The Heinz type inequality, Bloch type theorem and Lipschitz characteristic of polyharmonic mappings

Abstract

Suppose that $f$ 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 $\Delta^{0}f=\varphi_{0},\Delta^{1}f=\varphi_{1},~\ldots,~\Delta^{m-1}f=\varphi_{m-1}$ on $\mathbb{S}^{n-1}$ ($\varphi_{j}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ for $j\in\{0,1,\ldots,m-1\}$ and $\mathbb{S}^{n-1}$ denotes the boundary of the unit ball $\mathbb{B}^{n}$), and $(3)$ $f(0)=0$, where $n\geq3$ and $m\geq1$ are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in \cite{CP-Hi}. Additionally, we show that if $f$ is a $K$-quasiconformal self-mapping of $\mathbb{B}^{n}$ satisfying the above polyharmonic equation, then $f$ is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as $K\to 1^{+}$ and $\|\varphi_{j}\|_{\infty}\to 0^{+}$ for $j\in\{1,\ldots,m\}$.