Pre-Schwarzian derivative for Logharmonic mapppings

TL;DR

This paper introduces a new pre-Schwarzian derivative concept for logharmonic mappings, establishing its fundamental properties and invariance characteristics, and characterizes cases with holomorphic derivatives.

Contribution

It defines a novel pre-Schwarzian derivative for logharmonic mappings and proves key properties including chain rule and invariance, expanding the theoretical framework.

Findings

Pre-Schwarzian derivative is stable only under rotations of the identity.

Characterization of logharmonic mappings with holomorphic pre-Schwarzian derivatives.

Basic properties like chain rule and invariance are established for the new operator.

Abstract

We introduce a new definition of pre-Schwarzian derivative for logharmonic mappings and basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzain is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic.