Preprint Article Version 1 This version is not peer-reviewed

General Three Points Inequalities for Weighted Riemann-Stieltjes Integral

Version 1 : Received: 4 September 2018 / Approved: 4 September 2018 / Online: 4 September 2018 (08:20:50 CEST)

How to cite: Dragomir, S. General Three Points Inequalities for Weighted Riemann-Stieltjes Integral. Preprints 2018, 2018090060 (doi: 10.20944/preprints201809.0060.v1). Dragomir, S. General Three Points Inequalities for Weighted Riemann-Stieltjes Integral. Preprints 2018, 2018090060 (doi: 10.20944/preprints201809.0060.v1).

Abstract

In this paper we provide amongst others some simple error bounds in approximating the weighted Riemann-Stieltjes integral $\int_{a}^{b}f\left(t\right) g\left(t\right) dv\left(t\right) $ by the use of three points formula \begin{equation*} f\left(b\right) \int_{c}^{b}g\left(s\right) dv\left(s\right) +f\left(a\right) \int_{a}^{d}g\left( s\right) dv\left( s\right) -f\left(x\right) \int_{c}^{d}g\left(t\right) dv\left(t\right) \end{equation*} where $x,$ $c,$ $d\in \left[a,b\right],$ $g,$ $v:\left[a,b\right] \rightarrow \mathbb{C}$ under bounded variation and Lipschitzian assumptions for the function $f$ and such that the involved Riemann-Stieltjes integrals exist.

Subject Areas

Riemann-Stieltjes integral; continuous functions; functions of bounded variation; Lipschitzian functions

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