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

Abstract

For two constants $K\ge 1$ and $K'\ge 0$, suppose that f is a $(K,K')$-quasiconformal self-mapping of the unit disk ${\mathbb {D}}$, which satisfies the following: (1) the inhomogeneous polyharmonic equation $\Delta ^{n}f=\Delta (\Delta ^{n-1} f)=\varphi _{n}$ in ${\mathbb {D}}$ $(\varphi _{n}\in {\mathcal {C}}(\overline{{\mathbb {D}}}))$, (2) the boundary conditions $\Delta ^{n-1}f=\varphi _{n-1},~\ldots ,~\Delta ^{1}f=\varphi _{1}$ on ${\mathbb {T}}$ ($\varphi _{j}\in {\mathcal {C}}({\mathbb {T}})$ for $j\in \{1,\ldots ,n-1\}$ and ${\mathbb {T}}$ denotes the unit circle), and (3) $f(0)=0$, where $n\ge 2$ is an integer. The main aim of this paper is to prove that f is Lipschitz continuous, and, further, it is bi-Lipschitz continuous when $\Vert \varphi _{j}\Vert _{\infty }$ are small enough for $j\in \{1,\ldots ,n\}$. Moreover, the estimates are asymptotically sharp as $K\rightarrow 1^{+}$, $K'\rightarrow 0^{+}$ and $\Vert \varphi _{j}\Vert _{\infty }\rightarrow 0^{+}$ for $j\in \{1,\ldots ,n\}$.