On radial behaviour and balanced Bloch functions.

Juan Jesús Donaire; Christian Pommerenke

Revista Matemática Iberoamericana (1999)

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

Abstract

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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 < ∞holds either for all ζ ∈ T or for none.

How to cite

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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 -

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