Differential inequality of the second derivative that leads to normality

TL;DR

This paper establishes a new differential inequality involving the second derivative of meromorphic functions, providing a criterion that ensures the normality of a family of such functions.

Contribution

It introduces a novel differential inequality condition that guarantees the normality of families of meromorphic functions, expanding the theoretical understanding of normality criteria.

Findings

The family is normal if {|f|/(1+|f|^3)} is bounded away from zero.

The differential inequality relates the second derivative to the function's magnitude.

The result applies to meromorphic functions in complex domains.

Abstract

Let F be a family of functions meromorphic in a domain D. If {|f|/(1+|f|^3):f in F} is locally uniformly bounded away from zero, then F is normal.