Differential Inequalities, Normality and Quasi-Normality

TL;DR

This paper establishes conditions under which families of meromorphic functions are normal or quasi-normal based on differential inequalities involving their derivatives and magnitudes.

Contribution

It proves that certain differential inequalities guarantee normality or quasi-normality of families of meromorphic functions in complex domains.

Findings

For alpha>1, the family is normal.

For alpha=1, the family is quasi-normal but not necessarily normal.

The results connect differential inequalities with normality properties.

Abstract

We prove that if D is a domain in C, alpha>1 and c>0, then the family F of functions meromorphic in D such that |f'(z)|/(1+|f(z)|^alpha)>c for every z in D is normalin D. For alpha=1, the same assumptions imply quasi-normality but not necessarily normality.