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

On some interrelations of generalized $q$-entropies and a generalized Fisher information, including a Cram\'er-Rao inequality