A Normal Criterion Concerning Sequence of Functions and their Differential Polynomials

TL;DR

This paper establishes a new normality criterion for sequences of meromorphic functions based on the behavior of their differential polynomials and auxiliary holomorphic functions within the unit disk.

Contribution

It introduces a novel normality criterion involving differential polynomials and sequences of holomorphic functions, extending previous results in complex analysis.

Findings

Sequence of meromorphic functions is normal under given zero distribution conditions.

Differential polynomial conditions ensure normality of the function sequence.

Results apply to functions with poles of bounded multiplicity.

Abstract

In this paper, a normality criterion concerning a sequence of meromorphic functions and their differential polynomials is obtained. Precisely, we have proved: Let $\left\{f_j\right\}$ be a sequence of meromorphic functions in the open unit disk $\mathbb{D}$ such that, for each $j,$ $f_j$ has poles of multiplicity at least $m,~m\in\mathbb{N}.$ Let $\left\{h_j\right\}$ be a sequence of holomorphic functions in $\mathbb{D}$ such that $h_j\rightarrow h$ locally uniformly in $\mathbb{D},$ where $h$ is holomorphic in $\mathbb{D}$ and $h\not\equiv 0.$ Let $Q[f_j]$ be a differential polynomial of $f_j$ having degree $\lambda_Q$ and weight $\mu_Q.$ If, for each $j,$ $f_j(z)\neq 0$ and $Q[f_j]-h_j$ has at most $\mu_Q + \lambda_Q(m-1)-1$ zeros, ignoring multiplicities, in $\mathbb{D},$ then $\left\{f_j\right\}$ is normal in $\mathbb{D}.$