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Some coefficient estimates on complex valued kernel $α$-harmonic mappings

Boyong Long

Abstract

We call a kind of mappings induced by a kind of weighted Laplace operator as complex valued kernel $α$-harmonic mappings. In this article, for this class of mappings, the Heinz type lemma is established, and the best Heinz type inequality is obtained. Next, the extremal function of Schwartz's Lemma is discussed. Finally, the coefficients are estimated for the subclass of complex valued kernel alpha harmonic mappings whose coefficients are real numbers.

Some coefficient estimates on complex valued kernel $α$-harmonic mappings

Abstract

We call a kind of mappings induced by a kind of weighted Laplace operator as complex valued kernel -harmonic mappings. In this article, for this class of mappings, the Heinz type lemma is established, and the best Heinz type inequality is obtained. Next, the extremal function of Schwartz's Lemma is discussed. Finally, the coefficients are estimated for the subclass of complex valued kernel alpha harmonic mappings whose coefficients are real numbers.
Paper Structure (2 sections, 7 theorems, 53 equations)

This paper contains 2 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introductions and main results
  2. Proofs and Remarks

Key Result

Theorem 1.1

Olofsson2013 A function $u$ in $\mathbb{D}$ is a complex-valued kernel $\alpha$-harmonic mapping if and only if it is given by a convergent power series expansion of the form for some sequence $\{c_{k}\}_{-\infty}^{\infty}$ of complex number satisfying where

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['theorem1.2']}
  • ...and 4 more