Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm

TL;DR

This paper explores uniformly locally univalent harmonic mappings in the unit disk, linking their pre-Schwarzian norm with hyperbolic radius, providing characterizations, distortion estimates, and subordination principles.

Contribution

It introduces new relationships between pre-Schwarzian norm and hyperbolic radius, and offers multiple characterizations and estimates for these harmonic mappings.

Findings

Relationship between pre-Schwarzian norm and hyperbolic radius

Eight characterizations of uniformly locally univalent harmonic mappings

Sharp distortion and growth estimates

Abstract

In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates.