Periodicity of Transcendental Entire Functions Sharing Set with their Shifts

TL;DR

This paper investigates conditions under which transcendental entire functions of hyper-order less than one are periodic, specifically when they share a certain set with their shifts, extending understanding of their periodicity properties.

Contribution

It establishes that such functions are periodic if they share a specific set with their shift, advancing the theory of value sharing and periodicity in complex analysis.

Findings

Functions of hyper-order less than one are periodic when sharing a set with their shift

Shares a certain set with weight two imply periodicity

Extends previous results on value sharing and periodicity

Abstract

This paper aims to study the periodicity of a transcendental entire function of hyper-order less than one. For a transcendental entire function of hyper order less than one and a non-zero complex constant $c$, $\mathfrak{f} (z) \equiv \mathfrak{f} (z + c)$ if they share a certain set with weight two.