Abstract
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)∈S o⇔ g(z) = zf'(z) ∈ 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 → 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 * → 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 < Re zf'(z)/ f(z) < b (0 < a < 1 < b), in the class K(γ) of functions, almost convex in order γ. The results of the article summarize or reinforce previously known effects.