Uniformly locally univalent harmonic mappings

TL;DR

This paper characterizes uniformly locally univalent harmonic mappings in the unit disk, establishes sharp distortion, growth, and covering theorems for a specific family, and explores their inclusion in Hardy spaces and coefficient growth.

Contribution

It provides a comprehensive characterization and new theorems for the family of uniformly locally univalent harmonic mappings, including their Hardy space inclusion and coefficient growth behavior.

Findings

Sharp distortion, growth, and covering theorems established.

Subclass of k-quasiconformal harmonic mappings contained in Hardy spaces.

Growth estimates for coefficients of harmonic mappings in the family.

Abstract

The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family ${\mathcal B}_{H}(\lambda)$ of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of $k$-quasiconformal harmonic mappings in ${\mathcal B}_{H}(\lambda)$ and the class ${\mathcal B}_{H}(\lambda)$ are contained in the Hardy space of a specific exponent depending on the $\lambda$, respectively, and we also discuss the growth of coefficients for harmonic mappings in ${\mathcal B}_{H}(\lambda)$.