Simpson’s type inequalities via the Katugampola fractional integrals for \(s\)-convex functions. (English) Zbl 1499.26134
Summary: In this paper, we introduce some Simpson’s type integral inequalities via the Katugampola fractional integrals for functions whose first derivatives at certain powers are \(s\)-convex (in the second sense). The Katugampola fractional integrals are generalizations of the Riemann-Liouville and Hadamard fractional integrals. Hence, our results generalize some results in the literature related to the Riemann-Liouville fractional integrals. Results related to the Hadamard fractional integrals could also be derived from our results.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 26A33 | Fractional derivatives and integrals |
| 26A51 | Convexity of real functions in one variable, generalizations |
Keywords:
Simpson-type inequalities; Hölder inequality; \(s\)-convexity; Katugampola fractional integrals; Riemann-Liouville fractional integrals; Hadamard fractional integralsReferences:
| [1] | M. Alomari and M. Darus,On some inequalities of Simpson-type via quasi-convex functions with applications, Tran. J. Math. Mech.2(2010), 15-24. |
| [2] | M. Alomari and S. Hussain,Two inequalities of Simpson type for quasiconvex functions and applications, Appl. Math. E-Notes11(2011), 110-117. · Zbl 1223.26039 |
| [3] | W. W. Breckner,Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen, Publ. Inst. Math. (Beograd)23(1978), 13-20. · Zbl 0416.46029 |
| [4] | H. Chen and U. N. Katugampola,Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl.446(2) (2017), 1274-1291. · Zbl 1351.26011 |
| [5] | J. Chen and X. Huang,Some new Inequalities of Simpson’s type fors-convex functions via fractional integrals, Filomat31(15) (2017), 4989-4997. · Zbl 1499.26107 |
| [6] | S. S. Dragomir,On Simpson’s quadrature formula for mappings of bounded variation and applications, Tamkang J. Math.30(1) (1999), 53-58. · Zbl 0989.26019 |
| [7] | S. S. Dragomir, R. P. Agarwal and P. Cerone,On Simpson’s inequality and applications, J. Inequal. Appl.5(2000), 533-579. · Zbl 0976.26012 |
| [8] | S. S. Dragomir, J. E. Pečarić and S. Wang,The unified treatment of trapezoid, Simpson and Ostrowski type inequalities for monotonic mappings and applications, J. Inequal. Appl.31(2000), · Zbl 1042.26507 |
| [9] | U.N. Katugampola,New approach to a generalized fractional integral, Appl. Math. Comput. 218(3) (2011), 860-865. · Zbl 1231.26008 |
| [10] | U. N. Katugampola,A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl.6(4) (2014), 1-15. · Zbl 1317.26008 |
| [11] | M. Matloka,Some inequalities of Simpson type forh-convex functions via fractional integrals, Abstr. Appl. Anal.2015(2015), Article ID 956850, 5 pages. · Zbl 1433.26016 |
| [12] | I. Podlubny,Fractional Differential Equations: Mathematics in Science and Engineering, Academic Press, San Diego, CA, 1999. · Zbl 0924.34008 |
| [13] | S. G. Samko, A. A. Kilbas and O. I. Marichev,Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993. · Zbl 0818.26003 |
| [14] | M. Z. Sarikaya, E. Set and M. E. Özdemir,On new inequalities of Simpson’s fors-convex functions, Comput. Math. Appl.60(2010), 2191-2199. · Zbl 1205.65132 |
| [15] | N. Ujević,Two sharp inequalities of Simpson type and applications, Georgian Math. J.1(11) (2004), 187-194. · Zbl 1061.26021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
