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Transactions of the American Mathematical Society

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



Convex hulls and extreme points of some families of univalent functions

Author: D. J. Hallenbeck
Journal: Trans. Amer. Math. Soc. 192 (1974), 285-292
MSC: Primary 30A32
MathSciNet review: 0338338
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Abstract: The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on $ ( - 1,1)$. We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on $ ( - 1,1)$. Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. We also obtain such a representation for the functions $ f(z)$ analytic in the unit disk, normalized and satisfying $ \operatorname{Re} f'(z) > \alpha $ for $ \alpha < 1$. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfying $ \operatorname{Re} f'(z) > \alpha $.

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Keywords: Univalent functions, starlike functions, starlike function with real coefficients, convex function, convex function with real coefficients, close-to-convex function, close-to-convex function with real coefficients, extreme point, integral representation, probability measures, subordination, continuous linear functional
Article copyright: © Copyright 1974 American Mathematical Society

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