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

Abstract

A sharp companion of Ostrowski’s inequality for the RiemannStieltjes integral R b f(t) du(t), where f is assumed to be of r-H-Hölder type a 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. 1. Introduction In [12], Dragomir has proved an Ostrowski inequality for the Riemann-Stieltjes integral, as follows: Theorem 1.1 Let f : [a, b] →R be a r-H–Hölder type mapping, that is, it satisfies the condition |f(x) −f(y)| ≤H|x −y|r, ∀x, y ∈[a, b], where, H > 0 and r ∈(0, 1] are given, and u: [a, b] →R is a mapping of bounded variation on [a, b]. Then we have the inequality Z b hb −a f(x)(u(b) −u(a)) − f(t) du(t) ≤H + x −a + b ir · b_ (u) a 2 2 a for all x ∈[a, b], where, Wba(u) denotes the total variation of u on [a, b]. Furthermore, the constant 12 is the best possible in the sense that it cannot be replaced by a smaller one, for all r ∈(0, 1].