On Some Riemann-Stieltjes Integral Inequalities of Generalized Trapezoid Type with Applications

In this paper we provide some bounds for the error in approximating the Riemann-Stieltjes integral \(\int_{a}^{b}f (t) du (t)\) by the generalized trapezoidal rule \begin{equation*} \left[ u\left( b\right) -u\left( x\right) \right] f\left( b\right) +\left[ u\left( x\right) -u\left( a\right) \right] f\left( a\right) \end{equation*} under various assumptions for the integrand \(f\) and the integrator \(u\) for which the above integral exists. Applications for continuous functions of selfadjoint operators and unitary operators in Hilbert spaces are provided as well.