Abstract
Sharp error bounds in approximating the RiemannStieltjes integral
MSC: 26D15, 26D10, 41A55.
Keywords:
RiemannStieltjes integral; trapezoid rule; integral inequalities; weighted integrals1 Introduction
In [1], in order to approximate the RiemannStieltjes integral
the authors considered the error functional
and proved that
provided that
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:
for any
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
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
2 Representation results
We consider the following error functional
If we consider the associated functions
and
then we observe that
The following representation result can be stated.
Theorem 1Let
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
The proof is now complete. □
In the case where u is an integral, the following identity can be stated.
Corollary 1Let
Proof Since p and h are continuous, the function
Integrating by parts, we have
Since
then, by the definition of
The second part of (2.6) follows by (2.3). □
Remark 1 In the particular case
3 Some inequalities for fconvex
The following result concerning the nonnegativity of the error functional
Theorem 2Ifuis monotonic nonincreasing and
then
A sufficient condition for (3.1) to hold is thatfis convex on
Proof The condition (3.1) is equivalent with the fact that
we deduce that
If f is convex, then
which shows that
Corollary 2Let
We are now able to provide some new results.
Theorem 3Assume thatpandhare continuous and synchronous (asynchronous) on
Iffsatisfies (3.1) and is Riemann integrable on
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
We also have the following theorem.
Theorem 4Assume that
and
where
Proof
Observe that
and the inequality (3.8) is proved.
Further, by the Hölder inequality, we also have
for
Remark 2 The above result can be useful for providing some error estimates in approximating
the weighted integral
as follows:
provided f satisfies (3.1) and is Riemann integrable (or sufficiently, convex on
If
with
Finally, we can state the following Jensen type inequality for the error functional
Theorem 5Assume
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
is a Grüss type functional introduced in [8], any sharp bound for
We can state the following result.
Theorem 6Let
(i) If there exist constantsn, Nsuch that
The constant
(ii) Iffis of bounded variation anduisSLipschitzian (
The constant
(iii) Iffis monotonic nondecreasing anduisSLipschitzian, then
where
The constant
(iv) Iffis monotonic nondecreasing anduis of bounded variation and such that the RiemannStieltjes integral
where
The inequality (4.4) is sharp.
(v) Iffis continuous and convex on
The constant
(vi) If
The constants 2 and
Proof The inequality (4.1) follows from the inequality (2.5) in [8] applied to
If we consider the error functional in approximating the weighted integral
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
The constant
(ii) Iffis of bounded variation and
The constant
(iii) Iffis monotonic nondecreasing and
where
(iv) Iffis monotonic nondecreasing and
where
(v) Iffis continuous and convex on
The constant
(vi) If
The first inequality in (4.13) is sharp (if
Proof We only prove the first inequality in (4.13).
Utilising the inequality (4.6) for
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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