关于伸张函数估计的一点注记

Notes on the Estimate of Dilatation Function

设h是实轴上同胚 ,h(±∞ ) =±∞ ,h的Beurling -Ahlfors扩张记作(z) ,其伸张函数为D ,当h的拟对称函数 ρ(x ,t)被递减函数 ρ(t)控制时 ,得到如下估计 : D≤ 2 ρ ,ρ ≥ 2D≤ 2 ρ +1 / 2 ,1≤ρ <2其中 ρ =ρ(y2 )

Let h be a homeomorphism of R onto itself and with ∞ fixed.Let (z) be the Beurling-Ahlfors extnsion of h,whose dilatation function is D.When the quasisymmetric function ρ(x,t) of h is controled by a decreasing function ρ(t),we get the follow result:D≤2ρ*, ρ*≥2 D≥2ρ*+12,1≤ρ*<2