CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS OF COMPOSITE ENTIRE FUNCTIONS FROM THE VIEW POINT OF L -ORDER. Tanmay Biswas
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1 J Koean Soc Math Educ Se B: Pue Appl Math ISSNPint ISSNOnline Volume 5, Numbe 3 August 018, Pages CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS OF COMPOSITE ENTIRE FUNCTIONS FROM THE VIEW POINT OF L -ORDER Tanmay Biswas Abstact In this pape, we discuss cental index oiented and slowly changing unction based some gowth popeties o composite entie unctions 1 Intoduction, Deinitions and Notations Let be an entie unction deined in the open complex plane C Fo entie = a n z n on z =, the maximum modulus symbolized as M, the maximum tem n=0 denoted as µ and the cental index indicated as ν ae espectively deined as max z, max a n n and max {m, µ = a m m } Theeoe, cental index z = n 0 ν o an entie unction is the geatest onent m such that a m m = µ Obviously M, µ and ν ae eal and inceasing unction o Fo anothe entie unction g, M g and µ g ae also deined and the atios M M when g as well as µ µ g as ae called the compaative gowth o with espect to g in tems o thei maximum moduli and the maximum tem espectively The pime object o the study o the gowth investigation o entie unctions has usually been done though thei maximum moduli and maximum tem Though ν is much weake than M and µ g in some sense, om anothe angle o view ν ν g as is also called the gowth o with espect to g whee ν g denotes the cental index o entie g Consideing this, hee we compae the cental index o composition o two entie unctions with thei coesponding let and ight actos unde the teatment o the theoies o slowly changing unctions which in act means Received by the editos Mach 05, 018 Accepted July 13, Mathematics Subject Classiication 30D30, 30D35 Key wods and phases entie unction, cental index, ode espectively, lowe ode, L -ode espectively, L -lowe ode, composition, gowth 193 c 018 Koean Soc Math Educ
2 194 Tanmay Biswas that L a L as o evey positive constant a ie, La L = 1 whee L L Actually in this pape we attempt to pove some esults elated to the gowth ates o composite entie unctions on the basis o cental index using the idea o L -ode espectively, L -lowe ode o an entie unction whee L is nothing but a weake assumption o LOu notations ae standad within the theoy o Nevanlinna s value distibution o entie unctions and theeoe we do not lain those in detail as those ae available in [8] To stat ou pape we just ecall the ollowing deinitions which will be needed in the sequel: Deinition 1 The ode ρ and lowe ode λ o an entie unction ae deine as ρ λ = in log log M log = in log [] M log Theeoe it seems easonable to state suitably an altenative deinition o ode and lowe ode o entie unction in tems o its cental index He and Xiao [3] intoduced such a deinition in the ollowing way: ρ λ = in log ν log Let L L be a positive continuous unction inceasing slowly ie, L a L as o evey positive constant a Consideing L = log and a = 10 0, one can easily show that La L = 1 Somasundaam and Thamizhaasi [6] intoduced the notions o L-ode espectively L-lowe ode o entie unctions The moe genealized concept o L-ode and L-lowe ode o entie unctions is L -ode and L -lowe ode whose deinition ae as ollows: Deinition [6] The L -ode ρ L ae deined as = in and L -lowe ode log log M log [ el] = in o an entie unction log [] M log [ e L] Taking z = z and L = log, one can easily veiy that ρ = λ = 1 and = = 1 In tems o cental index o entie unctions, Deinition can be eomulated as: Deinition 3 The gowth indicatos as: = and in o an entie unction ae deined log ν log [ e L]
3 CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS 195 The concept o p, q-φ ode o entie unction was intoduced by Shen et al [5] whee p q 1 and φ : [0, + 0, + be a non-deceasing unbounded unction Shen et al [5] also established the equivalence o the deinition o p, q-φ ode o entie unction in tems o maximum modulus and cental index unde some cetain condition Fo details about it, one may see [5] Fo paticula i we conside p = 1, q = 1 and φ = e L, then in view o Poposition 1 o [5], we can wite that = in log log M log [ el] = Lemmas in log ν log [ e L] In this section we pesent some lemmas which will be needed in the sequel Lemma 1 [3, Theoems 19 and 110] o [4, Satz 43 and 44] Let be any entie unction, then and o < R, log µ = log a ν t dt whee a 0 0, t { M < µ ν R + R } R Lemma [1] Let and g ae any two entie unctions with g 0 = 0 Also let β satisy 0 < β < 1 and c β = 1 β 4β Then o all suiciently lage values o, M c β M g β M g M M g In addition i β = 1, then o all suiciently lage values o, 1 M g M 8 M g Lemma 3 Let be an entie unction with 0 < λ ρ < Also let g be an entie unction with non zeo inite lowe ode suiciently lage values o, ν g > ν α I 0 < α < λ g, then o all Poo Fo any constant E, we get om the second pat o Lemma 1, that log M < ν log ν + E {c [] }
4 196 Tanmay Biswas Theeoe om above we obtain that 1 ie, log M log M < ν log + ν + E ie, log M < ν 1 + E ie, log M < ν log e + E ie, log M < ν log e + E < ν log e 1 + E ν log e In view o 1 we get o all suiciently lage values o that log ν g > log [] M g log[] e E log 1 + ν g log e ie, in log ν g log ν α ie, in in log [] e log ν α log ν g log ν α log [] M g log ν α log in E 1 + ν g loge log ν α log [] M g log ν α Futhe in view o Lemma, we obtain o all suiciently lage values o that 1 log [] M g log [] M 8 M g 4 3 ie, log [] M g λ ε 1 λg ε 8 + λ ε 4 whee we choose ε in such a way that 0 < ε < min λ, λ g Again om the deinition o λ o entie unction in tems o cental index, we obtain o all suiciently lage values o that 4 log ν α ρ + ε log α ie, log ν α ρ + ε α Now om 3 and 4 it ollows o all suiciently lage values o that log [] M g log ν α λ ε λ ε λg ε 4 ρ + ε α
5 5 in CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS 197 log [] M g log ν α in As α < λ g we can choose ε > 0 in such a way that 6 α < λ g ε Thus om 5 and 6 we get that log [] M g 7 log ν α = Theeoe om and 7 we obtain that ie, λ ε λ ε λg ε 4 ρ + ε α log ν g log ν α = So om above we obtain o all suiciently lage values o and K > 1 that log ν g > K log ν α ie, log ν g > log {ν α } K ie, ν g ν α This poves the theoem In the line o Lemma 3, one can easily veiy the ollowing coollay and theeoe its poo is omitted Coollay 1 Let be an entie unction with non zeo lowe ode Also let g be an entie unction with 0 < λ g ρ g < I 0 < α < λ g, then o all suiciently lage values o, ν g > ν g α 3 Results In this section we pesent the main esults o the pape Theoem 4 Let be an entie unction with non zeo inite ode and lowe ode and g be an entie unction with non zeo inite lowe ode I 0 < then o any A > 0 log [] ν g log ν α + K, A; L =, <,
6 198 Tanmay Biswas 0 i α = o { L α A} whee 0 < α < λ g and K, A; L = as L α A othewise Poo Let 0 < α < α < λ g Now om the deinition o L -lowe ode we obtain in view o Lemma 3, o o all suiciently lage values o that log ν g log ν A α ie, log ν g { ε log A α L A } α ie, log ν g { ε α + L A } α ie, log ν g ε L A α α 1 + α ie, log [] ν g O 1 + α log A L A α 1 + α ie, log [] ν g O 1 + α A ie, log [] ν g O 1 + α A L A α 1 + α [1 + L α A ] α A ie, log [] ν g O 1 + α A + L α A log [ { L α A}] [1 + L α A ] α A
7 CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS 199 ie, log [] ν g O 1 + α A + L α A [ α A + L α A ] {L α } α A 8 ie, log [] ν g O 1 + α A α α + L α A Again we have o all suiciently lage values o that log ν α ρ L + ε log { } α e Lα ie, log ν α + ε {log α + L α } ie, log ν α ρ L + ε { α + L α } 9 ie, log ν α + ε + ε L α α 10 Now om 8 and 9 it ollows o all suiciently lage values o that log [] ν g α A α [ ] O 1 + log ν α + ε L α + ε + L α A 11 ie, log [] ν g log ν α + µ A α + ε 1 L α A + O 1 log ν α ρ L + ε L α log ν α Again om 10 we get o all suiciently lage values o that log [] ν g log ν α + L α O 1 α A µ L α log ν α + L α µ A α log ν + +ε α log ν α + L α + L α log ν α + L α
8 00 Tanmay Biswas 1 ie, log [] ν g log ν α + L α log ν α + µ A α +ε 1 + Lα log ν α + O1 α A µ L α Lα log ν α Lα ν α Lα Case I I α = o { L α A} then it ollows om 11 that log [] ν g log ν α = Case II α o { L α A} then two sub cases may aise Sub case a I L α A = o {log ν α }, then we get om 1 that log [] ν g log ν α + L α = Sub case b I L α A log ν α then and we obtain om 1 that L { α A} log ν α = 1 log [] ν g log ν α + L α = Combining Case I and Case II we may obtain that log [] ν g log ν α + K, A; L =, 0 i µ = o { L α A} whee K, A; L = as L α A othewise This poves the theoem Theoem 5 Let be an entie unction with non zeo inite ode and lowe ode and g be an entie unction with non zeo inite lowe ode I then o any A > 0 log [] ν g log ν g α + K, A; L =, > 0 and g <
9 CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS 01 0 i α = o { L α A} whee 0 < α < λ g and K, A; L = as L α A othewise The poo is omitted because it can be caied out in the line o Theoem 4 Reeences 1 J Clunie: The composition o entie and meomophic unctions Mathematical Essays dedicated to A J Macintye, Ohio Univesity Pess 1970, 75-9 ZX Chen & CC Yang: Some uthe esults on the zeos and gowths o entie solutions o second ode linea dieential equations Kodai Math Jounal 1999, YZ He & XZ Xiao: Algeboid unctions and odinay dieential equations Science Pess, 1988 in Chinese 4 G Jank & L Volkmann: Meomophic Funktionen und Dieentialgleichungen, Bikhause, Xia Shen, Jin Tu & Hong Yan Xu: Complex oscillation o a second-ode linea dieential equation with entie coeicients o [p, q] φ ode Adv Dieence Equ 014, 014:00, 14 pages 6 D Somasundaam & R Thamizhaasi: A note on the entie unctions o L-bounded index and L-type Indian J Pue ApplMath 19 Mach 1988, no 3, AP Singh & MS Baloia: On the maximum modulus and maximum tem o composition o entie unctions Indian J Pue Appl Math 1991, no 1, G Valion: Lectues on the Geneal Theoy o Integal Functions Chelsea Publishing Company, 1949 Rajbai, Rabindapalli, R N Tagoe Road, PO Kishnaga, Dist-Nadia, PIN , West Bengal, India addess: tanmaybiswas math@edimailcom
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