Radii of convexity of integral operators

TL;DR

This paper investigates the radius of convexity for two classes of integral operators involving products of derivatives of analytic functions, providing new bounds and insights into their geometric properties.

Contribution

It introduces new results on the radius of convexity for specific integral operators with complex parameters, extending existing theories in geometric function theory.

Findings

Derived bounds for the radius of convexity of the operators

Extended known results to more general classes of functions

Provided conditions under which the operators are convex within certain radii

Abstract

The object of the present paper is to study of radius of convexity two certain integral operators as follows \begin{equation*} F(z):=\int_{0}^{z}\prod_{i=1}^{n}\left(f'_i(t)\right)^{\gamma_i}{\rm d}t \end{equation*} and \begin{equation*} J(z):=\int_{0}^{z}\prod_{i=1}^{n}\left(f'_i(t)\right)^{\gamma_i}\prod_{j=1}^{m} \left(\frac{g_j(z)}{z}\right)^{\lambda_j}{\rm d}t, \end{equation*} where $\gamma_i, \lambda_i\in\mathbb{C}$, $f_i$ $(1\leq i\leq n)$ and $g_j$ $(1\leq j\leq m)$ belong to the certain subclass of analytic functions.