关于解析函数的一个唯一性定理及其应用

<正> §1.在近代解析函数论中边界值的唯一性定理有许多的研究,其中有我们所习知著名的(?)氏唯一性定理,即:若 D 是某一可求长约当曲线Γ所范围的内域,而 f(z)是 D 内的半纯函数,如在Γ上存在某一测度大于零的集 E_z,对 E_z 任一点 z_0 上,f(z)的角形边界值为零.则必致f(z)≡0于 D 内.同时,卢洵(?)与普里瓦洛夫还指出:存在有单位圆域内非常数的解析函数,而在一个正测度的集(?)上具有等于零的射形边界值.除此之外还有一个很有用的 Koebe 氏定理.即:

ON THE UNIQUENESS THEOREM OF ANALYTIC FUNCTIONS AND ITS APPLICATIONS

The main purpose of:the present paper is to investigate some theorems as following:Let f(z)be an analytic function,regular in a region D,which is an interior regionbounded by a certain rectifiable Jordan curve Γ,composed of two curves Γ′and Γ″,takinga b as its two end points.Let f(z)be cintinuous on Γ′(including the end points).Funthermore,we assume there e.xists a Sequence of simple rectifiable curves {λ_n},whichconverges to curve Γ″,belonging to D except its end poiuts z′_n and z″_n belonging to Γ″, and such that(?)then f(z)≡0 in D.Assume the same assumptions of D,г,г′,г″and {λ_n},and assum{f_n(z)} besequence of analytic functions,regular in D,and each f_n(z)be continuous and boundedon Γ′uniformly,i.e.(?)Furthermore suppose,that there exists a certain sequence of positive.numbers {ε_n}→0,as n→+∞,and that for any fixed ε_k and λ_k,there exists a positive integer number N,for m,n≥N,we havethen {f_n(z)} is uniformly convergent in any closed region within D.