Meromorphic functions sharing 1CM+1IM concerning periodicities and shifts
Bull. Korean Math. Soc. 2019 Vol. 56, No. 1, 45-56
Published online 2019 Jan 31
Xiao-Hua Cai, Jun-Fan Chen
Fujian Normal University; Fujian Province University
Abstract : The aim of this paper is to investigate the problems of meromorphic functions sharing values concerning periodicities and shifts. In this paper we prove the following result: Let $f(z)$ and $g(z)$ be two nonconstant entire functions, let $c\in\mathbb{C}\backslash\{0\}$, and let $a_1$, $a_2$ be two distinct finite complex numbers. Suppose that $\mu\left(f\right)\neq1$, $\rho_2\left(f\right)<1$, and $f(z)=f(z+c)$ for all $z\in\mathbb{C}$. If $f(z)$ and $g(z)$ share $a_1$ CM, $a_2$ IM, then $f(z)\equiv g(z)$. Moreover, examples are given to show that all the conditions are necessary.
Keywords : meromorphic function, shared value, periodicity, shift, unique\-ness
MSC numbers : 30D35, 30D30
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