International Journal of Nonlinear Analysis and Applications

New Ostrowski type conformable fractional inequalities concerning differentiable generalized relative semi-$(r; m, p, q, h_1, h_2)$-preinvex mappings

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Technical Science, University “Ismail Qemali”, 9400, Vlora, Albania

Abstract

In this article, we first presented a new integral identity concerning differentiable mappings defined on m-invex set. By using the notion of generalized relative semi-$(r; m, p, q, h_1, h_2)$-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Ostrowski type conformable fractional integral inequalities are established. It is pointed out that some new special cases can be deduced from main results of the article.

Keywords

[1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015) 57–66.
[2] R.P. Agarwal, M.J. Luo and R.K. Raina, On Ostrowski type inequalities, Fasc. Math. 204 (2016) 5–27.
[3] M. Ahmadmir and R. Ullah, Some inequalities of Ostrowski and Gruss type for triple integrals on time scales, Tamkang J. Math. 42(4) (2011) 415–426.
[4] M. Alomari, M. Darus, S.S. Dragomir and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s–convex in the second sense, Appl. Math. Lett. 23 (2010) 1071–1076.
[5] T. Antczak, Mean value in invexity analysis, Nonlinear Anal. 60 (2005) 1473–1484.
[6] S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Math. Ineq. & Appl. 1(2) (1998).
[7] S.S. Dragomir, The Ostrowski integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl. 38 (1999) 33–37.
[8] S.S. Dragomir, Some perturbed Ostrowski type inequalities for absolutely continuous functions (I), Acta Univ. M. Belii, Ser. Math. 23 (2015) 71–86.
[9] S.S. Dragomir, Ostrowski-type inequalities for Lebesgue integral: A survey of recent results, Aust. J. Math. Anal. Appl. 14(1) (2017) 1–287.
[10] S.S. Dragomir and S. Wang, An inequality of Ostrowski–Gruss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl. 13(11) (1997) 15–20.
[11] S.S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in L1–norm and applications to some special means and to some numerical quadrature rules, Tamkang J. Math. 28 (1997) 239–244.
[12] T.S. Du, J.G. Liao and Y.J. Li, Properties and integral inequalities of Hadamard-Simpson type for the generalized (s, m)–preinvex functions, J. Nonlinear Sci. Appl. 9 (2016) 3112–3126.
[13] T.S. Du, J.G. Liao, L.Z. Chen, M.U. Awan, Properties and Riemann–Liouville fractional Hermite–Hadamard inequalities for the generalized (α, m)–preinvex functions, J. Inequal. Appl. 2016 (2016), Article ID 306 pp. 24.
[14] G. Farid, Some new Ostrowski type inequalities via fractional integrals, Int. J. Anal. App. 14(1) (2017) 64–68.
 [15] H. Hudzik and L. Maligranda, Some remarks on s–convex functions, Aequationes Math. 48 (1994) 100–111.
[16] A. Kashuri and R. Liko, Generalizations of Hermite–Hadamard and Ostrowski type inequalities for MTm–preinvex functions, Proyecciones 36(1) (2017) 45–80.
[17] A. Kashuri and R. Liko, Ostrowski type fractional integral inequalities for generalized (s, m, ϕ)–preinvex functions, Aust. J. Math. Anal. Appl. 13(1) (2016) Article 16 1–11.
[18] A. Kashuri and R. Liko, Hermite–Hadamard type fractional integral inequalities for generalized (r; s, m, ϕ)– preinvex functions, Eur. J. Pure Appl. Math. 10(3) (2017) 495–505.
[19] R. Khalil, M.A Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014) 65–70.
[20] Z. Liu, Some Ostrowski–Gruss type inequalities and applications, Comput. Math. Appl. 53 (2007) 73–79.
[21] Z. Liu, Some companions of an Ostrowski type inequality and applications, J. Inequal. Pure  Appl. Math 10(2) (2009) Art. 52 pp. 12.
[22] W. Liu, W. Wen and J. Park, Ostrowski type fractional integral inequalities for MT–convex functions, Miskolc Math. Notes 16(1) (2015) 249–256.
[23] W. Liu, W. Wen, J. Park, Hermite–Hadamard type inequalities for MT–convex, functions via classical integrals and fractional integrals, J. Nonlinear Sci. Appl. 9 (2016) 766–777.
[24] M. Mat loka, Inequalities for h–preinvex functions, Appl. Math. Comput. 234 (2014) 52–57.
[25] M. Matloka, Ostrowski type inequalities for functions whose derivatives are h–convex via fractional integrals, J. Sci. Res. Rep. 3(12) (2014) 1633–1641.
[26] D.S. Mitrinovic, J.E. Pecaric and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
[27] O. Omotoyinbo and A. Mogbodemu, Some new Hermite–Hadamard integral inequalities for convex functions, Int. J. Sci. Innov. Tech. 1(1) (2014) 1–12.
[28] M.E. Ozdemir, H. Kavurmacı and E. Set, Ostrowski’s type inequalities for (α, m)–convex functions, Kyungpook Math. J. 50 (2010) 371–378.
[29] B.G. Pachpatte, On an inequality of Ostrowski type in three independent variables, J. Math. Anal. Appl. 249 (2000) 583–591.
[30] B.G. Pachpatte, On a new Ostrowski type inequality in two independent variables, Tamkang J. Math. 32(1) (2001) 45–49.
[31] B.G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll. 6 (2003).
[32] C. Peng, C. Zhou and T.S. Du, Riemann–Liouville fractional Simpson’s inequalities through generalized (m, h1, h2) preinvexity, Ital. J. Pure Appl. Math. 38 (2017) 345–367.
[33] R. Pini, Invexity and generalized convexity, Optimization 22 (1991) 513–525.
[34] F. Qi and B.Y. Xi, Some integral inequalities of Simpson type for GA–∊–convex functions, Georgian Math. J. 20(5) (2013) 775–788.
[35] A. Rafiq, N.A. Mir and F. Ahmad, Weighted Cebysev–Ostrowski type inequalities, Appl. Math. Mech.  (English Edition) 28(7) (2007) 901–906.
[36] M.Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae 79(1) (2010) 129–134.
[37] E. Set, A.O. Akdemir and I. Mumcu, Ostrowski type inequalities for functions whose derivatives are convex via conformable fractional integrals, https://www.researchgate.net/publication/303382132 (2016) pp. 11.
[38] E. Set, A.O. Akdemir and I. Mumcu, Chebyshev type inequalities for conformable fractional integrals, Miskolc Math. Notes 20(2) (2019) 1227–1236.
[39] E. Set, M.Z. Sarikaya and A. Gozpinar, Some Hermite–Hadamard type inequalities for convex functions via conformable fractional integrals and related inequalities, Creat. Math. Inf. 26(2) (2017) 221–229.
[40] E. Set and I. Mumcu, Hermite–Hadamard–Fejer type inequities for conformable fractional integrals, Miskolc Math. Notes 20(1) (2019) 475–488.
[41] M. Tunc, Ostrowski type inequalities for functions whose derivatives are MT–convex, J. Comput. Anal. Appl. 17(4) (2014) 691–696.
[42] M. Tunc, E. Gov and U. Sanal, On tgs–convex function and their inequalities, Facta Univ. Ser. Math. Inf. 30(5) (2015) 679–691.
[43] N. Ujevic, Sharp inequalities of Simpson type and Ostrowski type, Comput. Math. Appl. 48 (2004) 145–151.
[44] S. Varosanec, On h–convexity, J. Math. Anal. Appl. 326(1) (2007) 303–311.
[45] X.M. Yang, X.Q. Yang and K.L. Teo, Generalized invexity and generalized invariant monotonicity, J. Optim. Theory Appl. 117 (2003) 607–625.
[46] C. Yildiz, M.E. Ozdemir and M.Z. Sarikaya, New generalizations of Ostrowski-like type inequalities for fractional integrals, Kyungpook Math. J. 56 (2016) 161–172.
[47] E.A. Youness, E–convex sets, E–convex functions, and E–convex programming, J. Optim. Theory Appl. 102 (1999) 439–450.
[48] Y. Zhang, T.S. Du, H. Wang, Y.J. Shen and A. Kashuri, Extensions of different type parameterized inequalities for generalized (m, h)–preinvex mappings via k–fractional integrals, J. Inequal. Appl. 2018(49) (2018) pp. 30.
[49] L. Zhongxue, On sharp inequalities of Simpson type and Ostrowski type in two independent variables, Comput. Math. Appl. 56 (2008) 2043–2047.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 945-962
  • Receive Date: 12 November 2017
  • Revise Date: 20 November 2018
  • Accept Date: 07 November 2019