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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extremal problems in classes of analytic univalent functions with quasiconformal extensions

Author: J. Olexson McLeavey
Journal: Trans. Amer. Math. Soc. 195 (1974), 327-343
MSC: Primary 30A60
MathSciNet review: 0346154
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Abstract: This work solves many of the classical extremal problems posed in the class of functions $ {\Sigma _{K(\rho )}}$, the class of functions in $ \Sigma $ with $ K(\rho )$-quasiconformal extensions into the interior of the unit disk where $ K(\rho )$ is a piecewise continuous function of bounded variation on $ [r,1],0 \leq r < 1$. The approach taken is a variational technique and results are obtained through a limiting procedure. In particular, sharp estimates are given for the Golusin distortion functional, the Grunsky quadratic form, the first coefficient, and the Schwarzian derivative. Some extremal problems in $ {S_{K(\rho )}}$, the subclass of functions in S with $ K(\rho )$-quasiconformal extensions to the exterior of the unit disk, are also solved.

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Keywords: Quasiconformal mapping, Schiffer's variational method, extremal problems, Golusin's distortion theorem, Grunsky inequalities, Schwarzian derivative
Article copyright: © Copyright 1974 American Mathematical Society

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