A Sharp Bound of the Čebyšev Functional for the Riemann-Stieltjes Integral and Applications

S. S. Dragomir School of Computer Science and Mathematics, Victoria University, P.O. Box 14428, Melbourne, Victoria 8001, Australia Correspondence should be addressed to S. S. Dragomir, sever.dragomir@vu.edu.au Received 13 December 2007; Accepted 29 February 2008 Recommended by Yeol Cho A new sharp bound of the Čebyšev functional for the Riemann-Stieltjes integral is obtained. Applications for quadrature rules including the trapezoid and midpoint rules are given. Copyright q 2008 S. S. Dragomir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
In order the generalise the classical Cebyšev functional, namely, 1.1 where f, g, and fg are integrable on a, b , which has been extensively studied in the literature see, e.g., the book 1 , the author has introduced in 2 the following functional for Riemann-Stieltjes integrals: provided that f and g are continuous, m ≤ f t ≤ M for each t ∈ a, b , and u is of bounded variation on a, b with the total variation ∨ b a u .The constant 1/2 is sharp in 1.3 in the sense that it cannot be replaced by a smaller quantity.
In the case that u is monotonic nondecreasing, for which the constant 1/2 is best possible 2 .
Finally, in the case where u is Lipschitzian with the constant L, and in this case we can have f and g Riemann integrable on a, b , the following result has been obtained as well 2 : Here 1/2 is also sharp.
The aim of the present paper is to establish a new sharp bound for the absolute value of the Cebyšev functional 1.2 .Applications for the trapezoid and midpoint inequality are pointed out.A general perturbed quadrature rule and error estimates are obtained as well.

The results
The following result concerning a sharp bound for the absolute value of the Cebyšev functional T f, g; h can be stated.

2.1
The constant C 1 in the right-hand side of 2.1 cannot be replaced by a smaller quantity.
Proof.We use the following result for the Riemann-Stieltjes integral obtained in 1, page 337 .Let u, v, w : a, b → R such that u is of bounded variation on a, b and v, w are bounded functions with the property that the Riemann-Stieltjes integrals

2.2
We also use the representation see also 2 and in both sides of 2.9 we get the same quantity b − a .

Journal of Inequalities and Applications
Remark 2.2.We observe that where Δ g, h; x, a, b is defined by With this notation, inequality 2.1 becomes

2.13
Now, if we assume that h a < h x < h b for any x ∈ a, b , then on utilising the elementary inequality αβ ≤ 1/4 α β 2 , α, β ∈ 0, ∞ , we have 2.14 and from 2.9 , we deduce the following simpler inequality:

2.15
The constant 1/4 is best possible in 2.15 .
A sufficient condition for h such that h a < h x < h b for any x ∈ a, b is that h is strictly increasing on a, b .The sharpness of the constant will follow from a particular case considered in Corollary 2.5 below.

2.16
The inequality is sharp.
The proof follows by Theorem 2.1 on choosing h x x a w s ds.

2.17
The constant 1/4 is best possible.

2.18
The constant 1/4 is best possible in 2.18 .
Proof.For the sharpness of the constant, consider g t where

2.22
Since sup x∈ a,b δ x 2, inequality 2.18 becomes, for g given above, If in this inequality we choose f t sgn t − a b /2 , then we obtain in both sides of 2.23 the same quantity b − a .

Applications for the trapezoid rule
The following result concerning the error estimate for the trapezoid rule can be stated as follows.

3.1
The constant 1/8 is best possible.
Proof.We use the identity see, e.g., 5 If we apply inequality 2.18 , then we can write that hence, by 3.2 and 3.3 , we deduce 3.1 .
For the sharpness of the constant we choose f t |t − a b /2|.For this function, we have and ∨ b a f 2. If we replace the above quantities in 3.1 , we get the same result b − a /4 in both sides.
The following result can be stated as well.

Journal of Inequalities and Applications
Proof.Applying inequality 2.18 , we can also write that which, together with the identity 3.2 , produces the desired inequality 3.6 .
For other results on the trapezoid rule, see 5 .

Applications for the midpoint rule
The following result concerning the error estimates for the midpoint rule can be stated.
The constant 1/8 is best possible.
Proof.We use the identity see, e.g., 6 where p : a, b → R is given by

4.4
We notice that b a p t dt 0, δ x : The following result can be stated as well.

4.7
Proof.Applying inequality 2.18 , we can write For other results on the midpoint rule and their applications, see 6-8 .

Applications for general quadrature rules
Let h : a, b → R be a Riemann integrable function.Suppose that h is n-time differentiable and that there exists the division a and the error term E n h satisfies the bound

5.3
The proof is obvious by 2.9 on choosing f h n and g K n .
The second natural possibility is incorporated in the following proposition.
Proposition 5.2.With the above assumption and if K n is of bounded variation on a, b , then the representation 5.2 holds and the error term E n h satisfies the bounds

5.4
The proof follows by inequality 2.18 on choosing f K n and g h n .
Remark 5.3.As noted in the previous section, in practical applications and for a large number of quadrature rules, the Peano kernel K n is available and the involved quantities in the error estimates 5.3 and 5.4 can be completely specified.In some cases, the new perturbed rules provide a better approximation than the original one.The details are left to the interested reader.
involved integrals exist and u b / u a .It has been shown in 2 that

Theorem 2 . 1 .
Let f : a, b → R be a function of bounded variation and let g, h : a, b → R be bounded functions with h a / h b such that the Stieltjes integrals b a f t g t dh t and b a g t dh t exist.Then v t dw t exist.Then

Corollary 2 . 3 .
Let f, g, w : a, b → R be such that f is of bounded variation and the Riemann integrals b a f t w t dt, b a g t w t dt, b a f t g t w t dt, and b a w t dt exist and b a w t dt / 0.Then, one has the inequality

Remark 2 . 4 .
In particular, if w s > 0 for s ∈ a, b , then h x x a w s ds is strictly decreasing on a, b and by 2.15 we deduce the inequality 1

Corollary 2 . 5 .
Let f, g : a, b → R be such that f is of bounded variation and the Riemann integrals b a g t dt and b a f t g t dt exist.Then λ has been defined in the proof of Theorem 2.
f of bounded variation on a, b .

Proposition 3 . 1 .
Assume that f : a, b → R is absolutely continuous and has the derivative f : a, b → R of bounded variation on a, b .Then

Proposition 4 . 1 .
Assume that f : a, b → R is absolutely continuous and has the derivative f : a, b → R of bounded variation on a, b .Then 4.4 , we deduce 4.1 .For the sharpness of the constant 1/8, observe that for the absolutely continuous function f t |t − a b /2|, we get in both sides of 4.1 the same quantity b − a /4.

Proposition 4 . 2 .
If f : a, b → R is absolutely continuous on a, b , then − a, we deduce from 4.8 the desired inequality 4.7 .
and the weights α 0 , . . ., α n such that : a, b → R is the Peano kernel associated with the quadrature rule A h : With the above assumptions and if h n is of bounded variation, then n