Pre-Schwarzian and Schwarzian derivatives of harmonic mappings

TL;DR

This paper introduces new definitions for the pre-Schwarzian and Schwarzian derivatives of harmonic mappings in the complex plane, extending classical concepts without restrictions on dilatation, and proves related theorems including a univalence criterion.

Contribution

It provides a novel, general definition of Schwarzian derivatives for harmonic mappings, enabling new univalence criteria and extending classical results.

Findings

Established a new definition for harmonic Schwarzian derivatives

Proved theorems analogous to classical univalence results

Derived a Becker-type univalence criterion for harmonic mappings

Abstract

In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping $f$ in the complex plane without assuming any additional condition on the (second complex) dilatation $\omega_f$ of $f$. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove analogous theorems to those by Chuaqui, Duren, and Osgood. Also, we obtain a Becker-type criterion for the univalence of harmonic mappings.