Estimates of Errors of Quadrature Formulas by Linear Combinations of the Uniform Norm and Oscillation of Derivatives with Both Sharp Constants

Abstract

We establish inequalities of the following form. If f ∈ C ⁽¹⁾[a,b], and x ∈ [a,b], then

$$|{\int\limits_a^b {f-(b-a)f(x)}}| \leqslant (\min {x-a, b-x})^2 \frac{{\omega (f')}}{2}+(b-a) |{x - \frac{{a+b}}{2}}| ||f'||,$$

where ω(f') = max f' – min f'. Moreover, it is impossible to decrease the sum of constants at the norm and half oscillation on the right-hand side of the inequality. We cannot decrease the constant at the norm even if we increase the constant at the oscillation. This inequality implies the known sharp Ostrowski inequality. Bibliography: 7 titles.