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Geometric properties of certain integral operators involving Hornich operations (2106.06495v2)
Published 11 Jun 2021 in math.CV
Abstract: In this article, we investigate some standard geometric properties of the integral operators $$ J_\alpha f= \int_{0}{z}\bigg(\frac{f(w)}{w}\bigg)\alpha dw, \,\,\, \alpha \in \mathbb{C} \text{ and } |z|<1, $$ and $$ I_\beta g= \int_{0}{z}\big(g'(w)\big)\beta dw, \,\,\, \beta \in \mathbb{C} \text{ and } |z|<1, $$ 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.