Fréchet differentiable functionals and support points for families of analytic functions

Cochrane, Paul; MacGregor, Thomas H.

doi:10.1090/S0002-9947-1978-0460611-7

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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