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

Abstract

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.