拟共形映照的一个极值问题

An Extremal Problem for Quasiconformal Mappings

本文证明:如果f(z)是拓广复平面到自身使得f(0)=0,f(1)=1和f(∞)=∞的一个Q拟共形映照。则对任何r,|z|≤r |f(z)|≤r,成立|f(z)-z|≤4/π rK(1/1+r)K(r/1+r)·logQ,其中K(t)=integral from n=0 to 1(dx/((1-x^2)(1-tx^2))^(1/2)。它是夏道行的一个定理的拓广。

In this paper the author proves: If f(z) is a quasiconformal mapping of the extended complex plane onto itself such that f(0)=0, f(1)=1 and f(∞)=∞, then |f(z)-z|≤~4_π rK (1/1+r)K(~r_(1+r))lig Q holds for any r, |z|≤r and |f(z)|≤r, where K (t)=integral from 1 to 0 dx/(1-x^2)(1-tx^2)^(1/2). It is the generalization a therem of shah Taoshing (see also [1]).