Lipschitz constants and logarithmic Sobolev inequality

We revisit two results of  [3]; they are a logarithmic Sobolev inequality on R n with Lipschitz constants and an expression of Lipschitz constants as the limit of a functional by the entropy. We have two goals in this paper. The first goal is to clarify when the strict inequality holds in this inequality. The second goal is to investigate the asymptotic behavior of this functional by the Abelian and Tauberian theorems of Laplace transforms.