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Some Weighted Inequalities for Riemann–Stieltjes Integral When a Function Is Bounded

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Differential and Integral Inequalities

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 151))

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Abstract

In this chapter we provide some simple ways to approximate the Riemann–Stieltjes integral of a product of two functions \(\int _{a}^{b}f\left ( t\right ) g\left ( t\right ) dv\left ( t\right )\) by the use of simpler quantities and under several assumptions for the functions involved, one of them satisfying the boundedness condition

$$\displaystyle \left \vert f\left ( t\right ) -\frac {\gamma +\Gamma }{2}\right \vert \leq \frac { 1}{2}\left \vert \Gamma -\gamma \right \vert \ \text{for each}\ t\in \left [ a,b \right ] , $$

where \(f:\left [ a,b\right ] \rightarrow \mathbb {C}\). Applications for continuous functions of selfadjoint operators and functions of unitary operators on Hilbert spaces are also given.

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Authors and Affiliations

  1. Mathematics College of Engineering and Science, Victoria University, Melbourne, MC, Australia

    Silvestru Sever Dragomir

  2. DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

    Silvestru Sever Dragomir

Authors
  1. Silvestru Sever Dragomir
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Correspondence to Silvestru Sever Dragomir .

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Editors and Affiliations

  1. Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania

    Dorin Andrica

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Dragomir, S.S. (2019). Some Weighted Inequalities for Riemann–Stieltjes Integral When a Function Is Bounded. In: Andrica, D., Rassias, T. (eds) Differential and Integral Inequalities. Springer Optimization and Its Applications, vol 151. Springer, Cham. https://doi.org/10.1007/978-3-030-27407-8_8

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