Barthe, F.; Roberto, C. Sobolev inequalities for probability measures on the real line. (English) Zbl 1072.60008 Stud. Math. 159, No. 3, 481-497 (2003). A probability measure \(\mu \) on \(R\) satisfies Poincaré inequality if there exists a constant \(C_{\text P}>0\) such that for every smooth function \(f:R\rightarrow R\) \[ \int f^2\,d\mu-\biggl(\int f\,d\mu\biggr)^2\leq C_{\text P}\int(f^{\prime})^2\,d\mu \] and a logarithmic Sobolev inequality if there exists a constant \(C_{\text{LS}}>0\) such that for every smooth function \(f:R\rightarrow R\) \[ \int f^2\log f^2\,d\mu-\biggl(\int f^2\,d\mu\biggr)\log \biggl(\int f^2\,d\mu\biggr)\leq C_{\text{LS}}\int (f^{\prime})^2\,d\mu. \]The authors give lower and upper bounds for the optimal values of \(C_{\text P}\) and \(C_{\text{LS}}\). Similar results are established for certain other classes of inequalities. Reviewer: Vladimir Vatutin (Moskva) Cited in 59 Documents MSC: 60E15 Inequalities; stochastic orderings 26D15 Inequalities for sums, series and integrals Keywords:concentration; Poincaré inequalities; optimal constants PDFBibTeX XMLCite \textit{F. Barthe} and \textit{C. Roberto}, Stud. Math. 159, No. 3, 481--497 (2003; Zbl 1072.60008) Full Text: DOI Link