# On the dilatation of extremal quasiconformal mappings of polygons

### Kurt Strebel

Zürich, Switzerland

## Abstract

A polygon PN is the unit disk ${\Bbb D}$ with $n$ distinguished boundary points, $4\le n \le N$. An extremal quasiconformal mapping $f_0\: {\Bbb D}_z\to {\Bbb D}_w$ maps each polygon $P_N$ inscribed in ${\Bbb D}_z$ onto a polygon $P_N'$ inscribed in ${\Bbb D}_w$. 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 $\sup K_N\le K_0$. 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.