When two functions are sharing four values and a set (Q1300186)

In this paper, the authors obtain a few of uniqueness theorems on meromorphic functions. The main result is a type of Nevanlinna five values theorem as follows: Let \(a_1,a_2,a_3\) and \(a_4\) be four distinct values in \(\mathbb C\) and let \(S\) be a subset of \(\mathbb C\) disjoining with \(\{a_1,\dots{},a_4\}\). If two nonconstant meromorphic functions \(f\) and \(g\) in \(\mathbb C\) satisfy the following conditions \[ f^{-1}(S)\subset g^{-1}(S),\quad f^{-1}(a_i)=g^{-1}(a_i)\qquad (i=1,\dots{},4), \] it follows that either \(f=g\) or \(f\) is a Möbius transformation of \(g\). Furthermore, if \(S\) contains an odd number of values, then \(f=g\).