Functions whose derivative has a positive real part
Functions whose derivative has a positive real part
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DOI：
10.1090/s0002-9947-1962-0140674-7
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发表时间：
1962-03
影响因子：
1.3
通讯作者：
T. Macgregor
中科院分区：
文献类型：
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作者：
T. Macgregor
1.导论.设R表示正则函数类，满足Re f '（z）> 0，j zI <1，且f（0）= 0和f'（0）= 1是正规函数。本文给出了R中函数的一些性质。在JW亚历山大的一篇论文[1，第18页]中可以找到满足条件Re f '（z）> 0的函数的早期考虑。他证明：如果f（z）在IzI <1中是正则的，并且f '（z）"将IzI <1映射到包含在由通过原点的直线所限定的半平面内的区域上"，则f（z）对于zI <1是施利希特。J. Wolff [11]证明了f（z）在Re z> 0中是施利希特，如果它在Re z> 0是正则的并且满足Ref '（z）> 0. K. Noshiro [6，p. 151]和S. Warschawski [10，p.3121]证明了Ref '（z）> 0是f（z）在任意凸域上的schlicht性的充分条件，而SR Tims [9]证明了对任意单连通非凸域D，存在D中正则函数f（z），使得Ref'（z）> 0且f（z）在D中不是施利希特.这个结果是F. Herzog和G. Piranian [2].它们决定了域D（不一定是单连通的）具有如下性质的充要条件：D中每个正则且满足Ref '（z）> 0的函数都在那里是施利希特.比满足Re f '（z）> 0的函数更一般的函数类是近于凸函数类。W. Kaplan [3]称函数f（z）在IzI <1中接近凸，条件是存在函数g（z）在IzI <1中解析、施利希特且凸，且Re f '（z）/g'（z）}> 0。IzI <1中的每一个接近凸的函数都有施利希特。
1. Introduction. Let R denote the class of functions which are regular and satisfy Re f'(z)> 0 for j zI< 1 and are normalized by f (0)= 0 and f'(0)= 1. This paper develops some properties of functions in R. An early consideration of functions satisfying the condition Re f'(z)> 0 can be found in a paper by JW Alexander [1, p. 18]. He proves: if f (z) is regular in IzI< 1 and f'(z)" maps IzI< 1 upon a region contained within a half-plane bounded by a straight line through the origin" then f (z) is schlicht for zI< 1. J. Wolff [11] showed that f (z) is schlicht in Re z> 0 if it is regular there and satisfies Ref'(z)> 0. K. Noshiro [6, p. 151] and S. Warschawski [10, p. 3121 each demonstrated that Ref'(z)> 0 is a sufficient condition for the schlichtness of f (z) in any convex domain.Conversely, SR Tims [9] proved that for each simply connected nonconvex domain D there is a function f (z) regular in D such that Re f'(z)> 0 and f (z) is not schlicht in D. This result is a particular consequence of some more general theorems contained in a paper by F. Herzog and G. Piranian [2]. They determine both necessary and sufficient conditions for a domain D-not necessarily simply connected-to have the property that every function regular and satisfying Ref'(z)> 0 in D is schlicht there. A more general class of functions than those satisfying Re f'(z)> 0 is the class of close-to-convex functions. W. Kaplan [3] calls a function f (z) close-to-convex in I zI< 1 providing there is a function g (z) analytic, schlicht and convex in I zI< 1 for which Re f'(z)/g'(z)}> 0. Each function close-toconvex in IzI< 1 is schlicht there.