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

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 $f\equiv\Delta_{\eta}^{n}f$.