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

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.