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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Fréchet differentiable functionals and support points for families of analytic functions
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by Paul Cochrane and Thomas H. MacGregor PDF
Trans. Amer. Math. Soc. 236 (1978), 75-92 Request permission

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
  • © Copyright 1978 American Mathematical Society
  • 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