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

  • Silvestru Sever DragomirEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 151)

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.

1991 Mathematics Subject Classification

26D15 26D10 26D07 26A33 

References

  1. 1.
    T.M. Apostol, Mathematical Analysis, 2nd edn. (Addison-Wesley, Boston, 1981)Google Scholar
  2. 2.
    N.S. Barnett, W.S. Cheung, S.S. Dragomir, A. Sofo, Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators. Comput. Math. Appl. 57(2), 195–201 (2009). Preprint RGMIA Res. Rep. Coll. 9(2006), No. 4, Article 9MathSciNetCrossRefGoogle Scholar
  3. 3.
    P. Cerone, S.S. Dragomir, Trapezoid type rules from an inequalities point of view, in Handbook of Analytic Computational Methods in Applied Mathematics, ed. by G. Anastassiou (CRC Press, New York, 2000), pp. 65–134Google Scholar
  4. 4.
    P. Cerone, S.S. Dragomir, A refinement of the Grüss inequality and applications. Tamkang J. Math. 38(1), 37–49 (2007). Preprint RGMIA Res. Rep. Coll.,5(2) (2002), Article 14MathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Cerone, S.S. Dragomir, C.E.M. Pearce, A generalised trapezoid inequality for functions of bounded variation. Turk. J. Math. 24(2), 147–163 (2000)zbMATHGoogle Scholar
  6. 6.
    P. Cerone, W.S. Cheung, S.S. Dragomir, On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation. Comput. Math. Appl. 54(2), 183–191 (2007). Preprint RGMIA Res. Rep. Coll.9(2006), No. 2, Article 14. http://rgmia.vu.edu.au/v9n2.html MathSciNetCrossRefGoogle Scholar
  7. 7.
    X.L. Cheng, J. Sun, A note on the perturbed trapezoid inequality. J. Inequal. Pure Appl. Math. 3(2) (2002), Art. 29Google Scholar
  8. 8.
    W.S. Cheung, S.S. Dragomir, Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions. Bull. Aust. Math. Soc. 75(2), 299–311 (2007). Preprint RGMIA Res. Rep. Coll.9(2006), No. 3, Article 8.MathSciNetCrossRefGoogle Scholar
  9. 9.
    W.S. Cheung, S.S. Dragomir, A survey on Ostrowski type inequalities for Riemann-Stieltjes integral, in Handbook of Functional Equations. Springer Optimization and Applications, vol. 95 (Springer, New York, 2014), pp. 75–104Google Scholar
  10. 10.
    S.S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation. Bull. Aust. Math. Soc. 60(3), 495–508 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    S.S. Dragomir, Ostrowski’s inequality for monotonous mappings and applications. J. KSIAM 3(1), 127–135 (1999)Google Scholar
  12. 12.
    S.S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications. Comput. Math. Appl. 38, 33–37 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    S.S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltjes integral. Korean J. Appl. Math. 7, 477–485 (2000)MathSciNetGoogle Scholar
  14. 14.
    S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications. Math. Inequal. Appl. 4(1), 59–66 (2001). Preprint: RGMIA Res. Rep. Coll.2 (1999), Art. 7.Google Scholar
  15. 15.
    S.S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltjes integral \(\int _{a}^{b}f\left ( t\right ) du\left ( t\right ) \) where f is of Hölder type and u is of bounded variation and applications. J. KSIAM 5(1), 35–45 (2001)Google Scholar
  16. 16.
    S.S. Dragomir, The median principle for inequalities and applications, in Functional Equations, Inequalities and Applications (Kluwer Academic Publishers, Dordrecht, 2003), pp. 21–37CrossRefGoogle Scholar
  17. 17.
    S.S. Dragomir, A companion of the Grüss inequality and applications. Appl. Math. Lett. 17(4), 429–435 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    S.S. Dragomir, Inequalities of Grüss type for the Stieltjes integral. Kragujevac J. Math. 26, 89–122 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    S.S. Dragomir, A generalisation of Cerone’s identity and applications. Tamsui Oxf. J. Math. Sci. 23(1), 79–90 (2007). Preprint RGMIA Res. Rep. Coll.8(2005), No. 2. Article 19Google Scholar
  20. 20.
    S.S. Dragomir, Inequalities for Stieltjes integrals with convex integrators and applications. Appl. Math. Lett. 20, 123–130 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    S.S. Dragomir, The perturbed median principle for integral inequalities with applications, in Nonlinear Analysis and Variational Problems. Springer Optimization and Applications, vol. 35 (Springer, New York, 2010), pp. 53–63Google Scholar
  22. 22.
    S.S. Dragomir, Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces. Acta Math. Vietnam. 39, 287–303 (2014). https://doi.org/10.1007/s40306-014-0061-4. Preprint RGMIA Res. Rep. Coll.15(2012), Art. 16MathSciNetCrossRefGoogle Scholar
  23. 23.
    S.S. Dragomir, Ostrowski type inequalities for Lebesgue integral: a survey of recent results. Aust. J. Math. Anal. Appl. 14(1), 283 (2017). Art. 1Google Scholar
  24. 24.
    S.S. Dragomir, I. Fedotov, An inequality of Grüss type for the Riemann-Stieltjes integral and applications for special means. Tamkang J. Math. 29(4), 287–292 (1998)MathSciNetzbMATHGoogle Scholar
  25. 25.
    S.S. Dragomir, I. Fedotov, A Grüss type inequality for mappings of bounded variation and applications for numerical analysis. Nonlinear Funct. Anal. Appl. 6(3), 425–433 (2001)MathSciNetzbMATHGoogle Scholar
  26. 26.
    S.S. Dragomir, C. Buşe, M.V. Boldea, L. Braescu, A generalisation of the trapezoidal rule for the Riemann-Stieltjes integral and applications. Nonlinear Anal. Forum (Korea) 6(2), 337–351 (2001)Google Scholar
  27. 27.
    G. Helmberg, Introduction to Spectral Theory in Hilbert Space (Wiley, New York, 1969)zbMATHGoogle Scholar
  28. 28.
    Z. Liu, Refinement of an inequality of Grüss type for Riemann-Stieltjes integral. Soochow J. Math. 30(4), 483–489 (2004)MathSciNetzbMATHGoogle Scholar
  29. 29.
    A. Ostrowski, Uber die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematics College of Engineering and ScienceVictoria UniversityMelbourneAustralia
  2. 2.DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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