A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications

A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral ∫abf(t)du(t) $\int_a^b {f(t)\;du(t)} $ , where f is assumed to be of r-H-Hölder type on [a, b] and u is of bounded variation on [a, b], is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.