Advertisement

Some New Hermite–Hadamard Type Integral Inequalities via Caputo k–Fractional Derivatives and Their Applications

  • Artion Kashuri
  • Rozana Liko
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 151)

Abstract

The authors discover a general integral identity concerning (n + 1)-differentiable mappings defined on m-invex set via Caputo k-fractional derivatives. By using the notion of generalized ((h1, h2);(η1, η2))-convex mappings and this integral equation as an auxiliary result, we derive some new estimates with respect to Hermite–Hadamard type inequalities via Caputo k-fractional derivatives. It is pointed out that some new special cases can be deduced from main results. At the end, some applications to special means for different positive real numbers are provided as well.

References

  1. 1.
    S.M. Aslani, M.R. Delavar, S.M. Vaezpour, Inequalities of Fejér type related to generalized convex functions with applications. Int. J. Anal. Appl. 16(1), 38–49 (2018)zbMATHGoogle Scholar
  2. 2.
    F. Chen, A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals. Ital. J. Pure Appl. Math. 33, 299–306 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Y.-M. Chu, G.D. Wang, X.H. Zhang, Schur convexity and Hadamard’s inequality. Math. Inequal. Appl. 13(4), 725–731 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Y.-M. Chu, M.A. Khan, T.U. Khan, T. Ali, Generalizations of Hermite-Hadamard type inequalities for MT-convex functions. J. Nonlinear Sci. Appl. 9(5), 4305–4316 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Y.-M. Chu, M.A. Khan, T. Ali, S.S. Dragomir, Inequalities for α-fractional differentiable functions. J. Inequal. Appl. 2017(93), 12 pp. (2017)Google Scholar
  6. 6.
    Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration. Ann. Funct. Anal. 1(1), 51–58 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    M.R. Delavar, M. De La Sen, Some generalizations of Hermite-Hadamard type inequalities. SpringerPlus 5, 1661 (2016)Google Scholar
  8. 8.
    M.R. Delavar, S.S. Dragomir, On η-convexity. Math. Inequal. Appl. 20, 203–216 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    S.S. Dragomir, J. Pečarić, L.E. Persson, Some inequalities of Hadamard type. Soochow J. Math. 21, 335–341 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    T.S. Du, J.G. Liao, Y.J. Li, Properties and integral inequalities of Hadamard-Simpson type for the generalized (s, m)-preinvex functions. J. Nonlinear Sci. Appl. 9, 3112–3126 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    G. Farid, A.U. Rehman, Generalizations of some integral inequalities for fractional integrals. Ann. Math. Sil. 31, 14 pp. (2017)Google Scholar
  12. 12.
    G. Farid, A. Javed, A.U. Rehman, On Hadamard inequalities for n-times differentiable functions which are relative convex via Caputo k-fractional derivatives. Nonlinear Anal. Forum 22, 17–28 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    C. Fulga, V. Preda, Nonlinear programming with φ-preinvex and local φ-preinvex functions. Eur. J. Oper. Res. 192, 737–743 (2009)MathSciNetzbMATHGoogle Scholar
  14. 14.
    M.E. Gordji, S.S. Dragomir, M.R. Delavar, An inequality related to η-convex functions (II). Int. J. Nonlinear Anal. Appl. 6(2), 26–32 (2016)Google Scholar
  15. 15.
    M.E. Gordji, M.R. Delavar, M. De La Sen, On φ-convex functions. J. Math. Inequal. Wiss. 10(1), 173–183 (2016)zbMATHGoogle Scholar
  16. 16.
    A. Iqbal, M.A. Khan, S. Ullah, Y.-M. Chu, A. Kashuri, Hermite-Hadamard type inequalities pertaining conformable fractional integrals and their applications. AIP Adv. 8(7), 18 pp. (2018)Google Scholar
  17. 17.
    A. Kashuri, R. Liko, On Hermite-Hadamard type inequalities for generalized (s, m, φ)-preinvex functions via k-fractional integrals. Adv. Inequal. Appl. 6, 1–12 (2017)zbMATHGoogle Scholar
  18. 18.
    A. Kashuri, R. Liko, Generalizations of Hermite-Hadamard and Ostrowski type inequalities for MT m-preinvex functions. Proyecciones 36(1), 45–80 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for generalized (r; s, m, φ)-preinvex functions. Eur. J. Pure Appl. Math. 10(3), 495–505 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for twice differentiable generalized (s, m, φ)-preinvex functions. Konuralp J. Math. 5(2), 228–238 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    A. Kashuri, R. Liko, Hermite-Hadamard type inequalities for generalized (s, m, φ)-preinvex functions via k-fractional integrals. Tbil. Math. J. 10(4), 73–82 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for MT (m,φ)-preinvex functions. Stud. Univ. Babeş-Bolyai Math. 62(4), 439–450 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for twice differentiable generalized beta-preinvex functions. J. Fract. Calc. Appl. 9(1), 241–252 (2018)MathSciNetzbMATHGoogle Scholar
  24. 24.
    A. Kashuri, R. Liko, Some different type integral inequalities pertaining generalized relative semi-m-(r; h 1, h 2)-preinvex mappings and their applications. Electron. J. Math. Anal. Appl. 7(1), 351–373 (2019)zbMATHGoogle Scholar
  25. 25.
    M.A. Khan, Y. Khurshid, T. Ali, N. Rehman, Inequalities for three times differentiable functions. J. Math. Punjab Univ. 48(2), 35–48 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    M.A. Khan, T. Ali, S.S. Dragomir, M.Z. Sarikaya, Hermite-Hadamard type inequalities for conformable fractional integrals. Rev. Real Acad. Cienc. Exact. Fís. Natur. A Mat. (2017). https://doi.org/10.1007/s13398-017-0408-5 MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M.A. Khan, Y.-M. Chu, T.U. Khan, J. Khan, Some new inequalities of Hermite-Hadamard type for s-convex functions with applications. Open Math. 15, 1414–1430 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    M.A. Khan, Y. Khurshid, T. Ali, Hermite-Hadamard inequality for fractional integrals via η-convex functions. Acta Math. Univ. Comenian. 79(1), 153–164 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    M.A. Khan, Y.-M. Chu, A. Kashuri, R. Liko, G. Ali, New Hermite-Hadamard inequalities for conformable fractional integrals. J. Funct. Spaces 2018, 9 pp. (2018). Article ID 6928130Google Scholar
  30. 30.
    M.A. Khan, Y.-M. Chu, A. Kashuri, R. Liko, Hermite-Hadamard type fractional integral inequalities for MT (r;g,m,φ)-preinvex functions. J. Comput. Anal. Appl. 26(8), 1487–1503 (2019)Google Scholar
  31. 31.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204 (Elsevier, New York, 2006)zbMATHCrossRefGoogle Scholar
  32. 32.
    W. Liu, W. Wen, J. Park, Ostrowski type fractional integral inequalities for MT-convex functions. Miskolc Math. Notes 16(1), 249–256 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    W. Liu, W. Wen, J. Park, Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals. J. Nonlinear Sci. Appl. 9, 766–777 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    C. Luo, T.S. Du, M.A. Khan, A. Kashuri, Y. Shen, Some k-fractional integrals inequalities through generalized λ ϕm-MT-preinvexity. J. Comput. Anal. Appl. 27(4), 690–705 (2019)Google Scholar
  35. 35.
    M. Matłoka, Inequalities for h-preinvex functions. Appl. Math. Comput. 234, 52–57 (2014)MathSciNetzbMATHGoogle Scholar
  36. 36.
    S. Mubeen, G.M. Habibullah, k-Fractional integrals and applications. Int. J. Contemp. Math. Sci. 7, 89–94 (2012)Google Scholar
  37. 37.
    M.A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2, 126–131 (2007)MathSciNetzbMATHGoogle Scholar
  38. 38.
    M.A. Noor, K.I. Noor, M.U. Awan, S. Khan, Hermite-Hadamard inequalities for s-Godunova-Levin preinvex functions. J. Adv. Math. Stud. 7(2), 12–19 (2014)MathSciNetzbMATHGoogle Scholar
  39. 39.
    O. Omotoyinbo, A. Mogbodemu, Some new Hermite-Hadamard integral inequalities for convex functions. Int. J. Sci. Innov. Technol. 1(1), 1–12 (2014)Google Scholar
  40. 40.
    C. Peng, C. Zhou, T.S. Du, Riemann-Liouville fractional Simpson’s inequalities through generalized (m, h 1, h 2)-preinvexity. Ital. J. Pure Appl. Math. 38, 345–367 (2017)MathSciNetzbMATHGoogle Scholar
  41. 41.
    R. Pini, Invexity and generalized convexity. Optimization 22, 513–525 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    E. Set, Some new generalized Hermite-Hadamard type inequalities for twice differentiable functions (2017). https://www.researchgate.net/publication/327601181 zbMATHGoogle Scholar
  43. 43.
    E. Set, S.S. Karataş, M.A. Khan, Hermite-Hadamard type inequalities obtained via fractional integral for differentiable m-convex and (α, m)-convex functions. Int. J. Anal. 2016, 8 pp. (2016). Article ID 4765691Google Scholar
  44. 44.
    E. Set, A. Gözpinar, J. Choi, Hermite-Hadamard type inequalities for twice differentiable m-convex functions via conformable fractional integrals. Far East J. Math. Sci. 101(4), 873–891 (2017)zbMATHGoogle Scholar
  45. 45.
    E. Set, M.Z. Sarikaya, A. Gözpinar, Some Hermite-Hadamard type inequalities for convex functions via conformable fractional integrals and related inequalities. Creat. Math. Inform. 26(2), 221–229 (2017)MathSciNetzbMATHGoogle Scholar
  46. 46.
    H.N. Shi, Two Schur-convex functions related to Hadamard-type integral inequalities. Publ. Math. Debrecen 78(2), 393–403 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    M. Tunç, E. Göv, Ü. Şanal, On tgs-convex function and their inequalities. Facta Univ. Ser. Math. Inform. 30(5), 679–691 (2015)MathSciNetzbMATHGoogle Scholar
  48. 48.
    S. Varošanec, On h-convexity. J. Math. Anal. Appl. 326(1), 303–311 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Y. Wang, S.H. Wang, F. Qi, Simpson type integral inequalities in which the power of the absolute value of the first derivative of the integrand is s-preinvex. Facta Univ. Ser. Math. Inform. 28(2), 151–159 (2013)MathSciNetzbMATHGoogle Scholar
  50. 50.
    H. Wang, T.S. Du, Y. Zhang, k-fractional integral trapezium-like inequalities through (h, m)-convex and (α, m)-convex mappings. J. Inequal. Appl. 2017(311), 20 pp. (2017)Google Scholar
  51. 51.
    T. Weir, B. Mond, Preinvex functions in multiple objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    X.M. Zhang, Y.-M. Chu, X.H. Zhang, The Hermite-Hadamard type inequality of GA-convex functions and its applications. J. Inequal. Appl. 2010, 11 pp. (2010). Article ID 507560Google Scholar
  53. 53.
    Y. Zhang, T.S. Du, H. Wang, Y.J. Shen, A. Kashuri, Extensions of different type parameterized inequalities for generalized (m, h)-preinvex mappings via k-fractional integrals. J. Inequal. Appl. 2018(49), 30 pp. (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Artion Kashuri
    • 1
  • Rozana Liko
    • 1
  1. 1.Department of Mathematics, Faculty of Technical ScienceUniversity Ismail Qemali of VloraVlorëAlbania

Personalised recommendations