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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Fréchet differentiable functionals and support points for families of analytic functions


Authors: Paul Cochrane and Thomas H. MacGregor
Journal: Trans. Amer. Math. Soc. 236 (1978), 75-92
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9947-1978-0460611-7
MathSciNet review: 0460611
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Abstract: Given a closed subset of the family $ {S^\ast}(\alpha )$ of functions starlike of order $ \alpha $ of a particular form, a continuous Fréchet differentiable functional, J, is constructed with this collection as the solution set to the extremal problem $ \max \operatorname{Re} J(f)$ over $ {S^\ast}(\alpha )$. Similar results are proved for families which can be put into one-to-one correspondence with $ {S^\ast}(\alpha )$.

The support points of $ {S^\ast}(\alpha )$ and $ K(\alpha )$, the functions convex of order $ \alpha $, are completely characterized and shown to coincide with the extreme points of their respective convex hulls. Given any finite collection of support points of $ {S^\ast}(\alpha )$ (or $ K(\alpha )$), a continuous linear functional, J, is constructed with this collection as the solution set to the extremal problem $ \max \operatorname{Re} J(f)$ over $ {S^\ast}(\alpha )$ (or $ K(\alpha )$).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0460611-7
Keywords: Analytic function, continuous linear functional, Fréchet differentiable functional, univalent function, starlike mapping, starlike function of order $ \alpha $, convex mapping, convex function of order $ \alpha $, variations of functions, bounded functions, support point, extreme point, convex hull, coefficient region, function with a positive real part, subordination
Article copyright: © Copyright 1978 American Mathematical Society