The Landau-type theorems for functions with logharmonic Laplacian and bounded length distortions

TL;DR

This work extends Landau-type univalence results to functions in the logharmonic-Laplacian class with bounded length distortions by studying F(z)=|z|^2 L(z)+K(z) where L is logharmonic and K is harmonic in the unit disk. It develops coefficient estimates for logharmonic mappings under length-distortion bounds and leverages a differential operator D to derive explicit univalence radii and schlicht disks, presenting four sharp Landau-type theorems with clear extremal examples. The results cover both Λ1>1 and Λ1=1 regimes and provide precise radii r1, r1', r2, r2' and disk radii ρ1, ρ1', ρ2, ρ2', including corresponding D(F) counterparts. Overall, the paper generalizes classical Landau/Bloch phenomena to the L_{Lh}() setting under bounded length distortion, with sharp constants and explicit extremals suitable for analytical and computational applications.

Abstract

In this study, we establish certain Landau-type theorems for functions with logharmonic Laplacian of the form $F(z)=|z|^2L(z)+K(z)$, $|z|<1$, where $L$ is logharmonic and $K$ is harmonic, with $L$ and $K$ having bounded length distortion in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. Furthermore, we examine the univalence of the mappings $D(F)$, where $D$ is a differential operator.

Theorems & Definitions (15)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 3.1
• proof
• Theorem 3.2