Ukrainian Mathematical Journal

, Volume 64, Issue 4, pp 491–510 | Cite as

A companion of Dragomir’s generalization of the Ostrowski inequality and applications to numerical integration

  • M. W. Alomari
Article
First Online:
Received:
Some analogs of Dragomir’s generalization of the Ostrowski integral inequality
$$ \left| {\left( {b - a\left[ {\lambda \frac{{f(a) + f(b)}}{2} + \left( {1 - \lambda } \right)f(x)} \right] - \int\limits_a^b {f(t)dt} } \right)} \right| \leqslant \left[ {\frac{{{{\left( {b - a} \right)}^2}}}{4}\left( {\lambda^2 + {{\left( {1 - \lambda } \right)}^2}} \right) + {{\left( {x - \frac{{a + b}}{2}} \right)}^2}} \right]{\left\| {f'} \right\|_\infty } $$
are established. Some sharp inequalities are proved. An application to the composite quadrature rule is provided.

Keywords

Bounded Variation Quadrature Formula Quasiconvex Function Sharp Inequality Hadamard Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. W. Alomari, M. Darus, and U. S. Kirmaci, “Some inequalities of Hermite–Hadamard type for s-convex functions,” Acta Math. Sci., 31B, No. 4, 1643–1652 (2011).MathSciNetGoogle Scholar
  2. 2.
    M. W. Alomari, “A companion of Ostrowski’s inequality with applications,” Trans. J. Math. Mech., 3, 9–14 (2011).MathSciNetMATHGoogle Scholar
  3. 3.
    M. Alomari and S. Hussain, “Two inequalities of Simpson type for quasiconvex functions and applications,” Appl. Math. E-Notes, 11, 110–117 (2011).MathSciNetMATHGoogle Scholar
  4. 4.
    M. Alomari, M. Darus, and U. Kirmaci, “Refinements of Hadamard-type inequalities for quasiconvex functions with applications to trapezoidal formula and to special means,” Comput. Math. Appl., 59, 225–232 (2010).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, “Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense,” Appl. Math. Lett., 23, 1071–1076 (2010).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    M. Alomari, M. Darus, and S. S. Dragomir, “New inequalities of Hermite–Hadamard type for functions whose second derivatives absolute values are quasiconvex,” Tamkang J. Math., 41, 353–359 (2010).MathSciNetMATHGoogle Scholar
  7. 7.
    M. Alomari and M. Darus, “On some inequalities of Simpson-type via quasiconvex functions and applications,” Trans. J. Math. Mech., 2, 15–24 (2010).MathSciNetGoogle Scholar
  8. 8.
    M. Alomari and M. Darus, “Some Ostrowski type inequalities for quasiconvex functions with applications to special means,” RGMIA Preprint, 13, No. 2, Article No. 3 (2010) [http://rgmia. org/papers/v13n2/quasi-convex. pdf].
  9. 9.
    N. S. Barnett, S. S. Dragomir, and I. Gomma, “A companion for the Ostrowski and the generalized trapezoid inequalities,” J. Math. Comput. Modelling, 50, 179–187 (2009).MATHCrossRefGoogle Scholar
  10. 10.
    P. Cerone and S. S. Dragomir, “Midpoint-type rules from an inequalities point of view,” in: G. Anastassiou (editor), Handb. Anal. Comput. Methods Appl. Math., CRC Press, New York (2000), pp. 135–200.Google Scholar
  11. 11.
    P. Cerone and S. S. Dragomir, “Trapezoidal-type rules from an inequalities point of view,” in: G. Anastassiou (editor), Handb. Anal. Comput. Methods Appl. Math., CRC Press, New York (2000), pp. 65–134.Google Scholar
  12. 12.
    A. Guessab and G. Schmeisser, “Sharp integral inequalities of the Hermite–Hadamard type,” J. Approxim. Theory, 115, 260–288 (2002).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    S. S. Dragomir and T. M. Rassias (editors), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer, Dordrecht (2002).MATHGoogle Scholar
  14. 14.
    S. S. Dragomir, “Some companions of Ostrowski’s inequality for absolutely continuous functions and applications,” Bull. Korean Math. Soc., 42, No. 2, 213–230 (2005).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    S. S. Dragomir, “A companion of Ostrowski’s inequality for functions of bounded variation and applications,” RGMIA Preprint, 5, Suppl, Article No. 28 (2002) [http://ajmaa.org/RGMIA/papers/v5e/COIFBVApp.pdf].
  16. 16.
    S. S. Dragomir, P. Cerone, and J. Roumeliotis, “A new generalization of Ostrowski integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means,” Appl. Math. Lett., 13, No. 1, 19–25 (2000).MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    S. S. Dragomir, R. P. Agarwal, and P. Cerone, “On Simpson’s inequality and applications,” J. Inequal. Appl., 5, 533–579 (2000).MathSciNetMATHGoogle Scholar
  18. 18.
    S. S. Dragomir and C. E. M. Pearce, “Selected topics on Hermite–Hadamard inequalities and applications,” RGMIA Monographs, Victoria Univ. (2000); online: [http://www.staff.vu.edu.au/RGMIA/monographs/hermite hadamard.html].
  19. 19.
    Z. Liu, “Some companions of an Ostrowski type inequality and applications,” J. Inequal. Pure and Appl. Math., 10, Issue 2, Article 52, (2009).Google Scholar
  20. 20.
    N. Ujević, “A generalization of Ostrowski’s inequality and applications in numerical integration,” Appl. Math. Lett., 17, 133–137 (2004).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • M. W. Alomari
    • 1
  1. 1.Jerash UniversityJerashJordan

Personalised recommendations