International Journal of Analysis and Applications ISSN 2291-8639 Volume 6, Number 2 (2014), 139-143 http://www.etamaths.com ESTIMATION OF COMPARATIVE GROWTH PROPERTIES OF ENTIRE AND MEROMORPHIC FUNCTIONS IN TERMS OF THEIR RELATIVE ORDER ARKOJYOTI BISWAS Abstract. In this paper we discuss some comparative growth properties of entire and meromorphic functions on the basis of their relative order which improve some earlier results. 1. Introduction, Definitions and Notations. Let f be a non constant entire function in the open complex plane C and Mf (r) = max{|f(z)| : |z| = r}.Then Mf (r) is strictly increasing,its inverse M−1f : (|f(0)| ,∞) → (0,∞) exists and is such that lim r→∞ M−1f (r) = ∞. Two entire functions f and g are said to be asymptotically equivalent if there exists l (0 < l < ∞) such that Mf (r) Mg(r) → l as r →∞ and in that case we write f ∼ g.Clearly if f ∼ g then g ∼ f. The order and lower order of an entire function are defined in the following way: Definition 1. The order ρf and lower order λf of an entire function f are defined as follows: ρf = lim sup r→∞ log log Mf (r) log r and λf = lim inf r→∞ log log Mf (r) log r . The function f is said to be of regular growth if ρf = λf. The notion of order of an entire function was much improved by the introduction of the relative order of two entire functions.In this connection Bernal [1] gave the following definition. 2010 Mathematics Subject Classification. 30D30, 30D35. Key words and phrases. Entire and meromorphic functions; growth; relative order; asymptotic properties. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 139 140 ARKOJYOTI BISWAS Definition 2. [1]Let f and g be two entire functions.The relative order ρg(f) of f with respect to g is defined as follows: ρg(f) = inf{µ > 0 : Mf (r) < Mg(rµ) for all sufficiently large values of r} = lim sup r→∞ log M−1g Mf (r) log r . If we take g (z) as exp z then we see that ρg(f) = ρf and this shows that the relative order generalised the concept of the order of an entire function. Similarly the relative lower order λg(f) is defined as λg(f) = lim inf r→∞ log M−1g Mf (r) log r . For the case of a meromorphic function this generalisation was due to Lahiri and Banerjee [5].They introduced the notion of relative order ρg(f) of f with respect to g where f is meromorphic as follows: ρg(f) = inf{µ > 0 : Tf (r) < Tg(rµ) for all sufficiently large values of r} = lim sup r→∞ log T−1g Tf (r) log r . Similarly the relative lower order λg(f) of f with respect to g is defined by λg(f) = lim inf r→∞ log T−1g Tf (r) log r . In a recent paper Datta and Biswas [2] studied some growth properties of entire functions using relative order.In this paper we discuss some comparative growth properties of entire and meromorphic functions in terms of their relative order which improves some results of Datta and Biswas [2]. 2. Lemmas. In this section we present two lemmas which will be needed in the sequel. Lemma 1. If g1 and g2 be two entire functions with property (A) such that g1 ∼ g2.If f be meromorphic then ρg1 (f) = ρg2 (f). Lemma 1 follows from Theorem 5 {cf.[3]} on putting L(r) ≡ 1. Lemma 2 ([4]). If f,g be two meromorphic function and g is of regular growth.Then ρg(f) = ρf ρg . 3. Theorems. In this section we present the main results of our paper. Theorem 1. Let f be meromorphic and g,h be two entire functions with non zero finite orders.Then lim inf r→∞ log T−1g Tf (r) log T−1h Tf (r) ≤ ρg(f) ρh(f) ≤ lim sup r→∞ log T−1g Tf (r) log T−1h Tf (r) . ESTIMATION OF COMPARATIVE GROWTH PROPERTIES 141 Proof. From the definition of relative order we get for arbitrary ε (> 0) and for all sufficiently large values of r that (1) log T−1g Tf (r) < (ρg(f) + ε) log r. Also for a sequence of values of r tending to infinity we get that (2) log T−1g Tf (r) > (ρg(f) −ε) log r. Again for arbitrary ε (> 0) and for all sufficiently large values of r we obtain that (3) log T−1h Tf (r) < (ρh(f) + ε) log r and for a sequence of values of r tending to infinity we get that (4) log T−1h Tf (r) > (ρh(f) −ε) log r. Now from (1) and (4) we get for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1h Tf (r) < (ρg(f) + ε) (ρh(f) −ε) . As ε(> 0) is arbitrary it follows that (5) lim inf r→∞ log T−1g Tf (r) log T−1h Tf (r) ≤ ρg(f) ρh(f) . Also from (2) and (3) we get for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1h Tf (r) > (ρg(f) −ε) (ρh(f) + ε) . As ε(> 0) is arbitrary it follows that (6) lim sup r→∞ log T−1g Tf (r) log T−1h Tf (r) ≥ ρg(f) ρh(f) . Thus from (5) and (6) Theorem 1 follows.This completes the proof. � Corollary 1. If g and h are of regular growths then using Lemma 2 we get from Theorem 1 that lim inf r→∞ log T−1g Tf (r) log T−1h Tf (r) ≤ ρh ρg ≤ lim sup r→∞ log T−1g Tf (r) log T−1h Tf (r) . Corollary 2. If g and h are of regular growths and g ∼ h then using Lemma 1 we get from Theorem 1 that lim inf r→∞ log T−1g Tf (r) log T−1h Tf (r) ≤ 1 ≤ lim sup r→∞ log T−1g Tf (r) log T−1h Tf (r) . Remark 1. The converse of Corollary 2 is not always true which is evident from the following example. Example 1. Let g (z) = exp z and h (z) = exp(2z) so that Mg(r) = e r and Mh(r) = e 2r .Now Mg(r) Mh(r) → 0 as r →∞ and so g1 � g2 .Also Tg(r) = r π and Th(r) = r π 142 ARKOJYOTI BISWAS and therefore T−1g (r) = πr and T −1 h (r) = π 2 r. But lim r→∞ log T−1g Tf (r) log T−1h Tf (r) = lim r→∞ log πTf (r) log π 2 Tf (r) = 1. Theorem 2. Let f,h be meromorphic and g be entire functions with non zero finite order.Then lim inf r→∞ log T−1g Tf (r) log T−1g Th(r) ≤ ρg(f) ρg(h) ≤ lim sup r→∞ log T−1g Tf (r) log T−1g Th(r) . Proof. From the definition of relative order we get for arbitrary ε (> 0) and for all sufficiently large values of r that (7) log T−1g Th(r) < (ρg(h) + ε) log r and for a sequence of values of r tending to infinity we get that (8) log T−1g Th(r) > (ρg(h) −ε) log r. Now from (1) and (8) we get for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1g Th(r) < (ρg(f) + ε) (ρg(h) −ε) . As ε(> 0) is arbitrary it follows that (9) lim inf r→∞ log T−1g Tf (r) log T−1g Th(r) ≤ ρg(f) ρg(h) . Also from (2) and (7) we get for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1g Th(r) > (ρg(f) −ε) (ρg(h) + ε) . As ε(> 0) is arbitrary it follows that (10) lim sup r→∞ log T−1g Tf (r) log T−1g Th(r) ≥ ρg(f) ρg(h) . From (9) and (10) we obtain Theorem 2.This completes the proof. � Corollary 3. If g and h are of regular growths then using Lemma 2 we get from Theorem 2 that lim inf r→∞ log T−1g Tf (r) log T−1g Th(r) ≤ ρf ρh ≤ lim sup r→∞ log T−1g Tf (r) log T−1g Th(r) . Theorem 3. Let f, h be meromorphic and g,k be entire functions with non zero finite orders.Then lim inf r→∞ log T−1g Tf (r) log T−1k Th(r) ≤ ρg(f) ρk(h) ≤ lim sup r→∞ log T−1g Tf (r) log T−1k Th(r) . ESTIMATION OF COMPARATIVE GROWTH PROPERTIES 143 Proof. From the definition of relative order we get for arbitrary ε (> 0) and for all sufficiently large values of r that (11) log T−1k Th(r) < (ρk(h) + ε) log r. Also for a sequence of values of r tending to infinity we get that (12) log T−1k Th(r) > (ρk(h) −ε) log r. Now from (1) and (12) we get for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1k Th(r) < (ρg(f) + ε) (ρk(h) −ε) . As ε(> 0) is arbitrary it follows that (13) lim inf r→∞ log T−1g Tf (r) log T−1k Th(r) ≤ ρg(f) ρk(h) . Also from (2) and (11) we get for a sequence of values of r tending to infinity that log T−1g Tf (r) log T−1k Th(r) > (ρg(f) −ε) (ρk(h) + ε) . As ε(> 0) is arbitrary it follows that (14) lim sup r→∞ log T−1g Tf (r) log T−1k Th(r) ≥ ρg(f) ρk(h) . From (13) and (14) we obtain Theorem 3.This completes the proof. � References [1] L.Bernal: Orden relativo de crecimiento de functiones enteras, Collectanea Mathematica, 39(1988), 209-229. [2] S.K.Datta and T.Biswas: Relative order of composite entire functions and some related growth properties, Bull. Cal. Math. Soc., 102 (2010), 259-266. [3] S.K.Datta and A.Biswas: Estimation of relative order of entire and meromorphic functions in terms of slowly changing functions, Int.J.Contemp.Math.Sciences, 6 (2011), 1175-1186. [4] S.K.Datta and A.Biswas: A note on relative order of entire and meromorphic function- s,International J. of Math. Sci. & Engg. Appls. 6 (2012), 413-421. [5] B.K. Lahiri and D.Banerjee: Relative order of entire and meromorphic functions, Proc.Nat.Acad.Sci.India, 69 (1999), 339-354. Ranaghat Yusuf Institution, P.O.-Ranaghat, Dist-Nadia, PIN-741201,West Bengal, India