Generalized Ostrowski-type inequalities involving second derivatives via the Katugampola fractional integrals
Volume 12, Issue 8, pp 509--522
http://dx.doi.org/10.22436/jnsa.012.08.02
Publication Date: March 18, 2019
Submission Date: December 27, 2018
Revision Date: February 20, 2019
Accteptance Date: February 28, 2019
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Authors
Seth Kermausuor
- Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA.
Abstract
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.
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ISRP Style
Seth Kermausuor, Generalized Ostrowski-type inequalities involving second derivatives via the Katugampola fractional integrals, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 8, 509--522
AMA Style
Kermausuor Seth, Generalized Ostrowski-type inequalities involving second derivatives via the Katugampola fractional integrals. J. Nonlinear Sci. Appl. (2019); 12(8):509--522
Chicago/Turabian Style
Kermausuor, Seth. "Generalized Ostrowski-type inequalities involving second derivatives via the Katugampola fractional integrals." Journal of Nonlinear Sciences and Applications, 12, no. 8 (2019): 509--522
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
MSC
References
-
[1]
M. Alomari, M. Darus, S. S. Dragomir, P. Cerone, Ostrowski type inequalities for the functions whose derivative are $s$-convex in second sense, Appl. Math. Lett., 23 (2010), 1071--1076
-
[2]
G. A. Anastassiou, Ostrowski type inequalities, Proc. Amer. Math. Soc., 123 (1995), 3775--3781
-
[3]
M. Bracamonte, J. Giménez, M. Vivas-Cortez, Hermite-Hadamard-Fejér type inequalities for strongly $(s,m)$-convex functions with modulus $c$, in second sense, Appl. Math. Inf. Sci., 10 (2016), 2045--2053
-
[4]
W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen (German), Publ. Inst. Math. (Beograd) (N.S.), 23 (1978), 13--20
-
[5]
H. Chen, U. N. Katugampola, Hermite--Hadamard and Hermite--Hadamard--Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274--1291
-
[6]
S. S. Dragomir, 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., 3 (2001), 343--360
-
[7]
S. S. Dragomir, 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., 255 (2001), 605--626
-
[8]
S. S. Dragomir, S. Wang, 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., 28 (1997), 239--244
-
[9]
S. S. Dragomir, S. Wang, A new inequality of Ostrowski’s type in $L_p$-norm, Indian J. Math., 40 (1998), 299--304
-
[10]
N. Eftekhari, Some remarks on $(s, m)$-convexity in the second sense, J. Math. Inequal., 8 (2014), 489--495
-
[11]
G. Farid, U. N. Katugampola, M. Usman, Ostrowski type fractional integral inequalities for $s$-Godunova--Levin functions via Katugampola fractional integrals, Open J. Math. Sci., 1 (2017), 97--110
-
[12]
G. Farid, U. N. Katugampola, M. Usman, Ostrowski-type fractional integral inequalities for mappings whose derivatives are $h$-convex via Katugampola fractional integrals, Stud. Univ. Babeş-Bolyai Math., 63 (2018), 465--474
-
[13]
G. Farid, M. Usman, Ostrowski type $k$-fractional integral inequalities for MT-convex and $h$-convex functions, Nonlinear funct. Anal. Appl., 22 (2017), 627--639
-
[14]
U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860--865
-
[15]
U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1--15
-
[16]
B. Meftaha, K. Boukerrioua, Some new Ostrowski type inequalities for functions whose second derivative are $h$-convex via Riemann--Liouville fractional, Malaya J. Mat., 2 (2014), 445--459
-
[17]
A. M. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmitelwert, Comment. Math. Helv., 10 (1938), 226--227
-
[18]
I. Podlubny, Fractional differential equations, Academic Press, San Diego (1999)
-
[19]
B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 72--75
-
[20]
S. G. Samko, A. A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and applications, Gordon and Breach Science Publishers, Yverdon (1993)
-
[21]
E. Set, New inequalities of Ostrowski type for mappings whose derivatives are $s$-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), 1147--1154
-
[22]
E. Set, I. Mumcu, Hermite--Hadamard Type Inequalities for Quasi-Convex Functions via Katugampola Fractional Integrals, Int. J. Anal. Appl., 16 (2018), 605--613
-
[23]
G. Toader, Some generalizations of the Convexity, Proc. Colloq. Approx. Optim. Cluj-Naploca (Cluj-Napoca, 1985), 1985 (1985), 329--338