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 Lipschitz continuity for solutions nonhomogeneous biharmonic equation: $$\Delta (\Delta f)=g$$ , where g : $$\overline{{\mathbb D}}\rightarrow {\mathbb {C}}$$ a continuous function D}}$$ denotes closure unit disk $${\mathbb D}$$ in complex plane . In fact, we establish following properties these solutions: First, show that f do not always satisfy Schwarz–Pick-type $$\begin{aligned} \frac{1-|z|^2}{1-|f(z)|^2} \le C, \end{aligned}$$ C constant. Second, general under certain conditions. Third, discuss f, as applications, get with respect distance ratio metric hyperbolic metric.