Bi-Lipschitz characteristic of quasiconformal self-mappings of the unit disk satisfying bi-harmonic equation

Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies following: $(1)$ biharmonic equation $\Delta(\Delta f)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) boundary condition $\Delta f=\varphi$ ($\varphi\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes circle), $(3)$ $f(0)=0$. The purpose this paper to prove Lipschitz continuos, and, further, it bi-Lipschitz continuous when $\|g\|_{\infty}$ $\|\varphi\|_{\infty}$ are small enough. Moreover, estimates asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ $\|\varphi\|_{\infty}\to 0$, thus, such mapping behaves almost like rotation for sufficiently $K$, $\|\varphi\|_{\infty}$.