A Uniqueness Question of Meromorphic Functions of Finite Order Sharing Four Values

In 1992, Gundersen proposed the following famous question: suppose that two distinct and non-constant meromorphic functions f and g in the complex plane share 0, 1, c IM and ∞ CM, where c is a finite complex number such that c∉0, 1. Do f and g share 0, 1, c, ∞ CM? Based upon the assumption that f and g are two distinct and non-constant meromorphic functions of finite order and the assumption that c satisfies certain conditions, we have studied this question and proved the following result in this paper: suppose that f and g are two distinct and non-constant meromorphic functions of finite order such that they share 0, 1, c, ∞ IM, the positive integer M is defined as M=max{|C|: C∈C₀}, where C₀ is a set of all the finite critical values of f(z), that is to say C₀={C∈C: there exists a z∈C, such that f(z)=C and f′(z)=0}. Then when |c|>M, f and g share all the four values 0, 1, c, ∞ CM. As a corollary of this result, we give an affirmative answer to the above Gundersen′s question under the assumption that f and g are two distinct and non-constant meromorphic functions of finite order and the assumption that c satisfies certain conditions.