JournalszaaVol. 3, No. 1pp. 19–31

Flächensätze für quasikonform fortsetzbare Abbildungen

  • Erich Hoy

    Friedberg, Germany
Flächensätze für quasikonform fortsetzbare Abbildungen cover
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In this paper an extension of the area principle to conformal mappings with a QjQ_j-quasiconformal continuation into the component Bj\mathcal B_j of the complement of a region G\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 wzˉ=μ(z)wzˉw_{\bar z} = \mu (z) \bar {w_z} with μ(z)\mu (z) being a piecewise constant function. These area theorems are applied to the estimations of the ranges of the coefficient for z1z^{-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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Erich Hoy, Flächensätze für quasikonform fortsetzbare Abbildungen. Z. Anal. Anwend. 3 (1984), no. 1, pp. 19–31

DOI 10.4171/ZAA/88