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

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 $\mathrm{\mid }z\mathrm{\mid }<1$ with $G(0)=H(0)=0$ and $H^{\mathrm{\prime }}(0)=1$. 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.