Sharp error bounds in approximating the Riemann-Stieltjes integral by a generalised trapezoid formula and applications
© Cerone and Dragomir; licensee Springer 2013
Received: 18 July 2012
Accepted: 22 January 2013
Published: 18 February 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.
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.
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
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
and the first equality in (2.3) is proved.
The second and third identity is obvious by the relation (2.2).
The proof is now complete. □
In the case where u is an integral, the following identity can be stated.
Proof Since p and h are continuous, the function is differentiable and for each .
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). □
3 Some inequalities for f-convex
The following result concerning the nonnegativity of the error functional can be stated.
A sufficient condition for (3.1) to hold is that f is convex on .
we deduce that .
which shows that , namely, the condition (3.1) is satisfied. □
We are now able to provide some new results.
If f satisfies (3.1) and is Riemann integrable on (or sufficiently, f is convex on ), then
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.
and the inequality (3.8) is proved.
for , , and the theorem is proved. □
provided f satisfies (3.1) and is Riemann integrable (or sufficiently, convex on ), which is continuous on .
with , .
Finally, we can state the following Jensen type inequality for the error functional .
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.
Theorem 6 Let be bounded functions on .
The constant is best possible in (4.1).
The constant is best possible in (4.2)
The constant is best possible in both inequalities.
The inequality (4.4) is sharp.
The constant is sharp (if and are finite).
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. □
then the following corollary provides various sharp bounds for the absolute value of .
Corollary 3 Assume that f and u are Riemann integrable on .
The constant is best possible in (4.8).
The constant is best possible in (4.9).
where is defined in Theorem 6. The constant is sharp in both inequalities.
where is defined in Theorem 6. The inequality (4.11) is sharp.
The constant is sharp (if and are finite).
The first inequality in (4.13) is sharp (if and are finite).
Proof We only prove the first inequality in (4.13).
The rest of the inequality is obvious. □
Most of the work for this article was undertaken while the first author was at Victoria University, Melbourne Australia.
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