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On some properties of the class \(\mathcal P(B,b,\alpha)\). (English) Zbl 0897.30002
Summary: Let \(\mathcal P\) denote the well known class of functions of the form \(p(z) = 1+q_1 z+\ldots \) holomorphic in the unit disc D and fulfilling the condition Re \(p(z) >0\) in D. Let \(0\leq b<1\), \(b<B\), \(0<\alpha <1\) be fixed real numbers. \(\mathcal P(B,b,\alpha)\) denotes the class of functions \(p\in \mathcal P\) such that there exists a measurable subset F of the unit circle T, of Lebesgue measure \(2\pi \alpha \), such that the function \(p\) fulfils Re \(p(e^{i\theta })\geq B\) a.e. on F and Re \(p(e^{i\theta })\geq b\) a.e. on T\(\setminus \)F. In this paper further properties of the class \(\mathcal P(B,b,\alpha)\) are examined. In particular, the investigations included in it constitute a direct continuation of earlier papers and concern mainly the form of the closed convex hull of the class \(\mathcal P(B,b,\alpha)\) as well as the estimate of the functional Re \(\{e^{i\lambda }p(x)\}\), \(0\not = z \in {\mathbf D}\), \(\lambda \in \langle -\pi ,\pi)\), \(p\in \mathcal P(B,b,\alpha)\). This article belongs to the series of papers where different classes of functions defined by conditions on the circle T were studied.
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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