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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A remark on angular complex dilatations of quasiconformal mappings


Author: Richard Fehlmann
Journal: Proc. Amer. Math. Soc. 104 (1988), 1071-1077
MSC: Primary 30C60; Secondary 30C75
MathSciNet review: 930251
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Abstract: By a theorem of Ortel an angular complex dilatation $ \kappa $ is extremal iff it is Teichmüllèr (quadratic differential with finite norm) or if it satisfies an integral condition involving the angular limits $ {\lambda _x}(\vartheta )$. We show that this second case occurs iff $ {\lambda _x}(\vartheta )$ can be given explicitly at a certain point $ x$, namely by $ {\lambda _x}(\vartheta ) = {\left\Vert \kappa \right\Vert _\infty }{e^{2i}}(\vartheta - {\vartheta _0})$. Moreover, we investigate this statement under the weaker condition of angularity when the uniformity part in its definition is dropped.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0930251-3
PII: S 0002-9939(1988)0930251-3
Article copyright: © Copyright 1988 American Mathematical Society