On Asymptotically Sharp Bi-Lipschitz Inequalities of Quasiconformal Mappings Satisfying Inhomogeneous Polyharmonic Equations

Abstract

Suppose that $f$ is a $K$-quasiconformal ($(K,K')$-quasiconformal resp.) self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the inhomogeneous polyharmonic equation $\Delta^{n}f=\Delta(\Delta^{n-1} f)=\varphi_{n}$ $(\varphi_{n}\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary conditions $\Delta^{n-1}f|_{\mathbb{T}}=\varphi_{n-1},~\ldots,~\Delta^{1}f|_{\mathbb{T}}=\varphi_{1}$ ($\varphi_{j}\in\mathcal{C}(\mathbb{T})$ for $j\in\{1,\ldots,n-1\}$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$, where $n\geq2$ is an integer and $K\geq1$ ($K'\geq0$ resp.). The main aim of this paper is to prove that $f$ is Lipschitz continuous, and,further, it is bi-Lipschitz continuous when $\|\varphi_{j}\|_{\infty}$ are small enough for $j\in\{1,\ldots,n\}$. Moreover, the estimates are asymptotically sharp as $K\to 1$ ($K'\to0$ resp.) and $\|\varphi_{j}\|_{\infty}\to 0$ for $j\in\{1,\ldots,n\}$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$ ($K'$ resp.) and $\|\varphi_{j}\|_{\infty}$ for $j\in\{1,\ldots,n\}$.