Variance inequalities for functions of Gaussian variables

LetX be a standard Gaussian random variable andG an absolutely continuous function. The inequality $$Var G(X) \leqslant E(G'(X))^2 $$ was proved in Nash and later rediscovered in Brascamp and Lieb as a special case of a general inequality and in Chernoff. All the proofs are based on properties of the Gaussian density. By using the characteristic function rather than the density, generalizations with higher order derivatives are obtained. The method also establishes potentially useful connections with Karlin's total positivity.