New Form of Newton-Type Inequalities for Multiplicative Conformable Fractional Integrals

Year 2025, Volume: 8 Issue: 2, 93 - 111, 28.06.2025

Hüseyin Budak , Büşra Betül Ergün

https://doi.org/10.33187/jmsm.1674511

Abstract

In this study, a new Newton-type inequality form for multiplicative convex functions is derived using multiplicative conformable fractional integrals. The developed new form presents an inequality that has not been encountered before in literature. To obtain the main results, an essential identity is first introduced, and this identity is combined with multiplicative conformable fractional integrals to create a new Newton-type inequality. This work not only provides a significant contribution to previous research on multiplicative convex functions but also offers a new perspective on the subject. Future research may aim to explore the application of this form to different fractional integral operators or function types.

Keywords

Conformable fractional integrals, Convex functions, Multiplicative calculus, Newton inequality

References

• [1] J. Gleick, Isaac Newton, Vintage, New York, 2007.
• [2] G. W. Leibniz, Gottfried Wilhelm Leibniz, Leibniz-Forschungsstelle d. Univ. Münster, 1991.
• [3] B. Belhoste, Augustin-Louis Cauchy: A Biography, Springer Science & Business Media, 2012.
• [4] R. Bölling, Karl Weierstrass and some basic notions of the calculus, In: The Second W. Killing and K. Weierstraß Colloquium, Braniewo (Poland), March 2010, pp. 24–26.
• [5]A. Plotnitsky, Bernhard Riemann, In: G. Jones (Ed.), Deleuze's Philosophical Lineage, Edinburgh University Press, 2009, pp. 190–208.
• [6] S. C. Malik, S. Arora, Mathematical Analysis, New Age International, 1992.
• [7] A. Browder, Mathematical Analysis: An Introduction, Springer Science & Business Media, 2012.
• [8] C. P. Niculescu, A new look at Newton’s inequalities, J. Inequal. Pure Appl. Math., 1(2) (2000), Article ID 17.
• [9] T. Sitthiwirattham, K. Nonlaopon, M. A. Ali, et al., Riemann–Liouville fractional Newton’s type inequalities for differentiable convex functions, Fractal Fract., 6(3) (2022), Article ID 175. https://doi.org/10.3390/fractalfract6030175
• [10] F. Hezenci, H. Budak, Fractional Newton-type integral inequalities by means of various function classes, Math. Methods Appl. Sci., 48(1) (2025), 1198–1215. https://doi.org/10.1002/mma.10378
• [11] F. Hezenci, H. Budak, P. Kösem, A new version of Newton’s inequalities for Riemann–Liouville fractional integrals, Rocky Mountain J. Math., 53(1) (2023), 49–64. https://doi.org/10.1216/rmj.2023.53.49
• [12] S. Iftikhar, P. Kumam, S. Erden, Newton’s-type integral inequalities via local fractional integrals, Fractals, 28(3) (2020), Article ID 2050037. https://doi.org/10.1142/S0218348X20500371
• [13] C. Ünal, F. Hezenci, H. Budak, Conformable fractional Newton-type inequalities with respect to differentiable convex functions, J. Inequal. Appl., 2023(1) (2023), Article ID 85. https://doi.org/10.1186/s13660-023-02996-0
• [14] H. Xu, M. U. Awan, B. Meftah, et al., On conformable fractional Newton-type inequalities, Fractals, (2025), Early Access. https://doi.org/10.1142/s0218348x25500458
• [15] F. Hezenci, H. Budak, Note on Newton-type inequalities involving tempered fractional integrals, Korean J. Math., 32(2) (2024), 349–364. https://doi.org/10.11568/kjm.2024.32.2.349
• [16] J. Soontharanon, M. A. Ali, H. Budak, et al., Some new generalized fractional Newton’s type inequalities for convex functions, J. Funct. Spaces, 2022(1) (2022), Article ID 6261970. https://doi.org/10.1155/2022/6261970
• [17] M. A. Noor, K. I. Noor, S. Iftikhar, Some Newton’s type inequalities for harmonic convex functions, J. Adv. Math. Stud., 9(1) (2016), 7-16.
• [18] S. Iftikhar, S. Erden, P. Kumam, et al., Local fractional Newton’s inequalities involving generalized harmonic convex functions, Adv. Differ. Equ., 2020 (2020), Article ID 185, 1–14. https://doi.org/10.1186/s13662-020-02637-6
• [19] W. Saleh, A. Lakhdari, T. Abdeljawad, et al., On fractional biparameterized Newton-type inequalities, J. Inequal. Appl., 2023(1) (2023), Article ID 122. https://doi.org/10.1186/s13660-023-03033-w
• [20] Y. Mahajan, H. Nagar, Fractional Newton-type integral inequalities for the Caputo fractional operator, Math. Methods Appl. Sci., 48(4) (2025), 5244–5254. https://doi.org/10.1002/mma.10600
• [21] M. A. Noor, K. I. Noor, S. Iftikhar, Newton inequalities for p-harmonic convex functions, Honam Math. J., 40(2) (2018), 239–250. https://doi.org/10.5831/HMJ.2018.40.2.239
• [22] D. Stanley, A multiplicative calculus, PRIMUS, 9(4) (1999), 310–326. https://doi.org/10.1080/10511979908965937
• [23] M. Grossman, An introduction to non-Newtonian calculus, Int. J. Math. Educ. Sci. Technol., 10(4) (1979), 525–528. https://doi.org/10.1080/0020739790100406
• [24] A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337(1) (2008), 36–48. https://doi.org/10.1016/j.jmaa.2007.03.081
• [25] M. A. Ali, M. Abbas, Z. Zhang, et al., On integral inequalities for product and quotient of two multiplicatively convex functions, Asian Res. J. Math., 12(3) (2019), 1–11. https://doi.org/10.9734/ARJOM/2019/V12I330084
• [26] M. A. Ali, H. Budak, M. Z. Sarikaya, et al., Ostrowski and Simpson type inequalities for multiplicative integrals, Proyecciones (Antofagasta), 40(3) (2021), 743–763. http://dx.doi.org/10.22199/issn.0717-6279-4136
• [27] S. Chasreechai, M. A. Ali, S. Naowarat, et al., On some Simpson’s and Newton’s type of inequalities in multiplicative calculus with applications, AIMS Math., 8(2) (2023), 3885–3896. https://doi.org/10.3934/math.2023193
• [28] X. Zhan, A. Mateen, M. Toseef, et al., Some Simpson- and Ostrowski-type integral inequalities for generalized convex functions in multiplicative calculus with their computational analysis, Mathematics, 12(11) (2024), Article ID 1721. https://doi.org/10.3390/math12111721
• [29] B. Meftah, H. Boulares, A. Khan, et al., Fractional multiplicative Ostrowski-type inequalities for multiplicative differentiable convex functions, Jordan J. Math. Stat., 17(1) (2024), 113–128.
• [30] H. Boulares, B. Meftah, A. Moumen, et al., Fractional multiplicative Bullen-type inequalities for multiplicative differentiable functions, Symmetry, 15(2) (2023), Article ID 451. https://doi.org/10.3390/sym15020451
• [31] A. Moumen, H. Boulares, B. Meftah, et al., Multiplicatively Simpson type inequalities via fractional integral, Symmetry, 15(2) (2023), Article ID 460. https://doi.org/10.3390/sym15020460
• [32] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-convex functions, Konuralp J. Math., 8(1) (2020), 158–164.
• [33] A. Kashuri, S. K. Sahoo, M. Aljuaid, et al., Some new Hermite–Hadamard type inequalities pertaining to generalized multiplicative fractional integrals, Symmetry, 15(4) (2023), Article ID 868. https://doi.org/10.3390/sym15040868
• [34] B. Meftah, Maclaurin type inequalities for multiplicatively convex functions, Proc. Amer. Math. Soc., 151 (2023), 2115–2125. https://doi.org/10.1090/proc/16292
• [35] T. Abdeljawad, M. Grossman, On geometric fractional calculus, J. Semigroup Theory Appl., 2016 (2016), Article ID 2, 14 pages.
• [36] H. Budak, K. Özçelik, On Hermite-Hadamard type inequalities for multiplicative fractional integrals, Miskolc Math. Notes, 21(1) (2020), 91–99. https://doi.org/10.18514/MMN.2020.3129
• [37] M. A. Ali, On Simpson’s and Newton’s type inequalities in multiplicative fractional calculus, Filomat, 37(30) (2023), 10133–10144. https://doi.org/10.2298/FIL2330133A
• [38] T. Du, Y. Long, The multi-parameterized integral inequalities for multiplicative Riemann–Liouville fractional integrals, J. Math. Anal. Appl., 541(1) (2025), Article ID 128692. https://doi.org/10.1016/j.jmaa.2024.128692
• [39] D. Ai, T. Du, A study on Newton-type inequalities bounds for twice differentiable functions under multiplicative Katugampola fractional integrals, Fractals, 33(5) (2025), Article ID 2550032. https://doi.org/10.1142/S0218348X2550032X
• [40] A. Lakhdari, D. C. Benchettah, B. Meftah, Fractional multiplicative Newton-type inequalities for multiplicative s-convex positive functions with application, J. Comput. Appl. Math., 465 (2025), Article ID 116600. https://doi.org/10.1016/j.cam.2025.116600
• [41] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-preinvex functions, Turk. J. Anal. Number Theory, 9(3) (2021), 65–70. https://doi.org/10.12691/tjant-9-3-5
• [42] S. Özcan, A. Urus, S. I. Butt, Hermite–Hadamard-type inequalities for multiplicative harmonic s-convex functions, Ukr. Math. J., 76(9) (2024), 1537–1558. https://doi.org/10.1007/s11253-025-02404-4
• [43] M. Umar, S. I. Butt, Y. Seol, Hybrid fractional integral inequalities in multiplicative calculus with applications, Fractals, 33(1) (2025), Article ID 2550019. https://doi.org/10.1142/S0218348X25500197
• [44] H. Budak, B. B. Ergün, On multiplicative conformable fractional integrals: Theory and applications, Bound. Value Probl., 2025 (2025), Article ID 30. https://doi.org/10.1186/s13661-025-02026-6
• [45] J. E. Pečarić, Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, 1992.
• [46] C. Niculescu, L. E. Persson, Convex Functions and Their Applications, Vol. 23, Springer, New York, 2006.
• [47] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204, Elsevier, Amsterdam, 2006.
• [48] F. Jarad, E. Ugurlu, T. Abdeljawad, et al., On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), Article ID 247. https://doi.org/10.1186/s13662-017-1306-z