Linear homeomorphisms of some classical families of univalent functions

Author:
Frederick W. Hartmann

Journal:
Proc. Amer. Math. Soc. **63** (1977), 265-272

MSC:
Primary 30A36

DOI:
https://doi.org/10.1090/S0002-9939-1977-0454002-7

MathSciNet review:
0454002

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Abstract: The extreme points of the closed convex hull of some classical families of univalent functions analytic on the open unit disk, e.g. the convex, *K*, and starlike, *St*, have recently been characterized. These characterizations are used to determine an explicit representation for the class of linear homeomorphisms of the extreme points of the closed convex hulls of *K* and *St* and thus of the hulls themselves. With the aid of these representations it is shown that every linear homeomorphism of *K* or *St* is a rotation, i.e. convolution with . In the way of a positive result: if is the convex set of analytic functions with positive real part and and is a linear homeomorphism of , then , but .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0454002-7

Keywords:
Continuous linear operator,
matrix transformation,
functions with positive real part,
extreme points,
closed convex hull,
linear homeomorphism,
univalent functions,
starlike mappings,
convex mappings,
Krein-Milman theorem

Article copyright:
© Copyright 1977
American Mathematical Society