More on the averaged midpoint-trapezoid type rules

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: Keywords: Simpson type rule Averaged midpoint-trapezoid rule Corrected averaged midpoint-trapezoid rule Sharp bound Absolutely continuous Bounded variation Lipschitzian mapping a b s t r a c t A generalization of the corrected averaged midpoint-trapezoid rule is derived. Various error bounds for this generalization are established. In [7–9], Ujevic´has considered a simple 3-point quadrature rule of the form Z b a f ðxÞdx À b À a 2 f a þ b 2 þ f ðaÞ þ f ðbÞ 2 ! ¼ Rðf Þ; ð1Þ which is a convex combination of the mid-point quadrature rule and the trapezoid quadrature rule, and finally has been called as an averaged midpoint-trapezoid rule. In [2,3], Dragomir et al. and the author have shown that the averaged midpoint-trapezoid rule has a better estimation of error than the well-known Simpson-type rule when we estimate the error in terms of the first derivative f 0 of integrand f as Z b a f ðxÞdx À b À a 2 f a þ b 2 þ f ðaÞ þ f ðbÞ 2 ! 6 ðb À aÞ 2 8 kf 0 k 1 ; where f : [a, b] ? R is absolutely continuous on [a, b] and the derivative f 0 2 L 1 [a, b], and Z b a f ðxÞdx À b À a 2 f a þ b 2 þ f ðaÞ þ f ðbÞ 2 ! 6 ðb À aÞ 2 16 ðC À cÞ; where f : [a, b] ? R is absolutely continuous on [a, b] and c, C 2 R are constants such that c 6 f 0 ðxÞ 6 C a.e. on [a, b]. Similar results have also been obtained in [8] in a different way. In [1], Cerone and Dragomir have proved for mappings that are of bounded variation, Lipschitzian or monotonic, the averaged midpoint-trapezoid rule (1) gives tighter bounds than a Simpson-type rule and produces the best bounds. In [4–6], the author has paid more attention on the averaged midpoint-trapezoid rule (1) and the following corrected (perturbed) averaged midpoint-trapezoid rule.