Journal of the
Korean Mathematical Society JKMS

We show amongst other that if $f,g:\left[ a,b\right] \rightarrow \mathbb{C}$ are two functions of bounded variation and such that the Riemann-Stieltjes integral $\int_{a}^{b}f\left( t\right) dg\left( t\right) $ exists, then for any continuous functions $h:\left[ a,b\right] \rightarrow \mathbb{C}$, the Riemann-Stieltjes integral $\int_{a}^{b}h\left( t\right) d\left( f\left( t\right) g\left( t\right) \right) $ exists and \begin{equation*} \int_{a}^{b}h\left( t\right) d\left( f\left( t\right) g\left( t\right) \right) =\int_{a}^{b}h\left( t\right) f\left( t\right) d\left( g\left( t\right) \right) +\int_{a}^{b}h\left( t\right) g\left( t\right) d\left( f\left( t\right) \right) . \end{equation*} Using this identity we then provide sharp upper bounds for the quantity \begin{equation*} \left\vert \int_{a}^{b}h\left( t\right) d\left( f\left( t\right) g\left( t\right) \right) \right\vert \end{equation*} and apply them for trapezoid and Ostrowski type inequalities. Some applications for continuous functions of selfadjoint operators on complex Hilbert spaces are given as well.

Keywords: Riemann-Stieltjes integral, functions of bounded variation, Trapezoid and midpoint inequalities, selfadjoint operators, functions of selfadjoint operators

MSC numbers: 26D15, 47A63