Extremal problems in classes of analytic univalent functions with quasiconformal extensions

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.