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 \(j\in \{1,\ldots ,n\}\).

对于两个常数\（K \ GE 1 \）和\（K '\ GE 0 \） ，假设˚F是\（（K，K'）\）单位圆盘的-quasiconformal自映射\（{\ mathbb {D}} \），它满足以下条件：（1）不均匀多谐方程\（\ Delta ^ {n} f = \ Delta（\ Delta ^ {n-1} f）= \ varphi _ {n} \）（\（{\ mathbb {D}} \）\）（（\ varphi _ {n} \ in {\ mathcal {C}}（\ overline {{\ mathbb {D}}}））））（） 2）边界条件\（\德尔塔^ {N-1} F = \ varphi _ {N-1}，〜\ ldots，〜\德尔塔^ {1} F = \ varphi _ {1} \）上\（ {\ mathbb {T}} \）（\（\ varphi _ {j} \ in {\ mathcal {C}}（{\ mathbb {T}}）\\） for \（j \ in \ {1，\ ldots ，n-1 \} \）和\（{\ mathbb {T}} \）表示单位圆），以及（3）\（f（0）= 0 \），其中\（n \ ge 2 \）是整数。本文的主要目的是证明f是Lipschitz连续的，而且，当\（\ Vert \ varphi _ {j} \ Vert _ {\ infty} \足够小时，它是bi-Lipschitz连续的。 （j \ in \ {1，\ ldots，n \} \）。此外，估计值渐近为\（K \ rightarrow 1 ^ {+} \），\（K'\ rightarrow 0 ^ {+} \）和\（\ Vert \ varphi _ {j} \ Vert _ {\ infty} \ RIGHTARROW 0 ^ {+} \）为\（j \在\ {1，\ ldots，正\} \）。