A Function-Sharing Criterion for Normal Functions

Abstract

In this paper, we present a function-sharing criterion for the normality of meromorphic functions. Let $f$ be a meromorphic function in the unit disc $\mathbb{D}\subset \mathbb{C}$, $\ddot{I}{ˆ}_1$, $\ddot{I}{ˆ}_2$, and $\ddot{I}{ˆ}_3$ be three meromorphic functions in the unit disc $\mathbb{D}$, continuous on $\partial \mathbb{D}:=\{z\in \mathbb{C}:|z|=1\}$, such that $\ddot{I}{ˆ}_i(z)\ne \ddot{I}{ˆ}_j(z)$ $(1\le i<j\le 3)$ $\partial \mathbb{D}$. We prove that, if $\ddot{I}{ˆ}_1$, $\ddot{I}{ˆ}_2$, and $\ddot{I}{ˆ}_3$ share the function $f$ on $\mathbb{D}$, then $f$ is normal. Building upon this, we further establish an additional criterion for normal functions.