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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the coefficients of Bazilevič functions
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by F. R. Keogh and Sanford S. Miller PDF
Proc. Amer. Math. Soc. 30 (1971), 492-496 Request permission

Abstract:

Let $B(\alpha )$ denote the class of normalized $(f(0) = 0,f’(0) = 1)$, Bazilevič functions of type $\alpha$ defined in $\Delta :|z| < 1,{\text {i.e.}}f(z) = {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}$ 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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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