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

Suppose that $f$ satisfies the following: $(1)$ the polyharmonic equation $Δ^{m}f=Δ(Δ^{m-1} f)$$=φ_{m}$ $(φ_{m}\in \mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n}))$, (2) the boundary conditions $Δ^{0}f=φ_{0},Δ^{1}f=φ_{1},~\ldots,~Δ^{m-1}f=φ_{m-1}$ on $\mathbb{S}^{n-1}$ ($φ_{j}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ 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 $\|φ_{j}\|_{\infty}\to 0^{+}$ for $j\in\{1,\ldots,m\}$.

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