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A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral [...] ∫abf(t) du(t) ${\int }_a^{b}f\left(t\right)du\left(t\right)$ , 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.