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Proceedings of the American Mathematical Society

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

 
 

 

Subordination by convex functions


Authors: D. J. Hallenbeck and Stephan Ruscheweyh
Journal: Proc. Amer. Math. Soc. 52 (1975), 191-195
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9939-1975-0374403-3
MathSciNet review: 0374403
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Abstract: The following theorem is proven: Let $ F(z)$ be convex and univalent in $ \Delta = \{ z:\vert z\vert < 1\} ,F(0) = 1$. Let $ f(z)$ be analytic in $ \Delta ,f(0) = 1,f'(0) = \ldots = {f^{(n - 1)}}(0) = 0$, and let $ f(z) \prec F(z)$ in $ \Delta $. Then for all $ \gamma \ne 0$, Re $ \gamma \geqslant 0$,

$\displaystyle {\gamma _z}^{ - \gamma }\int_0^z {{\tau ^{\gamma - 1}}f(\tau )d\t... ...^{ - \gamma /n}}\int_0^{{z^{1/n}}} {{\tau ^{\gamma - 1}}F({\tau ^n})d\tau .} } $

This theorem, in combination with a method of D. Styer and D. Wright, leads to the following Corollary. Let $ f(z),g(z)$ be convex univalent in $ \Delta ,f(0) = f''(0) = g(0) = g''(0) = 0$. Then $ f(z) + g(z)$ is starlike univalent in $ \Delta $. Other applications of the theorem are concerned with the subordination of $ f(z)/z$ where $ f(z)$ belongs to certain classes of convex univalent functions.

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DOI: https://doi.org/10.1090/S0002-9939-1975-0374403-3
Keywords: Subordination, convolution, convex
Article copyright: © Copyright 1975 American Mathematical Society

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