On the angular limits of Bloch functions
type: Item
instance of: scholarly article
title: On the angular limits of Bloch functions
zbMATH Open document ID: 0655.30021
DOI: 10.5565/PUBLMAT_32288_07
published in: Publicacions Matemàtiques
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41056
review text: A function f analytic in the unit disk \({\mathbb{D}}\) belongs to the space \({\mathcal B}\) of Bloch functions if \((1-| z|)| f'(z)|\) is bounded; it belongs to \({\mathcal B}_ 0\) if this tends to 0 if \(| z| \to 1\). In the paper, the power series expansion of f is divided into overlapping blocks such that these blocks are uniformly bounded in \({\mathbb{D}}\) if and only if \(f\in {\mathcal B}\). This process is used to assign to every \(f\in {\mathcal B}_ 0\) a ``larger'' function \(g\in {\mathcal B}\) such that, if \[ \sup_{0<r<1}| g(re^{it})| <\infty \] for some t then \(\lim_{r\to 1}f(re^{it})\neq \infty\) exists. A recent result of C. J. Bishop and P. W. Jones (preprint 1989) therefore implies that every function in \({\mathcal B}_ 0\) has finite angular limits on a set of Hausdorff dimension one.
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 4069273
zbMATH Keywords: lacunary
zbMATH Keywords: Bloch functions
zbMATH Keywords: bounded
zbMATH Keywords: angular limits
zbMATH Keywords: Hausdorff dimension
author: Joan Josep Carmona
author: Julià Cufí
author: Christian Pommerenke
reviewed by: Christian Pommerenke
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4069273