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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 (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:
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