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A note on conformable fractional Newton-type inequalities via functions of bounded variation

Year 2025, Volume: 1 Issue: 1, 11 - 21, 28.11.2025

Abstract

In this paper, we establish an equality in order to obtain conformable fractional Newton-type inequalities. Moreover, we prove some Newton-type inequalities associated with conformable fractional operators for functions of bounded variation. Furthermore, some results are presented by using special choices of the obtained inequalities.

References

  • [1] S. Gao, W. Shi (2012), On new inequalities of Newton’s type for functions whose second derivatives absolute values are convex, Int. J. Pure Appl. Math., 74(1), 33–41.
  • [2] S. Erden, S. Iftikhar, P. Kumam, P. Thounthong (2020), On error estimations of Simpson’s second type quadrature formula, Math. Methods Appl. Sci., 47(13), 11232–11244. https://doi.org/10.1002/mma.7019
  • [3] M.A. Ali, H. Budak, Z. Zhang (2022), A new extension of quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions, Math. Methods Appl. Sci., 45(4), 1845–1863. https://doi.org/10.1002/mma.7889
  • [4] M.A. Noor, K.I. Noor, S. Iftikhar (2016), Some Newton’s type inequalities for harmonic convex functions, J. Adv. Math. Stud., 9(1), 07–16.
  • [5] M.A. Noor, K.I. Noor, S. Iftikhar (2018), Newton inequalities for p-harmonic convex functions, Honam Math. J., 40(2), 239–250. https://dx.doi.org/10.5831/HMJ.2018.40.2.239
  • [6] S. Iftikhar, P. Kumam, S. Erden (2020), Newton’s-type integral inequalities via local fractional integrals, Fractals, 28(03), 2050037. https://doi.org/10.1142/S0218348X20500371
  • [7] S. Iftikhar, S. Erden, M.A. Ali, J. Baili, H. Ahmad (2022), Simpson’s second-type inequalities for co-ordinated convex functions and applications for cubature formulas, Fractal Fract., 6(1), 33. https://doi.org/10.3390/fractalfract6010033
  • [8] M.A. Noor, K.I. Noor, M.U. Awan (2015), Some Newton’s type inequalities for geometrically relative convex functions, Malaysian J. Math. Sci., 9(3), 491–502.
  • [9] M.Z. Sarıkaya, E. Set, H. Yaldız, N. Başak (2013), Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57, 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048
  • [10] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Springer Verlag, Wien, 1997.
  • [11] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
  • [12] T. Sitthiwirattham, K. Nonlaopon, M.A. Ali, H. Budak (2022), Riemann-Liouville fractional Newton’s type inequalities for differentiable convex functions, Fractal Fract. 6(3), 175. https://doi.org/10.3390/fractalfract6030175
  • [13] S. Erden, S. Iftikhar, P. Kumam, M.U. Awan (2020), Some Newton’s like inequalities with applications, RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat., 114(4), 1–13. https://doi.org/10.1007/s13398-020-00926-z
  • [14] F. Hezenci, H. Budak (2025), Fractional Newton-type integral inequalities by means of various function classes, Math. Methods Appl. Sci., 48(1), 1198–1215. https://doi.org/10.1002/mma.10378
  • [15] S. Iftikhar, S. Erden, P. Kumam, M.U. Awan (2020), Local fractional Newton’s inequalities involving generalized harmonic convex functions, Adv. Difference Equ., 2020(1), 1–14. https://doi.org/10.1186/s13662-020-02637-6
  • [16] F. Hezenci, H. Budak, P. Kösem (2023), A new version of Newton’s inequalities for Riemann-Liouville fractional integrals, Rocky Mountain J. Math., 53(1), 49–64. https://doi.org/10.1216/rmj.2023.53.49
  • [17] F. Hezenci, H. Budak (2023), Some perturbed Newton type inequalities for Riemann-Liouville fractional integrals, Rocky Mountain J. Math., 53(4), 1117–1127. https://doi.org/10.1216/rmj.2023.53.1117
  • [18] V.V. Uchaikin, Fractional derivatives for physicists and engineers, Springer: Berlin/Heidelberg, Germany, 2013.
  • [19] G.A. Anastassiou, Generalized fractional calculus: New advancements and applications, Springer: Switzerland, 2021.
  • [20] N. Attia, A. Akgül, D. Seba, A. Nour (2020), An efficient numerical technique for a biological population model of fractional order, Chaos, Solitons & Fractals, 141, 110349. https://doi.org/10.1016/j.chaos.2020.110349
  • [21] A. Gabr, A.H. Abdel Kader, M.S. Abdel Latif (2021), The effect of the parameters of the generalized fractional derivatives on the behavior of linear electrical circuits, Int. J. Appl. Comput. Math., 7, 247. https://doi.org/10.1007/s40819-021-01160-w
  • [22] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh (2014), A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65–70. https://doi.org/10.1016/j.cam.2014.01.002
  • [23] A.A. Abdelhakim (2019), The flaw in the conformable calculus: It is conformable because it is not fractional, Fract. Calc. Appl. Anal., 22, 242–254. https://doi.org/10.1515/fca-2019-0016
  • [24] D. Zhao, M. Luo (2017), General conformable fractional derivative and its physical interpretation, Calcolo, 54, 903–917. https://doi.org/10.1007/s10092-017-0213-8
  • [25] A. Hyder, A.H. Soliman (2020), A new generalized q-conformable calculus and its applications in mathematical physics, Phys. Scr., 96, 015208. https://doi.org/10.1088/1402-4896/abc6d9
  • [26] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu (2017), On a new class of fractional operators, Adv. Difference Equ., 2017, 247. https://doi.org/10.1186/s13662-017-1306-z
  • [27] T. Abdeljawad (2015), On conformable fractional calculus, J. Comput. Appl. Math., 279, 57–66. https://doi.org/10.1016/j.cam.2014.10.016
  • [28] H. Budak, C. Ünal, F. Hezenci (2024), A study on error bounds for Newton-type inequalities in conformable fractional integrals, Math. Slovaca, 74(2), 313–330. https://doi.org/10.1515/ms-2024-0024
  • [29] F. Hezenci, P. Karagözoglu, H. Budak, Some error bounds for Newton formula in conformable fractional operators, submitted.
  • [30] M.W. Alomari (2012), A companion of Dragomir’s generalization of Ostrowski’s inequality and applications in numerical integration, Ukrainian Math. J., 64, 435–450.

Year 2025, Volume: 1 Issue: 1, 11 - 21, 28.11.2025

Abstract

References

  • [1] S. Gao, W. Shi (2012), On new inequalities of Newton’s type for functions whose second derivatives absolute values are convex, Int. J. Pure Appl. Math., 74(1), 33–41.
  • [2] S. Erden, S. Iftikhar, P. Kumam, P. Thounthong (2020), On error estimations of Simpson’s second type quadrature formula, Math. Methods Appl. Sci., 47(13), 11232–11244. https://doi.org/10.1002/mma.7019
  • [3] M.A. Ali, H. Budak, Z. Zhang (2022), A new extension of quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions, Math. Methods Appl. Sci., 45(4), 1845–1863. https://doi.org/10.1002/mma.7889
  • [4] M.A. Noor, K.I. Noor, S. Iftikhar (2016), Some Newton’s type inequalities for harmonic convex functions, J. Adv. Math. Stud., 9(1), 07–16.
  • [5] M.A. Noor, K.I. Noor, S. Iftikhar (2018), Newton inequalities for p-harmonic convex functions, Honam Math. J., 40(2), 239–250. https://dx.doi.org/10.5831/HMJ.2018.40.2.239
  • [6] S. Iftikhar, P. Kumam, S. Erden (2020), Newton’s-type integral inequalities via local fractional integrals, Fractals, 28(03), 2050037. https://doi.org/10.1142/S0218348X20500371
  • [7] S. Iftikhar, S. Erden, M.A. Ali, J. Baili, H. Ahmad (2022), Simpson’s second-type inequalities for co-ordinated convex functions and applications for cubature formulas, Fractal Fract., 6(1), 33. https://doi.org/10.3390/fractalfract6010033
  • [8] M.A. Noor, K.I. Noor, M.U. Awan (2015), Some Newton’s type inequalities for geometrically relative convex functions, Malaysian J. Math. Sci., 9(3), 491–502.
  • [9] M.Z. Sarıkaya, E. Set, H. Yaldız, N. Başak (2013), Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, 57, 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048
  • [10] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Springer Verlag, Wien, 1997.
  • [11] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
  • [12] T. Sitthiwirattham, K. Nonlaopon, M.A. Ali, H. Budak (2022), Riemann-Liouville fractional Newton’s type inequalities for differentiable convex functions, Fractal Fract. 6(3), 175. https://doi.org/10.3390/fractalfract6030175
  • [13] S. Erden, S. Iftikhar, P. Kumam, M.U. Awan (2020), Some Newton’s like inequalities with applications, RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat., 114(4), 1–13. https://doi.org/10.1007/s13398-020-00926-z
  • [14] F. Hezenci, H. Budak (2025), Fractional Newton-type integral inequalities by means of various function classes, Math. Methods Appl. Sci., 48(1), 1198–1215. https://doi.org/10.1002/mma.10378
  • [15] S. Iftikhar, S. Erden, P. Kumam, M.U. Awan (2020), Local fractional Newton’s inequalities involving generalized harmonic convex functions, Adv. Difference Equ., 2020(1), 1–14. https://doi.org/10.1186/s13662-020-02637-6
  • [16] F. Hezenci, H. Budak, P. Kösem (2023), A new version of Newton’s inequalities for Riemann-Liouville fractional integrals, Rocky Mountain J. Math., 53(1), 49–64. https://doi.org/10.1216/rmj.2023.53.49
  • [17] F. Hezenci, H. Budak (2023), Some perturbed Newton type inequalities for Riemann-Liouville fractional integrals, Rocky Mountain J. Math., 53(4), 1117–1127. https://doi.org/10.1216/rmj.2023.53.1117
  • [18] V.V. Uchaikin, Fractional derivatives for physicists and engineers, Springer: Berlin/Heidelberg, Germany, 2013.
  • [19] G.A. Anastassiou, Generalized fractional calculus: New advancements and applications, Springer: Switzerland, 2021.
  • [20] N. Attia, A. Akgül, D. Seba, A. Nour (2020), An efficient numerical technique for a biological population model of fractional order, Chaos, Solitons & Fractals, 141, 110349. https://doi.org/10.1016/j.chaos.2020.110349
  • [21] A. Gabr, A.H. Abdel Kader, M.S. Abdel Latif (2021), The effect of the parameters of the generalized fractional derivatives on the behavior of linear electrical circuits, Int. J. Appl. Comput. Math., 7, 247. https://doi.org/10.1007/s40819-021-01160-w
  • [22] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh (2014), A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65–70. https://doi.org/10.1016/j.cam.2014.01.002
  • [23] A.A. Abdelhakim (2019), The flaw in the conformable calculus: It is conformable because it is not fractional, Fract. Calc. Appl. Anal., 22, 242–254. https://doi.org/10.1515/fca-2019-0016
  • [24] D. Zhao, M. Luo (2017), General conformable fractional derivative and its physical interpretation, Calcolo, 54, 903–917. https://doi.org/10.1007/s10092-017-0213-8
  • [25] A. Hyder, A.H. Soliman (2020), A new generalized q-conformable calculus and its applications in mathematical physics, Phys. Scr., 96, 015208. https://doi.org/10.1088/1402-4896/abc6d9
  • [26] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu (2017), On a new class of fractional operators, Adv. Difference Equ., 2017, 247. https://doi.org/10.1186/s13662-017-1306-z
  • [27] T. Abdeljawad (2015), On conformable fractional calculus, J. Comput. Appl. Math., 279, 57–66. https://doi.org/10.1016/j.cam.2014.10.016
  • [28] H. Budak, C. Ünal, F. Hezenci (2024), A study on error bounds for Newton-type inequalities in conformable fractional integrals, Math. Slovaca, 74(2), 313–330. https://doi.org/10.1515/ms-2024-0024
  • [29] F. Hezenci, P. Karagözoglu, H. Budak, Some error bounds for Newton formula in conformable fractional operators, submitted.
  • [30] M.W. Alomari (2012), A companion of Dragomir’s generalization of Ostrowski’s inequality and applications in numerical integration, Ukrainian Math. J., 64, 435–450.
There are 30 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Research Article
Authors

Mehmet Sarıışık 0009-0008-9659-665X

Fatih Hezenci 0000-0003-1008-5856

Hüseyin Budak 0000-0001-8843-955X

Publication Date November 28, 2025
Submission Date July 28, 2025
Acceptance Date August 23, 2025
Published in Issue Year 2025 Volume: 1 Issue: 1

Cite

APA Sarıışık, M., Hezenci, F., & Budak, H. (2025). A note on conformable fractional Newton-type inequalities via functions of bounded variation. Düzce Mathematical Research, 1(1), 11-21.
AMA Sarıışık M, Hezenci F, Budak H. A note on conformable fractional Newton-type inequalities via functions of bounded variation. Düzce Mathematical Research. November 2025;1(1):11-21.
Chicago Sarıışık, Mehmet, Fatih Hezenci, and Hüseyin Budak. “A Note on Conformable Fractional Newton-Type Inequalities via Functions of Bounded Variation”. Düzce Mathematical Research 1, no. 1 (November 2025): 11-21.
EndNote Sarıışık M, Hezenci F, Budak H (November 1, 2025) A note on conformable fractional Newton-type inequalities via functions of bounded variation. Düzce Mathematical Research 1 1 11–21.
IEEE M. Sarıışık, F. Hezenci, and H. Budak, “A note on conformable fractional Newton-type inequalities via functions of bounded variation”, Düzce Mathematical Research, vol. 1, no. 1, pp. 11–21, 2025.
ISNAD Sarıışık, Mehmet et al. “A Note on Conformable Fractional Newton-Type Inequalities via Functions of Bounded Variation”. Düzce Mathematical Research 1/1 (November2025), 11-21.
JAMA Sarıışık M, Hezenci F, Budak H. A note on conformable fractional Newton-type inequalities via functions of bounded variation. Düzce Mathematical Research. 2025;1:11–21.
MLA Sarıışık, Mehmet et al. “A Note on Conformable Fractional Newton-Type Inequalities via Functions of Bounded Variation”. Düzce Mathematical Research, vol. 1, no. 1, 2025, pp. 11-21.
Vancouver Sarıışık M, Hezenci F, Budak H. A note on conformable fractional Newton-type inequalities via functions of bounded variation. Düzce Mathematical Research. 2025;1(1):11-2.