Research article

Revisiting the Hermite-Hadamard fractional integral inequality via a Green function

  • Received: 11 May 2020 Accepted: 20 July 2020 Published: 28 July 2020
  • MSC : 26A51, 26D15, 26E60, 41A55

  • The Hermite-Hadamard inequality by means of the Riemann-Liouville fractional integral operators is already known in the literature. In this paper, it is our purpose to reconstruct this inequality via a relatively new method called the green function technique. In the process, some identities are established. Using these identities, we obtain loads of new results for functions whose second derivative is convex, monotone and concave in absolute value. We anticipate that the method outlined in this article will stimulate further investigation in this direction.

    Citation: Arshad Iqbal, Muhammad Adil Khan, Noor Mohammad, Eze R. Nwaeze, Yu-Ming Chu. Revisiting the Hermite-Hadamard fractional integral inequality via a Green function[J]. AIMS Mathematics, 2020, 5(6): 6087-6107. doi: 10.3934/math.2020391

    Related Papers:

  • The Hermite-Hadamard inequality by means of the Riemann-Liouville fractional integral operators is already known in the literature. In this paper, it is our purpose to reconstruct this inequality via a relatively new method called the green function technique. In the process, some identities are established. Using these identities, we obtain loads of new results for functions whose second derivative is convex, monotone and concave in absolute value. We anticipate that the method outlined in this article will stimulate further investigation in this direction.


    加载中


    [1] M. A. Khan, S. H. Wu, H. Ullah, et al. Discrete majorization type inequalities for convex functions on rectangles, J. Inequal. Appl., 2019 (2019), 1-18. doi: 10.1186/s13660-019-1955-4
    [2] S. Z. Ullah, M. A. Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019), 1-10. doi: 10.1186/s13660-019-1955-4
    [3] T. H. Zhao, L. Shi, Y. M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM, 114 (2020), 1-14. doi: 10.1007/s13398-019-00732-2
    [4] M. K. Wang, H. H. Chu, Y. M. Li, et al. Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind, Appl. Anal. Discrete Math., 14 (2020), 255-271.
    [5] P. Agarwal, M. Kadakal, İ. İşcan, et al. Better approaches for n-times differentiable convex functions, Mathematics, 8 (2020), 1-11.
    [6] S. Khan, M. A. Khan, Y. M. Chu, Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Method. Appl. Sci., 43 (2020), 2577-2587. doi: 10.1002/mma.6066
    [7] S. Rafeeq, H. Kalsoom, S. Hussain, et al. Delay dynamic double integral inequalities on time scales with applications, Adv. Differ. Equ., 2020 (2020), 1-32. doi: 10.1186/s13662-019-2438-0
    [8] M. A. Khan, J. Pečarić, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Mathematics, 5 (2020), 4931-4945. doi: 10.3934/math.2020315
    [9] M. K. Wang, Z. Y. He, Y. M. Chu, Sharp power mean inequalities for the generalized elliptic integral of the first kind, Comput. Meth. Funct. Th., 20 (2020), 111-124. doi: 10.1007/s40315-020-00298-w
    [10] S. Rashid, M. A. Noor, K. I. Noor, et al. Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions, AIMS Mathematics, 5 (2020), 2629-2645. doi: 10.3934/math.2020171
    [11] T. H. Zhao, M. K. Wang, Y. M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Mathematics, 5 (2020), 4512-4528. doi: 10.3934/math.2020290
    [12] Z. H. Yang, W. M. Qian, W. Zhang, et al. Notes on the complete elliptic integral of the first kind, Math. Inequal. Appl., 23 (2020), 77-93.
    [13] B. Wang, C. L. Luo, S. H. Li, et al. Sharp one-parameter geometric and quadratic means bounds for the Sándor-Yang means, RACSAM, 114 (2020), 1-10. doi: 10.1007/s13398-019-00732-2
    [14] M. K. Wang, M. Y. Hong, Y. F. Xu, et al. Inequalities for generalized trigonometric and hyperbolic functions with one parameter, J. Math. Inequal., 14 (2020), 1-21.
    [15] M. K. Wang, H. H. Chu, Y. M. Chu, Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals, J. Math. Anal. Appl., 480 (2019), 1-9.
    [16] W. M. Qian, W. Zhang, Y. M. Chu, Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means, Miskolc Math. Notes, 20 (2019), 1157-1166. doi: 10.18514/MMN.2019.2334
    [17] S. S. Zhou, S. Rashid, F. Jarad, et al. New estimates considering the generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ., 2020 (2020), 1-15. doi: 10.1186/s13662-019-2438-0
    [18] S. Rashid, F. Jarad, Y. M. Chu, A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function, Math. Probl. Eng., 2020 (2020), 1-12.
    [19] S. Rashid, F. Jarad, H. Kalsoom, et al. On Pólya-Szegö and Ćebyšev type inequalities via generalized k-fractional integrals, Adv. Differ. Equ., 2020 (2020), 1-18. doi: 10.1186/s13662-019-2438-0
    [20] Y. Khurshid, M. Adil Khan, Y. M. Chu, Conformable fractional integral inequalities for GG- and GA-convex function, AIMS Mathematics, 5 (2020), 5012-5030. doi: 10.3934/math.2020322
    [21] I. Abbas Baloch, Y. M. Chu, Petrović-type inequalities for harmonic h-convex functions, J. Funct. Space., 2020 (2020), 1-7.
    [22] M. A. Latif, S. Rashid, S. S. Dragomir, et al. Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl., 2019 (2019), 1-33. doi: 10.1186/s13660-019-1955-4
    [23] M. U. Awan, N. Akhtar, A. Kashuri, et. al. 2D approximately reciprocal ρ-convex functions and associated integral inequalities, AIMS Mathematics, 5 (2020), 4662-4680. doi: 10.3934/math.2020299
    [24] S. Rashid, R. Ashraf, M. A. Noor, et al. New weighted generalizations for differentiable exponentially convex mapping with application, AIMS Mathematics, 5 (2020), 3525-3546. doi: 10.3934/math.2020229
    [25] M. U. Awan, N. Akhtar, S. Iftikhar, et al. 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
    [26] S. Rashid, İ. İşcan, D. Baleanu, et al. Generation of new fractional inequalities via n polynomials s-type convexixity with applications, Adv. Differ. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-019-2438-0
    [27] M. Adil Khan, M. Hanif, Z. A. Khan, et al. Association of Jensen's inequality for s-convex function with Csiszár divergence, J. Inequal. Appl., 2019 (2019), 1-14. doi: 10.1186/s13660-019-1955-4
    [28] P. O. Mohammed, M. Z. Sarikaya, Hermite-Hadamard type inequalities for F-convex function involving fractional integrals, J. Inequal. Appl., 2018 (2018), 1-33. doi: 10.1186/s13660-017-1594-6
    [29] F. Qi, P. O. Mohammed, J. C. Yao, et al. Generalized fractional integral inequalities of Hermite-Hadamard type for (α, m)-convex functions, J. Inequal. Appl., 2019 (2019), 1-17. doi: 10.1186/s13660-019-1955-4
    [30] M. Adil Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-019-2438-0
    [31] A. Iqbal, M. Adil Khan, S. Ullah, et al. Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Space., 2020 (2020), 1-18.
    [32] M. U. Awan, S. Talib, Y. M. Chu, et al. Some new refinements of Hermite-Hadamard-type inequalities involving Ψk-Riemann-Liouville fractional integrals and applications, Math. Probl. Eng., 2020 (2020), 1-10.
    [33] T. Abdeljawad, P. O. Mohammed, A. Kashuri, New modified conformable fractional integral inequalities of Hermite-Hadamard type with applications, J. Funct. Space., 2020 (2020), 1-14.
    [34] M. Z. Sarikaya, E. Set, H. Yaldiz, et al. Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403-2407. doi: 10.1016/j.mcm.2011.12.048
    [35] M. Adil Khan, A. Iqbal, M. Suleman, et al. Hermite-Hadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl., 2018 (2018), 1-15. doi: 10.1186/s13660-017-1594-6
    [36] R. P. Agarwal, P. J. Y. Patricia, Error Inequalities in Polynomial Interpolation and Their Applications, Springer Science & Business Media, 1993.
    [37] N. Mehmood, R. P. Agarwal, S. I. Butt, et al. New generalizations of Popoviciu-type inequalities via new Green's functions and Montgomery identity, J. Inequal. Appl., 2017 (2017), 1-17. doi: 10.1186/s13660-016-1272-0
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2961) PDF downloads(278) Cited by(18)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog