Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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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.

Meromorphic functions sharing 1CM+1IM concerning periodicities and shifts

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

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)