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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Convex hulls and extreme points of families of starlike and convex mappings

Authors: L. Brickman, D. J. Hallenbeck, T. H. Macgregor and D. R. Wilken
Journal: Trans. Amer. Math. Soc. 185 (1973), 413-428
MSC: Primary 30A32
MathSciNet review: 0338337
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Abstract: The closed convex hull and extreme points are obtained for the starlike functions of order $ \alpha $ and for the convex functions of order $ \alpha $. More generally, this is determined for functions which are also k-fold symmetric. Integral representations are given for the hulls of these and other families in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to or majorized by some function which is starlike of order $ \alpha $. Also, the lower bound on $ \operatorname{Re} \{ f(z)/z\} $ is found for each $ z\;(\vert z\vert < 1)$ where f varies over the convex functions of order $ \alpha $ and $ \alpha \geq 0$.

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Keywords: Univalent function, starlike function, starlike function of order $ \alpha $, convex function, convex function of order $ \alpha $, convex hull, extreme point, integral representation, probability measures, subordination, majorization, k-fold symmetric function, continuous linear functional
Article copyright: © Copyright 1973 American Mathematical Society