Quasi-normality induced by differential inequalities

TL;DR

This paper proves that a family of meromorphic functions satisfying a specific differential inequality is quasi-normal, using the Zalcman-Pang rescaling method, contributing to the understanding of function families in complex analysis.

Contribution

It introduces a new class of functions constrained by a differential inequality and proves their quasi-normality, expanding the theory of normal families.

Findings

The family ${ m extbf{ extit{F}}}_k$ is quasi-normal.

The proof utilizes the Zalcman-Pang rescaling method.

The result applies to all meromorphic functions satisfying the inequality.

Abstract

We show that the family ${\cal F}_k$ of all meromorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\ge C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. The proof relies mainly on the Zalcman-Pang rescaling method.