ABOUT SOME SUBCLASSES OF CLASSES OF CLOSE-TO-CONVEX AND DOUBLY CLOSE-TO-CONVEX FUNCTIONS

It is known that the class  close-to-convex functions is defined by the condition of positivity of the functional  with a starlike function . Replacing the starlike function  with a convex one leads to a known subclass  of the class . In this article, we introduce a generalization of the class  to the case when the set of values  is contained in a region of a special type, which, in a particular case, can coincide with a half-plane. The generalization is also associated with the extension of the class  to a certain subclass of the class of normalized doubly close-to-convex functions. The diversity of special cases of a domain of a special type and the transition to doubly close-to-convex functions allows us to obtain both new original results and generalizations of previously known results. The main research of this article is aimed at proving theorems on distortion, finding the radii of convexity of the considered classes of functions and justifying the accuracy of the obtained results. A connection has also been established between the introduced class of functions and a certain new class of doubly close-to-starlike functions, special cases of which have been actively studied in recent years. For this class and its subclasses, new results have also been obtained in the form of theorems on growth and radius of starlikeness and generalization of previously known results.

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