Arabian Journal of Mathematics

, Volume 7, Issue 2, pp 77–90 | Cite as

Weighted majorization inequalities for n-convex functions via extension of Montgomery identity using Green function

  • Andrea Aglić Aljinović
  • Asif R. Khan
  • Josip E. Pečarić
Open Access


New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using Čebyšev type inequalities. Mean value theorems are also discussed for functional related to new results.

Mathematics Subject Classification

26A51 26D15 26D20 



  1. 1.
    Aglić Aljinović, A.; Pečarić, J.; Vukelić, A.: On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula II. Tamkang J. Math. 36(4), 279–301 (2005)MathSciNetMATHGoogle Scholar
  2. 2.
    Aglić Aljinović, A.; Khan, A.R.; Pečarić, J.E.: Weighted majorization theorems via generalization of Taylor’s formula. J. Inequal. Appl. 2015(196), 1–22 (2015)MathSciNetMATHGoogle Scholar
  3. 3.
    Cerone, P.; Dragomir, S.S.: Some new Owstrowski-type bounds for the Čebyšev functional and applications. J. Math. Inequal. 8(1), 159–170 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fuchs, L.: A new proof of an inequality of Hardy–Littlewood–Polya. Mat. Tidsskr B, 53–54 (1947)MATHGoogle Scholar
  5. 5.
    Hardy, G.H.; Littlewood, J.E.; Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1978)MATHGoogle Scholar
  6. 6.
    Jakšetić, J.; Pečarić, J.: Exponential convexity method. J. Convex Anal. 20(1), 181–197 (2013)MathSciNetMATHGoogle Scholar
  7. 7.
    Jensen, J.L.W.V.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30, 175–193 (1906)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Khan, A.R.; Pecaric, J.; Praljak, M.: Popoviciu type inequalities for \(n\) convex functions via extension of Montgomery identity. Analele Stiintifice ale Universitatii Ovidius Constanta (to appear) Google Scholar
  9. 9.
    Khan, A.R.; Latif, N.; Pečarić, J.E.: Exponential convexity for majorization. J. Inequal. Appl. 2012(105), 1–13 (2012)MathSciNetMATHGoogle Scholar
  10. 10.
    Khan, A.R.; Pečarić, J.E.; Varošanec, S.: Popoviciu type characterization of positivity of sums and integrals for convex functions of higher order. J. Math. Inequal. 7(2), 195–212 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Marshall, A.W.; Olkin, I.; Arnold, B.C.: Inequalities: Theory of Majorization and its Applications, 2nd edn. Springer, New York (2011)CrossRefMATHGoogle Scholar
  12. 12.
    Mitrinović, D.S.: Analytic Inequalities. Springer, Berlin (1970). (In cooperation with P. M. Vasic)CrossRefMATHGoogle Scholar
  13. 13.
    Mitrinović, D.S.; Pečarić, J.E.; Fink, A.M.: Inequalities for Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht (1994)MATHGoogle Scholar
  14. 14.
    Niculescu, C.P.; Persson, L.E.: Convex Functions and Their Applications. A Contemporary Approach, CMS Books in Mathematics, vol. 23. Springer, New York (2006)CrossRefMATHGoogle Scholar
  15. 15.
    Pečarić, J.: On the Čebyšev inequality. Bul. Inst. Politehn. Timisoara 25(39), 10–11 (1980)MATHGoogle Scholar
  16. 16.
    Pečarić, J.E.: On some inequalities for functions with nondecreasing increments. J. Math. Anal. Appl. 98(1), 188–197 (1984)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pečarić, J.: On Jessenâs inequality for convex functions, III. J. Math. Anal. Appl. 156, 231–239 (1991)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pečarić, J.; Perić, J.: Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means. An. Univ. Craiova Ser. Mat. Inform. 39(1), 65–75 (2012)MathSciNetMATHGoogle Scholar
  19. 19.
    Pečarić, J.E.; Proschan, F.; Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Academic Press, New York (1992)MATHGoogle Scholar
  20. 20.
    Popoviciu, T.: Notes sur les fonctions convexes d’orde superieur III. Mathematica (Cluj) 16, 74–86 (1940)MathSciNetMATHGoogle Scholar
  21. 21.
    Popoviciu, T.: Notes sur les fonctions convexes d’orde superieur IV. Disqusitiones Math. 1, 163–171 (1940)Google Scholar
  22. 22.
    Popoviciu, T.: Les fonctions convexes. Herman and Cie, Editeurs, Paris (1944)MATHGoogle Scholar
  23. 23.
    Robert, A.W.; Varberg, D.E.: Convex Functions. Academic Press, New York (1973)MATHGoogle Scholar
  24. 24.
    Widder, D.V.: Completely convex function and Lidstone series. Trans. Am. Math. Soc. 51, 387–398 (1942)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia
  2. 2.Department of Mathematics, Faculty of ScienceUniversity of KarachiKarachiPakistan
  3. 3.Faculty of Textile TechnologyUniversity of ZagrebZagrebCroatia

Personalised recommendations