Normality Criteria of Meromorphic Functions That Share a Holomorphic Function

Abstract

Let $\mathcal{F}$ be a family of meromorphic functions defined in $D$, let $\psi \left(\cancel{\not\equiv }0\right)$, $a_0,a_1,...,a_{k-1}$ be holomorphic functions in $D$, and let $k$ be a positive integer. Suppose that, for every function $f\in \mathcal{F}$, $f\ne 0$, $P\left(f\right)=f^{\left(k\right)}+a_{k-1}{f}^{(k-1)}+\cdots +a_{1}f\text{'}+a_{0}f\ne 0$ and, for every pair functions $(f,g)\in \mathcal{F}$, $P\left(f\right)$, $P\left(g\right)$ share $\psi$, then $\mathcal{F}$ is normal in $D$.