A sufficient condition for generalized Lorenz order (Q1977404)

Let \(X\) and \(Y\) be two non-negative random variables with finite means, having distribution functions \(F\) and \(G\), respectively, with supports \(\text{supp}(F)\subseteq \text{supp}(G) \equiv [a,b]\), where \(0\leq a < b\leq \infty\). Assume that \(F\) and \(G\) are continuous and strictly increasing on their supports. The authors observe that if \(E[X] < E[Y]\), and if \(F\) crosses \(G\) at most once, and if so, from above, then the generalized Lorenz curve of \(F\) is pointwise below the generalized Lorenz curve of \(G\).