On the generalization of some classes of close-to-convex

F. F. Maiyer ; M. G. Tastanov ; A. A. Utemissova ; A. T. Baimankulov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 85 (2023), pp. 5-21

Résumé

he paper introduces the class $C(\lambda,\alpha,\gamma)=\left\{f(z) :\left| (1-\lambda z^2)f'(z)^{1/\gamma}-a\right|\leqslant a\right\}$, $0\leqslant\lambda\leqslant 1$, $0\leqslant\gamma\leqslant 1$, $a>1/2$, almost convex order for functions, generalizing classes of functions with limited rotation $(a\to+\infty, \lambda=0)$ and functions convex of order $\gamma$ in the direction of the imaginary axis $(a\to+\infty, \lambda=1)$. For the class $C(\lambda, a, \gamma)$ and its subclasses, unimprovable distortion theorems and exact convexity radii are found, and similar results are obtained in a class generalizing the class of typically real functions.

Détail

Keywords: geometric theory of functions, single-leaf functions, estimates of analytic functions, typically real functions, radii of convexity.