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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 univalence of some classes of regular functions


Authors: R. J. Libera and A. E. Livingston
Journal: Proc. Amer. Math. Soc. 30 (1971), 327-336
MSC: Primary 30.42
MathSciNet review: 0288244
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Abstract: Let $ F(z)$ be regular in the unit disk $ \Delta = \{ z:\vert z\vert < 1\} $ and normalized by the conditions $ F(0) = 0$ and $ F'(0) = 1$ and let $ 2f(z) = [zF(z)]'$. The paper deals with the mapping properties of $ f(z)$ when $ F(z)$ is known. For example, if $ F(z)$ is starlike of order $ \alpha, 0 \leq \alpha < 1$, then the disk in which $ f(z)$ is always starlike of order $ \beta, \alpha \leq \beta < 1$, is determined. All results are sharp.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0288244-5
PII: S 0002-9939(1971)0288244-5
Keywords: Univalent function, starlike function of order $ \alpha $, radius of starlikeness of order $ \alpha $, function with positive real part
Article copyright: © Copyright 1971 American Mathematical Society