Volterra type integral operator and analytic function spaces

TL;DR

The paper analyzes the geometric properties of Volterra-type integral operators on spaces of analytic functions, focusing on sharp convexity (and thus univalence) radii when the input functions belong to Carathéodory and Janowski-type classes. It develops a systematic framework to compute radii via sharp Janowski bounds and Robertson estimates, and extends the theory to higher-order Volterra-type operators with a normalized form that facilitates convexity analysis. Convolution representations and extremal constructions are used to prove sharpness of the radii, and the work is extended to study the asymptotics and scaling behavior of convexity radii for higher-order operators, culminating in open questions about universal $1/n$ scaling. The results provide concrete radius criteria across a spectrum of function classes and highlight nontrivial scaling phenomena when moving to higher-order operators.

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.

Figures (1)

• Figure 1: Plot of $F(r)$. The curve remains positive for $0<r<1$, confirms no root exists in the interval $(0,1)$.

Theorems & Definitions (36)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Corollary 2.1
• Theorem 3.1
• proof
• Remark 3.1
• Theorem 3.2