Volterra type integral operator and analytic function spaces

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.