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

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.