Entire Functions and Their Derivatives Share Two Finite Sets

Abstract

Abstract. In this paper, we study the uniqueness of entire functions and prove thefollowing theorem. Let n(5), k be positive integers, and let S 1 = fz : z n = 1g,S 2 = fa 1 ;a 2 ; ;a m g, where a 1 ;a 2 ; ;a m are distinct nonzero constants. If two non-constant entire functions f and g satisfy E f (S 1 ;2) = E g (S 1 ;2) and E f (k) (S 2 ;1) =E g (k) (S 2 ;1), then one of the following cases must occur: (1) f = tg, fa 1 ;a 2 ; ;a m g=tfa 1 ;a 2 ; cz;a m g, where t is a constant satisfying t n = 1; (2) f(z) = de , g(z) = td e cz ,fa 1 ;a 2 ; 2;a m g= ( 1) k c k tf 1a 1 ; ; 1a m g, where t, c, d are nonzero constants and t n = 1.The results in this paper improve the result given by Fang (M.L. Fang, Entire functionsand their derivatives share two nite sets, Bull. Malaysian Math. Sc. Soc. 24(2001),7-16). 1. Introduction, de nitions and resultsLet f and gbe two nonconstant meromorphic functions de ned in the opencomplex plane C. If for some a2C[f1g, fand ghave the same set of a-pointswith the same multiplicities then we say that fand gshare the value aCM (count-ing multiplicities). If we do not take the multiplicities into account, f and garesaid to share the value aIM (ignoring multiplicities). We assume that the readeris familiar with the notations of Nevanlinna theory that can be found, for instance,in [5] or [9].Let Sbe a set of distinct elements of C[f1gand E