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Subordination families and extreme points. (English) Zbl 0651.30019

Let F be a univalent in the unit disk \(\Delta\), let s(F) denote the family of functions in \(\Delta\) that are subordinate to F, and let \(B_ 0=s(id).\)
The authors prove that if \(f=F\circ \phi\) is an extreme point of s(F) then \(\Phi\) must be extreme in \(B_ 0\), therefore solving a conjecture made by the first author in [Proc. Am. Math. Soc. 87, 439-443 (1983; Zbl 0519.30023)]. If \(\lambda\) (w,\(\partial F(\Delta))\) denotes the distance between w and \(\partial F(\Delta)\) it is also shown that for each extreme point \(\phi\) of \(B_ 0\) the relation \[ \int^{2\pi}_{0}\log \lambda (F(e^{i\theta}\phi (e^{it})),\quad \partial F(\Delta))dt=-\infty \] holds for almost all \(\theta\).
Reviewer: W.Koepf

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C55 General theory of univalent and multivalent functions of one complex variable

Citations:

Zbl 0519.30023
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References:

[1] Yusuf Abu-Muhanna, On extreme points of subordination families, Proc. Amer. Math. Soc. 87 (1983), no. 3, 439 – 443. · Zbl 0519.30023
[2] 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. · Zbl 0227.30013
[3] Peter L. Duren, Theory of \?^{\?} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.
[4] David J. Hallenbeck, Extreme points of subordination families with univalent majorants, Proc. Amer. Math. Soc. 91 (1984), no. 1, 54 – 58. · Zbl 0512.30013
[5] D. J. Hallenbeck and T. H. MacGregor, Linear problems and convexity techniques in geometric function theory, Monographs and Studies in Mathematics, vol. 22, Pitman (Advanced Publishing Program), Boston, MA, 1984. · Zbl 0581.30001
[6] Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. · Zbl 0102.29401
[7] Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. · Zbl 0298.30014
[8] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005
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