Unicity on meromorphic function sharing three small functions CM with its higher-order difference operators

TL;DR

This paper investigates the uniqueness of meromorphic functions sharing three small functions with their higher-order difference operators, establishing conditions under which the functions are identical.

Contribution

It proves a new uniqueness theorem for meromorphic functions sharing three small functions CM with their difference operators, extending previous results.

Findings

If a meromorphic function with hyperorder less than 1 shares three small functions CM with its difference operator, then they are identical.

The result applies to non-constant meromorphic functions with specific growth conditions.

The theorem generalizes known uniqueness results to higher-order difference operators.

Abstract

In this paper, we study the uniqueness of the shift of meromorphic functions. We prove: Let $f$ be a non-constant meromorphic function satisfying $\rho_{2}(f)<1$, let $\eta$ be a non-zero complex number, and let $a,b,c\in\hat{S}(f)$ be three distinct small functions. If $f$ and $\Delta^{n}_{\eta}f$ share $a,b,c$ CM, then $f\equiv \Delta^{n}_{\eta}f$.