  A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ is called
plurisubharmonic (plush) if its nc complex Hessian takes only positive
semidefinite values on an nc neighborhood of 0. The main result of this paper
shows that an nc rational function is plush if and only if it is a composite of
a convex rational function with an analytic (no $x_j^*$) rational function. The
proof is entirely constructive. Further, a simple computable necessary and
sufficient condition for an nc rational function to be plush is given in terms
of its minimal realization.
