On the generalization of some classes of close-to-convex and typically real functions
Об обобщении некоторых классов почти выпуклых и типично вещественных функций
Maiyer F.F. Tastanov M.G. Utemissova A.A. Baimankulov A.T.
2023Tomsk State University
Vestnik Tomskogo Gosudarstvennogo Universiteta, Matematika i Mekhanika
2023Issue 855 - 21 pp.
The paper introduces a class C(, a,) of functions f (z), analytic in the unit disk E = z : z 1, having a power series expansion of the form f (z) = z + a2z2 + a3z3 + ..., z E, and satisfying the condition (1 − z2) f (z) 1 − a a, where 0 1, 0 1, and a 1 . It is proved that all functions of class C(, a,) are close-to-2 convex of the order of γ. Class C(, a,) generalizes classes of functions with bounded turning Re f (z) 0 (as a → +, = 0) and functions f (z) convex in the direction of the imaginary axis Re[(1 − z2 ) f (z)] 0 (a → +, = 1) and creates a simple parametric passage from one class to another. Based on the subordination method in class C(, a,) and its subclasses, exact estimates are obtained for f (z), f (z), zf (z) / f (z) and the exact radii of the convexity. In particular cases, they yield previously known results for functions with bounded turning and functions convex in the direction of the imaginary axis. Using the relationship f (z) C(, a,) F(z) = zf (z) T(, a,), the article introduces the class Т(, a,) = F(z): (1 − z2 )F(z) / z 1 − a a functions generalizing the class of typically real functions and the class of functions satisfying the condition Re[F(z) / z] 0. In the class Т(, a,) and its subclasses, exact estimates of F(z), zF(z) / F(z) are found and the exact radii of starlikeness are determined, which generalizes the classical results for typically real functions.
estimates of analytic functions , geometric theory of functions , radii of convexity , single-leaf functions , typically real functions
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Kostanay Regional University named after A. Baitursynov, Kostanay, Kazakhstan
Kostanay Regional University named after A. Baitursynov
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