Extending Cercignani's conjecture results from Botzmann to Boltzmann-Fermi-Dirac equation

We establish a connection between the relative Classical entropy and the relative Fermi-Dirac entropy, allowing to transpose, in the context of the Boltzmann or Landau equation, any entropy-entropy production inequality from one case to the other; therefore providing entropy-entropy production inequalities for the Boltzmann-Fermi-Dirac operator, similar to the ones of the Classical Boltzmann operator. We also provide a generalized version of the Csisz{á}r-Kullback-Pinsker inequality to weighted Lp norms, 1 $\le$ p $\le$ 2 and a wide class of entropies.