On the solutions to the weighted biharmonic equation in the unit disk

Abstract

In this paper, we investigate the solutions of the weighted biharmonic differential equation \(\Delta \big ((1-|z|^2)^{-1}\Delta \big ) \Phi =0\) in the unit disk \(|z|<1\), where \(\Delta =4\frac{\partial ^2}{\partial z\partial \overline{z}}\) denotes the Laplacian. The primary aim of the paper is to establish counterparts of several important results in the classical geometric function theory for this class of mappings. The main results include Schwarz type lemma and Landau type theorem. A continuous increasing function \(\omega :\, [0, \infty )\rightarrow [0, \infty )\) with \(\omega (0)=0\) and \(\omega (t)/t\) is non-increasing for \(t>0\) is called a fast majorant if for some \(\delta _0>0\) and \(0<\delta <\delta _0\), the inequality

$$\begin{aligned} \int ^{\delta }_{0}\frac{\omega (t)}{t}dt\le C\omega (\delta ), \end{aligned}$$

holds for some positive constant C. Then we obtain \(\omega \)-Lipschitz continuity for the solutions to the weighted biharmonic equation, when \(\omega \) is a fast majorant.