International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 343191, 12 pages

doi:10.1155/2012/343191

Research Article

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

J. Pečarić¹ and M. Ribičić Penava²

¹Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

²Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia

Received 28 March 2012; Accepted 10 July 2012

Academic Editor: Marianna Shubov

Copyright © 2012 J. Pečarić and M. Ribičić Penava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Proof. We apply (3.4) with and

for and

To obtain the second inequality we take

If , we have

By an elementary calculation we get

for . The function , , is decreasing on and increasing on if is even, and decreasing on if is odd. Thus

Finally,

and for

3.2. ,  

Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Lobatto formula reads

where

Corollary 3.4. 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 , , , , , , for .

The first and the second inequality are sharp.

Proof. Applying (3.4) with , and and carrying out the same analysis as in Corollay 3.3 we obtain the above inequalities.

References

D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, vol. 53 of Mathematics and its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.

J. E. Pečarić, “On the Čebyšev inequality,” Buletinul Ştiinţific al Universităţii Politehnica din Timişoara. Seria Matematică-Fizică, vol. 25, no. 39, pp. 10–11, 1980.

A. Aglić Aljinović and J. Pečarić, “On some Ostrowski type inequalities via Montgomery identity and Taylor's formula,” Tamkang Journal of Mathematics, vol. 36, no. 3, pp. 199–218, 2005.

A. Aglić Aljinović, J. Pečarić, and M. Ribičić Penava, “Sharp integral inequalities based on general two-point formulae via an extension of Montgomery's identity,” The ANZIAM Journal, vol. 51, no. 1, pp. 67–101, 2009.