Some new generalizations of Hadamard--type Midpoint inequalities involving fractional integrals

B. Bayraktar

Problemy analiza, Tome 9 (2020) no. 3, pp. 66-82.

Résumé

In this study, we formulate the identity and obtain some generalized inequalities of the Hermite–Hadamard type by using fractional Riemann–Liouville integrals for functions whose absolute values of the second derivatives are convex. The results are obtained by uniformly dividing a segment $[a,b]$ into $n$ equal sub-intervals. Using this approach, the absolute error of a Midpoint inequality is shown to decrease approximately $n^{2}$ times. A dependency between accuracy of the absolute error ($\varepsilon $) of the upper limit of the Hadamard inequality and the number ($n$) of lower intervals is obtained.

Détail

Keywords: convexity, Hadamard inequality, Holder's inequality, Power-mean inequality, Riemann-Liouville fractional integrals.

Bibliographie

[1] Bayraktar B., “Some integral inequalities Of Hermite–Hadamard type for differentiable $(s, m)$–convex functions via fractional integrals”, TWMS J. App. Eng. Math., 10:3 (2020), 625–637 | MR

[2] Boglaev Yu. P., Computational Mathematics and Programming, Higher School, M., 1990

[3] Butkovskii A. G., Postnov S. S., Postnova E. A., “Fractional integro–differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation”, Automation and Remote Control, 74:4 (2013), 543–574 | DOI | MR | Zbl

[4] Dragomir S. S., Pearce C. E. M., Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000 www.sta.vu.edu.au/RGMIA/monographs/hermite-hadamard.html

[5] Ekinci A., Őzdemir M. E., “Some new integral inequalities via Riemann–Liuouville integral operators”, Appl. Comput. Math., 18:3 (2019), 288–295 | MR | Zbl

[6] Erdem S, Sarıkaya M. Z., Çelik N., “Some generalized inequalities involving local fractional integrals and their applications for random variables and numerical integration”, Journal of Applied Mathematics, Statistics and Informatics, 12 (2016), 49–65 | DOI | MR

[7] Gürbüz M., Öztürk O., “Inequalities generated with Riemann – Liouville fractional integral operator”, TWMS Journal of Applied and Engineering Mathematics, 9:1 (2019), 91–100

[8] Hadamard J., “Étude sur les propriétés des fonctions entières en particulier d'une fonction considérée par Riemann”, J. Math. Pures Appl., 58 (1893), 171–215

[9] Latif M. A., Dragomir S. S., “New inequalities of Hermite–Hadamard type for functions whose derivatives in absolute value are convex with applications”, Acta Univ. M. Belii, Ser. Math., 21 (2013), 27–42 | MR | Zbl

[10] Matłoka M., “Hermite–Hadamard type inequalities for fractional integrals”, RGMIA Res. Rep. Coll., 20 (2017), 69, 11 pp. | Zbl

[11] Miller S., Ross B., An introduction to the fractional calculus and fractional differential equations, John Wiley Sons, USA, 1993 | MR | Zbl

[12] Mitrinovic D. S., Peĉariĉ J., Fink A. M., Classical and new inequalities in analysis, Kluwer Academic, Dordrecth, 1993 | MR | Zbl

[13] Mohammed P. O., Hamasalh F. K., “New conformable fractional integral inequalities of Hermite–Hadamard type for convex functions”, Symmetry, 11:2 (2019), 263 | DOI | MR | Zbl

[14] Nakhushev A. M., Fractional calculus and its application, Monographs, Fizmatlit, M., 2003 (in Russian)

[15] Napoles V. J. E., Rodriguez J. M., Sigarreta J. M., “New Hermite – Hadamard type inequalities involving non – conformable integral operators”, Symmetry, 11:9 (2019), 1108 | DOI

[16] Őzdemir M. E., Ekinci A., Akdemir A. O., “Some new integral inequalities for functions whose derivatives of absolute values are convex and concave”, TWMS J. Pure Appl. Math., 10:2 (2019), 212–224 | MR

[17] Sarıkaya M. Z., Aktan N., “On the generalization of some integral inequalities and their applications”, Math. Comput. Modelling, 54 (2011), 2175–2182 | DOI | MR | Zbl