Geometric properties of certain integral operators involving Hornich operations

TL;DR

This paper studies the geometric properties of specific integral operators involving Hornich operations, focusing on their preservation of properties, and provides sharp estimates of their pre-Schwarzian norms.

Contribution

It introduces new analyses of integral operators with Hornich operations, including preservation properties and sharp norm estimates, expanding understanding of their geometric behavior.

Findings

Operators preserve certain geometric properties.

Derived sharp pre-Schwarzian norm bounds.

Analyzed the Hornich sum of the operators.

Abstract

In this article, we investigate some standard geometric properties of the integral operators $$ J_\alpha [f](z)= \int_{0}^{z}\bigg(\frac{f(w)}{w}\bigg)^\alpha dw, \,\,\, \alpha \in \mathbb{C} \text{ and } |z|<1, $$ and $$ I_\beta [g](z)= \int_{0}^{z}\big(g'(w)\big)^\beta dw, \,\,\, \beta \in \mathbb{C} \text{ and } |z|<1, $$ where $f$ and $g$ are elements of certain classical families of normalized analytic functions defined on the unit disk. In particular, preserving properties of the Hornich sum of the operators $J_\alpha$ and $I_\beta$ will be studied. Moreover, we also present sharp pre-Schwarzian norm estimate of such integrals.