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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the coefficients of Bazilevič functions


Authors: F. R. Keogh and Sanford S. Miller
Journal: Proc. Amer. Math. Soc. 30 (1971), 492-496
MSC: Primary 30.43
DOI: https://doi.org/10.1090/S0002-9939-1971-0283191-7
MathSciNet review: 0283191
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Abstract: Let $ B(\alpha )$ denote the class of normalized $ (f(0) = 0,f'(0) = 1)$, Bazilevič functions of type $ \alpha $ defined in $ \Delta :\vert z\vert < 1,{\text{i.e.}}f(z) = {[\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$. Let $ {B_m}(\alpha )$ denote the subclass of $ B(\alpha )$ which is m-fold symmetric $ (f({e^{2\pi i/m}}z) = {e^{2\pi i/m}}f(z),m = 1,2, \cdots )$. Functions in $ B(\alpha )$ have been shown to be univalent. The authors obtain sharp coefficient inequalities for functions in $ {B_m}(1/N)$ where N is a positive integer. In addition an example of a Bazilevič function which is not close-to-convex is given.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0283191-7
Keywords: Univalent, Bazilevič functions, close-to-convex functions, m-fold symmetric, Bieberbach conjecture, majorization
Article copyright: © Copyright 1971 American Mathematical Society