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

Peijin Li, Yaxiang Li, Saminathan Ponnusamy
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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.

关于单位盘中加权双谐方程的解
本文研究了加权双谐微分方程 \(\Delta \big ((1-|z|^2)^{-1}\Delta \big ) 的解。\Phi =0\) in the unit disk \(|z|<1\),其中 \(\Delta =4\frac{partial ^2}\{partial z\partial \overline{z}}\) 表示拉普拉斯函数。本文的主要目的是为这一类映射建立经典几何函数理论中几个重要结果的对应关系。主要结果包括 Schwarz 型 Lemma 和 Landau 型定理。连续增函数 \(\omega :\(\omega(0)=0\)并且\(\omega(t)/t\)对于\(t>;如果对于某个 \(\delta _0>0\) 和 \(0<\delta <\delta _0\),不等式 $$\begin{aligned} 被称为快速大数。}\int ^{delta }_{0}frac{omega (t)}{t}dt\le C\omega (\delta ), \end{aligned}$$对于某个正常数 C 成立。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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