On the Pre-Schwarzian Norm and Starlikeness of Certain Logharmonic Mappings

TL;DR

This paper investigates the properties of certain logharmonic mappings in the unit disk, establishing sharp bounds on their pre-Schwarzian norm, growth behavior, and conditions for starlikeness, including examples of hereditarily starlike functions.

Contribution

It provides new sharp bounds for the pre-Schwarzian norm and introduces coefficient conditions for hereditarily starlike logharmonic mappings.

Findings

Sharp bound of pre-Schwarzian norm for logharmonic mappings

Growth theorem for these mappings

Coefficient conditions for hereditarily starlikeness

Abstract

In this note, we consider certain logharmonic mappings in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}.$ Next, we obtain sharp bound of pre-Schwarzian norm of such logharmonic mappings in the unit disk. Then we discuss growth theorem for the mappings. Moreover, we discuss starlikeness of logharmonic mappings and compute sufficient coefficient condition of hereditarily starlikeness. At the end, we present some example of logharmonic hereditarily starlike function.