Convex hulls and extreme points of some families of univalent functions

Hallenbeck, D. J.

doi:10.1090/S0002-9947-1974-0338338-8

Convex hulls and extreme points of some families of univalent functions
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by D. J. Hallenbeck PDF

Trans. Amer. Math. Soc. 192 (1974), 285-292 Request permission

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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Analytic functions

with

and applications

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