Q-subdifferential of Jensen-convex functions

A real function is called radially Q-differentiable at the point x if, for every real number h, the finite limit dQf(x,h) of the ratio (f(x+rh)−f(x))/r exists whenever r tends to zero through the positive rationals. We establish that, in particular, Jensen-convex functions are everywhere radially Q-differentiable. Moreover, if f is Jensen-convex, then, for each x, the mapping h↦dQf(x,h) is subadditive, and it is an upper bound for any additive mapping A satisfying the inequality f(x)+A(y−x)⩽f(y) for every y. We also characterize all set-valued mappings built up from additive solutions A of this inequality with some Jensen-convex function f.