JournalscmhVol. 74 , No. 1DOI 10.1007/s000140050080

On the dilatation of extremal quasiconformal mappings of polygons

  • Kurt Strebel

    Zürich, Switzerland
On the dilatation of extremal quasiconformal mappings of polygons cover

Abstract

A polygon PN is the unit disk D{\Bbb D} with nn distinguished boundary points, 4nN4\le n \le N. An extremal quasiconformal mapping f0DzDwf_0\: {\Bbb D}_z\to {\Bbb D}_w maps each polygon PNP_N inscribed in Dz{\Bbb D}_z onto a polygon PNP_N' inscribed in Dw{\Bbb D}_w. Let fN be the extremal quasiconformal mapping of PN onto P'N. Let KN be its dilatation and let K0 be the maximal dilatation of f0. Then, evidently supKNK0\sup K_N\le K_0. The problem is, when equality holds. This is completely answered, if f0 does not have any essential boundary points. For quadrilaterals Q and Q' = f0(Q) the problem is sup(M'/M) = K0, with M and M' the moduli of Q and Q' respectively.