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映像面积为有界的解析函数
FUNCTIONS WITH IMAGES OF BOUNDED MEASURE

祝颂和;
CHU SOON-HU

复旦大学,
(Fuhtan University,Shanghai

收稿日期: 1956-12-25
出版日期: 1956-11-15

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摘要  1.引言设单位圆|z|<1上的正则函数 w=f(z)=a_0+a_1z+a_2z~2+…(1)将单位圆映入w平面上的区域D,D的面积|D|——当D在某处有m层时按m次计算——不超过M,即|D|≤M。记这样的函数(1)的全体为S_M。设f(z)∈S_M,f′(o)≠0;这种f(z)成S_M之一子族S_M~'。此子族中的函数在原点之某一环境中是单叶的。如果这个环境符合於单位圆,这种函数的全体又成s_M~'之一子族s_M~"。
Abstract:This note is concerned with the class SM of functions f(z)=a_0+a_1z+…,|z|<1, such that the relation w=f(z) maps|z|<1 in D(f) on the w-plane, with the measure less than or equal to M. The surface D(f) may be of several sheets, the multiplicity is to be taken according to the number of sheets. Functions of S_M satisfying the condition f'(0)≠0 forms a subclass S_M~'We denote by S_M~" the sub-class of S_M~'of which each f(z) is schlicht in the unit circle. For the function f(z) of S_π the coefficients satisfies the inequalityn~(1/2)|a_n|≤1 (n=1, 2, …), and the equality n~(1/)2|a_n|=1 holds when and only when f(z) =a_0+ε/n~(1/)z~n,|ε|=1.The distortion theorem for f(z) ∈Sπruns that |f'(z)|≤ 1/1-r~2,|z|=r<1, (1) where the sign of equality holds only for the function f(z)=a_0+εa+z/a+az, |ε|=1 tvith some proper a, |a|<1. It follows from (1) that |f(z)-f(0)|
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