Logarithmic Sobolev inequalities for Moebius measures on spheres

In this paper, using the method in [1], i.e., reduce Moebius measures {\mu_{x}^{n}} indexed by {|x|<1} on spheres {S^{n-1}} ( {n\geq 3} ) to one-dimensional diffusions on {[0,\pi]} , we obtain that the optimal Poincaré constant is not greater than {\frac{2}{n-2}} and the optimal logarithmic Sobolev constant denoted by {C_{\rm LS}(\mu_{x}^{n})} behaves like {\frac{1}{n}\log(1+\frac{1}{1{-}|x|})} . As a consequence, we claim that logarithmic Sobolev inequalities are strictly stronger than {L^{2}} -transportation-information inequalities.