Uniqueness of entire functions sharing two pairs of values with its difference operator

TL;DR

This paper proves that certain entire functions of hyper-order less than one cannot share two specific pairs of values with their difference operators unless they are trivial, advancing the understanding of value sharing in complex analysis.

Contribution

It establishes new uniqueness results for entire functions sharing two pairs of values with their difference operators, under hyper-order constraints.

Findings

No non-constant entire function of hyper-order less than 1 shares two pairs of values with its difference operator.

Provides conditions under which entire functions sharing values with their difference operators must be trivial.

Extends previous results on value sharing to the context of difference operators and hyper-order constraints.

Abstract

In this paper, we investigate the sharing values problem that entire function $f(z)$ and its first order difference operator $\Delta_{\eta}f(z)$ share two distinct pairs of finite values IM. We prove: Let $f(z)$ be a non-constant entire function of hyper-order less than $1$, let $\eta$ be a non-zero complex number, and let $a$ be a nonzero finite number. Then there exists no such entire function so that $ f(z)$ and $\Delta_{\eta}f(z)$ share $(0,0)$ and $(a,-a)$ IM. Furthermore, using a result in Wang-Chen-Hu \cite{wch}, we obtain some uniqueness results that when $ f(z)$ and $\Delta_{\eta}f(z)$ share $a\neq0$ and $-a$ IM.