Unique additive information measures-Boltzmann-Gibbs-Shannon, Fisher and beyond

It is proved that the only additive and isotropic information measure that can depend on the probability distribution and also on its first derivative is a linear combination of the Boltzmann-Gibbs-Shannon and Fisher information measures. Power-law equilibrium distributions are found as a result of the interaction of the two terms. The case of second order derivative dependence is investigated and a corresponding additive information measure is given.