Bull. Korean Math. Soc. 2019; 56(1): 45-56
Online first article July 12, 2018 Printed January 31, 2019
https://doi.org/10.4134/BKMS.b180076
Copyright © The Korean Mathematical Society.
Xiao-Hua Cai, Jun-Fan Chen
Fujian Normal University; Fujian Province University
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
Supported by: Project supported by the National Natural Science Foundation of China (Grant No. 11301076), the Natural Science Foundation of Fujian Province, China (Grant No. 2018J01658) and Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (NO. SX201801)
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