DOUBLY CLOSE-TO-STARLIKE FUNCTIONS BASED ON STARLIKE FUNCTIONS OF A CERTAIN ORDER - Maiyer, Ф.Ф.; Tastanov , М.Г.; Utemissova , A.; Kalakov, Б.А. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2025)
The paper studies the properties of some classes of doubly close-to-starlike and doubly close-to-convex functions. The objective of this paper is to introduce and study the class of doubly close-to-starlike functions, which includes as special cases both the classical classes of close-to-starlike functions, studied at the early stages of the development of this problem, and some classes of doubly close-to-starlike functions, studied in recent years. The research methodology is based on the principle of subordination of analytic functions. The objectives of the study include obtaining theorems of growth (distortion) and radii of starlikeness (convexity) of classes of doubly close-to-starlike (doubly close-to-convex) functions. This is achieved due to the fact that the function , a starlike function of order , is used as the base function , which allows us to consider subclasses of this class as special cases when is not only a starlike function, but also a convex function. In the article, a growth theorem and a of starlikeness radius of order are obtained. introduced class of doubly close-to-starlike functions, special cases are considered. All results are exact, in special cases they lead to both new and previously known results. Using a certain substitution, a transition is made from the introduced class of doubly close-to-starlike functions to a new class of doubly close-to-convex functions, for which distortion theorems and the radius of convexity are obtained and special cases are considered.