### 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 language | English |
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Pages (from-to) | 429-449 |

Journal | Revista Matematica Iberoamericana |

Volume | 15 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jan 1999 |

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## Cite this

Donaire, J. J., & Pommerenke, C. (1999). On radial behaviour and balanced Bloch functions.

*Revista Matematica Iberoamericana*,*15*(3), 429-449. https://doi.org/10.4171/RMI/261