Flächensätze für quasikonform fortsetzbare Abbildungen

Abstract

In this paper an extension of the area principle to conformal mappings with a $Q_j$-quasiconformal continuation into the component ${\mathcal{B}}_j$ of the complement of a region $\mathcal{G}$ is given. A generalized area-theorem is proved for these mappings. The inequalities are sharp; the extrernal functions are connected with the solution of the equation $w_{\bar{z}}=\mu (z)\overline{w_z}$ with $\mu (z)$ being a piecewise constant function. These area theorems are applied to the estimations of the ranges of the coefficient for $z^{-1}$ of the Laurent expansion in the neighbourhood of infinity, the Schwarzian derivative and Golusin’s functional. Finally the possibility of an extension to conformal mappings with a quasiconformal continuation is shown. For Grunsky’s regions these inequalities are asymptotically sharp, if the restriction of the dilatation converges to a constant.