Summary: We establish several new functional inequalities comparing Boltzmann’s entropy production functional with the relative

$H$ functional. First we prove a longstanding conjecture by Cercignani under the nonphysical assumption that the Boltzmann collision kernel is superquadratic at infinity. The proof rests on the method introduced in [

*G. Toscani* and

*C. Villani*, Commun. Math. Phys. 203, 667–706 (1999;

Zbl 0944.35066)] combined with a novel use of the Blachman-Stam inequality. If the superquadraticity assumption is not satisfied, then it is known that Cercignani’s conjecture is not true; however we establish a slightly weakened version of it for all physically relevant collision kernels, thus extending previous results from the paper cited above. Finally, we consider the entropy-entropy production version of Kac’s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension. The first two results are sharp in some sense, and the third one is likely to be, too; they contain all previously known entropy estimates as particular cases. This gives a first coherent picture of the study of entropy production, according to a program started by

*E. Carlen* and

*M. Carvalho* [J. Stat. Phys. 67, 575–608 (1992;

Zbl 0899.76317)] ten years ago. These entropy inequalities are one step in our study of the trend to equilibrium for the Boltzmann equation, in both its spatially homogeneous and spatially inhomogeneous versions.