On the existence of aggregation functions with given super-additive and sub-additive transformations

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, A↦A⁎ and A↦A⁎, of an aggregation function A. We prove that if A⁎ has a slightly stronger property of being strictly directionally convex, then A=A⁎ and A⁎ is linear; dually, if A⁎ is strictly directionally concave, then A=A⁎ and A⁎ is linear. This implies, for example, the existence of pairs of functions f≤g sub-additive and super-additive on [0,∞[n, respectively, with zero value at the origin and satisfying relatively mild extra conditions, for which there exists no aggregation function A on [0,∞[n such that A⁎=f and A⁎=g.