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

The Hardy class of a Bazilevič function and its derivative


Author: Sanford S. Miller
Journal: Proc. Amer. Math. Soc. 30 (1971), 125-132
MSC: Primary 30.42
MathSciNet review: 0288246
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Abstract: The Bazilevič function $ f(z)$ defined in $ \Delta :\vert z\vert < 1$ by $ f(z) \equiv {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}$ where $ g(\zeta )$ is starlike in $ \Delta $, $ P(\zeta )$ is regular with Re $ P(\zeta ) > 0$ in $ \Delta $ and $ \alpha > 0$ is univalent. The class of such functions contains many of the special classes of univalent functions. The author determines the Hardy classes to which $ f(z)$ and $ f'(z)$ belong. In addition if $ f(z) = \sum\nolimits_0^\infty {{a_n}{z^n}} $ the limiting value of $ \vert{a_n}\vert/n$ is obtained.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0288246-9
PII: S 0002-9939(1971)0288246-9
Keywords: Univalent, Bazilevič functions, close-to-convex, Hardy class, coefficient growth condition, Cesaro summability
Article copyright: © Copyright 1971 American Mathematical Society