On some applications of harmonic measure in the geometric theory of analytic functions. (English) Zbl 0805.30010

Let \({\mathfrak p}\) denote the class of functions of the form \(p(z) = 1 + q_ 1z + \cdots + q_ nz^ n + \dots\) holomorphic in the disc \(\mathbb{D} = \{z; | z | < 1\}\) and fulfilling the condition \(\text{Re} p(z)>0\) in \(\mathbb{D}\). Let \(0 \leq b<1\), \(b<B\), \(0< \alpha<1\), be fixed numbers and \(\mathbb{F}\) a given measurable subset of the circle \(\mathbb{T} = \{z; | z | = 1\}\) of Lebesgue measure \(2 \pi \alpha\). For each \(\tau \in < - \pi, \pi)\), denote \(\mathbb{F}_ \tau = \{\xi \in \mathbb{T}; e^{-i \tau} \xi \in \mathbb{F}\}\). Denote by \({\mathfrak p} (B,b, \alpha; \mathbb{F})\) the class of functions \(p \in {\mathfrak p}\) satisfying the following condition: there exists \(\tau \in < - \pi, \pi)\) such that \(\text{Re} p(e^{i \theta}) \geq B\) a.e. on \(\mathbb{F}_ \tau\) and \(\text{Re} p(e^{i \theta}) \geq b\) a.e. on \(\mathbb{T} \backslash \mathbb{F}_ \tau\). In the paper, the properties of the class \({\mathfrak p} (B,b, \alpha; \mathbb{F})\) for different values of the parameters \(B,b,\alpha\) and the measurable sets \(\mathbb{F}\) are examined. This article belongs to a series of papers [the authors: 1) Ann. Pol. Math. 55, No. 19, 109-115 (1991; Zbl 0755.30023)], 2) in: Current topics in analytic function theory, World Sci. Publ. Company, 94- 105 (1992)), 3) Czech. acad. sci., math inst., preprint 72, 1-9 (1992)] where different classes of functions defined by conditions on the circle \(\mathbb{T}\) where studied. The results of papers 2), 3) are generalized.
Reviewer: J.Fuka


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)


Zbl 0755.30023
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