Skip to main content
SpringerLink
Log in

On Pompeiu–Chebyshev Functional and Its Generalization

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this work, a generalization of Chebyshev functional is presented. New inequalities of Grüss type via Pompeiu’s mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Grüss inequality is elaborated. Some remarks to further generalization of Chebyshev functional are presented. As applications, bounds for the reverse of CBS inequality are deduced. Hardy type inequalities on bounded real interval \(\left[ a,b\right] \) under some other circumstances are introduced. Other related ramified inequalities for differentiable functions are also given.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acu, A.M., Gonska, H., Raşa, I.: Grüss-type and Ostrowski-type inequalities in approximation theory. Ukr. Math. J. 63(6), 843–864 (2011)

    Article  Google Scholar 

  2. Acu, A.M., Sofonea, F.D.: On an inequality of Ostrowski type. J. Sci. Arts 3(16), 281–287 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Acu, A.M., Baboş, A., Sofonea, F.D.: The mean value theorems and inequalities of Ostrowski type. Sci. Stud. Res. Ser. Math. Inf. 21(1), 5–16 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Alomari, M.W.: On Beesack–Wirtinger inequality. Results Math. 72(3), 1213–1225 (2017)

    Article  MathSciNet  Google Scholar 

  5. Barnett, N.S., Dragomir, S.S.: An additive reverse of the Cauchy–Bunyakovsky–Schwarz integral inequality. Appl. Math. Lett. 21, 388–393 (2008)

    Article  MathSciNet  Google Scholar 

  6. Boggio, T.: Sur une proposition de M. Pompeiu. Mathematica (Cluj) 23, 101–102 (1947)

    MathSciNet  Google Scholar 

  7. Čebyšev, P.L.: Sur les expressions approximatives des intègrals dèfinis par les outres prises entre les même limites. Proc. Math. Soc. Charkov 2, 93–98 (1882)

    Google Scholar 

  8. Dragomir, S.S.: Ostrowski type inequalities for Lebesgue integral: a survey of recent results. Aust. J. Math. Anal. Appl. 14(1), 1–287 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Dragomir, S.S.: Some Grüss-type results via Pompeiu’s-like inequalities. Arab. J. Math. 4, 159–170 (2015)

    Article  MathSciNet  Google Scholar 

  10. Dragomir, S.S.: An inequality of Ostrowski type via Pompeiu’s mean value theorem. J. Inequal. Pure Appl. Math. 6(3), Article 83 (2005)

  11. Grüss, G.: Über das maximum des absoluten Betrages von \( \frac{1}{{b - a}}\int _a^b {f\left( x \right)g\left( x \right)dx} - \frac{1}{{(b - a)^2}}\int _a^b {f\left( x \right)dx} \int _a^b {g\left( x \right)dx}\). Math. Z. 39, 215–226 (1935)

    Article  MathSciNet  Google Scholar 

  12. Hardy, G.H.: Notes on some points in the integral calculus, XLI. On the convergence of certain integrals and series. Messenger Math. 45, 163–166 (1915)

    Google Scholar 

  13. Hardy, G.H.: Note on a theorem of Hilbert. Math. Z. 6(3–4), 314–317 (1920)

    Article  MathSciNet  Google Scholar 

  14. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1967)

    MATH  Google Scholar 

  15. Kufner, A., Maligranda, L., Persson, L.-E.: The prehistory of the Hardy inequality. Am. Math. Month. 113(8), 715–732 (2006)

    Article  MathSciNet  Google Scholar 

  16. Kufner, A., Maligranda, L., Persson, L.E.: The Hardy Inequality-About Its History and Some Related Results, Research Report. Luleȧ University of Technology, Luleȧ (2006)

    MATH  Google Scholar 

  17. Kufner, A., Persson, L.E.: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003)

    Book  Google Scholar 

  18. Opic, B., Kufner, A.: Hardy-Type Inequalities. Pitman Research Notes in Mathematics, No. 219. Long man Scientific & Technical, Harlow (1990)

    MATH  Google Scholar 

  19. Lupaş, A.: The best constant in an integral inequality. Mathematica, (Cluj, Romania) 15 (38)(2), 219–222 (1973)

    MathSciNet  MATH  Google Scholar 

  20. Matić, M., Ungar, Š.: More on the two-point Ostrolwski inequaluty. J. Math. Ineq. 3(3), 417–426 (2009)

    Article  Google Scholar 

  21. Milovanović, G.V., Pečarić, J.E.: On generalization of the inequality of A. Ostrowski and some related applications. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544–576, 155–158 (1976)

    MathSciNet  MATH  Google Scholar 

  22. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis, vol. 61 of Mathematics and Its Applications (East European Series), Kluwer Academic Publishers Group, Dordrecht, The Netherlands (1993)

  23. Ostrowski, A.M.: On an integral inequality. Aequat. Math. 4, 358–373 (1970)

    Article  Google Scholar 

  24. Pachpatte, B.G.: On Grüss like integral inequalities via Pompeiu’s mean value theorem. J. Inequal. Pure Appl. Math. 6(3), 132 (2005)

    MATH  Google Scholar 

  25. Pečarić, J., Ungar, Š.: On an inequality of Ostrowski type. J. Inequal. Pure Appl. Math. 7(4), Article 151 (2006)

  26. Pompeiu, D.: Sur une proposition analogue au theoreme des accroissements finis. Mathematica (Cluj) 22, 143–146 (1946)

    MathSciNet  MATH  Google Scholar 

  27. Popa, E.C.: An inequality of Ostrowski type via a mean value theorem. Gen. Math. 15(1), 93–100 (2007)

    MathSciNet  MATH  Google Scholar 

  28. Sahoo, P., Riedel, T.: Mean Value Theorems and Functional Equations. World Scientific Publishing Co., Pte. Ltd., Singapore (1998)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics Faculty of Science and Information Technology, Irbid National University, P.O. Box 2600 , Irbid, 21110, Jordan

    Mohammad W. Alomari

Authors
  1. Mohammad W. Alomari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad W. Alomari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alomari, M.W. On Pompeiu–Chebyshev Functional and Its Generalization. Results Math 74, 56 (2019). https://doi.org/10.1007/s00025-019-0977-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-0977-z

Mathematics Subject Classification

Keywords

Search

Navigation