We prove that for any pair of i.i.d. random vectors X,Y in (formula presented) and any real-valued continuous negative definite function (formula presented) the inequality(formula presented) holds. In particular, for (formula presented) and the Euclidean norm (formula presented) one has (formula presented) The latter inequality is due to A. Buja et al. [4] where it is used for some applications in multivariate statistics. We show a surprising connection with bifractional Brownian motion and provide some related counter-examples.