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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Marx-Strohhäcker differential subordination systems


Authors: Sanford S. Miller and Petru T. Mocanu
Journal: Proc. Amer. Math. Soc. 99 (1987), 527-534
MSC: Primary 30C80; Secondary 30C45
MathSciNet review: 875392
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Abstract: Let $ f(z) = z + {a_2}{z^2} + \cdots $ be analytic in the unit disc $ U$ and let $ k(z) = z/(1 - z)$. The classic Marx-Strohhäcker result, that a convex (univalent) function $ f$ is starlike of order $ \frac{1}{2}$, can be written in terms of differential subordinations as

$\displaystyle zf''(z)/f'(z) \prec zk''(z)/k'(z) \Rightarrow zf'(z)/f(z) \prec zk'(z)/k(z).$

The authors determine general conditions on $ k$ for which this relation holds. They also determine a different set of general conditions on $ k$ for which

$\displaystyle zf'(z)/f(z) \prec zk'(z)/k(z) \Rightarrow f(z)/z \prec k(z)/z.$

Finally, differential subordinations with starlike superordinate functions are considered.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0875392-3
PII: S 0002-9939(1987)0875392-3
Keywords: Differential subordination, superordinate, convex function, starlike function, univalent function
Article copyright: © Copyright 1987 American Mathematical Society