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