Meromorphic Functions Sharing a Nonzero Polynomial IM

Abstract

abstract. We study the uniqueness of meromorphic functions concerning nonlinear dif-ferential polynomials sharing a nonzero polynomial IM. Though the main concern of thepaper is to improve a recent result of the present author [12], as a consequence of the mainresult we also generalize two recent results of X. M. Li and L. Gao [11]. 1. Introduction, De nitions and ResultsIn this paper, by meromorphic functions we will always mean meromorphicfunctions in the complex plane. We adopt the standard notations in the Nevan-linna theory of meromorphic functions as explained in [7], [15] and [16]. For anonconstant meromorphic function h, we denote by T(r;h) the Nevanlinna charac-teristic of hand by S(r;h) any quantity satisfying S(r;h) = ofT(r;h)gas r!1possibly outside a set of nite linear measure. A meromorphic function a(z)(61)is called a small function with respect to f, provided that T(r;a) = S(r;f).Let f and gbe two nonconstant meromorphic functions, and let abe a nitevalue. We say that fand gshare the value aCM (counting multiplicities), providedthat f aand g ahave the same set of zeros with the same multiplicities. Similarly,we say that fand gshare aIM (ignoring multiplicities), provided that f aandg ahave the same set of zeros ignoring multiplicities.In 1959, W. K. Hayman (see [6], Corollary of Theorem 9) proved the followingtheorem:Theorem A. Let f be a transcendental meromorphic function and n(3) is aninteger. Then f