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 $\mathbb{D}\subset \mathbb{C}$, $\psi_1$, $\psi_2$, and $\psi_3$ be three meromorphic functions in the unit disc $\mathbb{D}$, continuous on $ \partial{\mathbb{D}}:=\{z\in\mathbb{C}\,:\,|z|=1\}$, such that $\psi_i(z)\neq\psi_j(z)$ $(1\leq i<j\leq 3)$ $\partial\mathbb{D}$. We prove that, if $\psi_1$, $\psi_2$, and $\psi_3$ share the function $f$ on $\mathbb{D}$, then $f$ is normal. Building upon this, we further establish an additional criterion for normal functions.