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Let $𝒜$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F\left(0\right)=0,F^{\text{'}}\left(0\right)=1$, whereas $A\subset 𝒜$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_{\alpha },\alpha \in C\{-1,-\frac{1}{2},...\}$, of functions of the form $f=F*k_{\alpha }$ are studied, where $F\in .A$, $k_{\alpha }\left(z\right)=k(z,\alpha )=z+\frac{1}{1+\alpha }{z}^2+...+\frac{1}{1+(n-1)\alpha }{z}^n+...$, 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]).