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Sharp error bounds in approximating the Riemann-Stieltjes integral by a generalised trapezoid formula and applications
Journal of Inequalities and Applications volume 2013, Article number: 53 (2013)
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.
In , in order to approximate the Riemann-Stieltjes integral by the generalised trapezoid formula
the authors considered the error functional
and proved that
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  in which they obtained the bound:
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 , while the case where one function was of Hölder type and the other was Lipschitzian was considered in . For other recent results in estimating the error for absolutely continuous integrands f and integrators u of bounded variation, see  and .
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:
If we consider the associated functions , and defined by
then we observe that
The following representation result can be stated.
Theorem 1 Let be bounded on and such that the Riemann-Stieltjes integral and the Riemann integral exist. Then we have the identities
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, )
The proof is now complete. □
In the case where u is an integral, the following identity can be stated.
Corollary 1 Let be continuous on and be Riemann integrable. Then we have the identity
Proof Since p and h are continuous, the function is differentiable and for each .
Integrating by parts, we have
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
3 Some inequalities for f-convex
The following result concerning the nonnegativity of the error functional can be stated.
Theorem 2 If u is monotonic nonincreasing and is such that the Riemann-Stieltjes integral exists and
then or, equivalently,
A sufficient condition for (3.1) to hold is that f is 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 2 Let be continuous on and be Riemann integrable. If for any and f satisfies (3.1) or, sufficiently, f is convex on , then
We are now able to provide some new results.
Theorem 3 Assume that p and h are continuous and synchronous (asynchronous) on , i.e.,
If f satisfies (3.1) and is Riemann integrable on (or sufficiently, f is convex on ), then
We use the Čebyšev inequality
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 4 Assume that is Riemann integrable and satisfies (3.1) (or sufficiently, f is concave on ). Then, for continuous, we have
where , . In particular, we have
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
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
with , .
Finally, we can state the following Jensen type inequality for the error functional .
Theorem 5 Assume is Riemann integrable and satisfies (3.1) (or sufficiently, f is convex on ), while is continuous. If is convex (concave), then
By the use of Jensen’s integral inequality, we have
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 , any sharp bound for will be a sharp bound for .
We can state the following result.
Theorem 6 Let be bounded functions on .
(i) If there exist constants n, N such that for any , u is Riemann integrable and f is K-Lipschitzian (), then
The constant is best possible in (4.1).
(ii) If f is of bounded variation and u is S-Lipschitzian (), then
The constant is best possible in (4.2)
(iii) If f is monotonic nondecreasing and u is S-Lipschitzian, then
The constant is best possible in both inequalities.
(iv) If f is monotonic nondecreasing and u is of bounded variation and such that the Riemann-Stieltjes integral exists, then
The inequality (4.4) is sharp.
(v) If f is continuous and convex on and u is of bounded variation on , then
The constant is sharp (if and are finite).
(vi) If is continuous and convex on and u is monotonic nondecreasing on , then
The constants 2 and are best possible in (4.6) (if and are finite).
Proof The inequality (4.1) follows from the inequality (2.5) in  applied to , while (4.2) comes from (1.3) of . The inequalities (4.3) and (4.4) follow from , while (4.5) and (4.6) are valid via the inequalities (2.8) and (2.1) from  applied to the functional . The details are omitted. □
If we consider the error functional in approximating the weighted integral by the generalised trapezoid formula,
namely (see also (2.7)),
then the following corollary provides various sharp bounds for the absolute value of .
Corollary 3 Assume that f and u are Riemann integrable on .
(i) If there exist constants γ, Γ such that for each , and f is K-Lipschitzian on , then
The constant is best possible in (4.8).
(ii) If f is of bounded variation and for each , then
The constant is best possible in (4.9).
(iii) If f is monotonic nondecreasing and , , then
where is defined in Theorem 6. The constant is sharp in both inequalities.
(iv) If f is monotonic nondecreasing and , then
where is defined in Theorem 6. The inequality (4.11) is sharp.
(v) If f is continuous and convex on and , then
The constant is sharp (if and are finite).
(vi) If is continuous and convex on and for , then
The first inequality in (4.13) is sharp (if and are finite).
Proof We only prove the first inequality in (4.13).
Utilising the inequality (4.6) for , we get
However, on integrating by parts, we have
The rest of the inequality is obvious. □
Dragomir SS, Buse C, Boldea MV, Braescu L: A generalization of the trapezoidal rule for the Riemann-Stieltjes integral and applications. Nonlinear Anal. Forum 2001, 6(2):337–351.
Cerone P, Dragomir SS: New bounds for the three-point rule involving the Riemann-Stieltjes integral. In Advances in Statistics, Combinatorics and Related Areas. World Scientific, River Edge; 2002:53–62.
Cheung W-S, Dragomir SS: Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions. Bull. Aust. Math. Soc. 2007, 75(2):299–311. 10.1017/S0004972700039228
Barnett NS, Cheung W-S, Dragomir SS, Sofo A: Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators. RGMIA Res. Rep. Collect. 2006., 9: Article ID 9. http://rgmia.org/v9n4.php
Cerone P, Dragomir SS: Approximation of the Stieltjes integral and applications in numerical integration. Appl. Math. 2006, 51(1):37–47. 10.1007/s10492-006-0003-0
Cerone P, Cheung WS, Dragomir SS: On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation. Comput. Math. Appl. 2007, 54(2):183–191. 10.1016/j.camwa.2006.12.023
Dragomir SS: Inequalities of Grüss type for the Stieltjes integral and applications. Kragujev. J. Math. 2004, 26: 89–122.
Dragomir SS, Fedotov IA: An inequality of Grüss’ type for Riemann-Stieltjes integral and applications for special means. Tamkang J. Math. 1998, 29(4):287–292.
Dragomir SS, Fedotov I: A Grüss type inequality for mappings of bounded variation and applications to numerical analysis. Nonlinear Funct. Anal. Appl. 2001, 6(3):425–438.
Dragomir SS: Inequalities for Stieltjes integrals with convex integrators and applications. Appl. Math. Lett. 2007, 20(2):123–130. 10.1016/j.aml.2006.02.027
Most of the work for this article was undertaken while the first author was at Victoria University, Melbourne Australia.
The authors declare that they have no competing interests.
PC and SSD have contributed to all parts of the article. Both authors read and approved the final manuscript.
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Cerone, P., Dragomir, S.S. Sharp error bounds in approximating the Riemann-Stieltjes integral by a generalised trapezoid formula and applications. J Inequal Appl 2013, 53 (2013). https://doi.org/10.1186/1029-242X-2013-53
- Riemann-Stieltjes integral
- trapezoid rule
- integral inequalities
- weighted integrals