On matrix variance inequalities

Olkin and Shepp (2005, J. Statist. Plann. Inference, vol. 130, pp. 351–358) presented a matrix form of Chernoff’s inequality for Normal and Gamma (univariate) distributions. We extend and generalize this result, proving Poincare-type and Bessel-type inequalities, for matrices of arbitrary order and for a large class of distributions.