Uniqueness of first derivatives and differences in meromorphic functions and the characterization of entire function periodicity

TL;DR

This paper investigates the uniqueness of meromorphic functions sharing specific values with their derivatives and differences, and provides conditions under which entire functions are periodic, advancing understanding in complex analysis.

Contribution

It resolves an open problem on meromorphic function uniqueness related to shared values and establishes new criteria for the periodicity of transcendental entire functions.

Findings

Meromorphic functions sharing two finite values with their difference operator are unique under certain conditions.

Provides a solution to an open problem for functions with finite hyper order.

Establishes sufficient conditions for entire functions to be periodic.

Abstract

The objective of the paper is twofold. The first objective is to study the uniqueness problem of meromorphic function $f(z)$ when $f^{(1)}(z)$ shares two distinct finite values $a_1$, $a_2$ and $\infty$ CM with $\Delta_cf(z)$. In this context, we provide a result that resolves the open problem posed by Qi et al. [Comput. Methods Funct. Theory, 18 (2018), 567-582] for the case when hyper order of the function is less than $\infty$. The second objective is to establish sufficient conditions for the periodicity of transcendental entire functions. In this direction, we obtain a result that affirms the question raised by Wei et al. [Anal. Math., 47 (2021), 695-708.]