A normality Criterion for a Family of Meromorphic Functions

TL;DR

This paper explores conditions under which a family of meromorphic functions is normal, focusing on the zeros of derivatives minus a holomorphic function, extending previous results on non-vanishing derivatives.

Contribution

It investigates the normality of meromorphic function families when their derivatives minus a holomorphic function have zeros, providing new insights beyond existing non-vanishing criteria.

Findings

Established conditions for normality based on zeros of derivatives minus holomorphic functions.

Extended previous non-vanishing derivative criteria to cases involving zeros.

Provided new characterizations of normal families in complex analysis.

Abstract

Schwick, in [6], states that let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and if for each $f\in\mathcal{F}$, $(f^n)^{(k)}\neq 1$, for $z\in D$, where $n, k$ are positive integers such that $n\geq k+3$, 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 $\mathcal{F}$?