VALUE DISTRIBUTION OF THE PRODUCT OF A MEROMORPHIC FUNCTION AND ITS DERIVATIVE. Abstract

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1 I. LAHIRI AND S. DEWAN KODAI MATH. J. 26 (2003), VALUE DISTRIBUTION OF THE PRODUCT OF A MEROMORPHIC FUNCTION AND ITS DERIVATIVE Indrajit Lahiri and Shyamali Dewan* Abstract In the paper we discuss the value distribution of the product of a meromorphic function and its derivative and we improve a recent result of K. W. Yu. 1. Introduction and definitions Let f be a transcendental meromorphic function defined in the open complex plane C. Hayman [5] proved the following theorem. Theorem A. If n ðb3þ is an integer then c ¼ f n f 0 assumes all finite values, except possibly zero, infinitely many times. He further conjectured [7] that Theorem A remains valid even if n ¼ 1or2. Mues [9] proved the result for n ¼ 2 and the case n ¼ 1 was proved by Bergweiler and Eremenko [1] and independently by Chen and Fang [3]. A natural question of investigating the value distribution of ff 0 a, where a ¼ aðzþ is a non-zero meromorphic function satisfying Tðr; aþ ¼ Sðr; f Þ, was raised and a number of researchers have worked on the problem. We call a meromorphic function a 1 aðzþ a small function of f if Tðr; aþ ¼ Sðr; f Þ. Following two theorems can be derived from two inequalities proved by Zhang {[12], see also [11]}. Theorem B. If dðy; f Þ > 7=9 then ff 0 a has infinitely many zeros, where a ð20; yþ is a small function of f. Theorem C. If 2dð0; f Þþdðy; f Þ > 1 then ff 0 a has infinitely many zeros, where a ð20; yþ is a small function of f. * The second author is thankful to C.S.I.R. for awarding her a junior research fellowship Mathematics Subject Classification: 30D35. Keywords and phrases: Meromorphic function, derivative, value distribution. Received May 17, 2002; revised July 19,

2 96 indrajit lahiri and shyamali dewan However in Theorem C the condition 2dð0; f Þþdðy; f Þ > 1 can easily be replaced by the weaker condition 2Yð0; f Þ þ Yðy; f Þ > 1. The following result of Bergweiler [2] is worth mentioning. Theorem D. If f is of finite order and a is a polynomial then ff 0 a has infinitely many zeros. In Theorem B and Theorem C we see that some conditions have to be imposed on f to achieve the desired result. On the other hand, though in Theorem D no restriction, except the order restriction, is imposed on f, the desired result is achieved only for polynomials in contrast to arbitrary small functions as the target. Recently Yu [11] treated the general case but instead of a single small function he achieved the result for a small function and its negative as a pair of targets. His result can be stated as follows. Theorem E. If a ð20; yþ is a small function of f then at least one of ff 0 a and ff 0 þ a has infinitely many zeros. In the paper we prove a result on the value distribution of ð f Þ n 0 ð f ðkþ Þ n 1, where n 0 ðb2þ, n 1, k are positive integers and as a consequence of this we improve Theorem E though most probably one should not expect any corresponding improvement of Theorem D because of the condition n 0 b 2. Throughout the paper we denote by f a transcendental meromorphic function defined in the open complex plane C. We do not explain the standard notations and definitions of the value distribution theory as those are available in [6]. Definition [8]. Let m be a positive integer. We denote by Nðr; a; f jamþ (Nðr; a; f jbmþ) the counting function of those a-points of f whose multiplicities are not greater (less) than m, where each a-point is counted according to its multiplicity. In a like manner we define Nðr; a; f j <mþ and Nðr; a; f j >mþ. Also Nðr; a; f jamþ, Nðr; a; f jbmþ, Nðr; a; f j <mþ and Nðr; a; f j >mþ are defined similarly where in counting the a-points of f we ignore the multiplicities. Finally we agree to take Nðr; a; f jayþ 1 Nðr; a; f Þ and Nðr; a; f jayþ 1 Nðr; a; f Þ. 2. Lemma In this section we prove a lemma which is required in the sequel. Lemma. If Nðr; 0; f ðkþ j f 0 0Þ denotes the counting function of those zeros of f ðkþ which are not the zeros of f, where a zero of f ðkþ is counted according to its multiplicity, then Nðr; 0; f ðkþ j f 0 0Þ a knðr; y; f ÞþNðr; 0; f j <kþþknðr; 0; f jbkþþsðr; f Þ:

3 Proof. By the first fundamental theorem and Milloux theorem {p. 55 [6]} we get! Nðr; 0; f ðkþ j f 0 0Þ a N r; 0; f ðkþ f!! a N r; f ðkþ þ m r; f ðkþ þ Oð1Þ f f This proves the lemma. value distribution of the product 97 a knðr; y; f ÞþNðr; 0; f j <kþþknðr; 0; f jbkþþsðr; f Þ: r 3. The main result In this section we discuss the main result of the paper. Theorem. Let c ¼ðfÞ n 0 ð f ðkþ Þ n 1, where n 0 ðb2þ, n 1 and k are positive integers such that n 0 ðn 0 1Þþð1þkÞðn 0 n 1 n 0 n 1 Þ > 0. Then 1 1 þ k n 0 þ k n 0 ð1 þ kþ Tðr; cþ a Nðr; a; cþþsðr; cþ ðn 0 þ kþfn 0 þð1þkþn 1 g for any small function a ð20; yþ of f. and Proof. First we note that {cf. [4, 10]} Tðr; f ÞþSðr; f Þ a CTðr; cþþsðr; cþ Tðr; cþ a fn 0 þð1 þ kþn 1 gtðr; f ÞþSðr; f Þ; where C is a constant. So it is clear that if a ð20; yþ is a small function of f then a is also a small function of c and vice-versa. Hence by Nevanlinna s three small functions theorem {p. 47 [6]} we get ð1þ Tðr; cþ a Nðr; 0; cþþnðr; y; cþþnðr; a; cþþsðr; cþ; where Nðr; a; cþ ¼Nðr; 0; c aþ. Now by the lemma we get ð2þ Nðr; 0; cþ a Nðr; 0; f ÞþNðr; 0; f ðkþ j f 0 0Þ a Nðr; 0; f ÞþkNðr; y; f ÞþNðr; 0; f j <kþ þ knðr; 0; f jbkþþsðr; f Þ a ð1 þ kþnðr; 0; f ÞþkNðr; y; f ÞþSðr; f Þ:

4 98 indrajit lahiri and shyamali dewan Again we see that Nðr; 0; cþ Nðr; 0; cþ b fð1 þ kþn 0 þ n 1 1gNðr; 0; f jb1 þ kþ Hence from (2) we get þðn 0 1ÞNðr; 0; f jakþ: Nðr; 0; cþ a ð1 þ kþnðr; 0; f jb1 þ kþþknðr; y; f Þ þ 1 þ k ½Nðr; 0; cþ Nðr; 0; cþ n 0 1 fð1þkþn 0 þ n 1 1gNðr; 0; f jb1 þ kþš þ Sðr; f Þ n 0 þ k 1 þ k Nðr; 0; cþ a Nðr; 0; cþþknðr; y; f Þ n 0 1 n 0 1 þ 1 þ k ð1 þ kþfð1 þ kþn 0 þ n 1 1g Nðr; 0; f jb1 þ kþ n 0 1 þ Sðr; f Þ a 1 þ k Nðr; 0; cþþknðr; y; f ÞþSðr; f Þ n 0 1 ð3þ Nðr; 0; cþ a 1 þ k n 0 þ k Nðr; 0; cþþkðn 0 1Þ Nðr; y; f ÞþSðr; f Þ: n 0 þ k If z 0 is a pole of f with multiplicity p then z 0 is a pole of c with multiplicity n 0 p þðpþkþn 1 b n 0 þð1þkþn 1. Hence ð4þ Nðr; y; cþ b fn 0 þð1þkþn 1 gnðr; y; cþ: Since Nðr; y; cþ ¼Nðr; y; f Þ and Sðr; cþ ¼Sðr; f Þ, from (1), (3) and (4) we get Tðr; cþ a 1 þ k Nðr; 0; cþþ n 0 þ k 1 þ kðn 0 1Þ Nðr; y; f ÞþNðr; a; cþþsðr; cþ n 0 þ k a 1 þ k n 0 ð1 þ kþ Nðr; 0; cþþ Nðr; y; cþþnðr; a; cþ n 0 þ k ðn 0 þ kþfn 0 þð1þkþn 1 g þ Sðr; cþ 1 1 þ k n 0 þ k n 0 ð1 þ kþ Tðr; cþ a Nðr; a; cþþsðr; cþ: ðn 0 þ kþfn 0 þð1þkþn 1 g This proves the theorem. r The following corollary improves Theorem E.

5 Corollary. Let F ¼ ff ðkþ, where k is a positive integer. Then for any small function a ð20; yþ of f 2 Yða; FÞþYð a; FÞ a 2 ð2 þ kþ 2 : Proof. Since a 2 is also a small function of f, we get from the theorem for n 0 ¼ n 1 ¼ 2 " # ð1 þ kþð3 þ kþ 1 ð2 þ kþ 2 Tðr; F 2 Þ a Nðr; a 2 ; F 2 ÞþSðr; FÞ " # ð1 þ kþð3 þ kþ 2 1 ð2 þ kþ 2 Tðr; FÞ a Nðr; a; FÞþNðr; a; FÞþSðr; FÞ; which shows that Yða; FÞþYð a; FÞ a This proves the corollary. value distribution of the product 99 r 2ð1 þ kþð3 þ kþ ð2 þ kþ 2 2 ¼ 2 ð2 þ kþ 2 : Authors are thankful to the referee for valuable com- Acknowledgement. ments. References [1] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, 11 (1995), [ 2] W. Bergweiler, On the product of a meromorphic function and its derivative, Bull. Hong Kong Math. Soc., 1 (1997), [3] H. H. Chen and M. L. Fang, The value distribution of f n f 0, Sci. China Ser. A, 38 (1995), [4] W. Doeringer, Exceptional values of di erential polynomials, Pacific J. Math., 98 (1982), [ 5] W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2), 70 (1959), [ 6] W. K. Hayman, Meromorphic Functions, Oxford Math. Monogr., Clarendon Press, Oxford, [7] W. K. Hayman, Research Problems in Function Theory, The Athlone Press University of London, London, [ 8] I. Lahiri, Value distribution of certain di erential polynomials, Int. J. Math. Math. Sci., 28 (2001), [ 9] E. Mues, Über ein Problem von Hayman, Math. Z., 164 (1979), [10] A. P. Singh, On order of homogeneous di erential polynomials, Indian J. Pure Appl. Math., 16 (1985),

6 100 indrajit lahiri and shyamali dewan [11] K.-W. Yu, A note on the product of meromorphic functions and its derivatives, Kodai Math. J., 24 (2001), [12] Q. D. Zhang, The value distribution of fðzþ f ðzþf 0 ðzþ, Acta Math. Sinica, 37 (1994), (in Chinese). Department of Mathematics University of Kalyani West Bengal India indrajit@cal2.vsnl.net.in Department of Mathematics University of Kalyani West Bengal India