Rotation bounds for H\"older continuous homeomorphisms with integrable distortion

TL;DR

This paper establishes optimal bounds on the rotation of certain complex homeomorphisms with integrable distortion and H"older continuous inverses, improving previous results and demonstrating sharpness through examples.

Contribution

It provides the first sharp rotation bounds for H"older continuous homeomorphisms with $L^p$-integrable distortion, extending prior work that did not assume H"older continuity.

Findings

Established sharp rotation bounds for the class of homeomorphisms with integrable distortion and H"older continuous inverses.

Improved existing bounds by incorporating H"older continuity assumptions.

Provided examples demonstrating the sharpness of the obtained bounds.

Abstract

We obtain sharp rotation bounds for the subclass of homeomorphisms $f:\mathbb{C}\to\mathbb{C}$ of finite distortion which have distortion function in $L^p_{loc}$, $p>1$, and for which a H\"older continuous inverse is available. The interest in this class is partially motivated by examples arising from fluid mechanics. Our rotation bounds hereby presented improve the existing ones, for which the H\"older continuity is not assumed. We also present examples proving sharpness.