| 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 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
119 |
| Issue:
|
1 |
| Year:
|
1994 |
| Pages:
|
57-74 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 |
| DOI:
|
10.21136/MB.1994.126198 |
| . |
| Date available:
|
2009-09-24T21:02:53Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126198 |
| . |
| 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 |
| . |
