Harmonic pre-Schwarzian and its applications

TL;DR

This paper extends inequalities related to the pre-Schwarzian derivative from analytic univalent functions to harmonic univalent mappings, improving bounds on Jacobians and connecting to conjectures in harmonic mapping theory.

Contribution

It introduces new inequalities for harmonic pre-Schwarzian derivatives and enhances existing results on Jacobian majorization for harmonic mappings.

Findings

Extended inequalities for harmonic pre-Schwarzian derivatives

Improved bounds on Jacobian majorization in harmonic mappings

Connected results to a conjecture on univalent harmonic mappings

Abstract

The primary aim of this article is to extend certain inequalities concerning the pre-Schwarzian derivatives from the case of analytic univalent functions to that of univalent harmonic mappings defined on certain domains. This is done in two different ways. One of the ways is to connect with a conjecture on the univalent harmonic mappings. Also, we improve certain known results on the majorization of the Jacobian of functions in the affine and linear invariant family of sense-preserving harmonic mappings. This is achieved as an application of a corresponding distortion theorem in terms of the harmonic pre-Schwarzian derivative.