We present two extended forms of Fisher information that t well in the nonextensive thermostatistics context. We show that there exists an in- terplay between these generalized Fisher information, the generalized Gaussian distributions and the q -entropies. q- The minimum of the general- ized Fisher information among distributions with a xed moment, or with a xed q -entropy is attained, in both cases, by a generalized q -Gaussian dis- This complements the fact that the q -Gaussians maximize the q -entropies subject to a moment constraint, and yields new variational char- acterizations of the generalized q -Gaussians. We show that the generalized tribution. Fisher information naturally pop up in the expression of the time derivative of the q -entropies, for distributions satisfying a certain nonlinear heat equa- tion. This result includes as a particular case the classical de Bruijn identity. We then study further properties of the generalized Fisher information and of their minimization. We show that, though non additive, the generalized Fisher information of a combined system is upper bounded. In the case of mixing, we show that the generalized Fisher information is convex for q ≥ 1. Finally, we show that the minimization of the generalized Fisher information subject to moment constraints satis es a Legendre structure analog to the Legendre structure of thermodynamics.