×

A companion of Dragomir’s generalization of the Ostrowski inequality and applications to numerical integration. (English. Russian original) Zbl 1257.26016

Ukr. Math. J. 64, No. 4, 491-510 (2012); translation from Ukr. Mat. Zh. 64, No. 4, 435-450 (2012).
Summary: Some analogs of Dragomir’s generalization of the Ostrowski integral inequality \[ \begin{split} \bigg| (b-a)\bigg[\lambda\frac{f(a)+f(b)}{2}+(1-\lambda)f(x)\bigg]-\int^b_af(t)dt\bigg| \\ \leq \bigg[\frac{(b-a)^2}{4}(\lambda^2+(1-\lambda)^2)+(x-\frac{a+b}{2})^2\bigg]\|f'\|_\infty \end{split} \] are established. Some sharp inequalities are proved. An application to the composite quadrature rule is provided.

MSC:

26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] M. W. Alomari, M. Darus, and U. S. Kirmaci, ”Some inequalities of Hermite–Hadamard type for s-convex functions,” Acta Math. Sci., 31B, No. 4, 1643–1652 (2011). · Zbl 1249.26042 · doi:10.1016/S0252-9602(11)60350-0
[2] M. W. Alomari, ”A companion of Ostrowski’s inequality with applications,” Trans. J. Math. Mech., 3, 9–14 (2011). · Zbl 1233.26006
[3] M. Alomari and S. Hussain, ”Two inequalities of Simpson type for quasiconvex functions and applications,” Appl. Math. E-Notes, 11, 110–117 (2011). · Zbl 1223.26039
[4] M. Alomari, M. Darus, and U. Kirmaci, ”Refinements of Hadamard-type inequalities for quasiconvex functions with applications to trapezoidal formula and to special means,” Comput. Math. Appl., 59, 225–232 (2010). · Zbl 1189.26037 · doi:10.1016/j.camwa.2009.08.002
[5] M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, ”Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense,” Appl. Math. Lett., 23, 1071–1076 (2010). · Zbl 1197.26021 · doi:10.1016/j.aml.2010.04.038
[6] M. Alomari, M. Darus, and S. S. Dragomir, ”New inequalities of Hermite–Hadamard type for functions whose second derivatives absolute values are quasiconvex,” Tamkang J. Math., 41, 353–359 (2010). · Zbl 1214.26003
[7] M. Alomari and M. Darus, ”On some inequalities of Simpson-type via quasiconvex functions and applications,” Trans. J. Math. Mech., 2, 15–24 (2010).
[8] M. Alomari and M. Darus, ”Some Ostrowski type inequalities for quasiconvex functions with applications to special means,” RGMIA Preprint, 13, No. 2, Article No. 3 (2010) [ http://rgmia . org/papers/v13n2/quasi-convex. pdf]. · Zbl 1189.26037
[9] N. S. Barnett, S. S. Dragomir, and I. Gomma, ”A companion for the Ostrowski and the generalized trapezoid inequalities,” J. Math. Comput. Modelling, 50, 179–187 (2009). · Zbl 1185.26038 · doi:10.1016/j.mcm.2009.04.005
[10] P. Cerone and S. S. Dragomir, ”Midpoint-type rules from an inequalities point of view,” in: G. Anastassiou (editor), Handb. Anal. Comput. Methods Appl. Math., CRC Press, New York (2000), pp. 135–200. · Zbl 0966.26015
[11] P. Cerone and S. S. Dragomir, ”Trapezoidal-type rules from an inequalities point of view,” in: G. Anastassiou (editor), Handb. Anal. Comput. Methods Appl. Math., CRC Press, New York (2000), pp. 65–134. · Zbl 0966.26014
[12] A. Guessab and G. Schmeisser, ”Sharp integral inequalities of the Hermite–Hadamard type,” J. Approxim. Theory, 115, 260–288 (2002). · Zbl 1012.26013 · doi:10.1006/jath.2001.3658
[13] S. S. Dragomir and T. M. Rassias (editors), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer, Dordrecht (2002). · Zbl 0992.26002
[14] S. S. Dragomir, ”Some companions of Ostrowski’s inequality for absolutely continuous functions and applications,” Bull. Korean Math. Soc., 42, No. 2, 213–230 (2005). · Zbl 1099.26015 · doi:10.4134/BKMS.2005.42.2.213
[15] S. S. Dragomir, ”A companion of Ostrowski’s inequality for functions of bounded variation and applications,” RGMIA Preprint, 5, Suppl, Article No. 28 (2002) [ http://ajmaa.org/RGMIA/papers/v5e/COIFBVApp.pdf ].
[16] S. S. Dragomir, P. Cerone, and J. Roumeliotis, ”A new generalization of Ostrowski integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means,” Appl. Math. Lett., 13, No. 1, 19–25 (2000). · Zbl 0946.26013 · doi:10.1016/S0893-9659(99)00139-1
[17] S. S. Dragomir, R. P. Agarwal, and P. Cerone, ”On Simpson’s inequality and applications,” J. Inequal. Appl., 5, 533–579 (2000). · Zbl 0976.26012
[18] S. S. Dragomir and C. E. M. Pearce, ”Selected topics on Hermite–Hadamard inequalities and applications,” RGMIA Monographs, Victoria Univ. (2000); online: [ http://www.staff.vu.edu.au/RGMIA/monographs/hermite hadamard.html].
[19] Z. Liu, ”Some companions of an Ostrowski type inequality and applications,” J. Inequal. Pure and Appl. Math., 10, Issue 2, Article 52, (2009). · Zbl 1168.26310
[20] N. Ujević, ”A generalization of Ostrowski’s inequality and applications in numerical integration,” Appl. Math. Lett., 17, 133–137 (2004). · Zbl 1057.26023 · doi:10.1016/S0893-9659(04)90023-7
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.