A normality criterion corresponding to the defect relations

TL;DR

This paper introduces a new normality criterion for families of meromorphic functions based on defect relations, unifying and extending many existing results in complex analysis.

Contribution

It provides a general sufficient condition for normality that encompasses numerous known criteria as special cases.

Findings

The criterion links local conditions to global normality via defect relations.

It generalizes previous normality criteria in complex analysis.

The approach uses the Zalcman Lemma and defect relations to establish normality.

Abstract

Let ${\cal F}$ be a family of meromorphic functions on a domain $D$. We present a quite general sufficient condition for ${\cal F}$ to be a normal family. This criterion contains many known results as special cases. The overall idea is that certain comparatively weak conditions on ${\cal F}$ locally lead to somewhat stronger conditions, which in turn lead to even stronger conditions on the limit function $g$ in the famous Zalcman Lemma. Ultimately, the defect relations for $g$ force normality of ${\cal F}$.