TAILIEUCHUNG - Uniqueness of derivatives of meromorphic functions sharing two or three sets
In the paper we consider the problem of uniqueness of derivatives of meromorphic functions when they share two or three sets and obtained five results which will improve all the existing results. | Turk J Math 34 (2010) , 21 – 34. ¨ ITAK ˙ c TUB doi: Uniqueness of derivatives of meromorphic functions sharing two or three sets Abhijit Banerjee and Pranab Bhattacharjee Abstract In the paper we consider the problem of uniqueness of derivatives of meromorphic functions when they share two or three sets and obtained five results which will improve all the existing results. Key word and phrases: Meromorphic functions, uniqueness, weighted sharing, derivative, shared set. 1. Introduction, definitions and results In this paper by meromorphic functions we will always mean meromorphic functions in the complex plane. It will be convenient to let E denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. For any non-constant meromorphic function h(z) we denote by S(r, h) any quantity satisfying S(r, h) = o(T (r, h)) (r −→ ∞, r ∈ E). Let f and g be two non-constant meromorphic functions and let a be a finite complex number. We say that f and g share a CM, provided that f − a and g − a have the same zeros with the same multiplicities. Similarly, we say that f and g share a IM, provided that f − a and g − a have the same zeros ignoring multiplicities. In addition, we say that f and g share ∞ CM, if 1/f and 1/g share 0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM. We denote by T (r) the maximum of T r, f (k) and T r, g(k) . The notation S(r) denotes any quantity satisfying S(r) = o(T (r)) (r −→ ∞, r ∈ E). Let S be a set of distinct elements of C∪ {∞} and Ef (S) = a∈S {z : f(z)−a = 0} , where each zero is counted according to its multiplicity. If we do not count the multiplicity the set Ef (S) = a∈S {z : f(z) − a = 0} is denoted by E f (S). If Ef (S) = Eg (S) we say that f and g share the set S CM. On the other hand, if E f (S) = E g (S), we say that f and g share the set S IM. F. Gross first considered the uniqueness of meromorphic functions that share sets of
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