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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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