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Zeitschrift für Analysis und ihre Anwendungen


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Volume 3, Issue 1, 1984, pp. 19–31
DOI: 10.4171/ZAA/88

Published online: 1984-02-29

Flächensätze für quasikonform fortsetzbare Abbildungen

Erich Hoy[1]

(1) Friedberg, Germany

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 $w_{\bar z} = \mu (z) \bar {w_z}$ 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.

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Hoy Erich: Flächensätze für quasikonform fortsetzbare Abbildungen. Z. Anal. Anwend. 3 (1984), 19-31. doi: 10.4171/ZAA/88