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Hülder Conditions and the Topology of Simply Connected Domains*

Published online by Cambridge University Press:  20 November 2018

Dov Aharonov
Dov Aharonov*
Affiliation:
Israel Inst. of Tech.Technion, Haifa, Israel
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Abstract

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Let ƒ be regular univalent and normalized in the unit disc U (i.e. ƒ ∊ S) and continuous on U ∈ T, where T denotes the boundary of U.

Recently Essén proved [5] a conjecture of Piranian [7] stating that if the derivative of ƒ ∊ S is bounded in U and ƒ(z1) = ƒ(z2) = … = ƒ(zn) for Zj ∊ T, 1 ≤ j ≤ n, then n ≤ 2. In fact, Essén proved a more general result, using a deep result on harmonic functions. The aim of the following article is to replace Essén's proof by a completely different proof which is based only on Goluzin's inequalities and is much more elementary.

Keywords

30C55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 2 , 01 June 1983 , pp. 189 - 191
Copyright
Copyright © Canadian Mathematical Society 1983

Footnotes

*

The research was supported by the Fund for the Promotion of Research at the Technion.

References

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