCT cana-4831 78 23 then then ADV cana-4831 78 24 𝑻𝒏(𝜢 𝑻𝒏(𝜢 ADJ cana-4831 78 25 , , PUNCT cana-4831 78 26 𝜷𝟐 𝜷𝟐 NOUN cana-4831 78 27 , , PUNCT cana-4831 78 28 𝜹 𝜹 X cana-4831 78 29 , , PUNCT cana-4831 78 30 𝝀 𝝀 NOUN cana-4831 78 31 ; ; PUNCT cana-4831 78 32 𝒒 𝒒 X cana-4831 78 33 ) ) PUNCT cana-4831 78 34 βŠ‚ βŠ‚ NOUN cana-4831 78 35 𝑻𝒏(𝜢 𝑻𝒏(𝜢 ADJ cana-4831 78 36 , , PUNCT cana-4831 78 37 𝜷𝟏 𝜷𝟏 NOUN cana-4831 78 38 , , PUNCT cana-4831 78 39 𝜹 𝜹 X cana-4831 78 40 , , PUNCT cana-4831 78 41 𝝀 𝝀 NOUN cana-4831 78 42 ; ; PUNCT cana-4831 78 43 𝒒 𝒒 X cana-4831 78 44 ) ) PUNCT cana-4831 78 45 . . PUNCT cana-4831 79 1 proof proof NOUN cana-4831 79 2 : : PUNCT cana-4831 79 3 from from ADP cana-4831 79 4 the the DET cana-4831 79 5 theorem theorem ADJ cana-4831 79 6 2.1 2.1 NUM cana-4831 79 7 , , PUNCT cana-4831 79 8 βˆ‘ βˆ‘ PUNCT cana-4831 79 9 π΅πœ† π΅πœ† PROPN cana-4831 79 10 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 79 11 , , PUNCT cana-4831 79 12 𝑐 𝑐 PROPN cana-4831 79 13 , , PUNCT cana-4831 79 14 𝑗 𝑗 NOUN cana-4831 79 15 , , PUNCT cana-4831 79 16 𝑛 𝑛 ADJ cana-4831 79 17 ; ; PUNCT cana-4831 79 18 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 80 1 [ [ X cana-4831 80 2 𝑗 𝑗 INTJ cana-4831 80 3 βˆ’ βˆ’ NOUN cana-4831 80 4 𝛼 𝛼 NOUN cana-4831 80 5 + + NOUN cana-4831 80 6 𝛼𝛽2 𝛼𝛽2 NOUN cana-4831 81 1 βˆ’ βˆ’ PROPN cana-4831 82 1 𝛼𝛽2𝑗 𝛼𝛽2𝑗 NOUN cana-4831 82 2 ] ] X cana-4831 82 3 ∞ ∞ PROPN cana-4831 82 4 𝑗=2 𝑗=2 PROPN cana-4831 82 5 ≀ ≀ NUM cana-4831 82 6 βˆ‘ βˆ‘ PUNCT cana-4831 82 7 π΅πœ† π΅πœ† PROPN cana-4831 82 8 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 82 9 , , PUNCT cana-4831 82 10 𝑐 𝑐 PROPN cana-4831 82 11 , , PUNCT cana-4831 82 12 𝑗 𝑗 NOUN cana-4831 82 13 , , PUNCT cana-4831 82 14 𝑛 𝑛 ADJ cana-4831 82 15 ; ; PUNCT cana-4831 82 16 π‘ž)π‘Žπ‘— π‘ž)π‘Žπ‘— PROPN cana-4831 83 1 [ [ X cana-4831 83 2 𝑗 𝑗 INTJ cana-4831 83 3 βˆ’ βˆ’ X cana-4831 83 4 𝛼 𝛼 SYM cana-4831 83 5 + + NOUN cana-4831 83 6 𝛼𝛽1 𝛼𝛽1 VERB cana-4831 83 7 βˆ’ βˆ’ NOUN cana-4831 84 1 𝛼𝛽1𝑗]∞ 𝛼𝛽1𝑗]∞ PROPN cana-4831 84 2 𝑗=2 𝑗=2 PROPN cana-4831 85 1 ≀ ≀ ADV cana-4831 85 2 1 1 NUM cana-4831 85 3 βˆ’ βˆ’ NOUN cana-4831 85 4 𝛼. 𝛼. NOUN cana-4831 85 5 for for ADP cana-4831 85 6 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 85 7 ) ) PUNCT cana-4831 85 8 ∈ ∈ NOUN cana-4831 85 9 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 85 10 , , PUNCT cana-4831 85 11 𝜷𝟐 𝜷𝟐 NOUN cana-4831 85 12 , , PUNCT cana-4831 85 13 𝜹 𝜹 X cana-4831 85 14 , , PUNCT cana-4831 85 15 𝝀 𝝀 NOUN cana-4831 85 16 ; ; PUNCT cana-4831 85 17 𝒒 𝒒 X cana-4831 85 18 ) ) PUNCT cana-4831 85 19 . . PUNCT cana-4831 86 1 hence hence ADV cana-4831 86 2 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 86 3 ) ) PUNCT cana-4831 86 4 ∈ ∈ NOUN cana-4831 86 5 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 86 6 , , PUNCT cana-4831 86 7 𝜷𝟏 𝜷𝟏 NOUN cana-4831 86 8 , , PUNCT cana-4831 86 9 𝜹 𝜹 X cana-4831 86 10 , , PUNCT cana-4831 86 11 𝝀 𝝀 NOUN cana-4831 86 12 ; ; PUNCT cana-4831 86 13 𝒒 𝒒 X cana-4831 86 14 ) ) PUNCT cana-4831 86 15 . . PUNCT cana-4831 87 1 theorem theorem VERB cana-4831 87 2 2.5 2.5 NUM cana-4831 87 3 : : PUNCT cana-4831 87 4 let let VERB cana-4831 87 5 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 87 6 ) ) PUNCT cana-4831 87 7 ∈ ∈ NOUN cana-4831 87 8 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 87 9 , , PUNCT cana-4831 87 10 𝜷 𝜷 NOUN cana-4831 87 11 , , PUNCT cana-4831 87 12 𝜹 𝜹 X cana-4831 87 13 , , PUNCT cana-4831 87 14 𝝀 𝝀 NOUN cana-4831 87 15 ; ; PUNCT cana-4831 87 16 𝒒 𝒒 X cana-4831 87 17 ) ) PUNCT cana-4831 87 18 . . PUNCT cana-4831 88 1 define define VERB cana-4831 88 2 𝑓1(𝑧 𝑓1(𝑧 NOUN cana-4831 88 3 ) ) PUNCT cana-4831 88 4 = = SYM cana-4831 88 5 𝑧 𝑧 PROPN cana-4831 88 6 and and CCONJ cana-4831 88 7 𝒇𝒋(𝒛 𝒇𝒋(𝒛 ADV cana-4831 88 8 ) ) PUNCT cana-4831 89 1 = = PUNCT cana-4831 90 1 𝒛 𝒛 NOUN cana-4831 90 2 + + NOUN cana-4831 90 3 1 1 NUM cana-4831 90 4 βˆ’ βˆ’ NUM cana-4831 90 5 𝛼 𝛼 PRON cana-4831 91 1 π΅πœ† π΅πœ† PROPN cana-4831 91 2 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 91 3 , , PUNCT cana-4831 91 4 𝑐 𝑐 PROPN cana-4831 91 5 , , PUNCT cana-4831 91 6 𝑗 𝑗 NOUN cana-4831 91 7 , , PUNCT cana-4831 91 8 𝑛 𝑛 PROPN cana-4831 91 9 ; ; PUNCT cana-4831 91 10 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 91 11 βˆ’ βˆ’ PROPN cana-4831 91 12 𝛼 𝛼 NOUN cana-4831 91 13 + + NOUN cana-4831 91 14 𝛼𝛽 𝛼𝛽 PROPN cana-4831 91 15 βˆ’ βˆ’ PROPN cana-4831 91 16 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 91 17 ] ] X cana-4831 91 18 𝑧𝑗 𝑧𝑗 INTJ cana-4831 91 19 , , PUNCT cana-4831 91 20 𝑗 𝑗 NOUN cana-4831 91 21 = = SYM cana-4831 91 22 2 2 NUM cana-4831 91 23 , , PUNCT cana-4831 91 24 3 3 NUM cana-4831 91 25 , , PUNCT cana-4831 91 26 β‹― β‹― PROPN cana-4831 91 27 , , PUNCT cana-4831 91 28 for for ADP cana-4831 91 29 some some DET cana-4831 91 30 𝛼 𝛼 NOUN cana-4831 91 31 , , PUNCT cana-4831 91 32 𝛽 𝛽 PROPN cana-4831 91 33 ( ( PUNCT cana-4831 91 34 0 0 NUM cana-4831 91 35 ≀ ≀ NUM cana-4831 91 36 𝛼 𝛼 NOUN cana-4831 91 37 , , PUNCT cana-4831 91 38 𝛽 𝛽 NOUN cana-4831 91 39 < < X cana-4831 91 40 1 1 NUM cana-4831 91 41 ) ) PUNCT cana-4831 91 42 , , PUNCT cana-4831 91 43 𝑛 𝑛 DET cana-4831 91 44 ∈ ∈ PROPN cana-4831 91 45 β„•0 β„•0 NOUN cana-4831 91 46 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN cana-4831 91 47 𝑧 𝑧 PROPN cana-4831 91 48 ∈ ∈ PROPN cana-4831 91 49 π‘ˆ. π‘ˆ. PROPN cana-4831 91 50 𝑓 𝑓 PRON cana-4831 91 51 ∈ ∈ NOUN cana-4831 91 52 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 91 53 , , PUNCT cana-4831 91 54 𝜷 𝜷 NOUN cana-4831 91 55 , , PUNCT cana-4831 91 56 𝜹 𝜹 X cana-4831 91 57 , , PUNCT cana-4831 91 58 𝝀 𝝀 NOUN cana-4831 91 59 ; ; PUNCT cana-4831 91 60 𝒒 𝒒 X cana-4831 91 61 ) ) PUNCT cana-4831 91 62 if if SCONJ cana-4831 91 63 and and CCONJ cana-4831 91 64 only only ADV cana-4831 91 65 if if SCONJ cana-4831 91 66 𝑓 𝑓 PRON cana-4831 91 67 can can AUX cana-4831 91 68 be be AUX cana-4831 91 69 expressed express VERB cana-4831 91 70 as as ADP cana-4831 91 71 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 91 72 ) ) PUNCT cana-4831 91 73 = = PUNCT cana-4831 91 74 βˆ‘ βˆ‘ PUNCT cana-4831 91 75 πœ‡π‘— πœ‡π‘— PROPN cana-4831 91 76 ∞ ∞ NUM cana-4831 91 77 𝑗=1 𝑗=1 PROPN cana-4831 91 78 𝑓𝑗(𝑧 𝑓𝑗(𝑧 PUNCT cana-4831 91 79 ) ) PUNCT cana-4831 91 80 where where SCONJ cana-4831 91 81 πœ‡π‘— πœ‡π‘— X cana-4831 91 82 β‰₯ β‰₯ X cana-4831 91 83 0 0 NUM cana-4831 91 84 and and CCONJ cana-4831 91 85 βˆ‘ βˆ‘ ADP cana-4831 91 86 πœ‡π‘— πœ‡π‘— NOUN cana-4831 91 87 ∞ ∞ NUM cana-4831 91 88 𝑗=1 𝑗=1 PUNCT cana-4831 91 89 = = PUNCT cana-4831 92 1 1 1 X cana-4831 92 2 . . X cana-4831 93 1 proof proof NOUN cana-4831 93 2 : : PUNCT cana-4831 93 3 if if SCONJ cana-4831 93 4 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 93 5 ) ) PUNCT cana-4831 93 6 = = PUNCT cana-4831 93 7 βˆ‘ βˆ‘ PUNCT cana-4831 93 8 πœ‡π‘— πœ‡π‘— PROPN cana-4831 93 9 ∞ ∞ NUM cana-4831 93 10 𝑗=1 𝑗=1 PROPN cana-4831 93 11 𝑓𝑗(𝑧 𝑓𝑗(𝑧 PUNCT cana-4831 93 12 ) ) PUNCT cana-4831 93 13 with with ADP cana-4831 93 14 βˆ‘ βˆ‘ ADV cana-4831 93 15 πœ‡π‘— πœ‡π‘— PROPN cana-4831 93 16 ∞ ∞ NUM cana-4831 93 17 𝑗=1 𝑗=1 PUNCT cana-4831 93 18 = = PUNCT cana-4831 94 1 1 1 NUM cana-4831 94 2 , , PUNCT cana-4831 94 3 πœ‡π‘— πœ‡π‘— X cana-4831 94 4 β‰₯ β‰₯ NOUN cana-4831 94 5 0 0 NUM cana-4831 94 6 , , PUNCT cana-4831 94 7 then then ADV cana-4831 94 8 βˆ‘ βˆ‘ PUNCT cana-4831 94 9 π΅πœ† π΅πœ† PROPN cana-4831 94 10 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 94 11 , , PUNCT cana-4831 94 12 𝑐 𝑐 PROPN cana-4831 94 13 , , PUNCT cana-4831 94 14 𝑗 𝑗 NOUN cana-4831 94 15 , , PUNCT cana-4831 94 16 𝑛 𝑛 PROPN cana-4831 94 17 ; ; PUNCT cana-4831 94 18 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 94 19 βˆ’ βˆ’ PROPN cana-4831 94 20 𝛼 𝛼 NOUN cana-4831 94 21 + + NOUN cana-4831 94 22 𝛼𝛽 𝛼𝛽 PROPN cana-4831 94 23 βˆ’ βˆ’ PROPN cana-4831 94 24 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 94 25 ] ] PUNCT cana-4831 94 26 πœ‡π‘— πœ‡π‘— X cana-4831 94 27 π΅πœ† π΅πœ† PROPN cana-4831 94 28 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 94 29 , , PUNCT cana-4831 94 30 𝑐 𝑐 PROPN cana-4831 94 31 , , PUNCT cana-4831 94 32 𝑗 𝑗 NOUN cana-4831 94 33 , , PUNCT cana-4831 94 34 𝑛 𝑛 PROPN cana-4831 94 35 ; ; PUNCT cana-4831 94 36 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 94 37 βˆ’ βˆ’ PROPN cana-4831 94 38 𝛼 𝛼 NOUN cana-4831 94 39 + + NOUN cana-4831 94 40 𝛼𝛽 𝛼𝛽 PROPN cana-4831 94 41 βˆ’ βˆ’ PROPN cana-4831 94 42 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 94 43 ] ] X cana-4831 94 44 ∞ ∞ PROPN cana-4831 94 45 𝑗=2 𝑗=2 X cana-4831 95 1 ( ( PUNCT cana-4831 95 2 1 1 NUM cana-4831 95 3 βˆ’ βˆ’ NOUN cana-4831 95 4 𝛼 𝛼 X cana-4831 95 5 ) ) PUNCT cana-4831 95 6 = = SYM cana-4831 95 7 βˆ‘ βˆ‘ PUNCT cana-4831 95 8 πœ‡π‘— πœ‡π‘— X cana-4831 95 9 ( ( PUNCT cana-4831 95 10 1 1 NUM cana-4831 95 11 βˆ’ βˆ’ NOUN cana-4831 95 12 𝛼 𝛼 NOUN cana-4831 95 13 ) ) PUNCT cana-4831 95 14 = = SYM cana-4831 95 15 ( ( PUNCT cana-4831 95 16 1 1 NUM cana-4831 95 17 βˆ’ βˆ’ NOUN cana-4831 95 18 πœ‡1)(1 πœ‡1)(1 NOUN cana-4831 95 19 βˆ’ βˆ’ NOUN cana-4831 95 20 𝛼 𝛼 NOUN cana-4831 95 21 ) ) PUNCT cana-4831 95 22 ≀ ≀ NOUN cana-4831 95 23 ( ( PUNCT cana-4831 95 24 1 1 NUM cana-4831 95 25 βˆ’ βˆ’ NOUN cana-4831 95 26 𝛼 𝛼 NOUN cana-4831 95 27 ) ) PUNCT cana-4831 95 28 . . PUNCT cana-4831 96 1 ∞ ∞ PROPN cana-4831 96 2 𝑗=2 𝑗=2 X cana-4831 96 3 hence hence ADV cana-4831 96 4 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 96 5 ) ) PUNCT cana-4831 96 6 ∈ ∈ NOUN cana-4831 96 7 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 96 8 , , PUNCT cana-4831 96 9 𝜷 𝜷 NOUN cana-4831 96 10 , , PUNCT cana-4831 96 11 𝜹 𝜹 X cana-4831 96 12 , , PUNCT cana-4831 96 13 𝝀 𝝀 NOUN cana-4831 96 14 ; ; PUNCT cana-4831 96 15 𝒒 𝒒 X cana-4831 96 16 ) ) PUNCT cana-4831 96 17 . . PUNCT cana-4831 97 1 conversely conversely ADV cana-4831 97 2 , , PUNCT cana-4831 97 3 let let VERB cana-4831 97 4 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 97 5 ) ) PUNCT cana-4831 98 1 = = SYM cana-4831 98 2 𝑧 𝑧 PRON cana-4831 98 3 βˆ’ βˆ’ PROPN cana-4831 98 4 βˆ‘ βˆ‘ SYM cana-4831 98 5 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-4831 98 6 𝑗=2 𝑗=2 PROPN cana-4831 98 7 ∈ ∈ PROPN cana-4831 98 8 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 98 9 , , PUNCT cana-4831 98 10 𝜷 𝜷 NOUN cana-4831 98 11 , , PUNCT cana-4831 98 12 𝜹 𝜹 X cana-4831 98 13 , , PUNCT cana-4831 98 14 𝝀 𝝀 NOUN cana-4831 98 15 ; ; PUNCT cana-4831 98 16 𝒒 𝒒 X cana-4831 98 17 ) ) PUNCT cana-4831 98 18 , , PUNCT cana-4831 98 19 define define VERB cana-4831 98 20 𝝁𝒋 𝝁𝒋 X cana-4831 98 21 = = PUNCT cana-4831 98 22 π΅πœ† π΅πœ† PROPN cana-4831 98 23 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[π‘—βˆ’π›Ό+π›Όπ›½βˆ’π›Όπ›½π‘—]|π‘Žπ‘—| 𝛿(π‘Ž,𝑐,𝑗,𝑛;π‘ž)[π‘—βˆ’π›Ό+π›Όπ›½βˆ’π›Όπ›½π‘—]|π‘Žπ‘—| PROPN cana-4831 98 24 , , PUNCT cana-4831 98 25 ( ( PUNCT cana-4831 98 26 πŸβˆ’πœΆ πŸβˆ’πœΆ NOUN cana-4831 98 27 ) ) PUNCT cana-4831 98 28 𝒋 𝒋 NOUN cana-4831 98 29 = = SYM cana-4831 98 30 𝟐 𝟐 PROPN cana-4831 98 31 , , PUNCT cana-4831 98 32 πŸ‘ πŸ‘ NUM cana-4831 98 33 β‹― β‹― NOUN cana-4831 98 34 , , PUNCT cana-4831 98 35 and and CCONJ cana-4831 98 36 define define VERB cana-4831 98 37 πœ‡1 πœ‡1 NOUN cana-4831 98 38 = = NOUN cana-4831 98 39 1 1 NUM cana-4831 98 40 βˆ’ βˆ’ PROPN cana-4831 98 41 βˆ‘ βˆ‘ PROPN cana-4831 98 42 πœ‡π‘—.∞ πœ‡π‘—.∞ NOUN cana-4831 98 43 𝑗=2 𝑗=2 PROPN cana-4831 98 44 from from ADP cana-4831 98 45 theorem theorem ADJ cana-4831 98 46 2.1 2.1 NUM cana-4831 98 47 , , PUNCT cana-4831 98 48 βˆ‘ βˆ‘ ADP cana-4831 98 49 πœ‡π‘— πœ‡π‘— ADP cana-4831 98 50 ≀ ≀ NUM cana-4831 98 51 1∞ 1∞ NUM cana-4831 98 52 𝑗=2 𝑗=2 NOUN cana-4831 98 53 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-4831 98 54 π‘ π‘œ π‘ π‘œ ADP cana-4831 98 55 πœ‡1 πœ‡1 PROPN cana-4831 98 56 β‰₯ β‰₯ NOUN cana-4831 98 57 0 0 NUM cana-4831 98 58 . . PUNCT cana-4831 99 1 since since SCONJ cana-4831 99 2 πœ‡π‘—π‘“π‘—(𝑧 πœ‡π‘—π‘“π‘—(𝑧 NUM cana-4831 99 3 ) ) PUNCT cana-4831 99 4 = = SYM cana-4831 99 5 πœ‡π‘—π‘“ πœ‡π‘—π‘“ PROPN cana-4831 100 1 + + CCONJ cana-4831 100 2 π‘Žπ‘— π‘Žπ‘— ADP cana-4831 100 3 𝑧𝑗 𝑧𝑗 INTJ cana-4831 100 4 , , PUNCT cana-4831 100 5 βˆ‘ βˆ‘ ADP cana-4831 100 6 πœ‡π‘— πœ‡π‘— NOUN cana-4831 100 7 𝑓𝑗(𝑧 𝑓𝑗(𝑧 ADV cana-4831 100 8 ) ) PUNCT cana-4831 101 1 = = SYM cana-4831 101 2 𝑧 𝑧 PRON cana-4831 101 3 βˆ’ βˆ’ PROPN cana-4831 101 4 βˆ‘ βˆ‘ SYM cana-4831 101 5 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-4831 101 6 𝑗=2 𝑗=2 PROPN cana-4831 101 7 = = SYM cana-4831 101 8 𝑓(𝑧).∞ 𝑓(𝑧).∞ NOUN cana-4831 101 9 𝑗=1 𝑗=1 NOUN cana-4831 101 10 communications communication NOUN cana-4831 101 11 on on ADP cana-4831 101 12 applied apply VERB cana-4831 101 13 nonlinear nonlinear ADJ cana-4831 101 14 analysis analysis NOUN cana-4831 101 15 issn issn NOUN cana-4831 101 16 : : PUNCT cana-4831 101 17 1074 1074 NUM cana-4831 101 18 - - PUNCT cana-4831 101 19 133x 133x NUM cana-4831 101 20 vol vol VERB cana-4831 101 21 32 32 NUM cana-4831 101 22 no no NOUN cana-4831 101 23 . . PUNCT cana-4831 102 1 10s 10 NOUN cana-4831 102 2 ( ( PUNCT cana-4831 102 3 2025 2025 NUM cana-4831 102 4 ) ) PUNCT cana-4831 102 5 386 386 NUM cana-4831 102 6 https://internationalpubls.com https://internationalpubls.com X cana-4831 102 7 theorem theorem VERB cana-4831 102 8 2.6 2.6 NUM cana-4831 102 9 : : PUNCT cana-4831 102 10 the the DET cana-4831 102 11 class class NOUN cana-4831 102 12 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 102 13 , , PUNCT cana-4831 102 14 𝜷 𝜷 NOUN cana-4831 102 15 , , PUNCT cana-4831 102 16 𝜹 𝜹 X cana-4831 102 17 , , PUNCT cana-4831 102 18 𝝀 𝝀 NOUN cana-4831 102 19 ; ; PUNCT cana-4831 102 20 𝒒 𝒒 X cana-4831 102 21 ) ) PUNCT cana-4831 102 22 is be AUX cana-4831 102 23 closed close VERB cana-4831 102 24 under under ADP cana-4831 102 25 convex convex ADJ cana-4831 102 26 linear linear ADJ cana-4831 102 27 combination combination NOUN cana-4831 102 28 . . PUNCT cana-4831 103 1 proof proof NOUN cana-4831 103 2 : : PUNCT cana-4831 103 3 let let VERB cana-4831 103 4 𝑓 𝑓 PRON cana-4831 103 5 , , PUNCT cana-4831 103 6 𝑔 𝑔 PROPN cana-4831 103 7 ∈ ∈ PROPN cana-4831 103 8 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 103 9 , , PUNCT cana-4831 103 10 𝜷 𝜷 NOUN cana-4831 103 11 , , PUNCT cana-4831 103 12 𝜹 𝜹 X cana-4831 103 13 , , PUNCT cana-4831 103 14 𝝀 𝝀 NOUN cana-4831 103 15 ; ; PUNCT cana-4831 103 16 𝒒 𝒒 X cana-4831 103 17 ) ) PUNCT cana-4831 103 18 and and CCONJ cana-4831 103 19 let let VERB cana-4831 103 20 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 103 21 ) ) PUNCT cana-4831 104 1 = = SYM cana-4831 104 2 𝑧 𝑧 PRON cana-4831 104 3 βˆ’ βˆ’ NOUN cana-4831 104 4 βˆ‘ βˆ‘ PUNCT cana-4831 104 5 π‘Žπ‘—π‘§π‘— π‘Žπ‘—π‘§π‘— VERB cana-4831 104 6 ∞ ∞ PROPN cana-4831 104 7 𝑗=2 𝑗=2 PROPN cana-4831 104 8 , , PUNCT cana-4831 104 9 𝑔(𝑧 𝑔(𝑧 PROPN cana-4831 104 10 ) ) PUNCT cana-4831 104 11 = = SYM cana-4831 105 1 𝑧 𝑧 PRON cana-4831 105 2 βˆ’ βˆ’ NOUN cana-4831 105 3 βˆ‘ βˆ‘ INTJ cana-4831 105 4 𝑏𝑗𝑧𝑗 𝑏𝑗𝑧𝑗 PROPN cana-4831 105 5 ∞ ∞ PROPN cana-4831 105 6 𝑗=2 𝑗=2 PROPN cana-4831 105 7 . . PUNCT cana-4831 106 1 for for ADP cana-4831 106 2 πœ‚ πœ‚ DET cana-4831 106 3 such such ADJ cana-4831 106 4 that that DET cana-4831 106 5 0 0 NUM cana-4831 106 6 ≀ ≀ NUM cana-4831 106 7 πœ‚ πœ‚ NOUN cana-4831 106 8 ≀ ≀ NUM cana-4831 106 9 1 1 NUM cana-4831 106 10 , , PUNCT cana-4831 106 11 it it PRON cana-4831 106 12 suffices suffice VERB cana-4831 106 13 to to PART cana-4831 106 14 show show VERB cana-4831 106 15 that that SCONJ cana-4831 106 16 the the DET cana-4831 106 17 function function NOUN cana-4831 106 18 defined define VERB cana-4831 106 19 by by ADP cana-4831 106 20 β„Ž(𝑧 β„Ž(𝑧 NOUN cana-4831 106 21 ) ) PUNCT cana-4831 106 22 = = PUNCT cana-4831 106 23 ( ( PUNCT cana-4831 106 24 1 1 NUM cana-4831 106 25 βˆ’ βˆ’ NOUN cana-4831 106 26 πœ‚)𝑓(𝑧 πœ‚)𝑓(𝑧 NOUN cana-4831 106 27 ) ) PUNCT cana-4831 106 28 + + NUM cana-4831 106 29 πœ‚π‘”(𝑧 πœ‚π‘”(𝑧 NOUN cana-4831 106 30 ) ) PUNCT cana-4831 106 31 , , PUNCT cana-4831 106 32 𝑧 𝑧 PROPN cana-4831 106 33 ∈ ∈ PROPN cana-4831 106 34 π‘ˆ π‘ˆ PROPN cana-4831 106 35 belongs belong VERB cana-4831 106 36 to to ADP cana-4831 106 37 𝑻𝒏(𝜢 𝑻𝒏(𝜢 ADJ cana-4831 106 38 , , PUNCT cana-4831 106 39 𝜷 𝜷 NOUN cana-4831 106 40 , , PUNCT cana-4831 106 41 𝜹 𝜹 X cana-4831 106 42 , , PUNCT cana-4831 106 43 𝝀 𝝀 NOUN cana-4831 106 44 ; ; PUNCT cana-4831 106 45 𝒒 𝒒 X cana-4831 106 46 ) ) PUNCT cana-4831 106 47 . . PUNCT cana-4831 107 1 now now ADV cana-4831 107 2 β„Ž(𝑧 β„Ž(𝑧 VERB cana-4831 107 3 ) ) PUNCT cana-4831 108 1 = = SYM cana-4831 108 2 𝑧 𝑧 PRON cana-4831 108 3 βˆ’ βˆ’ NOUN cana-4831 108 4 βˆ‘[(1 βˆ‘[(1 PUNCT cana-4831 108 5 βˆ’ βˆ’ PROPN cana-4831 109 1 πœ‚)π‘Žπ‘— πœ‚)π‘Žπ‘— PROPN cana-4831 109 2 + + NUM cana-4831 109 3 πœ‚π‘π‘—]𝑧𝑗 πœ‚π‘π‘—]𝑧𝑗 NUM cana-4831 109 4 , , PUNCT cana-4831 109 5 ∞ ∞ PROPN cana-4831 109 6 𝑗=2 𝑗=2 AUX cana-4831 109 7 applying apply VERB cana-4831 109 8 theorem theorem NOUN cana-4831 109 9 2.1 2.1 NUM cana-4831 109 10 , , PUNCT cana-4831 109 11 to to ADP cana-4831 109 12 𝑓 𝑓 PRON cana-4831 109 13 , , PUNCT cana-4831 109 14 𝑔 𝑔 PROPN cana-4831 109 15 ∈ ∈ PROPN cana-4831 109 16 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 109 17 , , PUNCT cana-4831 109 18 𝜷 𝜷 NOUN cana-4831 109 19 , , PUNCT cana-4831 109 20 𝜹 𝜹 X cana-4831 109 21 , , PUNCT cana-4831 109 22 𝝀 𝝀 NOUN cana-4831 109 23 ; ; PUNCT cana-4831 109 24 𝒒 𝒒 X cana-4831 109 25 ) ) PUNCT cana-4831 109 26 we we PRON cana-4831 109 27 have have AUX cana-4831 109 28 βˆ‘ βˆ‘ ADV cana-4831 109 29 π΅πœ† π΅πœ† PROPN cana-4831 109 30 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 109 31 , , PUNCT cana-4831 109 32 𝑐 𝑐 PROPN cana-4831 109 33 , , PUNCT cana-4831 109 34 𝑗 𝑗 NOUN cana-4831 109 35 , , PUNCT cana-4831 109 36 𝑛 𝑛 PROPN cana-4831 109 37 ; ; PUNCT cana-4831 109 38 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 109 39 βˆ’ βˆ’ PROPN cana-4831 109 40 𝛼 𝛼 NOUN cana-4831 109 41 + + PROPN cana-4831 109 42 𝛼𝛽 𝛼𝛽 NOUN cana-4831 109 43 βˆ’ βˆ’ PROPN cana-4831 109 44 𝛼𝛽𝑗][(1 𝛼𝛽𝑗][(1 PROPN cana-4831 109 45 βˆ’ βˆ’ PROPN cana-4831 109 46 πœ‚)π‘Žπ‘— πœ‚)π‘Žπ‘— PROPN cana-4831 109 47 + + CCONJ cana-4831 109 48 πœ‚π‘π‘— πœ‚π‘π‘— NOUN cana-4831 109 49 ] ] X cana-4831 110 1 ∞ ∞ NOUN cana-4831 110 2 𝑗=2 𝑗=2 PUNCT cana-4831 111 1 = = PUNCT cana-4831 111 2 ( ( PUNCT cana-4831 111 3 1 1 NUM cana-4831 111 4 βˆ’ βˆ’ PROPN cana-4831 111 5 πœ‚ πœ‚ X cana-4831 111 6 ) ) PUNCT cana-4831 111 7 βˆ‘ βˆ‘ PUNCT cana-4831 111 8 π΅πœ† π΅πœ† PROPN cana-4831 111 9 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 111 10 , , PUNCT cana-4831 111 11 𝑐 𝑐 PROPN cana-4831 111 12 , , PUNCT cana-4831 111 13 𝑗 𝑗 NOUN cana-4831 111 14 , , PUNCT cana-4831 111 15 𝑛 𝑛 PROPN cana-4831 111 16 ; ; PUNCT cana-4831 111 17 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 111 18 βˆ’ βˆ’ PROPN cana-4831 111 19 𝛼 𝛼 NOUN cana-4831 111 20 + + NOUN cana-4831 111 21 𝛼𝛽 𝛼𝛽 PROPN cana-4831 111 22 βˆ’ βˆ’ PROPN cana-4831 111 23 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 111 24 ] ] X cana-4831 111 25 π‘Žπ‘— π‘Žπ‘— ADP cana-4831 111 26 + + NOUN cana-4831 111 27 πœ‚ πœ‚ NOUN cana-4831 111 28 βˆ‘ βˆ‘ PUNCT cana-4831 111 29 π΅πœ† π΅πœ† PROPN cana-4831 111 30 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 111 31 , , PUNCT cana-4831 111 32 𝑐 𝑐 PROPN cana-4831 111 33 , , PUNCT cana-4831 111 34 𝑗 𝑗 NOUN cana-4831 111 35 , , PUNCT cana-4831 111 36 𝑛 𝑛 PROPN cana-4831 111 37 ; ; PUNCT cana-4831 111 38 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 111 39 βˆ’ βˆ’ PROPN cana-4831 111 40 𝛼 𝛼 NOUN cana-4831 111 41 + + NOUN cana-4831 111 42 𝛼𝛽 𝛼𝛽 PROPN cana-4831 111 43 βˆ’ βˆ’ PROPN cana-4831 111 44 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 111 45 ] ] PUNCT cana-4831 111 46 𝑏𝑗 𝑏𝑗 PROPN cana-4831 112 1 ∞ ∞ PROPN cana-4831 112 2 𝑗=2 𝑗=2 PROPN cana-4831 113 1 ∞ ∞ NUM cana-4831 113 2 𝑗=2 𝑗=2 PROPN cana-4831 113 3 ≀ ≀ NUM cana-4831 113 4 ( ( PUNCT cana-4831 113 5 1 1 NUM cana-4831 113 6 βˆ’ βˆ’ NOUN cana-4831 113 7 πœ‚)(1 πœ‚)(1 NUM cana-4831 114 1 βˆ’ βˆ’ ADP cana-4831 114 2 𝛼 𝛼 X cana-4831 114 3 ) ) PUNCT cana-4831 114 4 + + NUM cana-4831 114 5 πœ‚ πœ‚ X cana-4831 114 6 ( ( PUNCT cana-4831 114 7 1 1 NUM cana-4831 114 8 βˆ’ βˆ’ NOUN cana-4831 114 9 𝛼 𝛼 NOUN cana-4831 114 10 ) ) PUNCT cana-4831 114 11 = = SYM cana-4831 114 12 1 1 NUM cana-4831 114 13 βˆ’ βˆ’ NOUN cana-4831 114 14 𝛼. 𝛼. NOUN cana-4831 114 15 this this PRON cana-4831 114 16 implies imply VERB cana-4831 114 17 that that SCONJ cana-4831 114 18 β„Ž β„Ž PROPN cana-4831 114 19 ∈ ∈ PROPN cana-4831 114 20 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 114 21 , , PUNCT cana-4831 114 22 𝜷 𝜷 NOUN cana-4831 114 23 , , PUNCT cana-4831 114 24 𝜹 𝜹 X cana-4831 114 25 , , PUNCT cana-4831 114 26 𝝀 𝝀 NOUN cana-4831 114 27 ; ; PUNCT cana-4831 114 28 𝒒 𝒒 X cana-4831 114 29 ) ) PUNCT cana-4831 114 30 . . PUNCT cana-4831 115 1 corollary corollary ADJ cana-4831 115 2 2.7 2.7 NUM cana-4831 115 3 : : PUNCT cana-4831 115 4 if if SCONJ cana-4831 115 5 𝑓1(𝑧 𝑓1(𝑧 PROPN cana-4831 115 6 ) ) PUNCT cana-4831 115 7 , , PUNCT cana-4831 115 8 𝑓2(𝑧 𝑓2(𝑧 PROPN cana-4831 115 9 ) ) PUNCT cana-4831 115 10 ∈ ∈ NOUN cana-4831 115 11 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 115 12 , , PUNCT cana-4831 115 13 𝜷 𝜷 NOUN cana-4831 115 14 , , PUNCT cana-4831 115 15 𝜹 𝜹 X cana-4831 115 16 , , PUNCT cana-4831 115 17 𝝀 𝝀 NOUN cana-4831 115 18 ; ; PUNCT cana-4831 115 19 𝒒 𝒒 X cana-4831 115 20 ) ) PUNCT cana-4831 115 21 then then ADV cana-4831 115 22 the the DET cana-4831 115 23 function function NOUN cana-4831 115 24 defined define VERB cana-4831 115 25 by by ADP cana-4831 115 26 π’ˆ(𝒛 π’ˆ(𝒛 NOUN cana-4831 115 27 ) ) PUNCT cana-4831 115 28 = = SYM cana-4831 115 29 𝟏 𝟏 NUM cana-4831 115 30 𝟐 𝟐 NUM cana-4831 116 1 [ [ X cana-4831 116 2 π’‡πŸ(𝒛 π’‡πŸ(𝒛 NUM cana-4831 116 3 ) ) PUNCT cana-4831 116 4 + + NUM cana-4831 116 5 π’‡πŸ(𝒛 π’‡πŸ(𝒛 NUM cana-4831 116 6 ) ) PUNCT cana-4831 116 7 ] ] PUNCT cana-4831 116 8 is be AUX cana-4831 116 9 also also ADV cana-4831 116 10 in in ADP cana-4831 116 11 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 116 12 , , PUNCT cana-4831 116 13 𝜷 𝜷 NOUN cana-4831 116 14 , , PUNCT cana-4831 116 15 𝜹 𝜹 X cana-4831 116 16 , , PUNCT cana-4831 116 17 𝝀 𝝀 NOUN cana-4831 116 18 ; ; PUNCT cana-4831 116 19 𝒒 𝒒 X cana-4831 116 20 ) ) PUNCT cana-4831 116 21 . . PUNCT cana-4831 117 1 theorem theorem VERB cana-4831 117 2 2.8 2.8 NUM cana-4831 117 3 : : PUNCT cana-4831 117 4 let let VERB cana-4831 117 5 for for ADP cana-4831 117 6 π‘š π‘š NOUN cana-4831 117 7 = = SYM cana-4831 117 8 1,2 1,2 NUM cana-4831 117 9 , , PUNCT cana-4831 117 10 β‹― β‹― PROPN cana-4831 117 11 , , PUNCT cana-4831 117 12 𝑗 𝑗 PROPN cana-4831 117 13 π‘“π‘š(𝑧 π‘“π‘š(𝑧 PROPN cana-4831 117 14 ) ) PUNCT cana-4831 117 15 = = SYM cana-4831 117 16 𝑧 𝑧 PRON cana-4831 117 17 βˆ’ βˆ’ PROPN cana-4831 117 18 βˆ‘ βˆ‘ SYM cana-4831 117 19 π‘Žπ‘—,π‘š π‘Žπ‘—,π‘š PUNCT cana-4831 117 20 π‘§π‘—βˆž π‘§π‘—βˆž PROPN cana-4831 117 21 𝑗=2 𝑗=2 PROPN cana-4831 117 22 ∈ ∈ PROPN cana-4831 117 23 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 117 24 , , PUNCT cana-4831 117 25 𝜷 𝜷 NOUN cana-4831 117 26 , , PUNCT cana-4831 117 27 𝜹 𝜹 X cana-4831 117 28 , , PUNCT cana-4831 117 29 𝝀 𝝀 NOUN cana-4831 117 30 ; ; PUNCT cana-4831 117 31 𝒒 𝒒 X cana-4831 117 32 ) ) PUNCT cana-4831 117 33 and and CCONJ cana-4831 117 34 0 0 NUM cana-4831 117 35 < < X cana-4831 117 36 π›½π‘š π›½π‘š X cana-4831 117 37 < < X cana-4831 117 38 1 1 NUM cana-4831 117 39 such such ADJ cana-4831 117 40 that that SCONJ cana-4831 117 41 βˆ‘ βˆ‘ PUNCT cana-4831 117 42 π›½π‘š π›½π‘š PROPN cana-4831 117 43 = = SYM cana-4831 117 44 1,∞ 1,∞ PROPN cana-4831 117 45 π‘š=2 π‘š=2 PUNCT cana-4831 117 46 then then ADV cana-4831 117 47 the the DET cana-4831 117 48 function function NOUN cana-4831 117 49 𝐹(𝑧 𝐹(𝑧 NUM cana-4831 117 50 ) ) PUNCT cana-4831 117 51 defined define VERB cana-4831 117 52 by by ADP cana-4831 117 53 𝐹(𝑧 𝐹(𝑧 ADP cana-4831 117 54 ) ) PUNCT cana-4831 117 55 = = SYM cana-4831 117 56 βˆ‘ βˆ‘ PUNCT cana-4831 117 57 π›½π‘šπ‘“π‘š(𝑧)𝑗 π›½π‘šπ‘“π‘š(𝑧)𝑗 NOUN cana-4831 117 58 π‘š=2 π‘š=2 PUNCT cana-4831 117 59 𝑖𝑠 𝑖𝑠 ADP cana-4831 117 60 π‘Žπ‘™π‘ π‘œ π‘Žπ‘™π‘ π‘œ NOUN cana-4831 117 61 𝑖𝑛 𝑖𝑛 PRON cana-4831 117 62 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 117 63 , , PUNCT cana-4831 117 64 𝜷 𝜷 NOUN cana-4831 117 65 , , PUNCT cana-4831 117 66 𝜹 𝜹 X cana-4831 117 67 , , PUNCT cana-4831 117 68 𝝀 𝝀 NOUN cana-4831 117 69 ; ; PUNCT cana-4831 117 70 𝒒 𝒒 X cana-4831 117 71 ) ) PUNCT cana-4831 117 72 . . PUNCT cana-4831 118 1 𝑷𝒓𝒐𝒐𝒇 𝑷𝒓𝒐𝒐𝒇 NOUN cana-4831 118 2 : : PUNCT cana-4831 118 3 for for SCONJ cana-4831 118 4 each each DET cana-4831 118 5 π‘š π‘š PROPN cana-4831 118 6 ∈ ∈ PROPN cana-4831 118 7 { { PUNCT cana-4831 118 8 1,2 1,2 NUM cana-4831 118 9 , , PUNCT cana-4831 118 10 β‹― β‹― PROPN cana-4831 118 11 , , PUNCT cana-4831 118 12 𝑗 𝑗 NOUN cana-4831 118 13 } } PUNCT cana-4831 118 14 we we PRON cana-4831 118 15 obtain obtain VERB cana-4831 118 16 βˆ‘ βˆ‘ PUNCT cana-4831 118 17 π΅πœ† π΅πœ† PROPN cana-4831 118 18 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 118 19 , , PUNCT cana-4831 118 20 𝑐 𝑐 PROPN cana-4831 118 21 , , PUNCT cana-4831 118 22 𝑗 𝑗 NOUN cana-4831 118 23 , , PUNCT cana-4831 118 24 𝑛 𝑛 PROPN cana-4831 118 25 ; ; PUNCT cana-4831 118 26 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 118 27 βˆ’ βˆ’ PROPN cana-4831 118 28 𝛼 𝛼 NOUN cana-4831 118 29 + + NOUN cana-4831 118 30 𝛼𝛽 𝛼𝛽 PROPN cana-4831 118 31 βˆ’ βˆ’ PROPN cana-4831 118 32 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 118 33 ] ] X cana-4831 118 34 |π‘Žπ‘—| |π‘Žπ‘—| NOUN cana-4831 118 35 < < X cana-4831 118 36 1 1 NUM cana-4831 118 37 βˆ’ βˆ’ PROPN cana-4831 118 38 𝛼. 𝛼. NOUN cana-4831 118 39 ∞ ∞ PROPN cana-4831 118 40 𝑗=2 𝑗=2 PROPN cana-4831 118 41 𝐹(𝑧 𝐹(𝑧 ADP cana-4831 118 42 ) ) PUNCT cana-4831 118 43 = = PUNCT cana-4831 118 44 βˆ‘ βˆ‘ PUNCT cana-4831 118 45 π›½π‘š π›½π‘š PROPN cana-4831 118 46 ( ( PUNCT cana-4831 118 47 𝑧 𝑧 PROPN cana-4831 118 48 βˆ’ βˆ’ PROPN cana-4831 118 49 βˆ‘ βˆ‘ SYM cana-4831 118 50 π‘Žπ‘—,π‘š π‘Žπ‘—,π‘š PUNCT cana-4831 118 51 π‘§π‘—βˆž π‘§π‘—βˆž PROPN cana-4831 118 52 𝑗=2 𝑗=2 PROPN cana-4831 118 53 ) ) PUNCT cana-4831 118 54 𝑗 𝑗 NOUN cana-4831 118 55 π‘š=1 π‘š=1 PUNCT cana-4831 118 56 = = SYM cana-4831 118 57 𝑧 𝑧 NOUN cana-4831 118 58 βˆ’ βˆ’ NOUN cana-4831 118 59 βˆ‘ βˆ‘ PUNCT cana-4831 118 60 ( ( PUNCT cana-4831 118 61 βˆ‘ βˆ‘ PROPN cana-4831 118 62 π›½π‘š π›½π‘š NOUN cana-4831 118 63 π‘Žπ‘—,π‘š π‘Žπ‘—,π‘š PUNCT cana-4831 118 64 𝑧𝑗 𝑧𝑗 PROPN cana-4831 118 65 𝑗 𝑗 NOUN cana-4831 118 66 π‘š=1 π‘š=1 PUNCT cana-4831 118 67 ) ) PUNCT cana-4831 118 68 ∞ ∞ NUM cana-4831 118 69 𝑗=2 𝑗=2 VERB cana-4831 118 70 since since SCONJ cana-4831 118 71 , , PUNCT cana-4831 118 72 βˆ‘ βˆ‘ PROPN cana-4831 118 73 π΅πœ† π΅πœ† PROPN cana-4831 118 74 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 118 75 , , PUNCT cana-4831 118 76 𝑐 𝑐 PROPN cana-4831 118 77 , , PUNCT cana-4831 118 78 𝑗 𝑗 NOUN cana-4831 118 79 , , PUNCT cana-4831 118 80 𝑛 𝑛 PROPN cana-4831 118 81 ; ; PUNCT cana-4831 118 82 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 118 83 βˆ’ βˆ’ PROPN cana-4831 118 84 𝛼 𝛼 NOUN cana-4831 118 85 + + NOUN cana-4831 118 86 𝛼𝛽 𝛼𝛽 PROPN cana-4831 118 87 βˆ’ βˆ’ PROPN cana-4831 118 88 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 118 89 ] ] X cana-4831 118 90 [ [ PUNCT cana-4831 118 91 βˆ‘ βˆ‘ PROPN cana-4831 118 92 π›½π‘š π›½π‘š NOUN cana-4831 118 93 π‘Žπ‘—,π‘š π‘Žπ‘—,π‘š X cana-4831 118 94 𝑗 𝑗 X cana-4831 118 95 π‘š=1 π‘š=1 X cana-4831 118 96 ] ] PUNCT cana-4831 119 1 ∞ ∞ NUM cana-4831 120 1 𝑗=2 𝑗=2 X cana-4831 121 1 < < X cana-4831 121 2 βˆ‘ βˆ‘ PUNCT cana-4831 121 3 𝛽𝑗(1 𝛽𝑗(1 PROPN cana-4831 121 4 βˆ’ βˆ’ NUM cana-4831 121 5 𝛼 𝛼 NOUN cana-4831 121 6 ) ) PUNCT cana-4831 121 7 < < X cana-4831 121 8 1 1 NUM cana-4831 121 9 βˆ’ βˆ’ NOUN cana-4831 121 10 𝛼 𝛼 PRON cana-4831 121 11 𝑗 𝑗 NOUN cana-4831 121 12 π‘š=1 π‘š=1 PUNCT cana-4831 121 13 . . PUNCT cana-4831 122 1 therefore therefore ADV cana-4831 122 2 , , PUNCT cana-4831 122 3 𝐹(𝑧 𝐹(𝑧 NUM cana-4831 122 4 ) ) PUNCT cana-4831 122 5 ∈ ∈ NOUN cana-4831 122 6 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 122 7 , , PUNCT cana-4831 122 8 𝜷 𝜷 NOUN cana-4831 122 9 , , PUNCT cana-4831 122 10 𝜹 𝜹 X cana-4831 122 11 , , PUNCT cana-4831 122 12 𝝀 𝝀 NOUN cana-4831 122 13 ; ; PUNCT cana-4831 122 14 𝒒 𝒒 X cana-4831 122 15 ) ) PUNCT cana-4831 122 16 . . PUNCT cana-4831 123 1 communications communication NOUN cana-4831 123 2 on on ADP cana-4831 123 3 applied apply VERB cana-4831 123 4 nonlinear nonlinear ADJ cana-4831 123 5 analysis analysis NOUN cana-4831 123 6 issn issn NOUN cana-4831 123 7 : : PUNCT cana-4831 123 8 1074 1074 NUM cana-4831 123 9 - - PUNCT cana-4831 123 10 133x 133x NUM cana-4831 123 11 vol vol VERB cana-4831 123 12 32 32 NUM cana-4831 123 13 no no NOUN cana-4831 123 14 . . PUNCT cana-4831 124 1 10s 10 NOUN cana-4831 124 2 ( ( PUNCT cana-4831 124 3 2025 2025 NUM cana-4831 124 4 ) ) PUNCT cana-4831 124 5 387 387 NUM cana-4831 124 6 https://internationalpubls.com https://internationalpubls.com X cana-4831 124 7 theorem theorem VERB cana-4831 124 8 2.9 2.9 NUM cana-4831 124 9 : : PUNCT cana-4831 124 10 let let VERB cana-4831 124 11 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 124 12 ) ) PUNCT cana-4831 124 13 ∈ ∈ NOUN cana-4831 124 14 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 124 15 , , PUNCT cana-4831 124 16 𝜷 𝜷 NOUN cana-4831 124 17 , , PUNCT cana-4831 124 18 𝜹 𝜹 X cana-4831 124 19 , , PUNCT cana-4831 124 20 𝝀 𝝀 NOUN cana-4831 124 21 ; ; PUNCT cana-4831 124 22 𝒒 𝒒 X cana-4831 124 23 ) ) PUNCT cana-4831 124 24 . . PUNCT cana-4831 125 1 komato komato NOUN cana-4831 125 2 operator operator NOUN cana-4831 125 3 of of ADP cana-4831 125 4 𝑓 𝑓 PROPN cana-4831 125 5 is be AUX cana-4831 125 6 defined define VERB cana-4831 125 7 by by ADP cana-4831 125 8 π‘˜(𝑧 π‘˜(𝑧 PROPN cana-4831 125 9 ) ) PUNCT cana-4831 126 1 = = SYM cana-4831 126 2 ∫ ∫ PROPN cana-4831 126 3 ( ( PUNCT cana-4831 126 4 𝑐 𝑐 NOUN cana-4831 126 5 + + NUM cana-4831 126 6 1)𝛾 1)𝛾 NUM cana-4831 126 7 Ξ³(𝛾 Ξ³(𝛾 NOUN cana-4831 126 8 ) ) PUNCT cana-4831 126 9 1 1 NUM cana-4831 126 10 0 0 NUM cana-4831 126 11 𝑑𝑐 𝑑𝑐 VERB cana-4831 126 12 ( ( PUNCT cana-4831 126 13 log log NOUN cana-4831 126 14 ( ( PUNCT cana-4831 126 15 1 1 NUM cana-4831 126 16 𝑑 𝑑 NOUN cana-4831 126 17 ) ) PUNCT cana-4831 126 18 ) ) PUNCT cana-4831 127 1 π›Ύβˆ’1 π›Ύβˆ’1 NOUN cana-4831 127 2 𝑓(𝑑𝑧 𝑓(𝑑𝑧 NOUN cana-4831 127 3 ) ) PUNCT cana-4831 127 4 𝑑 𝑑 PROPN cana-4831 127 5 𝑑𝑑 𝑑𝑑 VERB cana-4831 127 6 , , PUNCT cana-4831 127 7 𝑐 𝑐 PROPN cana-4831 127 8 > > X cana-4831 127 9 βˆ’1 βˆ’1 NOUN cana-4831 127 10 , , PUNCT cana-4831 127 11 𝛾 𝛾 ADP cana-4831 127 12 β‰₯ β‰₯ NOUN cana-4831 127 13 0 0 NUM cana-4831 128 1 then then ADV cana-4831 128 2 π‘˜(𝑧 π‘˜(𝑧 PROPN cana-4831 128 3 ) ) PUNCT cana-4831 128 4 ∈ ∈ PROPN cana-4831 128 5 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 128 6 , , PUNCT cana-4831 128 7 𝜷 𝜷 NOUN cana-4831 128 8 , , PUNCT cana-4831 128 9 𝜹 𝜹 X cana-4831 128 10 , , PUNCT cana-4831 128 11 𝝀 𝝀 NOUN cana-4831 128 12 ; ; PUNCT cana-4831 128 13 𝒒 𝒒 X cana-4831 128 14 ) ) PUNCT cana-4831 128 15 . . PUNCT cana-4831 129 1 𝑷𝒓𝒐𝒐𝒇 𝑷𝒓𝒐𝒐𝒇 NOUN cana-4831 129 2 : : PUNCT cana-4831 129 3 we we PRON cana-4831 129 4 have have VERB cana-4831 129 5 ∫ ∫ PROPN cana-4831 129 6 𝑑𝑐 𝑑𝑐 ADJ cana-4831 129 7 ( ( PUNCT cana-4831 129 8 log log X cana-4831 129 9 ( ( PUNCT cana-4831 129 10 1 1 NUM cana-4831 129 11 𝑑 𝑑 NOUN cana-4831 129 12 ) ) PUNCT cana-4831 129 13 ) ) PUNCT cana-4831 130 1 π›Ύβˆ’1 π›Ύβˆ’1 PROPN cana-4831 130 2 𝑑𝑑 𝑑𝑑 ADP cana-4831 130 3 𝟏 𝟏 NUM cana-4831 130 4 𝟎 𝟎 PROPN cana-4831 130 5 = = SYM cana-4831 130 6 πšͺ(𝜸 πšͺ(𝜸 NUM cana-4831 130 7 ) ) PUNCT cana-4831 130 8 ( ( PUNCT cana-4831 130 9 𝒄 𝒄 PROPN cana-4831 130 10 + + CCONJ cana-4831 130 11 𝟏)𝜸 𝟏)𝜸 NUM cana-4831 130 12 ∫ ∫ NOUN cana-4831 130 13 𝑑𝑗+π‘βˆ’1 𝑑𝑗+π‘βˆ’1 PROPN cana-4831 130 14 ( ( PUNCT cana-4831 130 15 log log NOUN cana-4831 130 16 ( ( PUNCT cana-4831 130 17 1 1 NUM cana-4831 130 18 𝑑 𝑑 NOUN cana-4831 130 19 ) ) PUNCT cana-4831 130 20 ) ) PUNCT cana-4831 131 1 π›Ύβˆ’1 π›Ύβˆ’1 PROPN cana-4831 131 2 𝑑𝑑 𝑑𝑑 ADP cana-4831 131 3 𝟏 𝟏 NUM cana-4831 131 4 𝟎 𝟎 PROPN cana-4831 131 5 = = SYM cana-4831 131 6 πšͺ(𝜸 πšͺ(𝜸 NUM cana-4831 131 7 ) ) PUNCT cana-4831 131 8 ( ( PUNCT cana-4831 131 9 𝒄 𝒄 PROPN cana-4831 131 10 + + CCONJ cana-4831 131 11 𝟏)𝜸 𝟏)𝜸 NOUN cana-4831 131 12 , , PUNCT cana-4831 131 13 𝒋 𝒋 X cana-4831 131 14 = = SYM cana-4831 131 15 𝟐 𝟐 PROPN cana-4831 131 16 , , PUNCT cana-4831 131 17 πŸ‘ πŸ‘ NUM cana-4831 131 18 , , PUNCT cana-4831 131 19 β‹― β‹― NOUN cana-4831 131 20 , , PUNCT cana-4831 131 21 π‘˜(𝑧 π‘˜(𝑧 PROPN cana-4831 131 22 ) ) PUNCT cana-4831 132 1 = = PRON cana-4831 132 2 ( ( PUNCT cana-4831 132 3 𝑐 𝑐 NOUN cana-4831 132 4 + + NUM cana-4831 132 5 1)𝛾 1)𝛾 NUM cana-4831 132 6 Ξ³(𝛾 Ξ³(𝛾 X cana-4831 132 7 ) ) PUNCT cana-4831 133 1 [ [ X cana-4831 133 2 ∫ ∫ X cana-4831 133 3 𝑑𝑐 𝑑𝑐 VERB cana-4831 133 4 ( ( PUNCT cana-4831 133 5 log log NOUN cana-4831 133 6 ( ( PUNCT cana-4831 133 7 1 1 NUM cana-4831 133 8 𝑑 𝑑 NOUN cana-4831 133 9 ) ) PUNCT cana-4831 133 10 ) ) PUNCT cana-4831 134 1 π›Ύβˆ’1 π›Ύβˆ’1 PROPN cana-4831 134 2 𝑧 𝑧 NOUN cana-4831 134 3 𝑑𝑑 𝑑𝑑 ADP cana-4831 134 4 βˆ’ βˆ’ PROPN cana-4831 134 5 βˆ‘ βˆ‘ PROPN cana-4831 134 6 𝑧𝑗 𝑧𝑗 PROPN cana-4831 134 7 ∫ ∫ PROPN cana-4831 134 8 π‘Žπ‘— π‘Žπ‘— PROPN cana-4831 134 9 𝑑𝑗+π‘βˆ’1 𝑑𝑗+π‘βˆ’1 PROPN cana-4831 134 10 ( ( PUNCT cana-4831 134 11 log log NOUN cana-4831 134 12 ( ( PUNCT cana-4831 134 13 1 1 NUM cana-4831 134 14 𝑑 𝑑 NOUN cana-4831 134 15 ) ) PUNCT cana-4831 134 16 ) ) PUNCT cana-4831 135 1 π›Ύβˆ’1 π›Ύβˆ’1 NOUN cana-4831 135 2 𝑑𝑑 𝑑𝑑 ADP cana-4831 135 3 1 1 NUM cana-4831 135 4 0 0 NUM cana-4831 135 5 ∞ ∞ NUM cana-4831 135 6 𝑗=2 𝑗=2 PROPN cana-4831 135 7 1 1 NUM cana-4831 135 8 0 0 NUM cana-4831 135 9 ] ] PUNCT cana-4831 136 1 = = PUNCT cana-4831 136 2 𝒛 𝒛 NOUN cana-4831 136 3 βˆ’ βˆ’ PROPN cana-4831 136 4 βˆ‘ βˆ‘ PUNCT cana-4831 136 5 ( ( PUNCT cana-4831 136 6 𝒄 𝒄 PROPN cana-4831 136 7 + + CCONJ cana-4831 136 8 𝟏 𝟏 NUM cana-4831 136 9 𝒄 𝒄 NOUN cana-4831 136 10 + + NUM cana-4831 136 11 𝒋 𝒋 X cana-4831 136 12 ) ) PUNCT cana-4831 136 13 𝜸 𝜸 X cana-4831 136 14 𝒂𝒋𝒛𝒋. 𝒂𝒋𝒛𝒋. ADJ cana-4831 136 15 ∞ ∞ NUM cana-4831 136 16 𝒋=𝟐 𝒋=𝟐 PROPN cana-4831 136 17 since since SCONJ cana-4831 136 18 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 136 19 ) ) PUNCT cana-4831 136 20 ∈ ∈ NOUN cana-4831 136 21 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 136 22 , , PUNCT cana-4831 136 23 𝜷 𝜷 NOUN cana-4831 136 24 , , PUNCT cana-4831 136 25 𝜹 𝜹 X cana-4831 136 26 , , PUNCT cana-4831 136 27 𝝀 𝝀 NOUN cana-4831 136 28 ; ; PUNCT cana-4831 136 29 𝒒 𝒒 X cana-4831 136 30 ) ) PUNCT cana-4831 136 31 and and CCONJ cana-4831 136 32 ( ( PUNCT cana-4831 136 33 𝒄+𝟏 𝒄+𝟏 NOUN cana-4831 136 34 𝒄+𝒋 𝒄+𝒋 NUM cana-4831 136 35 ) ) PUNCT cana-4831 136 36 𝜸 𝜸 X cana-4831 136 37 < < X cana-4831 136 38 𝟏 𝟏 NUM cana-4831 136 39 , , PUNCT cana-4831 136 40 we we PRON cana-4831 136 41 have have VERB cana-4831 136 42 βˆ‘ βˆ‘ ADV cana-4831 136 43 π΅πœ† π΅πœ† PROPN cana-4831 136 44 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 136 45 , , PUNCT cana-4831 136 46 𝑐 𝑐 PROPN cana-4831 136 47 , , PUNCT cana-4831 136 48 𝑗 𝑗 NOUN cana-4831 136 49 , , PUNCT cana-4831 136 50 𝑛 𝑛 PROPN cana-4831 136 51 ; ; PUNCT cana-4831 136 52 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 136 53 βˆ’ βˆ’ PROPN cana-4831 136 54 𝛼 𝛼 NOUN cana-4831 136 55 + + NOUN cana-4831 136 56 𝛼𝛽 𝛼𝛽 PROPN cana-4831 136 57 βˆ’ βˆ’ PROPN cana-4831 136 58 𝛼𝛽𝑗 𝛼𝛽𝑗 NOUN cana-4831 136 59 ] ] X cana-4831 136 60 ( ( PUNCT cana-4831 136 61 𝒄 𝒄 PROPN cana-4831 137 1 + + CCONJ cana-4831 137 2 𝟏 𝟏 NUM cana-4831 137 3 𝒄 𝒄 NOUN cana-4831 137 4 + + NUM cana-4831 137 5 𝒋 𝒋 X cana-4831 137 6 ) ) PUNCT cana-4831 137 7 𝜸 𝜸 X cana-4831 137 8 π‘Žπ‘— π‘Žπ‘— ADP cana-4831 137 9 < < X cana-4831 137 10 ( ( PUNCT cana-4831 137 11 1 1 NUM cana-4831 137 12 βˆ’ βˆ’ NOUN cana-4831 137 13 𝛼 𝛼 NOUN cana-4831 137 14 ) ) PUNCT cana-4831 137 15 . . PUNCT cana-4831 138 1 ∞ ∞ PROPN cana-4831 138 2 𝑗=2 𝑗=2 PROPN cana-4831 138 3 theorem theorem VERB cana-4831 138 4 2.10 2.10 NUM cana-4831 138 5 : : PUNCT cana-4831 138 6 let let VERB cana-4831 138 7 𝑓(𝑧 𝑓(𝑧 NUM cana-4831 138 8 ) ) PUNCT cana-4831 138 9 ∈ ∈ NOUN cana-4831 138 10 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 138 11 , , PUNCT cana-4831 138 12 𝜷 𝜷 NOUN cana-4831 138 13 , , PUNCT cana-4831 138 14 𝜹 𝜹 X cana-4831 138 15 , , PUNCT cana-4831 138 16 𝝀 𝝀 NOUN cana-4831 138 17 ; ; PUNCT cana-4831 138 18 𝒒 𝒒 X cana-4831 138 19 ) ) PUNCT cana-4831 138 20 , , PUNCT cana-4831 138 21 then then ADV cana-4831 138 22 for for ADP cana-4831 138 23 every every DET cana-4831 138 24 0 0 NUM cana-4831 138 25 ≀ ≀ NUM cana-4831 138 26 𝜁 𝜁 NOUN cana-4831 138 27 < < X cana-4831 138 28 1 1 NUM cana-4831 138 29 the the DET cana-4831 138 30 function function NOUN cana-4831 138 31 π‘―πœ»(𝒛 π‘―πœ»(𝒛 NOUN cana-4831 138 32 ) ) PUNCT cana-4831 138 33 = = SYM cana-4831 138 34 ( ( PUNCT cana-4831 138 35 𝟏 𝟏 NUM cana-4831 138 36 βˆ’ βˆ’ NUM cana-4831 138 37 𝜻)𝒇(𝒛 𝜻)𝒇(𝒛 NUM cana-4831 138 38 ) ) PUNCT cana-4831 138 39 + + NUM cana-4831 138 40 𝜻 𝜻 PRON cana-4831 138 41 ∫ ∫ PROPN cana-4831 138 42 𝒇(𝒕 𝒇(𝒕 PROPN cana-4831 138 43 ) ) PUNCT cana-4831 138 44 𝒕 𝒕 PROPN cana-4831 138 45 𝒛 𝒛 NOUN cana-4831 138 46 𝟎 𝟎 NUM cana-4831 138 47 𝒅𝒕. 𝒅𝒕. NOUN cana-4831 138 48 𝑷𝒓𝒐𝒐𝒇 𝑷𝒓𝒐𝒐𝒇 PROPN cana-4831 138 49 : : PUNCT cana-4831 138 50 we we PRON cana-4831 138 51 have have VERB cana-4831 138 52 𝐻𝜁(𝑧 𝐻𝜁(𝑧 NOUN cana-4831 138 53 ) ) PUNCT cana-4831 139 1 = = SYM cana-4831 139 2 𝑧 𝑧 PRON cana-4831 139 3 βˆ’ βˆ’ NOUN cana-4831 139 4 βˆ‘ βˆ‘ INTJ cana-4831 139 5 ( ( PUNCT cana-4831 139 6 1 1 NUM cana-4831 139 7 + + CCONJ cana-4831 139 8 𝜁 𝜁 PROPN cana-4831 139 9 𝑗 𝑗 PRON cana-4831 139 10 βˆ’ βˆ’ PROPN cana-4831 139 11 𝜁 𝜁 NOUN cana-4831 139 12 ) ) PUNCT cana-4831 139 13 π‘Žπ‘—π‘§π‘— π‘Žπ‘—π‘§π‘— PROPN cana-4831 139 14 .∞ .∞ PROPN cana-4831 139 15 𝑗=2 𝑗=2 PROPN cana-4831 139 16 since since SCONJ cana-4831 139 17 ( ( PUNCT cana-4831 139 18 1 1 NUM cana-4831 139 19 + + CCONJ cana-4831 139 20 𝜁 𝜁 PROPN cana-4831 139 21 𝑗 𝑗 PRON cana-4831 139 22 βˆ’ βˆ’ PROPN cana-4831 139 23 𝜁 𝜁 NOUN cana-4831 139 24 ) ) PUNCT cana-4831 139 25 < < X cana-4831 139 26 1 1 NUM cana-4831 139 27 , , PUNCT cana-4831 139 28 𝑗 𝑗 PRON cana-4831 139 29 β‰₯ β‰₯ NOUN cana-4831 139 30 2 2 NUM cana-4831 139 31 , , PUNCT cana-4831 139 32 π‘ π‘œ π‘ π‘œ PRON cana-4831 139 33 𝑏𝑦 𝑏𝑦 NOUN cana-4831 139 34 theorem theorem VERB cana-4831 139 35 2.1 2.1 NUM cana-4831 139 36 , , PUNCT cana-4831 139 37 βˆ‘ βˆ‘ ADP cana-4831 139 38 ( ( PUNCT cana-4831 139 39 1 1 NUM cana-4831 139 40 + + CCONJ cana-4831 139 41 𝜁 𝜁 PROPN cana-4831 139 42 𝑗 𝑗 PRON cana-4831 139 43 βˆ’ βˆ’ PROPN cana-4831 139 44 𝜁 𝜁 PROPN cana-4831 139 45 ) ) PUNCT cana-4831 139 46 π΅πœ† π΅πœ† PROPN cana-4831 139 47 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 139 48 , , PUNCT cana-4831 139 49 𝑐 𝑐 PROPN cana-4831 139 50 , , PUNCT cana-4831 139 51 𝑗 𝑗 NOUN cana-4831 139 52 , , PUNCT cana-4831 139 53 𝑛 𝑛 PROPN cana-4831 139 54 ; ; PUNCT cana-4831 139 55 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 139 56 βˆ’ βˆ’ PROPN cana-4831 139 57 𝛼 𝛼 NOUN cana-4831 139 58 + + PROPN cana-4831 139 59 𝛼𝛽 𝛼𝛽 NOUN cana-4831 139 60 βˆ’ βˆ’ NUM cana-4831 139 61 𝛼𝛽𝑗]π‘Žπ‘— 𝛼𝛽𝑗]π‘Žπ‘— NUM cana-4831 140 1 ∞ ∞ PROPN cana-4831 140 2 𝑗=2 𝑗=2 X cana-4831 140 3 < < X cana-4831 140 4 βˆ‘ βˆ‘ PUNCT cana-4831 140 5 π΅πœ† π΅πœ† PROPN cana-4831 140 6 𝛿(π‘Ž 𝛿(π‘Ž PROPN cana-4831 140 7 , , PUNCT cana-4831 140 8 𝑐 𝑐 PROPN cana-4831 140 9 , , PUNCT cana-4831 140 10 𝑗 𝑗 NOUN cana-4831 140 11 , , PUNCT cana-4831 140 12 𝑛 𝑛 PROPN cana-4831 140 13 ; ; PUNCT cana-4831 140 14 π‘ž)[𝑗 π‘ž)[𝑗 X cana-4831 140 15 βˆ’ βˆ’ PROPN cana-4831 140 16 𝛼 𝛼 NOUN cana-4831 140 17 + + PROPN cana-4831 140 18 𝛼𝛽 𝛼𝛽 NOUN cana-4831 140 19 βˆ’ βˆ’ NUM cana-4831 140 20 𝛼𝛽𝑗]π‘Žπ‘— 𝛼𝛽𝑗]π‘Žπ‘— NOUN cana-4831 140 21 < < X cana-4831 140 22 1 1 NUM cana-4831 140 23 βˆ’ βˆ’ PROPN cana-4831 141 1 𝛼. 𝛼. NOUN cana-4831 141 2 ∞ ∞ PROPN cana-4831 141 3 𝑗=2 𝑗=2 PROPN cana-4831 142 1 therefore therefore ADV cana-4831 142 2 , , PUNCT cana-4831 142 3 𝐻𝜁(𝑧 𝐻𝜁(𝑧 NOUN cana-4831 142 4 ) ) PUNCT cana-4831 142 5 ∈ ∈ PROPN cana-4831 142 6 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 142 7 , , PUNCT cana-4831 142 8 𝜷 𝜷 NOUN cana-4831 142 9 , , PUNCT cana-4831 142 10 𝜹 𝜹 X cana-4831 142 11 , , PUNCT cana-4831 142 12 𝝀 𝝀 NOUN cana-4831 142 13 ; ; PUNCT cana-4831 142 14 𝒒 𝒒 X cana-4831 142 15 ) ) PUNCT cana-4831 142 16 . . PUNCT cana-4831 143 1 3 3 X cana-4831 143 2 . . X cana-4831 143 3 conclusion conclusion NOUN cana-4831 143 4 : : PUNCT cana-4831 143 5 here here ADV cana-4831 143 6 , , PUNCT cana-4831 143 7 in in ADP cana-4831 143 8 our our PRON cana-4831 143 9 present present ADJ cana-4831 143 10 investigation investigation NOUN cana-4831 143 11 , , PUNCT cana-4831 143 12 we we PRON cana-4831 143 13 have have AUX cana-4831 143 14 successfully successfully ADV cana-4831 143 15 introduced introduce VERB cana-4831 143 16 a a DET cana-4831 143 17 new new ADJ cana-4831 143 18 subclass subclass NOUN cana-4831 143 19 of of ADP cana-4831 143 20 analytic analytic ADJ cana-4831 143 21 functions function NOUN cana-4831 143 22 𝑻𝒏(𝜢 𝑻𝒏(𝜢 NOUN cana-4831 143 23 , , PUNCT cana-4831 143 24 𝜷 𝜷 NOUN cana-4831 143 25 , , PUNCT cana-4831 143 26 𝜹 𝜹 X cana-4831 143 27 , , PUNCT cana-4831 143 28 𝝀 𝝀 NOUN cana-4831 143 29 ; ; PUNCT cana-4831 143 30 𝒒 𝒒 X cana-4831 143 31 ) ) PUNCT cana-4831 143 32 using use VERB cana-4831 143 33 the the DET cana-4831 143 34 linear linear ADJ cana-4831 143 35 differential differential NOUN cana-4831 143 36 operator operator NOUN cana-4831 143 37 . . PUNCT cana-4831 144 1 many many ADJ cana-4831 144 2 properties property NOUN cana-4831 144 3 and and CCONJ cana-4831 144 4 characteristics characteristic NOUN cana-4831 144 5 of of ADP cana-4831 144 6 this this DET cana-4831 144 7 newly newly ADV cana-4831 144 8 defined define VERB cana-4831 144 9 function function NOUN cana-4831 144 10 class class NOUN cana-4831 144 11 such such ADJ cana-4831 144 12 as as ADP cana-4831 144 13 coefficient coefficient NOUN cana-4831 144 14 estimates estimate NOUN cana-4831 144 15 , , PUNCT cana-4831 144 16 distortion distortion NOUN cana-4831 144 17 theorem theorem ADJ cana-4831 144 18 , , PUNCT cana-4831 144 19 extreme extreme ADJ cana-4831 144 20 points point NOUN cana-4831 144 21 , , PUNCT cana-4831 144 22 integral integral ADJ cana-4831 144 23 theorem theorem NOUN cana-4831 144 24 have have AUX cana-4831 144 25 been be AUX cana-4831 144 26 studied study VERB cana-4831 144 27 . . PUNCT cana-4831 145 1 communications communication NOUN cana-4831 145 2 on on ADP cana-4831 145 3 applied apply VERB cana-4831 145 4 nonlinear nonlinear ADJ cana-4831 145 5 analysis analysis NOUN cana-4831 145 6 issn issn NOUN cana-4831 145 7 : : PUNCT cana-4831 145 8 1074 1074 NUM cana-4831 145 9 - - PUNCT cana-4831 145 10 133x 133x NUM cana-4831 145 11 vol vol VERB cana-4831 145 12 32 32 NUM cana-4831 145 13 no no NOUN cana-4831 145 14 . . PUNCT cana-4831 146 1 10s 10 NOUN cana-4831 146 2 ( ( PUNCT cana-4831 146 3 2025 2025 NUM cana-4831 146 4 ) ) PUNCT cana-4831 146 5 388 388 NUM cana-4831 146 6 https://internationalpubls.com https://internationalpubls.com X cana-4831 146 7 4 4 NUM cana-4831 146 8 . . PUNCT cana-4831 146 9 references reference NOUN cana-4831 146 10 : : PUNCT cana-4831 147 1 [ [ X cana-4831 147 2 1 1 X cana-4831 147 3 ] ] X cana-4831 147 4 h h NOUN cana-4831 147 5 s s NOUN cana-4831 147 6 almiri almiri PRON cana-4831 147 7 , , PUNCT cana-4831 147 8 on on ADP cana-4831 147 9 ruscheweyh ruscheweyh NOUN cana-4831 147 10 derivatives derivative NOUN cana-4831 147 11 , , PUNCT cana-4831 147 12 ann ann PROPN cana-4831 147 13 . . PROPN cana-4831 147 14 , , PUNCT cana-4831 147 15 polon polon ADJ cana-4831 147 16 math math NOUN cana-4831 147 17 . . PUNCT cana-4831 147 18 , , PUNCT cana-4831 147 19 38 38 NUM cana-4831 147 20 ( ( PUNCT cana-4831 147 21 1980 1980 NUM cana-4831 147 22 ) ) PUNCT cana-4831 147 23 , , PUNCT cana-4831 147 24 87 87 NUM cana-4831 147 25 - - SYM cana-4831 147 26 94 94 NUM cana-4831 147 27 . . PUNCT cana-4831 148 1 [ [ X cana-4831 148 2 2 2 NUM cana-4831 148 3 ] ] X cana-4831 148 4 f f PROPN cana-4831 148 5 m m PROPN cana-4831 148 6 al al PROPN cana-4831 148 7 - - PUNCT cana-4831 148 8 oboudi oboudi NOUN cana-4831 148 9 , , PUNCT cana-4831 148 10 on on ADP cana-4831 148 11 univalent univalent ADJ cana-4831 148 12 functions function NOUN cana-4831 148 13 defined define VERB cana-4831 148 14 by by ADP cana-4831 148 15 a a DET cana-4831 148 16 generalized generalize VERB cana-4831 148 17 salagean salagean ADJ cana-4831 148 18 operator operator NOUN cana-4831 148 19 , , PUNCT cana-4831 148 20 ind ind NOUN cana-4831 148 21 . . PROPN cana-4831 148 22 , , PUNCT cana-4831 148 23 j. j. PROPN cana-4831 148 24 math math PROPN cana-4831 148 25 . . PUNCT cana-4831 148 26 math math NOUN cana-4831 148 27 . . PUNCT cana-4831 149 1 sci sci PROPN cana-4831 149 2 . . PROPN cana-4831 149 3 , , PUNCT cana-4831 149 4 2004 2004 NUM cana-4831 149 5 , , PUNCT cana-4831 149 6 no no INTJ cana-4831 149 7 . . NOUN cana-4831 149 8 25 25 NUM cana-4831 149 9 - - SYM cana-4831 149 10 28 28 NUM cana-4831 149 11 , , PUNCT cana-4831 149 12 1429 1429 NUM cana-4831 149 13 - - SYM cana-4831 149 14 1436 1436 NUM cana-4831 149 15 . . PUNCT cana-4831 150 1 [ [ X cana-4831 150 2 3 3 NUM cana-4831 150 3 ] ] X cana-4831 150 4 o o NOUN cana-4831 150 5 altintas altintas PROPN cana-4831 150 6 and and CCONJ cana-4831 150 7 s s VERB cana-4831 150 8 owa owa PROPN cana-4831 150 9 , , PUNCT cana-4831 150 10 on on ADP cana-4831 150 11 subclasses subclass NOUN cana-4831 150 12 of of ADP cana-4831 150 13 univalent univalent ADJ cana-4831 150 14 functions function NOUN cana-4831 150 15 with with ADP cana-4831 150 16 negative negative ADJ cana-4831 150 17 coefficients coefficient NOUN cana-4831 150 18 , , PUNCT cana-4831 150 19 pusam pusam NOUN cana-4831 150 20 kyonganma kyonganma NOUN cana-4831 150 21 math math PROPN cana-4831 150 22 . . PUNCT cana-4831 150 23 , , PUNCT cana-4831 150 24 j. j. PROPN cana-4831 150 25 , , PUNCT cana-4831 150 26 4 4 NUM cana-4831 150 27 ( ( PUNCT cana-4831 150 28 1988 1988 NUM cana-4831 150 29 ) ) PUNCT cana-4831 150 30 , , PUNCT cana-4831 150 31 41 41 NUM cana-4831 150 32 - - SYM cana-4831 150 33 56 56 NUM cana-4831 150 34 . . PUNCT cana-4831 151 1 [ [ X cana-4831 151 2 4 4 X cana-4831 151 3 ] ] X cana-4831 151 4 annapoorna annapoorna NOUN cana-4831 151 5 s s PROPN cana-4831 151 6 and and CCONJ cana-4831 151 7 dileep dileep PROPN cana-4831 151 8 l l PROPN cana-4831 151 9 , , PUNCT cana-4831 151 10 applications application NOUN cana-4831 151 11 of of ADP cana-4831 151 12 linear linear PROPN cana-4831 151 13 differential differential NOUN cana-4831 151 14 operator operator NOUN cana-4831 151 15 on on ADP cana-4831 151 16 varying vary VERB cana-4831 151 17 arguments argument NOUN cana-4831 151 18 , , PUNCT cana-4831 151 19 communications communication NOUN cana-4831 151 20 in in ADP cana-4831 151 21 mathematics mathematic NOUN cana-4831 151 22 and and CCONJ cana-4831 151 23 applications application NOUN cana-4831 151 24 , , PUNCT cana-4831 151 25 vol.15 vol.15 NOUN cana-4831 151 26 , , PUNCT cana-4831 151 27 no no INTJ cana-4831 151 28 . . NOUN cana-4831 151 29 2 2 NUM cana-4831 151 30 , , PUNCT cana-4831 151 31 pp pp ADV cana-4831 151 32 791 791 NUM cana-4831 151 33 - - SYM cana-4831 151 34 799 799 NUM cana-4831 151 35 , , PUNCT cana-4831 151 36 2024 2024 NUM cana-4831 151 37 , , PUNCT cana-4831 151 38 doi:10.26713 doi:10.26713 PROPN cana-4831 151 39 / / SYM cana-4831 151 40 cam.v15i2.2735 cam.v15i2.2735 PROPN cana-4831 151 41 . . PUNCT cana-4831 152 1 [ [ X cana-4831 152 2 5 5 NUM cana-4831 152 3 ] ] SYM cana-4831 152 4 b b PROPN cana-4831 152 5 c c PROPN cana-4831 152 6 carlson carlson PROPN cana-4831 152 7 and and CCONJ cana-4831 152 8 shaffer shaffer PROPN cana-4831 152 9 , , PUNCT cana-4831 152 10 starlike starlike NOUN cana-4831 152 11 and and CCONJ cana-4831 152 12 pre pre ADJ cana-4831 152 13 - - ADJ cana-4831 152 14 starlike starlike ADJ cana-4831 152 15 hypergeometric hypergeometric ADJ cana-4831 152 16 functions function NOUN cana-4831 152 17 , , PUNCT cana-4831 152 18 siam siam PROPN cana-4831 152 19 j. j. PROPN cana-4831 152 20 , , PUNCT cana-4831 152 21 on on ADP cana-4831 152 22 math math NOUN cana-4831 152 23 . . PUNCT cana-4831 153 1 analysis analysis NOUN cana-4831 153 2 , , PUNCT cana-4831 153 3 15(1984 15(1984 NUM cana-4831 153 4 ) ) PUNCT cana-4831 153 5 , , PUNCT cana-4831 153 6 737 737 NUM cana-4831 153 7 - - SYM cana-4831 153 8 745 745 NUM cana-4831 153 9 . . PUNCT cana-4831 154 1 [ [ X cana-4831 154 2 6 6 NUM cana-4831 154 3 ] ] X cana-4831 154 4 dileep dileep PROPN cana-4831 154 5 l l PROPN cana-4831 154 6 and and CCONJ cana-4831 154 7 s s VERB cana-4831 154 8 latha latha PROPN cana-4831 154 9 , , PUNCT cana-4831 154 10 a a DET cana-4831 154 11 note note NOUN cana-4831 154 12 on on ADP cana-4831 154 13 salagean salagean ADJ cana-4831 154 14 type type NOUN cana-4831 154 15 functions function NOUN cana-4831 154 16 , , PUNCT cana-4831 154 17 global global ADJ cana-4831 154 18 journal journal NOUN cana-4831 154 19 of of ADP cana-4831 154 20 mathematical mathematical ADJ cana-4831 154 21 sciences science NOUN cana-4831 154 22 , , PUNCT cana-4831 154 23 theory theory NOUN cana-4831 154 24 and and CCONJ cana-4831 154 25 practical practical ADJ cana-4831 154 26 , , PUNCT cana-4831 154 27 vol.2 vol.2 PROPN cana-4831 154 28 , , PUNCT cana-4831 154 29 no.1 no.1 NUM cana-4831 154 30 , , PUNCT cana-4831 154 31 2010 2010 NUM cana-4831 154 32 , , PUNCT cana-4831 154 33 pp pp ADV cana-4831 154 34 29 29 NUM cana-4831 154 35 - - SYM cana-4831 154 36 35 35 NUM cana-4831 154 37 . . PUNCT cana-4831 155 1 [ [ X cana-4831 155 2 7 7 X cana-4831 155 3 ] ] X cana-4831 155 4 dileep dileep PROPN cana-4831 155 5 l l PROPN cana-4831 155 6 and and CCONJ cana-4831 155 7 s s VERB cana-4831 155 8 latha latha ADJ cana-4831 155 9 , , PUNCT cana-4831 155 10 certain certain ADJ cana-4831 155 11 subclasses subclass NOUN cana-4831 155 12 of of ADP cana-4831 155 13 analytic analytic ADJ cana-4831 155 14 functions function NOUN cana-4831 155 15 involving involve VERB cana-4831 155 16 salagean salagean ADJ cana-4831 155 17 - - PUNCT cana-4831 155 18 rushceweyh rushceweyh NOUN cana-4831 155 19 operator operator NOUN cana-4831 155 20 , , PUNCT cana-4831 155 21 vietnam vietnam PROPN cana-4831 155 22 journal journal NOUN cana-4831 155 23 of of ADP cana-4831 155 24 mathematics mathematics PROPN cana-4831 155 25 , , PUNCT cana-4831 155 26 vol vol NOUN cana-4831 155 27 . . PUNCT cana-4831 155 28 38(4 38(4 NUM cana-4831 155 29 ) ) PUNCT cana-4831 155 30 , , PUNCT cana-4831 155 31 403 403 NUM cana-4831 155 32 - - SYM cana-4831 155 33 412 412 NUM cana-4831 155 34 , , PUNCT cana-4831 155 35 2010 2010 NUM cana-4831 155 36 . . PUNCT cana-4831 156 1 [ [ X cana-4831 156 2 8 8 NUM cana-4831 156 3 ] ] X cana-4831 156 4 dileep dileep PROPN cana-4831 156 5 l l PROPN cana-4831 156 6 and and CCONJ cana-4831 156 7 s s VERB cana-4831 156 8 latha latha PROPN cana-4831 156 9 , , PUNCT cana-4831 156 10 a a DET cana-4831 156 11 note note NOUN cana-4831 156 12 on on ADP cana-4831 156 13 salagean salagean ADJ cana-4831 156 14 -carlson -carlson PROPN cana-4831 156 15 shaffer shaffer NOUN cana-4831 156 16 operator operator NOUN cana-4831 156 17 , , PUNCT cana-4831 156 18 int int NOUN cana-4831 156 19 . . PUNCT cana-4831 157 1 j. j. PROPN cana-4831 157 2 math math PROPN cana-4831 157 3 . . PUNCT cana-4831 158 1 archive archive NOUN cana-4831 158 2 , , PUNCT cana-4831 158 3 2(2 2(2 NUM cana-4831 158 4 ) ) PUNCT cana-4831 158 5 , , PUNCT cana-4831 158 6 272 272 NUM cana-4831 158 7 - - SYM cana-4831 158 8 279 279 NUM cana-4831 158 9 , , PUNCT cana-4831 158 10 2011 2011 NUM cana-4831 158 11 . . PUNCT cana-4831 159 1 [ [ X cana-4831 159 2 9 9 NUM cana-4831 159 3 ] ] X cana-4831 159 4 dileep dileep PROPN cana-4831 159 5 l l PROPN cana-4831 159 6 and and CCONJ cana-4831 159 7 mallige mallige PROPN cana-4831 159 8 rajeev rajeev PROPN cana-4831 159 9 , , PUNCT cana-4831 159 10 convolution convolution NOUN cana-4831 159 11 operators operator NOUN cana-4831 159 12 in in ADP cana-4831 159 13 geometric geometric ADJ cana-4831 159 14 function function NOUN cana-4831 159 15 theory theory NOUN cana-4831 159 16 , , PUNCT cana-4831 159 17 gis gis PROPN cana-4831 159 18 science science NOUN cana-4831 159 19 journal journal PROPN cana-4831 159 20 , , PUNCT cana-4831 159 21 vol vol NOUN cana-4831 159 22 . . PROPN cana-4831 159 23 7 7 NUM cana-4831 159 24 , , PUNCT cana-4831 159 25 issue issue NOUN cana-4831 159 26 7 7 NUM cana-4831 159 27 , , PUNCT cana-4831 159 28 2020 2020 NUM cana-4831 159 29 , , PUNCT cana-4831 159 30 472 472 NUM cana-4831 159 31 - - SYM cana-4831 159 32 483 483 NUM cana-4831 159 33 . . PUNCT cana-4831 160 1 [ [ X cana-4831 160 2 10 10 NUM cana-4831 160 3 ] ] X cana-4831 160 4 p p X cana-4831 160 5 l l PROPN cana-4831 160 6 duren duren PROPN cana-4831 160 7 , , PUNCT cana-4831 160 8 univalent univalent ADJ cana-4831 160 9 functions function NOUN cana-4831 160 10 , , PUNCT cana-4831 160 11 springer springer NOUN cana-4831 160 12 - - PUNCT cana-4831 160 13 verlag verlag PROPN cana-4831 160 14 , , PUNCT cana-4831 160 15 1983 1983 NUM cana-4831 160 16 . . PUNCT cana-4831 161 1 [ [ X cana-4831 161 2 11 11 NUM cana-4831 161 3 ] ] SYM cana-4831 161 4 s s X cana-4831 161 5 s s NOUN cana-4831 161 6 eker eker ADJ cana-4831 161 7 and and CCONJ cana-4831 161 8 s s NOUN cana-4831 161 9 owa owa PROPN cana-4831 161 10 , , PUNCT cana-4831 161 11 certain certain ADJ cana-4831 161 12 classes class NOUN cana-4831 161 13 of of ADP cana-4831 161 14 analytic analytic ADJ cana-4831 161 15 functions function NOUN cana-4831 161 16 involving involve VERB cana-4831 161 17 salagean salagean ADJ cana-4831 161 18 operator operator NOUN cana-4831 161 19 , , PUNCT cana-4831 161 20 j. j. PROPN cana-4831 161 21 inequal inequal PROPN cana-4831 161 22 . . PUNCT cana-4831 162 1 pure pure ADJ cana-4831 162 2 appl appl PROPN cana-4831 162 3 . . PUNCT cana-4831 162 4 math math PROPN cana-4831 162 5 . . PUNCT cana-4831 163 1 , , PUNCT cana-4831 164 1 [ [ X cana-4831 164 2 12 12 NUM cana-4831 164 3 ] ] X cana-4831 164 4 s s X cana-4831 164 5 s s X cana-4831 164 6 eker eker ADJ cana-4831 164 7 and and CCONJ cana-4831 164 8 s s NOUN cana-4831 164 9 owa owa PROPN cana-4831 164 10 , , PUNCT cana-4831 164 11 new new ADJ cana-4831 164 12 applications application NOUN cana-4831 164 13 of of ADP cana-4831 164 14 classes class NOUN cana-4831 164 15 of of ADP cana-4831 164 16 analytic analytic ADJ cana-4831 164 17 functions function NOUN cana-4831 164 18 involving involve VERB cana-4831 164 19 salagean salagean ADJ cana-4831 164 20 operator operator NOUN cana-4831 164 21 , , PUNCT cana-4831 164 22 int int NOUN cana-4831 164 23 . . PUNCT cana-4831 165 1 symposium symposium NOUN cana-4831 165 2 on on ADP cana-4831 165 3 complex complex ADJ cana-4831 165 4 function function NOUN cana-4831 165 5 theory theory NOUN cana-4831 165 6 and and CCONJ cana-4831 165 7 applications application NOUN cana-4831 165 8 , , PUNCT cana-4831 165 9 brasov brasov NOUN cana-4831 165 10 , , PUNCT cana-4831 165 11 romania romania PROPN cana-4831 165 12 , , PUNCT cana-4831 165 13 15 15 NUM cana-4831 165 14 , , PUNCT cana-4831 165 15 ( ( PUNCT cana-4831 165 16 2006 2006 NUM cana-4831 165 17 ) ) PUNCT cana-4831 165 18 . . PUNCT cana-4831 166 1 [ [ X cana-4831 166 2 13 13 NUM cana-4831 166 3 ] ] SYM cana-4831 166 4 s s PART cana-4831 166 5 s s NOUN cana-4831 166 6 eker eker ADJ cana-4831 166 7 and and CCONJ cana-4831 166 8 b b NOUN cana-4831 166 9 seker seker NOUN cana-4831 166 10 , , PUNCT cana-4831 166 11 on on ADP cana-4831 166 12 a a DET cana-4831 166 13 class class NOUN cana-4831 166 14 of of ADP cana-4831 166 15 multivalent multivalent NOUN cana-4831 166 16 functions function NOUN cana-4831 166 17 defined define VERB cana-4831 166 18 by by ADP cana-4831 166 19 salagean salagean ADJ cana-4831 166 20 operator operator NOUN cana-4831 166 21 , , PUNCT cana-4831 166 22 general general ADJ cana-4831 166 23 mathematics mathematic NOUN cana-4831 166 24 , , PUNCT cana-4831 166 25 vol vol NOUN cana-4831 166 26 . . PROPN cana-4831 166 27 15 15 NUM cana-4831 166 28 , , PUNCT cana-4831 166 29 n-.2 n-.2 NOUN cana-4831 166 30 - - PUNCT cana-4831 166 31 3(2007 3(2007 NUM cana-4831 166 32 ) ) PUNCT cana-4831 166 33 , , PUNCT cana-4831 166 34 154 154 NUM cana-4831 166 35 - - SYM cana-4831 166 36 163 163 NUM cana-4831 166 37 . . PUNCT cana-4831 167 1 [ [ X cana-4831 167 2 14 14 NUM cana-4831 167 3 ] ] X cana-4831 167 4 j j PROPN cana-4831 167 5 e e PROPN cana-4831 167 6 littlewood littlewood PROPN cana-4831 167 7 , , PUNCT cana-4831 167 8 on on ADP cana-4831 167 9 inequalities inequality NOUN cana-4831 167 10 in in ADP cana-4831 167 11 the the DET cana-4831 167 12 theory theory NOUN cana-4831 167 13 of of ADP cana-4831 167 14 functions function NOUN cana-4831 167 15 , , PUNCT cana-4831 167 16 proc proc NOUN cana-4831 167 17 . . PROPN cana-4831 167 18 , , PUNCT cana-4831 167 19 london london PROPN cana-4831 167 20 math math PROPN cana-4831 167 21 . . PUNCT cana-4831 168 1 soc soc PROPN cana-4831 168 2 . . PUNCT cana-4831 168 3 , , PUNCT cana-4831 168 4 23(1925 23(1925 NUM cana-4831 168 5 ) ) PUNCT cana-4831 168 6 , , PUNCT cana-4831 168 7 481 481 NUM cana-4831 168 8 - - SYM cana-4831 168 9 519 519 NUM cana-4831 168 10 . . PUNCT cana-4831 169 1 [ [ X cana-4831 169 2 15 15 NUM cana-4831 169 3 ] ] X cana-4831 169 4 a a DET cana-4831 169 5 o o PROPN cana-4831 169 6 mostafa mostafa PROPN cana-4831 169 7 , , PUNCT cana-4831 169 8 a a DET cana-4831 169 9 study study NOUN cana-4831 169 10 on on ADP cana-4831 169 11 starlike starlike NOUN cana-4831 169 12 and and CCONJ cana-4831 169 13 convex convex NOUN cana-4831 169 14 properties property NOUN cana-4831 169 15 for for ADP cana-4831 169 16 hypergeometric hypergeometric ADJ cana-4831 169 17 functions function NOUN cana-4831 169 18 , , PUNCT cana-4831 169 19 jipam jipam NOUN cana-4831 169 20 , , PUNCT cana-4831 169 21 vol vol NOUN cana-4831 169 22 . . PUNCT cana-4831 169 23 10(2009 10(2009 NUM cana-4831 169 24 ) ) PUNCT cana-4831 169 25 , , PUNCT cana-4831 169 26 issue issue NOUN cana-4831 169 27 3 3 NUM cana-4831 169 28 , , PUNCT cana-4831 169 29 87 87 NUM cana-4831 169 30 . . PUNCT cana-4831 170 1 [ [ X cana-4831 170 2 16 16 NUM cana-4831 170 3 ] ] X cana-4831 170 4 m m PROPN cana-4831 170 5 s s PROPN cana-4831 170 6 robertson robertson PROPN cana-4831 170 7 , , PUNCT cana-4831 170 8 on on ADP cana-4831 170 9 the the DET cana-4831 170 10 theory theory NOUN cana-4831 170 11 of of ADP cana-4831 170 12 univalent univalent ADJ cana-4831 170 13 functions function NOUN cana-4831 170 14 , , PUNCT cana-4831 170 15 annals annal NOUN cana-4831 170 16 of of ADP cana-4831 170 17 math math NOUN cana-4831 170 18 . . PUNCT cana-4831 170 19 , , PUNCT cana-4831 170 20 37 37 NUM cana-4831 170 21 ( ( PUNCT cana-4831 170 22 1936 1936 NUM cana-4831 170 23 ) ) PUNCT cana-4831 170 24 , , PUNCT cana-4831 170 25 pp pp ADP cana-4831 170 26 374406 374406 NUM cana-4831 170 27 . . PUNCT cana-4831 171 1 [ [ X cana-4831 171 2 17 17 NUM cana-4831 171 3 ] ] X cana-4831 171 4 g g PROPN cana-4831 171 5 s s PROPN cana-4831 171 6 salagean salagean ADJ cana-4831 171 7 , , PUNCT cana-4831 171 8 on on ADP cana-4831 171 9 some some DET cana-4831 171 10 classes class NOUN cana-4831 171 11 of of ADP cana-4831 171 12 univalent univalent ADJ cana-4831 171 13 functions function NOUN cana-4831 171 14 , , PUNCT cana-4831 171 15 seminar seminar NOUN cana-4831 171 16 of of ADP cana-4831 171 17 geometric geometric ADJ cana-4831 171 18 function function NOUN cana-4831 171 19 theory theory NOUN cana-4831 171 20 , , PUNCT cana-4831 171 21 cluj cluj NOUN cana-4831 171 22 - - PUNCT cana-4831 171 23 napoca napoca NOUN cana-4831 171 24 , , PUNCT cana-4831 171 25 1983 1983 NUM cana-4831 171 26 . . PUNCT cana-4831 172 1 [ [ X cana-4831 172 2 18 18 NUM cana-4831 172 3 ] ] X cana-4831 172 4 g g PROPN cana-4831 172 5 s s PROPN cana-4831 172 6 salagean salagean ADJ cana-4831 172 7 , , PUNCT cana-4831 172 8 subclasses subclass NOUN cana-4831 172 9 of of ADP cana-4831 172 10 univalent univalent ADJ cana-4831 172 11 functions function NOUN cana-4831 172 12 , , PUNCT cana-4831 172 13 lecture lecture NOUN cana-4831 172 14 notes note NOUN cana-4831 172 15 in in ADP cana-4831 172 16 mathe mathe PROPN cana-4831 172 17 . . PUNCT cana-4831 172 18 springerverlag springerverlag PROPN cana-4831 172 19 , , PUNCT cana-4831 172 20 2013 2013 NUM cana-4831 172 21 ( ( PUNCT cana-4831 172 22 1983 1983 NUM cana-4831 172 23 ) ) PUNCT cana-4831 172 24 , , PUNCT cana-4831 172 25 362 362 NUM cana-4831 172 26 - - SYM cana-4831 172 27 372 372 NUM cana-4831 172 28 . . PUNCT cana-4831 173 1 [ [ X cana-4831 173 2 19 19 NUM cana-4831 173 3 ] ] PUNCT cana-4831 173 4 s s X cana-4831 173 5 shams sham NOUN cana-4831 173 6 , , PUNCT cana-4831 173 7 s s NOUN cana-4831 173 8 r r NOUN cana-4831 173 9 kulakukarni kulakukarni NOUN cana-4831 173 10 and and CCONJ cana-4831 173 11 j j PROPN cana-4831 173 12 m m PROPN cana-4831 173 13 jahangiri jahangiri ADV cana-4831 173 14 , , PUNCT cana-4831 173 15 classes class NOUN cana-4831 173 16 of of ADP cana-4831 173 17 uniformly uniformly ADJ cana-4831 173 18 starlike starlike NOUN cana-4831 173 19 and and CCONJ cana-4831 173 20 functions function NOUN cana-4831 173 21 , , PUNCT cana-4831 173 22 int int NOUN cana-4831 173 23 . . PUNCT cana-4831 174 1 j. j. PROPN cana-4831 174 2 math math PROPN cana-4831 174 3 . . PUNCT cana-4831 175 1 and and CCONJ cana-4831 175 2 math math NOUN cana-4831 175 3 . . PUNCT cana-4831 176 1 si si PROPN cana-4831 176 2 . . PROPN cana-4831 176 3 , , PUNCT cana-4831 176 4 55 55 NUM cana-4831 176 5 ( ( PUNCT cana-4831 176 6 2004 2004 NUM cana-4831 176 7 ) ) PUNCT cana-4831 176 8 , , PUNCT cana-4831 176 9 2959 2959 NUM cana-4831 176 10 - - SYM cana-4831 176 11 2961 2961 NUM cana-4831 176 12 . . PUNCT cana-4831 177 1 [ [ X cana-4831 177 2 20 20 NUM cana-4831 177 3 ] ] PUNCT cana-4831 177 4 h h NOUN cana-4831 177 5 silverman silverman NOUN cana-4831 177 6 , , PUNCT cana-4831 177 7 univalent univalent ADJ cana-4831 177 8 functions function NOUN cana-4831 177 9 with with ADP cana-4831 177 10 varying vary VERB cana-4831 177 11 arguments argument NOUN cana-4831 177 12 , , PUNCT cana-4831 177 13 huston huston PROPN cana-4831 177 14 journal journal PROPN cana-4831 177 15 of of ADP cana-4831 177 16 math math NOUN cana-4831 177 17 . . PUNCT cana-4831 177 18 , , PUNCT cana-4831 177 19 vol.7 vol.7 PROPN cana-4831 177 20 , , PUNCT cana-4831 177 21 no.2 no.2 PROPN cana-4831 177 22 ( ( PUNCT cana-4831 177 23 1981 1981 NUM cana-4831 177 24 ) ) PUNCT cana-4831 177 25 . . PUNCT