A note on spherical derivatives and normal families

TL;DR

This paper investigates conditions under which families of meromorphic functions are normal based on their spherical derivatives, establishing bounds and properties related to their behavior in the unit disk.

Contribution

It proves that families with spherical derivatives bounded away from zero are normal and establishes an upper bound for the infimum of the spherical derivative of any meromorphic function in the disk.

Findings

Families with spherical derivatives bounded away from zero are normal.

For any meromorphic function, the infimum of its spherical derivative in the disk is at most 1/2.

The spherical derivative's behavior characterizes normality in the family.

Abstract

We show that a family of meromorphic functions in the unit disk $\dk$ whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each $f$ meromorphic in $\dk$ we have $\inf_{z\in\dk} f^#(z)\le \frac{1}{2}$$ where $f^#$ denotes the spherical derivative of $f$.