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

TL;DR

This paper investigates Schwarz-Pick type inequalities and Lipschitz continuity for solutions to nonhomogeneous biharmonic equations, establishing conditions under which these properties hold and exploring their behavior with respect to various metrics.

Contribution

It provides new insights into the Schwarz-Pick type inequalities and Lipschitz continuity for biharmonic solutions, including conditions for their validity and applications to different metrics.

Findings

Solutions do not always satisfy the classical Schwarz-Pick inequality.

A generalized Schwarz-Pick inequality is established under certain conditions.

Lipschitz continuity is proved with respect to the distance ratio and hyperbolic metrics.

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.