拟共形映照的掩盖定理

Covering theorems on quasi-conformal mappings

得到了关于拟共形映照的几个掩盖定理 ,推广了有关文献中的结果 .主要结果如下 :设 f(z)是 |z|≤ 1上的拟共形映照 ,f(0 ) =0 ,且存在常数β >0 ,M≥ 0 ,使limz→∞|f (z) ||z|β =α,∫10β - 1D(r)drr ≤ M,其中 D(r)是 f (z)的伸缩商 ,D(r) =12π∫2π0 D(r exp(iθ) ) dθ,则 f的像区域必包含圆盘 |w|<(α/ 4) e- M.

Several covering theorems on quasi- conformal mappings are proved in this paper, which generalize the results in other articles.The main result is following: Suppose thatf(z) ,withf(0 ) =0 ,is a quasi- conformal mapping in|z|<1,and there exist such constantsβ >0 ,M≥ 0 ,that lim z→∞ |f (z) | |z|β =α,∫1 0 β - 1 D(r) 1 rdr≤ M, where D(z) is the dilatation off and D(z) =1 2π∫2π 0 D(r exp(iθ) ) dθ, then,the image region off contains the disk|w|<(α/ 4) e- M .