A concavity property for the reciprocal of Fisher information and its consequences on Costa's EPI

We prove that the reciprocal of Fisher information of a logconcave probability density is concave in t with respect to the addition
of a Gaussian noise of variance t. As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability
density is nonnegative in t with respect to the addition of a Gaussian noise of variance t.
For log-concave densities this improves the well-known Costa’s concavity property of the entropy power.