On the dilatation of extremal quasiconformal mappings of polygons

Abstract

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