Geometric properties of the Bernatsky integral operator

F. F. Mayer ; M. G. Tastanov ; A. A. Utemisova

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 14 (2022) no. 4, pp. 12-19

Résumé

In the geometric theory of complex variable functions, the study of mapping of classes of regular functions using various operators has now become an independent trend. The connection $f(z)\in S^{o}\Leftrightarrow g(z) = zf'(z) \in S^*$ of the classes $S^{o}$ and $S^*$ of convex and star-shaped functions can be considered as mapping using the differential operator $G[f](x) = zf'(z)$ of class $S^{o}$ to class $S^*$, that is, $G: S^{o} \to S^*$ or $G(S^{o}) = S^*$. The impetus for studying this range of issues was M. Bernatsky's assumption that the inverse operator $G^{-1}[f](x)$, which translates $S^* \to S^{o}$ and thereby “improves” the properties of functions, maps the entire class $S$ of single-leaf functions into itself. At present, a number of articles have been published which study the various integral operators. In particular, they establish sets of values of indicators included in these operators where operators map class $S$ or its subclasses to themselves or to other subclasses. This paper determines the values of the Bernatsky parameter included in the generalized integral operator, at which this operator transforms a subclass of star-shaped functions allocated by the condition $a \mathrm{Re}\, zf'(z)/f(z) b$ ($0 a 1 b$), in the class $K(\gamma)$ of functions, almost convex in order $\gamma$. The results of the article summarize or reinforce previously known effects.

Détail

Keywords: geometric theory of functions of a complex variable, single-leaf functions, Bernatsky integral operator, star-shaped and almost convex functions.
Mots-clés : convex