关于Grzch问题的一个注记及万有Teichmüller空间一性质的证明

A Remark for Schwarz lemma of Quasi-conformal Mapping

朱华成等在"Grzch问题的域内特征"一文中给出了拟共形映射的Schwarz型引理:设f(z)是单位圆上的K-拟共形自同胚,若f(0)=0,limz→G|f(z)|/|z|1/K=1,则:f(z)=eiθz|z|1/K-1,θ是实数.原文等价证明部分对θ是实数的证明未说明关键点hn(θ)跟r无关(其中z=reiθ),本文做了补充研究;另给出了文献[3]中定理2.1的简洁证明.

Zhu Hua cheng,Zhou Ze min and He Cheng qi have given the characterization of Grtzsch＇s problem： Suppose that f（z） is K-quasiconformal in the unit disk △.If and f（0）=0,and lim z→G｜f（z）｜/｜z｜1/K=1 then f（z）=eiθz｜z｜1/K-1,θ∈R,（z=reiθ）.But during the proof of problem,it is a non-complete proof that is real.In this paper,we give a complete the proof of it,and offen the concise proof.For theorem 2.1 in Wang zhe＇s paper ＂ The distance between different component of the universal Teichmüller space ＂.