  An inequality of Brascamp and Lieb provides a bound on the covariance of two
functions with respect to log-concave measures. The bound estimates the
covariance by the product of the $L^2$ norms of the gradients of the functions,
where the magnitude of the gradient is computed using an inner product given by
the inverse Hessian matrix of the potential of the log-concave measure. Menz
and Otto \cite{OM} proved a variant of this with the two $L^2$ norms replaced
by $L^1$ and $L^\infty$ norms, but only for $\R^1$. We prove a generalization
of both by extending these inequalities to $L^p$ and $L^q$ norms and on $\R^n$,
for any $n\geq 1$. We also prove an inequality for integrals of divided
differences of functions in terms of integrals of their gradients.

Functional Analysis