Skip to main content
SpringerLink
Log in

Some Inequalities for Riemann-Stieltjes Integral and Applications

  • Chapter
Optimization and Related Topics

Part of the book series: Applied Optimization ((APOP,volume 47))

Abstract

In this survey paper we present some recent results of the author concerning certain inequalities of Trapezoid type, Ostrowski type and Grüss type for Riemann-Stieltjes integrals and their natural application to the problem of approximating the Riemann-Stieltjes integral. As many problems in the applications of Optimization Theory may require the Riemann-Stieltjes integral, we believe that our results here can provide a valuable tool in the numerical aspect of these problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Barnett, N.S. and Dragomir, S.S. (a), An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. of Math, accepted

    Google Scholar 

  • Barnett, N.S., Cerone, P., Dragomir, S.S., Roumeliotis, J. and Sofo, A., A survey on Ostrowski type inequalities for twice differentiable mappings and application, Proc. Canadian Math. Soc, accepted

    Google Scholar 

  • Barnett, N.S., Dragomir, S.S. and Pearce, C.E.M., A quasi-trapezoid inequality for double integrals, J. Austal. Math. Soc,B, submitted

    Google Scholar 

  • Barnett, N.S. and Dragomir, S.S. (1999a), An inequality of Ostrowski’s type for cumulative distribution functions, Kyungpook Math. J, 39, 303–311.

    MathSciNet  MATH  Google Scholar 

  • Bartle, R.G. (1976), The Elements of Real Analysis, Second Edition, John Wiley & Sons Inc.

    MATH  Google Scholar 

  • Cerone, P. and Dragomir, S.S. (b), Three point quadrature rules involving, at most, a first derivative, submitted

    Google Scholar 

  • Cerone, P., Dragomir, S.S. and Roumeliotis, J. (1999a), An inequality of Ostrowski–Grüss type for twice differentiable mappings and applications, Kyungp000k Math. J, 39, 333–341.

    MathSciNet  MATH  Google Scholar 

  • Cerone, P., Dragomir, S.S. and Roumeliotis, J. (1999b), An inequality of Ostrowski type for mappings whose second derivatives belong to L1 and applications, Hanam Math. J, 21, No. 1, 127–137.

    MathSciNet  MATH  Google Scholar 

  • Cerone, P., Dragomir, S.S. and Roumeliotis, J. (1999c), An inequality of Ostrowski type for mappings whose second derivatives are bounded and applications, East Asian J. of Math, 15, No. 1, 7–9.

    MathSciNet  Google Scholar 

  • Cerone, P., Dragomir, S.S. and Roumeliotis, J., (a), An Ostrowski type inequality for mappings whose second derivatives belong to L P and applications, submitted

    Google Scholar 

  • Cerone, P., Dragomir, S.S. and Roumeliotis, J. (1999d), Some Ostrowski type inequalities for n-time differentiable mappings and applications, Demonstratio Math, 32, No. 4, 697–712.

    MathSciNet  MATH  Google Scholar 

  • Cerone, P. and Dragomir, S.S. (a), Lobato type quadrature rules for functions with bounded derivatives, submitted

    Google Scholar 

  • Cerone, P. and Dragomir, S.S. (2000), On a weighted generalization of iyengar type inequalities involving bounded first derivative, Math. Ineq. é4 Appl, 3, No. 1, 35–44.

    MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S. (a), On the trapezoid inequality for absolutely continuous mappings, submitted

    Google Scholar 

  • Dragomir, S.S. (b), On the trapezoid quadrature formula for mappings of bounded variation and applications, Extracta Math, accepted

    Google Scholar 

  • Dragomir, S.S. (1999a), On the trapezoid formula for Lipschitzian mappings and applications, Tamkang J. of Math, 30, No. 2, 133–138.

    MATH  Google Scholar 

  • Dragomir, S.S., Cerone, P. and Sofo, A., Some remarks on the trapezoid rule in numerical integration, Indian J. of Pure and Appl. Math, in press

    Google Scholar 

  • Dragomoir, S.S. and Peachey, T.C. New estimation of the remainder in the trapezoidal formula with applications, Studia math. Babes-Bolyai, accepted

    Google Scholar 

  • Dragomir, S.S., Cerone, P. and Pearce, C.E.M., Generalizations of the trapezoid inequality for mappings of bounded variation and applications, submitted

    Google Scholar 

  • Dragomir, S.S. and McAndrew, A., On trapezoid inequality via a Grüss type result and applications, Tamkang J. of Math, accepted

    Google Scholar 

  • Dragomir, S.S., Kucera, A. and Roumeliotis, J., A trapezoid formula for Riemann - Stieltjes integral, in preparation

    Google Scholar 

  • Dragomir, S.S. (c), On the trapezoid inequality for the Riemann-Stieltjes integral and applications, submitted

    Google Scholar 

  • Dragomir, S.S. and Wang, S. (1998), Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett, 11, 105–109.

    Article  MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S. and Wang, S. (1997), A new inequality of Ostrowski’s type in L1 norm and applications to some special means and some numerical quadrature rules, Tamkang J. of Math, 28, 239–244.

    MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S. (1999b), Ostrowski’s inequality for mappings of bounded variation and applications, RGMIA Res. Rep. Coll, 1, 2, 103–110.

    Google Scholar 

  • Dragomir, S.S. (d), On the Ostrowski inequality for the Riemann-Stieltjes integral (I), in preparation

    Google Scholar 

  • Dragomir, S.S. (e), On the Ostrowski inequality for the Riemann-Stieltjes integral (II) in preparation

    Google Scholar 

  • Dragomir, S.S. and Wang, S. (a), A new inequality of Ostrowski’s type in Lp norm, Indian Journal of Mathematics.

    Google Scholar 

  • Dragomir, S.S. (1999c), Ostrowski’s inequality for monotonic mapping and applications, J. KSIAM, 3, No. 1, 127–135.

    Google Scholar 

  • Dragomir, S.S. (1999d), On the Ostrowski’s inequality for mappings of bounded variation, RGMIA Res. Rep. Coll, 2, No. 1, 73–80.

    Google Scholar 

  • Dragomir, S.S., Barnett, N.S. and Wang, S. (1999), An Ostrowski type inequality for a random variable whose probability density function belongs to L r [a, b], p 1, Mathematical Inequalities and Appl, 2, No. 4, 501–508.

    Article  MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S. (1999e), Grüss inequality in inner product spaces, The Australian Math Soc. Gazette, Vol. 26, (2), 66–70.

    MATH  Google Scholar 

  • Dragomir, S.S. (1999f), A generalization of Grüss’ inequality in inner product spaces and applications, J. Math. Anal. Appl., 237, 74–82.

    Article  MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S.(f), A Grüss type integral inequality for mappings of r-Hölder’s type and applications for trapezoid formula, Tamkang J. of Math, accepted

    Google Scholar 

  • Dragomir, S.S. (1999g), Some discrete inequalities of Grüss type and applications in guessing theory, Honam Math. J, 21, No. 1, 115–126.

    MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S.(g), Some integral inequalities of Grüss type, Indian J. of Pureand Appl. Math, accepted

    Google Scholar 

  • Dragomir, S.S. and Booth, G.L. On a Grüss-Lupa§ type inequality and its applications for the estimation of p-moments of guessing mappings, Mathematical Communications, accepted

    Google Scholar 

  • Dragomir, S.S. and Fedotov, I. (1998), An inequality of Grüss’ type for RiemannStieltjes integral and applications for special means, Tamkang J. of Math, 29 (4), 286–292.

    MathSciNet  Google Scholar 

  • Dragomir, S.S. and I. Fedotov, I.(a), A Grüss type inequality for mappings of bounded variation and applications to numerical analysis, submitted

    Google Scholar 

  • Dragomir, S.S. and Sofo, A. (1999), An estimation for ln k, Australian Math. Gazette, 26, No. 5, 227–231.

    MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S. (1998), On Simpson’s quadrature formula for differentiable mappings whose derivatives belong to 4-spaces and applications, JKSIAM, 2, No. 2, 57–65.

    Google Scholar 

  • Dragomir, S.S. (h), On Simpson’s quadrature formula and applications, submitted.

    Google Scholar 

  • Dragomir, S.S. (1999h), On Simpson’s quadrature formula for mappings with bounded variation and application, Soochow J. of Math, 25, No. 2, 175–180.

    MATH  Google Scholar 

  • Dragomir, S.S. (1999i), On Simpson’s quadrature formula for Lipschitzian mappings and applications, Tamkang J. of Math, 30, No. 1, 53–58.

    MATH  Google Scholar 

  • Dragomir, S.S., Agarwal, R.P. and Cerone, P. On Simpson’s inequality and applications, Journal of Inequalities and Applications, accepted

    Google Scholar 

  • Dragomir, S.S., Pecaric, J.E. and Wang, S., The unified treatment of trapezoid, Simpson and Ostrowski type inequality for monotonic mappings and applications, Computer and Math. with Appl, accepted

    Google Scholar 

  • Dragomir, S.S. (1999i), On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Bull. Austral. Math. Soc, 60, 495–508.

    Article  MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S. (1999j), On the Ostrowski integral inequality for Lipschitzian mappings and applications, Computer and Math. with Appl, 38, 33–37.

    Article  MATH  Google Scholar 

  • Dragomir, S.S. (i), A generalization of Ostrowski integral inequality for mappings whose derivatives belong to Loo and applications in numerical integration, submitted.

    Google Scholar 

  • Dragomir, S.S. (j), A generalization of Ostrowski integral inequality for mappings whose derivatives belong to Lr[a, b] and applications in numerical analysis, submitted.

    Google Scholar 

  • Dragomir, S.S. (k), A generalization of Ostrowski integral inequality for mappings of bounded variation and applications in numerical integration, submitted.

    Google Scholar 

  • Dragomir, S.S. (1), Ostrowski’s inequality for monotonous mappings and applications, J. KSIAM, in press

    Google Scholar 

  • Dragomir, S.S. (m), A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L1 [a, b] and applications in numerical analysis, J. of Computational Analysis and Appl., in press.

    Google Scholar 

  • Dragomir, S.S. (n), Some quadrature formulae for absolutely continuous mappings via Ostrowski type inequalities, submitted

    Google Scholar 

  • Dragomir, S.S., Agarwal, R.P. and Barnett, N.S., Inequalities for beta and gamma functions via some classical and new integral inequalities, Journal of Inequalities and Applications, accepted.

    Google Scholar 

  • Dragomir, S.S. and Barnett, N.S., An Ostrowski type inequality for mappings whose second derivatives are bounded and applicationsKyungp000k Math. J., accepted.

    Google Scholar 

  • Dragomir, S.S., Barnett, N.S. and Cerone, P. (a), An n-dimensional version of Ostrowski’s inequality for mappings of the Hölder’s type, Kyungp000k Math. J., accepted.

    Google Scholar 

  • Dragomir, S.S., Barnett, N.S. and Cerone, P. (b), An Ostrowski type inequality for double integrals in terms of Lr norms and applications in numerical integration, Anal. Num. Theor. Approx., accepted.

    Google Scholar 

  • Dragomir, S.S., Cerone, P., Barnett, N.S. and Roumeliotis, J., An inequality of the Ostrowski type for double integrals and applications for cubature formulae, submitted

    Google Scholar 

  • Dragomir, S.S., Cerone, P. and Roumeliotis, J. (2000), A new generalization of Ostrowski’s integral inequality for mappings whose derivatives are bounded and applications in numerical integration, Appl. Math. Lett, 13, 19–25.

    Article  MathSciNet  MATH  Google Scholar 

  • Dragomir, S.S., Cerone, P., Roumeliotis, J. and Wang, S. A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math. Soc. Math. Romania, accepted

    Google Scholar 

  • Dragomir, S.S. and Sofo, A., An integral inequality for twice differentiable mappings and applications, Tamkang J. of Math, accepted

    Google Scholar 

  • Fink, A.M., A treatise on Grüss’ inequality, submitted

    Google Scholar 

  • Grüss, G. (1935), Über das Maximum des absoluten Betrages von ó 1a fâ f (x)g(x)dx —(6 1Q)2 fâ f(x)dx fa g(x)dx, Math. Z, 39, 215–226.

    Article  MathSciNet  Google Scholar 

  • Mitrinovié, D.S., Pecaric, J.E. and Fink, A.M. (1993), Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht.

    Book  Google Scholar 

  • Mitrinovié, D.S., Pecaric, J.E. and Fink, A.M., (1994), Inequalities for Functions and Their Integrals and Derivatives,Kluwer Academic Publishers,Dordrecht.

    Google Scholar 

  • Peachey, T.C., McAndrew, A. and Dragomir, S.S. (1999), The best constant in an inequality of Ostrowski type, Tamkang J. of Math, 30, No. 3, 219–222.

    MathSciNet  MATH  Google Scholar 

  • Roumeliotis, J., Cerone, P. and Dragomir, S.S. (1999), An Ostrowski type inequality for weighted mappings with bounded second derivatives, J. KSIAM, 3, No. 2, 107–119.

    MathSciNet  Google Scholar 

  • Tortorella, M. (1990), Closed Newton-Cotes quadrature rules for Stieltjes integrals and numerical convolution of life distributions,Siam J. Sci. Stat. Comput, 11, No. 4, 732–748.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Communications and Informatics, Victoria University of Technology, Canada

    S. S. Dragomir

Authors
  1. S. S. Dragomir
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Information Technology & Mathematical Sciences, University of Ballarat, Victoria, Australia

    Alexander Rubinov

  2. School of Mathematics and Statistics, Curtin University of Technology, WA, Australia

    Barney Glover

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Dragomir, S.S. (2001). Some Inequalities for Riemann-Stieltjes Integral and Applications. In: Rubinov, A., Glover, B. (eds) Optimization and Related Topics. Applied Optimization, vol 47. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6099-6_13

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-6099-6_13

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4844-1

  • Online ISBN: 978-1-4757-6099-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation