On the maximal dilation of quasiconformal mappings

Abstract

1. Let G and G' be two plane open sets and w(z) a topological mapping of G onto G'. By Q we denote any quadrilateral in G, i.e. the topological image of a closed square with a distinguished pair of opposite sides. The conformal modulus m of Q is the ratio m = a/b of the sides of a conformally equivalent rectangle R, the distinguished sides of Q corresponding to the sides of length b. We call this essentially unique conformal mapping of Q onto R the canonical mapping of Q. The modulus m is equal to the extremal distance of the two distinguished sides of Q with respect to Q. The maximal dilation of the mapping w(z) on G is the number