A normality Criterion for a Family of Meromorphic Functions

Abstract

Schwick, in [6], states that let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and if for each $f\in \mathcal{F}$, $(f^n{)}^{(k)}\ne 1$, for $z\in D$, where $n,k$ are positive integers such that $n\ge k+3$, then $\mathcal{F}$ is a normal family in $D$. In this paper, we investigate the opposite view that if for each $f\in \mathcal{F}$, $(f^n{)}^{(k)}(z)-\psi (z)$ has zeros in $D$, where $\psi (z)$ is a holomorphic function in $D$, then what can be said about the normality of the family $\mathcal{F}$?