Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 2, Article 13, 2002
A GENERALIZATION OF THE PRE-GRÜSS INEQUALITY AND APPLICATIONS
TO SOME QUADRATURE FORMULAE
NENAD UJEVIĆ
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF S PLIT
T ESLINA 12/III
21000 S PLIT, C ROATIA .
ujevic@pmfst.hr
Received 03 May, 2001; accepted 26 October, 2001.
Communicated by P. Cerone
A BSTRACT. A generalization of the pre-Grüss inequality is presented. It is applied to estimations of remainders of some quadrature formulas.
Key words and phrases: Pre-Grüss inequality, Generalization, Quadrature formulae.
2000 Mathematics Subject Classification. 26D10, 41A55.
1. I NTRODUCTION
In recent years a number of authors have written about generalizations of Ostrowski’s inequality. For example, this topic is considered in [1], [2], [5], [7], [9] and [12]. In this way
some new types of inequalities are formed, such as inequalities of Ostrowski-Grüss type, inequalities of Ostrowski-Chebyshev type, etc. An important role in forming these inequalities is
played by the pre-Grüss inequality. This paper develops a new approach to the topic obtaining
better results than the approach using the pre-Grüss inequality. It presents new, improved versions of the mid-point and trapezoidal inequality. The mid-point inequality is considered in [1],
[2], [3], [7] and [9], while the trapezoidal inequality is considered in [4], [5], [7] and [9].
In [11] we can find the pre-Grüss inequality:
(1.1)
T (f, g)2 ≤ T (f, f )T (g, g),
where f, g ∈ L2 (a, b) and T (f, g) is the Chebyshev functional:
Z b
Z b
Z b
1
1
(1.2)
T (f, g) =
f (t)g(t)dt −
f (t)dt
g(t)dt.
b−a a
(b − a)2 a
a
If there exist constants γ, δ, Γ, ∆ ∈ R such that
δ ≤ f (t) ≤ ∆ and γ ≤ g(t) ≤ Γ, t ∈ [a, b]
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
038-01
2
N ENAD U JEVI Ć
then, using (1.1), we get the Grüss inequality:
|T (f, g)| ≤
(1.3)
(∆ − δ)(Γ − γ)
.
4
Specially, we have
(∆ − δ)2
.
4
Using the above inequalities we get the following inequalities:
"
2 # 12
Z b
2
f (b) − f (a)
a+b
(b − a)
1
2
√
(1.5) f
(b − a) −
f (t)dt ≤
kf 0 k2 −
2
b−a
b−a
2 3
a
T (f, f ) ≤
(1.4)
(b − a)2
√ (Γ − γ)
4 3
where f : [a, b] → R is an absolutely continuous function whose derivative f 0 ∈ L2 (a, b) and
γ ≤ f 0 (t) ≤ Γ, t ∈ [a, b] . As usual, k·k2 is the norm in L2 (a, b). Further,
"
2 # 12
Z b
f (a) + f (b)
(b − a)2
1
f
(b)
−
f
(a)
2
√
(b − a) −
f (t)dt ≤
kf 0 k2 −
(1.6) 2
b
−
a
b−a
2
3
a
≤
≤
(b − a)2
√ (Γ − γ)
4 3
and
(1.7)
Z b
f (a) + 2f a+b + f (b)
2
(b − a) −
f (t)dt
4
a
"
2 # 12
2
f (b) − f (a)
(b − a)
1
2
√
≤
kf 0 k2 −
b−a
b−a
4 3
(b − a)2
√ (Γ − γ)
8 3
where the function f satisfies the above conditions. The inequalities (1.5)-(1.7) are considered
(and proved) in [2], [9] and [12].
In this paper we generalize (1.1). We use the generalization to improve the above inequalities.
≤
2. M AIN R ESULTS
Lemma 2.1. Let f, g, Ψi ∈ L2 (a, b), i = 0, 1, 2, ..., n, where Ψ0i = Ψi (t)/ kΨi k2 are orthonormal functions. If Sn (f, g) is defined by
Z b
Z b
n Z b
X
0
Sn (f, g) =
f (t)g(t)dt −
f (s)Ψi (s)ds
g(s)Ψ0i (s)ds
a
i=0
a
then we have
a
1
1
|Sn (f, g)| ≤ Sn (f, f ) 2 Sn (g, g) 2 .
The proof follows by the known inequality holding in inner product spaces (H, h·, ·i)
2
!
!
n
n
n
X
X
X
2
2
2
2
hx, li i hli , yi ≤ kxk −
|hx, li i|
kyk −
|hli , yi| ,
hx, yi −
i=0
J. Inequal. Pure and Appl. Math., 3(2) Art. 13, 2002
i=0
i=0
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G ENERALIZATION OF PRE -G RÜSS INEQUALITY
3
where x, y ∈ H and {li }i=0,n is an orthonormal family in H, i.e., (li , lj ) = δij for i, j ∈
{0, . . . , n}.
√
We here use only the case n = 1. We choose Ψ00 (t) = 1/ b − a, Ψ1 (t) = Ψ(t) and denote
S1 (g, h) = SΨ (g, h) such that
Z b
Z b
Z b
1
g(t)dt
h(t)dt
(2.1) SΨ (g, h) =
g(t)h(t)dt −
b−a a
a
a
Z b
Z b
−
g(t)Ψ0 (t)dt
h(t)Ψ0 (t)dt
a
a
where g, h, Ψ ∈ L2 (a, b), Ψ0 (t) = Ψ(t)/ kΨk2 and
Z b
(2.2)
Ψ(t)dt = 0.
a
Lemma 2.2. With the above notations we have
1
1
|SΨ (g, h)| ≤ SΨ (g, g) 2 SΨ (h, h) 2 .
(2.3)
It is obvious that
b
Z
(2.4)
SΨ (g, h) = (b − a)T (g, h) −
Z
g(t)Ψ0 (t)dt
b
h(t)Ψ0 (t)dt
a
a
so that SΨ (g, h) is a generalization of the Chebyshev functional.
We also define the functions:

, t ∈ a, a+b
 t − 2a+b
3
2
(2.5)
Φ(t) =

t − a+2b
, t ∈ a+b
,b
3
2
and
(2.6)
χ(t) =

 t−
5a+b
,
6
t ∈ a, a+b
2
t−
a+5b
,
6
t∈

a+b
,b
2
.
It is not difficult to verify that
Z
b
Z
b
Φ(t)dt =
(2.7)
a
χ(t)dt = 0
a
and
(2.8)
kΦk22 = kχk22 =
(b − a)3
.
36
We define
(2.9)
Φ0 (t) =
Φ(t)
χ(t)
, χ0 (t) =
.
kΦk2
kχk2
Integrating by parts, we have
Z b
(2.10)
Q(f ; a, b) =
Φ0 (t)f 0 (t)dt
a
Z b
2
a+b
3
= √
f (a) + f
+ f (b) −
f (t)dt
2
b−a a
b−a
J. Inequal. Pure and Appl. Math., 3(2) Art. 13, 2002
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4
N ENAD U JEVI Ć
and
Z
(2.11)
b
χ0 (t)f 0 (t)dt
a
Z b
a+b
6
1
= √
f (a) + 4f
+ f (b) −
f (t)dt .
2
b−a a
b−a
P (f ; a, b) =
Remark 2.3. It is obvious that
Z
SΨ (g, g) = (b − a)T (g, g) −
(2.12)
b
2
g(t)Ψ0 (t)dt ≤ (b − a)T (g, g).
a
Theorem 2.4. (Mid-point inequality) Let I ⊂ R be a closed interval and a, b ∈ Int I, a < b. If
f : I → R is an absolutely continuous function whose derivative f 0 ∈ L2 (a, b) then we have
Z b
(b − a) 32
a+b
√ C1 ,
(2.13)
(b − a) −
f (t)dt ≤
f
2
2 3
a
where
(
2
kf 0 k2
C1 =
(2.14)
[f (b) − f (a)]2
−
− [Q(f ; a, b)]2
b−a
) 12
and Q(f ; a, b) is defined by (2.10).
Proof. We define
p(t) =
(2.15)

 t − a, t ∈ a, a+b
2
a+b
,b
2
t − b, t ∈

.
Then we have
b
Z
p(t)dt = 0
(2.16)
a
and
kpk22
(2.17)
b
Z
p(t)2 dt =
=
a
We now calculate
Z b
Z
(2.18)
p(t)Φ(t)dt =
a+b
2
a
a
2a + b
(t − a) t −
3
(b − a)3
.
12
b
Z
a + 2b
dt +
(t − b) t −
a+b
3
2
dt = 0.
Integrating by parts, we have
Z
b
a+b
2
Z
0
p(t)f (t)dt =
(2.19)
a
0
= f
J. Inequal. Pure and Appl. Math., 3(2) Art. 13, 2002
(t − b)f 0 (t)dt
(t − a)f (t)dt +
a+b
2
a
b
Z
a+b
2
Z
(b − a) −
b
f (t)dt.
a
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G ENERALIZATION OF PRE -G RÜSS INEQUALITY
5
Using (2.16), (2.18) and (2.19) we get
Z b
Z b
Z b
1
0
0
SΦ (p, f ) =
p(t)f (t)dt −
(2.20)
p(t)dt
f 0 (t)dt
b
−
a
a
a
a
Z b
Z b
−
f 0 (t)Φ0 (t)dt
p(t)Φ0 (t)dt
a
a
Z b
a+b
(b − a) −
f (t)dt.
= f
2
a
From (2.20) and (2.3) it follows that
Z b
1
1
a+b
(2.21)
(b − a) −
f (t)dt ≤ SΦ (f 0 , f 0 ) 2 SΦ (p, p) 2 .
f
2
a
From (2.16)-(2.18) we get
(2.22)
1
SΦ (p, p) =
−
b−a
(b − a)3
=
.
12
kpk22
Z
b
2 Z b
2
p(t)dt −
p(t)Φ0 (t)dt
a
a
We also have
C12 = SΦ (f 0 , f 0 ).
(2.23)
From (2.21)-(2.23) we easily find that (2.13) holds.
Remark 2.5. It is not difficult to see that (2.13) is better than the first estimation in (1.5).
Theorem 2.6. (Trapezoidal inequality) Under the assumptions of Theorem 2.4 we have
Z b
f (a) + f (b)
(b − a) 32
√ C2 ,
(2.24)
(b − a) −
f (t)dt ≤
2
2 3
a
where
(
) 12
2
[f
(b)
−
f
(a)]
2
2
(2.25)
C2 = kf 0 k2 −
− [P (f ; a, b)]
b−a
and P (f ; a, b) is defined by (2.11).
Proof. Let p(t) be defined by (2.15). We calculate
Z b
Z a+b
Z b
2
5a + b
a + 5b
(2.26)
p(t)χ(t)dt =
(t − a) t −
dt +
(t − b) t −
dt
a+b
6
6
a
a
2
(b − a)3
.
24
Integrating by parts, we have
Z b
Z a+b Z b 2
5a + b
a + 5b
0
0
t−
(2.27)
f (t)χ(t)dt =
f (t)dt +
t−
f 0 (t)dt
a+b
6
6
a
a
2
Z
b
f (a) + 4f a+b
+ f (b)
2
=
(b − a) −
f (t)dt.
6
a
=
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6
N ENAD U JEVI Ć
Using (2.16), (2.19), (2.26), (2.27) and (2.8) we get
Z b
Z b
Z b
1
0
0
0
Sχ (f , p) =
p(t)f (t)dt −
(2.28)
f (t)dt
p(t)dt
b−a a
a
a
Z b
Z b
−
p(t)χ0 (t)dt
f 0 (t)χ0 (t)dt
a
a
Z b
a+b
(b − a) −
f (t)dt
=f
2
a
"
#
Z b
)
+
f
(b)
3 f (a) + 4f ( a+b
2
−
(b − a) −
f (t)dt
2
6
a
Z
f (a) + f (b) 1 b
1
= − (b − a)
+
f (t)dt.
2
2
2 a
From (2.3) and (2.28) it follows that
Z b
f (a) + f (b)
1
1
(2.29)
(b − a) −
f (t)dt ≤ 2Sχ (f 0 , f 0 ) 2 Sχ (p, p) 2 .
2
a
We have
(2.30)
1
Sχ (p, p) =
−
b−a
3
(b − a)
=
48
kpk22
Z
b
2 Z b
2
p(t)dt −
p(t)χ0 (t)dt
a
a
and
C22 = Sχ (f 0 , f 0 ).
(2.31)
From (2.29)-(2.31) we easily get (2.24).
Remark 2.7. We see that (2.24) is better than the first estimation in (1.6).
We now consider a simple quadrature rule of the form
Z b
f (a) + 2f a+b
+ f (b)
2
(2.32)
(b − a) −
f (t)dt
4
a
Z b
1
a+b
f (a) + f (b)
=
f
+
(b − a) −
f (t)dt = R(f ).
2
2
2
a
It is not difficult to see that (2.32) is a convex combination of the mid-point quadrature rule and
the trapezoidal quadrature rule. In [5] it is shown that (2.32) has a better estimation of error
than the well-known Simpson quadrature rule (when we estimate the error in terms of the first
derivative f 0 of integrand f ). We here have a similar case.
Theorem 2.8. Under the assumptions of Theorem 2.4 we have
Z b
f (a) + 2f a+b + f (b)
(b − a) 32
2
√ C3 ,
(2.33)
(b − a) −
f (t)dt ≤
4
4 3
a
where
"
(2.34)
C3 =
2
kf 0 k2
[f (b) − f (a)]2
1
−
−
b−a
b−a
J. Inequal. Pure and Appl. Math., 3(2) Art. 13, 2002
f (a) − 2f
a+b
2
2 # 12
+ f (b)
.
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G ENERALIZATION OF PRE -G RÜSS INEQUALITY
7
Proof. We define
η(t) =
(2.35)



1, t ∈ a, a+b
2
−1, t ∈
η0 (t) =
(2.36)
,
a+b
,b
2
η(t)
.
kηk2
We easily find that
Z
b
(2.37)
η(t)dt = 0, kηk22 = b − a.
a
Let p(t) be defined by (2.15). Then we have
Z b
Z a+b
Z
2
(2.38)
p(t)η(t)dt =
(t − a)dt −
a
b
(t − b)dt =
a+b
2
a
(b − a)2
.
4
We also have
Z
b
0
f (t)η(t)dt = −f (a) + 2f
(2.39)
a
a+b
2
− f (b).
From (2.37)-(2.39) we get
Z b
Z b
Z b
1
0
0
0
(2.40)
Sη (f , p) =
p(t)f (t)dt −
f (t)dt
p(t)dt
b−a a
a
a
Z b
Z b
−
p(t)η0 (t)dt
f 0 (t)η0 (t)dt
a
a
Z b
a+b
(b − a) −
f (t)dt
= f
2
a
b−a
a+b
−
−f (a) + 2f
− f (b)
4
2
Z b
f (a) + 2f a+b
+ f (b)
2
=
(b − a) −
f (t)dt.
4
a
From (2.3) and (2.40) it follows that
Z b
f (a) + 2f a+b + f (b)
1
1
2
(2.41)
(b − a) −
f (t)dt ≤ Sη (f 0 , f 0 ) 2 Sη (p, p) 2 .
4
a
We now calculate
(2.42)
1
Sη (p, p) =
−
b−a
(b − a)3
=
.
48
kpk22
Z
b
2 Z b
2
p(t)dt −
p(t)η0 (t)dt
a
a
We also have
(2.43)
C32 = Sη (f 0 , f 0 ).
From (2.41)-(2.43) we easily get (2.33).
Remark 2.9. It is not difficult to see that (2.33) is better than the first estimation in (1.7).
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8
N ENAD U JEVI Ć
Finally, in [12] we can find the next inequality
Z b
f (a) + 4f a+b + f (b)
(b − a)2
2
(2.44)
(b − a) −
(Γ − γ),
f (t)dt ≤
6
12
a
where f : I → R, (I ⊂ R is an open interval, a < b, a, b ∈ I) is a differentiable function, f 0 is
integrable and there exist constants γ, Γ ∈ R such that γ ≤ f 0 (t) ≤ Γ, t ∈ [a, b].
Inequality (2.44) is a variant of the Simpson’s inequality. On the other hand, we have
Z b
f (a) + 2f a+b + f (b)
(b − a)2
2
√ (Γ − γ).
(b − a) −
f (t)dt ≤
(2.45)
4
8 3
a
Inequality (2.45) follows from (2.33), since
0
0
Sη (f , f ) ≤ (b − a)
(2.46)
Γ−γ
2
2
and (2.46) follows from (2.4) and (1.4).
Form (2.44) and (2.45) we see that the simple 3-point quadrature rule (2.32) has a better estimation of error than the well-known 3-point Simpson quadrature rule. Note that the estimations
are expressed in terms of the first derivative f 0 of integrand.
Finally, the following remark is valid.
Remark 2.10. The considered case n = 1 illustrates how to apply Lemma 2.1 to quadrature
formulas. It is also shown that the derived results are better than some recently obtained results.
We can use Lemma 2.1 to derive further improvements of the obtained results. However, in
such a case we must require
Z b
g(t)Ψ0i (t)dt = 0, i = 0, 1, 2, ..., n.
a
n
Thus, the construction of such a finite sequence {Ψ0i }0 can be complicated. However, if we
really need better error bounds, without taking into account possible complications, then we
can apply the procedure described in this section.
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