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.