On a Generalization of One Subclass of the Class of Close-To-Convex Functions
Maiyer F.F. Baimankulov A.T. Tastanov M.G. Utemissova A.A. Zhuaspayev T.A.
2025World Scientific and Engineering Academy and Society
WSEAS Transactions on Mathematics
2025#24431 - 442 pp.
The article introduces a generalization of the subclass c′ class of close-to-convex functions, which is defined by the condition of positivity of the functional Re(zf′(z)/g(z)), where g(z) is an arbitrary convex function. The generalization of this class is related to both the membership of the values of the functional zf′(z)/g(z) region of a special type contained in the right half-plane relative to the imaginary axis, and using starlike functions g(z) of arbitrary order. In this case, functions with gaps in the expansion in a series are considered. In a generalized subclass of the class of close-to-convex functions, theorems of distortion and rotation, as well as the radius of convexity of this class, are established. All results are exact. The breadth of the introduced class of close-to-convex functions allows us to obtain a generalization of a number of results both for various subclasses of the class of close-to-convex functions, including the entire class of close-to-convex functions, and for the class of close-to-starlike functions and its subclasses. In particular, new results are obtained for classes of functions convex in the direction of the imaginary axis or in the positive direction of the real axis, as well as for typically real functions.It is also shown that using starlike functions of order 1/2 instead of convex functions leads to the same distortion, rotation, and convexity radius theorems as for the classical subclass c′ close-to-convex functions.
close-to-convex functions , close-to-starlike functions , distortion theorems , growth theorems , radii of convexity , radii of starlikeness , starlike functions , univalent functions
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Department of Mathematics and Physics, Kostanay Regional University named after. A. Baitursynuly, Astana, Kostanay, Kazakhstan
The Department of Information Technology and Automation, Kostanay University of Engineering and Economics named after M.Dulatov, Astana, Kostanay, Kazakhstan
Department of Mathematics and Physics
The Department of Information Technology and Automation
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