The Fractional Integral Inequalities Involving Kober and Saigo–Maeda Operators

Year 2023, Volume: 6 Issue: 3, 135 - 141, 17.09.2023

Kalpana Rajput Rajshree Mıshra Deepak Kumar Jain Altaf Ahmad Bhat , Farooq Ahmad

https://doi.org/10.33434/cams.1275523

Abstract

This work uses the Marichev-Saigo-Maeda (MSM) fractional integral operator to achieve certain special fractional integral inequalities for synchronous functions. Compared to the previously mentioned classical inequalities, the inequalities reported in this study are more widespread. We also looked at several unique instances of these inequalities involving the fractional operators of the Saigo, Erdelyi, and Kober, and Riemann-Liouville types.

Keywords

Fractional Integral Inequalities, Saigo–Maeda Operators, Synchronous functions

References

• [1] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, (1993).
• [2] V. Kiryakova, Generalized Fractional Calculus and Applications, Wiley, New York, (1994).
• [3] J.L. Lavoie, T.J. Osler, R. Tremblay, Fractional derivatives and special functions, 18(2) (1976), 240-268.
• [4] V. Kiryakova, All the special functions are fractional differ integrals of elementary functions, J. Physics A: Math. and Gen., 30 (1997), 5085-5103.
• [5] V. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Computers and Mathematics with Applications, 59(5) (2010), 1885-1895.
• [6] M. Ciesielskia, M. Klimekb, T. Blaszczykb, The Fractional Sturm-Liouville Problem - Numerical Approximation and Application in Fractional Diffusion, J. Comput. Appl. Math., 317 (2017), 573-588.
• [7] V. Kiryakova, Fractional calculus operators of special functions? The result is well predictable, Chaos Solitons Fractals, 102 (2017), 1-14.
• [8] M. Saigo, R.K. Saxena, J. Ram, On the fractional calculus operator associated with the H- function, Gonita Sandesh, 6(1) (1992).
• [9] S. Jahanshahi, E. Babolian, D.F.M. Torres, A.R. Vahidi, A fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives, Chaos Solitons Fractals, 102 (2017), 295-304.
• [10] D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics, 7(9) (2019), 830.
• [11] A. Ekinci, M. Ozdemir, Some new integral inequalities via Riemann Liouville integral operators,Appl. Comput. Math., 18(3) (2019), 288-295.
• [12] S.I. Butt, M. Nadeem, G. Farid, On Caputo fractional derivatives via exponential s-convex functions, Turkish J. Sci., 5(2) (2020), 140-146.
• [13] S. Kizil, M.A. Ardi, Inequalities for strongly convex functions via Atangana Baleanu integral operators, Turkish J. Sci., 6(2) (2021), 96-109.
• [14] H. Kalsoom, M.A. Ali, M. Abbas, H. Budak, G. Murtaza, Generalized quantum Montgomery identity and Ostrowski type inequalities for preinvex functions, TWMS J. Pure Appl. Math., 13(1) (2022), 72-90.
• [15] J.A. Canavati, The Riemann-Liouville integral, Nieuw Archief Voor Wiskunde, 5(1) (1987), 53-75.
• [16] S.D. Purohit, R.K. Raina, Chebyshev type inequalities for the Saigo fractional integral and their q-analogues, J. Math. Inequal., 7(2) (2013), 239–249.
• [17] L. Curiel, L. Galue, A generalization of the integral operators involving the Gauss hypergeometric function, Revista Tecnica de la Facultad de Ingenierıa Universidad del Zulia, 19(1) (1996), 17–22.
• [18] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9(4) (2010), 493–497.
• [19] T.J. Olser, Leibniz rule for Fractional Derivatives Generalized and an Application to Infinite Series, SIAM. J. Appl. Math., 18 (1970), 658-678.
• [20] R. Herrmann, Towards a geometric interpretation of generalized fractional integrals-Erdelyi-Kober type integrals on R N, as an example, Fract. Calc. Appl. Anal., 17(2) (2014), 361-370.
• [21] S.S. Zhou, S. Rashid, S. Parveen, A.O. Akdemir, Z. Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, AIMS Math, 6(5) (2021), 4507-4525.
• [22] M. Saigo, A Remark on Integral Operators involving the Gauss hypergeometric functions, Math. Rep. College General Ed.Kyushu Univ., 11 (1978), 135-143.
• [23] M. Saigo, A certain boundary value problem for the Euler-Darboux equation, Math. Japonica, 24(4) (1979), 377-385.
• [24] S. Jain, R. Goyal, P. Agarwal, S. Momani, Certain Saigo type fractional integral inequalities and their q-analogues, An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 13(1) (2023), 1-9.