Levinson type inequalitis and their extensions via convexity on time scales

  • S. H. Saker
  • M. M. Osman
  • D. O’Regan
  • R. P. AgarwalEmail author
Original Paper


In this paper, we prove some new inequalities of Levinson-type on time scales. Also we will prove some new extensions of these inequalities via convexity.


Hardy’s inequality Jensen’s inequality Levinson’s inequality Convex functions Time scales 

Mathematics Subject Classification

26A15 26D10 26D15 39A13 34A40 34N05 


  1. 1.
    Agarwal, R.P., O’Regan, D., Saker, S.H.: Dynamic Inequalities on Time Scales. Springer, Cham (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bennett, G.: Some elementary inequalities. Q. J. Math. Oxf 38(2), 401–425 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bohner, M., Nosheen, A., Pečarić, J., Younus, A.: Some Dynamic Hardy-type inequalities with general kernel. J. Math. Inequal. 8(1), 185–199 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Čižmešija, A., Pečarić, J., Persson, L.-E.: On strenghtened Hardy and Pólya-Knopp’s inequalities. J. Approx. Theory 125, 74–83 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Copson, E.T.: Note on series of positive terms. J. Lond. Math. Soc. 2, 9–12 (1927)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Copson, E.T.: Note on series of positive terms. J. Lond. Math. Soc. 3, 49–51 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Copson, E.T.: Some integral inequalities. Proc. R. Soc. Edinb. 75A(13), 157–163 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hardy, G.H.: Note on a theorem of Hilbert. Mathematische Z. 6, 314–317 (1920)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hardy, G.H.: Notes on some points in the integral calculus (LX): an inequality between integrals. Messenger Math. 54, 150–156 (1925)Google Scholar
  12. 12.
    Hardy, G.H.: Notes on some points in the integral calculus (LXIV). Messenger Math. 57, 12–16 (1928)Google Scholar
  13. 13.
    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)zbMATHGoogle Scholar
  14. 14.
    Kaijser, S., Persson, L.-E., Öberg, A.: On Carleman and Knopp’s inequalities. J. Approx. Theory 117, 140–151 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaijser, S., Nikolova, L., Persson, L.-E., Wedestig, A.: Hardy-type inequalities via convexity. Math. Inequal. App. 8, 403–417 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Krantz, S.G.: Jensen’s inequality, Handbook of Complex Variables, p. 118. Birkhauser, Boston (1999)CrossRefGoogle Scholar
  17. 17.
    Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific Publishing Co., London (2003)CrossRefzbMATHGoogle Scholar
  18. 18.
    Levinson, N.: Generalization of an inequality of Hardy. Duke Math. J. 31, 389–393 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Opic, B., Kufner, A.: Hardy-Type Inequalities, Pitman Research Notes in Mathematics, vol. 219. Longman Scientific and Technical, Harlow (1990)zbMATHGoogle Scholar
  20. 20.
    Oguntuase, J.A., Persson, L.-E., Essel, E.K.: Multidimensional Hardy-type inequalities with general kernels. J. Math. Anal. Appl. 348(1), 411–418 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Oguntuase, J.A., Persson, L.-E.: Time scales Hardy-type inequalities via superquadracity. Ann. Funct. Anal. 5(2), 61–73 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Özkan, U.M., Yildirim, H.: Hardy–Knopp-type inequalities on time scales. Dyn. Syst. Appl. 17, 477–486 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Özkan, U.M., Yildirim, H.: Time scale Hardy–Knopp type integral inequalities. Commun. Math. Anal. 6(1), 36–41 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pachpatte, B.G.: A generalization of an inequality of Hardy. Indian J. pure appl. Math. 21(7), 617–620 (1990)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Pachpatte, B.G.: Mathematical Inequalities, North-Holland Mathematical Library, vol. 67. Elsevier, Amsterdam (2005)Google Scholar
  26. 26.
    Řehák, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 5, 495–507 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Saker, S.H., Graef, J.: A new class of dynamic inequalities of Hardy’s type on time scales. Dyn. Syst. Appl. 23, 83–100 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Saker, S.H., O’Regan, D.: Hardy and Littlewood inequalities on time scales, Bull. Malays. Math. Sci. Soc. doi: 10.1007/s40840-015-0300-4
  29. 29.
    Saker, S.H., O’Regan, D., Agarwal, R.P.: Dynamic inequalities of Hardy and Copson types on time scales. Anal.: Int. Math. J. Anal. Appl. 34(4), 391–402 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Saker, S.H., O’Regan, D., Agarwal, R.P.: Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales. Math. Nachr. 287(5–6), 687–698 (2014)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Saker, S.H., O’Regan, D., Agarwal, R.P.: Littlewood and Bennett inequalities on time scales. Mediterr. J. Math. 8, 1–15 (2014)zbMATHGoogle Scholar
  32. 32.
    Wong, F.H., Yeh, C.C., Lian, W.C.: An extension of Jensen’s inequality on time scales. Adv. Dyn. Syst. Appl. 1(1), 113–120 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2017

Authors and Affiliations

  • S. H. Saker
    • 1
  • M. M. Osman
    • 1
  • D. O’Regan
    • 2
  • R. P. Agarwal
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt
  2. 2.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland
  3. 3.Department of MathematicsTexas A and M University-KingsvilleKingsvilleUSA

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