Mutual bounds for Jensen-type operator inequalities related to higher order convexity

  • Mario KrnićEmail author
  • Rozarija Mikić
  • Josip Pečarić


The main objective of this article is to establish mutual bounds for the Jensen operator inequality related to convex functions of higher order. First we give several mutual bounds for the operator version of the Lah-Ribarič inequality which hold for a class of n-convex functions. By virtue of the established estimates, we then derive several mutual bounds for the Jensen operator inequality which are also related to n-convex functions.


Jensen operator inequality Lah-Ribarič operator inequality n-Convexity Mutual bounds 

Mathematics Subject Classification

Primary 47A63 Secondary 26A51 


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The research of the third author was supported by the Ministry of Education and Science of the Russian Federation (The Agreement No. 02.a03.21.0008.)


  1. 1.
    Agarwal, R.P., Wong, P.J.Y.: Error Inequalities in Polynomial Interpolation and Their Applications. Kluwer Academic Publishers, Dordrecht (1993)CrossRefGoogle Scholar
  2. 2.
    Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)CrossRefGoogle Scholar
  3. 3.
    Hansen, F., Pedersen, G.: Jensen’s operator inequality. Bull. Lond. Math. Soc. 35, 553–564 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hansen, F., Pečarić, J., Perić, I.: Jensen’s operator inequality and its converses. Math. Scand. 100, 61–73 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jakšić, R., Krnić, M., Pečarić, J.: On some new converses of the Jensen and the Lah-Ribarič operator inequality. Hacet. J. Math. Stat. 44, 1045–1055 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Jakšić, R., Krnić, M., Pečarić, J.: More precise estimates for the Jensen operator inequality obtained via the Lah-Ribarič inequality. Appl. Math. Comput. 249, 346–355 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Krnić, M., Mikić, R., Pečarić, J.: Double precision of the Jensen-type operator inequalities for bounded and Lipschitzian functions. Aequat. Math. (2018).
  8. 8.
    Mikić, R., Pečarić, Đ., Pečarić, J.: Inequalities of the Edmundson-Lah-Ribarič Type for \(n\)-Convex Functions with Applications. arXiv:1809.08813 (Submitted for publication)
  9. 9.
    Pečarić, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Academic Press Inc., San Diego (1992)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mario Krnić
    • 1
    Email author
  • Rozarija Mikić
    • 2
  • Josip Pečarić
    • 3
  1. 1.Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia
  2. 2.Faculty of Textile TechnologyUniversity of ZagrebZagrebCroatia
  3. 3.RUDN UniversityMoscowRussia

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