Hardy spaces of harmonic quasiconformal mappings and Baernstein's theorem

TL;DR

This paper investigates the properties of harmonic quasiconformal mappings within Hardy spaces, deriving extremal and integral mean estimates, and refines existing results on their inclusion in Hardy spaces.

Contribution

It establishes Baernstein type extremal results for harmonic quasiconformal mappings and determines the range of p for Hardy space inclusion, advancing the understanding of these mappings.

Findings

Derived Baernstein type extremal results for harmonic quasiconformal mappings.

Obtained integral means estimates for subclasses of these mappings.

Identified the range of p for Hardy space inclusion, refining previous results.

Abstract

Let $\mathcal{S}_H^0(K)$, $K\ge 1$, be the class of normalized $K$-quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses of $\mathcal{S}_H^0(K)$. We then apply these results to obtain integral means estimates for the respective classes. Furthermore, we find the range of $p>0$ such that these geometric classes of harmonic quasiconformal mappings are contained in the Hardy space $h^p$, thereby refining some earlier results of Nowak.