General Euler-Ostrowski formulae and applications to quadratures

The aim of this paper is to generalize inequality |f(x) - 1/b-a ∫ab f(s1) ds1 - f(b) - f(a)/b-a(x - a+b/2) -f'(b) - f'(a)/2(b-a).(x2-(a + b)x + a2 + b2 + 4ab/6)| ≤ ||f'''||∞.(b-a)3/6I(x-a/b-a) obtained in [A. Aglic Aljinovic, M. Matic, J. Pecaric, Improvements of some Ostrowski type inequalities, J. Comput. Anal. Appl., in Press], and therefore obtain a generalization and improvement of inequality |f(x) - 1/b-a ∫ab f(s1) ds1 - f(b)-f(a)/b-a (x - a+b/2) -f'(b)-f'(a)/2(b - a)ċ(x2 - (a + b)x + a2 + b2 + 4ab/6)| ≤ ||f'''||∞ċA(x)/(b-a)3 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. Dedic, M. Matic, J. Pecaric, 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.