Boundary behaviour of inner functions and holomorphic mappings
type: Item
instance of: scholarly article
title: Boundary behaviour of inner functions and holomorphic mappings
zbMATH Open document ID: 0890.30021
author: Artur Nicolau
author: J. L. Fernández
published in: Mathematische Annalen
publication date: 1998-07-14T00:00:00Z
review text: Let \(f\) be a holomorphic function in the unit disc omitting a set of values \(A\) of the complex plane. If \(A\) has positive logarithmic capacity, R. Nevanlinna proved that \(f\) has radial limit at almost every point of the unit circle, while if a closed set \(A\) has zero logarithmic capacity, then there is \(f\) omitting \(A\) with radial limits almost nowhere in the unit circle. Nonetheless, we prove that if \(A\) is an infinite set, then \(f\) has limit along a set of radii which has Hausdorff dimension 1. A localization technique reduces this result to the following theorem concerning inner functions. Let \(I\) be an inner function omitting a set of values \(B\) in the unit disc, then for any accumulation point \(b\) of \(B\) in the unit disc, there exists a set of Hausdorff dimension 1 of radii along which the function \(I\) has limit \(b\).
Mathematics Subject Classification ID: 30D40
Mathematics Subject Classification ID: 30C99
zbMATH DE Number: 1143008
zbMATH Keywords: logarithmic capacity
zbMATH Keywords: accumulation point
zbMATH Keywords: Hausdorff dimension
reviewed by: J. L. Fernández
MaRDI profile type: Publication
full work available at URL: s002080050155
OpenAlex ID: W1966929067
DOI: 10.1007/S002080050155
description: scientific article; zbMATH DE number 1143008