Document Type : Research Paper


1 UNNE, FaCENA, Ave. Libertad 5450, Corrientes 3400, Argentina.

2 UTN-FRRE, French 414, Resistencia, Chaco 3500, Argentina.

3 Bursa Uludag University, Faculty of Education Gorukle Capus, Bursa, Turkey.


New variants of the Hermite - Hadamard inequality within the framework of generalized fractional integrals for $(h,m,s)$-convex modified second type functions have been obtained in this article. To achieve these results, we used the Holder inequality and another form of it - power means. Some of the known results described in the literature can be considered as particular cases of the results obtained in our study.


Main Subjects

[1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), pp. 57-66.
[2] A. Akkurt, M.E. Yildirim and H. Yildirim, On some integral inequalities for (k, h)−Riemann-Liouville fractional integral, New Trends Math. Sci., 4 (1) (2016), pp. 138-146.
[3] B. Bayraktar and J.E. Nápoles Vald́es, New generalized integral inequalities via (h,m)−convex modified functions, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 60 (2022), pp. 3-15.
[4] B. Bayraktar and J.E. Nápoles Vald́es, Integral inequalities for mappings whose derivatives are (h, m, s)-convex modified of second type via Katugampola integrals, Annals of the University of Craiova, Mathematics and Computer Science Series, 49 (2) (2022), pp. 371-383.
[5] B. Bayraktar and M.E. Özdemir, Generalization Of Hadamard-Type Trapezoid Inequalities For Fractional Integral Operators, Ufa Math. J., 13 (1) (2021), pp. 119-130.
[6] H. Budak, E. Pehlivan and P. Kosem, On New Extensions of Hermite-Hadamard Inequalities for Generalized Fractional Integrals, Sahand Commun. Math. Anal., 18 (1) (2021), pp. 73-88.
[7] S.I. Butt, S. Yousaf, A.O. Akdemir and M.A. Dokuyucu, New Hadamard-type integral inequalities via a general form of fractional integral operators, Chaos Solitons Fractals, 148 (2013), 111025.
[8] S.I. Butt, S. Yousaf, A. Asghar, K.A. Khan and H.R. Moradi, New Fractional Hermite-Hadamard-Mercer Inequalities for Harmonically Convex Function, J. Funct. Spaces, 2021 (2021), Article ID 5868326.
[9] P.L Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2 (1882), pp. 93–98.
[10] H. Chen and U.N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2) 2017, pp. 1274-1291.
[11] R. Díaz and E. Pariguan, On hypergeometric functions and Pochhammer k−symbol, Divulg. Mat., 15 (2) (2007), pp. 179-192.
[12] S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to
[13] T.S. Du, C.Y. Luo and Z.J. Cao, On the Bullen-type inequalities via generalized fractional integrals and their applications, Fractals, 29 (7) (2021), Article ID 2150188, 20 pages.
[14] T.S. Du and T.C. Zhou, On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings, Chaos Solitons and Fractals, 156 (2022), Article ID 111846, 19 pages.
[15] G. Farid, A.U. Rehman and M. Zahra, On Hadamard inequalities for k−fractional integ, Nonlinear Funct. Anal. Appl., 21 (3) (2016), pp. 463-478.
[16] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, New York, 2006.
[17] N. Mehreen and M. Anwar, Integral inequalities for some convex functions via generalized fractional integrals, J. Inequal. Appl., (2018) 2018:208.
[18] S. Mubeen and G.M. Habibullah, k-fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), pp. 89-94.
[19] A.M. Nakhushev, Fractional Calculus and Its Application (Fizmatlit, Moscow, 2003) [in Russian].
[20] J.E. Nápoles Valdés, On the Hermite-Hadamard type inequalities involving generalized integrals, Contrib. Math., 5 (2022), pp. 45-51.
[21] J.E. Nápoles and B. Bayraktar, On The Generalized Inequalities Of The Hermite - Hadamard Type, Filomat, 35 (14) (2021), pp. 4917-4924.
[22] J.E. Nápoles V.B. Bayraktar and S. Butt, New integral inequalities of Hermite-Hadamard type in a generalized context, J. Math., Punjab Univ., 53 (11) (2021), pp. 765-777.
[23] J.E. Nápoles, F. Rabossi and A.D. Samaniego, Convex functions: Ariadne’s thread or Charlotte’s spiderweb?, Advanced Mathematical Models and Applications, 5 (2) (2020), pp.176-191.
[24] J.E. Nápoles, J.M. Rodríguez and J.M. Sigarreta, New Hermite-Hadamard Type Inequalities Involving Non-Conformable Integral Operators, Symmetry, 11 (2019), 1108.
[25] F. Qi, S. Habib, S. Mubeen and M.N. Naeem, Generalized kfractional conformable integrals and related inequalities, AIMS Math., 4 (3) 2019, pp.343–358.
[26] M.E. Özdemir, B. Bayraktar, S.I. Butt and A.O. Akdemir, Some New Hermite-Hadamard Type Inequalities Via Non-Conformable Fractional Integrals, Turkish Journal Of Inequalities, 5 (2) (2021), pp. 48-60.
[27] E.D. Rainville, Special Functions, Macmillan Co., New York, 1960.
[28] S. Rashid, Z. Hammouch, H. Kalsoom, R. Ashraf and Y.M. Chu, New Investigation on the Generalized k−Fractional Integral Operators, Front. Phys., 8 (2020), Article 25
[29] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications, Nauka i Tekhnika, Minsk, 1987 [in Russian].
[30] M.Z. Sarikaya and H. Budak, Generalized Hermite-Hadamard Type Integral Inequalities for Fractional Integrals, Filomat, 30 (5) (2016), pp. 1315-1326.
[31] M.Z. Sarikaya, Z. Dahmani, M.E. Kiris and F. Ahmad, (k, s)−Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45 (1) (2016), pp. 77-89.
[32] M.Z. Sarikaya, E. Set, H. Yaldiz and N. Başak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mat. Strukt. Model., 57 (9-10) (2013), pp. 2403-2407.
[33] J. Wang, X. Li and Y. Zhou, Hermite-Hadamard Inequalities Involving Riemann-Liouville Fractional Integrals via s-convex Functions and Applications to Special Means, Filomat, 30 (5) (2016), pp. 1143-1150.
[34] T.C. Zhou, Z.R. Yuan and T.S. Du, On the fractional integral inclusions having exponential kernels for interval-valued convex functions, Math. Sci., 17 (2) (2023), pp. 107–120.