New Upper and Lower Bounds for the Trapezoid Inequality of Absolutely Continuous Functions and Applications

Year 2019, Volume: 7 Issue: 2, 319 - 323, 15.10.2019

Mohammad W. Alomari

Abstract

In this paper, new upper and lower bounds for the Trapezoid inequality of absolutely continuous functions are obtained. Applications to some special means are provided as well.

Keywords

Hermite--Hadamard inequality, Trapezoid inequality

References

• [1] M.W. Alomari, A companion of the generalized trapezoid inequality and applications, Journal of Math. Appl., 36 (2013), 5–15.
• [2] M.W. Alomari, New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann–Stieltjes integrals and applications, Ukrainian Mathematical Journal, 65 (7) (2013), 995–1018.
• [3] M.W. Alomari, M. Darus and U.S. Kirmaci, Some inequalities of Hermite-Hadamard type for s-convex functions, Acta Mathematica Scientia, 31 B(4) (2011) : 1643–1652.
• [4] M.W. Alomari, M. Darus and U. Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comp. Math. Appl., 59 (2010), 225–232.
• [5] M. Alomari and M. Darus, On the Hadamard’s inequality for log-convex functions on the coordinates, J. Ineq. Appl., 2009, Article ID 283147, 13 pages, doi:10.1155/2009/283147.
• [6] H. Budak, F. Usta and M.Z. Sarikaya, New upper bounds of ostrowski type integral inequalities utilizing Taylor expansion, Hacettepe Journal of Mathematics and Statistics, 47 (3) (2018), 567–578.
• [7] H. Budak, F. Usta, M.Z. Sarikaya and M.E. Ozdemir, On generalization of midpoint type inequalities with generalized fractional integral operators, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 113(2) (2019), 769–790.
• [8] P. S. Bullen, D. S. Mitrinovi´c and M. Vasi´c”, Means and Their Inequalities, Dordrecht: Kluwer Academic, 1988.
• [9] P. S. Bullen, Handbook of Means and Their Inequalities, Dordrecht: Kluwer Academic, 2003.
• [10] S.S. Dragomir, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl., 167 (1992) 49–56.
• [11] S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998) 91–95.
• [12] S. S. Dragomir and C. E. M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications,” RGMIA Monographs, Victoria University, 2000, http://www.staff.vu.edu.au/RGMIA/ monographs/hermite hadamard.html.
• [13] S.S. Dragomir, Y.J. Cho and S.S. Kim, Inequalities of Hadamard’s type for Lipschitzian mappings and their applicaitions, J. Math. Anal. Appl., 245 (2000), 489–501.
• [14] D.A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Comp. Sci. Ser., 34 (2007), 82–87.
• [15] U.S. Kirmaci, Inequalities for differentiable mappings and applicatios to special means of real numbers to midpoint formula, Appl. Math. Comp., 147 (2004), 137–146.
• [16] U.S. Kirmaci and M.E. O¨ zdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153 (2004), 361–368.
• [17] M.E. O¨ zdemir, A theorem on mappings with bounded derivatives with applications to quadrature rules and means, Appl. Math. Comp., 138 (2003), 425–434.
• [18] C.E.M. Pearce and J. Peˇcari´c, Inequalities for differentiable mappings with application to special means and quadrature formula, Appl. Math. Lett., 13 (2000) 51–55.
• [19] G.S. Yang, D.Y. Hwang and K.L. Tseng, Some inequalities for differentiable convex and concave mappings, Comp. Math. Appl., 47 (2004), 207–216.
• [20] F. Usta, H. Budak, M.Z. Sarikaya and E. Set, On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators, Filomat 32 (6), 2153–2171.
• [21] F. Usta, H. Budak and M.Z. Sarikaya, Montgomery identities and Ostrowski type inequalities for fractional integral operators, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 113 (2) (2019), 1059–1080.