Certain generalized Riemann-Liouville fractional integrals inequalities based on exponentially \((h, m)\)-preinvexity. (English) Zbl 07762462
Summary: In this article, we define a new preinvex mapping known as the exponentially \((h, m)\)-preinvex. We explore the properties of this mapping, and derive several \(\Phi_k\)-Riemann-Liouville fractional integrals inequalities through exponentially \((h, m)\)-preinvexity. Using these results, we present certain estimates for Hermite-Hadamard-type inequalities, and also deduce several inequalities for the fractional integrals of \(k\)-Riemann-Liouville, which are special cases of the main results. During this period, we give several numerical examples to further demonstrate the correctness of our main results. As applications, we propose a few inequalities according to special means.
MSC:
| 26Axx | Functions of one variable |
| 26Dxx | Inequalities in real analysis |
| 34Axx | General theory for ordinary differential equations |
Keywords:
Riemann-Liouville fractional integrals; exponentially \((h, m)\)-preinvex mapping; Hermite-Hadamard inequalityReferences:
| [1] | Ali, M. A.; Nápoles Valdés, J. E.; Kashuri, A.; Zhang, Z. Y., Fractional non conformable Hermite-Hadamard inequalities for generalized \(φ\)-convex functions. Fasc. Math., 1, 5-16 (2020) |
| [2] | Butt, S. I.; Tariq, M.; Aslam, A.; Ahmad, H.; Nofal, T. A., Hermite-Hadamard type inequalities via generalized harmonic exponential convexity and applications. J. Funct. Spaces (2021) · Zbl 1460.26027 |
| [3] | Dragomir, S. S.; Gomm, I., Some Hermite-Hadamard type inequalities for functions whose exponentials are convex. Stud. Univ. Babeş-Bolyai, Math., 4, 527-534 (2015) · Zbl 1389.26038 |
| [4] | Du, T. S.; Liao, J. G.; Li, Y. J., Properties and integral inequalities of Hadamard-Simpson type for the generalized \((s, m)\)-preinvex functions. J. Nonlinear Sci. Appl., 3112-3126 (2016) · Zbl 1345.26034 |
| [5] | Farid, G.; Rehman, A.; Zahra, M., On Hadamard-type inequalities for \(k\)-fractional integrals. Konuralp J. Math., 2, 79-86 (2016) · Zbl 1355.26019 |
| [6] | Iqbal, S.; Mubeen, S.; Tomar, M., On Hadamard \(k\)-fractional integrals. J. Fract. Calc. Appl., 2, 255-267 (2018) · Zbl 1488.26016 |
| [7] | Katugampola, U. N., Approach to a generalized fractional integral. Appl. Math. Comput., 860-865 (2011) · Zbl 1231.26008 |
| [8] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Teory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies (2006), Elsevier: Elsevier New York, NY, USA · Zbl 1092.45003 |
| [9] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Application of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [10] | Kwun, Y. C.; Farid, G.; Nazeer, W.; Ullah, S.; Kang, S. M., Generalized Riemann-Liouville \(k\)-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities. IEEE Access, 64946-64953 (2018) |
| [11] | Liu, K.; Wang, J. R.; O’Regan, D., On the Hermite-Hadamard type inequality for Ψ-Riemann-Liouville fractional integrals via convex functions. J. Inequal. Appl. (2019) · Zbl 1499.26144 |
| [12] | Mehreen, N.; Anwar, M., Hermite-Hadamard type inequalities via exponentially \((p, h)\)-convex functions. IEEE Access, 37589-37595 (2020) |
| [13] | Miao, C.; Farid, G.; Yasmeen, H.; Bian, Y., Generalized Hadamard fractional integral inequalities for strongly \((s, m)\)-convex functions. J. Math. (2021) · Zbl 1477.26010 |
| [14] | Mohammed, P. O.; Abdeljawad, T.; Alqudah, M. A.; Jarad, F., New discrete inequalities of Hermite-Hadamard type for convex functions. Adv. Differ. Equ. (2021) · Zbl 1494.26054 |
| [15] | Mohan, S. R.; Neogy, S. K., On invex sets and preinvex functions. J. Math. Anal. Appl., 3, 901-908 (1995) · Zbl 0831.90097 |
| [16] | Mubeen, S.; Habibullah, G. M., \(k\)-Fractional integrals and applications. Int. J. Contemp. Math. Sci., 2, 89-94 (2012) · Zbl 1248.33005 |
| [17] | Mumcu, İ.; Set, E.; Akdemir, A. O.; Jarad, F., New extensions of Hermite-Hadamard inequalities via generalized proportional fractional integral. Numer. Methods Partial Differ. Equ., 1-12 (2021) |
| [18] | Noor, M. A.; Noor, K. I.; Rashid, S., Some new classes of preinvex functions and inequalities. Mathematics, 29, 1-16 (2019) |
| [19] | Podlubny, I., Fractional Differential Equations: Mathematics in Science and Engineering (1999), Academic Press: Academic Press San Diego · Zbl 0924.34008 |
| [20] | Rahman, G.; Nisar, K. S.; Ghanbari, B.; Abdeljawad, T., On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals. Adv. Differ. Equ. (2020) · Zbl 1485.26047 |
| [21] | Rashid, S.; Noor, M. A.; Noor, K. I., Some generalize Riemann-Liouville fractional estimates involving functions having exponentially convexity property. Punjab Univ. J. Math., 11, 1-15 (2019) |
| [22] | Rashid, S.; Abdeljawad, T.; Jarad, F.; Noor, M. A., Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications. Mathematics, 9, 1-18 (2019) |
| [23] | Rashid, S.; Noor, M. A.; Noor, K. I., Fractional exponentially \(m\)-convex functions and inequalities. Int. J. Anal. Appl., 3, 464-478 (2019) · Zbl 1438.26089 |
| [24] | Rashid, S.; Noor, M. A.; Noor, K. I., Some new estimates for exponentially \((ħ, m)\)-convex functions via extended generalized fractional integral operators. Korean J. Math., 4, 843-860 (2019) · Zbl 1436.26023 |
| [25] | Rashid, S.; Noor, M. A.; Noor, K. I.; Chu, Y. M., Ostrowski type inequalities in the sense of generalized \(\mathcal{K} \)-fractional integral operator for exponentially convex functions. AIMS Math., 3, 2629-2645 (2020) · Zbl 1484.26038 |
| [26] | Sarikaya, M. Z.; Yaldiz, H., On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals. Miskolc Math. Notes, 2, 1049-1059 (2017) · Zbl 1389.26051 |
| [27] | Simić, S.; Bin-Mohsin, B., Some generalizations of the Hermite-Hadamard integral inequality. J. Inequal. Appl. (2021) · Zbl 1504.26067 |
| [28] | Sun, W. B., Hermite-Hadamard type local fractional integral inequalities for generalized \(s\)-preinvex functions and their generalization. Fractals, 4 (2021) · Zbl 1487.26055 |
| [29] | Wu, S. H.; Iqbal, S.; Aamir, M.; Samraiz, M.; Younus, A., On some Hermite-Hadamard inequalities involving \(k\)-fractional operators. J. Inequal. Appl. (2021) · Zbl 1504.26071 |
| [30] | Zhang, Y.; Du, T. S.; Wang, H.; Shen, Y. J.; Kashuri, A., Extensions of different type parameterized inequalities for generalized \((m, h)\)-preinvex mappings via \(k\)-fractional integrals. J. Inequal. Appl. (2018) · Zbl 1404.26011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
