Geometric properties of the Bernatsky integral operator

In the geometric theory of complex variable functions, the study of mapping of classes of regular functions using various operators has now become an independent trend. The connection $f(z)\in S^{o}\Leftrightarrow g(z) = zf'(z) \in S^*$ of the classes $S^{o}$ and $S^*$ of convex and star-shaped functions can be considered as mapping using the differential operator $G[f](x) = zf'(z)$ of class $S^{o}$ to class $S^*$, that is, $G: S^{o} \to S^*$ or $G(S^{o}) = S^*$. The impetus for studying this range of issues was M. Bernatsky's assumption that the inverse operator $G^{-1}[f](x)$, which translates $S^* \to S^{o}$ and thereby “improves” the properties of functions, maps the entire class $S$ of single-leaf functions into itself.
At present, a number of articles have been published which study the various integral operators. In particular, they establish sets of values of indicators included in these operators where operators map class $S$ or its subclasses to themselves or to other subclasses.
This paper determines the values of the Bernatsky parameter included in the generalized integral operator, at which this operator transforms a subclass of star-shaped functions allocated by the condition $a < \mathrm{Re}\, zf'(z)/f(z) < b$ ($0 < a < 1 < b$), in the class $K(\gamma)$ of functions, almost convex in order $\gamma$. The results of the article summarize or reinforce previously known effects.