%0 Journal Article
%T Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity
%A J. Pe£¿ari£¿
%A M. Ribi£¿i£¿ Penava
%J International Journal of Mathematics and Mathematical Sciences
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/343191
%X We consider families of general four-point quadrature formulae using a generalization of the Montgomery identity via Taylor¡¯s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong to spaces. Generalizations of Simpson¡¯s 3/8 formula and the Lobatto four-point formula with related inequalities are considered as special cases. 1. Introduction The most elementary quadrature rules in four nodes are Simpson¡¯s rule based on the following four point formula where , and Lobatto rule based on the following four point formula where . Formula (1.1) is valid for any function with a continuous fourth derivative on and formula (1.2) is valid for any function with a continuous sixth derivative on . Let be differentiable on and integrable on . Then the Montgomery identity holds (see [1]) where the Peano kernel is In [2], Pe£¿ari£¿ proved the following weighted Montgomery identity where is some probability density function, that is, integrable function, satisfying , and for , for and for and is the weighted Peano kernel defined by Now, let us suppose that is an open interval in , , is such that is absolutely continuous for some , is a probability density function. Then the following generalization of the weighted Montgomery identity via Taylor¡¯s formula states (given by Agli£¿ Aljinovi£¿ and Pe£¿ari£¿ in [3]) where and If we take , , equality (1.7) reduces to where and For , (1.9) reduces to the Montgomery identity (1.3). In this paper, we generalize the results from [4]. Namely, we use identities (1.7) and (1.9) to establish for each number a general four-point quadrature formula of the type where is the remainder and is a real function. The obtained formula is used to prove a number of inequalities which give error estimates for the general four-point formula for functions whose derivatives are from -spaces. These inequalities are generally sharp. As special cases of the general non-weighted four-point quadrature formula, we obtain generalizations of the well-known Simpson¡¯s 3/8 formula and Lobatto four-point formula with related inequalities. 2. General Weighted Four-Point Formula Let be such that exists on for some . We introduce the following notation for each : In the next theorem we establish the general weighted four-point formula. Theorem 2.1. Let be an open interval in , , and let be some probability density function. Let be such that is absolutely continuous for some . Then for each the following identity holds Proof. We put and in (1.7) to obtain four new formulae. After multiplying these four formulae by ,
%U http://www.hindawi.com/journals/ijmms/2012/343191/