  In this paper, we establish some Schwarz type lemmas for mappings $\Phi$
satisfying the inhomogeneous biharmonic Dirichlet problem $ \Delta
(\Delta(\Phi)) = g$ in $\mathbb{D}$, $\Phi=f$ on $\mathbb{T}$ and $\partial_n
\Phi=h$ on $\mathbb{T}$, where $g$ is a continuous function on
$\overline{\mathbb{D}}$, $f,h$ are continuous functions on $\mathbb{T}$, where
$\mathbb{D}$ is the unit disc of the complex plane $\mathbb{C}$ and
$\mathbb{T}=\partial \mathbb{D}$ is the unit circle. To reach our aim, we start
by investigating some properties of $T_2$-harmonic functions. Finally, we prove
a Landau-type theorem.
