# Sharp error bounds in approximating the Riemann-Stieltjes integral by a generalised trapezoid formula and applications

Pietro Cerone1* and Sever S Dragomir23

Author Affiliations

1 Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC, 3086, Australia

2 Mathematics, College of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne, MC, 8001, Australia

3 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, Wits, 2050, South Africa

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Journal of Inequalities and Applications 2013, 2013:53  doi:10.1186/1029-242X-2013-53

 Received: 18 July 2012 Accepted: 22 January 2013 Published: 18 February 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Sharp error bounds in approximating the Riemann-Stieltjes integral with the generalised trapezoid formula are given for various pairs of functions. Applications for weighted integrals are also provided.

MSC: 26D15, 26D10, 41A55.

##### Keywords:
Riemann-Stieltjes integral; trapezoid rule; integral inequalities; weighted integrals

### 1 Introduction

In [1], in order to approximate the Riemann-Stieltjes integral by the generalised trapezoid formula

(1.1)

the authors considered the error functional

(1.2)

and proved that

(1.3)

provided that is of bounded variation on and u is of r-H-Hölder type, that is, satisfies the condition for any , where and are given.

The dual case, namely, when f is of q-K-Hölder type and u is of bounded variation, has been considered by the authors in [2] in which they obtained the bound:

(1.4)

for any .

The case where f is monotonic and u is of r-H-Hölder type, which provides a refinement for (1.3), and respectively the case where u is monotonic and f of q-K-Hölder type were considered by Cheung and Dragomir in [3], while the case where one function was of Hölder type and the other was Lipschitzian was considered in [4]. For other recent results in estimating the error for absolutely continuous integrands f and integrators u of bounded variation, see [5] and [6].

The main aim of the present paper is to investigate the error bounds in approximating the Stieltjes integral by a different generalised trapezoid rule than the one from (1.1) in which the value , is replaced with the integral mean . Applications in approximating the weighted integrals are also provided.

### 2 Representation results

We consider the following error functional in approximating the Riemann-Stieltjes integral by the generalised trapezoid formula:

(2.1)

If we consider the associated functions , and defined by

and

then we observe that

(2.2)

The following representation result can be stated.

Theorem 1Letbe bounded onand such that the Riemann-Stieltjes integraland the Riemann integralexist. Then we have the identities

(2.3)
where

(2.4)

Proof

Integrating the Riemann-Stieltjes integral by parts, we have

and the first equality in (2.3) is proved.

The second and third identity is obvious by the relation (2.2).

For the last equality, we use the fact that for any bounded functions for which the Riemann-Stieltjes integral and the Riemann integral exist, we have the representation (see, for instance, [7])

(2.5)

The proof is now complete. □

In the case where u is an integral, the following identity can be stated.

Corollary 1Letbe continuous onandbe Riemann integrable. Then we have the identity

(2.6)

Proof Since p and h are continuous, the function is differentiable and for each .

Integrating by parts, we have

Since

then, by the definition of in (2.1), we deduce the first part of (2.6).

The second part of (2.6) follows by (2.3). □

Remark 1 In the particular case , , we have the equality

(2.7)

### 3 Some inequalities for f-convex

The following result concerning the nonnegativity of the error functional can be stated.

Theorem 2Ifuis monotonic nonincreasing andis such that the Riemann-Stieltjes integralexists and

(3.1)

thenor, equivalently,

(3.2)

A sufficient condition for (3.1) to hold is thatfis convex on.

Proof The condition (3.1) is equivalent with the fact that for any and then, by the equality

we deduce that .

If f is convex, then

which shows that , namely, the condition (3.1) is satisfied. □

Corollary 2Letbe continuous onandbe Riemann integrable. Iffor anyandfsatisfies (3.1) or, sufficiently, fis convex on, then

(3.3)

We are now able to provide some new results.

Theorem 3Assume thatpandhare continuous and synchronous (asynchronous) on, i.e.,

(3.4)

Iffsatisfies (3.1) and is Riemann integrable on (or sufficiently, fis convex on), then

(3.5)
where

(3.6)

Proof

We use the Čebyšev inequality

(3.7)

which holds for synchronous (asynchronous) functions p, h and nonnegative α for which the involved integrals exist.

Now, on applying the Čebyšev inequality (3.7) for and utilising the representation result (2.6), we deduce the desired inequality (3.5). □

We also have the following theorem.

Theorem 4Assume thatis Riemann integrable and satisfies (3.1) (or sufficiently, fis concave on). Then, forcontinuous, we have

(3.8)

and

(3.9)

where, . In particular, we have

(3.10)

Proof

Observe that

and the inequality (3.8) is proved.

Further, by the Hölder inequality, we also have

for , , and the theorem is proved. □

Remark 2 The above result can be useful for providing some error estimates in approximating the weighted integral by the generalised trapezoid rule

as follows:

(3.11)

provided f satisfies (3.1) and is Riemann integrable (or sufficiently, convex on ), which is continuous on .

If , , then for some f, we also have

(3.12)

with , .

Finally, we can state the following Jensen type inequality for the error functional .

Theorem 5Assumeis Riemann integrable and satisfies (3.1) (or sufficiently, fis convex on), whileis continuous. Ifis convex (concave), then

(3.13)

Proof

By the use of Jensen’s integral inequality, we have

(3.14)

Since, by the identity (2.6), we have

then (3.14) is equivalent with the desired result (3.13). □

### 4 Sharp bounds via Grüss type inequalities

Due to the identity (2.3), in which the error bound can be represented as , where

is a Grüss type functional introduced in [8], any sharp bound for will be a sharp bound for .

We can state the following result.

Theorem 6Letbe bounded functions on.

(i) If there exist constantsn, Nsuch thatfor any, uis Riemann integrable andfisK-Lipschitzian (), then

(4.1)

The constantis best possible in (4.1).

(ii) Iffis of bounded variation anduisS-Lipschitzian (), then

(4.2)

The constantis best possible in (4.2)

(iii) Iffis monotonic nondecreasing anduisS-Lipschitzian, then

(4.3)

where

The constantis best possible in both inequalities.

(iv) Iffis monotonic nondecreasing anduis of bounded variation and such that the Riemann-Stieltjes integralexists, then

(4.4)

where

The inequality (4.4) is sharp.

(v) Iffis continuous and convex onanduis of bounded variation on, then

(4.5)

The constantis sharp (ifandare finite).

(vi) Ifis continuous and convex onanduis monotonic nondecreasing on, then

(4.6)

The constants 2 andare best possible in (4.6) (ifandare finite).

Proof The inequality (4.1) follows from the inequality (2.5) in [8] applied to , while (4.2) comes from (1.3) of [9]. The inequalities (4.3) and (4.4) follow from [7], while (4.5) and (4.6) are valid via the inequalities (2.8) and (2.1) from [10] applied to the functional . The details are omitted. □

If we consider the error functional in approximating the weighted integral by the generalised trapezoid formula,

(4.7)

then the following corollary provides various sharp bounds for the absolute value of .

Corollary 3Assume thatfanduare Riemann integrable on.

(i) If there exist constantsγ, Γ such thatfor each, andfisK-Lipschitzian on, then

(4.8)

The constantis best possible in (4.8).

(ii) Iffis of bounded variation andfor each, then

(4.9)

The constantis best possible in (4.9).

(iii) Iffis monotonic nondecreasing and, , then

(4.10)

whereis defined in Theorem 6. The constantis sharp in both inequalities.

(iv) Iffis monotonic nondecreasing and, then

(4.11)

whereis defined in Theorem 6. The inequality (4.11) is sharp.

(v) Iffis continuous and convex onand, then

(4.12)

The constantis sharp (ifandare finite).

(vi) Ifis continuous and convex onandfor, then

(4.13)

The first inequality in (4.13) is sharp (ifandare finite).

Proof We only prove the first inequality in (4.13).

Utilising the inequality (4.6) for , we get

(4.14)

However, on integrating by parts, we have

The rest of the inequality is obvious. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

PC and SSD have contributed to all parts of the article. Both authors read and approved the final manuscript.

### Acknowledgements

Most of the work for this article was undertaken while the first author was at Victoria University, Melbourne Australia.

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