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

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: $\mathrm{\Delta }(\mathrm{\Delta }f)=g$, where $g:$ $\overline{\text{ID}}\to \mathbb{C}$ is a continuous function and $\overline{\text{ID}}$ denotes the closure of the unit disk $\text{ID}$ 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 $$\frac{1-\mathrm{\mid }z{\mathrm{\mid }}^2}{1-\mathrm{\mid }f(z){\mathrm{\mid }}^2}\le C,$$ 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.