Upper bounds for Stein-type operators

We present sharp bounds on the supremum norm of DjSh for j>1, where D is the differential operator and S the Stein operator for the standard normal distribution. The same method is used to give analogous bounds for the exponential, Poisson and geometric distributions, with D replaced by the forward difference operator in the discrete case. We also discuss applications of these bounds to the central limit theorem, simple random sampling, Poisson-Charlier approximation and geometric approximation using stochastic orderings.