S. L. Krushkal

A Remark on Polygonal Quasiconformal Maps

Given a quasisymmetric map $h : \ {\hat {\text{\bf R}}} \to {\hat {\text{\bf R}}}$, let $f_0$ be an \eq \ \ext \ of $h$ onto the upper half-plane $U = \{z \in \C : \ \Im z > 0\}$ whose dilatation $k(f_0) = \inf \{k(f) : \ f|\partial U = h_0\} =: k(h)$. Let $k_n$ be the minimal dilatation of polygonal quasiconformal maps $f : \ U \to U$ satisfying $f(x_j) = h(x_j), j = 1, 2, \dots , n$, for any $n$ points of ${\hat {\text{\bf R}}}$ (vertices of $n$-gons). Already a long time ago, the question was posed whether $k(h) = \sup k_n$, where the supremum is taken over all possible $n$-gons of such kind. The answer is obtained (in the negative) for the case of quadrilaterals $(n = 4)$. We show that in the case of pentagons $(n = 5)$ the answer is also negative, i.e., there are quasisymmetric $h$ with $k(h) > \sup k_5$.