Research article

Certain novel estimates within fractional calculus theory on time scales

  • Received: 18 May 2020 Accepted: 15 July 2020 Published: 24 July 2020
  • MSC : 26D15, 26A33, 26E70

  • The key purpose of this study is to suggest a delta Riemann-Liouville (RL) fractional integral operators for deriving certain novel refinements of Pólya-Szegö and Čebyšev type inequalities on time scales. Some new Pólya-Szegö, Čebyšev and extended Čebyšev inequalities via delta-RL fractional integral operator on a time scale that captures some continuous and discrete analogues in the relative literature. New explicit bounds for unknown functions concerned are obtained due to the presented inequalities.

    Citation: Jian-Mei Shen, Saima Rashid, Muhammad Aslam Noor, Rehana Ashraf, Yu-Ming Chu. Certain novel estimates within fractional calculus theory on time scales[J]. AIMS Mathematics, 2020, 5(6): 6073-6086. doi: 10.3934/math.2020390

    Related Papers:

  • The key purpose of this study is to suggest a delta Riemann-Liouville (RL) fractional integral operators for deriving certain novel refinements of Pólya-Szegö and Čebyšev type inequalities on time scales. Some new Pólya-Szegö, Čebyšev and extended Čebyšev inequalities via delta-RL fractional integral operator on a time scale that captures some continuous and discrete analogues in the relative literature. New explicit bounds for unknown functions concerned are obtained due to the presented inequalities.


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    [1] Y. Khurshid, M. Adil Khan, Y. M. Chu, Conformable fractional integral inequalities for GG- and GA-convex function, AIMS Math., 5 (2020), 5012-5030. doi: 10.3934/math.2020322
    [2] S. Rashid, M. A. Noor, K. I. Noor, et al. Ostrowski type inequalities in the sense of generalized $\mathcal{K}$-fractional integral operator for exponentially convex functions, AIMS Math., 5 (2020), 2629-2645. doi: 10.3934/math.2020171
    [3] 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
    [4] 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.
    [5] 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.
    [6] 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
    [7] 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.
    [8] 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
    [9] 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
    [10] G. C. Wu, D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dynam., 75 (2014), 283-287. doi: 10.1007/s11071-013-1065-7
    [11] F. M. Atici, P. W. Eloe, Discerete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I, 3 (2009), 1-12.
    [12] T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), 1-12.
    [13] T. Abdeljawad, F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), 1-13.
    [14] T. Abdeljawad, Fractional difference operators with discrete generalized Mittag-Leffler kernels, Chaos Solitons Fractals, 126 (2019), 315-324. doi: 10.1016/j.chaos.2019.06.012
    [15] T. Abdeljawad, Different type kernel h-fractional differences and their fractional h-sums, Chaos Solitons Fractals, 116 (2018), 146-156. doi: 10.1016/j.chaos.2018.09.022
    [16] T. Abdeljawad, S. Banerjee, G. C. Wu, Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption, Optik, 218 (2020), Article 163698, Available from: https://doi.org/10.1016/j.ijleo.2019.163698.
    [17] S. Hilger, Ein Mabkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität Würzburg, 1988.
    [18] N. R. O. Bastos, Fractional Calculus on Time Scales, Ph.D. thesis, Instituto Politecnico de Viseu (Portugal), 2012.
    [19] S. S. Haider, M. ur Rehman, On substantial fractional difference operator, Adv. Differ. Equ., 2020 (2020), 1-18. doi: 10.1186/s13662-019-2438-0
    [20] B. T. Holm-Hansen, R. X. Gao, Time-scale analysis adapted for bearing diagnostics, Proc. SPIE 3833, Intelligent Systems in Design and Manufacturing II, (20 August 1999), Available from: https://doi.org/10.1117/12.359515.
    [21] G. C. Wu, Z. G. Deng, D. Baleanu, et. al. New variable-order fractional chaotic systems for fast image encryption, Chaos, 29 (2019), 1-11.
    [22] J. Zhu, L. Wu, Fractional Cauchy problem with Caputo nabla derivative on time scales, Abstr. Appl. Anal., 2015 (2015), 1-23.
    [23] S. L. Gao, Fractional time scale in calcium ion channels model, Int. J. Biomath., 6 (2013), 1-11.
    [24] J. J. Mohan, Variation of parameters for nabla fractional difference equations, Novi. Sad J. Math., 44 (2014), 149-159.
    [25] A. G. Radwan, On some generalized discrete logistic maps, J. Adv. Res., 4 (2013), 163-171. doi: 10.1016/j.jare.2012.05.003
    [26] S. Khan, M. Adil 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
    [27] M. Adil Khan, J. Pečarić, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Math., 5 (2020), 4931-4945. doi: 10.3934/math.2020315
    [28] 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
    [29] M. U. Awan, N. Akhtar, S. Iftikhar, et al. New Hermite-Hadamard type inequalities for npolynomial harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1-12. doi: 10.1186/s13660-019-2265-6
    [30] 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
    [31] S. Rashid, R. Ashraf, M. A. Noor, et al. New weighted generalizations for differentiable exponentially convex mapping with application, AIMS Math., 5 (2020), 3525-3546. doi: 10.3934/math.2020229
    [32] 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
    [33] T. H. Zhao, M. K. Wang, Y. M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Math., 5 (2020), 4512-4528. doi: 10.3934/math.2020290
    [34] 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.
    [35] 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.
    [36] 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
    [37] W. M. Qian, Z. Y. He, Y. M. Chu, Approximation for the complete elliptic integral of the first kind, RACSAM, 114 (2020), 1-12. doi: 10.1007/s13398-019-00732-2
    [38] 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
    [39] M. U. Awan, N. Akhtar, A. Kashuri, et. al. 2D approximately reciprocal ρ-convex functions and associated integral inequalities, AIMS Math., 5 (2020), 4662-4680. doi: 10.3934/math.2020299
    [40] S. Zaheer Ullah, M. Adil Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019), 1-10. doi: 10.1186/s13660-019-1955-4
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