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
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 1Let 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 1Let 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 2Ifuis monotonic nonincreasing and is such that the Riemann-Stieltjes integral exists and
then or, equivalently,
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 2Let be continuous on and be Riemann integrable. If for any andfsatisfies (3.1) or, sufficiently, fis convex on , then
We are now able to provide some new results.
Theorem 3Assume thatpandhare continuous and synchronous (asynchronous) on , i.e.,
Iffsatisfies (3.1) and is Riemann integrable on (or sufficiently, fis 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 4Assume that is Riemann integrable and satisfies (3.1) (or sufficiently, fis 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 5Assume is Riemann integrable and satisfies (3.1) (or sufficiently, fis 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 6Let be bounded functions on .
(i) If there exist constantsn, Nsuch that for any , uis Riemann integrable andfisK-Lipschitzian ( ), then
The constant is best possible in (4.1).
(ii) Iffis of bounded variation anduisS-Lipschitzian ( ), then
The constant is best possible in (4.2)
(iii) Iffis monotonic nondecreasing anduisS-Lipschitzian, then
The constant is best possible in both inequalities.
(iv) Iffis monotonic nondecreasing anduis of bounded variation and such that the Riemann-Stieltjes integral exists, then
The inequality (4.4) is sharp.
(v) Iffis continuous and convex on anduis of bounded variation on , then
The constant is sharp (if and are finite).
(vi) If is continuous and convex on anduis 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 3Assume thatfanduare Riemann integrable on .
(i) If there exist constantsγ, Γ such that for each , andfisK-Lipschitzian on , then
The constant is best possible in (4.8).
(ii) Iffis of bounded variation and for each , then
The constant is best possible in (4.9).
(iii) Iffis monotonic nondecreasing and , , then
where is defined in Theorem 6. The constant is sharp in both inequalities.
(iv) Iffis monotonic nondecreasing and , then
where is defined in Theorem 6. The inequality (4.11) is sharp.
(v) Iffis 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. □
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.
Most of the work for this article was undertaken while the first author was at Victoria University, Melbourne Australia.
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