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Fractional inequalities of the Hermite–Hadamard type for $ m $-polynomial convex and harmonically convex functions

  • Received: 21 October 2020 Accepted: 27 November 2020 Published: 03 December 2020
  • MSC : 26D15, 26A51, 26D10

  • In this paper, it is our purpose to establish some new fractional inequalities of the Hermite–Hadamard type for the $ m $-polynomial convex and harmonically convex functions. Our results involve the Caputo–Fabrizio and $ \zeta $-Riemann–Liouville fractional integral operators. They generalize, complement and extend existing results in the literature. By taking $ m\geq 2 $, we deduce loads of new and interesting inequalities. We expect that the thought laid out in this work will provoke advance examinations in this course.

    Citation: Eze R. Nwaeze, Muhammad Adil Khan, Ali Ahmadian, Mohammad Nazir Ahmad, Ahmad Kamil Mahmood. Fractional inequalities of the Hermite–Hadamard type for $ m $-polynomial convex and harmonically convex functions[J]. AIMS Mathematics, 2021, 6(2): 1889-1904. doi: 10.3934/math.2021115

    Related Papers:

  • In this paper, it is our purpose to establish some new fractional inequalities of the Hermite–Hadamard type for the $ m $-polynomial convex and harmonically convex functions. Our results involve the Caputo–Fabrizio and $ \zeta $-Riemann–Liouville fractional integral operators. They generalize, complement and extend existing results in the literature. By taking $ m\geq 2 $, we deduce loads of new and interesting inequalities. We expect that the thought laid out in this work will provoke advance examinations in this course.


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    [1] T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27. doi: 10.1016/S0034-4877(17)30059-9
    [2] M. U. Awan, N. Akhtar, S. Iftikhar, M. A. Noor, Y. M. Chu, New Hermite-Hadamard type inequalities for $n$-polynomial harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1-12. doi: 10.1186/s13660-019-2265-6
    [3] Y. M. Chu, M. Adil Khan, T. U. Khan, T. Ali, Generalizations of Hermite-Hadamard type inequalities for $MT$-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 4305-4316. doi: 10.22436/jnsa.009.06.72
    [4] M. Adil 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 (2017), 1414-1430. doi: 10.1515/math-2017-0121
    [5] M. R. Delavar, M. De La Sen, Some generalizations of Hermite-Hadamard type inequalities, SpringerPlus, 5 (2016), 1-9.
    [6] A. Guessab, G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115 (2002), 260-288. doi: 10.1006/jath.2001.3658
    [7] M. Gürbüz, A. O. Akdemir, S. Rashid, E. Set, Hermite-Hadamard inequality for fractional integrals of Caputo-Fabrizio type and related inequalities, J. Inequal. Appl., 2020 (2020), 1-10. doi: 10.1186/s13660-019-2265-6
    [8] İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.
    [9] İ. İşcan, S. Wu, Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237-244.
    [10] A. Iqbal, M. Adil Khan, S. Ullah, Y. M. Chu, Some new Hermite-Hadamard type inequalities associated with conformable fractional integrals and their applications, J. Funct. Space., 2020 (2020), 1-18.
    [11] A. Iqbal, M. Adil Khan, N. Mohammad, E. R. Nwaeze, Revisiting the Hermite-Hadamard fractional integral inequality via a Green function, AIMS Math., 5 (2020), 6087-6107. doi: 10.3934/math.2020391
    [12] A. Iqbal, M. Adil Khan, M. Suleman, Y. M. Chu, The right Riemann-Liouville fractional Hermite-Hadamard type inequalities derived from Green's function, AIP Adv., 10 (2020), 1-10.
    [13] M. Adil Khan, Y. Khurshid, T. Ali, Hermite-Hadamard Inequality for fractional integrals via $\eta$-convex functions, Acta Math. Univ. Comenianae., 86 (2017), 153-164.
    [14] M. Adil Khan, T. Ali, T. U. Khan, Hermite-Hadamard Type Inequalities with Applications, Fasciculi Mathematici, 59 (2017), 57-74. doi: 10.1515/fascmath-2017-0017
    [15] M. Adil Khan, N. Mohammad, E. R. Nwaeze, Y. M. Chu, Quantum Hermite-Hadamard inequality by means of a green function, Adv. Diff. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-019-2438-0
    [16] T. U. Khan, M. Adil Khan, Hermite-Hadamard inequality for new generalized conformable fractional operators, AIMS Math., 6 (2020), 23-38.
    [17] P. O. Mohammed, I. Brevik, A new version of the Hermite-Hadamard inequality for Riemann-Liouville fractional integrals, Symmetry, 12 (2020), 1-11.
    [18] P. O. Mohammed, M. Z. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math., 372 (2020), 1-15.
    [19] P. O. Mohammed, New integral inequalities for preinvex functions via generalized beta function, J. Interdiscip. Math., 22 (2019), 539-549. doi: 10.1080/09720502.2019.1643552
    [20] A. Fernandez, P. O. Mohammed, Hermite-Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels, Math. Meth. Appl. Sci., (2020), 1-18.
    [21] S. Mubeen, G. M. Habibullah, k-Fractional Integrals and Aplication, Int. J. Contemp. Math. Sci., 7 (2012), 89-94.
    [22] E. R. Nwaeze, Inequalities of the Hermite-Hadamard type for Quasi-convex functions via the $(k, s)$-Riemann-Liouville fractional integrals, Fractional Differ. Calc., 8 (2018), 327-336.
    [23] E. R. Nwaeze, D. F. M. Torres, Novel results on the Hermite-Hadamard kind inequality for $\eta$-convex functions by means of the $(k, r)$-fractional integral operators. In: S. S. Dragomir, P. Agarwal, M. Jleli, B, Samet (eds.) Advances in Mathematical Inequalities and Applications (AMIA), Trends in Mathematics, Birkhäuser, Singapore, 2018,311-321.
    [24] E. R. Nwaeze, M. Adil Khan, Y. M. Chu, Fractional inclusions of the Hermite-Hadamard type for $m$-polynomial convex interval-valued functions, Adv. Diff. Equ., 2020 (2020), 1-17. doi: 10.1186/s13662-019-2438-0
    [25] G. Rahman, K. S. Nisar, S. Rashid, T. Abdeljawad, Certain Gruss-type inequalities via tempered fractional integrals concerning another function, J. Inequal. Appl., 2020 (2020), 1-18. doi: 10.1186/s13660-019-2265-6
    [26] J. Sun, B. Y. Xi, F. Qi, Some new inequalities of the Hermite-Hadamard type for extended $s$-convex functions, J. Comput. Anal. Appl., 26 (2019), 985-996.
    [27] S. Salahshour, A. Ahmadian, C. S. Chan, Successive approximation method for Caputo q-fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 153-158. doi: 10.1016/j.cnsns.2014.12.014
    [28] M. R. B. Shahriyar, F. Ismail, S. Aghabeigi, A. Ahmadian, S. Salahshour, An eigenvalue-eigenvector method for solving a system of fractional differential equations with uncertainty, Math. Probl. Eng., 2013 (2013), 1-10.
    [29] A. Ahmadian, S. Salahshour, M. Ali-Akbari, F. Ismail, D. Baleanu, A novel approach to approximate fractional derivative with uncertain conditions, Chaos, Solitons Fractals, 104 (2017), 68-76. doi: 10.1016/j.chaos.2017.07.026
    [30] A. Ahmadian, C. S. Chan, S. Salahshour, V. Vembarasan, FTFBE: A numerical approximation for fuzzy time-fractional Bloch equation, IEEE international conference on fuzzy systems, 2014,418-423.
    [31] T. Toplu, M. Kadakal, İ. İşcan, On $n$-Polynomial convexity and some related inequalities, AIMS Math., 5 (2020), 1304-1318.
    [32] S. S. Zhou, S. Rashid, M. A. Noor, F. Safdar, New Hermite-Hadamard type inequalities for exponentially convex functions and applications, AIMS Math., 5 (2020), 6874-6901. doi: 10.3934/math.2020441
    [33] S. S. Zhou, S. Rashid, F. Jarad, H. Kalsoom, New estimates considering the generalized proportional Hadamard fractional integral operators, Adv. Diff. Equ., 2020 (2020), 1-15. doi: 10.1186/s13662-019-2438-0
    [34] S. S. Zhou, S. Rashid, S. S. Dragomir, M. A. Latif, Some New inequalities involving $k$-fractional integral for certain classes of functions and their applications, J. Funct. Space., 2020 (2020), 1-14.
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