Q-subdiffferential of Jensen-convex functions

摘要

A real function is called radially Q-differentiable at the point x if, for every real number h, the finite limit d(Q)f (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 -> d(Q)f (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. (c) 2005 Elsevier Inc. All rights reserved.