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Hardy-Littlewood theorems and the Bergman distance

Marijan Markovic

Abstract

We obtain non-Euclidean versions of classical theorems due to Hardy and Littlewood concerning smoothness of the boundary function of an analytic mapping on the unit disk with an appropriate growth condition.

Hardy-Littlewood theorems and the Bergman distance

Abstract

We obtain non-Euclidean versions of classical theorems due to Hardy and Littlewood concerning smoothness of the boundary function of an analytic mapping on the unit disk with an appropriate growth condition.

Paper Structure

This paper contains 5 sections, 8 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two generalizations of the Yamashita result
  4. Growth of the derivative and the boundary function
  5. Hardy and Littlewood theorems and invariant metrics

Key Result

Proposition \oldthetheorem

An analytic mapping $f$ on $\mathbb{D}$ satisfies $|f'|(z)=\mathcal{O}(1-|z|)^{\alpha-1}$, $|z|\to 1$, $\alpha\in (0,1]$, if and only if it has continuous extension on $\overline{\mathbb{D}}$ and $f_b\in \Lambda_\alpha$.

Theorems & Definitions (14)

  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof : Proof of Theorem \ref{['TH.1']}
  • proof : Proof of Theorem \ref{['TH.2']}
  • ...and 4 more