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Proceedings of the American Mathematical Society

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

 

 

Starlike functions


Author: Carl P. McCarty
Journal: Proc. Amer. Math. Soc. 43 (1974), 361-366
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1974-0333147-3
MathSciNet review: 0333147
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Abstract: Let $ {\mathcal{S}^\ast }[\alpha ]$ denote the class of functions $ f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}} $ analytic in $ \vert z\vert < 1$ and for which $ \vert zf'(z)/f(z) - 1\vert < 1 - \alpha $ for $ \vert z\vert < 1$ and $ \alpha \in [0,1)$. Sharp results concerning coefficients, distortion, and the radius of convexity are obtained. Furthermore, it is shown that $ \sum\nolimits_{n = 2}^\infty {[(n - \alpha )/(1 - \alpha )]\vert{a_n}\vert < 1} $ is a sufficient condition for $ f(z) \in {\mathcal{S}^\ast }[\alpha ]$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0333147-3
Keywords: Schlicht functions, starlike functions, radius of convexity
Article copyright: © Copyright 1974 American Mathematical Society