id sid tid token lemma pos cana-5356 1 1 communications communication NOUN cana-5356 1 2 on on ADP cana-5356 1 3 applied apply VERB cana-5356 1 4 nonlinear nonlinear ADJ cana-5356 1 5 analysis analysis NOUN cana-5356 1 6 issn issn NOUN cana-5356 1 7 : : PUNCT cana-5356 1 8 1074 1074 NUM cana-5356 1 9 - - PUNCT cana-5356 1 10 133x 133x NUM cana-5356 1 11 vol vol VERB cana-5356 1 12 32 32 NUM cana-5356 1 13 no no NOUN cana-5356 1 14 . . PUNCT cana-5356 2 1 10s 10 NOUN cana-5356 2 2 ( ( PUNCT cana-5356 2 3 2025 2025 NUM cana-5356 2 4 ) ) PUNCT cana-5356 2 5 1872 1872 NUM cana-5356 2 6 https://internationalpubls.com https://internationalpubls.com X cana-5356 2 7 a a DET cana-5356 2 8 new new ADJ cana-5356 2 9 subclass subclass NOUN cana-5356 2 10 of of ADP cana-5356 2 11 meromorphic meromorphic ADJ cana-5356 2 12 functions function NOUN cana-5356 2 13 associated associate VERB cana-5356 2 14 with with ADP cana-5356 2 15 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 2 16 operator operator NOUN cana-5356 2 17 debasmita debasmita NOUN cana-5356 2 18 dash dash NOUN cana-5356 2 19 1 1 NUM cana-5356 2 20 , , PUNCT cana-5356 2 21 saumya saumya VERB cana-5356 2 22 singh singh PROPN cana-5356 2 23 2 2 NUM cana-5356 2 24 * * SYM cana-5356 2 25 1 1 NUM cana-5356 2 26 , , PUNCT cana-5356 2 27 2 2 NUM cana-5356 2 28 department department NOUN cana-5356 2 29 of of ADP cana-5356 2 30 mathematics mathematic NOUN cana-5356 2 31 , , PUNCT cana-5356 2 32 o. o. PROPN cana-5356 2 33 p. p. PROPN cana-5356 2 34 jindal jindal PROPN cana-5356 2 35 university university PROPN cana-5356 2 36 , , PUNCT cana-5356 2 37 raigarh raigarh ADJ cana-5356 2 38 , , PUNCT cana-5356 2 39 496109 496109 NUM cana-5356 2 40 india india NOUN cana-5356 2 41 ; ; PUNCT cana-5356 2 42 * * PUNCT cana-5356 2 43 correspondence correspondence NOUN cana-5356 2 44 should should AUX cana-5356 2 45 be be AUX cana-5356 2 46 addressed address VERB cana-5356 2 47 to to PART cana-5356 2 48 saumya saumya VERB cana-5356 2 49 singh singh PROPN cana-5356 2 50 ; ; PUNCT cana-5356 2 51 emailsaumya.singh@opju.ac.in emailsaumya.singh@opju.ac.in NUM cana-5356 2 52 article article NOUN cana-5356 2 53 history history NOUN cana-5356 2 54 : : PUNCT cana-5356 2 55 received receive VERB cana-5356 2 56 : : PUNCT cana-5356 2 57 12 12 NUM cana-5356 2 58 - - SYM cana-5356 2 59 01 01 NUM cana-5356 2 60 - - PUNCT cana-5356 2 61 2025 2025 NUM cana-5356 2 62 revised revise VERB cana-5356 2 63 : : PUNCT cana-5356 2 64 15 15 NUM cana-5356 2 65 - - NUM cana-5356 2 66 02 02 NUM cana-5356 2 67 - - PUNCT cana-5356 2 68 2025 2025 NUM cana-5356 2 69 accepted accept VERB cana-5356 2 70 : : PUNCT cana-5356 2 71 01 01 NUM cana-5356 2 72 - - SYM cana-5356 2 73 03 03 NUM cana-5356 2 74 - - PUNCT cana-5356 2 75 2025 2025 NUM cana-5356 2 76 abstract abstract NOUN cana-5356 2 77 : : PUNCT cana-5356 2 78 in in ADP cana-5356 2 79 this this DET cana-5356 2 80 paper paper NOUN cana-5356 2 81 , , PUNCT cana-5356 2 82 we we PRON cana-5356 2 83 propose propose VERB cana-5356 2 84 a a DET cana-5356 2 85 novel novel ADJ cana-5356 2 86 subclass subclass NOUN cana-5356 2 87 of of ADP cana-5356 2 88 meromorphic meromorphic ADJ cana-5356 2 89 functions function NOUN cana-5356 2 90 within within ADP cana-5356 2 91 the the DET cana-5356 2 92 class class NOUN cana-5356 2 93 π’œπ‘ π’œπ‘ PROPN cana-5356 2 94 βˆ— βˆ— NOUN cana-5356 2 95 which which PRON cana-5356 2 96 includes include VERB cana-5356 2 97 functions function NOUN cana-5356 2 98 of of ADP cana-5356 2 99 the the DET cana-5356 2 100 form form NOUN cana-5356 2 101 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 2 102 ) ) PUNCT cana-5356 2 103 = = PRON cana-5356 2 104 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 2 105 + + NUM cana-5356 2 106 βˆ‘ βˆ‘ PROPN cana-5356 2 107 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 2 108 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 2 109 , , PUNCT cana-5356 2 110 where where SCONJ cana-5356 2 111 𝑧 𝑧 PRON cana-5356 2 112 ∈ ∈ PROPN cana-5356 2 113 β–³ β–³ NOUN cana-5356 2 114 βˆ— βˆ— NOUN cana-5356 2 115 and and CCONJ cana-5356 2 116 𝑝 𝑝 NOUN cana-5356 2 117 ∈ ∈ PROPN cana-5356 2 118 β„•. β„•. PROPN cana-5356 2 119 the the DET cana-5356 2 120 study study NOUN cana-5356 2 121 specifically specifically ADV cana-5356 2 122 focuses focus VERB cana-5356 2 123 on on ADP cana-5356 2 124 using use VERB cana-5356 2 125 the the DET cana-5356 2 126 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 2 127 operator operator NOUN cana-5356 2 128 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 2 129 𝑛 𝑛 DET cana-5356 2 130 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 2 131 ) ) PUNCT cana-5356 2 132 to to PART cana-5356 2 133 established establish VERB cana-5356 2 134 this this DET cana-5356 2 135 new new ADJ cana-5356 2 136 subclass subclass NOUN cana-5356 2 137 denoted denote VERB cana-5356 2 138 as as ADP cana-5356 2 139 β„³(𝛼 β„³(𝛼 X cana-5356 2 140 , , PUNCT cana-5356 2 141 𝛽 𝛽 NOUN cana-5356 2 142 , , PUNCT cana-5356 2 143 𝛾 𝛾 PROPN cana-5356 2 144 , , PUNCT cana-5356 2 145 π‘˜ π‘˜ NOUN cana-5356 2 146 , , PUNCT cana-5356 2 147 πœ† πœ† NOUN cana-5356 2 148 , , PUNCT cana-5356 2 149 𝑛 𝑛 PROPN cana-5356 2 150 , , PUNCT cana-5356 2 151 𝑝 𝑝 NOUN cana-5356 2 152 ) ) PUNCT cana-5356 2 153 which which PRON cana-5356 2 154 is be AUX cana-5356 2 155 defined define VERB cana-5356 2 156 by by ADP cana-5356 2 157 the the DET cana-5356 2 158 parameters parameter NOUN cana-5356 2 159 0 0 NUM cana-5356 2 160 ≀ ≀ NUM cana-5356 2 161 𝛼 𝛼 X cana-5356 2 162 < < X cana-5356 2 163 1 1 NUM cana-5356 2 164 , , PUNCT cana-5356 2 165 0 0 NUM cana-5356 2 166 ≀ ≀ NOUN cana-5356 3 1 𝛽 𝛽 NOUN cana-5356 3 2 < < X cana-5356 3 3 1 1 NUM cana-5356 3 4 , , PUNCT cana-5356 3 5 0 0 NUM cana-5356 3 6 ≀ ≀ NOUN cana-5356 3 7 𝛾 𝛾 ADP cana-5356 3 8 < < X cana-5356 3 9 1 1 NUM cana-5356 3 10 , , PUNCT cana-5356 3 11 π‘˜ π‘˜ DET cana-5356 3 12 β‰₯ β‰₯ NOUN cana-5356 3 13 0 0 NUM cana-5356 3 14 , , PUNCT cana-5356 3 15 0 0 PUNCT cana-5356 3 16 < < X cana-5356 3 17 πœ† πœ† X cana-5356 3 18 ≀ ≀ ADV cana-5356 3 19 1 1 NUM cana-5356 3 20 and and CCONJ cana-5356 3 21 𝑛 𝑛 DET cana-5356 3 22 ∈ ∈ PROPN cana-5356 3 23 β„•0 β„•0 NOUN cana-5356 3 24 . . PUNCT cana-5356 4 1 𝑅𝑒 𝑅𝑒 VERB cana-5356 4 2 ( ( PUNCT cana-5356 4 3 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 4 4 𝑛 𝑛 PRON cana-5356 4 5 𝑓(𝑧))β€²βˆ’(𝐷𝑝,πœ† 𝑓(𝑧))β€²βˆ’(𝐷𝑝,πœ† PROPN cana-5356 4 6 𝑛 𝑛 PRON cana-5356 4 7 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 4 8 ) ) PUNCT cana-5356 4 9 ) ) PUNCT cana-5356 5 1 𝛼𝑧(𝐷𝑝,πœ† 𝛼𝑧(𝐷𝑝,πœ† NOUN cana-5356 5 2 𝑛 𝑛 PRON cana-5356 5 3 𝑓(𝑧))β€²+(1βˆ’π‘Ÿ)(𝐷𝑝,πœ† 𝑓(𝑧))β€²+(1βˆ’π‘Ÿ)(𝐷𝑝,πœ† NOUN cana-5356 5 4 𝑛 𝑛 PRON cana-5356 5 5 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 5 6 ) ) PUNCT cana-5356 5 7 ) ) PUNCT cana-5356 5 8 ) ) PUNCT cana-5356 6 1 > > X cana-5356 7 1 π‘˜ π‘˜ PRON cana-5356 7 2 | | ADV cana-5356 7 3 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 7 4 𝑛 𝑛 ADP cana-5356 7 5 𝑓(𝑧))β€²βˆ’(𝐷𝑝,πœ† 𝑓(𝑧))β€²βˆ’(𝐷𝑝,πœ† PROPN cana-5356 7 6 𝑛 𝑛 PRON cana-5356 7 7 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 7 8 ) ) PUNCT cana-5356 7 9 ) ) PUNCT cana-5356 8 1 𝛼𝑧(𝐷𝑝,πœ† 𝛼𝑧(𝐷𝑝,πœ† NOUN cana-5356 8 2 𝑛 𝑛 PRON cana-5356 8 3 𝑓(𝑧))β€²+(1βˆ’π‘Ÿ)(𝐷𝑝,πœ† 𝑓(𝑧))β€²+(1βˆ’π‘Ÿ)(𝐷𝑝,πœ† NOUN cana-5356 8 4 𝑛 𝑛 PRON cana-5356 8 5 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 8 6 ) ) PUNCT cana-5356 8 7 ) ) PUNCT cana-5356 8 8 βˆ’ βˆ’ ADP cana-5356 9 1 1| 1| NUM cana-5356 10 1 + + CCONJ cana-5356 10 2 𝛽 𝛽 NOUN cana-5356 10 3 by by ADP cana-5356 10 4 the the DET cana-5356 10 5 given give VERB cana-5356 10 6 condition condition NOUN cana-5356 10 7 for for ADP cana-5356 10 8 this this DET cana-5356 10 9 class class NOUN cana-5356 10 10 we we PRON cana-5356 10 11 will will AUX cana-5356 10 12 next next VERB cana-5356 10 13 to to PART cana-5356 10 14 prove prove VERB cana-5356 10 15 this this DET cana-5356 10 16 class class NOUN cana-5356 10 17 of of ADP cana-5356 10 18 univalent univalent ADJ cana-5356 10 19 function function NOUN cana-5356 10 20 then then ADV cana-5356 10 21 we we PRON cana-5356 10 22 derive derive VERB cana-5356 10 23 the the DET cana-5356 10 24 related related ADJ cana-5356 10 25 results result NOUN cana-5356 10 26 . . PUNCT cana-5356 11 1 the the DET cana-5356 11 2 mathematical mathematical ADJ cana-5356 11 3 formulation formulation NOUN cana-5356 11 4 of of ADP cana-5356 11 5 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 11 6 𝑛 𝑛 PRON cana-5356 11 7 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 11 8 ) ) PUNCT cana-5356 11 9 , , PUNCT cana-5356 11 10 where where SCONJ cana-5356 11 11 the the DET cana-5356 11 12 operator operator NOUN cana-5356 11 13 influences influence VERB cana-5356 11 14 the the DET cana-5356 11 15 series series NOUN cana-5356 11 16 expansion expansion NOUN cana-5356 11 17 of of ADP cana-5356 11 18 the the DET cana-5356 11 19 meromorphic meromorphic ADJ cana-5356 11 20 functions function NOUN cana-5356 11 21 , , PUNCT cana-5356 11 22 is be AUX cana-5356 11 23 also also ADV cana-5356 11 24 elaborated elaborate VERB cana-5356 11 25 in in ADP cana-5356 11 26 the the DET cana-5356 11 27 study study NOUN cana-5356 11 28 . . PUNCT cana-5356 12 1 keywords keyword NOUN cana-5356 12 2 : : PUNCT cana-5356 12 3 meromorphic meromorphic ADJ cana-5356 12 4 functions function NOUN cana-5356 12 5 , , PUNCT cana-5356 12 6 function function NOUN cana-5356 12 7 class class NOUN cana-5356 12 8 π’œπ‘ π’œπ‘ PROPN cana-5356 12 9 βˆ— βˆ— NOUN cana-5356 12 10 , , PUNCT cana-5356 12 11 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 12 12 operator operator NOUN cana-5356 12 13 , , PUNCT cana-5356 12 14 starlikeness starlikeness NOUN cana-5356 12 15 , , PUNCT cana-5356 12 16 convexity convexity NOUN cana-5356 12 17 , , PUNCT cana-5356 12 18 subclass subclass NOUN cana-5356 12 19 of of ADP cana-5356 12 20 meromorphic meromorphic ADJ cana-5356 12 21 functions function NOUN cana-5356 12 22 , , PUNCT cana-5356 12 23 series series NOUN cana-5356 12 24 expansion expansion NOUN cana-5356 12 25 , , PUNCT cana-5356 12 26 analytic analytic ADJ cana-5356 12 27 function function NOUN cana-5356 12 28 properties property NOUN cana-5356 12 29 . . PUNCT cana-5356 13 1 1 1 X cana-5356 13 2 . . X cana-5356 13 3 introduction introduction NOUN cana-5356 13 4 the the DET cana-5356 13 5 study study NOUN cana-5356 13 6 of of ADP cana-5356 13 7 meromorphic meromorphic ADJ cana-5356 13 8 functions function NOUN cana-5356 13 9 has have AUX cana-5356 13 10 been be AUX cana-5356 13 11 a a DET cana-5356 13 12 significant significant ADJ cana-5356 13 13 area area NOUN cana-5356 13 14 of of ADP cana-5356 13 15 interest interest NOUN cana-5356 13 16 in in ADP cana-5356 13 17 complex complex ADJ cana-5356 13 18 analysis analysis NOUN cana-5356 13 19 , , PUNCT cana-5356 13 20 particularly particularly ADV cana-5356 13 21 in in ADP cana-5356 13 22 the the DET cana-5356 13 23 context context NOUN cana-5356 13 24 of of ADP cana-5356 13 25 geometric geometric ADJ cana-5356 13 26 function function NOUN cana-5356 13 27 theory theory NOUN cana-5356 13 28 . . PUNCT cana-5356 14 1 these these DET cana-5356 14 2 functions function NOUN cana-5356 14 3 , , PUNCT cana-5356 14 4 which which PRON cana-5356 14 5 are be AUX cana-5356 14 6 analytic analytic ADJ cana-5356 14 7 except except SCONJ cana-5356 14 8 for for ADP cana-5356 14 9 isolated isolated ADJ cana-5356 14 10 poles pole NOUN cana-5356 14 11 , , PUNCT cana-5356 14 12 play play VERB cana-5356 14 13 a a DET cana-5356 14 14 crucial crucial ADJ cana-5356 14 15 role role NOUN cana-5356 14 16 in in ADP cana-5356 14 17 various various ADJ cana-5356 14 18 applied apply VERB cana-5356 14 19 & & CCONJ cana-5356 14 20 mathematical mathematical ADJ cana-5356 14 21 fields field NOUN cana-5356 14 22 . . PUNCT cana-5356 15 1 within within ADP cana-5356 15 2 the the DET cana-5356 15 3 class class NOUN cana-5356 15 4 of of ADP cana-5356 15 5 meromorphic meromorphic ADJ cana-5356 15 6 functions function NOUN cana-5356 15 7 , , PUNCT cana-5356 15 8 researchers researcher NOUN cana-5356 15 9 have have AUX cana-5356 15 10 explored explore VERB cana-5356 15 11 different different ADJ cana-5356 15 12 subclasses subclass NOUN cana-5356 15 13 defined define VERB cana-5356 15 14 by by ADP cana-5356 15 15 specific specific ADJ cana-5356 15 16 function function NOUN cana-5356 15 17 properties property NOUN cana-5356 15 18 and and CCONJ cana-5356 15 19 operator operator NOUN cana-5356 15 20 - - PUNCT cana-5356 15 21 induced induce VERB cana-5356 15 22 transformations transformation NOUN cana-5356 15 23 . . PUNCT cana-5356 16 1 one one NUM cana-5356 16 2 such such ADJ cana-5356 16 3 transformation transformation NOUN cana-5356 16 4 involves involve VERB cana-5356 16 5 differential differential ADJ cana-5356 16 6 and and CCONJ cana-5356 16 7 integral integral ADJ cana-5356 16 8 operators operator NOUN cana-5356 16 9 that that PRON cana-5356 16 10 influence influence VERB cana-5356 16 11 the the DET cana-5356 16 12 function function NOUN cana-5356 16 13 's 's PART cana-5356 16 14 behaviour behaviour NOUN cana-5356 16 15 in in ADP cana-5356 16 16 the the DET cana-5356 16 17 punctured punctured ADJ cana-5356 16 18 unit unit NOUN cana-5356 16 19 disk disk NOUN cana-5356 16 20 . . PUNCT cana-5356 17 1 the the DET cana-5356 17 2 role role NOUN cana-5356 17 3 of of ADP cana-5356 17 4 multiplier multipli ADJ cana-5356 17 5 transformations transformation NOUN cana-5356 17 6 in in ADP cana-5356 17 7 defining define VERB cana-5356 17 8 new new ADJ cana-5356 17 9 function function NOUN cana-5356 17 10 classes class NOUN cana-5356 17 11 has have AUX cana-5356 17 12 been be AUX cana-5356 17 13 studied study VERB cana-5356 17 14 extensively extensively ADV cana-5356 17 15 . . PUNCT cana-5356 18 1 for for ADP cana-5356 18 2 some some DET cana-5356 18 3 basic basic ADJ cana-5356 18 4 concepts concept NOUN cana-5356 18 5 like like ADP cana-5356 18 6 univalent univalent ADJ cana-5356 18 7 functions function NOUN cana-5356 18 8 , , PUNCT cana-5356 18 9 which which PRON cana-5356 18 10 are be AUX cana-5356 18 11 holomorphic holomorphic ADJ cana-5356 18 12 and and CCONJ cana-5356 18 13 injective injective ADJ cana-5356 18 14 , , PUNCT cana-5356 18 15 are be AUX cana-5356 18 16 fundamental fundamental ADJ cana-5356 18 17 in in ADP cana-5356 18 18 geometric geometric ADJ cana-5356 18 19 function function NOUN cana-5356 18 20 theory theory NOUN cana-5356 18 21 for for ADP cana-5356 18 22 a a DET cana-5356 18 23 comprehensive comprehensive ADJ cana-5356 18 24 study study NOUN cana-5356 18 25 , , PUNCT cana-5356 18 26 ( ( PUNCT cana-5356 18 27 see see VERB cana-5356 18 28 [ [ X cana-5356 18 29 7 7 NUM cana-5356 18 30 ] ] PUNCT cana-5356 18 31 , , PUNCT cana-5356 18 32 [ [ X cana-5356 18 33 8 8 NUM cana-5356 18 34 ] ] NUM cana-5356 18 35 ) ) PUNCT cana-5356 18 36 . . PUNCT cana-5356 19 1 alharayzeh alharayzeh PROPN cana-5356 19 2 and and CCONJ cana-5356 19 3 ghanim ghanim NOUN cana-5356 19 4 ( ( PUNCT cana-5356 19 5 2022 2022 NUM cana-5356 19 6 ) ) PUNCT cana-5356 19 7 recently recently ADV cana-5356 19 8 introduced introduce VERB cana-5356 19 9 a a DET cana-5356 19 10 k k ADJ cana-5356 19 11 - - ADJ cana-5356 19 12 uniformly uniformly ADV cana-5356 19 13 univalent univalent ADJ cana-5356 19 14 subclass subclass NOUN cana-5356 19 15 with with ADP cana-5356 19 16 negative negative ADJ cana-5356 19 17 coefficients coefficient NOUN cana-5356 19 18 ( ( PUNCT cana-5356 19 19 see see VERB cana-5356 19 20 also also ADV cana-5356 19 21 [ [ X cana-5356 19 22 1 1 NUM cana-5356 19 23 ] ] NUM cana-5356 19 24 ) ) PUNCT cana-5356 19 25 , , PUNCT cana-5356 19 26 proving prove VERB cana-5356 19 27 key key ADJ cana-5356 19 28 coefficient coefficient NOUN cana-5356 19 29 estimates estimate NOUN cana-5356 19 30 and and CCONJ cana-5356 19 31 geometric geometric ADJ cana-5356 19 32 properties property NOUN cana-5356 19 33 . . PUNCT cana-5356 20 1 building build VERB cana-5356 20 2 upon upon SCONJ cana-5356 20 3 their their PRON cana-5356 20 4 findings finding NOUN cana-5356 20 5 , , PUNCT cana-5356 20 6 this this DET cana-5356 20 7 paper paper NOUN cana-5356 20 8 introduces introduce VERB cana-5356 20 9 a a DET cana-5356 20 10 new new ADJ cana-5356 20 11 subclass subclass NOUN cana-5356 20 12 incorporating incorporate VERB cana-5356 20 13 additional additional ADJ cana-5356 20 14 parameters parameter NOUN cana-5356 20 15 , , PUNCT cana-5356 20 16 leading lead VERB cana-5356 20 17 to to ADP cana-5356 20 18 broader broad ADJ cana-5356 20 19 applications application NOUN cana-5356 20 20 in in ADP cana-5356 20 21 function function NOUN cana-5356 20 22 theory theory NOUN cana-5356 20 23 . . PUNCT cana-5356 21 1 in in ADP cana-5356 21 2 ( ( PUNCT cana-5356 21 3 2016 2016 NUM cana-5356 21 4 ) ) PUNCT cana-5356 21 5 , , PUNCT cana-5356 21 6 amourah amourah VERB cana-5356 21 7 and and CCONJ cana-5356 21 8 darus darus NOUN cana-5356 21 9 defined define VERB cana-5356 21 10 a a DET cana-5356 21 11 new new ADJ cana-5356 21 12 class class NOUN cana-5356 21 13 of of ADP cana-5356 21 14 univalent univalent ADJ cana-5356 21 15 functions function NOUN cana-5356 21 16 by by ADP cana-5356 21 17 applying apply VERB cana-5356 21 18 a a DET cana-5356 21 19 generalized generalize VERB cana-5356 21 20 differential differential NOUN cana-5356 21 21 operator operator NOUN cana-5356 21 22 that that PRON cana-5356 21 23 includes include VERB cana-5356 21 24 negative negative ADJ cana-5356 21 25 coefficients coefficient NOUN cana-5356 21 26 , , PUNCT cana-5356 21 27 highlighting highlight VERB cana-5356 21 28 its its PRON cana-5356 21 29 geometric geometric ADJ cana-5356 21 30 and and CCONJ cana-5356 21 31 analytic analytic ADJ cana-5356 21 32 properties property NOUN cana-5356 21 33 in in ADP cana-5356 21 34 paper paper NOUN cana-5356 21 35 [ [ X cana-5356 21 36 3 3 NUM cana-5356 21 37 ] ] PUNCT cana-5356 21 38 . . PUNCT cana-5356 22 1 recent recent ADJ cana-5356 22 2 advancements advancement NOUN cana-5356 22 3 in in ADP cana-5356 22 4 geometric geometric ADJ cana-5356 22 5 function function NOUN cana-5356 22 6 theory theory NOUN cana-5356 22 7 have have AUX cana-5356 22 8 led lead VERB cana-5356 22 9 to to ADP cana-5356 22 10 the the DET cana-5356 22 11 development development NOUN cana-5356 22 12 of of ADP cana-5356 22 13 new new ADJ cana-5356 22 14 subclasses subclass NOUN cana-5356 22 15 of of ADP cana-5356 22 16 analytic analytic ADJ cana-5356 22 17 functions function NOUN cana-5356 22 18 . . PUNCT cana-5356 23 1 in in ADP cana-5356 23 2 ( ( PUNCT cana-5356 23 3 2024 2024 NUM cana-5356 23 4 ) ) PUNCT cana-5356 23 5 , , PUNCT cana-5356 23 6 carroll carroll NOUN cana-5356 23 7 discusses discuss VERB cana-5356 23 8 key key ADJ cana-5356 23 9 techniques technique NOUN cana-5356 23 10 in in ADP cana-5356 23 11 paper paper NOUN cana-5356 23 12 [ [ X cana-5356 23 13 4 4 NUM cana-5356 23 14 ] ] PUNCT cana-5356 23 15 , , PUNCT cana-5356 23 16 for for ADP cana-5356 23 17 analyzing analyze VERB cana-5356 23 18 these these DET cana-5356 23 19 classes class NOUN cana-5356 23 20 , , PUNCT cana-5356 23 21 which which PRON cana-5356 23 22 are be AUX cana-5356 23 23 the the DET cana-5356 23 24 foundation foundation NOUN cana-5356 23 25 of of ADP cana-5356 23 26 our our PRON cana-5356 23 27 investigation investigation NOUN cana-5356 23 28 . . PUNCT cana-5356 24 1 in in ADP cana-5356 24 2 ( ( PUNCT cana-5356 24 3 2010 2010 NUM cana-5356 24 4 ) ) PUNCT cana-5356 24 5 , , PUNCT cana-5356 24 6 ali ali PROPN cana-5356 24 7 and and CCONJ cana-5356 24 8 ravichandran ravichandran NOUN cana-5356 24 9 defined define VERB cana-5356 24 10 a a DET cana-5356 24 11 subclass subclass NOUN cana-5356 24 12 of of ADP cana-5356 24 13 meromorphic meromorphic ADJ cana-5356 24 14 functions function NOUN cana-5356 24 15 based base VERB cana-5356 24 16 on on ADP cana-5356 24 17 𝛼convexity 𝛼convexity NOUN cana-5356 24 18 and and CCONJ cana-5356 24 19 examined examine VERB cana-5356 24 20 its its PRON cana-5356 24 21 analytic analytic ADJ cana-5356 24 22 and and CCONJ cana-5356 24 23 mailto:saumya.singh@opju.ac.in mailto:saumya.singh@opju.ac.in PROPN cana-5356 24 24 communications communication NOUN cana-5356 24 25 on on ADP cana-5356 24 26 applied apply VERB cana-5356 24 27 nonlinear nonlinear ADJ cana-5356 24 28 analysis analysis NOUN cana-5356 24 29 issn issn NOUN cana-5356 24 30 : : PUNCT cana-5356 24 31 1074 1074 NUM cana-5356 24 32 - - PUNCT cana-5356 24 33 133x 133x NUM cana-5356 24 34 vol vol VERB cana-5356 24 35 32 32 NUM cana-5356 24 36 no no NOUN cana-5356 24 37 . . PUNCT cana-5356 25 1 10s 10 NOUN cana-5356 25 2 ( ( PUNCT cana-5356 25 3 2025 2025 NUM cana-5356 25 4 ) ) PUNCT cana-5356 25 5 1873 1873 NUM cana-5356 26 1 https://internationalpubls.com https://internationalpubls.com X cana-5356 26 2 geometric geometric ADJ cana-5356 26 3 properties property NOUN cana-5356 26 4 in in ADP cana-5356 26 5 paper paper NOUN cana-5356 26 6 [ [ X cana-5356 26 7 2 2 NUM cana-5356 26 8 ] ] PUNCT cana-5356 26 9 . . PUNCT cana-5356 27 1 darus darus NOUN cana-5356 27 2 et et PROPN cana-5356 27 3 al al PROPN cana-5356 27 4 . . PROPN cana-5356 27 5 established establish VERB cana-5356 27 6 univalence univalence NOUN cana-5356 27 7 criteria criterion NOUN cana-5356 27 8 in in ADP cana-5356 27 9 paper paper NOUN cana-5356 27 10 [ [ X cana-5356 27 11 5 5 NUM cana-5356 27 12 ] ] PUNCT cana-5356 27 13 , , PUNCT cana-5356 27 14 for for ADP cana-5356 27 15 analytic analytic ADJ cana-5356 27 16 functions function NOUN cana-5356 27 17 using use VERB cana-5356 27 18 a a DET cana-5356 27 19 generalized generalize VERB cana-5356 27 20 differential differential NOUN cana-5356 27 21 operator operator NOUN cana-5356 27 22 . . PUNCT cana-5356 28 1 building build VERB cana-5356 28 2 on on ADP cana-5356 28 3 their their PRON cana-5356 28 4 findings finding NOUN cana-5356 28 5 , , PUNCT cana-5356 28 6 this this DET cana-5356 28 7 paper paper NOUN cana-5356 28 8 explores explore VERB cana-5356 28 9 new new ADJ cana-5356 28 10 subclasses subclass NOUN cana-5356 28 11 by by ADP cana-5356 28 12 extending extend VERB cana-5356 28 13 the the DET cana-5356 28 14 operator operator NOUN cana-5356 28 15 ’s ’s PART cana-5356 28 16 properties property NOUN cana-5356 28 17 of of ADP cana-5356 28 18 generalized generalized ADJ cana-5356 28 19 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 28 20 operator operator NOUN cana-5356 28 21 and and CCONJ cana-5356 28 22 derived derive VERB cana-5356 28 23 new new ADJ cana-5356 28 24 results result NOUN cana-5356 28 25 . . PUNCT cana-5356 29 1 for for ADP cana-5356 29 2 computational computational ADJ cana-5356 29 3 purposes purpose NOUN cana-5356 29 4 we we PRON cana-5356 29 5 refer refer VERB cana-5356 29 6 to to ADP cana-5356 29 7 paper paper NOUN cana-5356 29 8 [ [ X cana-5356 29 9 6 6 NUM cana-5356 29 10 ] ] PUNCT cana-5356 29 11 , , PUNCT cana-5356 29 12 which which PRON cana-5356 29 13 is be AUX cana-5356 29 14 based base VERB cana-5356 29 15 on on ADP cana-5356 29 16 the the DET cana-5356 29 17 convexity convexity NOUN cana-5356 29 18 criterion criterion NOUN cana-5356 29 19 established establish VERB cana-5356 29 20 by by ADP cana-5356 29 21 deniz deniz PROPN cana-5356 29 22 and and CCONJ cana-5356 29 23 erhan erhan ADP cana-5356 29 24 in in ADP cana-5356 29 25 ( ( PUNCT cana-5356 29 26 2013 2013 NUM cana-5356 29 27 ) ) PUNCT cana-5356 29 28 . . PUNCT cana-5356 30 1 for for ADP cana-5356 30 2 further further ADJ cana-5356 30 3 study study NOUN cana-5356 30 4 on on ADP cana-5356 30 5 univalence univalence NOUN cana-5356 30 6 criteria criterion NOUN cana-5356 30 7 , , PUNCT cana-5356 30 8 we we PRON cana-5356 30 9 refer refer VERB cana-5356 30 10 also also ADV cana-5356 30 11 papers paper NOUN cana-5356 30 12 [ [ X cana-5356 30 13 11 11 NUM cana-5356 30 14 ] ] PUNCT cana-5356 30 15 and and CCONJ cana-5356 30 16 [ [ X cana-5356 30 17 12 12 NUM cana-5356 30 18 ] ] PUNCT cana-5356 30 19 . . PUNCT cana-5356 31 1 in in ADP cana-5356 31 2 ( ( PUNCT cana-5356 31 3 2009 2009 NUM cana-5356 31 4 ) ) PUNCT cana-5356 31 5 , , PUNCT cana-5356 31 6 lee lee PROPN cana-5356 31 7 et et PROPN cana-5356 31 8 al al PROPN cana-5356 31 9 . . PROPN cana-5356 31 10 established establish VERB cana-5356 31 11 coefficient coefficient NOUN cana-5356 31 12 bounds bound NOUN cana-5356 31 13 in in ADP cana-5356 31 14 paper paper NOUN cana-5356 32 1 [ [ X cana-5356 32 2 9 9 NUM cana-5356 32 3 ] ] PUNCT cana-5356 32 4 for for ADP cana-5356 32 5 such such ADJ cana-5356 32 6 functions function NOUN cana-5356 32 7 , , PUNCT cana-5356 32 8 providing provide VERB cana-5356 32 9 key key ADJ cana-5356 32 10 insights insight NOUN cana-5356 32 11 into into ADP cana-5356 32 12 their their PRON cana-5356 32 13 structure structure NOUN cana-5356 32 14 and and CCONJ cana-5356 32 15 meromorphic meromorphic ADJ cana-5356 32 16 properties property NOUN cana-5356 32 17 . . PUNCT cana-5356 33 1 definition definition NOUN cana-5356 33 2 : : PUNCT cana-5356 33 3 let let VERB cana-5356 33 4 the the DET cana-5356 33 5 class class NOUN cana-5356 33 6 π’œp π’œp PROPN cana-5356 33 7 βˆ— βˆ— NOUN cana-5356 33 8 consists consist VERB cana-5356 33 9 of of ADP cana-5356 33 10 all all DET cana-5356 33 11 analytic analytic ADJ cana-5356 33 12 functions function NOUN cana-5356 33 13 in in ADP cana-5356 33 14 the the DET cana-5356 33 15 punctured punctured ADJ cana-5356 33 16 unit unit NOUN cana-5356 33 17 disk disk NOUN cana-5356 33 18 β–³ β–³ PROPN cana-5356 33 19 βˆ— βˆ— NOUN cana-5356 33 20 , , PUNCT cana-5356 33 21 of of ADP cana-5356 33 22 the the DET cana-5356 33 23 form form NOUN cana-5356 33 24 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 33 25 ) ) PUNCT cana-5356 33 26 = = PRON cana-5356 33 27 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 33 28 + + NUM cana-5356 33 29 βˆ‘ βˆ‘ PROPN cana-5356 33 30 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 33 31 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 33 32 , , PUNCT cana-5356 33 33 𝑧 𝑧 PROPN cana-5356 33 34 ∈ ∈ PROPN cana-5356 33 35 β–³ β–³ NOUN cana-5356 33 36 βˆ— βˆ— NOUN cana-5356 33 37 and and CCONJ cana-5356 33 38 𝑝 𝑝 NOUN cana-5356 33 39 ∈ ∈ PROPN cana-5356 33 40 β„• β„• PROPN cana-5356 33 41 . . PUNCT cana-5356 34 1 for for ADP cana-5356 34 2 the the DET cana-5356 34 3 function function NOUN cana-5356 34 4 𝑓 𝑓 DET cana-5356 34 5 ∈ ∈ NOUN cana-5356 34 6 π’œp π’œp PROPN cana-5356 34 7 βˆ— βˆ— NOUN cana-5356 34 8 , , PUNCT cana-5356 34 9 here here ADV cana-5356 34 10 we we PRON cana-5356 34 11 introduced introduce VERB cana-5356 34 12 a a DET cana-5356 34 13 new new ADJ cana-5356 34 14 subclass subclass NOUN cana-5356 34 15 for for ADP cana-5356 34 16 0 0 NUM cana-5356 34 17 ≀ ≀ NUM cana-5356 34 18 𝛼 𝛼 X cana-5356 34 19 < < X cana-5356 34 20 1 1 NUM cana-5356 34 21 , , PUNCT cana-5356 34 22 0 0 NUM cana-5356 34 23 ≀ ≀ NOUN cana-5356 34 24 𝛽 𝛽 NOUN cana-5356 34 25 < < X cana-5356 34 26 1 1 NUM cana-5356 34 27 , , PUNCT cana-5356 34 28 0 0 NUM cana-5356 34 29 ≀ ≀ NOUN cana-5356 34 30 𝛾 𝛾 ADP cana-5356 34 31 < < X cana-5356 34 32 1 1 NUM cana-5356 34 33 , , PUNCT cana-5356 34 34 π‘˜ π‘˜ DET cana-5356 34 35 β‰₯ β‰₯ NOUN cana-5356 34 36 0 0 NUM cana-5356 34 37 , , PUNCT cana-5356 34 38 0 0 PUNCT cana-5356 34 39 < < X cana-5356 34 40 πœ† πœ† X cana-5356 34 41 ≀ ≀ NUM cana-5356 34 42 1 1 NUM cana-5356 34 43 , , PUNCT cana-5356 34 44 𝑝 𝑝 PROPN cana-5356 34 45 ∈ ∈ PROPN cana-5356 34 46 β„• β„• PROPN cana-5356 34 47 , , PUNCT cana-5356 34 48 𝑛 𝑛 DET cana-5356 34 49 ∈ ∈ NOUN cana-5356 34 50 β„•0 β„•0 NOUN cana-5356 34 51 = = SYM cana-5356 34 52 { { PUNCT cana-5356 34 53 0,1,2 0,1,2 NUM cana-5356 34 54 , , PUNCT cana-5356 34 55 … … PUNCT cana-5356 34 56 } } PUNCT cana-5356 34 57 , , PUNCT cana-5356 34 58 by by ADP cana-5356 34 59 using use VERB cana-5356 34 60 the the DET cana-5356 34 61 operator operator NOUN cana-5356 34 62 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 34 63 𝑛 𝑛 PRON cana-5356 34 64 by by ADP cana-5356 34 65 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 34 66 0 0 NUM cana-5356 34 67 = = SYM cana-5356 34 68 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 34 69 ) ) PUNCT cana-5356 34 70 which which PRON cana-5356 34 71 is be AUX cana-5356 34 72 introduced introduce VERB cana-5356 34 73 in in ADP cana-5356 34 74 paper paper NOUN cana-5356 34 75 [ [ X cana-5356 34 76 1 1 X cana-5356 34 77 ] ] PUNCT cana-5356 34 78 let let VERB cana-5356 34 79 β„³(𝛼 β„³(𝛼 PRON cana-5356 34 80 , , PUNCT cana-5356 34 81 𝛽 𝛽 NOUN cana-5356 34 82 , , PUNCT cana-5356 34 83 𝛾 𝛾 PROPN cana-5356 34 84 , , PUNCT cana-5356 34 85 π‘˜ π‘˜ NOUN cana-5356 34 86 , , PUNCT cana-5356 34 87 πœ† πœ† NOUN cana-5356 34 88 , , PUNCT cana-5356 34 89 𝑛 𝑛 PROPN cana-5356 34 90 , , PUNCT cana-5356 34 91 𝑝 𝑝 NOUN cana-5356 34 92 ) ) PUNCT cana-5356 35 1 𝑅𝑒 𝑅𝑒 PROPN cana-5356 35 2 ( ( PUNCT cana-5356 35 3 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 35 4 𝑛 𝑛 DET cana-5356 35 5 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 35 6 βˆ’ βˆ’ NOUN cana-5356 35 7 ( ( PUNCT cana-5356 35 8 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 35 9 𝑛 𝑛 ADP cana-5356 35 10 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 35 11 ) ) PUNCT cana-5356 35 12 ) ) PUNCT cana-5356 36 1 𝛼𝑧(𝐷𝑝,πœ† 𝛼𝑧(𝐷𝑝,πœ† NOUN cana-5356 36 2 𝑛 𝑛 PRON cana-5356 36 3 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 36 4 + + CCONJ cana-5356 36 5 ( ( PUNCT cana-5356 36 6 1 1 NUM cana-5356 36 7 βˆ’ βˆ’ PROPN cana-5356 36 8 π‘Ÿ)(𝐷𝑝,πœ† π‘Ÿ)(𝐷𝑝,πœ† NOUN cana-5356 36 9 𝑛 𝑛 PRON cana-5356 36 10 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 36 11 ) ) PUNCT cana-5356 36 12 ) ) PUNCT cana-5356 36 13 ) ) PUNCT cana-5356 37 1 > > X cana-5356 38 1 π‘˜ π‘˜ PRON cana-5356 38 2 | | ADV cana-5356 38 3 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 38 4 𝑛 𝑛 DET cana-5356 38 5 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 38 6 βˆ’ βˆ’ NOUN cana-5356 38 7 ( ( PUNCT cana-5356 38 8 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 38 9 𝑛 𝑛 ADP cana-5356 38 10 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 38 11 ) ) PUNCT cana-5356 38 12 ) ) PUNCT cana-5356 39 1 𝛼𝑧(𝐷𝑝,πœ† 𝛼𝑧(𝐷𝑝,πœ† NOUN cana-5356 39 2 𝑛 𝑛 PRON cana-5356 39 3 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 39 4 + + CCONJ cana-5356 39 5 ( ( PUNCT cana-5356 39 6 1 1 NUM cana-5356 39 7 βˆ’ βˆ’ PROPN cana-5356 39 8 π‘Ÿ)(𝐷𝑝,πœ† π‘Ÿ)(𝐷𝑝,πœ† NOUN cana-5356 39 9 𝑛 𝑛 PRON cana-5356 39 10 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 39 11 ) ) PUNCT cana-5356 39 12 ) ) PUNCT cana-5356 40 1 βˆ’ βˆ’ ADP cana-5356 40 2 1| 1| NUM cana-5356 41 1 + + CCONJ cana-5356 41 2 𝛽 𝛽 NOUN cana-5356 41 3 ( ( PUNCT cana-5356 41 4 1 1 NUM cana-5356 41 5 ) ) PUNCT cana-5356 42 1 where where SCONJ cana-5356 42 2 , , PUNCT cana-5356 42 3 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 42 4 𝑛 𝑛 PRON cana-5356 42 5 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 42 6 ) ) PUNCT cana-5356 42 7 = = PRON cana-5356 42 8 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 42 9 + + CCONJ cana-5356 42 10 βˆ‘ βˆ‘ PUNCT cana-5356 42 11 { { PUNCT cana-5356 42 12 ( ( PUNCT cana-5356 42 13 1 1 NUM cana-5356 42 14 βˆ’ βˆ’ NOUN cana-5356 42 15 πœ† πœ† X cana-5356 42 16 ) ) PUNCT cana-5356 42 17 ( ( PUNCT cana-5356 42 18 1 1 NUM cana-5356 42 19 + + CCONJ cana-5356 42 20 𝑗 𝑗 PROPN cana-5356 42 21 𝑝 𝑝 NOUN cana-5356 42 22 ) ) PUNCT cana-5356 42 23 π‘Žπ‘—π‘§π‘— π‘Žπ‘—π‘§π‘— NOUN cana-5356 42 24 } } PUNCT cana-5356 42 25 𝑛 𝑛 PROPN cana-5356 42 26 , , PUNCT cana-5356 42 27 𝑧 𝑧 PROPN cana-5356 42 28 ∈ ∈ PROPN cana-5356 42 29 β–³ β–³ NOUN cana-5356 42 30 βˆ— βˆ— NOUN cana-5356 42 31 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN cana-5356 42 32 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 42 33 > > X cana-5356 42 34 0 0 PUNCT cana-5356 43 1 ∞ ∞ NUM cana-5356 43 2 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 43 3 here here ADV cana-5356 43 4 𝐷𝑝,πœ† 𝐷𝑝,πœ† PRON cana-5356 43 5 𝑛 𝑛 PRON cana-5356 43 6 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 43 7 ) ) PUNCT cana-5356 43 8 be be AUX cana-5356 43 9 the the DET cana-5356 43 10 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 43 11 operator operator NOUN cana-5356 43 12 . . PUNCT cana-5356 44 1 2 2 X cana-5356 44 2 . . X cana-5356 44 3 objectives objective VERB cana-5356 44 4 the the DET cana-5356 44 5 primary primary ADJ cana-5356 44 6 objective objective NOUN cana-5356 44 7 of of ADP cana-5356 44 8 this this DET cana-5356 44 9 study study NOUN cana-5356 44 10 is be AUX cana-5356 44 11 to to PART cana-5356 44 12 investigate investigate VERB cana-5356 44 13 and and CCONJ cana-5356 44 14 analyze analyze VERB cana-5356 44 15 various various ADJ cana-5356 44 16 newly newly ADV cana-5356 44 17 defined define VERB cana-5356 44 18 subclasses subclass NOUN cana-5356 44 19 of of ADP cana-5356 44 20 analytic analytic ADJ cana-5356 44 21 functions function NOUN cana-5356 44 22 within within ADP cana-5356 44 23 the the DET cana-5356 44 24 unit unit NOUN cana-5356 44 25 disc disc NOUN cana-5356 44 26 . . PUNCT cana-5356 45 1 in in ADP cana-5356 45 2 this this DET cana-5356 45 3 study study NOUN cana-5356 45 4 we we PRON cana-5356 45 5 introduced introduce VERB cana-5356 45 6 a a DET cana-5356 45 7 new new ADJ cana-5356 45 8 subclass subclass NOUN cana-5356 45 9 of of ADP cana-5356 45 10 meromorphic meromorphic ADJ cana-5356 45 11 functions function NOUN cana-5356 45 12 β„³(𝛼 β„³(𝛼 PRON cana-5356 45 13 , , PUNCT cana-5356 45 14 𝛽 𝛽 NOUN cana-5356 45 15 , , PUNCT cana-5356 45 16 𝛾 𝛾 PROPN cana-5356 45 17 , , PUNCT cana-5356 45 18 π‘˜ π‘˜ NOUN cana-5356 45 19 , , PUNCT cana-5356 45 20 πœ† πœ† NOUN cana-5356 45 21 , , PUNCT cana-5356 45 22 𝑛 𝑛 PROPN cana-5356 45 23 , , PUNCT cana-5356 45 24 𝑝 𝑝 NOUN cana-5356 45 25 ) ) PUNCT cana-5356 45 26 , , PUNCT cana-5356 45 27 which which PRON cana-5356 45 28 lies lie VERB cana-5356 45 29 within within ADP cana-5356 45 30 the the DET cana-5356 45 31 class class NOUN cana-5356 45 32 π’œπ‘ π’œπ‘ PROPN cana-5356 45 33 βˆ— βˆ— NOUN cana-5356 45 34 and and CCONJ cana-5356 45 35 consists consist VERB cana-5356 45 36 of of ADP cana-5356 45 37 functions function NOUN cana-5356 45 38 of of ADP cana-5356 45 39 the the DET cana-5356 45 40 form form NOUN cana-5356 45 41 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 45 42 ) ) PUNCT cana-5356 45 43 = = PRON cana-5356 45 44 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 45 45 + + NUM cana-5356 45 46 βˆ‘ βˆ‘ PROPN cana-5356 45 47 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 45 48 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 45 49 , , PUNCT cana-5356 45 50 where where SCONJ cana-5356 45 51 𝑧 𝑧 PRON cana-5356 45 52 ∈ ∈ PROPN cana-5356 45 53 β–³ β–³ NOUN cana-5356 45 54 βˆ— βˆ— NOUN cana-5356 45 55 and and CCONJ cana-5356 45 56 𝑝 𝑝 NOUN cana-5356 45 57 ∈ ∈ PROPN cana-5356 45 58 β„•. β„•. PROPN cana-5356 45 59 this this DET cana-5356 45 60 study study NOUN cana-5356 45 61 specifically specifically ADV cana-5356 45 62 focuses focus VERB cana-5356 45 63 on on ADP cana-5356 45 64 using use VERB cana-5356 45 65 the the DET cana-5356 45 66 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 45 67 operator operator NOUN cana-5356 45 68 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 45 69 𝑛 𝑛 DET cana-5356 45 70 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 45 71 ) ) PUNCT cana-5356 45 72 to to PART cana-5356 45 73 established establish VERB cana-5356 45 74 the the DET cana-5356 45 75 foundational foundational ADJ cana-5356 45 76 properties property NOUN cana-5356 45 77 of of ADP cana-5356 45 78 this this DET cana-5356 45 79 class class NOUN cana-5356 45 80 , , PUNCT cana-5356 45 81 proving prove VERB cana-5356 45 82 its its PRON cana-5356 45 83 univalence univalence NOUN cana-5356 45 84 , , PUNCT cana-5356 45 85 and and CCONJ cana-5356 45 86 analyzing analyze VERB cana-5356 45 87 the the DET cana-5356 45 88 influence influence NOUN cana-5356 45 89 of of ADP cana-5356 45 90 the the DET cana-5356 45 91 operator operator NOUN cana-5356 45 92 on on ADP cana-5356 45 93 the the DET cana-5356 45 94 function function NOUN cana-5356 45 95 ’s ’s PART cana-5356 45 96 geometric geometric ADJ cana-5356 45 97 behavior behavior NOUN cana-5356 45 98 . . PUNCT cana-5356 46 1 the the DET cana-5356 46 2 mathematical mathematical ADJ cana-5356 46 3 formulation formulation NOUN cana-5356 46 4 of of ADP cana-5356 46 5 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 46 6 𝑛 𝑛 PRON cana-5356 46 7 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 46 8 ) ) PUNCT cana-5356 46 9 , , PUNCT cana-5356 46 10 where where SCONJ cana-5356 46 11 the the DET cana-5356 46 12 operator operator NOUN cana-5356 46 13 influences influence VERB cana-5356 46 14 on on ADP cana-5356 46 15 the the DET cana-5356 46 16 series series NOUN cana-5356 46 17 expansion expansion NOUN cana-5356 46 18 of of ADP cana-5356 46 19 the the DET cana-5356 46 20 meromorphic meromorphic ADJ cana-5356 46 21 functions function NOUN cana-5356 46 22 , , PUNCT cana-5356 46 23 is be AUX cana-5356 46 24 also also ADV cana-5356 46 25 elaborated elaborate VERB cana-5356 46 26 in in ADP cana-5356 46 27 the the DET cana-5356 46 28 study study NOUN cana-5356 46 29 . . PUNCT cana-5356 47 1 3 3 X cana-5356 47 2 . . X cana-5356 47 3 methods method NOUN cana-5356 47 4 in in ADP cana-5356 47 5 this this DET cana-5356 47 6 paper paper NOUN cana-5356 47 7 , , PUNCT cana-5356 47 8 we we PRON cana-5356 47 9 introduced introduce VERB cana-5356 47 10 the the DET cana-5356 47 11 subclass subclass NOUN cana-5356 47 12 β„³(𝛼 β„³(𝛼 PRON cana-5356 47 13 , , PUNCT cana-5356 47 14 𝛽 𝛽 NOUN cana-5356 47 15 , , PUNCT cana-5356 47 16 𝛾 𝛾 PROPN cana-5356 47 17 , , PUNCT cana-5356 47 18 π‘˜ π‘˜ NOUN cana-5356 47 19 , , PUNCT cana-5356 47 20 πœ† πœ† NOUN cana-5356 47 21 , , PUNCT cana-5356 47 22 𝑛 𝑛 PROPN cana-5356 47 23 , , PUNCT cana-5356 47 24 𝑝 𝑝 NOUN cana-5356 47 25 ) ) PUNCT cana-5356 47 26 of of ADP cana-5356 47 27 meromorphic meromorphic ADJ cana-5356 47 28 functions function NOUN cana-5356 47 29 defined define VERB cana-5356 47 30 in in ADP cana-5356 47 31 the the DET cana-5356 47 32 punctured punctured ADJ cana-5356 47 33 unit unit NOUN cana-5356 47 34 disk disk NOUN cana-5356 47 35 β–³ β–³ PROPN cana-5356 47 36 βˆ— βˆ— NOUN cana-5356 47 37 , , PUNCT cana-5356 47 38 associated associate VERB cana-5356 47 39 with with ADP cana-5356 47 40 the the DET cana-5356 47 41 generalized generalize VERB cana-5356 47 42 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 47 43 operator operator NOUN cana-5356 47 44 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 47 45 𝑛 𝑛 PRON cana-5356 47 46 which which PRON cana-5356 47 47 is be AUX cana-5356 47 48 defined define VERB cana-5356 47 49 as as ADP cana-5356 47 50 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 47 51 𝑛 𝑛 PRON cana-5356 47 52 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 47 53 ) ) PUNCT cana-5356 47 54 = = PRON cana-5356 47 55 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 47 56 + + CCONJ cana-5356 47 57 βˆ‘ βˆ‘ PUNCT cana-5356 47 58 { { PUNCT cana-5356 47 59 ( ( PUNCT cana-5356 47 60 1 1 NUM cana-5356 47 61 βˆ’ βˆ’ NOUN cana-5356 47 62 πœ† πœ† X cana-5356 47 63 ) ) PUNCT cana-5356 47 64 ( ( PUNCT cana-5356 47 65 1 1 NUM cana-5356 47 66 + + CCONJ cana-5356 47 67 𝑗 𝑗 PROPN cana-5356 47 68 𝑝 𝑝 NOUN cana-5356 47 69 ) ) PUNCT cana-5356 47 70 π‘Žπ‘—π‘§π‘— π‘Žπ‘—π‘§π‘— NOUN cana-5356 47 71 } } PUNCT cana-5356 47 72 𝑛 𝑛 PRON cana-5356 47 73 ∞ ∞ NUM cana-5356 47 74 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 47 75 where where SCONJ cana-5356 47 76 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 47 77 > > X cana-5356 47 78 0 0 X cana-5356 47 79 . . PUNCT cana-5356 48 1 coefficient coefficient NOUN cana-5356 48 2 estimates estimate NOUN cana-5356 48 3 are be AUX cana-5356 48 4 obtained obtain VERB cana-5356 48 5 by by ADP cana-5356 48 6 applying apply VERB cana-5356 48 7 a a DET cana-5356 48 8 differential differential ADJ cana-5356 48 9 subordination subordination NOUN cana-5356 48 10 condition condition NOUN cana-5356 48 11 for for ADP cana-5356 48 12 function function NOUN cana-5356 48 13 belongs belong VERB cana-5356 48 14 to to ADP cana-5356 48 15 the the DET cana-5356 48 16 subclass subclass NOUN cana-5356 48 17 β„³(𝛼 β„³(𝛼 PRON cana-5356 48 18 , , PUNCT cana-5356 48 19 𝛽 𝛽 NOUN cana-5356 48 20 , , PUNCT cana-5356 48 21 𝛾 𝛾 PROPN cana-5356 48 22 , , PUNCT cana-5356 48 23 π‘˜ π‘˜ NOUN cana-5356 48 24 , , PUNCT cana-5356 48 25 πœ† πœ† NOUN cana-5356 48 26 , , PUNCT cana-5356 48 27 𝑛 𝑛 PROPN cana-5356 48 28 , , PUNCT cana-5356 48 29 𝑝 𝑝 NOUN cana-5356 48 30 ) ) PUNCT cana-5356 48 31 . . PUNCT cana-5356 49 1 convexity convexity NOUN cana-5356 49 2 of of ADP cana-5356 49 3 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 49 4 ) ) PUNCT cana-5356 49 5 is be AUX cana-5356 49 6 established establish VERB cana-5356 49 7 under under ADP cana-5356 49 8 the the DET cana-5356 49 9 condition condition NOUN cana-5356 50 1 𝑅𝑒 𝑅𝑒 PROPN cana-5356 50 2 ( ( PUNCT cana-5356 50 3 1 1 NUM cana-5356 50 4 + + CCONJ cana-5356 50 5 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† AUX cana-5356 50 6 𝑛 𝑛 ADP cana-5356 50 7 𝑓(𝑧))β€²β€² 𝑓(𝑧))β€²β€² PRON cana-5356 50 8 ( ( PUNCT cana-5356 50 9 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 50 10 𝑛 𝑛 DET cana-5356 50 11 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 50 12 ) ) PUNCT cana-5356 50 13 > > X cana-5356 50 14 0 0 NUM cana-5356 50 15 , , PUNCT cana-5356 50 16 communications communication NOUN cana-5356 50 17 on on ADP cana-5356 50 18 applied apply VERB cana-5356 50 19 nonlinear nonlinear ADJ cana-5356 50 20 analysis analysis NOUN cana-5356 50 21 issn issn NOUN cana-5356 50 22 : : PUNCT cana-5356 50 23 1074 1074 NUM cana-5356 50 24 - - PUNCT cana-5356 50 25 133x 133x NUM cana-5356 50 26 vol vol VERB cana-5356 50 27 32 32 NUM cana-5356 50 28 no no NOUN cana-5356 50 29 . . PUNCT cana-5356 51 1 10s 10 NOUN cana-5356 51 2 ( ( PUNCT cana-5356 51 3 2025 2025 NUM cana-5356 51 4 ) ) PUNCT cana-5356 51 5 1874 1874 NUM cana-5356 52 1 https://internationalpubls.com https://internationalpubls.com X cana-5356 52 2 and and CCONJ cana-5356 52 3 univalence univalence NOUN cana-5356 52 4 criteria criterion NOUN cana-5356 52 5 is be AUX cana-5356 52 6 ensured ensure VERB cana-5356 52 7 by by ADP cana-5356 52 8 the the DET cana-5356 52 9 condition condition NOUN cana-5356 52 10 | | ADV cana-5356 52 11 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 52 12 𝑛 𝑛 ADP cana-5356 52 13 𝑓(𝑧))β€²β€² 𝑓(𝑧))β€²β€² PRON cana-5356 52 14 ( ( PUNCT cana-5356 52 15 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 52 16 𝑛 𝑛 PRON cana-5356 52 17 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 52 18 | | NOUN cana-5356 52 19 < < X cana-5356 52 20 1 1 NUM cana-5356 52 21 . . PUNCT cana-5356 53 1 these these DET cana-5356 53 2 methods method NOUN cana-5356 53 3 provide provide VERB cana-5356 53 4 conditions condition NOUN cana-5356 53 5 on on ADP cana-5356 53 6 the the DET cana-5356 53 7 coefficients coefficient NOUN cana-5356 53 8 and and CCONJ cana-5356 53 9 geometric geometric ADJ cana-5356 53 10 criteria criterion NOUN cana-5356 53 11 for for ADP cana-5356 53 12 functions function NOUN cana-5356 53 13 defined define VERB cana-5356 53 14 in in ADP cana-5356 53 15 the the DET cana-5356 53 16 above above ADJ cana-5356 53 17 subclass subclass NOUN cana-5356 53 18 . . PUNCT cana-5356 54 1 4 4 X cana-5356 54 2 . . X cana-5356 54 3 results result NOUN cana-5356 54 4 we we PRON cana-5356 54 5 begin begin VERB cana-5356 54 6 by by ADP cana-5356 54 7 proving prove VERB cana-5356 54 8 several several ADJ cana-5356 54 9 theorems theorem NOUN cana-5356 54 10 based base VERB cana-5356 54 11 on on ADP cana-5356 54 12 the the DET cana-5356 54 13 operator operator NOUN cana-5356 54 14 classification classification NOUN cana-5356 54 15 . . PUNCT cana-5356 55 1 theorem theorem VERB cana-5356 55 2 : : PUNCT cana-5356 55 3 1 1 NUM cana-5356 55 4 let let VERB cana-5356 55 5 0 0 NUM cana-5356 55 6 ≀ ≀ NOUN cana-5356 55 7 𝛼 𝛼 X cana-5356 55 8 < < X cana-5356 55 9 1 1 NUM cana-5356 55 10 , , PUNCT cana-5356 55 11 0 0 NUM cana-5356 55 12 ≀ ≀ NOUN cana-5356 56 1 𝛽 𝛽 NOUN cana-5356 56 2 < < X cana-5356 56 3 1 1 NUM cana-5356 56 4 , , PUNCT cana-5356 56 5 0 0 NUM cana-5356 56 6 ≀ ≀ NOUN cana-5356 56 7 𝛾 𝛾 ADP cana-5356 56 8 < < X cana-5356 56 9 1 1 NUM cana-5356 56 10 , , PUNCT cana-5356 56 11 π‘˜ π‘˜ DET cana-5356 56 12 β‰₯ β‰₯ NOUN cana-5356 56 13 0 0 NUM cana-5356 56 14 , , PUNCT cana-5356 56 15 0 0 PUNCT cana-5356 56 16 < < X cana-5356 56 17 πœ† πœ† X cana-5356 56 18 ≀ ≀ NUM cana-5356 56 19 1 1 NUM cana-5356 56 20 , , PUNCT cana-5356 56 21 𝑝 𝑝 PROPN cana-5356 56 22 ∈ ∈ PROPN cana-5356 56 23 β„• β„• PROPN cana-5356 56 24 , , PUNCT cana-5356 56 25 𝑛 𝑛 DET cana-5356 56 26 ∈ ∈ NOUN cana-5356 56 27 β„•0 β„•0 NOUN cana-5356 56 28 = = SYM cana-5356 56 29 { { PUNCT cana-5356 56 30 0,1,2 0,1,2 NUM cana-5356 56 31 , , PUNCT cana-5356 56 32 … … PUNCT cana-5356 56 33 } } PUNCT cana-5356 56 34 . . PUNCT cana-5356 57 1 a a DET cana-5356 57 2 function function NOUN cana-5356 57 3 𝑓 𝑓 PRON cana-5356 57 4 given give VERB cana-5356 57 5 by by ADP cana-5356 57 6 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 57 7 ) ) PUNCT cana-5356 57 8 = = PRON cana-5356 57 9 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 57 10 + + CCONJ cana-5356 57 11 βˆ‘ βˆ‘ PART cana-5356 57 12 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 57 13 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 57 14 be be AUX cana-5356 57 15 an an DET cana-5356 57 16 analytic analytic ADJ cana-5356 57 17 function function NOUN cana-5356 57 18 belonging belong VERB cana-5356 57 19 to to ADP cana-5356 57 20 the the DET cana-5356 57 21 class class NOUN cana-5356 57 22 β„³(𝛼 β„³(𝛼 ADP cana-5356 57 23 , , PUNCT cana-5356 57 24 𝛽 𝛽 NOUN cana-5356 57 25 , , PUNCT cana-5356 57 26 𝛾 𝛾 PROPN cana-5356 57 27 , , PUNCT cana-5356 57 28 π‘˜ π‘˜ NOUN cana-5356 57 29 , , PUNCT cana-5356 57 30 πœ† πœ† NOUN cana-5356 57 31 , , PUNCT cana-5356 57 32 𝑛 𝑛 PROPN cana-5356 57 33 , , PUNCT cana-5356 57 34 𝑝 𝑝 NOUN cana-5356 57 35 ) ) PUNCT cana-5356 57 36 , , PUNCT cana-5356 57 37 then then ADV cana-5356 57 38 𝑓 𝑓 DET cana-5356 57 39 satisfies satisfy VERB cana-5356 57 40 the the DET cana-5356 57 41 condition condition NOUN cana-5356 57 42 : : PUNCT cana-5356 58 1 𝑅𝑒 𝑅𝑒 PROPN cana-5356 58 2 ( ( PUNCT cana-5356 58 3 𝑧 𝑧 X cana-5356 58 4 ( ( PUNCT cana-5356 58 5 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 58 6 𝑛 𝑛 ADP cana-5356 58 7 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 58 8 ) ) PUNCT cana-5356 58 9 ) ) PUNCT cana-5356 58 10 β€² β€² NUM cana-5356 58 11 βˆ’ βˆ’ NOUN cana-5356 59 1 ( ( PUNCT cana-5356 59 2 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 59 3 𝑛 𝑛 ADP cana-5356 59 4 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 59 5 ) ) PUNCT cana-5356 59 6 ) ) PUNCT cana-5356 60 1 𝛼𝑧 𝛼𝑧 X cana-5356 60 2 ( ( PUNCT cana-5356 60 3 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 60 4 𝑛 𝑛 ADP cana-5356 60 5 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 60 6 ) ) PUNCT cana-5356 60 7 ) ) PUNCT cana-5356 61 1 β€² β€² PUNCT cana-5356 62 1 + + CCONJ cana-5356 62 2 ( ( PUNCT cana-5356 62 3 1 1 NUM cana-5356 62 4 βˆ’ βˆ’ NOUN cana-5356 62 5 π‘Ÿ π‘Ÿ NOUN cana-5356 62 6 ) ) PUNCT cana-5356 62 7 ( ( PUNCT cana-5356 62 8 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 62 9 𝑛 𝑛 ADP cana-5356 62 10 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 62 11 ) ) PUNCT cana-5356 62 12 ) ) PUNCT cana-5356 62 13 ) ) PUNCT cana-5356 63 1 > > X cana-5356 64 1 π‘˜ π‘˜ PRON cana-5356 64 2 | | ADV cana-5356 64 3 𝑧 𝑧 X cana-5356 64 4 ( ( PUNCT cana-5356 64 5 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 64 6 𝑛 𝑛 ADP cana-5356 64 7 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 64 8 ) ) PUNCT cana-5356 64 9 ) ) PUNCT cana-5356 65 1 β€² β€² NUM cana-5356 65 2 βˆ’ βˆ’ NOUN cana-5356 66 1 ( ( PUNCT cana-5356 66 2 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 66 3 𝑛 𝑛 ADP cana-5356 66 4 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 66 5 ) ) PUNCT cana-5356 66 6 ) ) PUNCT cana-5356 67 1 𝛼𝑧 𝛼𝑧 X cana-5356 67 2 ( ( PUNCT cana-5356 67 3 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 67 4 𝑛 𝑛 ADP cana-5356 67 5 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 67 6 ) ) PUNCT cana-5356 67 7 ) ) PUNCT cana-5356 68 1 β€² β€² PUNCT cana-5356 69 1 + + CCONJ cana-5356 69 2 ( ( PUNCT cana-5356 69 3 1 1 NUM cana-5356 69 4 βˆ’ βˆ’ NOUN cana-5356 69 5 π‘Ÿ π‘Ÿ NOUN cana-5356 69 6 ) ) PUNCT cana-5356 69 7 ( ( PUNCT cana-5356 69 8 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 69 9 𝑛 𝑛 ADP cana-5356 69 10 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 69 11 ) ) PUNCT cana-5356 69 12 ) ) PUNCT cana-5356 70 1 βˆ’ βˆ’ ADP cana-5356 70 2 1| 1| NUM cana-5356 71 1 + + CCONJ cana-5356 71 2 𝛽 𝛽 NOUN cana-5356 71 3 where where SCONJ cana-5356 71 4 , , PUNCT cana-5356 71 5 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 71 6 𝑛 𝑛 PRON cana-5356 71 7 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 71 8 ) ) PUNCT cana-5356 71 9 = = PRON cana-5356 71 10 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 71 11 + + CCONJ cana-5356 71 12 βˆ‘ βˆ‘ PUNCT cana-5356 71 13 { { PUNCT cana-5356 71 14 ( ( PUNCT cana-5356 71 15 1 1 NUM cana-5356 71 16 βˆ’ βˆ’ NOUN cana-5356 71 17 πœ† πœ† X cana-5356 71 18 ) ) PUNCT cana-5356 71 19 ( ( PUNCT cana-5356 71 20 1 1 NUM cana-5356 71 21 + + CCONJ cana-5356 71 22 𝑗 𝑗 PROPN cana-5356 71 23 𝑝 𝑝 NOUN cana-5356 71 24 ) ) PUNCT cana-5356 71 25 π‘Žπ‘—π‘§π‘— π‘Žπ‘—π‘§π‘— NOUN cana-5356 71 26 } } PUNCT cana-5356 71 27 𝑛 𝑛 PROPN cana-5356 71 28 , , PUNCT cana-5356 71 29 𝑧 𝑧 PROPN cana-5356 71 30 ∈ ∈ PROPN cana-5356 71 31 β–³ β–³ NOUN cana-5356 71 32 βˆ— βˆ— NOUN cana-5356 71 33 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN cana-5356 72 1 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 72 2 > > X cana-5356 72 3 0 0 PUNCT cana-5356 73 1 ∞ ∞ NUM cana-5356 73 2 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 73 3 then then ADV cana-5356 73 4 the the DET cana-5356 73 5 coefficients coefficient NOUN cana-5356 73 6 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 73 7 of of ADP cana-5356 73 8 the the DET cana-5356 73 9 taylor taylor PROPN cana-5356 73 10 series series PROPN cana-5356 73 11 expansion expansion NOUN cana-5356 73 12 satisfy satisfy VERB cana-5356 73 13 the the DET cana-5356 73 14 inequality inequality NOUN cana-5356 73 15 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 73 16 ≀ ≀ NUM cana-5356 73 17 ( ( PUNCT cana-5356 73 18 1 1 NUM cana-5356 73 19 βˆ’ βˆ’ NOUN cana-5356 73 20 𝛽)(𝛼𝑗 𝛽)(𝛼𝑗 NUM cana-5356 73 21 + + CCONJ cana-5356 73 22 1 1 NUM cana-5356 73 23 βˆ’ βˆ’ NUM cana-5356 73 24 π‘Ÿ π‘Ÿ NOUN cana-5356 73 25 ) ) PUNCT cana-5356 73 26 ( ( PUNCT cana-5356 73 27 π‘˜ π‘˜ PROPN cana-5356 73 28 + + NOUN cana-5356 73 29 1)|2 1)|2 NUM cana-5356 73 30 βˆ’ βˆ’ NOUN cana-5356 73 31 𝛾 𝛾 ADP cana-5356 73 32 βˆ’ βˆ’ PROPN cana-5356 74 1 𝑛(1 𝑛(1 PROPN cana-5356 74 2 βˆ’ βˆ’ PROPN cana-5356 74 3 𝛼)|(1 𝛼)|(1 PROPN cana-5356 74 4 + + CCONJ cana-5356 74 5 𝑗 𝑗 NOUN cana-5356 74 6 ) ) PUNCT cana-5356 74 7 , , PUNCT cana-5356 74 8 π‘“π‘œπ‘Ÿ π‘“π‘œπ‘Ÿ ADV cana-5356 74 9 π‘Žπ‘™π‘™ π‘Žπ‘™π‘™ VERB cana-5356 74 10 𝑗 𝑗 PRON cana-5356 74 11 β‰₯ β‰₯ NUM cana-5356 74 12 2 2 NUM cana-5356 74 13 this this DET cana-5356 74 14 result result NOUN cana-5356 74 15 establishes establish VERB cana-5356 74 16 an an DET cana-5356 74 17 upper upper ADJ cana-5356 74 18 bound bind VERB cana-5356 74 19 on on ADP cana-5356 74 20 the the DET cana-5356 74 21 growth growth NOUN cana-5356 74 22 of of ADP cana-5356 74 23 the the DET cana-5356 74 24 coefficients coefficient NOUN cana-5356 74 25 of of ADP cana-5356 74 26 functions function NOUN cana-5356 74 27 in in ADP cana-5356 74 28 this this DET cana-5356 74 29 subclass subclass NOUN cana-5356 74 30 β„³(𝛼 β„³(𝛼 PRON cana-5356 74 31 , , PUNCT cana-5356 74 32 𝛽 𝛽 NOUN cana-5356 74 33 , , PUNCT cana-5356 74 34 𝛾 𝛾 PROPN cana-5356 74 35 , , PUNCT cana-5356 74 36 π‘˜ π‘˜ NOUN cana-5356 74 37 , , PUNCT cana-5356 74 38 πœ† πœ† NOUN cana-5356 74 39 , , PUNCT cana-5356 74 40 𝑛 𝑛 PROPN cana-5356 74 41 , , PUNCT cana-5356 74 42 𝑝 𝑝 NOUN cana-5356 74 43 ) ) PUNCT cana-5356 74 44 . . PUNCT cana-5356 75 1 proof proof NOUN cana-5356 75 2 : : PUNCT cana-5356 75 3 given give VERB cana-5356 75 4 that that PRON cana-5356 75 5 β„³(𝛼 β„³(𝛼 ADP cana-5356 75 6 , , PUNCT cana-5356 75 7 𝛽 𝛽 NOUN cana-5356 75 8 , , PUNCT cana-5356 75 9 𝛾 𝛾 PROPN cana-5356 75 10 , , PUNCT cana-5356 75 11 π‘˜ π‘˜ NOUN cana-5356 75 12 , , PUNCT cana-5356 75 13 πœ† πœ† NOUN cana-5356 75 14 , , PUNCT cana-5356 75 15 𝑛 𝑛 PROPN cana-5356 75 16 , , PUNCT cana-5356 75 17 𝑝 𝑝 NOUN cana-5356 75 18 ) ) PUNCT cana-5356 75 19 if if SCONJ cana-5356 76 1 and and CCONJ cana-5356 76 2 only only ADV cana-5356 76 3 if if SCONJ cana-5356 76 4 the the DET cana-5356 76 5 condition condition NOUN cana-5356 76 6 ( ( PUNCT cana-5356 76 7 1 1 X cana-5356 76 8 ) ) PUNCT cana-5356 76 9 is be AUX cana-5356 76 10 satisfied satisfied ADJ cana-5356 76 11 , , PUNCT cana-5356 76 12 let let VERB cana-5356 76 13 πœ” πœ” VERB cana-5356 76 14 = = VERB cana-5356 76 15 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 76 16 𝑛 𝑛 DET cana-5356 76 17 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 76 18 βˆ’ βˆ’ NOUN cana-5356 76 19 ( ( PUNCT cana-5356 76 20 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 76 21 𝑛 𝑛 ADP cana-5356 76 22 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 76 23 ) ) PUNCT cana-5356 76 24 ) ) PUNCT cana-5356 77 1 𝛼𝑧(𝐷𝑝,πœ† 𝛼𝑧(𝐷𝑝,πœ† NOUN cana-5356 77 2 𝑛 𝑛 PRON cana-5356 77 3 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 77 4 + + CCONJ cana-5356 77 5 ( ( PUNCT cana-5356 77 6 1 1 NUM cana-5356 77 7 βˆ’ βˆ’ PROPN cana-5356 77 8 π‘Ÿ)(𝐷𝑝,πœ† π‘Ÿ)(𝐷𝑝,πœ† NOUN cana-5356 77 9 𝑛 𝑛 PRON cana-5356 77 10 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 77 11 ) ) PUNCT cana-5356 77 12 ) ) PUNCT cana-5356 77 13 subject subject ADJ cana-5356 77 14 to to ADP cana-5356 77 15 the the DET cana-5356 77 16 condition condition NOUN cana-5356 77 17 that that SCONJ cana-5356 77 18 , , PUNCT cana-5356 77 19 𝑅𝑒(πœ” 𝑅𝑒(πœ” PROPN cana-5356 77 20 ) ) PUNCT cana-5356 77 21 β‰₯ β‰₯ NOUN cana-5356 77 22 π‘˜|πœ” π‘˜|πœ” PUNCT cana-5356 78 1 βˆ’ βˆ’ VERB cana-5356 78 2 1| 1| NUM cana-5356 79 1 + + NOUN cana-5356 79 2 𝛽 𝛽 NOUN cana-5356 79 3 if if SCONJ cana-5356 79 4 and and CCONJ cana-5356 79 5 only only ADV cana-5356 79 6 if if SCONJ cana-5356 79 7 ( ( PUNCT cana-5356 79 8 π‘˜ π‘˜ X cana-5356 79 9 + + NOUN cana-5356 79 10 1)|πœ” 1)|πœ” NUM cana-5356 79 11 βˆ’ βˆ’ NUM cana-5356 79 12 1| 1| NUM cana-5356 79 13 ≀ ≀ NUM cana-5356 79 14 1 1 NUM cana-5356 79 15 βˆ’ βˆ’ NOUN cana-5356 79 16 𝛽 𝛽 NOUN cana-5356 79 17 now now ADV cana-5356 79 18 ( ( PUNCT cana-5356 79 19 π‘˜ π‘˜ PROPN cana-5356 79 20 + + NOUN cana-5356 79 21 1)|πœ” 1)|πœ” NUM cana-5356 79 22 βˆ’ βˆ’ NUM cana-5356 79 23 1| 1| NUM cana-5356 79 24 = = SYM cana-5356 79 25 ( ( PUNCT cana-5356 79 26 π‘˜ π‘˜ PROPN cana-5356 79 27 + + NOUN cana-5356 79 28 1 1 NUM cana-5356 79 29 ) ) PUNCT cana-5356 79 30 | | ADV cana-5356 79 31 ( ( PUNCT cana-5356 79 32 1 1 NUM cana-5356 79 33 βˆ’ βˆ’ NOUN cana-5356 80 1 𝑝)π‘§βˆ’π‘ 𝑝)π‘§βˆ’π‘ NOUN cana-5356 81 1 + + CCONJ cana-5356 81 2 βˆ‘ βˆ‘ PUNCT cana-5356 81 3 { { PUNCT cana-5356 81 4 ( ( PUNCT cana-5356 81 5 1 1 NUM cana-5356 81 6 βˆ’ βˆ’ NOUN cana-5356 81 7 πœ† πœ† X cana-5356 81 8 ) ) PUNCT cana-5356 81 9 ( ( PUNCT cana-5356 81 10 1 1 NUM cana-5356 81 11 + + NUM cana-5356 81 12 𝑗 𝑗 PROPN cana-5356 81 13 𝑝 𝑝 NOUN cana-5356 81 14 ) ) PUNCT cana-5356 81 15 } } PUNCT cana-5356 81 16 𝑛 𝑛 X cana-5356 81 17 ( ( PUNCT cana-5356 81 18 1 1 NUM cana-5356 81 19 + + CCONJ cana-5356 81 20 𝑗)π‘Žπ‘—π‘§π‘—βˆž 𝑗)π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 81 21 𝑗=2 𝑗=2 PROPN cana-5356 81 22 ( ( PUNCT cana-5356 81 23 1 1 NUM cana-5356 81 24 βˆ’ βˆ’ NOUN cana-5356 81 25 π‘Ÿ π‘Ÿ NOUN cana-5356 81 26 βˆ’ βˆ’ X cana-5356 81 27 𝛼𝑝)π‘§βˆ’π‘ 𝛼𝑝)π‘§βˆ’π‘ PROPN cana-5356 82 1 + + CCONJ cana-5356 82 2 βˆ‘ βˆ‘ PUNCT cana-5356 82 3 ( ( PUNCT cana-5356 82 4 𝛼𝑗 𝛼𝑗 NOUN cana-5356 82 5 + + CCONJ cana-5356 82 6 1 1 NUM cana-5356 82 7 βˆ’ βˆ’ NUM cana-5356 82 8 π‘Ÿ π‘Ÿ NOUN cana-5356 82 9 ) ) PUNCT cana-5356 82 10 { { PUNCT cana-5356 82 11 ( ( PUNCT cana-5356 82 12 1 1 NUM cana-5356 82 13 βˆ’ βˆ’ NOUN cana-5356 82 14 πœ† πœ† X cana-5356 82 15 ) ) PUNCT cana-5356 82 16 ( ( PUNCT cana-5356 82 17 1 1 NUM cana-5356 82 18 + + NUM cana-5356 82 19 𝑗 𝑗 PROPN cana-5356 82 20 𝑝 𝑝 NOUN cana-5356 82 21 ) ) PUNCT cana-5356 82 22 } } PUNCT cana-5356 82 23 𝑛 𝑛 DET cana-5356 82 24 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 82 25 𝑗=2 𝑗=2 PROPN cana-5356 83 1 βˆ’ βˆ’ PROPN cana-5356 83 2 1| 1| NUM cana-5356 83 3 ≀ ≀ NUM cana-5356 83 4 1 1 NUM cana-5356 83 5 βˆ’ βˆ’ NOUN cana-5356 84 1 𝛽 𝛽 NOUN cana-5356 84 2 is be AUX cana-5356 84 3 equivalent equivalent ADJ cana-5356 84 4 to to ADP cana-5356 84 5 ( ( PUNCT cana-5356 84 6 π‘˜ π‘˜ PROPN cana-5356 84 7 + + NOUN cana-5356 84 8 1 1 NUM cana-5356 84 9 ) ) PUNCT cana-5356 84 10 | | ADV cana-5356 84 11 ( ( PUNCT cana-5356 84 12 1 1 NUM cana-5356 84 13 βˆ’ βˆ’ NOUN cana-5356 84 14 𝑝)π‘§βˆ’π‘ 𝑝)π‘§βˆ’π‘ NOUN cana-5356 85 1 + + CCONJ cana-5356 85 2 βˆ‘ βˆ‘ PUNCT cana-5356 85 3 { { PUNCT cana-5356 85 4 ( ( PUNCT cana-5356 85 5 1 1 NUM cana-5356 85 6 βˆ’ βˆ’ NOUN cana-5356 85 7 πœ† πœ† X cana-5356 85 8 ) ) PUNCT cana-5356 85 9 ( ( PUNCT cana-5356 85 10 1 1 NUM cana-5356 85 11 + + NUM cana-5356 85 12 𝑗 𝑗 PROPN cana-5356 85 13 𝑝 𝑝 NOUN cana-5356 85 14 ) ) PUNCT cana-5356 85 15 } } PUNCT cana-5356 85 16 𝑛 𝑛 X cana-5356 85 17 ( ( PUNCT cana-5356 85 18 1 1 NUM cana-5356 85 19 + + CCONJ cana-5356 85 20 𝑗)π‘Žπ‘—π‘§π‘—βˆž 𝑗)π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 85 21 𝑗=2 𝑗=2 PROPN cana-5356 85 22 ( ( PUNCT cana-5356 85 23 1 1 NUM cana-5356 85 24 βˆ’ βˆ’ NUM cana-5356 85 25 𝛼𝑝 𝛼𝑝 INTJ cana-5356 85 26 βˆ’ βˆ’ PROPN cana-5356 85 27 π‘Ÿ)π‘§βˆ’π‘ π‘Ÿ)π‘§βˆ’π‘ NUM cana-5356 85 28 βˆ’ βˆ’ PROPN cana-5356 85 29 βˆ‘ βˆ‘ PUNCT cana-5356 85 30 ( ( PUNCT cana-5356 85 31 𝛼𝑗 𝛼𝑗 PROPN cana-5356 85 32 + + CCONJ cana-5356 85 33 1 1 NUM cana-5356 85 34 βˆ’ βˆ’ NUM cana-5356 85 35 π‘Ÿ π‘Ÿ NOUN cana-5356 85 36 ) ) PUNCT cana-5356 85 37 { { PUNCT cana-5356 85 38 ( ( PUNCT cana-5356 85 39 1 1 NUM cana-5356 85 40 βˆ’ βˆ’ NOUN cana-5356 85 41 πœ† πœ† X cana-5356 85 42 ) ) PUNCT cana-5356 85 43 ( ( PUNCT cana-5356 85 44 1 1 NUM cana-5356 85 45 + + NUM cana-5356 85 46 𝑗 𝑗 PROPN cana-5356 85 47 𝑝 𝑝 NOUN cana-5356 85 48 ) ) PUNCT cana-5356 85 49 } } PUNCT cana-5356 85 50 𝑛 𝑛 DET cana-5356 85 51 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 85 52 𝑗=2 𝑗=2 PROPN cana-5356 85 53 βˆ’ βˆ’ PROPN cana-5356 86 1 1| 1| NUM cana-5356 86 2 ≀ ≀ NUM cana-5356 87 1 1 1 NUM cana-5356 87 2 βˆ’ βˆ’ NOUN cana-5356 87 3 𝛽 𝛽 NOUN cana-5356 87 4 communications communication NOUN cana-5356 87 5 on on ADP cana-5356 87 6 applied apply VERB cana-5356 87 7 nonlinear nonlinear ADJ cana-5356 87 8 analysis analysis NOUN cana-5356 87 9 issn issn NOUN cana-5356 87 10 : : PUNCT cana-5356 87 11 1074 1074 NUM cana-5356 87 12 - - PUNCT cana-5356 87 13 133x 133x NUM cana-5356 87 14 vol vol VERB cana-5356 87 15 32 32 NUM cana-5356 87 16 no no NOUN cana-5356 87 17 . . PUNCT cana-5356 88 1 10s 10 NOUN cana-5356 88 2 ( ( PUNCT cana-5356 88 3 2025 2025 NUM cana-5356 88 4 ) ) PUNCT cana-5356 88 5 1875 1875 NUM cana-5356 88 6 https://internationalpubls.com https://internationalpubls.com X cana-5356 89 1 so so ADV cana-5356 89 2 ( ( PUNCT cana-5356 89 3 π‘˜ π‘˜ PROPN cana-5356 89 4 + + NOUN cana-5356 89 5 1 1 NUM cana-5356 89 6 ) ) PUNCT cana-5356 89 7 | | ADV cana-5356 89 8 ( ( PUNCT cana-5356 89 9 1 1 NUM cana-5356 89 10 βˆ’ βˆ’ NOUN cana-5356 90 1 𝑝)π‘§βˆ’π‘ 𝑝)π‘§βˆ’π‘ NOUN cana-5356 91 1 + + CCONJ cana-5356 91 2 βˆ‘ βˆ‘ SYM cana-5356 91 3 |2 |2 NUM cana-5356 91 4 βˆ’ βˆ’ NOUN cana-5356 91 5 𝛾 𝛾 NOUN cana-5356 91 6 βˆ’ βˆ’ PROPN cana-5356 91 7 𝑛(1 𝑛(1 PROPN cana-5356 91 8 βˆ’ βˆ’ PROPN cana-5356 91 9 𝛼)| 𝛼)| PROPN cana-5356 91 10 { { PUNCT cana-5356 91 11 ( ( PUNCT cana-5356 91 12 1 1 NUM cana-5356 91 13 βˆ’ βˆ’ NOUN cana-5356 91 14 πœ† πœ† X cana-5356 91 15 ) ) PUNCT cana-5356 91 16 ( ( PUNCT cana-5356 91 17 1 1 NUM cana-5356 91 18 + + CCONJ cana-5356 91 19 𝑗 𝑗 PROPN cana-5356 91 20 𝑝 𝑝 NOUN cana-5356 91 21 ) ) PUNCT cana-5356 91 22 } } PUNCT cana-5356 91 23 𝑛 𝑛 X cana-5356 91 24 ( ( PUNCT cana-5356 91 25 1 1 NUM cana-5356 91 26 + + NUM cana-5356 91 27 𝑗)π‘Žπ‘—π‘§π‘— 𝑗)π‘Žπ‘—π‘§π‘— PRON cana-5356 91 28 βˆ’ βˆ’ PROPN cana-5356 91 29 ( ( PUNCT cana-5356 91 30 1 1 NUM cana-5356 91 31 βˆ’ βˆ’ NUM cana-5356 91 32 𝛼𝑝 𝛼𝑝 NUM cana-5356 91 33 βˆ’ βˆ’ PROPN cana-5356 91 34 π‘Ÿ)π‘§βˆ’π‘βˆž π‘Ÿ)π‘§βˆ’π‘βˆž PROPN cana-5356 91 35 𝑗=2 𝑗=2 X cana-5356 91 36 ( ( PUNCT cana-5356 91 37 1 1 NUM cana-5356 91 38 βˆ’ βˆ’ NUM cana-5356 91 39 𝛼𝑝 𝛼𝑝 INTJ cana-5356 91 40 βˆ’ βˆ’ PROPN cana-5356 91 41 π‘Ÿ)π‘§βˆ’π‘ π‘Ÿ)π‘§βˆ’π‘ NUM cana-5356 91 42 βˆ’ βˆ’ PROPN cana-5356 91 43 βˆ‘ βˆ‘ PUNCT cana-5356 91 44 ( ( PUNCT cana-5356 91 45 𝛼𝑗 𝛼𝑗 PROPN cana-5356 91 46 + + CCONJ cana-5356 91 47 1 1 NUM cana-5356 91 48 βˆ’ βˆ’ NUM cana-5356 91 49 π‘Ÿ π‘Ÿ NOUN cana-5356 91 50 ) ) PUNCT cana-5356 91 51 { { PUNCT cana-5356 91 52 ( ( PUNCT cana-5356 91 53 1 1 NUM cana-5356 91 54 βˆ’ βˆ’ NOUN cana-5356 91 55 πœ† πœ† X cana-5356 91 56 ) ) PUNCT cana-5356 91 57 ( ( PUNCT cana-5356 91 58 1 1 NUM cana-5356 91 59 + + CCONJ cana-5356 91 60 𝑗 𝑗 PROPN cana-5356 91 61 𝑝 𝑝 NOUN cana-5356 91 62 ) ) PUNCT cana-5356 91 63 } } PUNCT cana-5356 91 64 𝑛 𝑛 DET cana-5356 91 65 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž NOUN cana-5356 91 66 𝑗=2 𝑗=2 PROPN cana-5356 92 1 | | ADV cana-5356 92 2 ≀ ≀ NUM cana-5356 92 3 1 1 NUM cana-5356 92 4 βˆ’ βˆ’ NOUN cana-5356 92 5 𝛽 𝛽 NOUN cana-5356 92 6 further far ADV cana-5356 92 7 the the DET cana-5356 92 8 above above ADJ cana-5356 92 9 inequality inequality NOUN cana-5356 92 10 ( ( PUNCT cana-5356 92 11 π‘˜ π‘˜ PROPN cana-5356 92 12 + + NOUN cana-5356 92 13 1 1 NUM cana-5356 92 14 ) ) PUNCT cana-5356 92 15 | | ADV cana-5356 92 16 ( ( PUNCT cana-5356 92 17 𝛼𝑝 𝛼𝑝 PROPN cana-5356 92 18 + + CCONJ cana-5356 92 19 π‘Ÿ π‘Ÿ NOUN cana-5356 92 20 βˆ’ βˆ’ NOUN cana-5356 92 21 𝑝)π‘§βˆ’π‘ 𝑝)π‘§βˆ’π‘ VERB cana-5356 93 1 + + NUM cana-5356 93 2 βˆ‘ βˆ‘ SYM cana-5356 93 3 |2 |2 NUM cana-5356 93 4 βˆ’ βˆ’ NOUN cana-5356 93 5 𝛾 𝛾 NOUN cana-5356 93 6 βˆ’ βˆ’ PROPN cana-5356 93 7 𝑛(1 𝑛(1 PROPN cana-5356 93 8 βˆ’ βˆ’ PROPN cana-5356 93 9 𝛼)| 𝛼)| PROPN cana-5356 93 10 { { PUNCT cana-5356 93 11 ( ( PUNCT cana-5356 93 12 1 1 NUM cana-5356 93 13 βˆ’ βˆ’ NOUN cana-5356 93 14 πœ† πœ† X cana-5356 93 15 ) ) PUNCT cana-5356 93 16 ( ( PUNCT cana-5356 93 17 1 1 NUM cana-5356 93 18 + + NUM cana-5356 93 19 𝑗 𝑗 PROPN cana-5356 93 20 𝑝 𝑝 NOUN cana-5356 93 21 ) ) PUNCT cana-5356 93 22 } } PUNCT cana-5356 93 23 𝑛 𝑛 X cana-5356 93 24 ( ( PUNCT cana-5356 93 25 1 1 NUM cana-5356 93 26 + + CCONJ cana-5356 93 27 𝑗)π‘Žπ‘—π‘§π‘—βˆž 𝑗)π‘Žπ‘—π‘§π‘—βˆž PROPN cana-5356 93 28 𝑗=2 𝑗=2 PROPN cana-5356 93 29 ( ( PUNCT cana-5356 93 30 1 1 NUM cana-5356 93 31 βˆ’ βˆ’ NUM cana-5356 93 32 𝛼𝑝 𝛼𝑝 INTJ cana-5356 93 33 βˆ’ βˆ’ PROPN cana-5356 93 34 π‘Ÿ)π‘§βˆ’π‘ π‘Ÿ)π‘§βˆ’π‘ NUM cana-5356 93 35 βˆ’ βˆ’ PROPN cana-5356 93 36 βˆ‘ βˆ‘ PUNCT cana-5356 93 37 ( ( PUNCT cana-5356 93 38 𝛼𝑗 𝛼𝑗 PROPN cana-5356 93 39 + + CCONJ cana-5356 93 40 1 1 NUM cana-5356 93 41 βˆ’ βˆ’ NUM cana-5356 93 42 π‘Ÿ π‘Ÿ NOUN cana-5356 93 43 ) ) PUNCT cana-5356 93 44 { { PUNCT cana-5356 93 45 ( ( PUNCT cana-5356 93 46 1 1 NUM cana-5356 93 47 βˆ’ βˆ’ NOUN cana-5356 93 48 πœ† πœ† X cana-5356 93 49 ) ) PUNCT cana-5356 93 50 ( ( PUNCT cana-5356 93 51 1 1 NUM cana-5356 93 52 + + NUM cana-5356 93 53 𝑗 𝑗 PROPN cana-5356 93 54 𝑝 𝑝 NOUN cana-5356 93 55 ) ) PUNCT cana-5356 93 56 } } PUNCT cana-5356 93 57 𝑛 𝑛 DET cana-5356 93 58 π‘Žπ‘—π‘§π‘—βˆž π‘Žπ‘—π‘§π‘—βˆž NOUN cana-5356 93 59 𝑗=2 𝑗=2 PROPN cana-5356 94 1 | | ADV cana-5356 94 2 ≀ ≀ NUM cana-5356 94 3 1 1 NUM cana-5356 94 4 βˆ’ βˆ’ NOUN cana-5356 94 5 𝛽 𝛽 NOUN cana-5356 94 6 by by ADP cana-5356 94 7 factoring factor VERB cana-5356 94 8 out out ADP cana-5356 94 9 the the DET cana-5356 94 10 term term NOUN cana-5356 94 11 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 94 12 ( ( PUNCT cana-5356 94 13 π‘˜ π‘˜ X cana-5356 94 14 + + NOUN cana-5356 94 15 1 1 NUM cana-5356 94 16 ) ) PUNCT cana-5356 94 17 | | ADV cana-5356 94 18 | | ADV cana-5356 94 19 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 94 20 ( ( PUNCT cana-5356 94 21 ( ( PUNCT cana-5356 94 22 𝛼𝑝 𝛼𝑝 PROPN cana-5356 94 23 + + NUM cana-5356 94 24 π‘Ÿ π‘Ÿ X cana-5356 94 25 βˆ’ βˆ’ PROPN cana-5356 94 26 𝑝 𝑝 NOUN cana-5356 94 27 ) ) PUNCT cana-5356 95 1 + + CCONJ cana-5356 95 2 βˆ‘ βˆ‘ ADP cana-5356 95 3 |2 |2 NUM cana-5356 95 4 βˆ’ βˆ’ NOUN cana-5356 95 5 𝛾 𝛾 NOUN cana-5356 95 6 βˆ’ βˆ’ PROPN cana-5356 95 7 𝑛(1 𝑛(1 PROPN cana-5356 95 8 βˆ’ βˆ’ PROPN cana-5356 95 9 𝛼)| 𝛼)| PROPN cana-5356 95 10 { { PUNCT cana-5356 95 11 ( ( PUNCT cana-5356 95 12 1 1 NUM cana-5356 95 13 βˆ’ βˆ’ NOUN cana-5356 95 14 πœ† πœ† X cana-5356 95 15 ) ) PUNCT cana-5356 95 16 ( ( PUNCT cana-5356 95 17 1 1 NUM cana-5356 95 18 + + NUM cana-5356 95 19 𝑗 𝑗 PROPN cana-5356 95 20 𝑝 𝑝 NOUN cana-5356 95 21 ) ) PUNCT cana-5356 95 22 } } PUNCT cana-5356 95 23 𝑛 𝑛 X cana-5356 95 24 ( ( PUNCT cana-5356 95 25 1 1 NUM cana-5356 95 26 + + CCONJ cana-5356 95 27 𝑗)π‘Žπ‘—π‘§π‘—+π‘βˆž 𝑗)π‘Žπ‘—π‘§π‘—+π‘βˆž PROPN cana-5356 95 28 𝑗=2 𝑗=2 PROPN cana-5356 95 29 ) ) PUNCT cana-5356 95 30 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 95 31 ( ( PUNCT cana-5356 95 32 ( ( PUNCT cana-5356 95 33 1 1 NUM cana-5356 95 34 βˆ’ βˆ’ NUM cana-5356 95 35 𝛼𝑝 𝛼𝑝 NUM cana-5356 95 36 βˆ’ βˆ’ PROPN cana-5356 95 37 π‘Ÿ π‘Ÿ NOUN cana-5356 95 38 ) ) PUNCT cana-5356 95 39 βˆ’ βˆ’ PROPN cana-5356 95 40 βˆ‘ βˆ‘ PUNCT cana-5356 95 41 ( ( PUNCT cana-5356 95 42 𝛼𝑗 𝛼𝑗 PROPN cana-5356 95 43 + + CCONJ cana-5356 95 44 1 1 NUM cana-5356 95 45 βˆ’ βˆ’ NUM cana-5356 95 46 π‘Ÿ π‘Ÿ NOUN cana-5356 95 47 ) ) PUNCT cana-5356 95 48 { { PUNCT cana-5356 95 49 ( ( PUNCT cana-5356 95 50 1 1 NUM cana-5356 95 51 βˆ’ βˆ’ NOUN cana-5356 95 52 πœ† πœ† X cana-5356 95 53 ) ) PUNCT cana-5356 95 54 ( ( PUNCT cana-5356 95 55 1 1 NUM cana-5356 95 56 + + NUM cana-5356 95 57 𝑗 𝑗 PROPN cana-5356 95 58 𝑝 𝑝 NOUN cana-5356 95 59 ) ) PUNCT cana-5356 95 60 } } PUNCT cana-5356 95 61 𝑛 𝑛 PROPN cana-5356 95 62 π‘Žπ‘—π‘§π‘—+π‘βˆž π‘Žπ‘—π‘§π‘—+π‘βˆž ADJ cana-5356 95 63 𝑗=2 𝑗=2 PROPN cana-5356 95 64 ) ) PUNCT cana-5356 96 1 | | ADV cana-5356 96 2 | | ADV cana-5356 96 3 ≀ ≀ NUM cana-5356 96 4 1 1 NUM cana-5356 96 5 βˆ’ βˆ’ NOUN cana-5356 97 1 𝛽 𝛽 NOUN cana-5356 97 2 cancel cancel NOUN cana-5356 97 3 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 97 4 from from ADP cana-5356 97 5 both both CCONJ cana-5356 97 6 numerator numerator NOUN cana-5356 97 7 and and CCONJ cana-5356 97 8 denominator denominator NOUN cana-5356 97 9 ( ( PUNCT cana-5356 97 10 π‘˜ π‘˜ PROPN cana-5356 97 11 + + NOUN cana-5356 97 12 1 1 NUM cana-5356 97 13 ) ) PUNCT cana-5356 98 1 | | ADV cana-5356 98 2 | | ADV cana-5356 98 3 ( ( PUNCT cana-5356 98 4 ( ( PUNCT cana-5356 98 5 𝛼𝑝 𝛼𝑝 PROPN cana-5356 98 6 + + NUM cana-5356 98 7 π‘Ÿ π‘Ÿ X cana-5356 98 8 βˆ’ βˆ’ PROPN cana-5356 98 9 𝑝 𝑝 NOUN cana-5356 98 10 ) ) PUNCT cana-5356 99 1 + + CCONJ cana-5356 99 2 βˆ‘ βˆ‘ ADP cana-5356 99 3 |2 |2 NUM cana-5356 99 4 βˆ’ βˆ’ NOUN cana-5356 99 5 𝛾 𝛾 NOUN cana-5356 99 6 βˆ’ βˆ’ PROPN cana-5356 99 7 𝑛(1 𝑛(1 PROPN cana-5356 99 8 βˆ’ βˆ’ PROPN cana-5356 99 9 𝛼)| 𝛼)| PROPN cana-5356 99 10 { { PUNCT cana-5356 99 11 ( ( PUNCT cana-5356 99 12 1 1 NUM cana-5356 99 13 βˆ’ βˆ’ NOUN cana-5356 99 14 πœ† πœ† X cana-5356 99 15 ) ) PUNCT cana-5356 99 16 ( ( PUNCT cana-5356 99 17 1 1 NUM cana-5356 99 18 + + NUM cana-5356 99 19 𝑗 𝑗 PROPN cana-5356 99 20 𝑝 𝑝 NOUN cana-5356 99 21 ) ) PUNCT cana-5356 99 22 } } PUNCT cana-5356 99 23 𝑛 𝑛 X cana-5356 99 24 ( ( PUNCT cana-5356 99 25 1 1 NUM cana-5356 99 26 + + CCONJ cana-5356 99 27 𝑗)π‘Žπ‘—π‘§π‘—+π‘βˆž 𝑗)π‘Žπ‘—π‘§π‘—+π‘βˆž PROPN cana-5356 99 28 𝑗=2 𝑗=2 PROPN cana-5356 99 29 ) ) PUNCT cana-5356 99 30 ( ( PUNCT cana-5356 99 31 ( ( PUNCT cana-5356 99 32 1 1 NUM cana-5356 99 33 βˆ’ βˆ’ NUM cana-5356 99 34 𝛼𝑝 𝛼𝑝 NUM cana-5356 99 35 βˆ’ βˆ’ PROPN cana-5356 99 36 π‘Ÿ π‘Ÿ NOUN cana-5356 99 37 ) ) PUNCT cana-5356 99 38 βˆ’ βˆ’ PROPN cana-5356 99 39 βˆ‘ βˆ‘ PUNCT cana-5356 99 40 ( ( PUNCT cana-5356 99 41 𝛼𝑗 𝛼𝑗 PROPN cana-5356 99 42 + + CCONJ cana-5356 99 43 1 1 NUM cana-5356 99 44 βˆ’ βˆ’ NUM cana-5356 99 45 π‘Ÿ π‘Ÿ NOUN cana-5356 99 46 ) ) PUNCT cana-5356 99 47 { { PUNCT cana-5356 99 48 ( ( PUNCT cana-5356 99 49 1 1 NUM cana-5356 99 50 βˆ’ βˆ’ NOUN cana-5356 99 51 πœ† πœ† X cana-5356 99 52 ) ) PUNCT cana-5356 99 53 ( ( PUNCT cana-5356 99 54 1 1 NUM cana-5356 99 55 + + NUM cana-5356 99 56 𝑗 𝑗 PROPN cana-5356 99 57 𝑝 𝑝 NOUN cana-5356 99 58 ) ) PUNCT cana-5356 99 59 } } PUNCT cana-5356 99 60 𝑛 𝑛 PROPN cana-5356 99 61 π‘Žπ‘—π‘§π‘—+π‘βˆž π‘Žπ‘—π‘§π‘—+π‘βˆž ADJ cana-5356 99 62 𝑗=2 𝑗=2 PROPN cana-5356 99 63 ) ) PUNCT cana-5356 100 1 | | ADV cana-5356 100 2 | | ADV cana-5356 100 3 ≀ ≀ NUM cana-5356 100 4 1 1 NUM cana-5356 100 5 βˆ’ βˆ’ NOUN cana-5356 101 1 𝛽 𝛽 NOUN cana-5356 101 2 then then ADV cana-5356 101 3 , , PUNCT cana-5356 101 4 ( ( PUNCT cana-5356 101 5 π‘˜ π‘˜ X cana-5356 101 6 + + NOUN cana-5356 101 7 1 1 NUM cana-5356 101 8 ) ) PUNCT cana-5356 102 1 [ [ X cana-5356 102 2 ( ( PUNCT cana-5356 102 3 𝛼𝑝 𝛼𝑝 PROPN cana-5356 102 4 + + NUM cana-5356 102 5 π‘Ÿ π‘Ÿ X cana-5356 102 6 βˆ’ βˆ’ PROPN cana-5356 102 7 𝑝 𝑝 NOUN cana-5356 102 8 ) ) PUNCT cana-5356 102 9 + + CCONJ cana-5356 102 10 βˆ‘ βˆ‘ ADP cana-5356 102 11 |2 |2 NUM cana-5356 102 12 βˆ’ βˆ’ NOUN cana-5356 102 13 𝛾 𝛾 NOUN cana-5356 102 14 βˆ’ βˆ’ PROPN cana-5356 102 15 𝑛(1 𝑛(1 PROPN cana-5356 102 16 βˆ’ βˆ’ PROPN cana-5356 102 17 𝛼)| 𝛼)| PROPN cana-5356 102 18 { { PUNCT cana-5356 102 19 ( ( PUNCT cana-5356 102 20 1 1 NUM cana-5356 102 21 βˆ’ βˆ’ NOUN cana-5356 102 22 πœ† πœ† X cana-5356 102 23 ) ) PUNCT cana-5356 102 24 ( ( PUNCT cana-5356 102 25 1 1 NUM cana-5356 102 26 + + CCONJ cana-5356 102 27 𝑗 𝑗 PROPN cana-5356 102 28 𝑝 𝑝 NOUN cana-5356 102 29 ) ) PUNCT cana-5356 102 30 } } PUNCT cana-5356 102 31 𝑛 𝑛 X cana-5356 102 32 ( ( PUNCT cana-5356 102 33 1 1 NUM cana-5356 102 34 + + CCONJ cana-5356 102 35 𝑗)π‘Žπ‘—π‘§π‘—+π‘βˆž 𝑗)π‘Žπ‘—π‘§π‘—+π‘βˆž PROPN cana-5356 102 36 𝑗=2 𝑗=2 X cana-5356 102 37 ] ] PUNCT cana-5356 102 38 ≀ ≀ NUM cana-5356 102 39 ( ( PUNCT cana-5356 102 40 1 1 NUM cana-5356 102 41 βˆ’ βˆ’ NUM cana-5356 102 42 𝛼𝑝 𝛼𝑝 INTJ cana-5356 102 43 βˆ’ βˆ’ NOUN cana-5356 102 44 π‘Ÿ)(1 π‘Ÿ)(1 NUM cana-5356 102 45 βˆ’ βˆ’ PROPN cana-5356 102 46 𝛽 𝛽 NOUN cana-5356 102 47 ) ) PUNCT cana-5356 102 48 βˆ’ βˆ’ PROPN cana-5356 103 1 ( ( PUNCT cana-5356 103 2 1 1 NUM cana-5356 103 3 βˆ’ βˆ’ PROPN cana-5356 103 4 𝛽 𝛽 NOUN cana-5356 103 5 ) ) PUNCT cana-5356 103 6 ( ( PUNCT cana-5356 103 7 βˆ‘(𝛼𝑗 βˆ‘(𝛼𝑗 PROPN cana-5356 103 8 + + CCONJ cana-5356 103 9 1 1 NUM cana-5356 103 10 βˆ’ βˆ’ NOUN cana-5356 103 11 π‘Ÿ π‘Ÿ NOUN cana-5356 103 12 ) ) PUNCT cana-5356 103 13 { { PUNCT cana-5356 103 14 ( ( PUNCT cana-5356 103 15 1 1 NUM cana-5356 103 16 βˆ’ βˆ’ NOUN cana-5356 103 17 πœ† πœ† X cana-5356 103 18 ) ) PUNCT cana-5356 103 19 ( ( PUNCT cana-5356 103 20 1 1 NUM cana-5356 103 21 + + CCONJ cana-5356 103 22 𝑗 𝑗 PROPN cana-5356 103 23 𝑝 𝑝 NOUN cana-5356 103 24 ) ) PUNCT cana-5356 103 25 } } PUNCT cana-5356 103 26 𝑛 𝑛 PROPN cana-5356 103 27 π‘Žπ‘—π‘§π‘—+𝑝 π‘Žπ‘—π‘§π‘—+𝑝 PROPN cana-5356 103 28 ∞ ∞ PROPN cana-5356 103 29 𝑗=2 𝑗=2 PROPN cana-5356 103 30 ) ) PUNCT cana-5356 103 31 now now ADV cana-5356 103 32 expand expand VERB cana-5356 103 33 the the DET cana-5356 103 34 terms term NOUN cana-5356 103 35 of of ADP cana-5356 103 36 left left ADJ cana-5356 103 37 - - PUNCT cana-5356 103 38 hand hand NOUN cana-5356 103 39 side side NOUN cana-5356 103 40 ( ( PUNCT cana-5356 103 41 lhs lhs PROPN cana-5356 103 42 ) ) PUNCT cana-5356 103 43 ( ( PUNCT cana-5356 103 44 π‘˜ π‘˜ PROPN cana-5356 103 45 + + NOUN cana-5356 103 46 1 1 NUM cana-5356 103 47 ) ) PUNCT cana-5356 104 1 [ [ X cana-5356 104 2 ( ( PUNCT cana-5356 104 3 𝛼𝑝 𝛼𝑝 PROPN cana-5356 104 4 + + NUM cana-5356 104 5 π‘Ÿ π‘Ÿ X cana-5356 104 6 βˆ’ βˆ’ PROPN cana-5356 104 7 𝑝 𝑝 NOUN cana-5356 104 8 ) ) PUNCT cana-5356 105 1 + + CCONJ cana-5356 106 1 βˆ‘|2 βˆ‘|2 NOUN cana-5356 106 2 βˆ’ βˆ’ NOUN cana-5356 106 3 𝛾 𝛾 NOUN cana-5356 106 4 βˆ’ βˆ’ PROPN cana-5356 106 5 𝑛(1 𝑛(1 PROPN cana-5356 106 6 βˆ’ βˆ’ PROPN cana-5356 106 7 𝛼)| 𝛼)| PROPN cana-5356 106 8 { { PUNCT cana-5356 106 9 ( ( PUNCT cana-5356 106 10 1 1 NUM cana-5356 106 11 βˆ’ βˆ’ NOUN cana-5356 106 12 πœ† πœ† X cana-5356 106 13 ) ) PUNCT cana-5356 106 14 ( ( PUNCT cana-5356 106 15 1 1 NUM cana-5356 106 16 + + CCONJ cana-5356 106 17 𝑗 𝑗 PROPN cana-5356 106 18 𝑝 𝑝 NOUN cana-5356 106 19 ) ) PUNCT cana-5356 106 20 } } PUNCT cana-5356 106 21 𝑛 𝑛 X cana-5356 106 22 ( ( PUNCT cana-5356 106 23 1 1 NUM cana-5356 106 24 + + NUM cana-5356 106 25 𝑗)π‘Žπ‘—π‘§π‘—+𝑝 𝑗)π‘Žπ‘—π‘§π‘—+𝑝 NOUN cana-5356 106 26 ∞ ∞ PROPN cana-5356 106 27 𝑗=2 𝑗=2 X cana-5356 106 28 ] ] PUNCT cana-5356 107 1 the the DET cana-5356 107 2 lhs lhs PROPN cana-5356 107 3 consists consist VERB cana-5356 107 4 of of ADP cana-5356 107 5 a a DET cana-5356 107 6 constant constant ADJ cana-5356 107 7 term term NOUN cana-5356 107 8 ( ( PUNCT cana-5356 107 9 π‘˜ π‘˜ PROPN cana-5356 107 10 + + ADJ cana-5356 107 11 1)(𝛼𝑝 1)(𝛼𝑝 NUM cana-5356 107 12 + + NUM cana-5356 107 13 π‘Ÿ π‘Ÿ X cana-5356 107 14 βˆ’ βˆ’ PROPN cana-5356 107 15 𝑝 𝑝 PROPN cana-5356 107 16 ) ) PUNCT cana-5356 107 17 and and CCONJ cana-5356 107 18 a a DET cana-5356 107 19 series series NOUN cana-5356 107 20 denoted denote VERB cana-5356 107 21 by by ADP cana-5356 107 22 𝑆1 𝑆1 NOUN cana-5356 107 23 , , PUNCT cana-5356 107 24 where where SCONJ cana-5356 107 25 𝑆1 𝑆1 NOUN cana-5356 107 26 = = PUNCT cana-5356 107 27 βˆ‘|2 βˆ‘|2 ADJ cana-5356 107 28 βˆ’ βˆ’ NOUN cana-5356 107 29 𝛾 𝛾 PROPN cana-5356 107 30 βˆ’ βˆ’ PROPN cana-5356 107 31 𝑛(1 𝑛(1 PROPN cana-5356 107 32 βˆ’ βˆ’ PROPN cana-5356 107 33 𝛼)| 𝛼)| PROPN cana-5356 107 34 { { PUNCT cana-5356 107 35 ( ( PUNCT cana-5356 107 36 1 1 NUM cana-5356 107 37 βˆ’ βˆ’ NOUN cana-5356 107 38 πœ† πœ† X cana-5356 107 39 ) ) PUNCT cana-5356 107 40 ( ( PUNCT cana-5356 107 41 1 1 NUM cana-5356 107 42 + + CCONJ cana-5356 107 43 𝑗 𝑗 PROPN cana-5356 107 44 𝑝 𝑝 NOUN cana-5356 107 45 ) ) PUNCT cana-5356 107 46 } } PUNCT cana-5356 107 47 𝑛 𝑛 X cana-5356 107 48 ( ( PUNCT cana-5356 107 49 1 1 NUM cana-5356 107 50 + + NUM cana-5356 107 51 𝑗)π‘Žπ‘—π‘§π‘—+𝑝 𝑗)π‘Žπ‘—π‘§π‘—+𝑝 PROPN cana-5356 107 52 ∞ ∞ PROPN cana-5356 107 53 𝑗=2 𝑗=2 PUNCT cana-5356 108 1 so so ADV cana-5356 108 2 , , PUNCT cana-5356 108 3 the the DET cana-5356 108 4 lhs lhs PROPN cana-5356 108 5 becomes become VERB cana-5356 108 6 ( ( PUNCT cana-5356 108 7 π‘˜ π‘˜ PROPN cana-5356 108 8 + + X cana-5356 108 9 1)(𝛼𝑝 1)(𝛼𝑝 NUM cana-5356 108 10 + + NUM cana-5356 108 11 π‘Ÿ π‘Ÿ X cana-5356 108 12 βˆ’ βˆ’ PROPN cana-5356 108 13 𝑝 𝑝 NOUN cana-5356 108 14 ) ) PUNCT cana-5356 109 1 + + CCONJ cana-5356 109 2 ( ( PUNCT cana-5356 109 3 π‘˜ π‘˜ PROPN cana-5356 109 4 + + NOUN cana-5356 109 5 1)𝑆1 1)𝑆1 NOUN cana-5356 109 6 communications communication NOUN cana-5356 109 7 on on ADP cana-5356 109 8 applied apply VERB cana-5356 109 9 nonlinear nonlinear ADJ cana-5356 109 10 analysis analysis NOUN cana-5356 109 11 issn issn NOUN cana-5356 109 12 : : PUNCT cana-5356 109 13 1074 1074 NUM cana-5356 109 14 - - PUNCT cana-5356 109 15 133x 133x NUM cana-5356 109 16 vol vol VERB cana-5356 109 17 32 32 NUM cana-5356 109 18 no no NOUN cana-5356 109 19 . . PUNCT cana-5356 110 1 10s 10 NOUN cana-5356 110 2 ( ( PUNCT cana-5356 110 3 2025 2025 NUM cana-5356 110 4 ) ) PUNCT cana-5356 110 5 1876 1876 NUM cana-5356 110 6 https://internationalpubls.com https://internationalpubls.com X cana-5356 110 7 right right ADJ cana-5356 110 8 - - PUNCT cana-5356 110 9 hand hand NOUN cana-5356 110 10 side side NOUN cana-5356 110 11 ( ( PUNCT cana-5356 110 12 rhs rhs PROPN cana-5356 110 13 ) ) PUNCT cana-5356 110 14 ( ( PUNCT cana-5356 110 15 1 1 NUM cana-5356 110 16 βˆ’ βˆ’ NUM cana-5356 110 17 𝛼𝑝 𝛼𝑝 INTJ cana-5356 110 18 βˆ’ βˆ’ NOUN cana-5356 110 19 π‘Ÿ)(1 π‘Ÿ)(1 NUM cana-5356 110 20 βˆ’ βˆ’ PROPN cana-5356 110 21 𝛽 𝛽 NOUN cana-5356 110 22 ) ) PUNCT cana-5356 110 23 βˆ’ βˆ’ PROPN cana-5356 110 24 ( ( PUNCT cana-5356 110 25 1 1 NUM cana-5356 110 26 βˆ’ βˆ’ PROPN cana-5356 110 27 𝛽 𝛽 NOUN cana-5356 110 28 ) ) PUNCT cana-5356 110 29 ( ( PUNCT cana-5356 110 30 βˆ‘(𝛼𝑗 βˆ‘(𝛼𝑗 PROPN cana-5356 110 31 + + CCONJ cana-5356 110 32 1 1 NUM cana-5356 110 33 βˆ’ βˆ’ NOUN cana-5356 110 34 π‘Ÿ π‘Ÿ NOUN cana-5356 110 35 ) ) PUNCT cana-5356 110 36 { { PUNCT cana-5356 110 37 ( ( PUNCT cana-5356 110 38 1 1 NUM cana-5356 110 39 βˆ’ βˆ’ NOUN cana-5356 110 40 πœ† πœ† X cana-5356 110 41 ) ) PUNCT cana-5356 110 42 ( ( PUNCT cana-5356 110 43 1 1 NUM cana-5356 110 44 + + CCONJ cana-5356 110 45 𝑗 𝑗 PROPN cana-5356 110 46 𝑝 𝑝 NOUN cana-5356 110 47 ) ) PUNCT cana-5356 110 48 } } PUNCT cana-5356 110 49 𝑛 𝑛 PROPN cana-5356 110 50 π‘Žπ‘—π‘§π‘—+𝑝 π‘Žπ‘—π‘§π‘—+𝑝 PROPN cana-5356 110 51 ∞ ∞ PROPN cana-5356 110 52 𝑗=2 𝑗=2 PROPN cana-5356 110 53 ) ) PUNCT cana-5356 111 1 the the DET cana-5356 111 2 rhs rhs PROPN cana-5356 111 3 also also ADV cana-5356 111 4 has have VERB cana-5356 111 5 a a DET cana-5356 111 6 constant constant ADJ cana-5356 111 7 term term NOUN cana-5356 111 8 ( ( PUNCT cana-5356 111 9 1 1 NUM cana-5356 111 10 βˆ’ βˆ’ NUM cana-5356 111 11 𝛼𝑝 𝛼𝑝 INTJ cana-5356 111 12 βˆ’ βˆ’ NOUN cana-5356 111 13 π‘Ÿ)(1 π‘Ÿ)(1 NUM cana-5356 111 14 βˆ’ βˆ’ PROPN cana-5356 111 15 𝛽 𝛽 NOUN cana-5356 111 16 ) ) PUNCT cana-5356 111 17 and and CCONJ cana-5356 111 18 a a DET cana-5356 111 19 series series NOUN cana-5356 111 20 denoted denote VERB cana-5356 111 21 by by ADP cana-5356 111 22 𝑆2 𝑆2 PROPN cana-5356 111 23 , , PUNCT cana-5356 111 24 where where SCONJ cana-5356 111 25 𝑆2 𝑆2 PROPN cana-5356 111 26 = = PUNCT cana-5356 112 1 βˆ‘(𝛼𝑗 βˆ‘(𝛼𝑗 PROPN cana-5356 113 1 + + CCONJ cana-5356 113 2 1 1 NUM cana-5356 113 3 βˆ’ βˆ’ NOUN cana-5356 113 4 π‘Ÿ π‘Ÿ NOUN cana-5356 113 5 ) ) PUNCT cana-5356 113 6 { { PUNCT cana-5356 113 7 ( ( PUNCT cana-5356 113 8 1 1 NUM cana-5356 113 9 βˆ’ βˆ’ NOUN cana-5356 113 10 πœ† πœ† X cana-5356 113 11 ) ) PUNCT cana-5356 113 12 ( ( PUNCT cana-5356 113 13 1 1 NUM cana-5356 113 14 + + CCONJ cana-5356 113 15 𝑗 𝑗 PROPN cana-5356 113 16 𝑝 𝑝 NOUN cana-5356 113 17 ) ) PUNCT cana-5356 113 18 } } PUNCT cana-5356 113 19 𝑛 𝑛 PROPN cana-5356 113 20 π‘Žπ‘—π‘§π‘—+𝑝 π‘Žπ‘—π‘§π‘—+𝑝 PROPN cana-5356 113 21 ∞ ∞ PROPN cana-5356 113 22 𝑗=2 𝑗=2 PROPN cana-5356 114 1 so so ADV cana-5356 114 2 , , PUNCT cana-5356 114 3 the the DET cana-5356 114 4 rhs rhs PROPN cana-5356 114 5 becomes become VERB cana-5356 114 6 ( ( PUNCT cana-5356 114 7 1 1 NUM cana-5356 114 8 βˆ’ βˆ’ NUM cana-5356 114 9 𝛼𝑝 𝛼𝑝 INTJ cana-5356 114 10 βˆ’ βˆ’ NOUN cana-5356 114 11 π‘Ÿ)(1 π‘Ÿ)(1 NUM cana-5356 115 1 βˆ’ βˆ’ PROPN cana-5356 115 2 𝛽 𝛽 NOUN cana-5356 115 3 ) ) PUNCT cana-5356 116 1 βˆ’ βˆ’ PROPN cana-5356 116 2 ( ( PUNCT cana-5356 116 3 1 1 NUM cana-5356 116 4 βˆ’ βˆ’ NOUN cana-5356 116 5 𝛽)𝑆2 𝛽)𝑆2 NOUN cana-5356 116 6 then then ADV cana-5356 116 7 using use VERB cana-5356 116 8 the the DET cana-5356 116 9 expressions expression NOUN cana-5356 116 10 for for ADP cana-5356 116 11 lhs lhs PROPN cana-5356 116 12 & & CCONJ cana-5356 116 13 rhs rhs PROPN cana-5356 116 14 ( ( PUNCT cana-5356 116 15 π‘˜ π‘˜ PROPN cana-5356 116 16 + + ADJ cana-5356 116 17 1)(𝛼𝑝 1)(𝛼𝑝 NUM cana-5356 116 18 + + NUM cana-5356 116 19 π‘Ÿ π‘Ÿ X cana-5356 116 20 βˆ’ βˆ’ PROPN cana-5356 116 21 𝑝 𝑝 NOUN cana-5356 116 22 ) ) PUNCT cana-5356 117 1 + + CCONJ cana-5356 117 2 ( ( PUNCT cana-5356 117 3 π‘˜ π‘˜ PROPN cana-5356 117 4 + + NOUN cana-5356 117 5 1)𝑆1 1)𝑆1 NOUN cana-5356 117 6 ≀ ≀ NUM cana-5356 117 7 ( ( PUNCT cana-5356 117 8 1 1 NUM cana-5356 117 9 βˆ’ βˆ’ NUM cana-5356 117 10 𝛼𝑝 𝛼𝑝 INTJ cana-5356 117 11 βˆ’ βˆ’ NOUN cana-5356 117 12 π‘Ÿ)(1 π‘Ÿ)(1 NUM cana-5356 118 1 βˆ’ βˆ’ PROPN cana-5356 118 2 𝛽 𝛽 NOUN cana-5356 118 3 ) ) PUNCT cana-5356 119 1 βˆ’ βˆ’ PROPN cana-5356 119 2 ( ( PUNCT cana-5356 119 3 1 1 NUM cana-5356 119 4 βˆ’ βˆ’ NOUN cana-5356 119 5 𝛽)𝑆2 𝛽)𝑆2 ADP cana-5356 119 6 rearranging rearrange VERB cana-5356 119 7 terms term NOUN cana-5356 119 8 ( ( PUNCT cana-5356 119 9 π‘˜ π‘˜ X cana-5356 119 10 + + X cana-5356 119 11 1)(𝛼𝑝 1)(𝛼𝑝 NUM cana-5356 119 12 + + NUM cana-5356 119 13 π‘Ÿ π‘Ÿ X cana-5356 119 14 βˆ’ βˆ’ PROPN cana-5356 119 15 𝑝 𝑝 PROPN cana-5356 119 16 ) ) PUNCT cana-5356 119 17 βˆ’ βˆ’ PROPN cana-5356 120 1 ( ( PUNCT cana-5356 120 2 1 1 NUM cana-5356 120 3 βˆ’ βˆ’ NUM cana-5356 120 4 𝛼𝑝 𝛼𝑝 INTJ cana-5356 120 5 βˆ’ βˆ’ NOUN cana-5356 120 6 π‘Ÿ)(1 π‘Ÿ)(1 NUM cana-5356 120 7 βˆ’ βˆ’ PROPN cana-5356 120 8 𝛽 𝛽 NOUN cana-5356 120 9 ) ) PUNCT cana-5356 120 10 ≀ ≀ NOUN cana-5356 120 11 βˆ’(π‘˜ βˆ’(π‘˜ NOUN cana-5356 120 12 + + CCONJ cana-5356 121 1 1)𝑆1 1)𝑆1 NUM cana-5356 122 1 βˆ’ βˆ’ NOUN cana-5356 123 1 ( ( PUNCT cana-5356 123 2 1 1 NUM cana-5356 123 3 βˆ’ βˆ’ NOUN cana-5356 123 4 𝛽)𝑆2 𝛽)𝑆2 NOUN cana-5356 123 5 then then ADV cana-5356 123 6 expand expand VERB cana-5356 123 7 and and CCONJ cana-5356 123 8 simplify simplify VERB cana-5356 123 9 the the DET cana-5356 123 10 constants constant NOUN cana-5356 123 11 ( ( PUNCT cana-5356 123 12 π‘˜ π‘˜ PROPN cana-5356 123 13 + + ADJ cana-5356 123 14 1)(𝛼𝑝 1)(𝛼𝑝 NUM cana-5356 123 15 + + NUM cana-5356 123 16 π‘Ÿ π‘Ÿ X cana-5356 123 17 βˆ’ βˆ’ PROPN cana-5356 123 18 𝑝 𝑝 PROPN cana-5356 123 19 ) ) PUNCT cana-5356 123 20 βˆ’ βˆ’ PROPN cana-5356 124 1 ( ( PUNCT cana-5356 124 2 1 1 NUM cana-5356 124 3 βˆ’ βˆ’ NUM cana-5356 124 4 𝛼𝑝 𝛼𝑝 INTJ cana-5356 124 5 βˆ’ βˆ’ NOUN cana-5356 124 6 π‘Ÿ)(1 π‘Ÿ)(1 NUM cana-5356 124 7 βˆ’ βˆ’ PROPN cana-5356 124 8 𝛽 𝛽 NOUN cana-5356 124 9 ) ) PUNCT cana-5356 124 10 distribute distribute VERB cana-5356 124 11 ( ( PUNCT cana-5356 124 12 1 1 NUM cana-5356 124 13 βˆ’ βˆ’ PROPN cana-5356 124 14 𝛽 𝛽 NOUN cana-5356 124 15 ) ) PUNCT cana-5356 124 16 in in ADP cana-5356 124 17 the the DET cana-5356 124 18 second second ADJ cana-5356 124 19 term term NOUN cana-5356 124 20 ( ( PUNCT cana-5356 124 21 π‘˜ π‘˜ PROPN cana-5356 124 22 + + ADJ cana-5356 124 23 1)(𝛼𝑝 1)(𝛼𝑝 NUM cana-5356 124 24 + + NUM cana-5356 124 25 π‘Ÿ π‘Ÿ X cana-5356 124 26 βˆ’ βˆ’ PROPN cana-5356 124 27 𝑝 𝑝 PROPN cana-5356 124 28 ) ) PUNCT cana-5356 124 29 βˆ’ βˆ’ PROPN cana-5356 125 1 ( ( PUNCT cana-5356 125 2 1 1 NUM cana-5356 125 3 βˆ’ βˆ’ NOUN cana-5356 125 4 𝛽)(1 𝛽)(1 NUM cana-5356 125 5 ) ) PUNCT cana-5356 126 1 + + CCONJ cana-5356 126 2 ( ( PUNCT cana-5356 126 3 1 1 NUM cana-5356 126 4 βˆ’ βˆ’ NOUN cana-5356 126 5 𝛽)(𝛼𝑝 𝛽)(𝛼𝑝 NOUN cana-5356 126 6 + + NUM cana-5356 126 7 π‘Ÿ π‘Ÿ X cana-5356 126 8 ) ) PUNCT cana-5356 126 9 combine combine VERB cana-5356 126 10 like like ADP cana-5356 126 11 terms term NOUN cana-5356 126 12 : : PUNCT cana-5356 126 13 π‘˜(𝛼𝑝 π‘˜(𝛼𝑝 X cana-5356 127 1 + + CCONJ cana-5356 127 2 π‘Ÿ π‘Ÿ X cana-5356 127 3 βˆ’ βˆ’ PROPN cana-5356 127 4 𝑝 𝑝 NOUN cana-5356 127 5 ) ) PUNCT cana-5356 128 1 + + CCONJ cana-5356 128 2 ( ( PUNCT cana-5356 128 3 𝛼𝑝 𝛼𝑝 PROPN cana-5356 128 4 + + NUM cana-5356 128 5 π‘Ÿ π‘Ÿ X cana-5356 128 6 βˆ’ βˆ’ PROPN cana-5356 128 7 𝑝 𝑝 PROPN cana-5356 128 8 ) ) PUNCT cana-5356 128 9 βˆ’ βˆ’ PROPN cana-5356 129 1 ( ( PUNCT cana-5356 129 2 1 1 NUM cana-5356 129 3 βˆ’ βˆ’ PROPN cana-5356 129 4 𝛽 𝛽 NOUN cana-5356 129 5 ) ) PUNCT cana-5356 130 1 + + CCONJ cana-5356 130 2 ( ( PUNCT cana-5356 130 3 1 1 NUM cana-5356 130 4 βˆ’ βˆ’ NOUN cana-5356 130 5 𝛽)(𝛼𝑝 𝛽)(𝛼𝑝 NOUN cana-5356 130 6 + + NUM cana-5356 130 7 π‘Ÿ π‘Ÿ X cana-5356 130 8 ) ) PUNCT cana-5356 130 9 simplify simplify VERB cana-5356 130 10 further far ADV cana-5356 130 11 , , PUNCT cana-5356 130 12 then then ADV cana-5356 130 13 we we PRON cana-5356 130 14 get get VERB cana-5356 130 15 ( ( PUNCT cana-5356 130 16 𝛼𝑝 𝛼𝑝 PROPN cana-5356 130 17 + + NUM cana-5356 130 18 π‘Ÿ π‘Ÿ X cana-5356 130 19 βˆ’ βˆ’ PROPN cana-5356 130 20 𝑝 𝑝 PROPN cana-5356 130 21 ) ) PUNCT cana-5356 131 1 + + CCONJ cana-5356 131 2 π‘˜(𝛼𝑝 π‘˜(𝛼𝑝 X cana-5356 131 3 + + CCONJ cana-5356 131 4 π‘Ÿ π‘Ÿ X cana-5356 131 5 βˆ’ βˆ’ PROPN cana-5356 131 6 𝑝 𝑝 NOUN cana-5356 131 7 ) ) PUNCT cana-5356 132 1 + + CCONJ cana-5356 132 2 ( ( PUNCT cana-5356 132 3 1 1 NUM cana-5356 132 4 βˆ’ βˆ’ NOUN cana-5356 132 5 𝛽)(𝛼𝑝 𝛽)(𝛼𝑝 NOUN cana-5356 132 6 + + NUM cana-5356 132 7 π‘Ÿ π‘Ÿ X cana-5356 132 8 βˆ’ βˆ’ PROPN cana-5356 132 9 1 1 NUM cana-5356 132 10 ) ) PUNCT cana-5356 132 11 for for ADP cana-5356 132 12 the the DET cana-5356 132 13 series series NOUN cana-5356 132 14 𝑆1 𝑆1 NOUN cana-5356 132 15 and and CCONJ cana-5356 132 16 𝑆2,the 𝑆2,the DET cana-5356 132 17 terms term NOUN cana-5356 132 18 of of ADP cana-5356 132 19 𝑧𝑗+𝑝. 𝑧𝑗+𝑝. PUNCT cana-5356 132 20 since since SCONJ cana-5356 132 21 𝑧𝑗+𝑝 𝑧𝑗+𝑝 NOUN cana-5356 132 22 terms term NOUN cana-5356 132 23 are be AUX cana-5356 132 24 independent independent ADJ cana-5356 132 25 of of ADP cana-5356 132 26 the the DET cana-5356 132 27 constant constant ADJ cana-5356 132 28 parts part NOUN cana-5356 132 29 , , PUNCT cana-5356 132 30 we we PRON cana-5356 132 31 can can AUX cana-5356 132 32 combine combine VERB cana-5356 132 33 the the DET cana-5356 132 34 inequalities inequality NOUN cana-5356 132 35 to to PART cana-5356 132 36 solve solve VERB cana-5356 132 37 for for ADP cana-5356 132 38 the the DET cana-5356 132 39 coefficients coefficient NOUN cana-5356 132 40 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 132 41 , , PUNCT cana-5356 132 42 given give VERB cana-5356 132 43 the the DET cana-5356 132 44 following follow VERB cana-5356 132 45 condition condition NOUN cana-5356 132 46 must must AUX cana-5356 132 47 hold hold VERB cana-5356 132 48 : : PUNCT cana-5356 132 49 ( ( PUNCT cana-5356 132 50 π‘˜ π‘˜ X cana-5356 132 51 + + NOUN cana-5356 132 52 1)|2 1)|2 NUM cana-5356 132 53 βˆ’ βˆ’ NOUN cana-5356 132 54 𝛾 𝛾 ADP cana-5356 132 55 βˆ’ βˆ’ PROPN cana-5356 132 56 𝑛(1 𝑛(1 PROPN cana-5356 132 57 βˆ’ βˆ’ PROPN cana-5356 132 58 𝛼)| 𝛼)| PROPN cana-5356 132 59 { { PUNCT cana-5356 132 60 ( ( PUNCT cana-5356 132 61 1 1 NUM cana-5356 132 62 βˆ’ βˆ’ NOUN cana-5356 132 63 πœ† πœ† X cana-5356 132 64 ) ) PUNCT cana-5356 132 65 ( ( PUNCT cana-5356 132 66 1 1 NUM cana-5356 132 67 + + CCONJ cana-5356 132 68 𝑗 𝑗 PROPN cana-5356 132 69 𝑝 𝑝 NOUN cana-5356 132 70 ) ) PUNCT cana-5356 132 71 } } PUNCT cana-5356 132 72 𝑛 𝑛 X cana-5356 132 73 ( ( PUNCT cana-5356 132 74 1 1 NUM cana-5356 132 75 + + NUM cana-5356 132 76 𝑗)π‘Žπ‘— 𝑗)π‘Žπ‘— PROPN cana-5356 132 77 ≀ ≀ NOUN cana-5356 132 78 ( ( PUNCT cana-5356 132 79 1 1 NUM cana-5356 132 80 βˆ’ βˆ’ NOUN cana-5356 132 81 𝛽)(𝛼𝑗 𝛽)(𝛼𝑗 NUM cana-5356 132 82 + + CCONJ cana-5356 132 83 1 1 NUM cana-5356 132 84 βˆ’ βˆ’ NUM cana-5356 132 85 π‘Ÿ π‘Ÿ NOUN cana-5356 132 86 ) ) PUNCT cana-5356 132 87 { { PUNCT cana-5356 132 88 ( ( PUNCT cana-5356 132 89 1 1 NUM cana-5356 132 90 βˆ’ βˆ’ NOUN cana-5356 132 91 πœ† πœ† X cana-5356 132 92 ) ) PUNCT cana-5356 132 93 ( ( PUNCT cana-5356 132 94 1 1 NUM cana-5356 132 95 + + CCONJ cana-5356 132 96 𝑗 𝑗 PROPN cana-5356 132 97 𝑝 𝑝 NOUN cana-5356 132 98 ) ) PUNCT cana-5356 132 99 } } PUNCT cana-5356 132 100 𝑛 𝑛 PRON cana-5356 132 101 π‘Žπ‘— π‘Žπ‘— VERB cana-5356 132 102 then then ADV cana-5356 132 103 dividing divide VERB cana-5356 132 104 both both DET cana-5356 132 105 sides side NOUN cana-5356 132 106 by by ADP cana-5356 132 107 { { PUNCT cana-5356 132 108 ( ( PUNCT cana-5356 132 109 1 1 NUM cana-5356 132 110 βˆ’ βˆ’ NOUN cana-5356 132 111 πœ† πœ† X cana-5356 132 112 ) ) PUNCT cana-5356 132 113 ( ( PUNCT cana-5356 132 114 1 1 NUM cana-5356 132 115 + + CCONJ cana-5356 132 116 𝑗 𝑗 PROPN cana-5356 132 117 𝑝 𝑝 NOUN cana-5356 132 118 ) ) PUNCT cana-5356 132 119 } } PUNCT cana-5356 132 120 𝑛 𝑛 PROPN cana-5356 132 121 , , PUNCT cana-5356 132 122 we we PRON cana-5356 132 123 get get VERB cana-5356 132 124 ( ( PUNCT cana-5356 132 125 π‘˜ π‘˜ X cana-5356 132 126 + + NOUN cana-5356 132 127 1)|2 1)|2 NUM cana-5356 132 128 βˆ’ βˆ’ NOUN cana-5356 132 129 𝛾 𝛾 ADP cana-5356 132 130 βˆ’ βˆ’ PROPN cana-5356 132 131 𝑛(1 𝑛(1 PROPN cana-5356 132 132 βˆ’ βˆ’ PROPN cana-5356 132 133 𝛼)|(1 𝛼)|(1 PROPN cana-5356 132 134 + + PROPN cana-5356 132 135 𝑗)π‘Žπ‘— 𝑗)π‘Žπ‘— PROPN cana-5356 132 136 ≀ ≀ NOUN cana-5356 132 137 ( ( PUNCT cana-5356 132 138 1 1 NUM cana-5356 132 139 βˆ’ βˆ’ NOUN cana-5356 132 140 𝛽)(𝛼𝑗 𝛽)(𝛼𝑗 NUM cana-5356 132 141 + + CCONJ cana-5356 132 142 1 1 NUM cana-5356 132 143 βˆ’ βˆ’ NOUN cana-5356 132 144 π‘Ÿ)π‘Žπ‘— π‘Ÿ)π‘Žπ‘— NOUN cana-5356 132 145 for for SCONJ cana-5356 132 146 nonzero nonzero PROPN cana-5356 132 147 π‘Žπ‘— π‘Žπ‘— PROPN cana-5356 132 148 , , PUNCT cana-5356 132 149 divide divide VERB cana-5356 132 150 through through ADP cana-5356 132 151 by by ADP cana-5356 132 152 π‘Žπ‘— π‘Žπ‘— PROPN cana-5356 132 153 ( ( PUNCT cana-5356 132 154 π‘˜ π‘˜ PROPN cana-5356 132 155 + + NOUN cana-5356 132 156 1)|2 1)|2 NUM cana-5356 132 157 βˆ’ βˆ’ NOUN cana-5356 132 158 𝛾 𝛾 ADP cana-5356 132 159 βˆ’ βˆ’ PROPN cana-5356 132 160 𝑛(1 𝑛(1 PROPN cana-5356 132 161 βˆ’ βˆ’ PROPN cana-5356 132 162 𝛼)|(1 𝛼)|(1 PROPN cana-5356 132 163 + + CCONJ cana-5356 132 164 𝑗 𝑗 NOUN cana-5356 132 165 ) ) PUNCT cana-5356 132 166 ≀ ≀ NOUN cana-5356 132 167 ( ( PUNCT cana-5356 132 168 1 1 NUM cana-5356 132 169 βˆ’ βˆ’ NOUN cana-5356 132 170 𝛽)(𝛼𝑗 𝛽)(𝛼𝑗 NUM cana-5356 132 171 + + CCONJ cana-5356 132 172 1 1 NUM cana-5356 132 173 βˆ’ βˆ’ NUM cana-5356 132 174 π‘Ÿ π‘Ÿ NOUN cana-5356 132 175 ) ) PUNCT cana-5356 132 176 communications communication NOUN cana-5356 132 177 on on ADP cana-5356 132 178 applied apply VERB cana-5356 132 179 nonlinear nonlinear ADJ cana-5356 132 180 analysis analysis NOUN cana-5356 132 181 issn issn NOUN cana-5356 132 182 : : PUNCT cana-5356 132 183 1074 1074 NUM cana-5356 132 184 - - PUNCT cana-5356 132 185 133x 133x NUM cana-5356 132 186 vol vol VERB cana-5356 132 187 32 32 NUM cana-5356 132 188 no no NOUN cana-5356 132 189 . . PUNCT cana-5356 133 1 10s 10 NOUN cana-5356 133 2 ( ( PUNCT cana-5356 133 3 2025 2025 NUM cana-5356 133 4 ) ) PUNCT cana-5356 133 5 1877 1877 NUM cana-5356 134 1 https://internationalpubls.com https://internationalpubls.com X cana-5356 134 2 then then ADV cana-5356 134 3 expand expand VERB cana-5356 134 4 and and CCONJ cana-5356 134 5 simplify simplify VERB cana-5356 134 6 ( ( PUNCT cana-5356 134 7 π‘˜ π‘˜ PROPN cana-5356 134 8 + + NOUN cana-5356 134 9 1)|2 1)|2 NUM cana-5356 134 10 βˆ’ βˆ’ NOUN cana-5356 134 11 𝛾 𝛾 ADP cana-5356 134 12 βˆ’ βˆ’ PROPN cana-5356 134 13 𝑛(1 𝑛(1 PROPN cana-5356 134 14 βˆ’ βˆ’ PROPN cana-5356 134 15 𝛼)| 𝛼)| PROPN cana-5356 134 16 ≀ ≀ NUM cana-5356 135 1 ( ( PUNCT cana-5356 135 2 1 1 NUM cana-5356 135 3 βˆ’ βˆ’ PROPN cana-5356 135 4 𝛽 𝛽 NOUN cana-5356 135 5 ) ) PUNCT cana-5356 135 6 ( ( PUNCT cana-5356 135 7 𝛼𝑗 𝛼𝑗 NOUN cana-5356 135 8 + + CCONJ cana-5356 135 9 1 1 NUM cana-5356 135 10 βˆ’ βˆ’ NUM cana-5356 135 11 π‘Ÿ π‘Ÿ NOUN cana-5356 135 12 ) ) PUNCT cana-5356 135 13 ( ( PUNCT cana-5356 135 14 1 1 NUM cana-5356 135 15 + + CCONJ cana-5356 135 16 𝑗 𝑗 X cana-5356 135 17 ) ) PUNCT cana-5356 135 18 this this PRON cana-5356 135 19 gives give VERB cana-5356 135 20 a a DET cana-5356 135 21 bound bind VERB cana-5356 135 22 for for ADP cana-5356 135 23 the the DET cana-5356 135 24 coefficients coefficient NOUN cana-5356 135 25 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 135 26 then then ADV cana-5356 135 27 the the DET cana-5356 135 28 solution solution NOUN cana-5356 135 29 depends depend VERB cana-5356 135 30 on on ADP cana-5356 135 31 analysing analyse VERB cana-5356 135 32 the the DET cana-5356 135 33 coefficients coefficient NOUN cana-5356 135 34 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 135 35 satisfying satisfy VERB cana-5356 135 36 : : PUNCT cana-5356 135 37 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 135 38 ≀ ≀ NUM cana-5356 135 39 ( ( PUNCT cana-5356 135 40 1 1 NUM cana-5356 135 41 βˆ’ βˆ’ NOUN cana-5356 135 42 𝛽)(𝛼𝑗 𝛽)(𝛼𝑗 NUM cana-5356 136 1 + + CCONJ cana-5356 136 2 1 1 NUM cana-5356 136 3 βˆ’ βˆ’ NUM cana-5356 136 4 π‘Ÿ π‘Ÿ NOUN cana-5356 136 5 ) ) PUNCT cana-5356 136 6 ( ( PUNCT cana-5356 136 7 π‘˜ π‘˜ PROPN cana-5356 136 8 + + NOUN cana-5356 136 9 1)|2 1)|2 NUM cana-5356 136 10 βˆ’ βˆ’ NOUN cana-5356 136 11 𝛾 𝛾 ADP cana-5356 136 12 βˆ’ βˆ’ PROPN cana-5356 136 13 𝑛(1 𝑛(1 PROPN cana-5356 136 14 βˆ’ βˆ’ PROPN cana-5356 136 15 𝛼)|(1 𝛼)|(1 PROPN cana-5356 136 16 + + CCONJ cana-5356 136 17 𝑗 𝑗 X cana-5356 136 18 ) ) PUNCT cana-5356 136 19 the the DET cana-5356 136 20 result result NOUN cana-5356 136 21 is be AUX cana-5356 136 22 bounded bound VERB cana-5356 136 23 by by ADP cana-5356 136 24 the the DET cana-5356 136 25 choice choice NOUN cana-5356 136 26 of of ADP cana-5356 136 27 parameters parameter NOUN cana-5356 136 28 𝛼 𝛼 VERB cana-5356 136 29 , , PUNCT cana-5356 136 30 𝛽 𝛽 NOUN cana-5356 136 31 , , PUNCT cana-5356 136 32 𝛾 𝛾 PROPN cana-5356 136 33 , , PUNCT cana-5356 136 34 π‘˜ π‘˜ NOUN cana-5356 136 35 , , PUNCT cana-5356 136 36 πœ† πœ† NOUN cana-5356 136 37 , , PUNCT cana-5356 136 38 𝑝 𝑝 NOUN cana-5356 136 39 and and CCONJ cana-5356 136 40 𝑛. 𝑛. NOUN cana-5356 136 41 corollary corollary NOUN cana-5356 136 42 : : PUNCT cana-5356 136 43 ( ( PUNCT cana-5356 136 44 boundedness boundedness NOUN cana-5356 136 45 of of ADP cana-5356 136 46 coefficients coefficient NOUN cana-5356 136 47 for for ADP cana-5356 136 48 special special ADJ cana-5356 136 49 cases case NOUN cana-5356 136 50 ) ) PUNCT cana-5356 136 51 if if SCONJ cana-5356 136 52 we we PRON cana-5356 136 53 take take VERB cana-5356 136 54 𝛼 𝛼 NOUN cana-5356 136 55 = = SYM cana-5356 136 56 0 0 NUM cana-5356 136 57 , , PUNCT cana-5356 136 58 𝛽 𝛽 NOUN cana-5356 136 59 = = SYM cana-5356 136 60 0 0 NUM cana-5356 136 61 , , PUNCT cana-5356 136 62 𝛾 𝛾 NOUN cana-5356 136 63 = = SYM cana-5356 136 64 0 0 NUM cana-5356 136 65 and and CCONJ cana-5356 136 66 π‘˜ π‘˜ X cana-5356 137 1 = = NOUN cana-5356 137 2 0 0 PUNCT cana-5356 138 1 then then ADV cana-5356 138 2 the the DET cana-5356 138 3 bound bind VERB cana-5356 138 4 on on ADP cana-5356 138 5 the the DET cana-5356 138 6 coefficients coefficient NOUN cana-5356 138 7 simplifies simplifie NOUN cana-5356 138 8 to to PART cana-5356 138 9 : : PUNCT cana-5356 138 10 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 138 11 ≀ ≀ ADV cana-5356 138 12 1 1 NUM cana-5356 138 13 |2 |2 NUM cana-5356 138 14 βˆ’ βˆ’ PROPN cana-5356 138 15 𝑛|(1 𝑛|(1 PROPN cana-5356 138 16 + + PROPN cana-5356 138 17 𝑗 𝑗 NOUN cana-5356 138 18 ) ) PUNCT cana-5356 138 19 for for ADP cana-5356 138 20 all all DET cana-5356 138 21 𝑗 𝑗 PRON cana-5356 138 22 β‰₯ β‰₯ NOUN cana-5356 138 23 2 2 NUM cana-5356 138 24 . . PUNCT cana-5356 139 1 this this PRON cana-5356 139 2 shows show VERB cana-5356 139 3 that that SCONJ cana-5356 139 4 in in ADP cana-5356 139 5 this this DET cana-5356 139 6 special special ADJ cana-5356 139 7 case case NOUN cana-5356 139 8 , , PUNCT cana-5356 139 9 the the DET cana-5356 139 10 coefficients coefficient NOUN cana-5356 139 11 decay decay VERB cana-5356 139 12 inversely inversely ADV cana-5356 139 13 with with ADP cana-5356 139 14 respect respect NOUN cana-5356 139 15 to to ADP cana-5356 139 16 ( ( PUNCT cana-5356 139 17 1 1 NUM cana-5356 139 18 + + CCONJ cana-5356 139 19 𝑗 𝑗 NOUN cana-5356 139 20 ) ) PUNCT cana-5356 139 21 , , PUNCT cana-5356 139 22 ensuring ensure VERB cana-5356 139 23 the the DET cana-5356 139 24 function function NOUN cana-5356 139 25 remains remain VERB cana-5356 139 26 in in ADP cana-5356 139 27 a a DET cana-5356 139 28 bounded bounded ADJ cana-5356 139 29 subclass subclass NOUN cana-5356 139 30 of of ADP cana-5356 139 31 analytic analytic ADJ cana-5356 139 32 functions function NOUN cana-5356 139 33 . . PUNCT cana-5356 140 1 theorem theorem VERB cana-5356 140 2 : : PUNCT cana-5356 140 3 2 2 NUM cana-5356 140 4 ( ( PUNCT cana-5356 140 5 convexity convexity NOUN cana-5356 140 6 condition condition NOUN cana-5356 140 7 ) ) PUNCT cana-5356 140 8 let let VERB cana-5356 140 9 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 140 10 ) ) PUNCT cana-5356 140 11 ∈ ∈ PROPN cana-5356 140 12 β„³(𝛼 β„³(𝛼 PRON cana-5356 140 13 , , PUNCT cana-5356 140 14 𝛽 𝛽 PROPN cana-5356 140 15 , , PUNCT cana-5356 140 16 𝛾 𝛾 PROPN cana-5356 140 17 , , PUNCT cana-5356 140 18 π‘˜ π‘˜ NOUN cana-5356 140 19 , , PUNCT cana-5356 140 20 πœ† πœ† NOUN cana-5356 140 21 , , PUNCT cana-5356 140 22 𝑛 𝑛 PROPN cana-5356 140 23 , , PUNCT cana-5356 140 24 𝑝 𝑝 NOUN cana-5356 140 25 ) ) PUNCT cana-5356 140 26 . . PUNCT cana-5356 141 1 if if SCONJ cana-5356 141 2 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 141 3 𝑛 𝑛 ADP cana-5356 141 4 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 141 5 ) ) PUNCT cana-5356 141 6 satisfies satisfy VERB cana-5356 141 7 the the DET cana-5356 141 8 following follow VERB cana-5356 141 9 condition condition NOUN cana-5356 141 10 : : PUNCT cana-5356 141 11 𝑅𝑒 𝑅𝑒 VERB cana-5356 141 12 ( ( PUNCT cana-5356 141 13 1 1 NUM cana-5356 141 14 + + CCONJ cana-5356 141 15 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 141 16 𝑛 𝑛 ADP cana-5356 141 17 𝑓(𝑧))β€²β€² 𝑓(𝑧))β€²β€² PRON cana-5356 141 18 ( ( PUNCT cana-5356 141 19 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 141 20 𝑛 𝑛 DET cana-5356 141 21 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 141 22 ) ) PUNCT cana-5356 141 23 > > X cana-5356 141 24 0 0 NUM cana-5356 141 25 , , PUNCT cana-5356 141 26 then then ADV cana-5356 141 27 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 141 28 ) ) PUNCT cana-5356 141 29 is be AUX cana-5356 141 30 convex convex ADJ cana-5356 141 31 in in ADP cana-5356 141 32 β–³ β–³ NOUN cana-5356 141 33 βˆ— βˆ— NOUN cana-5356 141 34 . . PUNCT cana-5356 142 1 proof proof NOUN cana-5356 142 2 : : PUNCT cana-5356 142 3 here here ADV cana-5356 142 4 the the DET cana-5356 142 5 operator operator NOUN cana-5356 142 6 𝐷𝑝,πœ† 𝐷𝑝,πœ† VERB cana-5356 142 7 𝑛 𝑛 PRON cana-5356 142 8 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 142 9 ) ) PUNCT cana-5356 142 10 is be AUX cana-5356 142 11 defined define VERB cana-5356 142 12 as as ADP cana-5356 142 13 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 142 14 𝑛 𝑛 PRON cana-5356 142 15 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 142 16 ) ) PUNCT cana-5356 142 17 = = PRON cana-5356 142 18 π‘§βˆ’π‘ π‘§βˆ’π‘ VERB cana-5356 142 19 + + CCONJ cana-5356 142 20 βˆ‘ βˆ‘ PUNCT cana-5356 142 21 { { PUNCT cana-5356 142 22 ( ( PUNCT cana-5356 142 23 1 1 NUM cana-5356 142 24 βˆ’ βˆ’ NOUN cana-5356 142 25 πœ† πœ† X cana-5356 142 26 ) ) PUNCT cana-5356 142 27 ( ( PUNCT cana-5356 142 28 1 1 NUM cana-5356 142 29 + + CCONJ cana-5356 142 30 𝑗 𝑗 PROPN cana-5356 142 31 𝑝 𝑝 NOUN cana-5356 142 32 ) ) PUNCT cana-5356 142 33 π‘Žπ‘—π‘§π‘— π‘Žπ‘—π‘§π‘— NOUN cana-5356 142 34 } } PUNCT cana-5356 142 35 𝑛 𝑛 PROPN cana-5356 142 36 , , PUNCT cana-5356 142 37 𝑧 𝑧 PROPN cana-5356 142 38 ∈ ∈ PROPN cana-5356 142 39 β–³ β–³ NOUN cana-5356 142 40 βˆ— βˆ— NOUN cana-5356 142 41 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN cana-5356 142 42 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 142 43 > > X cana-5356 142 44 0 0 PUNCT cana-5356 143 1 ∞ ∞ NUM cana-5356 143 2 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 143 3 for for ADP cana-5356 143 4 simplicity simplicity NOUN cana-5356 143 5 , , PUNCT cana-5356 143 6 let let VERB cana-5356 143 7 𝑔(𝑧 𝑔(𝑧 PRON cana-5356 143 8 ) ) PUNCT cana-5356 143 9 = = PUNCT cana-5356 144 1 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 144 2 𝑛 𝑛 PRON cana-5356 144 3 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 144 4 ) ) PUNCT cana-5356 144 5 compute compute NOUN cana-5356 144 6 the the DET cana-5356 144 7 1st 1st ADJ cana-5356 144 8 and and CCONJ cana-5356 144 9 2nd 2nd ADJ cana-5356 144 10 derivative derivative NOUN cana-5356 144 11 of of ADP cana-5356 144 12 𝑔(𝑧 𝑔(𝑧 NOUN cana-5356 144 13 ) ) PUNCT cana-5356 144 14 , , PUNCT cana-5356 144 15 as as SCONJ cana-5356 144 16 follows follow VERB cana-5356 144 17 then then ADV cana-5356 144 18 we we PRON cana-5356 144 19 get get VERB cana-5356 144 20 𝑔′(𝑧 𝑔′(𝑧 PRON cana-5356 144 21 ) ) PUNCT cana-5356 144 22 = = VERB cana-5356 145 1 βˆ’π‘π‘§βˆ’π‘βˆ’1 βˆ’π‘π‘§βˆ’π‘βˆ’1 ADJ cana-5356 146 1 + + CCONJ cana-5356 146 2 βˆ‘ βˆ‘ PROPN cana-5356 146 3 𝑛 𝑛 DET cana-5356 146 4 ∞ ∞ NUM cana-5356 146 5 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 147 1 [ [ X cana-5356 147 2 ( ( PUNCT cana-5356 147 3 1 1 NUM cana-5356 147 4 βˆ’ βˆ’ NOUN cana-5356 147 5 πœ† πœ† X cana-5356 147 6 ) ) PUNCT cana-5356 147 7 ( ( PUNCT cana-5356 147 8 1 1 NUM cana-5356 147 9 + + CCONJ cana-5356 147 10 𝑗 𝑗 PROPN cana-5356 147 11 𝑝 𝑝 ADJ cana-5356 147 12 ) ) PUNCT cana-5356 147 13 π‘Žπ‘—π‘§π‘—βˆ’1 π‘Žπ‘—π‘§π‘—βˆ’1 PROPN cana-5356 147 14 ] ] PUNCT cana-5356 147 15 ( ( PUNCT cana-5356 147 16 1 1 NUM cana-5356 147 17 + + CCONJ cana-5356 147 18 𝑗 𝑗 PROPN cana-5356 147 19 𝑝 𝑝 NOUN cana-5356 147 20 ) ) PUNCT cana-5356 147 21 𝑔′′(𝑧 𝑔′′(𝑧 NOUN cana-5356 147 22 ) ) PUNCT cana-5356 147 23 = = PUNCT cana-5356 148 1 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 148 2 + + CCONJ cana-5356 148 3 1)π‘§βˆ’π‘βˆ’2 1)π‘§βˆ’π‘βˆ’2 NUM cana-5356 148 4 + + CCONJ cana-5356 148 5 βˆ‘ βˆ‘ PROPN cana-5356 148 6 𝑛 𝑛 PROPN cana-5356 148 7 ( ( PUNCT cana-5356 148 8 𝑛 𝑛 PROPN cana-5356 148 9 βˆ’ βˆ’ PROPN cana-5356 148 10 1 1 NUM cana-5356 148 11 ) ) PUNCT cana-5356 148 12 ∞ ∞ NUM cana-5356 148 13 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 149 1 [ [ X cana-5356 149 2 ( ( PUNCT cana-5356 149 3 1 1 NUM cana-5356 149 4 βˆ’ βˆ’ NOUN cana-5356 149 5 πœ† πœ† X cana-5356 149 6 ) ) PUNCT cana-5356 149 7 ( ( PUNCT cana-5356 149 8 1 1 NUM cana-5356 149 9 + + CCONJ cana-5356 149 10 𝑗 𝑗 PROPN cana-5356 149 11 𝑝 𝑝 NOUN cana-5356 149 12 ) ) PUNCT cana-5356 149 13 π‘Žπ‘—π‘§π‘—βˆ’2 π‘Žπ‘—π‘§π‘—βˆ’2 PROPN cana-5356 149 14 ] ] X cana-5356 149 15 ( ( PUNCT cana-5356 149 16 1 1 NUM cana-5356 149 17 + + CCONJ cana-5356 149 18 𝑗 𝑗 PROPN cana-5356 149 19 𝑝 𝑝 NOUN cana-5356 149 20 ) ) PUNCT cana-5356 149 21 2 2 NUM cana-5356 149 22 substitute substitute NOUN cana-5356 149 23 these these DET cana-5356 149 24 derivatives derivative NOUN cana-5356 149 25 into into ADP cana-5356 149 26 the the DET cana-5356 149 27 condition condition NOUN cana-5356 149 28 from from ADP cana-5356 149 29 𝑅𝑒 𝑅𝑒 PROPN cana-5356 149 30 ( ( PUNCT cana-5356 149 31 1 1 NUM cana-5356 149 32 + + NUM cana-5356 149 33 𝑧𝑔′′(𝑧 𝑧𝑔′′(𝑧 PROPN cana-5356 149 34 ) ) PUNCT cana-5356 149 35 𝑔′(𝑧 𝑔′(𝑧 NOUN cana-5356 149 36 ) ) PUNCT cana-5356 149 37 ) ) PUNCT cana-5356 150 1 > > X cana-5356 150 2 0 0 NUM cana-5356 151 1 communications communication NOUN cana-5356 151 2 on on ADP cana-5356 151 3 applied apply VERB cana-5356 151 4 nonlinear nonlinear ADJ cana-5356 151 5 analysis analysis NOUN cana-5356 151 6 issn issn NOUN cana-5356 151 7 : : PUNCT cana-5356 151 8 1074 1074 NUM cana-5356 151 9 - - PUNCT cana-5356 151 10 133x 133x NUM cana-5356 151 11 vol vol VERB cana-5356 151 12 32 32 NUM cana-5356 151 13 no no NOUN cana-5356 151 14 . . PUNCT cana-5356 152 1 10s 10 NOUN cana-5356 152 2 ( ( PUNCT cana-5356 152 3 2025 2025 NUM cana-5356 152 4 ) ) PUNCT cana-5356 152 5 1878 1878 NUM cana-5356 152 6 https://internationalpubls.com https://internationalpubls.com X cana-5356 152 7 β‡’ β‡’ PROPN cana-5356 153 1 𝑅𝑒 𝑅𝑒 PROPN cana-5356 153 2 ( ( PUNCT cana-5356 153 3 1 1 NUM cana-5356 153 4 + + CCONJ cana-5356 153 5 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 153 6 𝑛 𝑛 ADP cana-5356 153 7 𝑓(𝑧))β€²β€² 𝑓(𝑧))β€²β€² PRON cana-5356 153 8 ( ( PUNCT cana-5356 153 9 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 153 10 𝑛 𝑛 DET cana-5356 153 11 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 153 12 ) ) PUNCT cana-5356 153 13 > > SYM cana-5356 153 14 0 0 PUNCT cana-5356 154 1 β‡’ β‡’ NOUN cana-5356 154 2 𝑅𝑒 𝑅𝑒 PROPN cana-5356 154 3 ( ( PUNCT cana-5356 154 4 1 1 NUM cana-5356 154 5 + + CCONJ cana-5356 154 6 𝑧 𝑧 X cana-5356 154 7 { { PUNCT cana-5356 154 8 𝑝(𝑝 𝑝(𝑝 NOUN cana-5356 154 9 + + CCONJ cana-5356 154 10 1)π‘§βˆ’π‘βˆ’2 1)π‘§βˆ’π‘βˆ’2 NUM cana-5356 154 11 + + CCONJ cana-5356 154 12 βˆ‘ βˆ‘ PROPN cana-5356 154 13 𝑛 𝑛 PROPN cana-5356 154 14 ( ( PUNCT cana-5356 154 15 𝑛 𝑛 PROPN cana-5356 154 16 βˆ’ βˆ’ PROPN cana-5356 154 17 1)∞ 1)∞ NUM cana-5356 155 1 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 156 1 [ [ X cana-5356 156 2 ( ( PUNCT cana-5356 156 3 1 1 NUM cana-5356 156 4 βˆ’ βˆ’ NOUN cana-5356 156 5 πœ† πœ† X cana-5356 156 6 ) ) PUNCT cana-5356 156 7 ( ( PUNCT cana-5356 156 8 1 1 NUM cana-5356 156 9 + + NUM cana-5356 156 10 𝑗 𝑗 PROPN cana-5356 156 11 𝑝 𝑝 NOUN cana-5356 156 12 ) ) PUNCT cana-5356 156 13 π‘Žπ‘—π‘§π‘—βˆ’2 π‘Žπ‘—π‘§π‘—βˆ’2 PROPN cana-5356 156 14 ] ] X cana-5356 156 15 ( ( PUNCT cana-5356 156 16 1 1 NUM cana-5356 156 17 + + NUM cana-5356 156 18 𝑗 𝑗 PROPN cana-5356 156 19 𝑝 𝑝 NOUN cana-5356 156 20 ) ) PUNCT cana-5356 156 21 2 2 NUM cana-5356 156 22 } } PUNCT cana-5356 156 23 βˆ’π‘π‘§βˆ’π‘βˆ’1 βˆ’π‘π‘§βˆ’π‘βˆ’1 NOUN cana-5356 157 1 + + CCONJ cana-5356 157 2 βˆ‘ βˆ‘ PROPN cana-5356 157 3 π‘›βˆž π‘›βˆž PROPN cana-5356 157 4 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 157 5 [ [ X cana-5356 157 6 ( ( PUNCT cana-5356 157 7 1 1 NUM cana-5356 157 8 βˆ’ βˆ’ NOUN cana-5356 157 9 πœ† πœ† X cana-5356 157 10 ) ) PUNCT cana-5356 157 11 ( ( PUNCT cana-5356 157 12 1 1 NUM cana-5356 157 13 + + NUM cana-5356 157 14 𝑗 𝑗 PROPN cana-5356 157 15 𝑝 𝑝 NOUN cana-5356 157 16 ) ) PUNCT cana-5356 157 17 π‘Žπ‘—π‘§π‘—βˆ’1 π‘Žπ‘—π‘§π‘—βˆ’1 PROPN cana-5356 157 18 ] ] PUNCT cana-5356 157 19 ( ( PUNCT cana-5356 157 20 1 1 NUM cana-5356 157 21 + + NUM cana-5356 157 22 𝑗 𝑗 PROPN cana-5356 157 23 𝑝 𝑝 NOUN cana-5356 157 24 ) ) PUNCT cana-5356 157 25 ) ) PUNCT cana-5356 157 26 > > X cana-5356 157 27 0 0 PUNCT cana-5356 158 1 in in ADP cana-5356 158 2 the the DET cana-5356 158 3 punctured punctured ADJ cana-5356 158 4 unit unit NOUN cana-5356 158 5 disk disk NOUN cana-5356 158 6 β–³ β–³ PROPN cana-5356 158 7 βˆ— βˆ— NOUN cana-5356 158 8 , , PUNCT cana-5356 158 9 |𝑧| |𝑧| VERB cana-5356 158 10 < < X cana-5356 158 11 1 1 NUM cana-5356 158 12 , , PUNCT cana-5356 158 13 so so ADV cana-5356 158 14 higher high ADJ cana-5356 158 15 - - PUNCT cana-5356 158 16 order order NOUN cana-5356 158 17 terms term NOUN cana-5356 158 18 𝑧𝑗 𝑧𝑗 VERB cana-5356 158 19 , , PUNCT cana-5356 158 20 becomes become VERB cana-5356 158 21 negligible negligible ADJ cana-5356 158 22 as as ADP cana-5356 158 23 |𝑧| |𝑧| PROPN cana-5356 158 24 β†’ β†’ SYM cana-5356 158 25 0 0 NUM cana-5356 158 26 for for ADP cana-5356 158 27 leading lead VERB cana-5356 158 28 order order NOUN cana-5356 158 29 terms term NOUN cana-5356 158 30 , , PUNCT cana-5356 158 31 then then ADV cana-5356 158 32 the the DET cana-5356 158 33 condition condition NOUN cana-5356 158 34 will will AUX cana-5356 158 35 be be AUX cana-5356 158 36 𝑅𝑒 𝑅𝑒 VERB cana-5356 158 37 ( ( PUNCT cana-5356 158 38 1 1 NUM cana-5356 158 39 + + CCONJ cana-5356 158 40 𝑧[𝑝(𝑝 𝑧[𝑝(𝑝 ADJ cana-5356 158 41 + + NOUN cana-5356 158 42 1)π‘§βˆ’π‘βˆ’2 1)π‘§βˆ’π‘βˆ’2 NUM cana-5356 158 43 ] ] PUNCT cana-5356 158 44 βˆ’π‘π‘§βˆ’π‘βˆ’1 βˆ’π‘π‘§βˆ’π‘βˆ’1 NUM cana-5356 158 45 ) ) PUNCT cana-5356 158 46 > > SYM cana-5356 158 47 0 0 PUNCT cana-5356 159 1 β‡’ β‡’ NOUN cana-5356 159 2 𝑅𝑒 𝑅𝑒 PROPN cana-5356 159 3 ( ( PUNCT cana-5356 159 4 1 1 NUM cana-5356 159 5 + + CCONJ cana-5356 159 6 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 159 7 + + CCONJ cana-5356 159 8 1)π‘§βˆ’π‘βˆ’1 1)π‘§βˆ’π‘βˆ’1 NUM cana-5356 159 9 βˆ’π‘π‘§βˆ’π‘βˆ’1 βˆ’π‘π‘§βˆ’π‘βˆ’1 NOUN cana-5356 159 10 ) ) PUNCT cana-5356 159 11 > > SYM cana-5356 159 12 0 0 PUNCT cana-5356 160 1 β‡’ β‡’ NOUN cana-5356 160 2 𝑅𝑒 𝑅𝑒 PROPN cana-5356 160 3 ( ( PUNCT cana-5356 160 4 1 1 NUM cana-5356 160 5 + + CCONJ cana-5356 160 6 ( ( PUNCT cana-5356 160 7 𝑝(𝑝 𝑝(𝑝 NOUN cana-5356 160 8 + + CCONJ cana-5356 160 9 1 1 X cana-5356 160 10 ) ) PUNCT cana-5356 160 11 βˆ’π‘ βˆ’π‘ NOUN cana-5356 160 12 ) ) PUNCT cana-5356 160 13 ) ) PUNCT cana-5356 161 1 > > SYM cana-5356 161 2 0 0 PUNCT cana-5356 162 1 β‡’ β‡’ PROPN cana-5356 162 2 𝑅𝑒(1 𝑅𝑒(1 NOUN cana-5356 162 3 βˆ’ βˆ’ PROPN cana-5356 163 1 ( ( PUNCT cana-5356 163 2 𝑝 𝑝 PROPN cana-5356 163 3 + + NOUN cana-5356 163 4 1 1 NUM cana-5356 163 5 ) ) PUNCT cana-5356 163 6 ) ) PUNCT cana-5356 163 7 > > SYM cana-5356 163 8 0 0 PUNCT cana-5356 164 1 β‡’ β‡’ PROPN cana-5356 164 2 𝑅𝑒(1 𝑅𝑒(1 NOUN cana-5356 164 3 βˆ’ βˆ’ PROPN cana-5356 164 4 𝑝 𝑝 NOUN cana-5356 164 5 βˆ’ βˆ’ PROPN cana-5356 164 6 1 1 NUM cana-5356 164 7 ) ) PUNCT cana-5356 164 8 > > SYM cana-5356 164 9 0 0 PUNCT cana-5356 164 10 β‡’ β‡’ NOUN cana-5356 164 11 𝑅𝑒(βˆ’π‘ 𝑅𝑒(βˆ’π‘ NOUN cana-5356 164 12 ) ) PUNCT cana-5356 164 13 > > X cana-5356 164 14 0 0 PUNCT cana-5356 165 1 this this PRON cana-5356 165 2 means mean VERB cana-5356 165 3 the the DET cana-5356 165 4 real real ADJ cana-5356 165 5 part part NOUN cana-5356 165 6 of of ADP cana-5356 165 7 βˆ’π‘ βˆ’π‘ NOUN cana-5356 165 8 , , PUNCT cana-5356 165 9 must must AUX cana-5356 165 10 be be AUX cana-5356 165 11 greater great ADJ cana-5356 165 12 than than ADP cana-5356 165 13 0 0 NUM cana-5356 165 14 . . PUNCT cana-5356 166 1 let let VERB cana-5356 166 2 𝑝 𝑝 NOUN cana-5356 166 3 = = PUNCT cana-5356 166 4 π‘Ž π‘Ž PROPN cana-5356 166 5 + + X cana-5356 166 6 𝑏𝑖 𝑏𝑖 ADP cana-5356 166 7 , , PUNCT cana-5356 166 8 where where SCONJ cana-5356 166 9 π‘Ž π‘Ž NOUN cana-5356 166 10 and and CCONJ cana-5356 166 11 𝑏 𝑏 PROPN cana-5356 166 12 be be AUX cana-5356 166 13 the the DET cana-5356 166 14 real real ADJ cana-5356 166 15 part part NOUN cana-5356 166 16 and and CCONJ cana-5356 166 17 𝑖 𝑖 AUX cana-5356 166 18 be be AUX cana-5356 166 19 the the DET cana-5356 166 20 imaginary imaginary ADJ cana-5356 166 21 part part NOUN cana-5356 166 22 of of ADP cana-5356 166 23 this this DET cana-5356 166 24 complex complex ADJ cana-5356 166 25 number number NOUN cana-5356 166 26 , , PUNCT cana-5356 166 27 then then ADV cana-5356 166 28 βˆ’π‘ βˆ’π‘ VERB cana-5356 167 1 = = NOUN cana-5356 168 1 βˆ’π‘Ž βˆ’π‘Ž NOUN cana-5356 168 2 βˆ’ βˆ’ PROPN cana-5356 168 3 𝑏𝑖 𝑏𝑖 ADP cana-5356 168 4 thus thus ADV cana-5356 168 5 , , PUNCT cana-5356 168 6 the the DET cana-5356 168 7 real real ADJ cana-5356 168 8 part part NOUN cana-5356 168 9 of of ADP cana-5356 168 10 βˆ’π‘ βˆ’π‘ NOUN cana-5356 168 11 is be AUX cana-5356 168 12 βˆ’π‘Ž βˆ’π‘Ž ADJ cana-5356 168 13 , , PUNCT cana-5356 168 14 the the DET cana-5356 168 15 inequality inequality NOUN cana-5356 168 16 becomes become VERB cana-5356 168 17 βˆ’π‘Ž βˆ’π‘Ž ADJ cana-5356 168 18 > > X cana-5356 168 19 0 0 NUM cana-5356 169 1 π‘Ž π‘Ž X cana-5356 169 2 < < X cana-5356 169 3 0 0 NUM cana-5356 169 4 therefore therefore ADV cana-5356 169 5 , , PUNCT cana-5356 169 6 𝑅𝑒 𝑅𝑒 PROPN cana-5356 169 7 ( ( PUNCT cana-5356 169 8 1 1 NUM cana-5356 169 9 + + NUM cana-5356 169 10 𝑧𝑔′′(𝑧 𝑧𝑔′′(𝑧 PROPN cana-5356 169 11 ) ) PUNCT cana-5356 169 12 𝑔′(𝑧 𝑔′(𝑧 NOUN cana-5356 169 13 ) ) PUNCT cana-5356 169 14 ) ) PUNCT cana-5356 170 1 > > X cana-5356 170 2 0 0 PUNCT cana-5356 171 1 this this DET cana-5356 171 2 inequality inequality NOUN cana-5356 171 3 holds hold VERB cana-5356 171 4 when when SCONJ cana-5356 171 5 the the DET cana-5356 171 6 real real ADJ cana-5356 171 7 part part NOUN cana-5356 171 8 of of ADP cana-5356 171 9 𝑝 𝑝 PROPN cana-5356 171 10 is be AUX cana-5356 171 11 negative negative ADJ cana-5356 171 12 . . PUNCT cana-5356 172 1 theorem theorem VERB cana-5356 172 2 : : PUNCT cana-5356 172 3 3 3 NUM cana-5356 172 4 ( ( PUNCT cana-5356 172 5 univalence univalence NOUN cana-5356 172 6 criterion criterion NOUN cana-5356 172 7 ) ) PUNCT cana-5356 172 8 let let VERB cana-5356 172 9 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 172 10 ) ) PUNCT cana-5356 172 11 ∈ ∈ PROPN cana-5356 172 12 β„³(𝛼 β„³(𝛼 PRON cana-5356 172 13 , , PUNCT cana-5356 172 14 𝛽 𝛽 PROPN cana-5356 172 15 , , PUNCT cana-5356 172 16 𝛾 𝛾 PROPN cana-5356 172 17 , , PUNCT cana-5356 172 18 π‘˜ π‘˜ NOUN cana-5356 172 19 , , PUNCT cana-5356 172 20 πœ† πœ† NOUN cana-5356 172 21 , , PUNCT cana-5356 172 22 𝑛 𝑛 PROPN cana-5356 172 23 , , PUNCT cana-5356 172 24 𝑝 𝑝 NOUN cana-5356 172 25 ) ) PUNCT cana-5356 172 26 , , PUNCT cana-5356 172 27 if if SCONJ cana-5356 172 28 the the DET cana-5356 172 29 following follow VERB cana-5356 172 30 inequality inequality NOUN cana-5356 172 31 holds hold VERB cana-5356 172 32 : : PUNCT cana-5356 172 33 | | ADV cana-5356 172 34 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 172 35 𝑛 𝑛 ADP cana-5356 172 36 𝑓(𝑧))β€²β€² 𝑓(𝑧))β€²β€² PRON cana-5356 172 37 ( ( PUNCT cana-5356 172 38 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 172 39 𝑛 𝑛 DET cana-5356 172 40 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 172 41 | | CCONJ cana-5356 172 42 < < X cana-5356 172 43 1 1 NUM cana-5356 172 44 , , PUNCT cana-5356 172 45 then then ADV cana-5356 172 46 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 172 47 ) ) PUNCT cana-5356 172 48 is be AUX cana-5356 172 49 univalent univalent ADJ cana-5356 172 50 in in ADP cana-5356 172 51 β–³ β–³ NOUN cana-5356 172 52 βˆ— βˆ— NOUN cana-5356 172 53 . . PUNCT cana-5356 173 1 proof proof NOUN cana-5356 173 2 : : PUNCT cana-5356 173 3 given give VERB cana-5356 173 4 the the DET cana-5356 173 5 form form NOUN cana-5356 173 6 of of ADP cana-5356 173 7 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 173 8 𝑛 𝑛 PRON cana-5356 173 9 𝑓(𝑧 𝑓(𝑧 NUM cana-5356 173 10 ) ) PUNCT cana-5356 173 11 , , PUNCT cana-5356 173 12 here here ADV cana-5356 173 13 the the DET cana-5356 173 14 1st 1st ADJ cana-5356 173 15 derivative derivative NOUN cana-5356 173 16 of of ADP cana-5356 173 17 this this DET cana-5356 173 18 operator operator NOUN cana-5356 173 19 will will AUX cana-5356 173 20 be be AUX cana-5356 173 21 communications communication NOUN cana-5356 173 22 on on ADP cana-5356 173 23 applied apply VERB cana-5356 173 24 nonlinear nonlinear ADJ cana-5356 173 25 analysis analysis NOUN cana-5356 173 26 issn issn NOUN cana-5356 173 27 : : PUNCT cana-5356 173 28 1074 1074 NUM cana-5356 173 29 - - PUNCT cana-5356 173 30 133x 133x NUM cana-5356 173 31 vol vol VERB cana-5356 173 32 32 32 NUM cana-5356 173 33 no no NOUN cana-5356 173 34 . . PUNCT cana-5356 174 1 10s 10 NOUN cana-5356 174 2 ( ( PUNCT cana-5356 174 3 2025 2025 NUM cana-5356 174 4 ) ) PUNCT cana-5356 174 5 1879 1879 NUM cana-5356 174 6 https://internationalpubls.com https://internationalpubls.com X cana-5356 174 7 ( ( PUNCT cana-5356 174 8 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 174 9 𝑛 𝑛 DET cana-5356 174 10 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 174 11 = = SYM cana-5356 175 1 βˆ’π‘π‘§βˆ’π‘βˆ’1 βˆ’π‘π‘§βˆ’π‘βˆ’1 ADJ cana-5356 176 1 + + CCONJ cana-5356 176 2 βˆ‘ βˆ‘ PROPN cana-5356 176 3 𝑛 𝑛 DET cana-5356 176 4 ∞ ∞ NUM cana-5356 176 5 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 177 1 [ [ X cana-5356 177 2 ( ( PUNCT cana-5356 177 3 1 1 NUM cana-5356 177 4 βˆ’ βˆ’ NOUN cana-5356 177 5 πœ† πœ† X cana-5356 177 6 ) ) PUNCT cana-5356 177 7 ( ( PUNCT cana-5356 177 8 1 1 NUM cana-5356 177 9 + + CCONJ cana-5356 177 10 𝑗 𝑗 PROPN cana-5356 177 11 𝑝 𝑝 ADJ cana-5356 177 12 ) ) PUNCT cana-5356 177 13 π‘Žπ‘—π‘§π‘—βˆ’1 π‘Žπ‘—π‘§π‘—βˆ’1 PROPN cana-5356 177 14 ] ] PUNCT cana-5356 177 15 ( ( PUNCT cana-5356 177 16 1 1 NUM cana-5356 177 17 + + CCONJ cana-5356 177 18 𝑗 𝑗 PROPN cana-5356 177 19 𝑝 𝑝 NOUN cana-5356 177 20 ) ) PUNCT cana-5356 177 21 2nd 2nd ADJ cana-5356 177 22 derivative derivative NOUN cana-5356 177 23 will will AUX cana-5356 177 24 be be AUX cana-5356 177 25 ( ( PUNCT cana-5356 177 26 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 177 27 𝑛 𝑛 PART cana-5356 177 28 𝑓(𝑧))β€²β€² 𝑓(𝑧))β€²β€² NOUN cana-5356 177 29 = = SYM cana-5356 177 30 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 177 31 + + CCONJ cana-5356 177 32 1)π‘§βˆ’π‘βˆ’2 1)π‘§βˆ’π‘βˆ’2 NUM cana-5356 177 33 + + CCONJ cana-5356 177 34 βˆ‘ βˆ‘ PROPN cana-5356 177 35 𝑛 𝑛 PROPN cana-5356 177 36 ( ( PUNCT cana-5356 177 37 𝑛 𝑛 PROPN cana-5356 177 38 βˆ’ βˆ’ PROPN cana-5356 177 39 1 1 NUM cana-5356 177 40 ) ) PUNCT cana-5356 177 41 ∞ ∞ NUM cana-5356 178 1 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 179 1 [ [ X cana-5356 179 2 ( ( PUNCT cana-5356 179 3 1 1 NUM cana-5356 179 4 βˆ’ βˆ’ NOUN cana-5356 179 5 πœ† πœ† X cana-5356 179 6 ) ) PUNCT cana-5356 179 7 ( ( PUNCT cana-5356 179 8 1 1 NUM cana-5356 179 9 + + CCONJ cana-5356 179 10 𝑗 𝑗 PROPN cana-5356 179 11 𝑝 𝑝 NOUN cana-5356 179 12 ) ) PUNCT cana-5356 179 13 π‘Žπ‘—π‘§π‘—βˆ’2 π‘Žπ‘—π‘§π‘—βˆ’2 PROPN cana-5356 179 14 ] ] X cana-5356 179 15 ( ( PUNCT cana-5356 179 16 1 1 NUM cana-5356 179 17 + + CCONJ cana-5356 179 18 𝑗 𝑗 PROPN cana-5356 179 19 𝑝 𝑝 NOUN cana-5356 179 20 ) ) PUNCT cana-5356 179 21 2 2 NUM cana-5356 179 22 substitute substitute NOUN cana-5356 179 23 these these DET cana-5356 179 24 derivatives derivative NOUN cana-5356 179 25 in in ADP cana-5356 179 26 to to ADP cana-5356 179 27 the the DET cana-5356 179 28 conditions condition NOUN cana-5356 179 29 we we PRON cana-5356 179 30 get get VERB cana-5356 179 31 , , PUNCT cana-5356 179 32 | | ADV cana-5356 179 33 𝑧(𝐷𝑝,πœ† 𝑧(𝐷𝑝,πœ† VERB cana-5356 179 34 𝑛 𝑛 ADP cana-5356 179 35 𝑓(𝑧))β€²β€² 𝑓(𝑧))β€²β€² PRON cana-5356 179 36 ( ( PUNCT cana-5356 179 37 𝐷𝑝,πœ† 𝐷𝑝,πœ† NOUN cana-5356 179 38 𝑛 𝑛 DET cana-5356 179 39 𝑓(𝑧))β€² 𝑓(𝑧))β€² NOUN cana-5356 179 40 | | CCONJ cana-5356 179 41 < < X cana-5356 179 42 1 1 NUM cana-5356 179 43 , , PUNCT cana-5356 179 44 = = PUNCT cana-5356 179 45 || || NOUN cana-5356 180 1 𝑧 𝑧 VERB cana-5356 180 2 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 180 3 + + CCONJ cana-5356 180 4 1)π‘§βˆ’π‘βˆ’2 1)π‘§βˆ’π‘βˆ’2 NUM cana-5356 180 5 + + CCONJ cana-5356 180 6 βˆ‘ βˆ‘ PROPN cana-5356 180 7 𝑛 𝑛 PROPN cana-5356 180 8 ( ( PUNCT cana-5356 180 9 𝑛 𝑛 PROPN cana-5356 180 10 βˆ’ βˆ’ PROPN cana-5356 180 11 1)∞ 1)∞ NUM cana-5356 180 12 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 181 1 [ [ X cana-5356 181 2 ( ( PUNCT cana-5356 181 3 1 1 NUM cana-5356 181 4 βˆ’ βˆ’ NOUN cana-5356 181 5 πœ† πœ† X cana-5356 181 6 ) ) PUNCT cana-5356 181 7 ( ( PUNCT cana-5356 181 8 1 1 NUM cana-5356 181 9 + + NUM cana-5356 181 10 𝑗 𝑗 PROPN cana-5356 181 11 𝑝 𝑝 NOUN cana-5356 181 12 ) ) PUNCT cana-5356 181 13 π‘Žπ‘—π‘§π‘—βˆ’1 π‘Žπ‘—π‘§π‘—βˆ’1 PROPN cana-5356 181 14 ] ] PUNCT cana-5356 181 15 ( ( PUNCT cana-5356 181 16 1 1 NUM cana-5356 181 17 + + NUM cana-5356 181 18 𝑗 𝑗 PROPN cana-5356 181 19 𝑝 𝑝 NOUN cana-5356 181 20 ) ) PUNCT cana-5356 181 21 2 2 NUM cana-5356 181 22 βˆ’π‘π‘§βˆ’π‘βˆ’1 βˆ’π‘π‘§βˆ’π‘βˆ’1 NOUN cana-5356 182 1 + + CCONJ cana-5356 182 2 βˆ‘ βˆ‘ PROPN cana-5356 182 3 π‘›βˆž π‘›βˆž PROPN cana-5356 182 4 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 182 5 [ [ X cana-5356 182 6 ( ( PUNCT cana-5356 182 7 1 1 NUM cana-5356 182 8 βˆ’ βˆ’ NOUN cana-5356 182 9 πœ† πœ† X cana-5356 182 10 ) ) PUNCT cana-5356 182 11 ( ( PUNCT cana-5356 182 12 1 1 NUM cana-5356 182 13 + + NUM cana-5356 182 14 𝑗 𝑗 PROPN cana-5356 182 15 𝑝 𝑝 NOUN cana-5356 182 16 ) ) PUNCT cana-5356 182 17 π‘Žπ‘—π‘§π‘—βˆ’1 π‘Žπ‘—π‘§π‘—βˆ’1 PROPN cana-5356 182 18 ] ] PUNCT cana-5356 182 19 ( ( PUNCT cana-5356 182 20 1 1 NUM cana-5356 182 21 + + NUM cana-5356 182 22 𝑗 𝑗 PROPN cana-5356 182 23 𝑝 𝑝 NOUN cana-5356 182 24 ) ) PUNCT cana-5356 182 25 || || NOUN cana-5356 183 1 < < X cana-5356 183 2 1 1 NUM cana-5356 183 3 , , PUNCT cana-5356 183 4 = = PUNCT cana-5356 183 5 || || NOUN cana-5356 183 6 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 183 7 + + CCONJ cana-5356 183 8 1)π‘§βˆ’π‘βˆ’1 1)π‘§βˆ’π‘βˆ’1 NUM cana-5356 183 9 + + CCONJ cana-5356 183 10 βˆ‘ βˆ‘ PROPN cana-5356 183 11 𝑛 𝑛 PROPN cana-5356 183 12 ( ( PUNCT cana-5356 183 13 𝑛 𝑛 PROPN cana-5356 183 14 βˆ’ βˆ’ PROPN cana-5356 184 1 1)∞ 1)∞ NUM cana-5356 185 1 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 186 1 [ [ X cana-5356 186 2 ( ( PUNCT cana-5356 186 3 1 1 NUM cana-5356 186 4 βˆ’ βˆ’ NOUN cana-5356 186 5 πœ† πœ† X cana-5356 186 6 ) ) PUNCT cana-5356 186 7 ( ( PUNCT cana-5356 186 8 1 1 NUM cana-5356 186 9 + + NUM cana-5356 186 10 𝑗 𝑗 PROPN cana-5356 186 11 𝑝 𝑝 NOUN cana-5356 186 12 ) ) PUNCT cana-5356 186 13 π‘Žπ‘—π‘§π‘—βˆ’2 π‘Žπ‘—π‘§π‘—βˆ’2 PROPN cana-5356 186 14 ] ] X cana-5356 186 15 ( ( PUNCT cana-5356 186 16 1 1 NUM cana-5356 186 17 + + NUM cana-5356 186 18 𝑗 𝑗 PROPN cana-5356 186 19 𝑝 𝑝 NOUN cana-5356 186 20 ) ) PUNCT cana-5356 186 21 2 2 NUM cana-5356 186 22 βˆ’π‘π‘§βˆ’π‘βˆ’1 βˆ’π‘π‘§βˆ’π‘βˆ’1 NOUN cana-5356 187 1 + + CCONJ cana-5356 187 2 βˆ‘ βˆ‘ PROPN cana-5356 187 3 π‘›βˆž π‘›βˆž PROPN cana-5356 187 4 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 187 5 [ [ X cana-5356 187 6 ( ( PUNCT cana-5356 187 7 1 1 NUM cana-5356 187 8 βˆ’ βˆ’ NOUN cana-5356 187 9 πœ† πœ† X cana-5356 187 10 ) ) PUNCT cana-5356 187 11 ( ( PUNCT cana-5356 187 12 1 1 NUM cana-5356 187 13 + + CCONJ cana-5356 187 14 𝑗 𝑗 PROPN cana-5356 187 15 𝑝 𝑝 ADJ cana-5356 187 16 ) ) PUNCT cana-5356 187 17 π‘Žπ‘—π‘§π‘—βˆ’1 π‘Žπ‘—π‘§π‘—βˆ’1 PROPN cana-5356 187 18 ] ] PUNCT cana-5356 187 19 ( ( PUNCT cana-5356 187 20 1 1 NUM cana-5356 187 21 + + CCONJ cana-5356 187 22 𝑗 𝑗 PROPN cana-5356 187 23 𝑝 𝑝 PROPN cana-5356 187 24 ) ) PUNCT cana-5356 187 25 || || PROPN cana-5356 188 1 < < X cana-5356 188 2 1 1 NUM cana-5356 188 3 , , PUNCT cana-5356 188 4 = = SYM cana-5356 188 5 || || PROPN cana-5356 188 6 π‘§βˆ’π‘βˆ’1 π‘§βˆ’π‘βˆ’1 PROPN cana-5356 188 7 { { PUNCT cana-5356 188 8 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 188 9 + + CCONJ cana-5356 188 10 1 1 X cana-5356 188 11 ) ) PUNCT cana-5356 188 12 + + CCONJ cana-5356 188 13 βˆ‘ βˆ‘ PROPN cana-5356 188 14 𝑛 𝑛 PROPN cana-5356 188 15 ( ( PUNCT cana-5356 188 16 𝑛 𝑛 PROPN cana-5356 188 17 βˆ’ βˆ’ PROPN cana-5356 188 18 1)∞ 1)∞ NUM cana-5356 189 1 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 190 1 [ [ X cana-5356 190 2 ( ( PUNCT cana-5356 190 3 1 1 NUM cana-5356 190 4 βˆ’ βˆ’ NOUN cana-5356 190 5 πœ† πœ† X cana-5356 190 6 ) ) PUNCT cana-5356 190 7 ( ( PUNCT cana-5356 190 8 1 1 NUM cana-5356 190 9 + + NUM cana-5356 190 10 𝑗 𝑗 PROPN cana-5356 190 11 𝑝 𝑝 NOUN cana-5356 190 12 ) ) PUNCT cana-5356 190 13 π‘Žπ‘—π‘§π‘—+𝑝+1 π‘Žπ‘—π‘§π‘—+𝑝+1 PROPN cana-5356 190 14 ] ] PUNCT cana-5356 190 15 ( ( PUNCT cana-5356 190 16 1 1 NUM cana-5356 190 17 + + NUM cana-5356 190 18 𝑗 𝑗 PROPN cana-5356 190 19 𝑝 𝑝 NOUN cana-5356 190 20 ) ) PUNCT cana-5356 190 21 2 2 NUM cana-5356 190 22 } } PUNCT cana-5356 190 23 π‘§βˆ’π‘βˆ’1 π‘§βˆ’π‘βˆ’1 PROPN cana-5356 190 24 { { PUNCT cana-5356 190 25 βˆ’π‘ βˆ’π‘ NOUN cana-5356 190 26 + + CCONJ cana-5356 190 27 βˆ‘ βˆ‘ PROPN cana-5356 190 28 π‘›βˆž π‘›βˆž PROPN cana-5356 190 29 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 191 1 [ [ X cana-5356 191 2 ( ( PUNCT cana-5356 191 3 1 1 NUM cana-5356 191 4 βˆ’ βˆ’ NOUN cana-5356 191 5 πœ† πœ† X cana-5356 191 6 ) ) PUNCT cana-5356 191 7 ( ( PUNCT cana-5356 191 8 1 1 NUM cana-5356 191 9 + + NUM cana-5356 191 10 𝑗 𝑗 PROPN cana-5356 191 11 𝑝 𝑝 NOUN cana-5356 191 12 ) ) PUNCT cana-5356 191 13 π‘Žπ‘—π‘§π‘—+𝑝+1 π‘Žπ‘—π‘§π‘—+𝑝+1 PROPN cana-5356 191 14 ] ] PUNCT cana-5356 191 15 ( ( PUNCT cana-5356 191 16 1 1 NUM cana-5356 191 17 + + NUM cana-5356 191 18 𝑗 𝑗 PROPN cana-5356 191 19 𝑝 𝑝 NOUN cana-5356 191 20 ) ) PUNCT cana-5356 191 21 } } PUNCT cana-5356 191 22 || || PUNCT cana-5356 192 1 < < X cana-5356 192 2 1 1 NUM cana-5356 192 3 , , PUNCT cana-5356 192 4 after after ADP cana-5356 192 5 factoring factor VERB cana-5356 192 6 π‘§βˆ’π‘βˆ’1 π‘§βˆ’π‘βˆ’1 PROPN cana-5356 192 7 out out ADP cana-5356 192 8 the the DET cana-5356 192 9 fraction fraction NOUN cana-5356 192 10 simplifies simplifie NOUN cana-5356 192 11 to to ADP cana-5356 192 12 , , PUNCT cana-5356 192 13 = = SYM cana-5356 192 14 || || NOUN cana-5356 192 15 { { PUNCT cana-5356 192 16 𝑝(𝑝 𝑝(𝑝 NOUN cana-5356 192 17 + + CCONJ cana-5356 192 18 1 1 X cana-5356 192 19 ) ) PUNCT cana-5356 192 20 + + CCONJ cana-5356 192 21 βˆ‘ βˆ‘ PROPN cana-5356 192 22 𝑛 𝑛 PROPN cana-5356 192 23 ( ( PUNCT cana-5356 192 24 𝑛 𝑛 PROPN cana-5356 192 25 βˆ’ βˆ’ PROPN cana-5356 192 26 1)∞ 1)∞ NUM cana-5356 192 27 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 193 1 [ [ X cana-5356 193 2 ( ( PUNCT cana-5356 193 3 1 1 NUM cana-5356 193 4 βˆ’ βˆ’ NOUN cana-5356 193 5 πœ† πœ† X cana-5356 193 6 ) ) PUNCT cana-5356 193 7 ( ( PUNCT cana-5356 193 8 1 1 NUM cana-5356 193 9 + + NUM cana-5356 193 10 𝑗 𝑗 PROPN cana-5356 193 11 𝑝 𝑝 NOUN cana-5356 193 12 ) ) PUNCT cana-5356 193 13 π‘Žπ‘—π‘§π‘—+𝑝+1 π‘Žπ‘—π‘§π‘—+𝑝+1 PROPN cana-5356 193 14 ] ] PUNCT cana-5356 193 15 ( ( PUNCT cana-5356 193 16 1 1 NUM cana-5356 193 17 + + NUM cana-5356 193 18 𝑗 𝑗 PROPN cana-5356 193 19 𝑝 𝑝 NOUN cana-5356 193 20 ) ) PUNCT cana-5356 193 21 2 2 NUM cana-5356 193 22 } } PUNCT cana-5356 193 23 { { PUNCT cana-5356 193 24 βˆ’π‘ βˆ’π‘ VERB cana-5356 193 25 + + CCONJ cana-5356 193 26 βˆ‘ βˆ‘ PROPN cana-5356 193 27 π‘›βˆž π‘›βˆž PROPN cana-5356 193 28 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 194 1 [ [ X cana-5356 194 2 ( ( PUNCT cana-5356 194 3 1 1 NUM cana-5356 194 4 βˆ’ βˆ’ NOUN cana-5356 194 5 πœ† πœ† X cana-5356 194 6 ) ) PUNCT cana-5356 194 7 ( ( PUNCT cana-5356 194 8 1 1 NUM cana-5356 194 9 + + NUM cana-5356 194 10 𝑗 𝑗 PROPN cana-5356 194 11 𝑝 𝑝 NOUN cana-5356 194 12 ) ) PUNCT cana-5356 194 13 π‘Žπ‘—π‘§π‘—+𝑝+1 π‘Žπ‘—π‘§π‘—+𝑝+1 PROPN cana-5356 194 14 ] ] PUNCT cana-5356 194 15 ( ( PUNCT cana-5356 194 16 1 1 NUM cana-5356 194 17 + + NUM cana-5356 194 18 𝑗 𝑗 PROPN cana-5356 194 19 𝑝 𝑝 NOUN cana-5356 194 20 ) ) PUNCT cana-5356 194 21 } } PUNCT cana-5356 194 22 || || PUNCT cana-5356 195 1 < < X cana-5356 195 2 1 1 NUM cana-5356 195 3 , , PUNCT cana-5356 195 4 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 195 5 + + CCONJ cana-5356 195 6 1 1 X cana-5356 195 7 ) ) PUNCT cana-5356 195 8 + + CCONJ cana-5356 195 9 βˆ‘ βˆ‘ PROPN cana-5356 195 10 𝑛 𝑛 PROPN cana-5356 195 11 ( ( PUNCT cana-5356 195 12 𝑛 𝑛 PROPN cana-5356 195 13 βˆ’ βˆ’ PROPN cana-5356 195 14 1 1 NUM cana-5356 195 15 ) ) PUNCT cana-5356 195 16 ∞ ∞ NUM cana-5356 195 17 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 196 1 [ [ X cana-5356 196 2 ( ( PUNCT cana-5356 196 3 1 1 NUM cana-5356 196 4 βˆ’ βˆ’ NOUN cana-5356 196 5 πœ† πœ† X cana-5356 196 6 ) ) PUNCT cana-5356 196 7 ( ( PUNCT cana-5356 196 8 1 1 NUM cana-5356 196 9 + + CCONJ cana-5356 196 10 𝑗 𝑗 PROPN cana-5356 196 11 𝑝 𝑝 NOUN cana-5356 196 12 ) ) PUNCT cana-5356 196 13 π‘Žπ‘—π‘§π‘—+𝑝+1 π‘Žπ‘—π‘§π‘—+𝑝+1 PROPN cana-5356 196 14 ] ] PUNCT cana-5356 196 15 ( ( PUNCT cana-5356 196 16 1 1 NUM cana-5356 196 17 + + CCONJ cana-5356 196 18 𝑗 𝑗 PROPN cana-5356 196 19 𝑝 𝑝 NOUN cana-5356 196 20 ) ) PUNCT cana-5356 196 21 2 2 NUM cana-5356 196 22 here here ADV cana-5356 196 23 the the DET cana-5356 196 24 leading lead VERB cana-5356 196 25 term term NOUN cana-5356 196 26 is be AUX cana-5356 196 27 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 196 28 + + CCONJ cana-5356 196 29 1 1 NUM cana-5356 196 30 ) ) PUNCT cana-5356 196 31 and and CCONJ cana-5356 196 32 the the DET cana-5356 196 33 remaining remain VERB cana-5356 196 34 part part NOUN cana-5356 196 35 is be AUX cana-5356 196 36 a a DET cana-5356 196 37 series series NOUN cana-5356 196 38 that that PRON cana-5356 196 39 depends depend VERB cana-5356 196 40 on on ADP cana-5356 196 41 𝑧. 𝑧. NOUN cana-5356 196 42 βˆ’π‘ βˆ’π‘ NOUN cana-5356 197 1 + + CCONJ cana-5356 197 2 βˆ‘ βˆ‘ PROPN cana-5356 197 3 𝑛 𝑛 DET cana-5356 197 4 ∞ ∞ NUM cana-5356 197 5 𝑗=𝑝 𝑗=𝑝 PROPN cana-5356 198 1 [ [ X cana-5356 198 2 ( ( PUNCT cana-5356 198 3 1 1 NUM cana-5356 198 4 βˆ’ βˆ’ NOUN cana-5356 198 5 πœ† πœ† X cana-5356 198 6 ) ) PUNCT cana-5356 198 7 ( ( PUNCT cana-5356 198 8 1 1 NUM cana-5356 198 9 + + CCONJ cana-5356 198 10 𝑗 𝑗 PROPN cana-5356 198 11 𝑝 𝑝 NOUN cana-5356 198 12 ) ) PUNCT cana-5356 198 13 π‘Žπ‘—π‘§π‘—+𝑝+1 π‘Žπ‘—π‘§π‘—+𝑝+1 PROPN cana-5356 198 14 ] ] PUNCT cana-5356 198 15 ( ( PUNCT cana-5356 198 16 1 1 NUM cana-5356 198 17 + + CCONJ cana-5356 198 18 𝑗 𝑗 PROPN cana-5356 198 19 𝑝 𝑝 NOUN cana-5356 198 20 ) ) PUNCT cana-5356 198 21 here here ADV cana-5356 198 22 the the DET cana-5356 198 23 leading lead VERB cana-5356 198 24 term term NOUN cana-5356 198 25 is be AUX cana-5356 198 26 βˆ’π‘ βˆ’π‘ VERB cana-5356 198 27 , , PUNCT cana-5356 198 28 with with ADP cana-5356 198 29 the the DET cana-5356 198 30 remaining remain VERB cana-5356 198 31 part part NOUN cana-5356 198 32 as as ADP cana-5356 198 33 a a DET cana-5356 198 34 series series NOUN cana-5356 198 35 . . PUNCT cana-5356 199 1 to to PART cana-5356 199 2 evaluate evaluate VERB cana-5356 199 3 the the DET cana-5356 199 4 modulus modulus NOUN cana-5356 199 5 at at ADP cana-5356 199 6 𝒛 𝒛 PROPN cana-5356 199 7 β†’ β†’ SYM cana-5356 199 8 𝟎 𝟎 PROPN cana-5356 199 9 , , PUNCT cana-5356 199 10 communications communication NOUN cana-5356 199 11 on on ADP cana-5356 199 12 applied apply VERB cana-5356 199 13 nonlinear nonlinear ADJ cana-5356 199 14 analysis analysis NOUN cana-5356 199 15 issn issn NOUN cana-5356 199 16 : : PUNCT cana-5356 199 17 1074 1074 NUM cana-5356 199 18 - - PUNCT cana-5356 199 19 133x 133x NUM cana-5356 199 20 vol vol VERB cana-5356 199 21 32 32 NUM cana-5356 199 22 no no NOUN cana-5356 199 23 . . PUNCT cana-5356 200 1 10s 10 NOUN cana-5356 200 2 ( ( PUNCT cana-5356 200 3 2025 2025 NUM cana-5356 200 4 ) ) PUNCT cana-5356 200 5 1880 1880 NUM cana-5356 200 6 https://internationalpubls.com https://internationalpubls.com X cana-5356 201 1 the the DET cana-5356 201 2 series series NOUN cana-5356 201 3 terms term NOUN cana-5356 201 4 involving involve VERB cana-5356 201 5 𝑧𝑗+𝑝+1 𝑧𝑗+𝑝+1 NOUN cana-5356 201 6 will will AUX cana-5356 201 7 reduce reduce VERB cana-5356 201 8 , , PUNCT cana-5356 201 9 making make VERB cana-5356 201 10 the the DET cana-5356 201 11 leading lead VERB cana-5356 201 12 terms term NOUN cana-5356 201 13 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 201 14 + + CCONJ cana-5356 201 15 1 1 X cana-5356 201 16 ) ) PUNCT cana-5356 201 17 in in ADP cana-5356 201 18 the the DET cana-5356 201 19 numerator numerator NOUN cana-5356 201 20 and and CCONJ cana-5356 201 21 βˆ’π‘ βˆ’π‘ VERB cana-5356 201 22 in in ADP cana-5356 201 23 the the DET cana-5356 201 24 denominator denominator NOUN cana-5356 201 25 dominant dominant NOUN cana-5356 201 26 . . PUNCT cana-5356 202 1 the the DET cana-5356 202 2 expression expression NOUN cana-5356 202 3 becomes become VERB cana-5356 202 4 approximately approximately ADV cana-5356 202 5 | | ADV cana-5356 202 6 𝑝(𝑝 𝑝(𝑝 PROPN cana-5356 202 7 + + CCONJ cana-5356 202 8 1 1 X cana-5356 202 9 ) ) PUNCT cana-5356 202 10 βˆ’π‘ βˆ’π‘ NOUN cana-5356 202 11 | | ADV cana-5356 202 12 = = SYM cana-5356 202 13 |βˆ’(𝑝 |βˆ’(𝑝 NOUN cana-5356 202 14 + + X cana-5356 203 1 1)| 1)| NUM cana-5356 203 2 = = SYM cana-5356 203 3 ( ( PUNCT cana-5356 203 4 𝑝 𝑝 NOUN cana-5356 203 5 + + NOUN cana-5356 203 6 1 1 NUM cana-5356 203 7 ) ) PUNCT cana-5356 203 8 < < X cana-5356 203 9 1 1 NUM cana-5356 203 10 this this PRON cana-5356 203 11 needs need VERB cana-5356 203 12 to to PART cana-5356 203 13 be be AUX cana-5356 203 14 less less ADJ cana-5356 203 15 than than ADP cana-5356 203 16 1 1 NUM cana-5356 203 17 , , PUNCT cana-5356 203 18 so so ADV cana-5356 203 19 for for ADP cana-5356 203 20 small small ADJ cana-5356 203 21 𝑧 𝑧 ADP cana-5356 203 22 this this DET cana-5356 203 23 inequality inequality NOUN cana-5356 203 24 holds hold VERB cana-5356 203 25 only only ADV cana-5356 203 26 if if SCONJ cana-5356 203 27 ( ( PUNCT cana-5356 203 28 𝑝 𝑝 NOUN cana-5356 203 29 + + NOUN cana-5356 203 30 1 1 NUM cana-5356 203 31 ) ) PUNCT cana-5356 203 32 < < X cana-5356 203 33 1 1 NUM cana-5356 203 34 , , PUNCT cana-5356 203 35 which which PRON cana-5356 203 36 suggests suggest VERB cana-5356 203 37 𝑝 𝑝 ADP cana-5356 203 38 < < X cana-5356 203 39 0 0 NUM cana-5356 203 40 . . PUNCT cana-5356 204 1 at at ADP cana-5356 204 2 𝒛 𝒛 PROPN cana-5356 204 3 β†’ β†’ SYM cana-5356 204 4 ∞ ∞ PROPN cana-5356 204 5 , , PUNCT cana-5356 204 6 the the DET cana-5356 204 7 series series NOUN cana-5356 204 8 terms term NOUN cana-5356 204 9 involving involve VERB cana-5356 204 10 higher high ADJ cana-5356 204 11 powers power NOUN cana-5356 204 12 of of ADP cana-5356 204 13 𝑧 𝑧 PROPN cana-5356 204 14 will will AUX cana-5356 204 15 dominate dominate VERB cana-5356 204 16 , , PUNCT cana-5356 204 17 and and CCONJ cana-5356 204 18 we we PRON cana-5356 204 19 need need VERB cana-5356 204 20 to to PART cana-5356 204 21 analyse analyse VERB cana-5356 204 22 the the DET cana-5356 204 23 growth growth NOUN cana-5356 204 24 rate rate NOUN cana-5356 204 25 of of ADP cana-5356 204 26 the the DET cana-5356 204 27 series series NOUN cana-5356 204 28 terms term NOUN cana-5356 204 29 relative relative ADJ cana-5356 204 30 to to ADP cana-5356 204 31 each each DET cana-5356 204 32 other other ADJ cana-5356 204 33 . . PUNCT cana-5356 205 1 for for ADP cana-5356 205 2 large large ADJ cana-5356 205 3 𝑧 𝑧 NOUN cana-5356 205 4 , , PUNCT cana-5356 205 5 we we PRON cana-5356 205 6 compare compare VERB cana-5356 205 7 the the DET cana-5356 205 8 asymptotic asymptotic ADJ cana-5356 205 9 behaviour behaviour NOUN cana-5356 205 10 of of ADP cana-5356 205 11 the the DET cana-5356 205 12 series series NOUN cana-5356 205 13 in in ADP cana-5356 205 14 both both CCONJ cana-5356 205 15 the the DET cana-5356 205 16 numerator numerator NOUN cana-5356 205 17 and and CCONJ cana-5356 205 18 denominator denominator NOUN cana-5356 205 19 . . PUNCT cana-5356 206 1 5 5 X cana-5356 206 2 . . X cana-5356 206 3 discussion discussion NOUN cana-5356 206 4 in in ADP cana-5356 206 5 this this DET cana-5356 206 6 work work NOUN cana-5356 206 7 , , PUNCT cana-5356 206 8 a a DET cana-5356 206 9 new new ADJ cana-5356 206 10 subclass subclass NOUN cana-5356 206 11 β„³(𝛼 β„³(𝛼 ADP cana-5356 206 12 , , PUNCT cana-5356 206 13 𝛽 𝛽 NOUN cana-5356 206 14 , , PUNCT cana-5356 206 15 𝛾 𝛾 PROPN cana-5356 206 16 , , PUNCT cana-5356 206 17 π‘˜ π‘˜ NOUN cana-5356 206 18 , , PUNCT cana-5356 206 19 πœ† πœ† NOUN cana-5356 206 20 , , PUNCT cana-5356 206 21 𝑛 𝑛 PROPN cana-5356 206 22 , , PUNCT cana-5356 206 23 𝑝 𝑝 NOUN cana-5356 206 24 ) ) PUNCT cana-5356 206 25 of of ADP cana-5356 206 26 meromorphic meromorphic ADJ cana-5356 206 27 functions function NOUN cana-5356 206 28 defined define VERB cana-5356 206 29 in in ADP cana-5356 206 30 the the DET cana-5356 206 31 punctured punctured ADJ cana-5356 206 32 unit unit NOUN cana-5356 206 33 disk disk NOUN cana-5356 206 34 β–³ β–³ NOUN cana-5356 206 35 βˆ— βˆ— NOUN cana-5356 206 36 = = SYM cana-5356 206 37 { { PUNCT cana-5356 206 38 𝑧 𝑧 NOUN cana-5356 206 39 ∈ ∈ NOUN cana-5356 206 40 β„‚ β„‚ PROPN cana-5356 206 41 ∢ ∢ NOUN cana-5356 206 42 0 0 PUNCT cana-5356 206 43 < < X cana-5356 206 44 |𝑧| |𝑧| X cana-5356 206 45 < < X cana-5356 206 46 1 1 NUM cana-5356 206 47 } } PUNCT cana-5356 206 48 is be AUX cana-5356 206 49 introduced introduce VERB cana-5356 206 50 using use VERB cana-5356 206 51 a a DET cana-5356 206 52 generalized generalize VERB cana-5356 206 53 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 206 54 - - PUNCT cana-5356 206 55 type type NOUN cana-5356 206 56 differential differential NOUN cana-5356 206 57 operator operator NOUN cana-5356 206 58 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 206 59 𝑛 𝑛 PROPN cana-5356 206 60 . . PUNCT cana-5356 207 1 the the DET cana-5356 207 2 functions function NOUN cana-5356 207 3 in in ADP cana-5356 207 4 this this DET cana-5356 207 5 class class NOUN cana-5356 207 6 are be AUX cana-5356 207 7 characterized characterize VERB cana-5356 207 8 by by ADP cana-5356 207 9 satisfying satisfy VERB cana-5356 207 10 a a DET cana-5356 207 11 specific specific ADJ cana-5356 207 12 real real ADJ cana-5356 207 13 - - PUNCT cana-5356 207 14 part part NOUN cana-5356 207 15 inequality inequality NOUN cana-5356 207 16 involving involve VERB cana-5356 207 17 the the DET cana-5356 207 18 parameters parameter NOUN cana-5356 207 19 𝛼 𝛼 VERB cana-5356 207 20 , , PUNCT cana-5356 207 21 𝛽 𝛽 NOUN cana-5356 207 22 , , PUNCT cana-5356 207 23 𝛾 𝛾 PROPN cana-5356 207 24 , , PUNCT cana-5356 207 25 π‘˜ π‘˜ NOUN cana-5356 207 26 , , PUNCT cana-5356 207 27 πœ† πœ† NOUN cana-5356 207 28 , , PUNCT cana-5356 207 29 𝑛 𝑛 PROPN cana-5356 207 30 , , PUNCT cana-5356 207 31 𝑝 𝑝 NOUN cana-5356 207 32 which which PRON cana-5356 207 33 controls control VERB cana-5356 207 34 their their PRON cana-5356 207 35 geometric geometric ADJ cana-5356 207 36 behaviour behaviour NOUN cana-5356 207 37 . . PUNCT cana-5356 208 1 a a DET cana-5356 208 2 sharp sharp ADJ cana-5356 208 3 upper upper ADJ cana-5356 208 4 bound bind VERB cana-5356 208 5 is be AUX cana-5356 208 6 derived derive VERB cana-5356 208 7 for for ADP cana-5356 208 8 the the DET cana-5356 208 9 coefficients coefficient NOUN cana-5356 208 10 π‘Žπ‘— π‘Žπ‘— ADP cana-5356 208 11 of of ADP cana-5356 208 12 functions function NOUN cana-5356 208 13 in in ADP cana-5356 208 14 this this DET cana-5356 208 15 class class NOUN cana-5356 208 16 . . PUNCT cana-5356 209 1 this this DET cana-5356 209 2 bound bind VERB cana-5356 209 3 depends depend VERB cana-5356 209 4 on on ADP cana-5356 209 5 the the DET cana-5356 209 6 involved involved ADJ cana-5356 209 7 parameters parameter NOUN cana-5356 209 8 and and CCONJ cana-5356 209 9 effectively effectively ADV cana-5356 209 10 restricts restrict VERB cana-5356 209 11 the the DET cana-5356 209 12 growth growth NOUN cana-5356 209 13 of of ADP cana-5356 209 14 the the DET cana-5356 209 15 series series NOUN cana-5356 209 16 coefficients coefficient NOUN cana-5356 209 17 . . PUNCT cana-5356 210 1 such such ADJ cana-5356 210 2 bounds bound NOUN cana-5356 210 3 are be AUX cana-5356 210 4 crucial crucial ADJ cana-5356 210 5 for for ADP cana-5356 210 6 understanding understand VERB cana-5356 210 7 the the DET cana-5356 210 8 geometric geometric ADJ cana-5356 210 9 structure structure NOUN cana-5356 210 10 and and CCONJ cana-5356 210 11 distortion distortion NOUN cana-5356 210 12 properties property NOUN cana-5356 210 13 of of ADP cana-5356 210 14 the the DET cana-5356 210 15 functions function NOUN cana-5356 210 16 . . PUNCT cana-5356 211 1 if if SCONJ cana-5356 211 2 the the DET cana-5356 211 3 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 211 4 operator operator NOUN cana-5356 211 5 𝐷𝑝,πœ† 𝐷𝑝,πœ† VERB cana-5356 211 6 𝑛 𝑛 PRON cana-5356 211 7 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 211 8 ) ) PUNCT cana-5356 211 9 satisfies satisfy VERB cana-5356 211 10 a a DET cana-5356 211 11 specific specific ADJ cana-5356 211 12 positivity positivity NOUN cana-5356 211 13 condition condition NOUN cana-5356 211 14 involving involve VERB cana-5356 211 15 its its PRON cana-5356 211 16 logarithmic logarithmic ADJ cana-5356 211 17 derivative derivative NOUN cana-5356 211 18 , , PUNCT cana-5356 211 19 then then ADV cana-5356 211 20 the the DET cana-5356 211 21 function function NOUN cana-5356 211 22 𝑓 𝑓 PROPN cana-5356 211 23 is be AUX cana-5356 211 24 convex convex ADJ cana-5356 211 25 in in ADP cana-5356 211 26 β–³ β–³ NOUN cana-5356 211 27 βˆ— βˆ— NOUN cana-5356 211 28 . . PUNCT cana-5356 212 1 this this DET cana-5356 212 2 result result NOUN cana-5356 212 3 connects connect VERB cana-5356 212 4 analytic analytic ADJ cana-5356 212 5 properties property NOUN cana-5356 212 6 of of ADP cana-5356 212 7 the the DET cana-5356 212 8 differential differential ADJ cana-5356 212 9 operator operator NOUN cana-5356 212 10 to to ADP cana-5356 212 11 the the DET cana-5356 212 12 geometric geometric ADJ cana-5356 212 13 shape shape NOUN cana-5356 212 14 ( ( PUNCT cana-5356 212 15 convexity convexity NOUN cana-5356 212 16 ) ) PUNCT cana-5356 212 17 of of ADP cana-5356 212 18 the the DET cana-5356 212 19 function function NOUN cana-5356 212 20 image image NOUN cana-5356 212 21 . . PUNCT cana-5356 213 1 theorem theorem NOUN cana-5356 213 2 3 3 NUM cana-5356 213 3 provides provide VERB cana-5356 213 4 a a DET cana-5356 213 5 sufficient sufficient ADJ cana-5356 213 6 condition condition NOUN cana-5356 213 7 for for ADP cana-5356 213 8 univalence univalence NOUN cana-5356 213 9 : : PUNCT cana-5356 213 10 if if SCONJ cana-5356 213 11 the the DET cana-5356 213 12 modulus modulus NOUN cana-5356 213 13 of of ADP cana-5356 213 14 the the DET cana-5356 213 15 logarithmic logarithmic ADJ cana-5356 213 16 derivative derivative NOUN cana-5356 213 17 of of ADP cana-5356 213 18 𝐷𝑝,πœ† 𝐷𝑝,πœ† PROPN cana-5356 213 19 𝑛 𝑛 PRON cana-5356 213 20 𝑓(𝑧 𝑓(𝑧 PROPN cana-5356 213 21 ) ) PUNCT cana-5356 213 22 remains remain VERB cana-5356 213 23 less less ADJ cana-5356 213 24 than than ADP cana-5356 213 25 one one NUM cana-5356 213 26 , , PUNCT cana-5356 213 27 then then ADV cana-5356 213 28 the the DET cana-5356 213 29 function function NOUN cana-5356 213 30 𝑓 𝑓 PROPN cana-5356 213 31 is be AUX cana-5356 213 32 univalent univalent ADJ cana-5356 213 33 ( ( PUNCT cana-5356 213 34 injective injective ADJ cana-5356 213 35 ) ) PUNCT cana-5356 213 36 in in ADP cana-5356 213 37 β–³ β–³ NOUN cana-5356 213 38 βˆ— βˆ— NOUN cana-5356 213 39 .this .this PRON cana-5356 213 40 ensures ensure VERB cana-5356 213 41 that that SCONJ cana-5356 213 42 the the DET cana-5356 213 43 function function NOUN cana-5356 213 44 preserves preserve VERB cana-5356 213 45 distinctness distinctness NOUN cana-5356 213 46 of of ADP cana-5356 213 47 points point NOUN cana-5356 213 48 under under ADP cana-5356 213 49 mapping mapping NOUN cana-5356 213 50 , , PUNCT cana-5356 213 51 a a DET cana-5356 213 52 foundational foundational ADJ cana-5356 213 53 property property NOUN cana-5356 213 54 in in ADP cana-5356 213 55 geometric geometric ADJ cana-5356 213 56 function function NOUN cana-5356 213 57 theory theory NOUN cana-5356 213 58 . . PUNCT cana-5356 214 1 overall overall ADV cana-5356 214 2 , , PUNCT cana-5356 214 3 these these DET cana-5356 214 4 theorems theorem NOUN cana-5356 214 5 collectively collectively ADV cana-5356 214 6 establish establish VERB cana-5356 214 7 important important ADJ cana-5356 214 8 structural structural ADJ cana-5356 214 9 properties property NOUN cana-5356 214 10 coefficient coefficient NOUN cana-5356 214 11 bounds bound NOUN cana-5356 214 12 , , PUNCT cana-5356 214 13 convexity convexity NOUN cana-5356 214 14 , , PUNCT cana-5356 214 15 and and CCONJ cana-5356 214 16 univalence univalence NOUN cana-5356 214 17 of of ADP cana-5356 214 18 the the DET cana-5356 214 19 functions function NOUN cana-5356 214 20 in in ADP cana-5356 214 21 the the DET cana-5356 214 22 class class NOUN cana-5356 214 23 β„³(𝛼 β„³(𝛼 ADP cana-5356 214 24 , , PUNCT cana-5356 214 25 𝛽 𝛽 NOUN cana-5356 214 26 , , PUNCT cana-5356 214 27 𝛾 𝛾 PROPN cana-5356 214 28 , , PUNCT cana-5356 214 29 π‘˜ π‘˜ NOUN cana-5356 214 30 , , PUNCT cana-5356 214 31 πœ† πœ† NOUN cana-5356 214 32 , , PUNCT cana-5356 214 33 𝑛 𝑛 PROPN cana-5356 214 34 , , PUNCT cana-5356 214 35 𝑝 𝑝 NOUN cana-5356 214 36 ) ) PUNCT cana-5356 214 37 , , PUNCT cana-5356 214 38 highlighting highlight VERB cana-5356 214 39 the the DET cana-5356 214 40 impact impact NOUN cana-5356 214 41 of of ADP cana-5356 214 42 the the DET cana-5356 214 43 sΔƒlΔƒgean sΔƒlΔƒgean ADJ cana-5356 214 44 operator operator NOUN cana-5356 214 45 on on ADP cana-5356 214 46 the the DET cana-5356 214 47 analytic analytic ADJ cana-5356 214 48 and and CCONJ cana-5356 214 49 geometric geometric ADJ cana-5356 214 50 behaviour behaviour NOUN cana-5356 214 51 of of ADP cana-5356 214 52 meromorphic meromorphic ADJ cana-5356 214 53 functions function NOUN cana-5356 214 54 . . PUNCT cana-5356 215 1 references reference NOUN cana-5356 215 2 [ [ X cana-5356 215 3 1 1 NUM cana-5356 215 4 ] ] X cana-5356 215 5 alharayzeh alharayzeh NOUN cana-5356 215 6 , , PUNCT cana-5356 215 7 ma’moun ma’moun PROPN cana-5356 215 8 iy iy PROPN cana-5356 215 9 , , PUNCT cana-5356 215 10 and and CCONJ cana-5356 215 11 firas firas PROPN cana-5356 215 12 ghanim ghanim PROPN cana-5356 215 13 . . PUNCT cana-5356 216 1 " " PUNCT cana-5356 216 2 new new ADJ cana-5356 216 3 subclass subclass NOUN cana-5356 216 4 of of ADP cana-5356 216 5 k‐uniformly k‐uniformly X cana-5356 216 6 univalent univalent ADJ cana-5356 216 7 analytic analytic ADJ cana-5356 216 8 functions function NOUN cana-5356 216 9 with with ADP cana-5356 216 10 negative negative ADJ cana-5356 216 11 coefficients coefficient NOUN cana-5356 216 12 defined define VERB cana-5356 216 13 by by ADP cana-5356 216 14 multiplier multipli ADJ cana-5356 216 15 transformation transformation NOUN cana-5356 216 16 . . PUNCT cana-5356 216 17 " " PUNCT cana-5356 217 1 in in ADP cana-5356 217 2 abstract abstract ADJ cana-5356 217 3 and and CCONJ cana-5356 217 4 applied apply VERB cana-5356 217 5 analysis analysis NOUN cana-5356 217 6 , , PUNCT cana-5356 217 7 vol vol NOUN cana-5356 217 8 . . PUNCT cana-5356 218 1 no no INTJ cana-5356 218 2 . . NOUN cana-5356 218 3 1 1 NUM cana-5356 218 4 ( ( PUNCT cana-5356 218 5 2022 2022 NUM cana-5356 218 6 ) ) PUNCT cana-5356 218 7 , , PUNCT cana-5356 218 8 p. p. NOUN cana-5356 218 9 4593799 4593799 NUM cana-5356 218 10 . . PUNCT cana-5356 219 1 hindawi hindawi ADJ cana-5356 219 2 . . PUNCT cana-5356 220 1 [ [ X cana-5356 220 2 2 2 NUM cana-5356 220 3 ] ] X cana-5356 220 4 ali ali PROPN cana-5356 220 5 , , PUNCT cana-5356 220 6 rosihan rosihan NOUN cana-5356 220 7 m. m. NOUN cana-5356 220 8 , , PUNCT cana-5356 220 9 and and CCONJ cana-5356 220 10 v. v. ADP cana-5356 220 11 ravichandran ravichandran NOUN cana-5356 220 12 . . PUNCT cana-5356 221 1 β€œ " PUNCT cana-5356 221 2 classes class NOUN cana-5356 221 3 of of ADP cana-5356 221 4 meromorphic meromorphic ADJ cana-5356 221 5 Ξ± Ξ± ADJ cana-5356 221 6 - - PUNCT cana-5356 221 7 convex convex ADJ cana-5356 221 8 functions function NOUN cana-5356 221 9 . . PUNCT cana-5356 221 10 ” " PUNCT cana-5356 222 1 taiwanese taiwanese ADJ cana-5356 222 2 journal journal NOUN cana-5356 222 3 of of ADP cana-5356 222 4 mathematics mathematics PROPN cana-5356 222 5 14 14 NUM cana-5356 222 6 , , PUNCT cana-5356 222 7 no no INTJ cana-5356 222 8 . . NOUN cana-5356 222 9 4 4 NUM cana-5356 222 10 ( ( PUNCT cana-5356 222 11 2010 2010 NUM cana-5356 222 12 ): ): PUNCT cana-5356 222 13 1479 1479 NUM cana-5356 222 14 - - SYM cana-5356 222 15 1490 1490 NUM cana-5356 222 16 . . PUNCT cana-5356 223 1 [ [ X cana-5356 223 2 3 3 NUM cana-5356 223 3 ] ] X cana-5356 223 4 amourah amourah PROPN cana-5356 223 5 , , PUNCT cana-5356 223 6 a. a. NOUN cana-5356 223 7 , , PUNCT cana-5356 223 8 and and CCONJ cana-5356 223 9 maslina maslina PROPN cana-5356 223 10 darus darus NOUN cana-5356 223 11 . . PUNCT cana-5356 224 1 β€œ " PUNCT cana-5356 224 2 some some DET cana-5356 224 3 properties property NOUN cana-5356 224 4 of of ADP cana-5356 224 5 a a DET cana-5356 224 6 new new ADJ cana-5356 224 7 class class NOUN cana-5356 224 8 of of ADP cana-5356 224 9 univalent univalent ADJ cana-5356 224 10 functions function NOUN cana-5356 224 11 involving involve VERB cana-5356 224 12 a a DET cana-5356 224 13 new new ADJ cana-5356 224 14 generalized generalized ADJ cana-5356 224 15 differential differential NOUN cana-5356 224 16 operator operator NOUN cana-5356 224 17 with with ADP cana-5356 224 18 negative negative ADJ cana-5356 224 19 coefficients coefficient NOUN cana-5356 224 20 . . PUNCT cana-5356 224 21 ” " PUNCT cana-5356 225 1 indian indian PROPN cana-5356 225 2 journal journal PROPN cana-5356 225 3 of of ADP cana-5356 225 4 science science NOUN cana-5356 225 5 and and CCONJ cana-5356 225 6 technology technology NOUN cana-5356 225 7 9 9 NUM cana-5356 225 8 , , PUNCT cana-5356 225 9 no no INTJ cana-5356 225 10 . . NOUN cana-5356 225 11 36 36 NUM cana-5356 225 12 ( ( PUNCT cana-5356 225 13 2016 2016 NUM cana-5356 225 14 ): ): PUNCT cana-5356 225 15 1 1 NUM cana-5356 225 16 - - SYM cana-5356 225 17 7 7 NUM cana-5356 225 18 . . PUNCT cana-5356 225 19 communications communication NOUN cana-5356 225 20 on on ADP cana-5356 225 21 applied apply VERB cana-5356 225 22 nonlinear nonlinear ADJ cana-5356 225 23 analysis analysis NOUN cana-5356 225 24 issn issn NOUN cana-5356 225 25 : : PUNCT cana-5356 225 26 1074 1074 NUM cana-5356 225 27 - - PUNCT cana-5356 225 28 133x 133x NUM cana-5356 225 29 vol vol VERB cana-5356 225 30 32 32 NUM cana-5356 225 31 no no NOUN cana-5356 225 32 . . PUNCT cana-5356 226 1 10s 10 NOUN cana-5356 226 2 ( ( PUNCT cana-5356 226 3 2025 2025 NUM cana-5356 226 4 ) ) PUNCT cana-5356 226 5 1881 1881 NUM cana-5356 226 6 https://internationalpubls.com https://internationalpubls.com X cana-5356 227 1 [ [ X cana-5356 227 2 4 4 NUM cana-5356 227 3 ] ] X cana-5356 227 4 carroll carroll PROPN cana-5356 227 5 , , PUNCT cana-5356 227 6 tom tom PROPN cana-5356 227 7 . . PUNCT cana-5356 227 8 geometric geometric ADJ cana-5356 227 9 function function NOUN cana-5356 227 10 theory theory NOUN cana-5356 227 11 . . PUNCT cana-5356 228 1 springer springer NOUN cana-5356 228 2 nature nature NOUN cana-5356 228 3 , , PUNCT cana-5356 228 4 ( ( PUNCT cana-5356 228 5 2024 2024 NUM cana-5356 228 6 ) ) PUNCT cana-5356 228 7 . . PUNCT cana-5356 229 1 [ [ X cana-5356 229 2 5 5 NUM cana-5356 229 3 ] ] PUNCT cana-5356 229 4 darus darus NOUN cana-5356 229 5 , , PUNCT cana-5356 229 6 maslina maslina NOUN cana-5356 229 7 , , PUNCT cana-5356 229 8 and and CCONJ cana-5356 229 9 rabha rabha PROPN cana-5356 229 10 w. w. PROPN cana-5356 229 11 ibrahim ibrahim PROPN cana-5356 229 12 . . PUNCT cana-5356 230 1 β€œ " PUNCT cana-5356 230 2 on on ADP cana-5356 230 3 univalence univalence NOUN cana-5356 230 4 criteria criterion NOUN cana-5356 230 5 for for ADP cana-5356 230 6 analytic analytic ADJ cana-5356 230 7 functions function NOUN cana-5356 230 8 defined define VERB cana-5356 230 9 by by ADP cana-5356 230 10 a a DET cana-5356 230 11 generalized generalize VERB cana-5356 230 12 differential differential NOUN cana-5356 230 13 operator operator NOUN cana-5356 230 14 . . PUNCT cana-5356 230 15 ” " PUNCT cana-5356 231 1 acta acta PROPN cana-5356 231 2 universitatis universitatis PROPN cana-5356 231 3 apulensis apulensis NOUN cana-5356 231 4 23 23 NUM cana-5356 231 5 ( ( PUNCT cana-5356 231 6 2010 2010 NUM cana-5356 231 7 ): ): PUNCT cana-5356 231 8 195 195 NUM cana-5356 231 9 - - SYM cana-5356 231 10 200 200 NUM cana-5356 231 11 . . PUNCT cana-5356 232 1 [ [ X cana-5356 232 2 6 6 NUM cana-5356 232 3 ] ] X cana-5356 232 4 deniz deniz NOUN cana-5356 232 5 , , PUNCT cana-5356 232 6 erhan erhan ADV cana-5356 232 7 . . PUNCT cana-5356 233 1 β€œ " PUNCT cana-5356 233 2 convexity convexity NOUN cana-5356 233 3 of of ADP cana-5356 233 4 integral integral ADJ cana-5356 233 5 operators operator NOUN cana-5356 233 6 involving involve VERB cana-5356 233 7 generalized generalized ADJ cana-5356 233 8 bessel bessel ADJ cana-5356 233 9 functions function NOUN cana-5356 233 10 . . PUNCT cana-5356 233 11 ” " PUNCT cana-5356 233 12 integral integral ADJ cana-5356 233 13 transforms transform NOUN cana-5356 233 14 and and CCONJ cana-5356 233 15 special special ADJ cana-5356 233 16 functions function NOUN cana-5356 233 17 24 24 NUM cana-5356 233 18 , , PUNCT cana-5356 233 19 no no INTJ cana-5356 233 20 . . NOUN cana-5356 233 21 3 3 NUM cana-5356 233 22 ( ( PUNCT cana-5356 233 23 2013 2013 NUM cana-5356 233 24 ): ): PUNCT cana-5356 233 25 201 201 NUM cana-5356 233 26 - - SYM cana-5356 233 27 216 216 NUM cana-5356 233 28 . . PUNCT cana-5356 234 1 [ [ X cana-5356 234 2 7 7 NUM cana-5356 234 3 ] ] X cana-5356 234 4 duren duren PROPN cana-5356 234 5 , , PUNCT cana-5356 234 6 peter peter PROPN cana-5356 234 7 l. l. PROPN cana-5356 234 8 univalent univalent PROPN cana-5356 234 9 functions function NOUN cana-5356 234 10 . . PUNCT cana-5356 235 1 vol vol NOUN cana-5356 235 2 . . PUNCT cana-5356 236 1 259 259 NUM cana-5356 236 2 . . X cana-5356 236 3 springer springer NOUN cana-5356 236 4 science science PROPN cana-5356 236 5 & & CCONJ cana-5356 236 6 business business NOUN cana-5356 236 7 media medium NOUN cana-5356 236 8 , , PUNCT cana-5356 236 9 ( ( PUNCT cana-5356 236 10 2001 2001 NUM cana-5356 236 11 ) ) PUNCT cana-5356 236 12 . . PUNCT cana-5356 237 1 [ [ X cana-5356 237 2 8 8 NUM cana-5356 237 3 ] ] X cana-5356 237 4 goodman goodman PROPN cana-5356 237 5 , , PUNCT cana-5356 237 6 adolph adolph PROPN cana-5356 237 7 w. w. PROPN cana-5356 237 8 β€œ " PUNCT cana-5356 237 9 univalent univalent ADJ cana-5356 237 10 functions function NOUN cana-5356 237 11 and and CCONJ cana-5356 237 12 nonanalytic nonanalytic ADJ cana-5356 237 13 curves curve NOUN cana-5356 237 14 . . PUNCT cana-5356 237 15 ” " PUNCT cana-5356 238 1 proceedings proceeding NOUN cana-5356 238 2 of of ADP cana-5356 238 3 the the DET cana-5356 238 4 american american PROPN cana-5356 238 5 mathematical mathematical PROPN cana-5356 238 6 society society NOUN cana-5356 238 7 8 8 NUM cana-5356 238 8 , , PUNCT cana-5356 238 9 no no INTJ cana-5356 238 10 . . NOUN cana-5356 238 11 3 3 NUM cana-5356 238 12 ( ( PUNCT cana-5356 238 13 1957 1957 NUM cana-5356 238 14 ): ): PUNCT cana-5356 238 15 598 598 NUM cana-5356 238 16 - - SYM cana-5356 238 17 601 601 NUM cana-5356 238 18 . . PUNCT cana-5356 239 1 [ [ X cana-5356 239 2 9 9 NUM cana-5356 239 3 ] ] SYM cana-5356 239 4 lee lee PROPN cana-5356 239 5 , , PUNCT cana-5356 239 6 see see VERB cana-5356 239 7 keong keong PROPN cana-5356 239 8 , , PUNCT cana-5356 239 9 v. v. ADP cana-5356 239 10 ravichandran ravichandran NOUN cana-5356 239 11 , , PUNCT cana-5356 239 12 and and CCONJ cana-5356 239 13 supramaniam supramaniam VERB cana-5356 239 14 shamani shamani NOUN cana-5356 239 15 . . PUNCT cana-5356 240 1 " " PUNCT cana-5356 240 2 coefficient coefficient NOUN cana-5356 240 3 bounds bound VERB cana-5356 240 4 for for ADP cana-5356 240 5 meromorphic meromorphic ADJ cana-5356 240 6 starlike starlike NOUN cana-5356 240 7 and and CCONJ cana-5356 240 8 convex convex NOUN cana-5356 240 9 functions function NOUN cana-5356 240 10 . . PUNCT cana-5356 240 11 " " PUNCT cana-5356 241 1 jipam jipam ADV cana-5356 241 2 . . PUNCT cana-5356 242 1 journal journal PROPN cana-5356 242 2 of of ADP cana-5356 242 3 inequalities inequality NOUN cana-5356 242 4 in in ADP cana-5356 242 5 pure pure ADJ cana-5356 242 6 & & CCONJ cana-5356 242 7 applied applied ADJ cana-5356 242 8 mathematics mathematic NOUN cana-5356 243 1 [ [ X cana-5356 243 2 electronic electronic ADJ cana-5356 243 3 only only ADV cana-5356 243 4 ] ] PUNCT cana-5356 243 5 10 10 NUM cana-5356 243 6 , , PUNCT cana-5356 243 7 no no INTJ cana-5356 243 8 . . NOUN cana-5356 243 9 3 3 NUM cana-5356 243 10 ( ( PUNCT cana-5356 243 11 2009 2009 NUM cana-5356 243 12 ): ): PUNCT cana-5356 243 13 paper paper NOUN cana-5356 243 14 - - PUNCT cana-5356 243 15 no no NOUN cana-5356 243 16 . . PUNCT cana-5356 244 1 [ [ X cana-5356 244 2 10 10 NUM cana-5356 244 3 ] ] SYM cana-5356 244 4 li li PROPN cana-5356 244 5 , , PUNCT cana-5356 244 6 ming ming PROPN cana-5356 244 7 - - PUNCT cana-5356 244 8 liang liang PROPN cana-5356 244 9 , , PUNCT cana-5356 244 10 lei lei PROPN cana-5356 244 11 shi shi PROPN cana-5356 244 12 , , PUNCT cana-5356 244 13 and and CCONJ cana-5356 244 14 zhi zhi PROPN cana-5356 244 15 - - PUNCT cana-5356 244 16 gang gang PROPN cana-5356 244 17 wang wang PROPN cana-5356 244 18 . . PUNCT cana-5356 245 1 β€œ " PUNCT cana-5356 245 2 on on ADP cana-5356 245 3 a a DET cana-5356 245 4 subclass subclass NOUN cana-5356 245 5 of of ADP cana-5356 245 6 meromorphic meromorphic PROPN cana-5356 245 7 close‐to close‐to PROPN cana-5356 245 8 [ [ X cana-5356 245 9 11 11 NUM cana-5356 245 10 ] ] PUNCT cana-5356 245 11 convex convex NOUN cana-5356 245 12 functions function NOUN cana-5356 245 13 . . PUNCT cana-5356 245 14 ” " PUNCT cana-5356 246 1 the the DET cana-5356 246 2 scientific scientific ADJ cana-5356 246 3 world world NOUN cana-5356 246 4 journal journal PROPN cana-5356 246 5 2014 2014 NUM cana-5356 246 6 , , PUNCT cana-5356 246 7 no no INTJ cana-5356 246 8 . . NOUN cana-5356 246 9 1 1 NUM cana-5356 246 10 ( ( PUNCT cana-5356 246 11 2014 2014 NUM cana-5356 246 12 ): ): PUNCT cana-5356 246 13 806168 806168 NUM cana-5356 246 14 . . PUNCT cana-5356 247 1 [ [ X cana-5356 247 2 12 12 NUM cana-5356 247 3 ] ] PUNCT cana-5356 247 4 park park NOUN cana-5356 247 5 , , PUNCT cana-5356 247 6 ji ji PROPN cana-5356 247 7 hyang hyang PROPN cana-5356 247 8 , , PUNCT cana-5356 247 9 hari hari PROPN cana-5356 247 10 mohan mohan PROPN cana-5356 247 11 srivastava srivastava PROPN cana-5356 247 12 , , PUNCT cana-5356 247 13 and and CCONJ cana-5356 247 14 nak nak PROPN cana-5356 247 15 eun eun PROPN cana-5356 247 16 cho cho PROPN cana-5356 247 17 . . PUNCT cana-5356 248 1 β€œ " PUNCT cana-5356 248 2 univalence univalence NOUN cana-5356 248 3 and and CCONJ cana-5356 248 4 convexity convexity NOUN cana-5356 248 5 [ [ X cana-5356 248 6 13 13 NUM cana-5356 248 7 ] ] ADJ cana-5356 248 8 conditions condition NOUN cana-5356 248 9 for for ADP cana-5356 248 10 certain certain ADJ cana-5356 248 11 integral integral ADJ cana-5356 248 12 operators operator NOUN cana-5356 248 13 associated associate VERB cana-5356 248 14 with with ADP cana-5356 248 15 the the DET cana-5356 248 16 lommel lommel ADJ cana-5356 248 17 function function NOUN cana-5356 248 18 of of ADP cana-5356 248 19 the the DET cana-5356 248 20 first first ADJ cana-5356 248 21 [ [ X cana-5356 248 22 14 14 NUM cana-5356 248 23 ] ] SYM cana-5356 248 24 kind kind NOUN cana-5356 248 25 . . PUNCT cana-5356 248 26 " " PUNCT cana-5356 248 27 aims aim VERB cana-5356 248 28 math math NOUN cana-5356 248 29 6 6 NUM cana-5356 248 30 , , PUNCT cana-5356 248 31 no no INTJ cana-5356 248 32 . . NOUN cana-5356 248 33 10 10 NUM cana-5356 248 34 ( ( PUNCT cana-5356 248 35 2021 2021 NUM cana-5356 248 36 ): ): PUNCT cana-5356 248 37 11380 11380 NUM cana-5356 248 38 - - SYM cana-5356 248 39 11402 11402 NUM cana-5356 248 40 . . PUNCT cana-5356 249 1 [ [ X cana-5356 249 2 15 15 NUM cana-5356 249 3 ] ] X cana-5356 249 4 silverman silverman NOUN cana-5356 249 5 , , PUNCT cana-5356 249 6 herb herb NOUN cana-5356 249 7 . . PUNCT cana-5356 250 1 " " PUNCT cana-5356 250 2 univalent univalent ADJ cana-5356 250 3 functions function NOUN cana-5356 250 4 with with ADP cana-5356 250 5 negative negative ADJ cana-5356 250 6 coefficients coefficient NOUN cana-5356 250 7 . . PUNCT cana-5356 250 8 " " PUNCT cana-5356 251 1 proceedings proceeding NOUN cana-5356 251 2 of of ADP cana-5356 251 3 the the DET cana-5356 251 4 [ [ X cana-5356 251 5 16 16 NUM cana-5356 251 6 ] ] PUNCT cana-5356 251 7 american american PROPN cana-5356 251 8 mathematical mathematical PROPN cana-5356 251 9 society society NOUN cana-5356 251 10 51 51 NUM cana-5356 251 11 , , PUNCT cana-5356 251 12 no no INTJ cana-5356 251 13 . . NOUN cana-5356 251 14 1 1 NUM cana-5356 251 15 ( ( PUNCT cana-5356 251 16 1975 1975 NUM cana-5356 251 17 ): ): PUNCT cana-5356 251 18 109 109 NUM cana-5356 251 19 - - SYM cana-5356 251 20 116 116 NUM cana-5356 251 21 . . PUNCT