  Motivated by classical notions of bilinear matrix inequalities (BMIs) and
partial convexity, this article investigates partial convexity for
noncommutative functions. It is shown that noncommutative rational functions
that are partially convex admit novel butterfly-type realizations that
necessitate square roots. The notion of xy-convexity, a strengthening of
partial convexity arising in connection with BMIs, is also considered. A
characterization of xy-convex polynomials is given.
