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The purpose of this paper is to study the Schwarz-Pick type inequality andthe Lipschitz continuity for the solutions to the nonhomogeneous biharmonicequation: $\Delta(\Delta f)=g$, where $g:$$\overline{\ID}\rightarrow\mathbb{C}$ is a continuous function and$\overline{\ID}$ denotes the closure of the unit disk $\ID$ in the complexplane $\mathbb{C}$. In fact, we establish the following properties for thesesolutions: Firstly, we show that the solutions $f$ do not always satisfy theSchwarz-Pick type inequality $$\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 thedistance ratio metric and the Lipschitz continuity with respect to thehyperbolic metric.

On Schwarz-Pick type inequality and Lipschitz continuity for solutions to nonhomogeneous biharmonic equations