There is a proof by the author of a stronger inequality.    
First follow the argument at the top of page 486 of <A HREF="https://cedricvillani.org/sites/dev/files/old_images/2012/08/P09.ICM_1.pdf">Hypocoercive diffusion operators</A> to arrive at    
<IMG SRC="https://ilorentz.org/beenakker/MO/Villani_1.png" WIDTH="500"/>    
Then follow the proof in <A HREF="https://arxiv.org/abs/math/0609050">arXiv:math/0609050</A>, page 152, where it is shown that the functional
<IMG SRC="https://ilorentz.org/beenakker/MO/Villani_2.png" WIDTH="500"/>    
evolves in time such that
<IMG SRC="https://ilorentz.org/beenakker/MO/Villani_3.png" WIDTH="500"/>