Some Further Results on Uniqueness of Entire Functions and Fixed Points

Abstract

Abstract. In this paper, we investigate the uniqueness problem on entire functionssharing fixed points (ignoring multiplicities). Our main results improve and generalizesome results due to Zhang [13], Qi-Yang [10] and Dou-Qi-Yang [1]. 1. IntroductionIn this paper, a meromorphic function will mean meromorphic in the wholecomplex plane. We assume that the reader is familiar with standard notations andfundamental results of Nevanlinna Theory as explained in [12].We say that two meromorphic functions f and g share a small function a(z) IM(ignoring multiplicities) when f −a and g−a have the same zeros. If f and g havethe same zeros with the same multiplicities, then we say that f and g share a(z)CM (counting multiplicities).Let p be a positive integer and a ∈ C. We denote by N p (r, 1f−a ) the countingfunction of the zeros of f − a where an m-fold zero is counted m times if m ≤ pand p times if m > p. We denote by N L (r, 1f−1 ) the counting function for 1-pointsof both f(z) and g(z) about which f(z) has a larger multiplicity than g(z), withmultiplicity not being counted. We say that a finite value z