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Sharp bounds for the growth and distortion of the analytic part of convex K-quasiconformal harmonic mappings

Peijin Li, Saminathan Ponnusamy

TL;DR

This paper extends sharp growth and distortion bounds for the analytic part of convex harmonic mappings from the zero-dilatation case to the convex $K$-quasiconformal setting. By constructing an auxiliary harmonic mapping in $\mathcal C_H(K)$ and employing a key lemma on positivity and a subordination framework, it derives explicit upper and lower bounds for the analytic part $h$ of $f=h+\overline{g}$ in $\mathcal C_H^0(K)$, with exact extremals given by $H_k^{\lambda_1}$ and $H_k^{\lambda_2}$. The main results recover the previously known bounds in the limit $k\to1^-$ (Theorem A) and identify sharp growth rates $A(k,|z|)$ and $B(k,|z|)$ as functions of the quasiconformal dilatation. The work advances the understanding of the geometric behavior of convex harmonic mappings under quasiconformal constraints and provides explicit, extremal constructions for growth and distortion of the analytic part.

Abstract

The main aim of this paper is to obtain the sharp upper and lower bounds for the growth and distortion of the analytic part $h$ of sense-preserving convex $K$-quasiconformal harmonic mappings.

Sharp bounds for the growth and distortion of the analytic part of convex K-quasiconformal harmonic mappings

TL;DR

This paper extends sharp growth and distortion bounds for the analytic part of convex harmonic mappings from the zero-dilatation case to the convex -quasiconformal setting. By constructing an auxiliary harmonic mapping in and employing a key lemma on positivity and a subordination framework, it derives explicit upper and lower bounds for the analytic part of in , with exact extremals given by and . The main results recover the previously known bounds in the limit (Theorem A) and identify sharp growth rates and as functions of the quasiconformal dilatation. The work advances the understanding of the geometric behavior of convex harmonic mappings under quasiconformal constraints and provides explicit, extremal constructions for growth and distortion of the analytic part.

Abstract

The main aim of this paper is to obtain the sharp upper and lower bounds for the growth and distortion of the analytic part of sense-preserving convex -quasiconformal harmonic mappings.

Paper Structure

This paper contains 6 sections, 1 theorem, 47 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Growth and distortion of convex analytic functions
  3. Growth and distortion of convex harmonic functions
  4. Main result and discussion
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 2.1

Let $f=h+\overline{g}\in{\mathcal{C}}_H^0(K)$. Then, for $z \in \mathbb{D}$, the growth and distortion of the analytic part $h$ of $f$ are subject, respectively, to the bounds and Here and The estimates are sharp: the upper bound equality holds in any of the inequalities if and only if $h$ equals $H_k^{\lambda_1}(z)$ defined by hkz1 and the lower bound equality holds in any of the inequalities

Theorems & Definitions (2)

  • Theorem 2.1
  • Remark 2.1