Extremal Problems in Classes of Analytic Univalent Functions with Quasiconformal Extensions

This work solves many of the classical extremal problems posed in the class of functions <!-- MATH ${\Sigma _{K(\rho )}}$ --> , the class of functions in with -quasiconformal extensions into the interior of the unit disk where is a piecewise continuous function of bounded variation on <!-- MATH $[r,1],0 \leq r < 1$ --> <img width="136" height="41" align="MIDDLE" border="0" src="images/img5.gif" alt="$ [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 <!-- MATH ${S_{K(\rho )}}$ --> , the subclass of functions in S with -quasiconformal extensions to the exterior of the unit disk, are also solved.