General Euler–Ostrowski formulae and applications to quadratures

The aim of this paper is to generalize inequalityf(x)-1b-a∫abf(s1)ds1-f(b)-f(a)b-ax-a+b2-f′(b)-f′(a)2(b-a)·x2-(a+b)x+a2+b2+4ab6⩽∥f‴∥∞·(b-a)36Ix-ab-aobtained 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 inequalityf(x)-1b-a∫abf(s1)ds1-f(b)-f(a)b-ax-a+b2-f′(b)-f′(a)2(b-a)·x2-(a+b)x+a2+b2+4ab6⩽∥f‴∥∞·A(x)(b-a)3obtained 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 Bn∗(x-mt) which enables us to obtain a vide variety of quadrature formulae such as trapezoid, midpoint, bitrapezoid, twopoint formulae and their multipoint generalizations.