Estimates and Radii of Convexity in Some Classes of Regular Functions
Maiyer F.F. Tastanov M.G. Utemissova A.A. Temirbekov N.M. Kenzhebekova D.S.
2024World Scientific and Engineering Academy and Society
WSEAS Transactions on Mathematics
2024#23446 - 457 pp.
A class Cn (λ, δ, a, γ ) is being introduced regular in the circle E = {z : ∣z∣ < 1} functions f (z), satisfying the condition | ((1- λ zn )(1- δ zn ) f (z))1/γ - a |≤ a, z ∊ E, where λ, δ ≥ 0, 0 < γ ≤ 1, a > 1/2, n ≥ 1. Class Cn (λ, δ, a, γ ) generalizes various subclasses of close-to-convex functions, including functions which are convex in a certain direction and functions with limited rotation. Estimates of the derivative and logarithmic derivative of the function f (z)∊ Cn (λ,δ, a, γ ) are found, and also the radii of the convexity of the class Cn (λ, δ, a, γ ). The case is also considered when the function f (z) has gaps in the expansion in a row. Similar results are formulated for the class Tn (λ,δ, a,γ ) of functions F(z), satisfying the condition | ((1 - λ zn)(1 - δ zn)F(z)/z)1/γ - a |≤ a, z∊ E, which generalizes classes of typically real and close-to-starlike functions. All results are accurate. With the appropriate selection of parameter values of λ,δ, a,γ, n both new and previously published results are obtained.
close-to-convex functions , close-to-starlike functions , estimates of regular functions , geometric theory of functions , radii of convexity , typically real functions
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Department of Mathematics and Physics, Kostanay Regional University named after. A. Baitursynuly, Kostanay, Kazakhstan
Department of Mathematics and Physics
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