Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

Abstract

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], Pečarić 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 Aglić Aljinović and Pečarić 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 , respectively, and adding, we get (2.2).

Remark 2.2. Identity (2.2) holds true in the case . It can also be obtained by taking and in (1.5), multiplying these four formulae by , respectively, and adding. In this special case we have where

Theorem 2.3. Suppose that all assumptions of Theorem 2.1 hold. Additionally, assume that is a pair of conjugate exponents, that is, , , let for some . Then for each we have Inequality (2.5) is sharp for .

Proof. By applying the Hölder inequality we have By using the above inequality from (2.2) we obtain estimate (2.5). Let us denote . For the proof of sharpness, we will find a function such that For , take to be such that where for we put

Remark 2.4. Inequality (2.5) for was proved by Aglić Aljinović et al. in [4].

3. Non-Weighted Four-Point Formula and Applications

Here we define

Theorem 3.1. Let be an open interval in , , and let be such that is absolutely continuous for some . Then for each the following identity holds

Proof. We take , in (2.2).

Theorem 3.2. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that is a pair of conjugate exponents, that is, , and for some . Then for each we have Inequality (3.4) is sharp for .

Proof. We take , in (2.5).

Now, we set This special choice of the function enables us to consider generalizations of the well-known Simpson’s formula (1.1) and Lobatto formula (1.2)

3.1.

Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Simpson’s formula reads where

In the next corollaries we will use the beta function and the incomplete beta function of Euler type defined by

Corollary 3.3. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that is a pair of conjugate exponents and . (a)If , then (b) If , then (c) If , then where , , and , for .
The first and the second inequality are sharp.