Abstract
Keywords:
RiemannStieltjes integral; trapezoid rule; integral inequalities; weighted integrals1 Introduction
In [1], in order to approximate the RiemannStieltjes 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 rHHölder type, that is, satisfies the condition for any , where and are given.
The dual case, namely, when f is of qKHölder type and u is of bounded variation, has been considered by the authors in [2] in which they obtained the bound:
The case where f is monotonic and u is of rHHölder type, which provides a refinement for (1.3), and respectively the case where u is monotonic and f of qKHölder 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 RiemannStieltjes integral by the generalised trapezoid formula:
If we consider the associated functions , and defined by
and
then we observe that
The following representation result can be stated.
Theorem 1Letbe bounded onand such that the RiemannStieltjes integraland the Riemann integralexist. Then we have the identities
where
Proof
Integrating the RiemannStieltjes 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 RiemannStieltjes integral and the Riemann integral exist, we have the representation (see, for instance, [7])
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
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
3 Some inequalities for fconvex
The following result concerning the nonnegativity of the error functional can be stated.
Theorem 2Ifuis monotonic nonincreasing andis such that the RiemannStieltjes integralexists and
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
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
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
where
Proof
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 thatis Riemann integrable and satisfies (3.1) (or sufficiently, fis concave on). Then, forcontinuous, we have
and
where, . In particular, we have
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:
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
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
Proof
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 [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 andfisKLipschitzian (), then
The constantis best possible in (4.1).
(ii) Iffis of bounded variation anduisSLipschitzian (), then
The constantis best possible in (4.2)
(iii) Iffis monotonic nondecreasing anduisSLipschitzian, then
where
The constantis best possible in both inequalities.
(iv) Iffis monotonic nondecreasing anduis of bounded variation and such that the RiemannStieltjes integralexists, then
where
The inequality (4.4) is sharp.
(v) Iffis continuous and convex onanduis of bounded variation on, then
The constantis sharp (ifandare finite).
(vi) Ifis continuous and convex onanduis monotonic nondecreasing on, then
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,
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 thatfor each, andfisKLipschitzian on, then
The constantis best possible in (4.8).
(ii) Iffis of bounded variation andfor each, then
The constantis best possible in (4.9).
(iii) Iffis monotonic nondecreasing and, , then
whereis defined in Theorem 6. The constantis sharp in both inequalities.
(iv) Iffis monotonic nondecreasing and, then
whereis defined in Theorem 6. The inequality (4.11) is sharp.
(v) Iffis continuous and convex onand, then
The constantis sharp (ifandare finite).
(vi) Ifis continuous and convex onandfor, then
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
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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