Characterizations of distributions by variance bounds

The distribution of a continuous r.v. X is characterized by the function w appearing in the lower bound [sigma]2E2[w(X)g'(X)] for the variance of a function g(X); for a discrete X, g'(x) is replaced by [Delta]g(x) = g(x + 1) - g(x). The same characterizations are obtained by considering the upper bound [sigma]2E{w(X)[g'(X)]2} [greater-or-equal, slanted] Var[g(X)]. The special case w(x) = 1 gives the normal, Borovkov and Utev (1983), and the Poisson, Prakasa Rao and Sreehari (1987). The results extend to independent random variables.