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

 

Distortion properties of $ p$-fold symmetric alpha-starlike functions


Authors: H. B. Coonce and S. S. Miller
Journal: Proc. Amer. Math. Soc. 44 (1974), 336-340
MSC: Primary 30A32
MathSciNet review: 0344440
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Abstract: Starlike functions $ f$ which are of Mocanu type $ \alpha $ and have power series of the form

$\displaystyle f(z) = z + {a_{p + 1}}{z^{p + 1}} + {a_{2p + 1}}{z^{2p + 1}} + \cdots ,$

where $ p = 1,2,3, \cdots $, are shown to satisfy the relation $ f(z) = {[g({z^p})]^{1/p}}$ where $ g$ is of Mocanu type $ p\alpha $ with power series $ g(z) = z + {b_2}{z^2} + {b_3}{z^3} + \cdots $. Distortion results dealing with the $ \tfrac{1}{4}$-theorem and bounds on $ \vert f(z)\vert$ are obtained.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0344440-2
PII: S 0002-9939(1974)0344440-2
Keywords: Starlike, convex, alpha-starlike, Mocanu type, $ p$-fold symmetric, distortion theorems
Article copyright: © Copyright 1974 American Mathematical Society



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