Revista Matemática Iberoamericana

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Volume 15, Issue 3, 1999, pp. 429–449
DOI: 10.4171/RMI/261

Published online: 1999-12-31

On radial behaviour and balanced Bloch functions

Juan J. Donaire[1] and Christian Pommerenke[2]

(1) Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain
(2) Technische Universität Berlin, Germany

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, $g(r \zeta) (r \longrightarrow 1)$ follows any prescribed curve at a bounded distance for $\zeta$ 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 $\lbrace |z| = r\rbrace$ except for small exceptional arcs. We show e.g. that $$\int^1_0|g'(r \zeta)|dr< \infty$$ holds either for all $\zeta \in \mathbb T$ or for none.

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Donaire Juan, Pommerenke Christian: On radial behaviour and balanced Bloch functions. Rev. Mat. Iberoam. 15 (1999), 429-449. doi: 10.4171/RMI/261