General Euler-Ostrowski formulae and applications to quadratures

The aim of this paper is to generalize inequality 0096-3 doi:10 * Co E-m f ðxÞ 1 b a Z b a f ðs1Þds1 f ðbÞ f ðaÞ b a x aþ b 2 f 0ðbÞ f 0ðaÞ 2ðb aÞ x 2 ðaþ bÞxþ a 2 þ b þ 4ab 6 6 kf k1 ðb aÞ 3 6 I x a b a obtained in [A. Aglić Aljinović, M. Matić, J. Pečarić, Improvements of some Ostrowski type inequalities, J. Comput. Anal. Appl., in press], and therefore obtain a generalization and improvement of inequality f ðxÞ 1 b a Z b a f ðs1Þds1 f ðbÞ f ðaÞ b a x aþ b 2 f 0ðbÞ f 0ðaÞ 2ðb aÞ x 2 ðaþ bÞxþ a 2 þ b þ 4ab 6 6 kf k1 AðxÞ ðb aÞ obtained in [G.A. Anastassiou, Univariate Ostrowski inequalities, Revisited, Monatsh. Math. 135 (2002) 175–189]. To do this, first we derive general Euler–Ostrowski formulae which generalize extended Euler formulae, obtained in [Lj. Dedić, M. Matić, J. Pečarić, On generalizations of Ostrowski inequality via some Euler-type identities, Math. Inequal. Appl. 3(3) (2000) 337–353]. The main novelty is that a remainder is expressed in terms of B nðx mtÞ which enables us to obtain a vide variety of quadrature formulae such as trapezoid, midpoint, bitrapezoid, twopoint formulae and their multipoint generalizations. 2005 Elsevier Inc. All rights reserved.