Derivation of Some Results on the Generalized Relative Orders of Meromorphic Functions

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1 DOI: /awutm Analele Universităţii de Vest, Timişoara Seria Matematică Inormatică LV, 1, 2017), Derivation o Some Results on te Generalized Relative Orders o Meromorpic Functions Sanjib Kumar Datta and Tanmay Biswas Abstract. In tis paper we intend to ind out relative order relative lower order) o a meromorpic unction wit respect to anoter entire unction g wen generalized relative order generalized relative lower order) o and generalized relative order generalized relative lower order) o g wit respect to anoter entire unction are given. AMS Subject Classiication 2000). 30D35, 30D30, 30D20 Keywords. relative order relative lower order), generalized relative order generalized relative lower order), growt 1 Introduction, Deinitions and Notations Let be an entire unction deined in te inite complex plane C. Te maximum modulus unction corresponding to entire is deined as M r) = max { z) : z = r}. Wile is meromorpic, one may deine a dierent unction T r) amous as Nevanlinna s Caracteristic unction o, playing same role as maximum modulus unction in te ollowing manner: T r) = N r) + m r), Unautenticated

2 52 S. K. Datta and T. Biswas An. U.V.T. ) N were te unction N r, a) r, a) points distinct a-points) o meromorpic is deined as N r, a) = r 0 known as counting unction o a- n t, a) n 0, a) dt + n 0, a) t r N n t, a) n 0, a) r, a) = dt + n 0, a), t 0 n ) moreover we denote by n r, a) r, a) te number o a-points distinct a-points) o in z r and an -point is a pole o. In many occasions N r, ) and N r, ) are denoted by N r) and N r) respectively. Also te unction m r, ) alternatively denoted by m r) known as te proximity unction o is deined as ollows: Also we may denote m m r) = 1 2π log + re iθ) dθ, 2π 0 log + x = max log x, 0) or all x 0. r, 1 a ) by m r, a). were I is entire unction, ten te Nevanlinna s Caracteristic unction T r) o is deined as T r) = m r). However, te study o comparative growt properties o entire and meromorpic unctions wic is one o te prominent branc o te value distribution teory o entire and meromorpic unctions is te prime concern o te paper. We do not explain te standard deinitions and notations in te teory o entire and meromorpic unctions as tose are available in [11] and [14]. In te sequel te ollowing two notations are used: ) log [k] x = log log [k] x or k = 1, 2, 3, ; and log [0] x = x exp [k] x = exp exp [k] x ) or k = 1, 2, 3, ; exp [0] x = x. Unautenticated

3 Vol. LV 2017) Derivation o Some Results on te Generalized Relative Orders...53 Taking tis into account te generalized order respectively, generalized lower order) o an entire unction as introduced by Sato [13] is given by: log [l] M r) = lim sup log log M exp z r) respectively log [l] M r) = lim in log log M exp z r) log [l] M r) = lim sup = lim in log [l] M r) were l 1. Wen is meromorpic unction, one can easily veriy tat = lim sup log [l] T r) log T exp z r) respectively = lim sup = lim in log [l] T r) log r π log [l] T r) log T exp z r) ) = lim sup = lim in ) log [l] T r) + O1) ) log [l] T r) + O1) were l 1. Tese deinitions extend te deinitions o order ρ and lower order λ o an entire or meromorpic unction since or l = 2, tese correspond to te particular case ρ [2] = ρ and λ [2] = λ. Given a non-constant entire unction g deined in te open complex plane C, its Nevanlinna s Caracteristic unction r) is strictly increasing and continuous unctions o r. Also te inverse : 0), ) 0, ) exists and is suc tat lim s) =. s Extending te idea o relative order o entire unctions as establised by Bernal {[1], [2]}, Lairi and Banerjee [12] introduced te deinition o relative order o a meromorpic unction wit respect to anoter entire unction g, denoted by ρ g ) to avoid comparing growt just wit exp z as ollows: ρ g ) = in {µ > 0 : T r) < r µ ) or all suiciently large r} = lim sup. Te deinition coincides wit te classical one i g z) = exp z {c. [12] }. Likewise, one can deine te relative lower order o a meromorpic unction wit respect to an entire unction g denoted by λ g ) as ollows : λ g ) = lim in. Unautenticated

4 54 S. K. Datta and T. Biswas An. U.V.T. Debnat et. al. [3] gave a more generalized concept o relative order a meromorpic unction wit respect to an entire unction in te ollowing way : Deinition 1.1. [3] Let be any meromorpic unction and g be any entire unction wit index-pairs m 1, q) and m 2, p) respectively were m 1 = m 2 = m and p, q, m are all positive integers suc tat m p and m q. Ten te relative p, q) t order o wit respect to g is deined as ρ p,q) g ) == lim sup log [p] log [q] r For details about index-pair o meromorpic unction, one may see [3]. Wen p = l 1 and q = 1, te above deinition reduces to te deinition o generalized relative order o a meromorpic unction wit respect to an entire unction g, denoted by g ) wic is as ollows g ) = lim sup log [l] Likewise one can deine te generalized relative lower order o a meromorpic unction wit respect to an entire unction g denoted by g ) as log [l] T g g ) = lim in. For entire and meromropic unctions, te notions o teir growt indicators suc as order is classical in complex analysis and during te past decades, several researcers ave already been exploring teir studies in te area o comparative growt properties o composite entire and meromorpic unctions in dierent directions using te classical growt indicators. But at tat time, te concepts o relative orders and consequently te generalized relative orders o entire and meromorpic unctions wit respect to anoter entire unction and as well as teir tecnical advantages o not comparing wit te growts o exp z are not at all known to te researcers o tis area. Tereore te growt o composite entire and meromorpic unctions needs to be modiied on te basis o teir relative order some o wic as been explored in [4], [5], [6], [7], [8], [9] and [10]. In tis paper we establis some newly developed results related to te growt rates o entire and meromorpic unctions on te basis o teir relative orders respectively relative lower orders) and generalized relative orders respectively generalized relative lower orders)... Unautenticated

5 Vol. LV 2017) Derivation o Some Results on te Generalized Relative Orders Teorem In tis section we present te main results o te paper. Teorem 2.1. Let be a meromorpic unction and g and be any two entire unctions suc tat 0 < ) ρ[l] ) < and 0 < λ[l] g) g) < were l 1. Ten ) g) λ g ) min { ) Proo. From te deinitions o large values o r tat } g), ) { g) max ) } g), ) g) T r) T [ exp [l] { ) + ε ) ρ g ) ρ[l] ) g). ) and λ[l] ), we ave or all suiciently }] }] T r) T [ exp [l] { ) ε ), 2.1) 2.2) and also or a sequence o values o r tending to ininity, we get tat [ { ) }] T r) T exp [l] ) ε, 2.3) [ { ) }] T r) T exp [l] ) + ε. 2.4) Similarly rom te deinitions o large values o r tat g) and λ[l] g), it ollows or all suiciently { ) } T r) exp [l] g) + ε [ { ) r) T exp [l] g) + ε T r) }] log [l] r ), 2.5) g) + ε { ) } T r) exp [l] g) ε log T r) [l] r ) 2.6) g) ε Unautenticated

6 56 S. K. Datta and T. Biswas An. U.V.T. and or a sequence o values o r tending to ininity we obtain tat { ) } T r) exp [l] g) ε log i.e. T r) [l] r ), 2.7) g) ε { ) } T r) exp [l] g) + ε log T r) [l] r ). 2.8) g) + ε Now rom 2.3) and in view o 2.5), we get or a sequence o values o r tending to ininity tat [ { ) }] T exp [l] ) ε { ) } log [l] exp [l] ) ε ) g) + ε ) ε = exp ) g) + ε As ε > 0 is arbitrary, it ollows tat ) ) ε ). g) + ε ρ g ) ρ[l] ). 2.9) g) Analogously rom 2.2) and in view o 2.8), it ollows or a sequence o values o r tending to ininity tat [ { ) }] T exp [l] ) ε Unautenticated

7 Vol. LV 2017) Derivation o Some Results on te Generalized Relative Orders...57 { ) } log [l] exp [l] ) ε ) g) + ε ) ε = exp ) g) + ε ) ) ε ). g) + ε Since ε > 0) is arbitrary, we get rom above tat ρ g ) λ[l] ). 2.10) g) Again in view o 2.6), we ave rom 2.1), or all suiciently large values o r tat [ { ) }] T exp [l] ) + ε { ) } log [l] exp [l] ) + ε ) g) ε ) + ε = exp ) g) ε Since ε > 0) is arbitrary, we obtain tat ) ) + ε ). g) ε ρ g ) ρ[l] ). 2.11) g) Again rom 2.2) and in view o 2.5), it ollows or all suiciently large values o r tat [ { ) }] T exp [l] ) ε Unautenticated

8 58 S. K. Datta and T. Biswas An. U.V.T. { ) } log [l] exp [l] ) ε ) g) + ε ) ε = exp ) g) + ε ) ) ε ). g) + ε Since ε > 0) is arbitrary, we get rom above tat λ g ) λ[l] ). 2.12) g) Also in view o 2.7), we get rom 2.1) or a sequence o values o r tending to ininity tat [ { ) }] T exp [l] ) + ε { ) } log [l] exp [l] ) + ε ) g) ε ) + ε = exp ) g) ε ) ) + ε ). g) ε Since ε > 0) is arbitrary, we get rom above tat λ g ) ρ[l] ). 2.13) g) Similarly rom 2.4) and in view o 2.6), it ollows or a sequence o values o r tending to ininity tat [ { ) }] T exp [l] ) + ε Unautenticated

9 Vol. LV 2017) Derivation o Some Results on te Generalized Relative Orders...59 { ) } log [l] exp [l] ) + ε ) g) ε ) + ε = exp ) g) ε ) ) + ε ). g) ε As ε > 0) is arbitrary, we obtain rom above tat λ g ) λ[l] ). 2.14) g) Te teorem ollows rom 2.9), 2.10), 2.11), 2.12), 2.13) and 2.14). Corollary 2.2. Let be a meromorpic unction and g and be any two entire unctions suc tat 0 < ) ρ[l] ) < and 0 < λ[l] g) = g) < were l 1. Ten In addition, i λ g ) = λ[l] ) ) = ρ[l] g), ten g) and ρ g ) = ρ ρ g ) = 1. [l] ) g). Corollary 2.3. Let be a meromorpic unction and g and be any two entire unctions suc tat 0 < ) = ρ[l] ) < and 0 < λ[l] g) = g) < were l 1. Ten λ g ) = ρ g ) = ρ[l] ) g). Corollary 2.4. Let be a meromorpic unction and g and be any two entire unctions suc tat 0 < ) = ρ[l] ) = λ[l] g) = ρ[l] g) < were l 1. Ten λ g ) = ρ g ) = 1. Unautenticated

10 60 S. K. Datta and T. Biswas An. U.V.T. Corollary 2.5. Let be a meromorpic unction and g and be any two entire unctions suc tat 0 < ) < ρ[l] ) < were l 1. Ten i) λ g ) = wen g) = 0, ii) ρ g ) = wen g) = 0, iii) λ g ) = 0 wen g) = and iv) ρ g ) = 0 wen g) =. Corollary 2.6. Let be a meromorpic unction and g and be any two entire unctions suc tat 0 < g) = ρ[l] g) < were l 1. Ten i) ρ g ) = 0 wen ) = 0, ii) λ g ) = 0 wen ) = 0, iii) ρ g ) = wen ) = and iv) λ g ) = wen ) =. Acknowledgement Te autors are tankul to reeree or is/er valuable suggestions towards te improvement o te paper. Reerences [1] L. Bernal, Crecimiento relativo de unciones enteras. Contribucion al estudio de lasunciones enteras con indice exponencial inito, 1984.) [2] L. Bernal, Orden relative de crecimiento de unciones enteras, Collect. Mat., 39, 1988), [3] L. Debnat, S. K. Datta, T. Biswas, and A. Kar, Growt o meromorpic unctions depending on p,q)-t relative order, Facta Univ. Ser. Mat. Inorm., 31, 2016), [4] S. K. Datta and T. Biswas, Growt o entire unctions based on relative order, Int. J. Pure Appl. Mat., 51, 2009), Unautenticated

11 Vol. LV 2017) Derivation o Some Results on te Generalized Relative Orders...61 [5] S. K. Datta and T. Biswas, Relative order o composite entire unctions and some related growt properties, Bull. Cal. Mat. Soc., 102, 2010)), [6] S. K. Datta, T. Biswas, and D. C. Pramanik, On relative order and maximum term-related comparative growt rates o entire unctions, Journal o Tripura Matematical Society, 14, 2012), [7] S. K. Datta, T. Biswas, and R. Biswas, On relative order based growt estimates o entire unctions, International J. o Mat. Sci. & Engg. Appls. IJMSEA), 7, 2013), [8] S. K. Datta, T. Biswas, and R. Biswas, Comparative growt properties o composite entire unctions in te ligt o teir relative order, Te Matematics Student, 82, 2013), [9] S. K. Datta, T. Biswas, and C. Biswas, Growt analysis o composite entire and meromorpic unctions in te ligt o teir relative orders, International Scolarly Researc Notices, 2014), 6 pages. [10] S. K. Datta, T. Biswas, and C. Biswas, Measure o growt ratios o composite entire and meromorpic unctions wit a ocus on relative order, International J. o Mat. Sci. & Engg. Appls. IJMSEA), 8, 2014), [11] W.K. Hayman, Meromorpic Functions, Te Clarendon Press, Oxord, [12] B. K. Lairi and D. Banerjee, Relative order o entire and meromorpic unctions, Proc. Nat. Acad. Sci. India Ser. A., 69A), 1999), [13] D. Sato, On te rate o growt o entire unctions o ast growt, Bull. Amer. Mat. Soc., 69, 1963), [14] G. Valiron, Lectures on te general teory o integral unctions, Celsea Publising Company, Sanjib Kumar Datta Department o Matematics University o Kalyani P.O.-Kalyani, Dist-Nadia PIN , West Bengal India sanjib kr datta@yaoo.co.in Tanmay Biswas Rajbari, Rabindrapalli R. N. Tagore Road P.O.-Krisnagar, Dist-Nadia PIN , West Bengal India tanmaybiswas mat@redimail.com Received: Accepted: Unautenticated