Meromorphic Function Sharing Two Small Functions with Its Derivative

Abstract

In this paper, we deal with the problem of uniqueness of meromorphic func- tions that share two small functions with their derivatives, and obtain the following result which improves a result of Yao and Li: Let f(z) be a nonconstant meromorphic function, k > 5 be an integer. If f(z) and g(z) = a1(z)f(z) + a2(z)f (k) (z) share the value 0 CM, and share b(z) IM, NE(r,f = 0 = f (k) ) = S(r), then f g, where a1(z), a2(z) and b(z) are small functions of f(z). 1. Introduction and main results In this paper, a meromorphic function will mean meromorphic in the whole complex plane. We say that two meromorphic functions f and g share a finite value a IM (ignoring multiplicities) when f a and g a have the same zeros. If f a and g a have the same zeros with the same multiplicities, then we say that f and g share the value a CM (counting multiplicities). Denote by N(r,f = b = g) the reduced counting function of the common zeros of f b and g b ignoring the multiplicities, and NE(r,f = b = g) the reduced counting function of the common zeros of f b and g b with the same multiplicities. We say that f and g share b IM provided that