In this paper an extension of the area principle to conformal mappings with a -quasiconformal continuation into the component of the complement of a region 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 with being a piecewise constant function. These area theorems are applied to the estimations of the ranges of the coefficient for 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.
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Erich Hoy, Flächensätze für quasikonform fortsetzbare Abbildungen. Z. Anal. Anwend. 3 (1984), no. 1, pp. 19–31DOI 10.4171/ZAA/88