On Heinz type inequality and Lipschitz characteristic for mappings satisfying polyharmonic equations

For $K\geq1$, suppose that $f$ is a $K$-quasiconformal self-mapping of the unit ball $\mathbb{B}^{n}$, which satisfies the following: $(1)$ the polyharmonic equation $\Delta^{m}f=\Delta(\Delta^{m-1} f)$$=\varphi_{m}$ $(\varphi_{m}\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n}))$, (2) the boundary conditions $\Delta^{m-1}f|_{\mathbb{S}^{n-1}}=\varphi_{m-1},~\ldots,~\Delta^{1}f|_{\mathbb{S}^{n-1}}=\varphi_{1}$ ($\varphi_{k}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ for $j\in\{1,\ldots,m-1\}$ and $\mathbb{S}^{n-1}$ denotes the unit sphere in $\mathbb{R}^{n}$), and $(3)$ $f(0)=0$, where $n\geq3$ and $m\geq2$ are integers. We first establish a Heinz type inequality on mappings satisfying the polyharmonic equation. Then we use the obtained results to show that $f$ is Lipschitz continuous, and the estimate is asymptotically sharp as $K\to 1$ and $\|\varphi_{j}\|_{\infty}\to 0$ for $j\in\{1,\ldots,m\}$.Comment: 28 page

Schwarz-Pick type estimates of pluriharmonic mappings in the unit polydisk

In this paper, we will give Schwarz-Pick type estimates of arbitrary order partial derivatives for bounded pluriharmonic mappings defined in the unit polydisk. Our main results are generalizations of results of Colonna for planar harmonic mappings in [Indiana Univ. Math. J. 38: 829--840, 1989].Comment: 9 page