“2CM+1IM” Theorem for Periodic Meromorphic Functions

Abstract

Generally, the concrete relations between two nonconstant meromorphic functions that share two values CM and one value IM are hard to determine. However, for the class \({\mathcal{F}}\) of all nonconstant meromorphic functions with the same period \({c\neq0}\), we prove a result in this paper that: let \({f(z), g(z) \in \mathcal{F}}\) such that the hyper-order \({\rho_2(f) < 1}\), if \({f(z), g(z)}\) share \({0, \infty}\) CM and 1 IM, then either \({f(z)\equiv g(z)}\) or \({f(z)=e^{az+b}g(z)}\) and \({\mu(f)=\mu(g)=1}\), where \({a=\frac{2k\pi i}{c}}\) and k is some integer. As an application of this result, we obtain an uniqueness theorem for elliptic meromorphic functions. Moreover, examples are given to illustrate that all the conditions are necessary and sharp.