Some Further Results on Weighted Sharing of Values for Meromorphic Functions Concerning a Result of Terglane

In this paper, we deal with the problem of meromorphic functions that havethree weighted sharing values, and obtain some uniqueness theorems which improve thosegiven by N. Terglane, Hong-Xun Yi & Xiao-Min Li, and others. Some examples are pro-vided to show that the results in this paper are best possible. 1. Introduction and main resultsIn this paper, by meromorphic functions we will always mean meromorphicfunctions in the complex plane. We adopt the standard notations in the Nevan-linna theory of meromorphic functions as explained in [3]. It will be convenient tolet E denote any set of positive real numbers of finite linear measure, not necessarilythe same at each occurrence. For any nonconstant meromorphic function h(z), wedenote by S(r,h) any quantity satisfying S(r,h) = o(T(r,h)) (r → ∞,r 6∈E).Let f(z) and g(z) be two nonconstant meromorphic functions, and let a ∈C ∪ {∞}, where C ∪ {∞} denotes the extended complex plane. We denote byN 0 (r,a,f,g) the counting function of the common zeros of f(z) − a and g(z) − a,and each point is counted only once, where f(z)−∞ means 1/f(z) (see [10]). We saythat f and g share the value a CM, provided that f and g have the same a−pointswith the same multiplicities. Similarly, we say that f and g share the value a IM,provided that f and g have the same a−points ignoring multiplicities (see [12]).Throughout this paper, we denote by N