  The purpose of this paper is to study the Schwarz-Pick type inequality and
the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic
equation: $\Delta(\Delta f)=g$, where $g:$
$\overline{\ID}\rightarrow\mathbb{C}$ is a continuous function and
$\overline{\ID}$ denotes the closure of the unit disk $\ID$ in the complex
plane $\mathbb{C}$. In fact, we establish the following properties for these
solutions: Firstly, we show that the solutions $f$ do not always satisfy the
Schwarz-Pick type inequality $$\frac{1-|z|^2}{1-|f(z)|^2}\leq C, $$ where $C$
is a constant. Secondly, we establish a general Schwarz-Pick type inequality of
$f$ under certain conditions. Thirdly, we discuss the Lipschitz continuity of
$f$, and as applications, we get the Lipschitz continuity with respect to the
distance ratio metric and the Lipschitz continuity with respect to the
hyperbolic metric.
