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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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 $ \{ \exp(in \theta ):n = 0,1, \ldots \} $. In the way of a positive result: if $ \mathcal{P}$ is the convex set of analytic functions with positive real part and $ f(0) = 1$ and $ \mathcal{L}$ is a linear homeomorphism of $ \mathcal{P}$, then $ \mathcal{L}(St) \subset St$, but $ \mathcal{L}(K) \not\subset K$.


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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