Extreme points method and univalent harmonic mappings

TL;DR

This paper establishes sufficient conditions on the analytic part and dilatation of harmonic mappings to ensure their univalence on the unit disk, advancing the understanding of harmonic univalent functions.

Contribution

It introduces new criteria linking the analytic component and dilatation to univalence, expanding the class of harmonic mappings known to be univalent.

Findings

Derived sufficient conditions for univalence of harmonic mappings.

Characterized the role of the second complex dilatation in univalence.

Extended existing theories on harmonic univalent functions.

Abstract

We consider the class of all sense-preserving complex-valued harmonic mappings $f=h+\bar {g}$ defined on the unit disk $\ID$ with the normalization $h(0)=h'(0)-1=0$ and $g(0)=g'(0)=0$ with the second complex dilatation $\omega:\,\ID\rightarrow \ID$, $g'(z)=\omega (z)h'(z)$. In this paper, the authors determine sufficient conditions on $h$ and $\omega$ that would imply the univalence of harmonic mappings $f=h+\bar {g}$ on $\ID$.