Some Uniqueness Results Related to Meromorphic Function That Share a Small Function with Its Derivative

In this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane C. We adopt the standard notations of the Nevanlinna theory of meromorphic functions as explained in [6]. It will be convenient to let E denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. For a non-constant meromorphic function h, we denote by T (r, h) the Nevanlinna characteristic of h and by S(r, h) any quantity satisfying S(r, h) = o{T (r, h)}, as r −→ ∞ and r 6∈ E. Let f and g be two non-constant meromorphic functions and let a be a complex number. We say that f and g share a CM, provided that f − a and g − a have the same zeros with the same multiplicities. Similarly, we say that f and g share a IM, provided that f−a and g−a have the same zeros ignoring multiplicities. In addition, we say that f and g share ∞ CM, if 1/f and 1/g share 0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM. A meromorphic function a is said to be a small function of f provided that T (r, a) = S(r, f), that is T (r, a) = o(T (r, f)) as r −→∞, r 6∈ E. In 1979 Mues and Steinmetz considered the following uniqueness result of an entire function and its derivative when they share two values IM.