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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

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