On radial behaviour and balanced Bloch functions

Juan Jesús Donaire, Christian Pommerenke

Research output: Contribution to journalArticleResearchpeer-review

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(r ζ) (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 ∫10|g′(r ζ)|dr < ∞ holds either for all ζ ∈ T or for none.
Original languageEnglish
Pages (from-to)429-449
JournalRevista Matematica Iberoamericana
Volume15
Issue number3
DOIs
Publication statusPublished - 1 Jan 1999

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