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Sharp pointwise estimate of $α-$harmonic functions

David Kalaj

Abstract

Let $α>-1$ and assume that $f$ is $α-$harmonic mapping defined in the unit disk that belongs to the Hardy class $h^p$ with $p\ge 1$. We obtain some sharp estimates of the type $|f(z)|\le g(|r|) \|f^\ast\|_p$ and $|Df(z)|\le h(|r|)\|f^\ast\|_p$. We also prove a Schwarz type lemma for the class of $α-$harmonic mappings of the unit disk onto itself fixing the origin.

Sharp pointwise estimate of $α-$harmonic functions

Abstract

Let and assume that is harmonic mapping defined in the unit disk that belongs to the Hardy class with . We obtain some sharp estimates of the type and . We also prove a Schwarz type lemma for the class of harmonic mappings of the unit disk onto itself fixing the origin.
Paper Structure (3 sections, 6 theorems, 93 equations)

This paper contains 3 sections, 6 theorems, 93 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Proofs
  3. A remark to the sharp form of Theorem \ref{['due']}

Key Result

Theorem 1.1

For $p\geqslant 1$, there is a function $B_{\alpha,p}(r)$ and a constant $b_{\alpha,p}=\max_r B_{\alpha,p}(r)$ defined in bealr and beal below, so that for $f^\ast\in L^p(\mathbb{T})$ and for $z\in\mathbb{D}$ we have and The function $B$ and the constant $b$ are sharp. In particular for every $z\in \mathbb{D}$,

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Schwarz lemma for $\alpha-$harmonic mappings
  • Remark 1.4
  • proof : Proof of Theorem \ref{['one']}
  • proof : Proof of Theorem \ref{['due']}
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['marte']}
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['bele']}
  • ...and 3 more