Landau-type theorems for certain bounded bi-analytic functions and biharmonic mappings

TL;DR

This paper develops new Landau-type theorems for bounded bi-analytic functions and biharmonic mappings, providing sharp bounds and generalizations that advance understanding of these complex functions.

Contribution

It introduces three new Landau-type theorems for bi-analytic functions, two of which are sharp, and extends results to subclasses of biharmonic mappings.

Findings

Two theorems are sharp bounds for bi-analytic functions.

One theorem generalizes or improves previous results.

New sharp Landau-type theorems for biharmonic mappings.

Abstract

In this paper, we establish three new versions of Landau-type theorems for bounded bi-analytic functions of the form $F(z)=\bar{z}G(z)+H(z)$, where $G$ and $H$ are analytic in the unit disk $|z|<1$ with $G(0)=H(0)=0$ and $H'(0)=1$. In particular, two of them are sharp while the other one either generalizes or improves the corresponding result of Abdulhadi and Hajj. As consequences, several new sharp versions of Landau-type theorems for certain subclasses of bounded biharmonic mappings are proved.