On convex to pseudoconvex mappings

In the works of Darboux and Walsh it was remarked that a one to one self mapping of ${\text{rr}}^3$ which sends convex sets to convex ones is affine. It can be remarked also that a ${\text{calc}}^2$-diffeomorphism $F:U\to U^{{}^{\mathrm{\prime }}}$ between two domains in ${\text{cc}}^n$, $n\ge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic. \smallskip In this note we are interested in the self mappings of ${\text{cc}}^n$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: {\it A ${\text{calc}}^2$ - diffeomorphism $F:U^{\mathrm{\prime }}\to U$ (where $U’, U\subset \cc^n$ are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map $\mathrm{\Phi }\text{deff}{F}^{-1}$ is weakly pluriharmonic, i.e. it satisfies some nice second order PDE very close to $\text{d}\bar{\text{d}}\mathrm{\Phi }=0$.} In fact all pluriharmonic $\mathrm{\Phi }$-s do satisfy this equation, but there are also other solutions.