On periodicity of a meromorphic function when sharing two sets IM

TL;DR

This paper establishes new sufficient conditions for the periodicity of meromorphic functions sharing two specific sets, improving previous results and demonstrating that such functions need not be of finite order.

Contribution

It provides improved criteria for meromorphic function periodicity when sharing two sets, extending prior work and including examples and open questions.

Findings

Meromorphic functions sharing two sets are periodic under certain conditions.

The sets involved cannot be replaced by arbitrary sets.

Finite order is not necessary for the functions to be periodic.

Abstract

In this paper, we have investigated the sufficient conditions for periodicity of meromorphic functions and obtained two results directly improving the result of \emph{Bhoosnurmath-Kabbur} \cite{Bho & Kab-2013}, \emph{Qi-Dou-Yang} \cite{Qi & Dou & Yan-ADE-2012} and \emph{Zhang} \cite{Zha-JMMA-2010}. Let $\mathcal{S}_{1}=\left\{z:\displaystyle\int_{0}^{z-a}(t-a)^n(t-b)^4dt+1=0\right\}$ and $\mathcal{S}_{2}=\bigg\{a,b\bigg\}$, where $n\geq 4(n\geq 3)$ be an integer.\emph{Let $f(z)$ be a non-constant meromorphic (entire) function satisfying $\ol E_{f(z)}(\mathcal{S}_j)=\ol E_{f(z+c)}(\mathcal{S}_j), (j=1,\;2)$ then $f(z)\equiv f(z+c)$.} Some examples have been exhibited to show that, it is not necessary that meromorphic function should be of finite order and also to show that the sets considered in the paper simply can't be replace by arbitrary sets. At the last section, we have posed an open question for the further improvement of the results of this paper.