单连通区域上的正规函数与Koebe弧序列

NORMAL FUNCTION IN A SIMPLY CONNECTED DOMAIN AND SEQUENCE OF KOEBE ARCS

将单位圆盘上正规函数的概念推广到扩充复平面,其边界点不止一个的单连通区域。证明了:若G(?)C是单连通区域,(?)G是所含不止一点的紧集,则G上任何非常数正规函数均无Koebe弧的充要条件,(?)G是局部连通的。

Let G be a simply connected domain whose boundary G is a compact set and has at least two points . A function g meromorphic in G is said to benormal in G if sup 1/λ(z) g#(z) < ∞ , where λ (z) is the hyperbolic metric ofG and S#(z) = |g'(z)|1+g(z)| . A sequence {γn} of Jordan arcs is called asequence of Koebe arcs with respect to a function g , if there is a bounded subdo-main G1 G , and some c ∈ C , such that γn G1 , diam γn>γ>0, (n=1,2,…), and Azn ∈ γn g (zn ) → c , (n → ∞ ) . It is obtiained that any non - constant normal function in G has no sequence of Koebe arcs if and only if G is locally connected .