Covariance inequalities for convex and log-concave functions

Extending results of Harg{é} and Hu for the Gaussian measure, we prove inequalities for the covariance Cov${}_{\mu }(f,g)$ where $\mu$ is a general product probability measure on ${\mathbb{R}}^d$ and $f,g:{\mathbb{R}}^d\to \mathbb{R}$ satisfy some convexity or log-concavity assumptions, with possibly some symmetries.