Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 824610, 11 pages
doi:10.1155/2008/824610
Research Article
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.
1. Introduction
In order the generalise the classical C̆ebyšev functional, namely,
T f, g :
1
b−a
b
fxgxdx −
a
1
b−a
b
fxdx ·
a
1
b−a
b
1.1
gxdx,
a
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 RiemannStieltjes integrals:
T f, g; u :
1
ub − ua
b
ftgtdut−
a
1
ub − ua
b
ftdut·
a
1
ub − ua
b
gtdut,
a
1.2
provided that the involved integrals exist and ub /
ua.
It has been shown in 2 that
b
b
1
1
T f, g; u ≤ 1 M − m · gsdus
u,
g−
ub − ua
2
ub − ua a
∞ a
1.3
2
Journal of Inequalities and Applications
provided that f and g are continuous, m ≤ ft ≤ M for each t ∈ a, b, and u is of bounded
variation on a, b with the total variation ∨ba 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,
1
T f, g; u ≤ 1 M − m · ub − ua
2
b gt −
a
1
ub − ua
b
a
gsdusdut,
1.4
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:
1
T f, g; u ≤ 1 LM − m · ub − ua
2
b gt −
a
1
ub − ua
b
a
gsdusdt.
1.5
Here 1/2 is also sharp.
For other results, see 3, 4.
The aim of the present paper is to establish a new sharp bound for the absolute value
of the C̆ebyšev 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.
2. The results
The following result concerning a sharp bound for the absolute value of the C̆ebyšev functional
T f, g; h can be stated.
Theorem 2.1. Let f : a, b → R be a function of bounded variation and let g, h : a, b → R be
b
b
bounded functions with ha /
hb such that the Stieltjes integrals a ftgtdht and a gtdht
exist. Then
x
b
1
hx − ha b
T f, g; h ≤ f sup gtdht −
gsdhs.
hb − ha
hb
−
ha
a
x∈a,b a
a
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
b
bounded functions with the property that the Riemann-Stieltjes integrals a vtdwt and
b
utvtdwt exist. Then
a
b
x
b
utvtdwt ≤ ub u sup vtdwt.
a
a
x∈a,b
a
2.2
S. S. Dragomir
3
We also use the representation see also 2
1
T f, g; h hb − ha
b
ft − γ gt −
a
1
hb − ha
b
gsdhs dht,
2.3
a
which holds for any γ ∈ R.
Now, if we choose γ fb, ut ft − fb,
vt gt −
1
hb − ha
b
gsdhs,
2.4
a
and wt ht, t ∈ a, b, then we get
b
x
b
hb − ha T f, g; h ≤ f sup gtdht − hx − ha
gsdhs
hb − ha a
x∈a,b a
a
2.5
and inequality 2.1 is proved.
For the sharpness of the inequality, assume that ht t and gt sgnt − a b/2,
t ∈ a, b. Then 2.1 becomes
x
b
b
ftsgn t − a b dt ≤ f sup sgn t − a b dt,
2
2
a
x∈a,b a
a
2.6
provided that f is of bounded variation on a, b.
Notice that, if we consider λx defined by
⎧
ab
⎪
⎪
,
⎪a − x, if x ∈ a,
x
⎨
2
ab
λx :
dt sgn t −
⎪
2
ab
⎪
a
⎪
,b ,
⎩x − b, if x ∈
2
2.7
b−a
sup λx .
2
x∈a,b
2.8
b
b
ftsgn t − a b dt ≤ b − a · f.
2
2
a
a
2.9
then
Therefore, 2.6 becomes
Now, if in 2.9 we choose ft sgnt − a b/2, then ∨ba f 2,
b
a
ab
dt b − a,
ftsgn t −
2
and in both sides of 2.9 we get the same quantity b − a.
2.10
4
Journal of Inequalities and Applications
Remark 2.2. We observe that
x
gtdht −
a
hx − ha
hb − ha
b
gsdhs
a
b
hx − ha x
gtdht −
gsdhs gsdhs
hb − ha a
a
x
x
b
hb − hx
hx − ha
·
· gsdhs
gsdhs −
hb − ha a
hb − ha x
hb − hx hx − ha
Δg, h; x, a, b,
hb − ha
x
2.11
where Δg, h; x, a, b is defined by
1
Δg, h; x, a, b hx − ha
x
a
1
gsdhs −
hb − hx
b
gsdhs,
2.12
x
provided that hx /
ha, hb for x ∈ a, b.
With this notation, inequality 2.1 becomes
b
hb − hx
hx − ha
1
T f, g; h ≤ f sup · Δg, h; x, a, b
hb − ha
hb − ha
x∈a,b
a
b
hb − hx
hx − ha
1
f sup ≤
sup Δg, h; x, a, b.
hb − ha
x∈a,b
hb − ha
x∈a,b
a
2.13
Now, if we assume that ha < hx < hb for any x ∈ a, b, then on utilising the elementary
inequality αβ ≤ 1/4α β2 , α, β ∈ 0, ∞, we have
1
2
hb − hx hx − ha ≤ hb − ha ,
4
2.14
and from 2.9, we deduce the following simpler inequality:
b
T f, g; h ≤ 1 · f sup Δg, h; x, a, b.
4 a
x∈a,b
2.15
The constant 1/4 is best possible in 2.15.
A sufficient condition for h such that ha < hx < hb 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.
S. S. Dragomir
5
Corollary 2.3. Let f, g, w : a, b → R be such that f is of bounded variation and the Riemann inb
b
b
b
b
tegrals a ftwtdt, a gtwtdt, a ftgtwtdt, and a wtdt exist and a wtdt / 0. Then,
one has the inequality
b
b
b
1
1
1
ftgtwtdt − b
ftwtdt · b
gtwtdt
b
wtdt a
a
a
wtdt
wtdt
a
a
a
x
b
b
wtdt b
1
a
gtwtdt.
≤ b
f sup gtwtdt − b
wtdt a
x∈a,b a
wtdt a
a
a
2.16
The inequality is sharp.
The proof follows by Theorem 2.1 on choosing hx x
wsds.
x
Remark 2.4. In particular, if ws > 0 for s ∈ a, b, then hx a wsds is strictly decreasing
on a, b and by 2.15 we deduce the inequality
a
b
b
b
1
1
1
ftgtwtdt − b
ftwtdt · b
gtwtdt
b
wtdt a
wtdt a
wtdt a
a
a
a
x
x
b
wsds b
1
≤ b
f sup gswsds − ab
gswsds
x∈a,b a
wsds a
wsds a
a
a
b
x
b
1
1 1
≤ · f sup x
gswsds − b
gswsds.
wsds a
4 a
x∈a,b
wsds a
a
2.17
x
The constant 1/4 is best possible.
Corollary 2.5. Let f, g : a, b → R be such that f is of bounded variation and the Riemann integrals
b
b
gtdt and a ftgtdt exist. Then
a
b
b
b
1
1
1
ftgtdt −
ftdt ·
gtdt
b − a
b
−
a
b
−
a
a
a
a
b
x
1 x−a b
≤
f sup gtdt −
gtdt
b−aa
b−a a
x∈a,b a
x
b
b
1
1 1
≤ · f sup gsds −
gsds.
4 a
x
−
a
b
−
x
a
x
x∈a,b
The constant 1/4 is best possible in 2.18.
2.18
6
Journal of Inequalities and Applications
Proof. For the sharpness of the constant, consider gt sgnt − a b/2, t ∈ a, b. If we
denote
x
b
1
1
ab
ab
dt −
dt, x ∈ a, b,
2.19
μx :
sgn t −
sgn t −
x−a a
2
b−x x
2
then
x
b
1
ab
ab
ab
dt −
dt −
dt
sgn t −
sgn t −
sgn t −
2
b−x
2
2
a
a
a
x
b−a
b−a
ab
dt sgn t −
· λx,
x − ab − x a
2
x − ab − x
2.20
μx 1
x−a
x
where λ has been defined in the proof of Theorem 2.1.
Therefore,
sup μx b − a sup δx,
x∈a,b
where
2.21
x∈a,b
⎧
1
ab
⎪
⎪
⎪
,
if
x
∈
a,
,
⎨b − x
2
δx ⎪
ab
1
⎪
⎪
, if x ∈
,b .
⎩
x−a
2
2.22
Since supx∈a,b δx 2, inequality 2.18 becomes, for g given above,
b
b
ftsgn t − a b dt ≤ 1 f,
2
2
a
a
2.23
for any function f of bounded variation on a, b.
If in this inequality we choose ft sgnt − a b/2, then we obtain in both sides of
2.23 the same quantity b − a.
3. Applications for the trapezoid rule
The following result concerning the error estimate for the trapezoid rule can be stated as follows.
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
b
b
1
fa fb 1
≤
ftdt
−
f .
b
−
a
b − a
8
2
a
a
The constant 1/8 is best possible.
3.1
S. S. Dragomir
7
Proof. We use the identity see, e.g., 5
fa fb
1
−
2
b−a
b
a
1
ftdt b−a
b ab t−
f tdt.
2
a
3.2
If we apply inequality 2.18, then we can write that
b
b
b 1
1
ab
ab
1
dt −
dt
t−
f t t −
f tdt ·
b − a
2
b−a a
b−a a
2
a
b
x
ab
ab
x−a b
1 t−
t−
dt −
dt.
f sup ≤
b−a a
2
b−a a
2
x∈a,b a
3.3
b x 1
ab 2
ab
ab
b−a 2
dt 0,
dt x−
t−
t−
,
−
2
2
2
2
2
a
a
x 1
ab 2
ab
b − a 2 b − a2
sup dt sup x −
,
t−
−
2
2 x∈a,b
2
2
8
x∈a,b a
3.4
Since
hence, by 3.2 and 3.3, we deduce 3.1.
For the sharpness of the constant we choose ft |t − a b/2|. For this function, we
have
1
b−a
b
a
ftdt 1
b−a
b a b
t −
dt b − a ,
2 4
a
fa fb b − a
,
2
2
⎧
ab
⎪
⎪
⎪
,
−1,
if
x
∈
a,
⎨
2
f t ⎪
ab
⎪
⎪
if x ∈
,b ,
⎩1,
2
3.5
and ∨ba 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.
Proposition 3.2. If f : a, b → R is absolutely continuous on a, b, then
b
1
fb − fa fa fb ftdt −
≤ sup fx − fa − x − a ·
b − a
2
b−a
a
x∈a,b
fx − fa fb − fx 1
.
≤ b − a · sup −
4
x−a
b−x
x∈a,b
3.6
8
Journal of Inequalities and Applications
Proof. Applying inequality 2.18, we can also write that
b
b
b 1
ab
1
ab
1
t−
dt −
dt
f t t −
f tdt ·
b − a
2
b
−
a
b
−
a
2
a
a
a
x
b
b
1 ab
x−a
≤
· sup f tdt −
·−
f tdt
b−a a
2
b−a a
x∈a,b a
3.7
b x b a f tdt
f tdt 1
ab
x
,
≤
·−
· sup −
4 a
2
x−a
b−x x∈a,b
which, together with the identity 3.2, produces the desired inequality 3.6.
For other results on the trapezoid rule, see 5.
4. Applications for the midpoint rule
The following result concerning the error estimates for the midpoint rule can be stated.
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
b
b
1
a b 1
≤ b − a
ftdt − f
f .
b − a
2
8
a
a
4.1
The constant 1/8 is best possible.
Proof. We use the identity see, e.g., 6
f
ab
2
−
1
b−a
b
ftdt a
1
b−a
b
ptf tdt,
4.2
a
where p : a, b → R is given by
⎧
ab
⎪
⎪
⎪
,
t
−
a,
if
t
∈
a,
⎨
2
pt ⎪
ab
⎪
⎪
,b .
⎩t − b, if t ∈
2
4.3
If we apply inequality 2.18, we can write that
b
b
x
b
1
1 b 1 b
≤ 1
ptdt− x − a ptdt.
f
tptdt−
f
tdt·
ptdt
f
sup
b−a
b−a a
b−a a
b−a a
b−a a
a
x∈a,b a
4.4
S. S. Dragomir
9
We notice that
b
ptdt 0,
a
x
δx :
ptdt
a
⎧ x
⎪
⎪
⎪ t − adt,
⎨
a
x
ab/2
⎪
⎪
⎪
t
−
adt
⎩
a
ab/2
t − bdt,
⎧
1
ab
⎪
2
⎪
⎪
x
−
a
,
,
if
t
∈
a,
⎨2
2
⎪
1
ab
⎪
2
⎪
,b ,
⎩ b − x , if t ∈
2
2
ab
,
if t ∈ a,
2
ab
,b ,
if t ∈
2
4.5
for x ∈ a, b.
Since
1
sup δx b − a2 ,
8
x∈a,b
4.6
then by 4.2 and 4.4, we deduce 4.1.
For the sharpness of the constant 1/8, observe that for the absolutely continuous function
ft |t − a b/2|, we get in both sides of 4.1 the same quantity b − a/4.
The following result can be stated as well.
Proposition 4.2. If f : a, b → R is absolutely continuous on a, b, then
b
1
a b fx − fa − x − a · fb − fa ≤
sup
ftdt
−
f
b − a
2
b−a
a
x∈a,b
fx − fa fb − fx 1
.
≤ b − a · sup −
4
x−a
b−x
x∈a,b
4.7
Proof. Applying inequality 2.18, we can write
b
b
b
1
1
1
ptf
tdt
−
ptdt
·
f
tdt
b − a
b−a a
b−a a
a
b x b
b
x a f tdt
1
f tdt 1 x−a b x
,
≤
−
p sup f tdt −
f tdt ≤
p sup b−a a
b−a a
4 a
x−a
b−x x∈a,b a
x∈a,b
4.8
and since ∨ba p b − a, we deduce from 4.8 the desired inequality 4.7.
For other results on the midpoint rule and their applications, see 6–8.
10
Journal of Inequalities and Applications
5. 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 x0 < x1 < · · · < xn−1 < xn b and the weights α0 , . . . , αn such
that
b
htdt a
n
i0
αi hxi b
Kn thn tdt,
5.1
a
where Kn : a, b → R is the Peano kernel associated with the quadrature rule Ah : ni0αi hxi .
Utilising inequality 2.18, we can produce a “perturbed quadrature rule” by approxib
mating the error terms a Kn thn tdt as follows.
Proposition 5.1. With the above assumptions and if hn is of bounded variation, then
b
htdt a
n
hn−1 b − hn−1 a b
· Kn tdt En h
αi h xi b−a
a
i0
5.2
and the error term En h satisfies the bound
x
b
n x−a b
En h ≤
Kn tdt
h
sup Kn tdt −
b
−
a
a
x∈a,b a
a
b
x
b
a Kn tdt
Kn tdt n 1
.
≤ · b − a
h
sup − x
4
x
−
a
b
−
x
x∈a,b
a
5.3
The proof is obvious by 2.9 on choosing f hn and g Kn .
The second natural possibility is incorporated in the following proposition.
Proposition 5.2. With the above assumption and if Kn is of bounded variation on a, b, then the
representation 5.2 holds and the error term En h satisfies the bounds
b
hn−1 b − hn−1 a En h ≤
Kn sup hn−1 x − hn−1 a − x − a ·
b−a
a
x∈a,b
n−1
b
h
1 x − hn−1 a hn−1 b − hn−1 x ≤ ·
−
Kn sup .
4 a
x−a
b−x
x∈a,b
5.4
The proof follows by inequality 2.18 on choosing f Kn and g hn .
Remark 5.3. As noted in the previous section, in practical applications and for a large number of quadrature rules, the Peano kernel Kn 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.
S. S. Dragomir
11
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