Bounding the Cebysev functional for the Riemann-Stieltjes integral via a Beesack inequality and applications

Lower and upper bounds of the μ Cebyšev functional for the RiemannStieltjes integral are given. Applications for the three point quadrature rules of functions that are n time di¤erentiable are also provided. 1. Introduction In 1975, P.R. Beesack [1] showed that, if y; v; w are real valued functions de…ned on a compact interval [a; b] ; where w is of bounded variation with total variation Wb a (w) ; and such that the Riemann-Stieltjes integrals R b a y (t) dv (t) and R b a w (t) y (t) dv (t) both exist, then m Z b a y (t) dv (t) + b _ a (w) inf a < b "Z y (t) dv (t) # (1.1) Z b a w (t) y (t) dv (t)