Classes of analytic functions subordinate to convex functions and extreme points. (English) Zbl 1025.30011

Suppose that \(\Delta\) is the unit disk, \(D\) is a convex domain, \(D\neq\mathbb{C}\), \(0\in D\), \(F\) is an analytic and univalent mapping of \(\Delta\) onto \(D\). Let \(s(F)\) denote the set of all analytic functions \(f\) subordinate to \(F\) on \(\Delta\) and let \(Es(F)\) be the set of extreme points. The set \(s(F)\) is known when \(D\) is an infinite wedge, a strip, a half plane or a domain with smooth boundary and piecewise positive curvature but a full description of \(Es(F)\) otherwise is an open problem. Considering the open question the authors of this paper established two new sufficient conditions for \(f\) to be an extreme point of \(s(F)\).


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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[1] Abu-Muhanna, Y.; MacGregor, T. H., Extreme points of families of analytic functions subordinate to convex mappings, Math. Z., 176, 511-519 (1981) · Zbl 0461.30018
[2] Gevirtz, J., On extreme points of families of analytic functions with values in a convex set, Math. Z., 193, 79-83 (1986) · Zbl 0603.30028
[3] Hallenbeck, D. J.; MacGregor, T. H., Linear problems and convexity techniques in geometric function theory, Monographs and Stud. in Math., 22 (1984), Pitman: Pitman New York · Zbl 0581.30001
[4] Milcetich, J. G., On the extreme points of some sets of analytic functions, Proc. Amer. Math. Soc., 45, 223-228 (1974) · Zbl 0296.30015
[5] Tkaczyńska, K., On extreme points of subordination families with a convex majorant, J. Math. Anal. Appl., 145, 216-231 (1990) · Zbl 0694.30022
[6] Tkaczyńska-Hallenbeck, K., On properties of extreme points of subordination families with a convex majorant, Math. Japon., 3, 537-543 (1995) · Zbl 0841.30017
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