Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals

Liu, Wenjun¹ ; Wen, Wangshu¹ ; Park, Jaekeun²

¹ College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
² Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 3, p. 766-777.

Résumé

Some inequalities of Hermite-Hadamard type for MT-convex functions via classical integrals and Riemann- Liouville fractional integrals are introduced, respectively, and applications for special means are given. Some error estimates for the trapezoidal formula are also obtained.

Détail

DOI : 10.22436/jnsa.009.03.05

Classification : 26A33, 33B15, 26A51, 26D10
Keywords: MT-convex function, Hermite-Hadamard inequality, Hölder inequality, fractional integral, trapezoidal formula.

Bibliographie

[1] Alomari, M.; Darus, M.; S. S. Dragomir Inequalities of Hermite-Hadamard's type for functions whose derivatives absolute values are quasi-convex, RGMIA Res. Rep. Coll., Volume 12 (2009), pp. 1-11

[2] Dahmani, Z.; Tabharit, L.; Taf, S. New generalisations of Gruss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., Volume 2 (2010), pp. 93-99

[3] Dragomir, S. S. On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., Volume 33 (2002), pp. 55-65

[4] Dragomir, S. S.; Agarwal, R. P. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula , Appl. Math. Lett., Volume 11 (1998), pp. 91-95

[5] Dragomir, S. S.; Pearce, C. E. M. Quasi-convex functions and Hadamard's inequality, Bull. Austral. Math. Soc., Volume 57 (1998), pp. 377-385

[6] o, R. Goren; Mainardi, F. Fractional calculus: integral and differential equations of fractional order, in Fractals and fractional calculus in continuum mechanics, Springer, Vienna, 1997

[7] Huy, V. N.; Chung, N. T. Some generalizations of the Fejér and Hermite-Hadamard inequalities in Hölder spaces, J. Appl. Math. Inform., Volume 29 (2011), pp. 859-868

[8] D. A. Ion Some estimates on the Hermite-Hadamard inequality through quasi-convex functions , An. Univ. Craiova Ser. Mat. Inform., Volume 34 (2007), pp. 83-88

[9] Kavurmaci, H.; Avci, M.; M. E. Özdemir New inequalities of Hermite-Hadamard type for convex functions with applications , J. Inequal. Appl., Volume 2011 (2011), pp. 1-11

[10] W. J. Liu New integral inequalities via (\(\alpha,m\))-convexity and quasi-convexity, Hacet. J. Math. Stat., Volume 42 (2013), pp. 289-297

[11] Liu, W. J. Some Ostrowski type inequalities via Riemann-Liouville fractional integrals for h-convex functions, J. Comput. Anal. Appl., Volume 16 (2014), pp. 998-1004

[12] W. J. Liu Some Simpson type inequalities for h-convex and (\(\alpha,m\))-convex functions, J. Comput. Anal. Appl., Volume 16 (2014), pp. 1005-1102

[13] Liu, W. J. Ostrowski type fractional integral inequalities for MT-convex functions, Miskolc Mathematical Notes, Volume 16 (2015), pp. 249-256

[14] Liu, W. J.; Gao, X. Y. Approximating the finite Hilbert transform via a companion of Ostrowski's inequality for function of bounded variation and applications, Appl. Math. Comput., Volume 247 (2014), pp. 373-385

[15] Liu, W. J.; Ngo, Q. A.; Chen, W. B. On new Ostrowski type inequalities for double integrals on time scales, Dynam. Systems Appl., Volume 19 (2010), pp. 189-198

[16] Liu, W. J.; Wen, W. S. Some generalizations of different type of integral inequalities for MT-convex functions, FILOMAT ( In press)

[17] Liu, W. J.; Wen, W. S.; J. Park A refinement of the difference between two integral means in terms of the cumulative variation and applications , J. Math. Inequal., Volume 10 (2016)

[18] Miller, K. S.; Ross, B. An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, Inc., New York, 1993

[19] Özdemir, M. E.; Avci, M.; Kavurmaci, Havva Hermite-Hadamard type inequalities for s-convex and s-concave functions via fractional integrals, arXiv, Volume 2012 (2012), pp. 1-9

[20] Özdemir, M. E.; Set, E.; M. Alomari Integral inequalities via several kinds of convexity, Creat. Math. Inform., Volume 20 (2011), pp. 62-73

[21] Podlubny, I. Fractional differential equations, Academic Press, Inc., San Diego, CA, 1999

[22] Qi, F.; Xi, B.-Y. Some integral inequalities of Simpson type for GA-\(\varepsilon\)-convex functions, Georgian Math. J., Volume 20 (2013), pp. 775-788

[23] M. Z. Sarikaya On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integral Transforms Spec. Funct., Volume 25 (2014), pp. 134-147

[24] Sarikaya, M. Z.; Saglam, A.; H. Yildirim On some Hadamard-type inequalities for h-convex functions, J. Math. Inequal., Volume 2 (2008), pp. 335-341

[25] Sarikaya, M. Z.; Set, E.; Özdemir, M. E. On some new inequalities of Hadamard type involving h-convex functions, Acta Math. Univ. Comenian. (N.S.), Volume 79 (2010), pp. 265-272

[26] Sarikaya, M. Z.; Set, E.; Yaldiz, H.; Basak, N. Hermite-Hadamards inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modelling, Volume 57 (2013), pp. 2403-2407

[27] M. Tunc On some integral inequalities via h-convexity , Miskolc Math. Notes, Volume 14 (2013), pp. 1041-1057

[28] Tunc, M. Ostrowski type inequalities for functions whose derivatives are MT-convex, J. Comput. Anal. Appl., Volume 17 (2014), pp. 691-696

[29] Tunc, M.; Subas, Y.; Karabayir, I. On some Hadamard type inequalities for MT-convex functions, Int. J. Open Probl. Comput. Sci. Math., Volume 6 (2013), pp. 102-113

[30] Tunc, M.; Yildirim, H. On MT-convexity, arXiv, Volume 2012 (2012), pp. 1-6