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Proceedings of the American Mathematical Society

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



Distortions properties of alpha-starlike functions

Author: Sanford S. Miller
Journal: Proc. Amer. Math. Soc. 38 (1973), 311-318
MSC: Primary 30A32
MathSciNet review: 0310222
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Abstract: Let $ \alpha $ be real and suppose that $ f(z) = z + \Sigma _2^\infty {a_n}{z^n}$ is regular in the unit disc $ D$ with $ f(z)f'(z) \ne 0$ in $ 0 < \vert z\vert < 1$. If $ \operatorname{Re} [(1 - \alpha )zf'(z)/f(z) + \alpha ((zf''(z)/f'(z)) + 1)] > 0$ for $ z \in D$, then $ f(z)$ is said to be an alpha-starlike function. These functions are univalent and they very naturally unify the classes of starlike $ (\alpha = 0)$ and convex $ (\alpha = 1)$ functions. The author obtains the $ \tfrac{1}{4}$-theorem, sharp bounds on $ \vert f(z)\vert$ and $ \vert f'(z)\vert$, and growth conditions on $ M(r)$.

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Keywords: Univalent, alpha-starlike, starlike, convex, Mocanu, distortion, $ \tfrac{1}{4}$-theorem
Article copyright: © Copyright 1973 American Mathematical Society

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