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Proceedings of the American Mathematical Society

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

 

 

Functions with real part greater than $ \alpha $


Author: Carl P. McCarty
Journal: Proc. Amer. Math. Soc. 35 (1972), 211-216
MSC: Primary 30A76
DOI: https://doi.org/10.1090/S0002-9939-1972-0298014-0
MathSciNet review: 0298014
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Abstract: Let $ {\mathcal{P}_b}(\alpha )$ denote the class of functions $ P(z) = 1 + b(1 - \alpha )z + \cdots $ which are analytic and satisfy $ \operatorname{Re} \{ P(z)\} > \alpha $ for $ \vert z\vert < 1$ where $ \alpha \in [0,1)$ and $ b \in [0,2]$. We demonstrate some inequalities involving $ \vert P'(z)\vert$ and $ \vert P'(z)/P(z)\vert$ dependent on b and $ \alpha $ which are subsequently applied to the class of functions whose derivative lies in $ {\mathcal{P}_b}(\alpha )$ to obtain distortion, covering, and radius of convexity properties.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0298014-0
Keywords: Functions with positive real part, functions starlike of order $ \alpha $, radius of convexity
Article copyright: © Copyright 1972 American Mathematical Society