Nonnegativity of odd functional moments of positive random variables with decreasing density

In this note we give some results on the nonnegativity of odd functional moments of random variables with a decreasing density. More precisely, we prove by purely elementary arguments, E phi(X - EX)greater than or equal to 0 for suitable functions phi that satisfy phi(x) = -phi(-x) for all x greater than or equal to 0 and random variables X greater than or equal to 0 with a decreasing Lebesgue density on (0,infinity) or counting density on N-0. The motivation came from a problem recently published in Statistica Neerlandica (Vol. 43, p. 66). We give a more specialized result in this paper.