The Relationships Of Functional Inequalities And Examples

Poincaré(PI)?transportation(W2H)and logarithmic Sobolev inequalities(L-SI)are important tools in the research of concentration of measures and ergodic theory.Furthermore,the relationship of above three inequalities are also the hot topic in the probability theory study.In this paper,based on the simple situation,we consider the relationships among them.We give an alterative proof about that LSI is stronger than W2H by using Moebius measures of the unit ring on the R2.In other words,the optimal constant of LSI will "blow up" at some special condition,but neither does the optimal constant of W2H.According to the literature[5,12,13],we are accessible to get the proof of W2H stronger than PI.Combining the related literature,we get that LSI stronger than W2H,and W2H stronger than PI.