Two-point distortion theorems for harmonic mappings

TL;DR

This paper develops two-point distortion theorems for sense-preserving harmonic mappings in the unit disk, including sharp bounds for convex functions and normalized mappings with linearly connected images.

Contribution

It introduces new two-point distortion theorems for harmonic mappings satisfying univalence criteria, extending classical results to harmonic and convex cases.

Findings

Established two-point distortion theorems for harmonic mappings

Derived sharp bounds for convex harmonic functions

Analyzed mappings with linearly connected images

Abstract

We establish two-point distortion theorems for sense-preserving planar harmonic mappings $f=h+\overline{g}$ which satisfies the univalence criteria in the unit disc such that, Becker's and Nehari`s harmonic version. In addition, we find the sharp two-point distortion theorem when $h$ is a convex function, and normalized mappings such that $h(\D)$ is a $c$-linearly connected domain. To do this, we use the order of this family.