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On singular boundary points of complex functions

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  26 February 2010

Miroslav Zelený
Miroslav Zelený
Affiliation:
KMA, Faculty of Mathematics and Physics, Charles University, Sokolovskà 83, Prague 186 00, Czech Republic. e-mail: zeleny@karlin.mff.cuni.cz
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Abstract

Let f be a complex valued function from the open upper halfplane E of the complex plane. We study the set of all z∈∂E such that there exist two Stoltz angles V1, V2 in E with vertices in z (i.e., Vi is a closed angle with vertex at z and Vi\{z} ⊂ E, i = 1, 2) such that the function f has different cluster sets with respect to these angles at z. E. P. Dolzhenko showed that this set of singular points is G∂σ and σ-porous for every f. He posed the question of whether each G∂σ σ-porous set is a set of such singular points for some f. We answer this question negatively. Namely, we construct a G∂ porous set, which is a set of such singular points for no function f.

MSC classification

Secondary: 30D40: Cluster sets, prime ends, boundary behavior
Type
Research Article
Information
Mathematika , Volume 45 , Issue 1 , June 1998 , pp. 119 - 133
Copyright
Copyright © University College London 1998

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References

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