On convex to pseudoconvex mappings

In the works of Darboux and Walsh, it was remarked that a one-to-one self-mapping of $\mathbb{R}^{3}$ which sends convex sets to convex ones is affine. It can be remarked also that a $\mathcal{C}^{2}$-diffeomorphism $F:U\to U^{'}$ between two domains in $\mathbb{C}^{n}$, $n\ge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic.

In this paper we are interested in the self-mappings of $\mathbb{C}^{n}$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: A $\mathcal{C}^{2}$- diffeomorphism $F:U'\to U$ (where $U', U\subset \mathbb{C}^{n}$ are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map $\Phi := F^{-1}$ is weakly pluriharmonic, i.e., if it satisfies some nice second order PDE very close to $\partial \bar{\partial}\Phi = 0$. In fact all pluriharmonic $\Phi$'s do satisfy this equation, but there are also other solutions.