A note on Schwarzian derivatives and normal families

TL;DR

This paper explores criteria for normality of families of analytic and meromorphic functions based on their derivatives and Schwarzian derivatives, providing new insights into their boundedness and normality conditions.

Contribution

It introduces new criteria linking the normality of function families to the boundedness of their derivatives and Schwarzian derivatives, advancing understanding in complex analysis.

Findings

Established a criterion for local boundedness and normality based on derivatives.

Analyzed the relationship between normality domains and Schwarzian derivatives.

Provided criteria for the normality of Schwarzian derivative families.

Abstract

We establish a criterion for local boundedness and hence normality of a family $\F$ of analytic functions on a domain $D$ in the complex plane whose corresponding family of derivatives is locally bounded. Furthermore we investigate the relation between domains of normality of a family $\F$ of meromorphic functions and its corresponding Schwarzian derivative family. We also establish some criterion for the Schwarzian derivative family of a family $\F$ of analytic functions on a domain $D$ in the complex plane to be a normal family.