Volterra type integral operator and analytic function spaces

TL;DR

This paper explores the geometric properties of Volterra-type integral operators on analytic function spaces, providing sharp estimates for convexity and univalence radii, and introduces higher-order operators with open questions on their scaling behavior.

Contribution

It offers new sharp estimates for convexity and univalence radii of Volterra operators and introduces higher-order variants with open questions on their properties.

Findings

Sharp convexity radius estimates for Volterra operators

Determination of univalence radius across classical function families

Introduction of higher-order Volterra-type operators and open questions

Abstract

We investigate the geometric properties of the Volterra-type integral operator \begin{equation*} T_g[f](z) = \int_{0}^{z} f(s)\, g'(s)\, ds, \quad |z|<1, \end{equation*} acting on various subclasses of analytic functions in the unit disk. Sharp estimates are obtained for the convexity radius of $T_g$, which simultaneously determine its univalence radius, across several classical function families. In addition, we introduce and study higher-order Volterra-type operators, establish their normalized forms, and propose an open question on the scaling behavior of their convexity radii.