圆环区域上的单叶性准则

The Singular-Leaf Criterion on the Circle Ring Domain

设H={z:1<|z|<R}是平面上的圆环区域,本文证明了H是(α,β)——一致区域。由此我们给出了多连通区域H上的一个局部单叶的解析函数为整体单叶的充分条件,即:设f(z)在圆环域H上解析,且f′(z)≠0,则当 ‖S_f‖H<1/8([(1+1/r)π]+5)^(-5)(log(2R/(1+R))~2,或 (?)|f″(z)/f′(z)|ρ_H^(-1)(z)<1/2([(1+1/r)π]+5)^(-3)log(2R/(1+R))时,f在H内单叶。

Let H={Z: 1<|z|<CR} be a circle ring domain in plane, in this paper, we prove that H is a (α,β)-uniform domain. From this, we obtain a sufficient condition that a partial univalent analytic function is whole univalent analytic in Multiply connected domain H, that is. let f(z) be analytic in circle ring domain H, and f'(z)≠0 for all z∈H. If either||Hf||H<1/8([(1+1/r)π]+5)-5(log2R/1+R)2 |f'(z)/f'(z)|-ρH-1(z)<1/2([1+1/r)π]+5)-5log2R/1+R,then f is injective in H.