Journal of Inequalities in Pure and
Applied Mathematics
NOTE ON THE PERTURBED TRAPEZOID INEQUALITY
XIAO-LIANG CHENG AND JIE SUN
Department of Mathematics
Zhejiang University,
Xixi Campus, Zhejiang 310028,
The People’s Republic of China.
EMail: xlcheng@mail.hz.zj.cn
volume 3, issue 2, article 29,
2002.
Received 21 May, 2001;
accepted 01 February, 2002.
Communicated by: N.S. Barnett
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
046-01
Quit
Abstract
In this paper, we utilize a variant of the Grüss inequality to obtain some new
perturbed trapezoid inequalities. We improve the error bound of the trapezoid
rule in numerical integration in some recent known results. Also we give a new
Iyengar’s type inequality involving a second order bounded derivative for the
perturbed trapezoid inequality.
In the literature [2], [4] – [8], [11], [12] on numerical integration, the following estimation is well known as the trapezoid inequality:
Z b
1
≤ 1 M2 (b − a)3 ,
(1)
f
(x)
dx
−
(b
−
a)(f
(a)
+
f
(b))
12
2
a
where the mapping f : [a, b] → R is supposed to be twice differentiable
on the interval (a, b), with the second derivative bounded on (a, b) by M2 =
supx∈(a,b) |f 00 (x)| < +∞. In [5], the authors derived the error bounds for the
trapezoid inequality (1) by different norm of mapping f . In [2, 7, 11], the authors obtained the trapezoid inequality by the difference of sup and inf bound
of the first derivative, that is,
Z b
1
1
f (x) dx − (b − a)(f (a) + f (b)) ≤ (Γ1 − γ1 )(b − a)2 ,
2
8
a
where Γ1 = supx∈(a,b) f 0 (x) < +∞ and γ1 = inf x∈(a,b) f 0 (x) > −∞.
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
For the perturbed trapezoid inequality, S. Dragomir et al. [5] obtained the
following inequality by an application of the Grüss inequality:
(2)
Z b
1
1
2
0
0
f
(x)
dx
−
(b
−
a)(f
(a)
+
f
(b))
+
(b
−
a)
(f
(b)
−
f
(a))
2
12
a
1
≤ (Γ2 − γ2 )(b − a)3 ,
32
where f is supposed to be twice differentiable on the interval (a, b), with the
second derivative bounded on (a, b) by Γ2 = supx∈(a,b) f 00 (x) < +∞ and γ2 =
1
inf x∈(a,b) f 00 (x) > −∞. The constant 32
is smaller than 6√1 5 given in [11] and
1√
given in [2].
18 3
1
In this note we first improve the constant 32
in the inequality (2) to the best
1√
possible one of 36 3 . Then we give two new perturbed trapezoid inequalities
for high-order differentiable mappings. We need the following variant of the
Grüss inequality:
Theorem 1. Let h, g : [a, b] → R be two integrable functions such that φ ≤
g(x) ≤ Φ for some constants φ, Φ for all x ∈ [a, b], then
(3)
Z b
Z b
Z b
1
1
h(x)g(x)
dx
−
h(x)
dx
g(x)
dx
b − a
2
(b
−
a)
a
a
a
Z b Z b
1
h(x) − 1
≤
h(y)dy dx (Φ − φ).
2
b−a a
a
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
Proof. We write the left hand of inequality (3) as
Z
a
b
1
h(x)g(x) dx −
b−a
Denote
+
Z
b
I =
a
Z
b
a
Z
b
h(x) dx
g(x) dx
a
Z b
Z b
1
h(y) dy)g(x) dx.
=
(h(x) −
b−a a
a
1
max(h(x) −
b−a
Z
1
b−a
Z
b
Littlewood’s Inequality for
p−Bimeasures
h(y) dy, 0) dx
a
Xiao-liang Cheng and Jie Sun
and
I− =
Z
b
min(h(x) −
a
b
h(y) dy, 0) dx.
a
Title Page
−
+
Obviously I + I = 0. For φ ≤ g(x) ≤ Φ, then
b
Z
a
and
Z
−
a
1
(h(x) −
b−a
b
1
(h(x) −
b−a
Contents
b
Z
h(y) dy)g(x) dx ≤ I + Φ + I − φ
a
Z
JJ
J
II
I
Go Back
b
+
−
h(y) dy)g(x) dx ≤ −I φ − I Φ
Close
a
and hence the obtained result (3) follows.
Theorem 2. Let f : [a, b] → R be a twice differentibale mapping on (a, b) with
Γ2 = supx∈(a,b) f 00 (x) < +∞ and γ2 = inf x∈(a,b) f 00 (x) > −∞, then we have
Quit
Page 4 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
the estimation
Z b
1
1
2
0
0
(4) f (x) dx − (b − a)(f (a) + f (b)) + (b − a) (f (b) − f (a))
2
12
a
1
≤ √ (Γ2 − γ2 )(b − a)3 ,
36 3
where the constant
by a smaller one.
1√
36 3
is the best one in the sense that it cannot be replaced
Proof. We choose in (3), h(x) = − 12 (x − a)(b − x) and g(x) = f 00 (x), we get
Z x2 Z Z b
1
1 b 1
2
h(x) −
h(y)dy dx = h(x) + (b − a) dx
2 a
b−a a
12
x1
1
√ (b − a)3 ,
=
36 3
√
3− 3
(b
6
√
3+ 3
(b
6
where x1 = a +
− a) and x2 = a +
− a). Thus from (3), we
derive
Z b
1
1
2
0
0
(b
−
a)(f
(a)
+
f
(b))
+
(b
−
a)
(f
(b)
−
f
(a))
f
(x)
dx
−
2
12
a
Z b
1
= − (x − a)(b − x)f 00 (x) dx
2
a
Z b
Z b
1
1
00
−
− (x − a)(b − x) dx
f (x) dx
b−a a 2
a
1
≤ √ (Γ2 − γ2 )(b − a)3 .
36 3
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
To explain the best constant 361√3 in the inequality (4), we can construct the
Rx Ry
function f (x) = a a j(z) dz dy to attain the inequality in (4),
√

3
3
−


(b − a),
γ2 , a ≤ x < x1 = a +


6




√
j(x) =
3+ 3

Γ2 , x1 ≤ x < x2 = a +
(b − a),


6





γ2 , x2 ≤ x ≤ b.
The proof is complete.
Theorem 3. Let f : [a, b] → R be a third-order differentibale mapping on (a, b)
with Γ3 = supx∈(a,b) f 000 (x) < +∞ and γ3 = inf x∈(a,b) f 000 (x) > −∞, then we
have the estimation
Z b
1
1
2
0
0
(5) f (x) dx − (b − a)(f (a) + f (b)) + (b − a) (f (b) − f (a))
2
12
a
1
≤
(Γ3 − γ3 )(b − a)4 ,
384
1
where the constant 384
is the best one in the sense that it cannot be replaced by
a smaller one.
1
Proof. We choose in (3), h(x) = 12
(x − a)(2x − a − b)(b − x), g(x) = f 000 (x),
to get
Z
Z b
Z a+b
2
1 b
1
1
|h(x) −
h(y) dy| dx =
h(x) dx =
(b − a)4 ,
2 a
b−a a
384
a
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
Thus from (3) in Theorem 1, we can derive
(5)
immediately.
R x the
R y inequality
Rz
Finally, we construct the function f (x) = a a
j(s) ds dz dy, where
and j(x) = γ3 for a+b
≤ x ≤ b, then the equality
j(x) = Γ3 for a ≤ x < a+b
2
2
holds in (5).
Theorem 4. Let f : [a, b] → R be a fourth-order differentibale mapping on
(a, b) with M4 = supx∈(a,b) |f (4) (x)| < +∞, then
(6)
Z b
1
1
2
0
0
f
(x)
dx
−
(b
−
a)(f
(a)
+
f
(b))
+
(b
−
a)
(f
(b)
−
f
(a))
2
12
a
1
≤
M4 (b − a)5 ,
720
1
720
where
is the best possible constant.
(7)
a
b
1
1
f (x) dx − (b − a)(f (a) + f (b)) + (b − a)2 (f 0 (b) − f 0 (a))
2
12
Z b
=
f (4) (x)k4 (x) dx,
a
where k4 (x) =
1
(x
24
Z
(8)
a
b
2
2
− a) (b − x) . Then we get
1
|k4 (x)| dx =
24
Z
0
Xiao-liang Cheng and Jie Sun
Title Page
Proof. We may write the remainder of the perturbed trapezoid inequality in the
kernel form
Z
Littlewood’s Inequality for
p−Bimeasures
1
x2 (1 − x)2 dx =
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 16
1
.
720
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
Then (7) – (8) imply (6). The equality holds for f (x) = x4 , a ≤ x ≤ b in
inequality (6).
Remark 0.1. We also can prove Theorem 2 and 3 in the kernel form
Z
(9)
a
b
1
1
f (x) dx − (b − a)(f (a) + f (b)) + (b − a)2 (f 0 (b) − f 0 (a))
2
12
Z b
=
f (n) (x)kn (x) dx,
a
1
1
where k2 (x) = − 12 (x−a)(b−x)+ 12
and k3 (x) = 12
(x−a)(2x−a−b)(b−x).
By the formula (7) and (9), and derive the perturbed trapezoid inequality for
different norms as shown in [5].
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Now we present the composite perturbed trapezoid quadrature for an equidistant partitioning of interval [a, b] into n subintervals. Applying Theorems 2 – 4,
we obtain
Z
b
f (x) dx = Tn (f ) + Rn (f ),
a
Contents
JJ
J
II
I
Go Back
where
n−1 b−aX
b−a
b−a
Tn (f ) =
f a+i
+ f a + (i + 1)
2n i=0
n
n
(b − a)2 0
−
(f (b) − f 0 (a)),
2
12n
Close
Quit
Page 8 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
and the remainder Rn (f ) satisfies the error estimate

(b − a)3


√
(Γ2 − γ2 ), if γ2 ≤ f 00 (x) ≤ Γ2 , ∀x ∈ (a, b),


2

36 3n





(b − a)4
(10) |Rn (f )| ≤
(Γ3 − γ3 ), if γ3 ≤ f 000 (x) ≤ Γ3 , ∀x ∈ (a, b)

3
384n






5


 (b − a) M4 ,
if |f (4) (x)| ≤ M4 , ∀x ∈ (a, b).
720n4
Then we can use (10) to get different error estimates of the composite perturbed
trapezoid quadrature.
As in [5], we may also apply the Theorems 2, 3 and 4 to special means.
In this case we may improve some of the bounds related to inequalities about
special means as given in [5, p. 492-494].
Furthermore, we discuss the Iyengar’s type inequality for the perturbed trapezoidal quadrature rule for functions whose first and second order derivatives are
bounded. In [1, 3, 9, 10] they proved the following interesting inequality involving bounded derivatives.
If f is a differentiable function on (a, b) and |f 0 (x)| ≤ M1 , then
Z b
M1 (b − a)2 (f (b) − f (a))2
1
≤
f
(x)
dx
−
(b
−
a)(f
(a)
+
f
(b))
−
.
2
4
4M1
a
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
If |f 00 (x)| ≤ M2 , x ∈ [a, b] for positive constant M2 ∈ R, then
Z b
1
1
2
0
0
f (x) dx − (b − a)(f (a) + f (b)) + (b − a) (f (b) − f (a))
2
8
a
3 !
M2
|∆|
≤
(b − a)3 −
,
24
M2
Z b
1
1
2
0
0
f
(x)
dx
−
(b
−
a)(f
(a)
+
f
(b))
+
(b
−
a)
(f
(b)
−
f
(a))
2
8
a
∆2 (b − a)
M2
≤
(b − a)3 − 1
,
24
16M2
where
(11)
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
a+b
+ f 0 (b),
2
(f (b) − f (a))
+ f 0 (b).
∆1 = f 0 (a) − 2
b−a
∆ = f 0 (a) − 2f 0
II
I
Go Back
Close
We will prove the following inequality.
Quit
Theorem 5. Let f : I → R, where I ⊆ R is an interval. Suppose that f
◦
◦
is twice differentiable in the interior I of I, and let a, b ∈ I with a < b. If
Page 10 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
|f 00 (x)| ≤ M2 , x ∈ [a, b] for positive constant M2 ∈ R. Then
Z b
1
1
2
0
0
(12) f (x) dx − (b − a)(f (a) + f (b)) + (b − a) (f (b) − f (a))
2
8
a
s
1
|∆1 |3 (b − a)3
,
≤ M2 (b − a)3 −
24
72M2
where ∆1 is defined as (11).
Proof. Denote
Z b
1
1
Jf =
f (x) dx − (b − a)(f (a) + f (b)) + (b − a)2 (f 0 (b) − f 0 (a)).
2
8
a
It is easy to see that
Xiao-liang Cheng and Jie Sun
Title Page
Contents
Z
Jf =
a
b
1
2
a+b
x−
2
2
f 00 (x) dx,
JJ
J
II
I
Go Back
and
(13)
Littlewood’s Inequality for
p−Bimeasures
2(f (b) − f (a))
∆1 = f 0 (a) −
+ f 0 (b)
b−a
Z b 1
a+b
=
2 x−
f 00 (x) dx.
b−a a
2
Close
Quit
Page 11 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
For any |ε| ≤ 18 , we get for |f 00 (x)| ≤ M2 , x ∈ [a, b],
Jf + ε(b − a)2 ∆1
2
!
Z b
a+b
a+b
1
+ 2ε(b − a) x −
f 00 (x) dx
=
x−
2
2
2
a
≤ F (ε)M2 (b − a)3 ,
where
2
Z b 1
a
+
b
a
+
b
1
F (ε) =
x−
+ 2ε(b − a) x −
dx
(b − a)3 a 2
2
2
2
Z 1 1 1
1
=
x−
+ 2ε x −
dx.
2
2 0 2
For the case 0 ≤ ε ≤ 18 , we have
2
Z 1 1
1
1
F (ε) =
x−
+ 2ε x −
dx
2
2
2
0
(Z 1
2
!
−4ε
2
1
1
1
=
x−
+ 2ε x −
dx
2
2
2
0
2
!
Z 1
2
1
1
1
−
x−
+ 2ε x −
dx
1
2
2
2
−4ε
2
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
1
Z
+
1
2
=
1
2
1
x−
2
2
1
+ 2ε x −
2
!
)
dx
32
1
+ ε3 .
24
3
For the case − 18 ≤ ε ≤ 0, we have similarly
F (ε) =
1
32
− ε3 .
24
3
Littlewood’s Inequality for
p−Bimeasures
We can prove |∆1 | ≤ 12 (b − a)M2 easily from (13). Thus we choose the parameter
s
1
|∆1 |
ε∗ = sign(∆1 )
,
|ε∗ | ≤ .
32(b − a)M2
8
1
(b − a)3 M2 −
24
s
1
= −Jf ≤ (b − a)3 M2 −
24
a)3 |∆1 |3
(b −
72M2
Replacing f with −f , we have
J−f
Title Page
Contents
By the above inequalities, we obtain
Jf ≤ F (ε∗ )M2 − ε∗ (b − a)2 ∆1 ≤
Xiao-liang Cheng and Jie Sun
.
JJ
J
II
I
Go Back
Close
s
(b − a)3 |∆1 |3
.
72M2
Thus we obtain bounds for |Jf | and prove the inequality (12).
Quit
Page 13 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
Remark 0.2. As |∆1 | ≤ 12 M2 (b − a), we have
s
|∆1 |
≥
M2
r
2 |∆1 |
,
b − a M2
Z b
1
1
2
0
0
f (x) dx − (b − a)(f (a) + f (b)) + (b − a) (f (b) − f (a))
2
8
a
M2
∆2 (b − a)
≤
(b − a)3 − 1
.
24
6M2
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
For the case f 0 (a) = f 0 (b) = 0, we have
Title Page
(14)
Z b
1
f
(x)
dx
−
(b
−
a)(f
(a)
+
f
(b))
2
a
M2
2 |f (b) − f (a)|2
≤
(b − a)3 −
.
24
3 M2 (b − a)
The inequality (14) is sharper than that stated in [9, p. 69].
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
References
[1] R.P. AGARWAL, V. ČULJAK AND J. PEČARIĆ, Some integral inequalities involving bounded higher order derivatives, Mathl. Comput. Modelling, 28(3) (1998), 51–57.
[2] X.L. CHENG, Improvement of some Ostrowski-Grüss type inequalities,
Computers Math. Applic., 42 (2001), 109–114.
[3] X.L. CHENG, The Iyengar type inequality, Appl. Math. Lett., 14 (2001),
975–978.
Littlewood’s Inequality for
p−Bimeasures
[4] S.S. DRAGOMIR AND R.P. AGARWAL, Two inequalities for differentiable mappings and applications to special means of real numbers and to
trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95.
Xiao-liang Cheng and Jie Sun
[5] S.S. DRAGOMIR, P. CERONE AND A. SOFO, Some remarks on the
trapezoid rule in numerical integration, Indian J. Pure Appl. Math., 31(5)
(2000), 475–494.
Contents
[6] S.S. DRAGOMIR, Y.J. CHO AND S.S. KIM, Inequalities of Hadamard’s
type for Lipschitizian mappings and their applications, J. Math. Anal.
Appl., 245 (2000), 489–501.
[7] S.S. DRAGOMIR AND S. WANG, An inequality of Ostrowski-Grüss’ type
and its applications to the estimation of error bounds for some special
means and for some numerical quadrature rules, Computers Math. Applic.,
33(11) (1997), 15–20.
Title Page
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au
[8] S.S. DRAGOMIR AND S. WANG, Applications of Ostrowski’ inequality
to the estimation of error bounds for some special means and for some
numerical quadrature rules, Appl. Math. Lett., 11(1) (1998), 105–109.
[9] N. ELEZOVIĆ AND J. PEČARIĆ, Steffensen’s inequality and estimates
of error in trapezoidal rule, Appl. Math. Lett., 11(6) (1998), 63–69.
[10] K.S.K. IYENGAR, Note on an inequality, Math. Student, 6 (1938), 75–
76.
[11] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Improvement and further generalization of inequalities of Ostrowski-Grüss type, Computers Math. Applic., 39(3-4) (2000), 161–175.
[12] D.S. MITRINOVIĆ, J. PEČARIĆ AND A.M. FINK, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers,
Dordrecht, (1994).
Littlewood’s Inequality for
p−Bimeasures
Xiao-liang Cheng and Jie Sun
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 16
J. Ineq. Pure and Appl. Math. 3(2) Art. 29, 2002
http://jipam.vu.edu.au