Praveen Agarwal, Shaher Momani, Shilpi JAIN, Rahul Goyal

Certain Saigo type fractional integral inequalities and their q-analogues

The main purpose of the present article is to introduce certain new Saigo fractional integral inequalities and their q-extensions. We also studied some special cases of these inequalities involving Riemann- Liouville and Erdelyi-Kober fractional integral operators.

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[1] Baleanu, D., and Fernandez, A. (2019). On fractional operators and their classifications. Mathematics, 7(9), 830.
[2] Ekinci, A., and Ozdemir, M. (2019). Some new integral inequalities via Riemann- Liouville integral operators. Applied and Computational Mathematics, 18(3).
[3] Butt, S.I., Nadeem, M., and Farid, G. (2020). On Caputo fractional derivatives via expo- nential s-convex functions. Turkish Journal of Science, 5(2), 140-146.
[4] Kizil, S ̧., and Ardi ̧c, M.A. (2021). Inequalities for strongly convex functions via Atangana- Baleanu integral operators. Turkish Journal of Science, 6(2), 96-109.
[5] Kalsoom, H., Ali, M.A., Abbas, M., Budak, H., and Murtaza G. (2022). Generalized quan- tum Montgomery identity and Ostrowski type inequalities for preinvex functions. TWMS Journal Of Pure And Applied Mathematics, 13(1), 72-90.
[6] Zhou, S.S., Rashid, S., Parveen, S., Akdemir, A.O., and Hammouch, Z. (2021). New com- putations for extended weighted function- als within the Hilfer generalized proportional fractional integral operators. AIMS Mathe- matics, 6(5), 4507-4525.
[7] Samko, S.G., Kilbas, A.A., Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach: New York, NY, USA.
[8] Sneddon, I.N. (1975). The use in mathe- matical physics of Erd ́elyi-Kober operators and of some of their generalizations. In Frac- tional Calculus and Its Applications (West Haven, CT, USA, 15–16 June 1974); Ross, B., Ed.; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 457, 37–79.
[9] Saigo, M. (1978). A remark on integral opera- tors involving the Gauss hypergeometric func- tions. Mathematical Repository Kyushu Uni- versity, 11, 135-143.
[10] Olver, W.J.F., Lozier, W.D., Boisvert, F.R., Clark, W.C. (2010). NIST Handbook of Math- ematical Functions. Cambridge University Press, New York, NY, USA.
[11] Rainville, E.D. (1960). Special Functions. Macmillan: New York, NY, USA.
[12] Kuang, J.C. (2004). Applied Inequalities. Shandong Science and Technology Press, Shandong, China.
[13] Mitrinovi ́c, D.S. (1970). Analytic Inequali- ties. Springer, Berlin, Germany.
[14] Chebyshev, P.L. (1882). Sur les expressions approximatives des integrales definies par les autres prises entre les mˆemes limites. Proceed- ings of Mathematical Society of Charkov, 2, 93-98.
[15] Anastassiou, G.A. (2011). Advances on Frac- tional Inequalities. Springer Science & Busi- ness Media.
[16] Belarbi, S., and Dahmani, Z. (2009). On some new fractional integral inequalities. Journal of Inequalities in Pure and Applied Mathematics, 10(3), 1-12.
[17] Dahmani, Z., Mechouar, O., and Brahami, S. (2011). Certain inequalities related to the Chebyshev’s functional involving a Riemann- Liouville operator. Bulletin of Mathematical Analysis and Applications, 3(4), 38-44.
[18] Dragomir, S.S. (1998). Some integral inequal- ities of Gr ̈uss type. RGMIA Research Report Collection, 1(2), 1998.
[19] Kalla, S.L. and Rao, A. (2011). On Gr ̈uss type inequality for a hypergeometric frac- tional integral. Le Matematiche, 66(1), 57-64.
[20] Lakshmikantham, V., and Vatsala, A.S. (2007). Theory of fractional differential in- equalities and applications. Communications in Applied Analysis, 11(3-4), 395-402.
[21] ̈O ̆g ̈unmez, H., and ̈Ozkan, U. (2011). Frac- tional quantum integral inequalities. Journal of Inequalities and Applications, 2011, 1-7.
[22] Sulaiman, W.T. (2011). Some new fractional integral inequalities. Journal of Mathematical Analysis, 2(2), 23–28.
[23] Baleanu, D., Purohit, S.D., and Agarwal, P. (2014). On fractional integral inequalities involving hypergeometric operators. Chinese Journal of Mathematics, 2014, 1-10.
[24] Jackson, F.H. (1908). On q-functions and a certain difference operator. Transactions of the Royal Society of Edinburgh, 46, 64–72.
[25] Al-Salam, W.A. and Verma A. (1975). A fractional Leibniz q-formula. Pacific Journal of Mathematics, 60, 1-9.
[26] Al-Salam, W.A. (1953). q-Analogues of Cauchy’s formula. Proceedings of the Amer- ican Mathematical Society, 17(3), 182-184.
[27] Al-Salam, W.A. (1969). Some fractional q- integrals and q-derivatives. Proceedings of the Edinburgh Mathematical Society, 15(2), 135-140.
[28] Agrawal, R.P. (1969). Certain fractional q-integrals and q-derivatives. Mathematical Proceedings of the Cambridge Philosophical Society, 66(2), 365-370.
[29] Rajkovic, P.M., Marinkovic, S.D., Stankovic, M.S. (2007). Fractional integrals and deriva- tives in q-calculus. Applicable Analysis and Discrete Mathematics, 1, 311-323,
[30] Isogawa, S., Kobachi, N. and Hamada, S. (2007). A q-analogue of Riemann-Liouville fractional derivative. Res. Rep. Yatsushiro Nat. Coll. Tech., 29, 59-68.
[31] Gasper, G. and Rahman, M. (1990). Ba- sic Hypergeometric Series. Cambridge Univ. Press, Cambridge.
[32] Garg, M. and Chanchkani, L. (2011). q- analogues of Saigo’s fractional calculus oper- ators. Bulletin of Mathemtical Analysis and Applications, 3(4), 169-179.
[33] Choi, J., and Agarwal, P. (2014). Some new Saigo type fractional integral inequalities and their-analogues. Abstract and Applied Analy- sis, 2014.