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Zbl 0976.26012
Dragomir, S.S.; Agarwal, R.P.; Cerone, P.
On Simpson's inequality and applications. (English)
[J] J. Inequal. Appl. 5, No.6, 533-579 (2000). ISSN 1029-242X/e

This is a survey paper on recent developments on Simpson's inequality, Simpson's quadrature formula and various related results. The following theorem is typical: Let $f: [a,b]\to\bbfR$ be of bounded variation on $[a,b]$. Then $$\Biggl|\int^b_a f(x) dx- {b-a\over 6} \Biggl[ f(a)+ 4f\Biggl({a+ b\over 2}\Biggr)+ f(b)\Biggr]\Biggr|\le {1\over 3} (b-a) \bigvee^b_a (f),$$ where $\bigvee^b_a(f)$ is the total variation of $f$ on $[a,b]$.\par Except two of the 38 cited papers, all are due to Dragomir et al. Each chapter in this paper is concluded with certain applications of the results for special means of two arguments. It is not mentioned that the first application to special means of Simpson's quadrature formula is due to the reviewer [Arch. Math. 56, No. 5, 471-473 (1991; Zbl 0693.26005); see also Aequationes Math. 40, No. 2/3, 261-270 (1990; Zbl 0717.26014)].
[József Sándor (Cluj-Napoca)]

MSC 2000:: *26D15 Inequalities for sums, series and integrals of real functions
26D20 Analytical inequalities involving real functions
41A55 Approximate quadratures
65D32 Quadrature formulas (numerical methods)

Keywords: Simpson's inequality; Simpson's quadrature formula; total variation; special means

Citations: Zbl 0693.26005; Zbl 0717.26014

Cited in: Zbl 1034.65018

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