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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A product theorem for $ \mathcal{F}_p$ classes and an application


Author: Kent Pearce
Journal: Proc. Amer. Math. Soc. 84 (1982), 509-515
MSC: Primary 30C45
MathSciNet review: 643739
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Abstract: For Re $ p > 0$ let $ {\mathcal{F}_p} = \{ f\vert f(z) = \int_{\vert x\vert = 1} {{{(1 - xz)}^{ - p}}d\mu (x)} $, $ \vert z\vert < 1$, $ \mu $ a probability measure on $ \vert x\vert = 1$ and let $ {\mathcal{F}_p} \cdot {\mathcal{F}_q} = \{ fg\vert f \in {\mathcal{F}_p},g \in {\mathcal{F}_q}\} $. Brickman, Hallenbeck, MacGregor and Wilken proved a product theorem for the $ {\mathcal{F}_p}$ classes; they showed that if $ p > 0$, $ q > 0$, then $ {\mathcal{F}_p} \cdot {\mathcal{F}_q} \subset {\mathcal{F}_{p + q}}$. We give an (essentially complete) converse for the result of Brickman et al., i.e., we show that if $ {\mathcal{F}_p}\cdot{\mathcal{F}_q} \subset {\mathcal{F}_{p + q}}$, then $ p > 0$, $ q > 0$ or else $ p = q = 1 + it$ for some $ t$ real. As an immediate consequence we disprove a conjecture about the extreme points of the closed convex hulls of the classes $ {\text{Sp(}}\gamma {\text{)}}$, $ 0 < \vert\gamma \vert < \pi /2$, of $ \gamma $-spirallike univalent functions, i.e., writing $ m = 1 + {e^{ - 2i\gamma }}$, we show $ \{ z/{(1 - xz)^m}\vert\vert x\vert = 1\} \mathop \subset \limits_ \ne \mathcal{E}\mathcal{K}{\text{Sp(}}\gamma {\text{)}}$, $ 0 < \vert\gamma \vert < \pi /2$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0643739-4
PII: S 0002-9939(1982)0643739-4
Keywords: Product theorem, extreme points, spirallike functions
Article copyright: © Copyright 1982 American Mathematical Society