Generalized Ostrowski-type inequalities involving second derivatives via the Katugampola fractional integrals

Kermausuor, Seth ¹

¹ Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA

Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 8, p. 509-522.

Résumé

In this paper, we provide some Ostrowski-type integral inequalities for functions whose second derivatives belongs to the Lebesgue $L_q$ spaces using the Katugampola fractional integrals. We also introduced some new inequalities of Ostrowski-type for functions whose second derivatives in absolute value at some powers are strongly $(s, m)$-convex with modulus $\mu\geq0$ (in the second sense). Our results are generalizations of some earlier results in the literature.

Détail

DOI : 10.22436/jnsa.012.08.02

Classification : 26A33, 26A51, 26D10, 26D15
Keywords: Ostrowski inequality, convex functions, strongly \((s, m)\)-convex functions, Riemann--Liouville fractional integrals, Hadamard fractional integrals, Katugampola fractional integrals, Hölder's inequality, power mean inequality

Bibliographie

[1] Alomari, M.; Darus, M.; Dragomir, S. S.; Cerone, P. Ostrowski type inequalities for the functions whose derivative are $s$-convex in second sense, Appl. Math. Lett., Volume 23 (2010), pp. 1071-1076 | DOI

[2] Anastassiou, G. A. Ostrowski type inequalities, Proc. Amer. Math. Soc., Volume 123 (1995), pp. 3775-3781 | DOI

[3] Bracamonte, M.; Giménez, J.; Vivas-Cortez, M. Hermite-Hadamard-Fejér type inequalities for strongly $(s,m)$-convex functions with modulus $c$, in second sense, Appl. Math. Inf. Sci., Volume 10 (2016), pp. 2045-2053

[4] Breckner, W. W. Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen (German), Publ. Inst. Math. (Beograd) (N.S.), Volume 23 (1978), pp. 13-20 | Zbl

[5] Chen, H.; Katugampola, U. N. Hermite--Hadamard and Hermite--Hadamard--Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., Volume 446 (2017), pp. 1274-1291 | Zbl | DOI

[6] Dragomir, S. S. A generalization of Ostrowski integral inequality for mappings whose derivatives belong to $L_1[a,b]$ and applications in numerical integration, J. Comput. Anal. Appl., Volume 3 (2001), pp. 343-360 | DOI

[7] Dragomir, S. S. A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to $L_p[a,b]$ and applications in numerical integration, J. Math. Anal. Appl., Volume 255 (2001), pp. 605-626 | DOI | Zbl

[8] Dragomir, S. S.; Wang, S. A new inequality of Ostrowski's type in $ L_1$-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. Math., Volume 28 (1997), pp. 239-244

[9] Dragomir, S. S.; Wang, S. A new inequality of Ostrowski’s type in $L_p$-norm, Indian J. Math., Volume 40 (1998), pp. 299-304

[10] Eftekhari, N. Some remarks on $(s, m)$-convexity in the second sense, J. Math. Inequal., Volume 8 (2014), pp. 489-495

[11] Farid, G.; Katugampola, U. N.; Usman, M. Ostrowski type fractional integral inequalities for $s$-Godunova--Levin functions via Katugampola fractional integrals, Open J. Math. Sci., Volume 1 (2017), pp. 97-110

[12] Farid, G.; Katugampola, U. N.; Usman, M. Ostrowski-type fractional integral inequalities for mappings whose derivatives are $h$-convex via Katugampola fractional integrals, Stud. Univ. Babeş-Bolyai Math., Volume 63 (2018), pp. 465-474

[13] Farid, G.; Usman, M. Ostrowski type $k$-fractional integral inequalities for MT-convex and $h$-convex functions, Nonlinear funct. Anal. Appl., Volume 22 (2017), pp. 627-639 | Zbl

[14] Katugampola, U. N. New approach to a generalized fractional integral, Appl. Math. Comput., Volume 218 (2011), pp. 860-865 | Zbl | DOI

[15] Katugampola, U. N. A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., Volume 6 (2014), pp. 1-15 | Zbl

[16] Meftaha, B.; Boukerrioua, K. Some new Ostrowski type inequalities for functions whose second derivative are $h$-convex via Riemann--Liouville fractional, Malaya J. Mat., Volume 2 (2014), pp. 445-459

[17] Ostrowski, A. M. Uber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmitelwert, Comment. Math. Helv., Volume 10 (1938), pp. 226-227

[18] Podlubny, I. Fractional differential equations, Academic Press, San Diego, 1999 | Zbl

[19] Polyak, B. T. Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., Volume 7 (1966), pp. 72-75 | Zbl

[20] Samko, S. G.; Kilbas, A. A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993 | Zbl

[21] Set, E. New inequalities of Ostrowski type for mappings whose derivatives are $s$-convex in the second sense via fractional integrals, Comput. Math. Appl., Volume 63 (2012), pp. 1147-1154 | Zbl | DOI

[22] Set, E.; Mumcu, I. Hermite--Hadamard Type Inequalities for Quasi-Convex Functions via Katugampola Fractional Integrals, Int. J. Anal. Appl., Volume 16 (2018), pp. 605-613 | Zbl

[23] Toader, G. Some generalizations of the Convexity, Proc. Colloq. Approx. Optim. Cluj-Naploca (Cluj-Napoca, 1985), Volume 1985 (1985), pp. 329-338 | Zbl