Uniqueness of meromorphic function sharing three small functions CM with its $n-$ exact difference

TL;DR

This paper proves that a non-constant meromorphic function of hyper-order less than one sharing three small functions CM with its n-th difference must be identical to its difference, under certain periodicity conditions.

Contribution

It establishes a uniqueness theorem for meromorphic functions sharing three small functions CM with their n-th difference, extending previous results in difference and value sharing theory.

Findings

If $f$ and $ riangle_{ ext{eta}}^{n}f$ share three small functions CM, then $f$ equals its n-th difference.

The result applies to functions of hyper-order less than one with two of the shared functions being periodic.

The theorem generalizes known uniqueness results to the setting of differences and small functions.

Abstract

In this paper, we study the uniqueness of the difference of meromorphic functions. We prove the following result: Let $f$ be a non-constant meromorphic function of hyper-order less than $1$, let $\eta$ be a non-zero complex number, $n\geq1$, an integer, and let $a,b,c\in\hat{S}(f)$ be three distinct small functions and two of them be periodic small functions with period $\eta$. If $f$ and $\Delta_{\eta}^{n}f$ share $a,b,c$ CM, then $f\equiv\Delta_{\eta}^{n}f$.