| Title:
|
On some properties of the class $\scr P(B,b,\alpha)$ (English) |
| Author:
|
Fuka, J. |
| Author:
|
Jakubowski, Z. J. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
122 |
| Issue:
|
2 |
| Year:
|
1997 |
| Pages:
|
197-220 |
| 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 $\bD$ and fulfilling the condition $\Re p(z)>0$ in $\bD$. Let $0\le b<1$, $b<B$, $0<\a<1$ be fixed real numbers. $\Cal P(B,b,\a)$ denotes the class of functions $p\in\Cal P$ such that there exists a measurable subset $\bF$ of the unit circle $\bT$, of Lebesgue measure $2\pi\a$, such that the function $p$ fulfils $\Re p(\ee^{\ii\theta})\ge B$ a.e. on $\bF$ and $\Re p(\ee^{\ii\theta})\ge b$ a.e. on $\bT\setminus\bF$. In this paper further properties of the class $\Cal P(B,b,\alpha)$ are examined. In particular, the investigations included in it constitute a direct continuation of papers [6]-[8] and concern mainly the form of the closed convex hull of the class $\Cal P(B,b,\alpha)$ as well as the estimates of the functional $\Re \{\ee^{\ii\lambda}p(z)\}$, $0\neq z\in\bD$, $\lambda\in\langle-\pi,\pi)$, $p\in\Cal P(B,b,\alpha)$. This article belongs to the series of papers ([1]-[8]) where different classes of functions defined by conditions on the circle $\bT$ were studied. (English) |
| Keyword:
|
Carathéodory functions |
| Keyword:
|
closed convex hull |
| Keyword:
|
estimates of functionals |
| MSC:
|
30C45 |
| idZBL:
|
Zbl 0897.30002 |
| idMR:
|
MR1460950 |
| DOI:
|
10.21136/MB.1997.125914 |
| . |
| Date available:
|
2009-09-24T21:25:11Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/125914 |
| . |
| Reference:
|
[1] J. Fuka Z. J. Jakubowski: On certain subclasses of bounded univalent functions.Ann. Polon. Math. 55 (1991), 109-115. MR 1141428, 10.4064/ap-55-1-109-115 |
| Reference:
|
[2] J. Fuka Z. J. Jakubowski: A certain class of Carathéodory functions defined by conditions on the unit circle.Current Topics in Analytic Function Theory, editors: H.M. Sгivastava, Shigeyoshi Owa, World Sci. Publ. Company, Singapore (1992), 94-105. MR 1232431 |
| Reference:
|
[3] J. Fuka Z. J. Jakubowski: On extreme points of some subclasses of Carathéodory functions.Czechoslovak Academy Sci. Math. Inst., Preprint 72 (1992), 1-9. MR 1232431 |
| Reference:
|
[4] J. Fuka Z. J. Jakubowski: On coefficient estimates in a class of Carathéodory functions with positive real part.Proc. of the 15-th Instr. Conf. on Complex Analysis and Geometry, Bronislawów 11-15.01.1993, Lódź (1994), 17-24. |
| Reference:
|
[5] J. Fuka Z. J. Jakubowski: The problem of convexity and compactness of some class of Carathéodory functions.Proc. of the 15-th Instr. Conf. on Complex Analysis and Geometry, Bronislawów, 11-15.01.1993, Lódź (1994), 25-30. |
| Reference:
|
[6] J. Fuka Z. J. Jakubowski: On some applications of harmonic measure in the geometric theory of analytic functions.Math. Bohem. 119 (1994), 57-74. MR 1303552 |
| Reference:
|
[7] J. Fuka Z. J. Jakubowski: On some closure of the class P(B,b,$\alpha$).Proc. of the 16-th Instr. Conf. on Complex Analysis and Geometry, Bronislawów, 10-14.01.1994, Lódź (1995), 9-11. |
| Reference:
|
[8] J. Fuka Z. J. Jakubowski: On estimates of functionals in some classes of functions with positive real part.Math. Slovaca 46 (1996), No. 2-3, 213-230. MR 1427006 |
| Reference:
|
[9] P. T. Mocanu: Une propriété de convexité generalisée dans la théorie de la représentation conforme.Mathematica (Cluj) 11 (З4) (1969), 127-133. Zbl 0195.36401, MR 0273000 |
| Reference:
|
[10] M. S. Robertson: Analytic functions star-like in one direction.Amer. J. Math. 58 (1936), 465-472. Zbl 0014.12002, MR 1507169, 10.2307/2370963 |
| . |
