Landau-Bloch type theorems of certain subclasses of biharmonic mappings

Abstract

In this paper, we first establish a Landau-Bloch type theorem for certain bounded and normalized biharmonic mappings $F(z)=|z|^2g(z)+h(z)$, where $g(z)$ and $h(z)$ are harmonic in the unit disk with $|g(z)|\leq M_1, |h(z)|\leq M_2$. In particular, our result is sharp when $M_1=M_2=1$. Then, we establish several new versions of Landau-Bloch type theorems for certain normalized biharmonic mappings with the coefficients condition in place of $|h(z)|\leq M_2$ or $|g(z)|\leq M_1$, and obtain several sharp results.