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


On the radius of starlikeness of certain analytic functions

Author: Hassoon S. Al-Amiri
Journal: Proc. Amer. Math. Soc. 42 (1974), 466-474
MSC: Primary 30A32
MathSciNet review: 0330431
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Abstract: Let $ F(z)$ be regular in the unit disk $ \Delta :\vert z\vert < 1$ and normalized by the conditions $ F(0) = 0$ and $ F'(0) = 1$. Let $ f(z) = \tfrac{1}{2}[zF(z)]'$. Recently Libera and Livingston have studied the mapping properties of $ f(z)$ when $ F(z)$ is known. In particular, they have determined the radius of starlikeness of order $ \beta $ for $ f(z)$ when $ F(z)$ is starlike of order $ \alpha ,0 \leqq \alpha \leqq \beta < 1$. The author extends this study to include the complementary case $ 0 \leqq \beta < \alpha $. Also, a different proof has been given to determine the disk in which $ \operatorname{Re} \{ f'(z)\} > \beta $ when $ \operatorname{Re} \{ F'(z)\} > \alpha ,0 \leqq \alpha < 1,0 \leqq \beta < 1$. All results are sharp.

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Additional Information

PII: S 0002-9939(1974)0330431-4
Keywords: Univalent functions, starlike functions of order $ \alpha $, radius of starlikeness of order $ \alpha $, functions with positive real part, extremal function
Article copyright: © Copyright 1974 American Mathematical Society

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