Hermite–Hadamard type inequalities for conformable fractional integrals


In this paper, first, we prove an identity for conformable fractional integrals. Second, by using this identity we will present some integral inequalities connected with the left hand side of the Hermite–Hadamard type inequalities for conformable fractional integrals. At the end, applications to some special means and error estimates for the midpoint formula are provided.

This is a preview of subscription content, access via your institution.


  1. 1.

    Adil Khan, M., Khurshid, Y., Ali, T., Rehman, N.: Inequalities for three times differentiable functions. Punjab Univ. J. Math. 48(2), 35–48 (2016)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Abdeljawad, T., Horani, M.A., Khalil, R.: Conformable fractional semigroups of operators. J. Semigr. Theory Appl. 2015, 1–9 (2015)

    Google Scholar 

  4. 4.

    Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Sys. App. 10(2), 109–137 (2015)

    MathSciNet  Google Scholar 

  5. 5.

    Anderson, D.R.: Taylor’s Formula and Integral Inequalities for Conformable Fractional Derivatives. Contributions in Mathematics and Engineering, Springer, New York, pp. 25–44 (2016)

    Chapter  Google Scholar 

  6. 6.

    Anderson, D.R.: Second-order self-adjoint differential equations using a proportional-derivative controller. Commun. Appl. Nonlinear Anal. 24(1), 17–48 (2017)

    MathSciNet  Google Scholar 

  7. 7.

    Bai, R.F., Qi, F., Xi, B.Y.: Hermite–Hadamard type inequalities for the \(m\)-and \(( \alpha, m)\)-logarithmically convex functions. Filomat 27(1), 1–7 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bombardelli, M., Varosanec, S.: Properties of h-convex functions related to the Hermite–Hadamard–Fejer inequalities. Comput. Math. Appl. 58, 1869–1877 (2009)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dragomir, S.S.: Two mappings in connection to Hadamard’s inequality. J. Math. Anal. Appl. 167, 49–56 (1992)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Dragomir, S.S., Agarwal, R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula. Appl. Math. Lett. 11(5), 91–95 (1998)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dragomir, S.S., Fitzpatrick, S.: The Hadamard inequalities for s-convex functions in the second sense. Demonstr. Math. 32(4), 687–696 (1999)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Dragomir, S.S., Mcandrew, A.: Refinment of the Hermite–Hadamard inequality for convex functions. J. Inequal. Pure Appl. Math. 6, 1–6 (2005)

    Google Scholar 

  13. 13.

    Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite-Hadamard Inequalities and Applications. Victoria University, Melbourne (2000)

    Google Scholar 

  14. 14.

    Dragomir, S.S., Pecaric, J., Persson, L.E.: Some inequalities of Hadamard type. Soochow J. Math. 21, 335–341 (1995)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Wu, Y., Qi, F., Niu, D.W.: Integral inequalities of Hermite–Hadamard type for the product of strongly logarithmically convex and other convex functions. Maejo Int. J. Sci. Technol. 9(3), 394–402 (2015)

    Google Scholar 

  16. 16.

    Hadamard, J.: Étude sur les propriétés des fonctions entières et en particulier d̀une fonction considérée par Riemann. J. Math. Pure. Appl. 58, 171–215 (1893)

    MATH  Google Scholar 

  17. 17.

    Hammad, M.A., Khalil, R.: Conformable fractional heat differential equations. Int. J. Pure. Appl. Mth. 94(2), 215–221 (2014)

    MATH  Google Scholar 

  18. 18.

    Hammad, M.A., Khalil, R.: Abel’s formula and wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl. 13(3), 177–183 (2014)

    MATH  Google Scholar 

  19. 19.

    Iyiola, O.S., Nwaeze, E.R.: Some new results on the new conformable fractional calculus with application using D. Alambert approach. Progr. Fract. Differ. Appl. 2(2), 115–122 (2016)

    Article  Google Scholar 

  20. 20.

    Khalil, R., Alhorani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Kirmaci, U.S.: Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147, 137–146 (2004)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Kirmaci, U.S., Bakula, M.K., Ozdemir, M.E., Pecaric, J.: Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193, 26–35 (2007)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Matloka, M.: On some Hadamard-type inequalities for (\(h_{1}\), \(h_{2}\))-preinvex functions on the co-ordinates. J. Inequal. Appl. 2013, 227 (2013)

    Article  Google Scholar 

  24. 24.

    Noor, M.A.: On Hermite–Hadamard integral inequalities for involving two log-preinvex functions. J. Inequal. Pure. Appl. Math. 3, 75–81 (2007)

    MATH  Google Scholar 

  25. 25.

    Noor, M.A., Noor, K.I., Awan, M.U.: Hermite–Hadamard inequalities for relative semi-convex functions and applications. Filomat 28(2), 221–230 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pachpatte, B.G.: On some inequalities for convex functions. RGMIA Res. Rep. Coll. 6(E), 1–9 (2003)

  27. 27.

    Pachpatte, B.G.: Mathematical Inequalities, vol. 67. North-Holland Library, Elsevier Science, Amsterdam, Holland (2005)

    Book  Google Scholar 

  28. 28.

    Sarikaya, M.Z., Saglam, A., Yildrim, H.: On some Hadamard-type inequalities for h-convex functons. J. Math. Inequal. 2, 335–341 (2008)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Modell. 57, 2403–2407 (2013)

    Article  Google Scholar 

  30. 30.

    Set, E., Akdemir, A.O., Mumcu, İ.: Hermite–Hadamard’s Inequality and its Extensions for Conformable Fractional Integrals of Any Order \(\alpha > 0\). (submitted)

  31. 31.

    Set, E., Gözpinar, A., Ekinci, A.: Hermite–Hadamard Type Inequalities via Conformable Fractional Integrals. Acta Math. Univ. Comenian. (2017)

Download references


The authors express their sincere thanks to the referees for careful reading of the manuscript and very helpful suggestions that improved the current manuscript substantially.

Author information



Corresponding author

Correspondence to M. Adil Khan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khan, M.A., Ali, T., Dragomir, S.S. et al. Hermite–Hadamard type inequalities for conformable fractional integrals. RACSAM 112, 1033–1048 (2018). https://doi.org/10.1007/s13398-017-0408-5

Download citation


  • Convex function
  • Hermite–Hadamard inequality
  • Conformable derivative
  • Conformable integrals
  • Special means
  • Midpoint formula

Mathematics Subject Classification

  • 26D15
  • 26A51
  • 26A33
  • 26A42