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Keywords:
Carathéodory functions; closed convex hull; estimates of functionals
Carathéodory functions; closed convex hull; estimates of functionals
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 $\bD$ and fulfilling the condition $\Re p(z)>0$ in $\bD$. Let $0\le b<1$, $b<B$, $0<\a<1$ be fixed real numbers. $\Cal P(B,b,\a)$ denotes the class of functions $p\in\Cal P$ such that there exists a measurable subset $\bF$ of the unit circle $\bT$, of Lebesgue measure $2\pi\a$, such that the function $p$ fulfils $\Re p(\ee^{\ii\theta})\ge B$ a.e. on $\bF$ and $\Re p(\ee^{\ii\theta})\ge b$ a.e. on $\bT\setminus\bF$. In this paper further properties of the class $\Cal P(B,b,\alpha)$ are examined. In particular, the investigations included in it constitute a direct continuation of papers [6]-[8] and concern mainly the form of the closed convex hull of the class $\Cal P(B,b,\alpha)$ as well as the estimates of the functional $\Re \{\ee^{\ii\lambda}p(z)\}$, $0\neq z\in\bD$, $\lambda\in\langle-\pi,\pi)$, $p\in\Cal P(B,b,\alpha)$. This article belongs to the series of papers ([1]-[8]) where different classes of functions defined by conditions on the circle $\bT$ were studied.
References:
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