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Marx-Strohhäcker Differential Subordination Systems

Sanford S. Miller and Petru T. Mocanu
Proceedings of the American Mathematical Society
Vol. 99, No. 3 (Mar., 1987), pp. 527-534
DOI: 10.2307/2046357
Stable URL: http://www.jstor.org/stable/2046357
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Marx-Strohhäcker Differential Subordination Systems
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Abstract

Let f(z) = z + a2z 2 + ⋯ 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 1/2, can be written in terms of differential subordinations as $zf''(z)/f'(z) \prec zk''(z)/k'(z) \Rightarrow zf'(z)/f(z) \prec zk'(z)/k(z).$ 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 $zf'(z)/f(z) \prec zk'(z)/k(z) \Rightarrow f(z)/z \prec k(z)/z.$ Finally, differential subordinations with starlike superordinate functions are considered.

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