  Extending results of Harg{\'e} 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.
