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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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Keywords: Analytic function, continuous linear functional, Fr&#233;chet differentiable functional, univalent function, starlike mapping, starlike function of order <IMG WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$\alpha$">, convex mapping, convex function of order <IMG WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\alpha$">, 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