On Opial-type inequality for a generalized fractional integral operator. (English) Zbl 1509.26016
Summary: This article is aimed at establishing some results concerning integral inequalities of the Opial type in the fractional calculus scenario. Specifically, a generalized definition of a fractional integral operator is introduced from a new Raina-type special function, and with certain results proposed in previous publications and the choice of the parameters involved, the established results in the work are obtained. In addition, some criteria are established to obtain the aforementioned inequalities based on other integral operators. Finally, a more generalized definition is suggested, with which interesting results can be obtained in the field of fractional integral inequalities.
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | D. Baleanu, P. O. Mohammed, M. Vivas-Cortez, and Y. Rangel-Oliveros, Some modifications in conformable fractional integral inequalities, Adv. Differ. Equ. 2020 (2020), no. 1, 374-380, . · Zbl 1485.26019 · doi:10.1186/s13662-020-02837-0 |
| [2] | T. Abdeljawad, P. O. Mohammed, and A. Kashuri, New modified conformable fractional integral inequalities of Hermite-Hadamard type with applications, J. Funct. Space 2020 (2020), no. 1, 357-435, . · Zbl 1448.26025 · doi:10.1155/2020/4352357 |
| [3] | P. O. Mohammed, Some integral inequalities of fractional quantum type, Malaya J. Mat. 4 (2016), no. 1, 93-99. |
| [4] | P. O. Mohammed and T. Abdeljawad, Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel, Adv. Differ. Equ. 2020 (2020), no. 1, 345-363, . · Zbl 1485.26020 · doi:10.1186/s13662-020-02825-4 |
| [5] | O. Bazighifan, An approach for studying asymptotic properties of solutions of neutral differential equations, Symmetry 12 (2020), no. 4, 1-20, . · doi:10.3390/sym12040555 |
| [6] | P. O. Mohammed and M. Z. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math. 372 (2020), no. 1, 1-15, . · Zbl 1435.26027 · doi:10.1016/j.cam.2020.112740 |
| [7] | M. J. Cloud, B. C. Drachman, and L. Lebedev, Inequalities with Applications to Engineering, Springer International Publishing, New York, 2014. · Zbl 1295.26002 |
| [8] | I. Ahmad, H. Ahmad, P. Thounthong, Y.-M. Chu, and C. Cesarano, Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method, Symmetry 12 (2020), no. 7, 1-20, . · doi:10.3390/sym12071195 |
| [9] | F. M. Atici and H. Yaldiz, Convex functions on discrete time domains, Canad. Math. Bull. 59 (2016), no. 1, 225-233, . · Zbl 1342.26035 · doi:10.4153/CMB-2015-065-6 |
| [10] | R. P. Agarwal, P. Y. M. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers, London, 1995 · Zbl 0821.26013 |
| [11] | D. S. Mitrinovic, J. E. Pećarić, and A. M. Fink, Opial’s inequality, in: Inequalities Involving Functions and Their Integrals and Derivatives: Mathematics and Its Applications, East European Series, vol. 53, Springer, Dordrecht, 1991. · Zbl 0744.26011 |
| [12] | J. Calvert, Some generalizations of Opial’s inequality, Proc. Amer. Math. Soc. 18 (1967), no. 1, 72-75, . · Zbl 0144.30801 · doi:10.1090/s0002-9939-1967-0204594-1 |
| [13] | C.-J. Zhao, On Opial’s type integral inequalities, Mathematics 7 (2019), no. 4, 375, 1-9, . · doi:10.3390/math7040375 |
| [14] | K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley & Sons, New York, 1993. · Zbl 0789.26002 |
| [15] | G. Farid, A. U. Rehman, and S. Ullah, Opial-type inequalities for convex functions and associated results in fractional calculus, Adv. Differ. Equ. 152 (2019), no. 1, 1-13, . · Zbl 1459.26030 · doi:10.1186/s13662-019-2089-1 |
| [16] | Z. Opial, Sur une inegalite, Ann. Polon. Math. 8 (1960), no. 1, 29-32. · Zbl 0089.27403 |
| [17] | J. H. He, A tutorial review on fractal space time and fractional calculus, Int. J. Theor. Phys. 53 (2014), 3698-3718, . · Zbl 1312.83028 · doi:10.1007/s10773-014-2123-8 |
| [18] | C.-H. He, C. Liu, J.-H. He, H. M. Sedighi, A. Shokri, and K. A. Gepreel, A fractal model for the internal temperature response of a porous concrete, Appl. Comput. Math. 21 (2022), no. 1, 71-77, . · Zbl 07601677 · doi:10.30546/1683-6154.21.1.2022.71 |
| [19] | D. Baleanu and A. Fernandez, On fractional operators and their classifications, Mathematics 7 (2019), no. 9, 1-10, . · doi:10.3390/math7090830 |
| [20] | R. Hilfer and Y. Luchko, Desiderata for fractional derivatives and integrals, Mathematics 7 (2019), no. 2, 1-8, . · doi:10.3390/math7020149 |
| [21] | U. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), no. 4, 1-15. · Zbl 1317.26008 |
| [22] | A. Atangana, Fractional Operators with Constant and Variable Order with Application to Geo-hydrology, Academic Press, New York, 2017. |
| [23] | J. Hristov, The Craft of Fractional Modelling in Science and Engineering, MDPI, Basel, 2018. |
| [24] | R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. · Zbl 1309.33001 |
| [25] | A. Wiman, Uber den Fundamentalsatz in der Theorie der Funktionen Eα(z), Acta Math. 29 (1905), no. 1, 191-201. · JFM 36.0471.01 |
| [26] | T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971), no. 1, 7-15. · Zbl 0221.45003 |
| [27] | T. O. Salim and A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl. 3 (2012), no. 5, 1-13. · Zbl 1488.33067 |
| [28] | R. K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operator, East Asian Math. J. 21 (2005), no. 2, 191-203. · Zbl 1101.33016 |
| [29] | R. P. Agarwal, M.-J. Luo, and R. K. Raina, On Ostrowski type inequalities, Fasc. Math. 56 (2016), no. 1, 5-27, . · Zbl 1352.26003 · doi:10.1515/fascmath-2016-0001 |
| [30] | S.-B. Chen, S. Rashid, Z. Hammouch, M. A. Noor, R. Ashraf, and Y.-M Chu, Integral inequalities via Raina’s fractional integrals operator with respect to a monotone function, Adv. Differ. Equ. 2020 (2020), no. 1, 1-20, . · Zbl 1487.26034 · doi:10.1186/s13662-020-03108-8 |
| [31] | J. Choi and P. Agarwal, Certain fractional integral inequalities involving hypergeometric operators, East Asian Math. J. 30 (2014), no. 3, 283-291, . · Zbl 1326.26033 · doi:10.7858/eamj.2014.018 |
| [32] | J. E. Hernández Hernández and M. Vivas-Cortez, Hermite-Hadamard inequalities type for Raina’s fractional integral operator using η-convex functions, Rev. Mat. Teor. Apl. 26 (2019), no. 1, 1-20, . · Zbl 1446.26014 · doi:10.15517/rmta.v26i1.36214 |
| [33] | M. Vivas-Cortez, A. Kashuri, and J. E. Hernández, Trapezium-type inequalities for Raina’s fractional integrals operator using generalized convex functions, Symmetry 12 (2020), no. 6, 1-17, . · doi:10.3390/sym12061034 |
| [34] | T. U. Khan and M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math. 346 (2019), no. 1, 378-389, . · Zbl 1401.26012 · doi:10.1016/j.cam.2018.07.018 |
| [35] | K. S. Nisar, G. Rahman, and K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl. 2019 (2019), no. 1, 1-9 · Zbl 1499.26154 |
| [36] | G. Farid, J. Pećarić, and Z. Tomovski, Opial-type inequalities for fractional integral operator involving Mittag-Leffler function, Fract. Differ. Calc. 5 (2015), no. 1, 93-106, . · Zbl 1412.26047 · doi:10.7153/fdc-05-09 |
| [37] | D. S. Mitrinovick and J. Pećarić, Generalizations of two inequalities of Godunova and Levin, Bull. Polish Acad. Sci. Math. 36 (1988), no. 1, 645-648, . · Zbl 0758.26011 · doi:10.4236/am.2014.53034 |
| [38] | A. Andrić, A. Barbir, G. Farid, and J. Pećarić, More on certain Opial-type inequality for fractional derivatives, Nonlinear Funct. Anal. Appl 19 (2014), no. 4, 565-583. · Zbl 1308.26009 |
| [39] | M. Andrić, A. Barbir, S. Iqbal, and J. Pećarić, An Opial-type inequality and exponentially convex functions, Fract. Differ. Calc. 5 (2015), no. 1, 25-42, . · Zbl 1412.26033 · doi:10.7153/fdc-05-03 |
| [40] | J. Pećarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, Inc., New York, 1992. · Zbl 0749.26004 |
| [41] | M. Andrić, A. Barbir, G. Farid, and J. Pećarić, Opial-type inequality due to Agarwal-Pang and fractional differential inequalities, Integral Transforms Spec. Funct. 25 (2014), no. 4, 324-335, . · Zbl 1288.26003 · doi:10.1080/10652469.2013.851079 |
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.
