A normality criterion for a family of meromorphic functions

TL;DR

This paper establishes a new normality criterion for families of meromorphic functions based on the behavior of a differential polynomial and its zeros, extending classical results in complex analysis.

Contribution

It introduces a novel normality criterion involving differential polynomials and zero multiplicity conditions for meromorphic function families.

Findings

Proves normality under specified zero multiplicity conditions.

Extends classical normality criteria to differential polynomial settings.

Provides conditions ensuring the family of functions is normal.

Abstract

We consider a family $\mathscr{F}$ of meromorphic functions defined in a domain $D$, a holomorphic function $\psi$ and a homogeneous differential polynomial $ P[f] $ of degree $d$ with weight $w$. In this paper, we prove the normality of $\mathscr{F}$ under certain conditions such as $f\neq 0$, $P[f]\neq 0$ and all the zeros of the function $P[f] - \psi^d$ have multipicity at least $\displaystyle{\frac{w+1}{w-1}}$, for each $f \in \mathscr{F}$.