On the pre-Schwarzian and Schwarzian derivatives of log-harmonic mappings

TL;DR

This work extends the theory of pre-Schwarzian and Schwarzian derivatives to locally univalent log-harmonic mappings in the unit disk, providing a unified treatment for both vanishing and non-vanishing cases. It defines $P_f$ and $S_f$ for $f=h\overline{g}$, derives explicit formulas in terms of $h,g$ and the second complex dilatation $\omega$, and establishes fundamental structural results: $P_f$ and $S_f$ are analytic (or harmonic) precisely when $\omega$ is constant; finite pre-Schwarzian norms are tightly connected to the analytic counterpart $hg$, with sharp bounds and Bloch-function conditions. The paper also analyzes how $P_f$ and $S_f$ behave under deformations $f_\varepsilon=hg^{\varepsilon}$, providing Becker-type univalence criteria and sharp constants, thereby enriching geometric function theory for log-harmonic mappings. Together, these results offer new tools for studying global univalence and geometric properties of log-harmonic mappings in the disk with direct links to analytic and harmonic theory.

Abstract

In this paper, we introduce definitions of the pre-Schwarzian and the Schwarzian derivatives for any locally univalent log-harmonic mappings defined in the unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$. We explore the properties and applications of these concepts in the context of geometric function theory, and we also establish a necessary and sufficient condition for a non-vanishing log-harmonic mapping having a finite pre-Schwarzian norm. Additionally, we establish a relationship between the pre-Schwarzian norm of a non-vanishing log-harmonic mapping and that of a certain analytic function in $\mathbb{D}$.

Figures (1)

• Figure 1: The image of $\mathbb{D}$ under the mapping $\omega(z)=(2-3z)/(3-2z)$

Theorems & Definitions (23)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.1
• proof