A note on one-dimensional Poincare inequalities by Stein-type integration

We study the weighted Poincare constant C(p, w) of a probability density p with weight function w using integra-tion methods inspired by Stein's method. We obtain a new version of the Chen-Wang variational formula which, as a byproduct, yields simple upper and lower bounds on C(p, w) in terms of the so-called Stein kernel of p. We also iterate these variational formulas so as to build sequences of nested intervals containing the Poincare constant, sequences of functions converging to said constant, as well as sequences of functions converging to the solutions of the corresponding spectral problem. Our results rely on the properties of a pseudo inverse operator of the classi-cal Sturm-Liouville operator. We illustrate our methods on a variety of examples: Gaussian functionals, weighted Gaussian, beta, gamma, Subbotin, and Weibull distributions.