Unicity of Meromorphic Function Sharing One Small Function with Its Derivative

In this article, a meromorphic function means meromorphic in the open complex plane. We assume that the reader is familiar with the Nevanlinna theory of meromorphic functions and the standard notations such as T r, f , m r, f , N r, f ,N r, f , and so on. Let f and g be two nonconstant meromorphic functions; a meromorphic function a z /≡∞ is called a small functions with respect to f provided that T r, a S r, f . Note that the set of all small function of f is a field. Let b z be a small function with respect to f and g.We say that f and g share b z CM IM provided that f − b and g − b have same zeros counting multiplicities ignoring multiplicities . Moreover, we use the following notations. Let k be a positive integer. We denote by Nk r, 1/ f − a the counting function for the zeros of f − a with multiplicity ≤ k and by Nk r, 1/ f − a the corresponding one for which the multiplicity is not counted. Let N k r, 1/ f − a be the counting function for the zeros of f − a with multiplicity ≥ k, and let N k r, 1/ f − a be the corresponding one for which the multiplicity is not counted. Set Nk r, 1/ f − a N r · 1/ f − a N 2 r, 1/ f − a · · · N k r, 1/ f − a . And we define