Mathematics

CLC number: O174.5

Meromorphic Functions Sharing Two Values

School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China

^† Corresponding author. E-mail: alexehuang@sina.com

Received: 25 April 2023

Abstract

Let $f$ and $g$ be two nonconstant meromorphic functions, and $n$ be a positive integer. If $f^n\left(z\right)$ and $g^n\left(z\right)$ share 1 CM (counting multiplicities), $f\left(z\right)$ and $g\left(z\right)$ share $\mathrm{\infty }$ IM (ignoring multiplicities), and $\overline{N}(r,\frac{\mathrm{1}}{f})+\overline{N}(r,\frac{\mathrm{1}}{g})=S(r,f)+S(r,g)$, then either $f\left(z\right)\equiv tg\left(z\right)$ or $f\left(z\right)g\left(z\right)\equiv t$, where $t^n=\mathrm{1}$. As an application, shared values problems of a meromorphic function related to its shifts and difference operators are also investigated.