# Article

Full entry | PDF   (1.3 MB)
Keywords:
Carathéodory functions; closed convex hull; estimates of functionals
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.
References:
[1] J. Fuka Z. J. Jakubowski: On certain subclasses of bounded univalent functions. Ann. Polon. Math. 55 (1991), 109-115. DOI 10.4064/ap-55-1-109-115 | MR 1141428
[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
[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
[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.
[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.
[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
[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.
[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
[9] P. T. Mocanu: Une propriété de convexité generalisée dans la théorie de la représentation conforme. Mathematica (Cluj) 11 (З4) (1969), 127-133. MR 0273000 | Zbl 0195.36401
[10] M. S. Robertson: Analytic functions star-like in one direction. Amer. J. Math. 58 (1936), 465-472. DOI 10.2307/2370963 | MR 1507169 | Zbl 0014.12002

Partner of