Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 21 August 2018 / Approved: 27 August 2018 / Online: 27 August 2018 (11:06:59 CEST)

How to cite: Dragomir, S. On Some Riemann-Stieltjes Integral Inequalities of Generalized Trapezoid Type with Applications. Preprints 2018, 2018080453 (doi: 10.20944/preprints201808.0453.v1). Dragomir, S. On Some Riemann-Stieltjes Integral Inequalities of Generalized Trapezoid Type with Applications. Preprints 2018, 2018080453 (doi: 10.20944/preprints201808.0453.v1).

## Abstract

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.