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 $$\frac{1-|z{|}^2}{1-|f(z){|}^2}\le C,$$ 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.

12 pages, To appear in Mediterranean Journal of Mathematics