# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by D. J. Hallenbeck and Stephan Ruscheweyh
Proc. Amer. Math. Soc. 52 (1975), 191-195 Request permission

## Abstract:

The following theorem is proven: Let $F(z)$ be convex and univalent in $\Delta = \{ z:|z| < 1\} ,F(0) = 1$. Let $f(z)$ be analytic in $\Delta ,f(0) = 1,fâ€™(0) = \ldots = {f^{(n - 1)}}(0) = 0$, and let $f(z) \prec F(z)$ in $\Delta$. Then for all $\gamma \ne 0$, Re $\gamma \geqslant 0$, ${\gamma _z}^{ - \gamma }\int _0^z {{\tau ^{\gamma - 1}}f(\tau )d\tau \prec \gamma {z^{ - \gamma /n}}\int _0^{{z^{1/n}}} {{\tau ^{\gamma - 1}}F({\tau ^n})d\tau .} }$ 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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