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

Suppose that $f$ satisfies the following: $\left(1\right)$ the polyharmonic equation ${\Delta }^{m}f=\Delta \left({\Delta }^{m-1}f\right)$ $={\phi }_m$ $({\phi }_m\in C(\overline{B^n},R^n))$, (2) the boundary conditions ${\Delta }^{0}f={\phi }_0,{\Delta }^{1}f={\phi }_1,\dots ,{\Delta }^{m-1}f={\phi }_{m-1}$ on $S^{n-1}$ (${\phi }_j\in C(S^{n-1},R^n)$ for $j\in \{0,1,\dots ,m-1\}$ and $S^{n-1}$ denotes the boundary of the unit ball $B^n$), and $\left(3\right)$ $f\left(0\right)=0$, where $n\ge 3$ and $m\ge 1$ 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 Chen and Ponnusamy, (2019). Additionally, we show that if $f$ is a $K$-quasiconformal self-mapping of $B^n$ satisfying the above polyharmonic equation, then $f$ is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as $K\to 1^{+}$ and ${\Vert {\phi }_j\Vert }_{\infty }\to 0^{+}$ for $j\in \{1,\dots ,m\}$.