On the pre-Schwarzian and Schwarzian derivatives of log-harmonic mappings

TL;DR

This paper extends the concepts of pre-Schwarzian and Schwarzian derivatives to log-harmonic mappings in the unit disk, exploring their properties, applications, and conditions for finite norms within geometric function theory.

Contribution

It introduces definitions of these derivatives for log-harmonic mappings and establishes key properties, including a criterion for finite pre-Schwarzian norm and relationships with analytic functions.

Findings

Defined pre-Schwarzian and Schwarzian derivatives for log-harmonic mappings.

Established a necessary and sufficient condition for finite pre-Schwarzian norm.

Linked the pre-Schwarzian norm of log-harmonic mappings to that of analytic functions.

Abstract

In this paper, we introduce definitions of the pre-Schwarzian and the Schwarzian derivatives for any locally univalent log-harmonic mappings defined in the unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$. We explore the properties and applications of these concepts in the context of geometric function theory, and we also establish a necessary and sufficient condition for a non-vanishing log-harmonic mapping having a finite pre-Schwarzian norm. Additionally, we establish a relationship between the pre-Schwarzian norm of a non-vanishing log-harmonic mapping and that of a certain analytic function in $\mathbb{D}$.