# On radial behaviour and balanced Bloch functions.

• Volume: 15, Issue: 3, page 429-449
• ISSN: 0213-2230

top

## Abstract

top
A Bloch function g is a function analytic in the unit disk such that (1 - |z|2) |g' (z)| is bounded. First we generalize the theorem of Rohde that, for every bad Bloch function, g(rζ) (r → 1) follows any prescribed curve at a bounded distance for ζ in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that |g'(z)| does not vary much on each circle {|z| = r} except for small exceptional arcs. We show e.g. that∫01 |g'(rζ)|dr &lt; ∞holds either for all ζ ∈ T or for none.

## How to cite

top

Donaire, Juan Jesús, and Pommerenke, Christian. "On radial behaviour and balanced Bloch functions.." Revista Matemática Iberoamericana 15.3 (1999): 429-449. <http://eudml.org/doc/39582>.

@article{Donaire1999,
abstract = {A Bloch function g is a function analytic in the unit disk such that (1 - |z|2) |g' (z)| is bounded. First we generalize the theorem of Rohde that, for every bad Bloch function, g(rζ) (r → 1) follows any prescribed curve at a bounded distance for ζ in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that |g'(z)| does not vary much on each circle \{|z| = r\} except for small exceptional arcs. We show e.g. that∫01 |g'(rζ)|dr &lt; ∞holds either for all ζ ∈ T or for none.},
author = {Donaire, Juan Jesús, Pommerenke, Christian},
journal = {Revista Matemática Iberoamericana},
keywords = {Funciones analíticas; Aplicación conforme; Bloch functions; conformal mappings},
language = {eng},
number = {3},
pages = {429-449},
title = {On radial behaviour and balanced Bloch functions.},
url = {http://eudml.org/doc/39582},
volume = {15},
year = {1999},
}

TY - JOUR
AU - Donaire, Juan Jesús
AU - Pommerenke, Christian
TI - On radial behaviour and balanced Bloch functions.
JO - Revista Matemática Iberoamericana
PY - 1999
VL - 15
IS - 3
SP - 429
EP - 449
AB - A Bloch function g is a function analytic in the unit disk such that (1 - |z|2) |g' (z)| is bounded. First we generalize the theorem of Rohde that, for every bad Bloch function, g(rζ) (r → 1) follows any prescribed curve at a bounded distance for ζ in a set of Hausdorff dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that |g'(z)| does not vary much on each circle {|z| = r} except for small exceptional arcs. We show e.g. that∫01 |g'(rζ)|dr &lt; ∞holds either for all ζ ∈ T or for none.
LA - eng
KW - Funciones analíticas; Aplicación conforme; Bloch functions; conformal mappings
UR - http://eudml.org/doc/39582
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.