Some Trapezoid-type Inequalities for Newly Defined Proportional Caputo-Hybrid Operator
DOI:
https://doi.org/10.63286/jima.2025.05Keywords:
Hermite-Hadamard type inequalities, trapezoid-type inequalities, convex functions, Riemann-Liouville fractional integrals, proportional Caputo-hybrid operatorAbstract
This study starts with the construction of a novel identity for the proportional Caputo-hybrid operator. Building on this identity, we develop several integral inequalities related to the right-hand side of Hermite-Hadamard-type inequalities in the context of the proportional Caputo-hybrid operator. Additionally, we demonstrate that the proposed results improve and generalize some previously established findings in the domain of integral inequalities. Lastly, in order to clarify and improve comprehension of the recently established inequalities, we provide numerous examples together with their graphical representations.
References
[1] S. S. Dragomir and C. E. M. Pearce, Selected topics on the Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University.
[2] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction considree par, Riemann, Journal de Mathématiques Pures et Appliquées, 58, 1893, 171–215.
[3] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Applied Mathematics Letters, 11(5), 1998, 91–95.
DOI: 10.1016/S0893-9659(98)00086-X
[4] U. S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Applied Mathematics and Computation, 147(5), 2004, 137–146.
DOI: 10.1016/S0096-3003(02)00657-4
[5] N. Alp, M. Z. Sarıkaya, M. Kunt, and İ. İ. Işcan, q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, Journal of King Saud University - Science, 30(2), 2018, 193–203.
DOI: 10.1016/j.jksus.2016.09.007
[6] H. Budak, T. Tunç, and M. Z. Sarıkaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, Proceedings of the American Mathematical Society, 148(2), 2020, 705–718.
DOI: 10.1090/proc/14741
[7] F. Hezenci, H. Kara, and H. Budak, Conformable fractional versions of Hermite-Hadamard-type inequalities for twice-differentiable functions, Boundary Value Problems, 2023(48), 2023, 16 pages.
DOI: 10.1186/s13661-023-01737-y
[8] M. Z. Javed, M. U. Awan, L. Ciurdariu, S. S. Dragomir, and Y. Almalki, On extended class of totally ordered interval-valued convex stochastic processes and applications, Fractal and Fractional, 8(10), 2024, 577.
DOI: 10.3390/fractalfract8100577
[9] M. Z. Javed, M. U. Awan, B. Bin-Mohsin, and S. Treanta, Upper bounds for the remainder term in Boole’s quadrature rule and applications to numerical analysis, Mathematics, 12(18), 2024, 2920.
DOI: 10.3390/math12182920
[10] M. Z. Javed, M. U. Awan, B. Bin-Mohsin, H. Budak, and S. S. Dragomir, Some classical inequalities associated with generic identity and applications, Axioms, 13(8), 2024, 533.
[11] B. Bin-Mohsin, M. Z. Javed, M. U. Awan and A. Kashuri, On some new AB-fractional inclusion relations, Axioms, 7(10), 2023, 725.
DOI: 10.3390/fractalfract7100725
[12] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1(2), 2015, 73–85.
DOI: 10.12785/pfda/010201
[13] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20(2), 2016, 763–769.
DOI: 10.48550/arXiv.1602.03408
[14] F. Sabzikar, M. M. Meerschaert, and J. Chen, Tempered fractional calculus, Journal of Computational Physics, 293, 2015, 14–28.
DOI: 10.1016/j.jcp.2014.04.024
[15] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
[16] M. Z. Sarıkaya and H. Yıldırım, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Mathematical Notes, 17(2), 2017, 1049–1059.
[17] M. Z. Sarıkaya, E. Set, H. Yaldız, and N. Basak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57(9–10), 2013, 2403–2407.
DOI: 10.1016/j.mcm.2011.12.048
[18] M. Iqbal, B. M. Iqbal, and K. Nazeer, Generalization of Inequalities Analogous to Hermite-Hadamard Inequality via Fractional Integrals, Bulletin of the Korean Mathematical Society, 52(3), 2015, 707–716.
DOI: 10.4134/BKMS.2015.52.3.707
[19] H. Budak, F. Ertuğral, and M. Z. Sarıkaya, New generalization of Hermite-Hadamard type inequalities via generalized fractional integrals, Analele Universitatii din Craiova. Seria Matematica-Informatica, 47(2), 2020, 369–386.
[20] P. O. Mohammed and I. Brevik, A new version of the Hermite–Hadamard inequality for Riemann–Liouville fractional integrals, Symmetry, 12(4), 2020, 11 pages.
DOI: 10.3390/sym12040610
[21] S. Samko, A. Kilbas, and O. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, London, 1993.
[22] K. Diethelm, The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, Springer, Berlin, Germany, 2010.
[23] D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Advances in Dynamical Systems and Applications, 10(2), 2015, 109–137.
[24] H. Günerhan, H. Dutta, M. A. Dokuyucu, and W. Adel, Analysis of a fractional HIV model with Caputo and constant proportional Caputo operators, Chaos, Solitons and Fractals, 139, 2020, 19 pages.
DOI: 10.1016/j.chaos.2020.110053
[25] R. Hajaj and Z. Odibat, Numerical solutions of fractional epidemic models with generalized Caputo-type derivatives, Physica Scripta, 98(4), 2023.
[26] G. Rahman, K. S. Nisar, and T. Abdeljawad, Certain Hadamard proportional fractional integral inequalities, Mathematics, 8(4), 2020.
DOI: 10.3390/math8040504
[27] D. Balenau, A. Ferrandez, and A. Akgül, On a fractional operator combining proportional and classical differintegrals, Mathematics, 8(3), 2020, 13 pages.
DOI: 10.3390/math8030360
[28] M. Z. Sarıkaya, On Hermite-Hadamard type inequalities for proportional Caputo-Hybrid Operator, Konuralp Journal of Mathematics, 11(1), 2023, 31–39.
[29] T. Tunç and İ. Demir, On a new version of Hermite-Hadamard-type inequality based on proportional Caputo-Hybrid operator, Boundary Value Problems, 2024(44), 2024.
DOI: 10.1186/s13661-024-01852-4
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Tuba Tunç, İzzettin Demir
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
