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

Abstract

Suppose that f satisfies the following: (1) the polyharmonic equation Δmf=Δ(Δm−1f)=φm(φm∈C(Bn¯,Rn)), (2) the boundary conditions Δ0f=φ0,Δ1f=φ1,…,Δm−1f=φm−1 on Sn−1 (φj∈C(Sn−1,Rn) for j∈{0,1,…,m−1} and Sn−1 denotes the boundary of the unit ball Bn), and (3)f(0)=0, where n≥3 and m≥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 Bn satisfying the above polyharmonic equation, then f is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as K→1+ and ‖φj‖∞→0+ for j∈{1,…,m}.