Sums of increasing convex and increasing concave functions (Q581222)

A necessary and sufficient condition is given for a function t(x) to be representable as a sum of an increasing convex function r(x) and an increasing concave function s(x). The representation is shown to be unique (i.e. \(r_ 1-r_ 2=s_ 1-s_ 2\) is an affine function for any two possible representations \((r_ 1\), \(r_ 2)\), \((s_ 1\), \(s_ 2))\) if a complementary slackness condition is required, which reduces to \(r''(x)s''(x)=0\) when both second derivatives of r(x) and s(x) exist. This function class has applications to the theory of majorization.