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Title: On some applications of harmonic measure in the geometric theory of analytic functions (English)
Author: Fuka, Jaroslav
Author: Jakubowski, Z. J.
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 119
Issue: 1
Year: 1994
Pages: 57-74
Summary lang: English
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Category: math
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Summary: Let $\Cal P$ denote the well-known class of functions of the form $p(z)=1+q_1z+\ldots$ holomorphic in the unit disc $\bold D$ and fulfilling the conditions $Rep(z)>0$ in $\bold D$. Let $0\leq b<1, b<B, 0<\alpha <1$, be fixed real numbers and $zbold F$ a given measurable subset of the unit circle $\bold T$ of Lebesgue measure $2\pi\alpha$. For each $r \in (-\pi,\pi)$, denote by $\bold F_r=\{\xi\in \bold T; e^{-iT}\xi \in \bold F\}$ the set arising by rotation of $\bold F$ through the angle $\tau$. Denote by $\Cal P(B,b,\alpha;\bold F)$ the class of functions $p\in \Cal P$ satisfying the following conditions: there exists $\tau \in (-\pi,\pi)$ such that Rep(^{i\theta})\geq b$ a.e. on $\bold T\ \bold F_r$. In the paper the properties of the class $\Cal P(B,b,\alpha;\bold F)$ for different values of the parameters $B, b, \alpha$ and measurable sets $\bold F$ are examined. This article belongs to the series of papers ([4], [5], [6]) where different classes of functions defined by conditions on the circle $\bold T$ were studied. The results of papers [5], [6] are generalized. (English)
Keyword: harmonic measure
Keyword: Carathéodory functions
Keyword: extreme points
Keyword: support points
Keyword: coefficient estimates
MSC: 30C45
MSC: 30C50
MSC: 30C85
idZBL: Zbl 0805.30010
idMR: MR1303552
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Date available: 2009-09-24T21:02:53Z
Last updated: 2015-09-03
Stable URL: http://hdl.handle.net/10338.dmlcz/126198
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Reference: [1] L. V. Ahlfors: Conformal invariants: Topics in geometric function theory.McGraw-Hill, New York, 1973. Zbl 0272.30012, MR 0357743
Reference: [2] C. Carathéodory: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen.Rend. Сirc. Math., Palermo 32 (1911), 193-217. 10.1007/BF03014795
Reference: [3] P. L. Duren: Univalent functions. Grundlehren der mathematischen Wissenchaften.259, 1983. MR 0708494
Reference: [4] J. Fuka Z.J. Jakubowski: On certain subclasses of bounded univalent functions.Ann. Рolon. Mth. 55 (1991), 109-115; Proc. of the XІ-th Instructional Сonference on the Theory of Extremal Problems (in Polish), Lódź, 1990, 20-27. MR 1141428
Reference: [5] J. Fuka Z. J. Jakubowski: On a certain class of Сarathéodory functions defined by conditions on the circle.in: Сurrent Topics in Analytic Function Theory, editors H.M. Srivastava, S. Owa, Woгld Sci. Publ. Сompany, 94-105; Proc. of the V-th Intern. Сonf. on Сomplex Analysis, Varna, September 15-21, 1991, p. 11; Proc. of the XIII-th Instr. Сonf. on the Theory of Extremal Problems (in Polish), Lódź, 1992, 9-13. MR 1232431
Reference: [6] J. Fuka Z. J. Jakubowski: On extreme points of some subclasses of Сarathéodory functions.Сzechoslovak Academy of Sci., Math. lnst., Preprint 12 (1992), 1-9. MR 1232431
Reference: [7] J. B. Garnett: Bounded analytic functions.Academic Press, 1981. Zbl 0469.30024, MR 0628971
Reference: [8] D. J. Hallenbeck T.H. MacGregor: Linear Problems and convexity techniques in geometric function theory.Pitman Advanced Publ. Program, 1984. MR 0768747
Reference: [9] M. S. Robertson: On the theory of univalent functions.Ann. Math. 57 (1936), 374-408. Zbl 0014.16505, 10.2307/1968451
Reference: [10] W. Rudin: Real and complex analysis.McGraw-Hill, New York, 1974. Zbl 0278.26001, MR 0344043
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