A Lower Bound on the Relative Entropy with Respect to a Symmetric Probability

Let $\rho$ and $\mu$ be two probability measures on $\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\mu \mathrm{\mid }\rho )$ the relative entropy of $\mu$ with respect to $\rho$. We prove that, if $\rho$ is symmetric and $\mu$ has a finite first moment, then $$H(\mu \mathrm{\mid }\rho )\ge \frac{{\displaystyle ({\int }_{\mathbb{R}}zd\mu (z){)}^2}}{{\displaystyle 2{\int }_{\mathbb{R}}{z}^{2}d\mu (z)}},$$ with equality if and only if $\mu =\rho$.