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

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

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

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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 $\operatorname {Re} f’(z) > \alpha$.

References

L. Brickman, D. J. Hallenbeck, T. H. MacGregor, and D. R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185 (1973), 413–428 (1974). MR 338337, DOI 10.1090/S0002-9947-1973-0338337-5
L. Brickman, T. H. MacGregor, and D. R. Wilken, Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc. 156 (1971), 91–107. MR 274734, DOI 10.1090/S0002-9947-1971-0274734-2
R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755–758. MR 178131, DOI 10.1090/S0002-9939-1965-0178131-2
T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104 (1962), 532–537. MR 140674, DOI 10.1090/S0002-9947-1962-0140674-7
Thomas H. MacGregor, The radius of convexity for starlike functions of order ${1\over 2}$, Proc. Amer. Math. Soc. 14 (1963), 71–76. MR 150282, DOI 10.1090/S0002-9939-1963-0150282-6
Thomas H. MacGregor, Applications of extreme-point theory to univalent functions, Michigan Math. J. 19 (1972), 361–376. MR 311885
Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0045823

On the theory of schlicht functions

Pasquale Porcelli, Linear spaces of analytic functions, Rand McNally & Co., Chicago, Ill., 1966. MR 0259581
Malcolm I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), no. 2, 374–408. MR 1503286, DOI 10.2307/1968451
M. S. Robertson, Analytic Functions Star-Like in One Direction, Amer. J. Math. 58 (1936), no. 3, 465–472. MR 1507169, DOI 10.2307/2370963
Werner Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (2) 48 (1943), 48–82. MR 8625, DOI 10.1112/plms/s2-48.1.48

Analytic functions

with

and applications

19

Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966