# Radial length, radial John disks and $K$-quasiconformal harmonic mappings

## TL;DR

This paper studies the boundary behavior of $K$-quasiconformal harmonic mappings, providing growth theorems, characterizations of radial John disks, and analyzing distortion and continuity properties.

## Contribution

It offers new sharp growth results, an alternative characterization of radial John disks, and insights into distortion and differential properties of these mappings.

## Key findings

- Established a sharp growth theorem for radial length of $K$-quasiconformal harmonic mappings.
- Provided an alternative characterization of radial John disks.
- Analyzed linear measure distortion and Lipschitz continuity of these mappings.

## Abstract

In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of $K$-quasiconformal harmonic mappings. Then we present an alternate characterization of radial John disks. In addition, we investigate the linear measure distortion and the Lipschitz continuity on $K$-quasiconformal harmonic mappings of the unit disk onto a radial John disk. Finally, using Pommerenke interior domains, we characterize certain differential properties of $K$-quasiconformal harmonic mappings.