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An inequality of Brascamp and Lieb provides a bound on the covariance of twofunctions with respect to log-concave measures. The bound estimates thecovariance 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 bythe inverse Hessian matrix of the potential of the log-concave measure. Menzand Otto \cite{OM} proved a variant of this with the two $L^2$ norms replacedby $L^1$ and $L^\infty$ norms, but only for $\R^1$. We prove a generalizationof 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 divideddifferences of functions in terms of integrals of their gradients.

Asymmetric Covariance Estimates of Brascamp-Lieb Type and Related Inequalities for Log-concave Measures