Error inequalities for an optimal 3-point quadrature formula of closed type

In recent years a number of authors have considered an error analysis for quadrature rules of Newton-Cotes type. In particular, the mid-point, trapezoid and Simpson rules have been investigated more recently ([2], [4], [5], [6], [11]) with the view of obtaining bounds on the quadrature rule in terms of a variety of norms involving, at most, the first derivative. In the mentioned papers explicit error bounds for the quadrature rules are given. These results are obtained from an inequalities point of view. The authors use Peano type kernels for obtaining a specific quadrature rule. Quadrature formulas can be formed in many different ways. For example, we can integrate a Lagrange interpolating polynomial of a function f to obtain a corresponding quadrature formula (Newton-Cotes formulas). We can also seek a quadrature formula such that it is exact for polynomials of maximal degree (Gauss formulas). Gauss-like quadrature formulas are considered in [12]. Here we present a new approach to this topic. Namely, we give a type of quadrature formula. We also give a way of estimation of its error and all parameters which appear in the estimation. Then we seek a quadrature formula of the given type such that the estimation of its error is best possible. Let us consider the above described procedure with more details. If we define