  In this communication, we describe some interrelations between generalized
$q$-entropies and a generalized version of Fisher information. In information
theory, the de Bruijn identity links the Fisher information and the derivative
of the entropy. We show that this identity can be extended to generalized
versions of entropy and Fisher information. More precisely, a generalized
Fisher information naturally pops up in the expression of the derivative of the
Tsallis entropy. This generalized Fisher information also appears as a special
case of a generalized Fisher information for estimation problems. Indeed, we
derive here a new Cram\'er-Rao inequality for the estimation of a parameter,
which involves a generalized form of Fisher information. This generalized
Fisher information reduces to the standard Fisher information as a particular
case. In the case of a translation parameter, the general Cram\'er-Rao
inequality leads to an inequality for distributions which is saturated by
generalized $q$-Gaussian distributions. These generalized $q$-Gaussians are
important in several areas of physics and mathematics. They are known to
maximize the $q$-entropies subject to a moment constraint. The Cram\'er-Rao
inequality shows that the generalized $q$-Gaussians also minimize the
generalized Fisher information among distributions with a fixed moment.
Similarly, the generalized $q$-Gaussians also minimize the generalized Fisher
information among distributions with a given $q$-entropy.
