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# Extremal Problems in Classes of Analytic Univalent Functions with Quasiconformal Extensions

J. Olexson McLeavey
Transactions of the American Mathematical Society
Vol. 195 (Aug., 1974), pp. 327-343
DOI: 10.2307/1996735
Stable URL: http://www.jstor.org/stable/1996735
Page Count: 17
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## Abstract

This work solves many of the classical extremal problems posed in the class of functions ΣK(ρ), the class of functions in Σ with K(ρ)- quasiconformal extensions into the interior of the unit disk where K(ρ) is a piecewise continuous function of bounded variation on $\lbrack r < 1 \rbrack, 0 \leq r < 1$. 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 SK(ρ), the subclass of functions in S with K(ρ)-quasiconformal extensions to the exterior of the unit disk, are also solved.

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