Meromorphic Functions Sharing four Values with their Difference Operators

Abstract

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and $f(z+\eta )$ share a, b, c IM, where $\eta$ is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then $f(z)=f(z+\eta )$ for all $z\in \mathbb {C}$. We prove that if a non-constant finite order entire function f and its n-th difference operator $\Delta ^n_{\eta }f$ share $a_1$, $a_2$, $a_3$ IM, where n is a positive integer, $\eta \ne 0$ is a finite complex value, while $a_1$, $a_2$, $a_3$ are three distinct finite complex values, then $f=\Delta ^n_{\eta }f$. The main results in this paper also improve Theorems 1.1 and 1.2 from Li and Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).