A matrix variance inequality

In the course of solving a variational problem Chernoff (Ann. Probab. 9 (1981) 533) obtained what appears to be a specialized inequality for a variance, namely, that for a standard normal variable X, Varg(X)]Eg^′(X)]². However, both the simplicity and usefulness of the inequality has generated a plethora of extensions, as well as alternative proofs. All previous papers have focused on a single function. We provide here an inequality for the covariance matrix of k functions, which leads to a matrix inequality in the sense of Loewner.