On a class of univalent, star shaped mappings
On a class of univalent, star shaped mappings
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关于一类单价星形映射
DOI：
10.1090/s0002-9939-1958-0095954-5
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发表时间：
1958
期刊：
影响因子：
--
通讯作者：
A. Schild
中科院分区：
文献类型：
--
作者：
A. Schild
1. Introduction. Among all functions w= f (z)= z+ E'a2 n regu-lar and univalent in the unit circle, two classes of functions have been discussed extensively: The class of functions mapping the unit circle onto star shaped regions, characterized by Re {zf'(z)/f (z)} _ 0 for zl< 1, and the class of functions mapping the unit circle onto convex regions, characterized by Re {zf"(z)/f'(z)}+ 1? 0 for Izl< 1.This short paper will examine some of the geometric and analytic properties of a class of functions w= f (z)= z+ En'= 2 a, zn which map the unit circle onto a region whose geometric nature is somewhat intermediate between star shaped and convex. The functions under consideration are to satisfy Re {zf'(z)/f (z)}? 1/2 for all Izl< I. Interest in functions of this type can be traced back to two papers by A. Marx [4] and E. Strohacker [8] who have shown that for any function w= f (z)= z+ fXn2 anzn, which maps the unit circle onto a convex region, we have Re {zf'(z)/f (z)}> 1/2, and as the function f (z)= z/(l+ z) shows, the constant 1/2 cannot be improved. It is also clear that the converse is not true, ie functions for which Re {zf'(z)/f (z)} _ 1/2 need not map the unit circle onto a convex region. An example of this type is given by the function w= f (z)= z-1/3 z2 for which Re {zf'(z)/f (z)}> 1/2,| z| 1, yet the image region is not convex.
1. Introduction. Among all functions w= f (z)= z+ E'a2 n regu-lar and univalent in the unit circle, two classes of functions have been discussed extensively: The class of functions mapping the unit circle onto star shaped regions, characterized by Re {zf'(z)/f (z)} _ 0 for zl< 1, and the class of functions mapping the unit circle onto convex regions, characterized by Re {zf"(z)/f'(z)}+ 1? 0 for Izl< 1.This short paper will examine some of the geometric and analytic properties of a class of functions w= f (z)= z+ En'= 2 a, zn which map the unit circle onto a region whose geometric nature is somewhat intermediate between star shaped and convex. The functions under consideration are to satisfy Re {zf'(z)/f (z)}? 1/2 for all Izl< I. Interest in functions of this type can be traced back to two papers by A. Marx [4] and E. Strohacker [8] who have shown that for any function w= f (z)= z+ fXn2 anzn, which maps the unit circle onto a convex region, we have Re {zf'(z)/f (z)}> 1/2, and as the function f (z)= z/(l+ z) shows, the constant 1/2 cannot be improved. It is also clear that the converse is not true, ie functions for which Re {zf'(z)/f (z)} _ 1/2 need not map the unit circle onto a convex region. An example of this type is given by the function w= f (z)= z-1/3 z2 for which Re {zf'(z)/f (z)}> 1/2,| z| 1, yet the image region is not convex.