On multiplicative Hermite-Hadamard- and Newton-type inequalities for multiplicatively \((P, m)\)-convex functions. (English) Zbl 07808105
Summary: We develop a fresh family of functions, called as multiplicatively \((P, m)\)-convex functions. In this direction, we study the properties of such type functions, and establish integer-order inequalities of Hermite-Hadamard type. After that, we introduce the multiplicative \(k\)-Riemann-Liouville fractional integrals and discuss their \({}^\ast\)integrability, continuity as well as commutativity. Based on the proposed integral operators and the fact that the function is multiplicatively \((P, m)\)-convex or \((P, m)\)-convex, we derive some \(k\)-Riemann-Liouville fractional Hermite-Hadamard- and Newton-type inequalities. In the meantime, we provide certain examples together with their graphical descriptions, from which support the correctness of the inequalities obtained here. Some applications in special means are provided as well.
MSC:
| 26Dxx | Inequalities in real analysis |
| 26Axx | Functions of one variable |
| 34Axx | General theory for ordinary differential equations |
Keywords:
multiplicatively \((P, m)\)-convex functions; multiplicative \(k\)-Riemann-Liouville fractional integrals; Hermite-Hadamard-type inequalities; Newton-type inequalitiesReferences:
| [1] | Abdeljawad, T.; Grossman, M., On geometric fractional calculus, J. Semigroup Theory Appl., 2016, Article 2 pp. (2016) |
| [2] | Agarwal, P.; Jleli, M.; Tomar, M., Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals, J. Inequal. Appl., 2017, Article 55 pp. (2017) · Zbl 1359.26005 |
| [3] | Agarwal, P.; Restrepo, J. E., An extension by means of ω-weighted classes of the generalized Riemann-Liouville k-fractional integral inequalities, J. Math. Inequal., 14, 1, 35-46 (2020) · Zbl 1437.26023 |
| [4] | Ali, M. A.; Abbas, M.; Zhang, Z. Y.; Sial, I. B.; Arif, R., On integral inequalities for product and quotient of two multiplicatively convex functions, Asian Res. J. Math., 12, 3, 1-11 (2019) |
| [5] | Ali, M. A.; Budak, H.; Sarikaya, M. Z.; Zhang, Z. Y., Ostrowski and Simpson type inequalities for multiplicative integrals, Proyecciones, 40, 3, 743-763 (2021) · Zbl 1478.26015 |
| [6] | Ali, M. A.; Budak, H.; Fečkan, M.; Patanarapeelert, N.; Sitthiwirattham, T., On some Newton’s type inequalities for differentiable convex functions via Riemann-Liouville fractional integrals, Filomat, 37, 11, 3427-3441 (2023) |
| [7] | Bashirov, A. E.; Kurpınar, E. M.; Özyapıcı, A., Multiplicative calculus and its applications, J. Math. Anal. Appl., 337, 36-48 (2008) · Zbl 1129.26007 |
| [8] | Boulares, H.; Meftah, B.; Moumen, A.; Shafqat, R.; Saber, H.; Alraqad, T.; Ali, E. E., Fractional multiplicative Bullen-type inequalities for multiplicative differentiable functions, Symmetry, 15, Article 451 pp. (2023) |
| [9] | Budak, H.; Özçelik, K., On Hermite-Hadamard type inequalities for multiplicative fractional integrals, Miskolc Math. Notes, 21, 1, 91-99 (2020) · Zbl 1463.26028 |
| [10] | Budak, H.; Erden, S.; Ali, M. A., Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Methods Appl. Sci., 44, 1, 378-390 (2020) · Zbl 1470.26023 |
| [11] | Butt, S. I.; Yousaf, S.; Akdemir, A. O.; Dokuyucu, M. A., New Hadamard-type integral inequalities via a general form of fractional integral operators, Chaos Solitons Fractals, 148, Article 111025 pp. (2021) · Zbl 1485.26023 |
| [12] | Chasreechai, S.; Ali, M. A.; Naowarat, S.; Sitthiwirattham, T.; Nonlaopon, K., On some Simpson’s and Newton’s type of inequalities in multiplicative calculus with applications, AIMS Math., 8, 2, 3885-3896 (2022) |
| [13] | Díaz, R.; Pariguan, E., On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat., 15, 2, 179-192 (2007) · Zbl 1163.33300 |
| [14] | Dragomir, S. S.; Pečarič, J.; Persson, L. E., Some inequalities of Hadamard type, Soochow J. Math., 21, 3, 335-341 (1995) · Zbl 0834.26009 |
| [15] | Dragomir, S. S.; Toader, G., Some inequalities for m-convex functions, Stud. Univ. Babeş-Bolyai, Math., 38, 1, 21-38 (1993) · Zbl 0829.26010 |
| [16] | Du, T. S.; Peng, Y., Hermite-Hadamard type inequalities for multiplicative Riemann-Liouville fractional integrals, J. Comput. Appl. Math., 440, Article 115582 pp. (2024) · Zbl 1541.26040 |
| [17] | Du, T. S.; Yuan, X. M., On the parameterized fractal integral inequalities and related applications, Chaos Solitons Fractals, 170, Article 113375 pp. (2023) |
| [18] | Erden, S.; Iftikhar, S.; Kumam, P.; Awan, M. U., Some Newton’s like inequalities with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 114, 4, Article 195 pp. (2020) · Zbl 1451.26026 |
| [19] | Ertuǧral, F.; Sarikaya, M. Z., Simpson type integral inequalities for generalized fractional integral, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 113, 4, 3115-3124 (2019) · Zbl 1426.26011 |
| [20] | Fu, H.; Peng, Y.; Du, T. S., Some inequalities for multiplicative tempered fractional integrals involving the λ-incomplete gamma functions, AIMS Math., 6, 7, 7456-7478 (2021) · Zbl 1485.26034 |
| [21] | Hezenci, F.; Budak, H.; Kösem, P., A new version of Newton’s inequalities for Riemann-Liouvlle fractional integrals, Rocky Mt. J. Math., 53, 1, 49-64 (2023) · Zbl 1519.26011 |
| [22] | İşcan, İ., Weighted Hermite-Hadamard-Mercer type inequalities for convex functions, Numer. Methods Partial Differ. Equ., 37, 1, 118-130 (2020) · Zbl 1540.26011 |
| [23] | Kadakal, H., Multiplicatively P-functions and some new inequalities, New Trends Math. Sci., 6, 4, 111-118 (2018) |
| [24] | Kadakal, M., Some Hermite-Hadamard type inequalities for \((P, m)\)-function and quasi m-convex functions, Int. J. Optim. Control, Theor. Appl., 10, 1, 78-84 (2020) |
| [25] | Kalsoom, H.; Khan, Z. A., Hermite-Hadamard-Fejér type inequalities with generalized k-fractional conformable integrals and their applications, Mathematics, 10, 3, Article 483 pp. (2022) |
| [26] | Kashuri, A.; Sahoo, S. K.; Aljuaid, M.; Tariq, M.; Sen, M. D.L., Some new Hermite-Hadamard type inequalities pertaining to generalized multiplicative fractional integrals, Symmetry, 15, 4, Article 868 pp. (2023) |
| [27] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [28] | Luangboon, W.; Nonlaopon, K.; Tariboon, J.; Ntouyas, S. K., Simpson- and Newton-type inequalities for convex functions via \((p, q)\)-calculus, Mathematics, 9, 12, Article 1338 pp. (2021) |
| [29] | Luo, C. Y.; Yu, B.; Zhang, Y.; Du, T. S., Certain bounds related to multi-parameterized k-fractional integral inequalities and their applications, IEEE Access, 7, 124662-124673 (2019) |
| [30] | Meftah, B., Maclaurin type inequalities for multiplicatively convex functions, Proc. Am. Math. Soc., 151, 5, 2115-2125 (2023) · Zbl 1526.26004 |
| [31] | Moumen, A.; Boulares, H.; Meftah, B.; Shafqat, R.; Alraqad, T.; Ali, E. E.; Khaled, Z., Multiplicatively Simpson type inequalities via fractional integral, Symmetry, 15, 2, Article 460 pp. (2023) |
| [32] | Mubeen, S.; Habibullah, G. M., k-fractional integrals and application, Int. J. Contemp. Math. Sci., 7, 2, 89-94 (2012) · Zbl 1248.33005 |
| [33] | Noor, M. A.; Noor, K. I.; Iftikhar, S., Newton inequalities for p-harmonic convex functions, Honam Math. J., 40, 2, 239-250 (2018) · Zbl 1396.26036 |
| [34] | Özcan, S., Hermite-Hadamard type inequalities for multiplicatively P-functions, Gümüşhane Üniv. Fen Bilim. Enst. Derg., 10, 2, 486-491 (2020) |
| [35] | Özcan, S., Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions, AIMS Math., 5, 2, 1505-1518 (2020) · Zbl 1484.26084 |
| [36] | Pečarić, J. E.; Proschan, F.; Tong, Y. L., Convex Functions, Partial Orderings and Statistical Applications (1992), Academic Press, Inc.: Academic Press, Inc. San Diego · Zbl 0749.26004 |
| [37] | Peng, Y.; Du, T. S., Fractional Maclaurin-type inequalities for multiplicatively convex functions and multiplicatively P-functions, Filomat, 37, 28, 9497-9509 (2023) |
| [38] | Peng, Y.; Fu, H.; Du, T. S., Estimations of bounds on the multiplicative fractional integral inequalities having exponential kernels, Commun. Math. Stat. (2022) |
| [39] | Promsakon, C.; Ali, M. A.; Budak, H.; Abbas, M.; Muhammad, F.; Sitthiwirattham, T., On generalizations of quantum Simpson’s and quantum Newton’s inequalities with some parameters, AIMS Math., 6, 12, 13954-13975 (2021) · Zbl 1525.26023 |
| [40] | Sarikaya, M. Z.; Karaca, A., On the k-Riemann-Liouville fractional integral and applications, Int. J. Stat. Math., 1, 3, 33-43 (2014) |
| [41] | Sarikaya, M. Z.; Set, E.; Yaldiz, H.; Başak, N., Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57, 9-10, 2403-2407 (2013) · Zbl 1286.26018 |
| [42] | Sarikaya, M. Z.; Yildirim, H., On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17, 2, 1049-1059 (2016) · Zbl 1389.26051 |
| [43] | Set, E.; Tomar, M.; Sarikaya, M. Z., On generalized Grüss type inequalities for k-fractional integrals, Appl. Math. Comput., 269, 29-34 (2015) · Zbl 1410.26046 |
| [44] | Sitthiwirattham, T.; Nonlaopon, K.; Ali, M. A.; Budak, H., Riemann-Liouville fractional Newton’s type inequalities for differentiable convex functions, Fractal Fract., 6, 3, Article 175 pp. (2022) |
| [45] | Soontharanon, J.; Ali, M. A.; Budak, H.; Kösem, P.; Nonlaopon, K.; Sitthiwirattham, T., Some new generalized fractional Newton’s type inequalities for convex functions, J. Funct. Spaces, 2022, Article 6261970 pp. (2022) · Zbl 1506.26038 |
| [46] | Toader, G., Some generalizations of the convexity, (Proc. Colloqu. Approx. Optim.. Proc. Colloqu. Approx. Optim., Cluj-Napoca (Romania) (1984)), 329-338 · Zbl 0582.26003 |
| [47] | Ünal, C.; Hezenci, F.; Budak, H., Conformable fractional Newton-type inequalities with respect to differentiable convex functions, J. Inequal. Appl., 2023, Article 85 pp. (2023) · Zbl 1532.26044 |
| [48] | Vivas-Cortez, M.; Bibi, M.; Muddassar, M.; Al-Sa’di, S., On local fractional integral inequalities via generalized \(( \widetilde{h_1}, \widetilde{h_2})\)-preinvexity involving local fractional integral operators with Mittag-Leffler kernel, Demonstr. Math., 56, Article 20220216 pp. (2023) · Zbl 1529.26022 |
| [49] | Xie, J. Q.; Ali, M. A.; Sitthiwirattham, T., Some new midpoint and trapezoidal type inequalities in multiplicative calculus with applications, Filomat, 37, 20, 6665-6675 (2023) |
| [50] | You, X. X.; Ali, M. A.; Budak, H.; Vivas-Cortez, M.; Qaisar, S., Some parameterized quantum Simpson’s and quantum Newton’s integral inequalities via quantum differentiable convex mappings, Math. Probl. Eng., 2021, Article 5526726 pp. (2021) |
| [51] | Zafar, F.; Mehmood, S.; Asiri, A., Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals, Demonstr. Math., 56, 1, Article 20220254 pp. (2023) · Zbl 07734707 |
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