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

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 $a\ne 0$ and $-a$ IM.

11 pages

References in corpus (1)

• Nevanlinna theory for the difference operator