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
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:
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
then we observe that
The following representation result can be stated.
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.
Integrating by parts, we have
The second part of (2.6) follows by (2.3). □
3 Some inequalities for f-convex
If f is convex, then
We are now able to provide some new results.
We use the Čebyšev inequality
which holds for synchronous (asynchronous) functions p, h and nonnegative α for which the involved integrals exist.
We also have the following theorem.
and the inequality (3.8) is proved.
Further, by the Hölder inequality, we also have
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
is a Grüss type functional introduced in , any sharp bound for will be a sharp bound for .
We can state the following result.
(iii) Iffis monotonic nondecreasing anduisS-Lipschitzian, then
The inequality (4.4) is sharp.
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. □
namely (see also (2.7)),
Proof We only prove the first inequality in (4.13).
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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