单位圆上解析函数的掩盖定理

Covering Theorems on the Analytic Functions in the Unit Disk

设w=f(z)是单位圆D上满足规范条件 f(0)=0,f′(0)=1的解析函数。众所周知,当f(z)是单叶解析函数时,有著名的Koebe的1/4掩盖定理。在去掉单叶性的假设时。

In this paper, Koebe's 1/4 covering theorem of univalent functions is extended to a similar result for analytic functions. Suppose that f(z) is analytic function in the unit disk D(|z|<1) with f(0)=0 and f'(0)=1, following covering properties are obtained. If E is an unbounded branch of C-f(D), then the distance between 0 and E e(0, E)≥1/4; if ?is a nondegenerate bounded branch of C-f(D), then p(0,E)≥MB, where MB is a constant which depends on E. Also if f(z) satisfies some additional condition which depends on the index n of f(z) and ?is a bounded branch of C-f(D), then it is proved that p(0, E)≥1/4n. Some spacial functions which indicated above results are sharp are constructed.