Landau-Type Theorems for Certain Biharmonic Mappings

Abstract

Let be a biharmonic mapping of the unit disk $𝔻$, where $g$ and $h$ are harmonic in $𝔻$. In this paper, the Landau-type theorems for biharmonic mappings of the form $L(F)$ are provided. Here $L$ represents the linear complex operator $L=\mathrm{(z}\partial /\partial \mathrm{z)}-\overline{\mathrm{(z}}\partial /\partial \overline{\mathrm{z)}}$ defined on the class of complex-valued $C^1$ functions in the plane. The results, presented in this paper, improve the related results of earlier authors.