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

TL;DR

This paper proves Landau-type theorems for a class of functions combining logharmonic and harmonic parts with bounded length distortions, and investigates their univalence properties within the unit disk.

Contribution

It introduces new Landau-type theorems for functions with logharmonic Laplacian and bounded length distortions, extending classical results to this broader class.

Findings

Established Landau-type theorems for these functions.

Analyzed univalence of the differential operator applied to these functions.

Extended classical distortion theorems to logharmonic and harmonic function classes.

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.