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

TL;DR

The paper addresses the pre-Schwarzian norm and starlikeness of logharmonic mappings in the unit disk, focusing on a structured class $L_{\mathcal{R}}$ built from $f=H\overline{G}$ with $H=e^h$, $h\in\mathcal{R}$ and dilatation $\omega=G'/(Gh')$. It proves a sharp bound $\|P_f\|\le 11$ for $f\in L_{\mathcal{R}}$, shows finiteness via Bloch-type estimates, and derives sharp growth bounds for $|f(z)|$; it also links $\log f$ to a uniformly locally univalent harmonic mapping. The work introduces the centered class $L^0_{\mathcal{R}}$ and a coefficient condition $|1-b_1|+\sum_{n=2}^\infty n|a_n-b_n|\le 1$ that suffices for hereditary starlikeness, illustrated by the explicit family $f_\alpha$ with $\omega(z)=z$ and $h(z)=z+\alpha z^2/2$ for $\alpha\in[0,1]$. These results extend pre-Schwarzian and starlikeness theory to logharmonic mappings, offering concrete tools for univalence and geometric properties in the unit disk.

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.

Figures (1)

• Figure 1: $f_\alpha(\mathbb{D})$ for certain values of $\alpha.$

Theorems & Definitions (12)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Remark 2.1
• Lemma 2.1
• Theorem 2.3
• proof
• Definition 2.1
• Example 2.1