Document Type : Research Paper


Department of Mathematics, 8 may 1945 University, Guelma 24000, Algeria.


In this paper, we introduce the class of strongly $m$--$MT$-convex functions  based on the identity given in [P. Cerone et al., 1999]. We establish new inequalities of the Ostrowski-type for functions whose $n^{th}$ derivatives are strongly $m$--$MT$-convex functions.


Main Subjects

[1] G. Anastassiou, A. Kashuri and R. Liko, Local fractional integrals involving generalized strongly $m$-convex mappings, Arab. J. Math. (Springer), 8 (2) (2019), pp. 95-107.
[2] P. Cerone, S.S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for $n$-time differentiable mappings and applications, Demonstratio Math., 32 (4) (1999), pp. 697-712.
[3] Z. Dahmani, New classes of integral inequalities of fractional order, Matematiche (Catania), 69 (1) (2014), pp. 237-247.
[4] S.S. Dragomir, New inequalities of Hermite-Hadamard type for $GG$-convex functions, Indian J. Math., 60 (1) (2018), pp. 1-21.
[5] G. Farid and M. Usman, Ostrowski type $k$-fractional integral inequalities for $MT$-convex and $h$-convex functions, Nonlinear Funct. Anal. Appl., 22 (3) (2017), pp. 627-639.
[6] A. Fernandez and P.O. Mohammed, Hermite-Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels, Math. Meth. Appl. Sci., 2020, pp. 1-18.
[7] M. Houas, Certain weighted integral inequalities involving the fractional hypergeometric operators, Sci. Ser. A Math. Sci., 27 (2016), pp. 87-97.
[8] A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized $(s,m,\varphi )$-preinvex functions, Aust. J. Math. Anal. Appl., 13 (1) (2016), 11 pp.
[9] A. Kashuri and R. Liko, Generalizations of Hermite-Hadamard and Ostrowski type inequalities for $MTm$-preinvex functions, Proyecciones, 36 (1) (2017), pp. 45-80.
[10] A. Kashuri, R. Liko and T. Du, Ostrowski type fractional integral operators for generalized beta $(r,g)$-preinvex functions, Khayyam J. Math., 4 (1) (2018), pp. 39-58.
[11] A. Kashuri, R. Liko and S.S. Dragomir, Some new refinement of Hermite-Hadamard type inequalities and their applications, Tbilisi Math. J., 12 (4) (2019), pp. 159-188.
[12] T. Lara, N. Merentes, R. Quintero and E. Rosales, On strongly $m$-convex functions, Mathematica Aeterna, 5 (3) (2015), pp. 521-535.
[13] W. Liu, Ostrowski type fractional integral inequalities for $MT$-convex functions, Miskolc Math. Notes, 16 (1) (2015), pp. 249-256.
[14] B. Meftah, Some new Ostrwoski's inequalities for functions whose $n^{th}$ derivatives are $r$-convex, Int. J. Anal., 2016, 7 pp.
[15] B. Meftah, New Ostrowski's inequalities, Rev. Colombiana Mat. 51 (1) (2017), pp. 57-69.
[16] B. Meftah, Some new Ostrowski's inequalities for $n$-times differentiable mappings which are quasi-convex, Facta Univ. Ser. Math. Inform., 32 (3) (2017), pp. 319-327.
[17] B. Meftah, Some new Ostrowski's inequalities for functions whose $n^{th}$ derivatives are logarithmically convex, Ann. Math. Sil., 32 (1) (2017), pp. 275-284.
[18] B. Meftah, Some Ostrowski's inequalities for functions whose $n^{th}$ derivatives are $s$-convex, An Univ Oradea Fasc. Mat., 25 (2) (2018), pp. 185-212.
[19] B. Meftah, M. Merad, N. Ouanas and A. Souahi, Some new Hermite-Hadamard type inequalities whose $n^{th}$ derivatives are convex, Acta Comment. Univ. Tartu. Math., 23 (2) (2019), pp. 163-178.
[20] G.V. Milovanovic and J.E. Pecaric, On generalization of the inequality of A. Ostrowski and some related applications, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 544-576 (1976), pp. 155-158.
[21] P.O. Mohammed and T. Abdeljawad, Modification of certain fractional integral inequalities for convex functions, Adv. Difference Equ., 2020, Paper No. 69.
[22] P.O. Mohammed and M.Z. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math., 372 (2020), 15 pp.
[23] O. Omotoyinbo and A. Mogbademu, Some New Hermite-Hadamard Integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1 (1) (2014), pp. 001-012.
[24] A.M. Ostrowski, Uber die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, (German) Comment. Math. Helv., 10 (1) (1937), pp. 226-227.
[25] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
[26] B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restictions, Soviet Math. Dokl., 7 (1966), pp. 72-75.
[27] F. Qi, P.O. Mohammed, J.-C. Yao and Y.H. Yao, Generalized fractional integral inequalities of Hermite-Hadamard type for $\left( \alpha,m\right) $-convex functions, J. Inequal. Appl., 2019, Paper No. 135, 17 pp.
[28] G. Toader, Some generalizations of the convexity, Proceedings of the colloquium on approximation and optimization (Cluj-Napoca, 1985), 329-338, Univ. Cluj-Napoca, Cluj-Napoca, 1985.
[29] M. Tunc, H. Yildirim, On $MT$-convexity, arXiv preprint, 2012 (2012), 7 pages.
[30] M. Tunc, Ostrowski type inequalities for functions whose derivatives are $MT$-convex, J. Comput. Anal. Appl., 17 (4) (2014), pp. 691-696.
[31] H. Yaldi z, M.Z. Sari kaya and Z. Dahmani, On the Hermite-Hadamard-Fejer-type inequalities for co-ordinated convex functions via fractional integrals, Int. J. Optim. Control. Theor. Appl. IJOCTA, 7 (2) (2017), pp. 205-215.