Bounds for the optimal constant of the Bakry-\'Emery $\Gamma_2$ criterion inequality on $ RP^{d-1}$

We prove upper and lower bounds on the optimal constant $\Lambda_d$ of the Bakry-\'Emery $\Gamma_2$ criterion for positive symmetric functions on the unit sphere $S^{d-1}$, which also can be identified as positive functions on the real projective space $RP^{d-1}$. The Bakry-\'Emery $\Gamma_2$ criterion inequality was crucially used to prove the monotonicty of the Fisher information for the Landau equation by Guillen and Silvestre recently. Therefore, a better bound on the optimal constant $\Lambda_d$ expands the range of interaction potentials that exhibits the monotonicity of the Fisher information. In particular, we compute that $\Lambda_3$ is between $5.5$ and $5.739$.