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On asymptotically sharp bi-Lipschitz inequalities of quasiconformal mappings satisfying inhomogeneous polyharmonic equations

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 $Δ^{n}f=Δ(Δ^{n-1} f)=φ_{n}$ $(φ_{n}\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary conditions $Δ^{n-1}f|_{\mathbb{T}}=φ_{n-1},~\ldots,~Δ^{1}f|_{\mathbb{T}}=φ_{1}$ ($φ_{j}\in\mathcal{C}(\mathbb{T})$ 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 $\|φ_{j}\|_{\infty}$ are small enough for $j\in\{1,\ldots,n\}$. Moreover, the estimates are asymptotically sharp as $K\to 1$ ($K'\to0$ resp.) and $\|φ_{j}\|_{\infty}\to 0$ for $j\in\{1,\ldots,n\}$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$ ($K'$ resp.) and $\|φ_{j}\|_{\infty}$ for $j\in\{1,\ldots,n\}$.