Univalence criterion for harmonic mappings and $\Phi$-like functions

TL;DR

This paper introduces a new, simpler criterion for univalent harmonic mappings in the unit disk, extending classical results and providing a method to construct such mappings using an improved distortion theorem.

Contribution

It presents a novel characterization for univalent harmonic mappings and a structural formula for associated $ ext{ extPhi}$-like functions, simplifying previous proofs and enabling new constructions.

Findings

New univalence criterion for harmonic mappings

Structural formula for $ ext{ extPhi}$-like functions

Method for constructing univalent harmonic mappings

Abstract

In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic $\Phi$-like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas'yanyuk \cite{Kas59} and Brickman \cite{Brick73} for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mappings with the help of an improved distortion theorem.