Geometric Properties of log-Polyharmonic Mappings

In this paper, a class of $$\log $$log-polyharmonic mappings $$\mathcal {L}_p\mathcal {H}$$LpH together with its subclass $$\mathcal {L}_p\mathcal {H}(G)$$LpH(G) in the unit disk $$\mathbb {D}=\{z: |z|<1\}$$D={z:|z|<1} is introduced, and several geometrical properties, such as the starlikeness, convexity and univalence are investigated. In particular, we consider the Goodman–Saff conjecture and prove that the conjecture is true in $$\mathcal {L}_p\mathcal {H}(G)$$LpH(G).