Geometric properties of certain integral operators involving Hornich operations

Abstract

In this article, we investigate some standard geometric properties of the integral operators $$J_{\alpha }[f](z)={\int }_0^z(\frac{f(w)}{w}{)}^{\alpha }dw,\alpha \in \mathbb{C}\text{ and }|z|<1,$$ and $$I_{\beta }[g](z)={\int }_0^z(g^{\prime }(w){)}^{\beta }dw,\beta \in \mathbb{C}\text{ and }|z|<1,$$ 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.