On the univalence of polyharmonic mappings

A 2p-times continuously differentiable complex valued function $f = u + iv$ in a simply connected domain is polyharmonic (or p-harmonic) if it satisfies the polyharmonic equation $\Delta^pF = 0$ . Every polyharmonic mapping f can be written as $f(z) =\sum_{k}^{p} |z|^{2(p-1)}G_{p-k+1}(z)$ where each $G_{p-k+1}$ 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 $G_p(z)$. The notions of stable univalence and logpolyharminc mappings are also considered.