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

DOI:
https://doi.org/10.1090/S0002-9947-1974-0338338-8

MathSciNet review:
0338338

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Abstract | References | Similar Articles | Additional Information

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 . We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on . 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 analytic in the unit disk, normalized and satisfying for . 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 .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0338338-8

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