On the univalence of polyharmonic mappings

A 2p-times continuously differentiable complex valued function f=u+iv in a simply connected domain Ω is polyharmonic (or p-harmonic) if it satisfies the polyharmonic equation △pF=0. Every polyharmonic mapping f can be written as f(z)=∑k=1p|z|2(p−1)Gp−k+1(z), where each Gp−k+1 is harmonic. In this paper we investigate the univalence of polyharmonic mappings on linearly connected domains and the relation between univalence of f(z) and that of Gp(z). The notion of stable univalence and logpolyharminc mappings are also considered.