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Keywords:
Hadamard product; typically real functions; class of type $A_\alpha$
Hadamard product; typically real functions; class of type $A_\alpha$
Summary:
Let $\Cal A$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F(0)=0, F'(0)=1$, whereas $A\subset \Cal A$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_\alpha, \alpha \in C \{-1,-\frac{1}{2},\ldots\}$, of functions of the form $f=F*k_\alpha$ are studied, where $F\in .A$, $k_\alpha(z)=k(z,\alpha)=z+\frac{1}{1+\alpha}z^2+\ldots + \frac{1}{1+(n-1)\alpha}z^n+\ldots$, and $F*k_\alpha$ denotes the Hadamard product of the functions $F$ and $k_\alpha$. Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3]).
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