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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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Subordination by convex functions
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by D. J. Hallenbeck and Stephan Ruscheweyh PDF
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 $\Delta ,f(0) = 1,f’(0) = \ldots = {f^{(n - 1)}}(0) = 0$, and let $f(z) \prec F(z)$ in $\Delta$. Then for all $\gamma \ne 0$, Re $\gamma \geqslant 0$, \[ {\gamma _z}^{ - \gamma }\int _0^z {{\tau ^{\gamma - 1}}f(\tau )d\tau \prec \gamma {z^{ - \gamma /n}}\int _0^{{z^{1/n}}} {{\tau ^{\gamma - 1}}F({\tau ^n})d\tau .} } \] 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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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