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A Survey of Perturbed Ostrowski Type Inequalities

  • Silvestru Sever Dragomir
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Part of the Springer Optimization and Its Applications book series (SOIA, volume 111)

Abstract

In this paper we survey a number of recent perturbed versions of Ostrowski inequality that have been obtained by the author and provide their connections with numerous classical results of interest.

Keywords

Ostrowski inequality Lebesgue integral Integral mean Integral inequalities 
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References

  1. 1.
    Apostol, T.M.: Mathematical Analysis, 2nd edn. Addison-Wesley, Reading (1975)Google Scholar
  2. 2.
    Dragomir, S.S.: On the Ostrowski’s integral inequality for mappings with bounded variation and applications. Math. Inequal. Appl. 4(1), 33–40 (2001). Preprint, RGMIA Res. Rep. Collect. 2(1), Article 7 (1999). http://rgmia.org/papers/v2n1/v2n1-7.pdf
  3. 3.
    Dragomir, S.S.: A refinement of Ostrowski’s inequality for absolutely continuous functions and applications. Acta Math. Vietnam. 27(2), 203–217 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dragomir, S.S.: A refinement of Ostrowski’s inequality for absolutely continuous functions whose derivatives belong to \(L_{\infty }\) and applications. Lib. Math. 22, 49–63 (2002)zbMATHGoogle Scholar
  5. 5.
    Dragomir, S.S.: An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3(2), Article 31, 8 pp. (2002)Google Scholar
  6. 6.
    Dragomir, S.S.: An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3(3), Article 35, 8 pp. (2002)Google Scholar
  7. 7.
    Dragomir, S.S.: Improvements of Ostrowski and generalised trapezoid inequality in terms of the upper and lower bounds of the first derivative. Tamkang J. Math. 34(3), 213–222 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dragomir, S.S.: Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation. Arch. Math. 91(5), 450–460 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dragomir, S.S.: A functional generalization of Ostrowski inequality via Montgomery identity. Preprint, Acta Math. Univ. Comenian. (2015), 84(1), 63–78. RGMIA Res. Rep. Collect. 16, Article 65, pp. 15 (2013). http://rgmia.org/papers/v16/v16a65.pdf
  10. 10.
    Dragomir, S.S.: Some perturbed Ostrowski type inequalities for functions of bounded variation. Preprint, Asian-Eur. J. Math. (2015), 8(4), 1550069, pp. 14 RGMIA Res. Rep. Collect. 16, Article 93 (2013). http://rgmia.org/papers/v16/v16a93.pdf
  11. 11.
    Dragomir, S.S.: Some perturbed Ostrowski type inequalities for absolutely continuous functions (I). Theory Appl. Math. Comput. Sci. (2015), 5(2), 132–147. Preprint, RGMIA Res. Rep. Collect. 16, Article 94 (2013). http://rgmia.org/papers/v16/v16a94.pdf
  12. 12.
    Dragomir, S.S.: Some perturbed Ostrowski type inequalities for absolutely continuous functions (II). Acta Univ. Apulensis Math. Inform. (43) (2015), 209–228. Preprint, RGMIA Res. Rep. Collect. 16, Article 95 (2013)Google Scholar
  13. 13.
    Dragomir, S.S.: Some perturbed Ostrowski type inequalities for absolutely continuous functions (III). Transylv. J. Math. Mech. (2015), 7(1), 31–43. Preprint RGMIA Res. Rep. Collect. 16, Article 96 (2013). http://rgmia.org/papers/v16/v16a96.pdf
  14. 14.
    Dragomir, S.S., Rassias, Th.M. (eds.): Ostrowski Type Inequalities and Applications in Numerical Integration. Springer, Dordrecht (2002). Print ISBN 978-90-481-5990-1Google Scholar
  15. 15.
    Dragomir, S.S., Wang, S.: A new inequality of Ostrowski’s type in L 1 norm and applications to some special means and to some numerical quadrature rules. Tamkang J. Math. 28, 239–244 (1997)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dragomir, S.S., Wang, S.: A new inequality of Ostrowski’s type in L p norm and applications to some special means and to some numerical quadrature rules. Indian J. Math. 40(3), 299–304 (1998)MathSciNetGoogle Scholar
  17. 17.
    Dragomir, S.S., Wang, S.: Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and for some numerical quadrature rules. Appl. Math. Lett. 11(1), 105–109 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fink, A.M.: Bounds on the derivative of a function from its averages. Czechoslov. Math. J. 42(117), 289–310 (1992)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Milovanović, G.V., Rassias, M.Th. (eds.): Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York (2014)zbMATHGoogle Scholar
  20. 20.
    Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Inequalities for Functions and Their Integrals and Derivatives. Kluwer Academic, Dordrecht (1994)zbMATHGoogle Scholar
  21. 21.
    Ostrowski, A.: Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Peachey, T.C., Mcandrew, A., Dragomir, S.S.: The best constant in an inequality of Ostrowski type. Tamkang J. Math. 30(3), 219–222 (1999)MathSciNetzbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Silvestru Sever Dragomir
    • 1
    • 2
  1. 1.Mathematics, College of Engineering & ScienceVictoria UniversityMelbourneAustralia
  2. 2.School of Computer Science & Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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