Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a problem of Pavlović involving harmonic quasiconformal mappings

Zhi Gang Wang, Xiao Yuan Wang, Antti Rasila, Jia Le Qiu

TL;DR

The paper advances the theory of harmonic $K$-quasiconformal mappings with bounded Schwarzian norm by constructing a harmonic Koebe extremal and leveraging Hardy-space theory to obtain the optimal order for membership in harmonic Hardy spaces, partially answering Pavlović’s 2014 question. It develops a framework linking coefficient bounds, Schwarzian norms, and AL-family structure, and formulates sharp conjectures for coefficient extremals and Hardy-space thresholds via the harmonic Koebe function $f_k$. The work also provides explicit pre-Schwarzian and Schwarzian norm estimates for a broad class of harmonic mappings, enriching the toolkit for geometric function theory of harmonic mappings and offering concrete targets for future sharpness results and conjectures.

Abstract

We obtain a sharp result on the order of harmonic quasiconformal mappings with bounded Schwarzian norm. This problem is motivated by the work of Chuaqui, Hernández and Martín [Math. Ann. 367: 1099--1122, 2017]. Firstly, for $K\ge1$, we construct a harmonic $K$-quasiconformal counterpart of the classical Koebe function and use it to formulate the corresponding conjectures. Then we consider Hardy spaces $H^p$ of harmonic quasiconformal mappings by applying results for quasiconformal mappings obtained by Astala and Koskela [Pure Appl. Math. Q. 7: 19--50, 2011]. In particular, we determine the optimal order of the family of harmonic quasiconformal mappings with bounded Schwarzian norm to belong to a harmonic Hardy space. This partially solves an open problem posed by Pavlović in 2014. Finally, we derive pre-Schwarzian and Schwarzian norm estimates of certain harmonic mappings.

On a problem of Pavlović involving harmonic quasiconformal mappings

TL;DR

The paper advances the theory of harmonic -quasiconformal mappings with bounded Schwarzian norm by constructing a harmonic Koebe extremal and leveraging Hardy-space theory to obtain the optimal order for membership in harmonic Hardy spaces, partially answering Pavlović’s 2014 question. It develops a framework linking coefficient bounds, Schwarzian norms, and AL-family structure, and formulates sharp conjectures for coefficient extremals and Hardy-space thresholds via the harmonic Koebe function . The work also provides explicit pre-Schwarzian and Schwarzian norm estimates for a broad class of harmonic mappings, enriching the toolkit for geometric function theory of harmonic mappings and offering concrete targets for future sharpness results and conjectures.

Abstract

We obtain a sharp result on the order of harmonic quasiconformal mappings with bounded Schwarzian norm. This problem is motivated by the work of Chuaqui, Hernández and Martín [Math. Ann. 367: 1099--1122, 2017]. Firstly, for , we construct a harmonic -quasiconformal counterpart of the classical Koebe function and use it to formulate the corresponding conjectures. Then we consider Hardy spaces of harmonic quasiconformal mappings by applying results for quasiconformal mappings obtained by Astala and Koskela [Pure Appl. Math. Q. 7: 19--50, 2011]. In particular, we determine the optimal order of the family of harmonic quasiconformal mappings with bounded Schwarzian norm to belong to a harmonic Hardy space. This partially solves an open problem posed by Pavlović in 2014. Finally, we derive pre-Schwarzian and Schwarzian norm estimates of certain harmonic mappings.
Paper Structure (14 sections, 12 theorems, 112 equations, 3 figures)

This paper contains 14 sections, 12 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction and statement of the main result
  2. Harmonic quasiconformal mappings
  3. Hardy spaces for harmonic and quasiconformal mappings
  4. The order of a Hardy space
  5. Affine and linear invariant families
  6. Harmonic pre-Schwarzian and Schwarzian derivatives
  7. A problem of Pavlović
  8. Preliminaries
  9. Estimation of $\left\vert a_2\right\vert$ of a univalent harmonic mapping
  10. Estimates of $\left\vert a_2\right\vert$ of harmonic $K$-quasiconformal mappings
  11. The order of an affine and linear invariant family $\mathcal{F}_{\lambda}$
  12. Harmonic quasiconformal Koebe function
  13. Proof of Theorem \ref{['t1']}
  14. Harmonic pre-Schwarzian and Schwarzian norms

Key Result

Theorem A

If $f$ is a $K$-quasiconformal mapping in $\hbox{$\mathbb{D}$}$, then $f\in H^p_K$ for and the constant $1/(2K)$ is sharp.

Figures (3)

  • Figure 1: Images of the unit disk under the mappings $f_{0}(z)$, $f_{1/5}(z)$, $f_{2/5}(z)$, $f_{3/5}(z)$, $f_{4/5}(z)$, and $\mathbb{K}(z)$.
  • Figure :
  • Figure :

Theorems & Definitions (28)

  • Theorem A
  • Theorem B
  • Remark 1
  • Proposition 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem C
  • Remark 4
  • Theorem D
  • ...and 18 more