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On Simpson’s inequality and applications. (English) Zbl 0976.26012
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 $[a,b]$. 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)].

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