Journal of Inequalities in Pure and
Applied Mathematics
A GENERALIZATION OF THE PRE-GRÜSS INEQUALITY AND
APPLICATIONS TO SOME QUADRATURE FORMULAE
volume 3, issue 1, article 13,
2002.
NENAD UJEVIĆ
Department of Mathematics
University of Split
Teslina 12/III
21000 Split, Croatia.
EMail: ujevic@pmfst.hr
Received 03 May, 2001;
accepted 26 October, 2001.
Communicated by: P. Cerone
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
038-01
Quit
Abstract
A generalization of the pre-Grüss inequality is presented. It is applied to estimations of remainders of some quadrature formulas.
2000 Mathematics Subject Classification: 26D10, 41A55.
Key words: Pre-Grüss inequality, Generalization, Quadrature formulae.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
1.
Introduction
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 preGrüss 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]
then, using (1.1), we get the Grüss inequality:
(1.3)
(∆ − δ)(Γ − γ)
|T (f, g)| ≤
.
4
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
Specially, we have
T (f, f ) ≤
(1.4)
(∆ − δ)2
.
4
Using the above inequalities we get the following inequalities:
Z b
a+b
(1.5)
(b − a) −
f (t)dt
f
2
a
"
2 # 12
f
(b)
−
f
(a)
(b − a)2
1
2
√
≤
kf 0 k2 −
b
−
a
b−a
2 3
≤
(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,
Z b
f (a) + f (b)
(1.6)
(b
−
a)
−
f
(t)dt
2
a
"
2 # 12
2
1
f (b) − f (a)
(b − a)
2
√
≤
kf 0 k2 −
b−a
b−a
2 3
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
2
≤
(b − a)
√ (Γ − γ)
4 3
Page 4 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
and
(1.7)
Z b
f (a) + 2f a+b + f (b)
2
(b − a) −
f (t)dt
4
a
"
2 # 12
f
(b)
−
f
(a)
(b − a)2
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.
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
2.
Main Results
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
X
hx, li i hli , yi
hx, yi −
i=0
!
!
n
n
X
X
kyk2 −
≤ kxk2 −
|hx, li i|2
|hli , yi|2 ,
i=0
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
i=0
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
(2.1) SΨ (g, h) =
g(t)h(t)dt −
g(t)dt
h(t)dt
b−a a
a
a
Z b
Z b
−
g(t)Ψ0 (t)dt
h(t)Ψ0 (t)dt
a
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 20
a
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
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
Z
(2.4)
SΨ (g, h) = (b − a)T (g, h) −
b
Z
g(t)Ψ0 (t)dt
a
b
h(t)Ψ0 (t)dt
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
, t ∈ a+b
,b .
6
2
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
It is not difficult to verify that
Z b
Z b
(2.7)
Φ(t)dt =
χ(t)dt = 0
Page 7 of 20
a
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
a
http://jipam.vu.edu.au
and
kΦk22 = kχk22 =
(2.8)
(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
a+b
3
2
=√
f (a) + f
+ f (b) −
f (t)dt
2
b−a a
b−a
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
and
Contents
(2.11)
P (f ; a, b)
Z b
=
χ0 (t)f 0 (t)dt
a
Z b
6
1
a+b
+ f (b) −
f (t)dt .
=√
f (a) + 4f
2
b−a a
b−a
Remark 2.1. It is obvious that
JJ
J
II
I
Go Back
Close
Quit
Z
(2.12) SΨ (g, g) = (b − a)T (g, g) −
2
b
g(t)Ψ0 (t)dt ≤ (b − a)T (g, g).
Page 8 of 20
a
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
Theorem 2.3. (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 ,
(b − a) −
f (t)dt ≤
(2.13)
f
2
2 3
a
where
(
(2.14)
C1 =
2
kf 0 k2
[f (b) − f (a)]2
−
− [Q(f ; a, b)]2
b−a
and Q(f ; a, b) is defined by (2.10).
) 12
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Proof. We define
Title Page
(2.15)
p(t) =

 t − a, t ∈ a, a+b
2

t − b, t ∈ a+b
,
b
.
2
Then we have
p(t)dt = 0
(2.16)
a
II
I
Close
Quit
and
(2.17)
JJ
J
Go Back
b
Z
Contents
kpk22
Z
=
a
b
(b − a)3
p(t) dt =
.
12
Page 9 of 20
2
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
We now calculate
Z b
(2.18)
p(t)Φ(t)dt
a
Z
=
a
a+b
2
2a + b
(t − a) t −
3
Integrating by parts, we have
Z b
Z
0
(2.19)
p(t)f (t)dt =
a+b
2
Z
b
a + 2b
dt +
(t − b) t −
a+b
3
2
0
= f
dt = 0.
b
Z
0
(t − a)f (t)dt +
(t − b)f (t)dt
a+b
2
a
a
a+b
2
Z
(b − a) −
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
b
f (t)dt.
Nenad Ujević
a
Using (2.16), (2.18) and (2.19) we get
Z b
Z b
Z b
1
0
0
(2.20) SΦ (p, f ) =
p(t)dt
p(t)f (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
a
+
b
f
≤ SΦ (f 0 , f 0 ) 12 SΦ (p, p) 12 .
(2.21)
(b
−
a)
−
f
(t)dt
2
a
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
From (2.16)-(2.18) we get
(2.22)
1
SΦ (p, p) =
−
b−a
3
(b − a)
=
.
12
kpk22
Z
b
2 Z b
2
p(t)dt −
p(t)Φ0 (t)dt
a
a
We also have
C12
(2.23)
0
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
0
= SΦ (f , f ).
From (2.21)-(2.23) we easily find that (2.13) holds.
Nenad Ujević
Remark 2.2. It is not difficult to see that (2.13) is better than the first estimation
in (1.5).
Theorem 2.4. (Trapezoidal inequality) Under the assumptions of Theorem 2.3
we have
Z b
(b − a) 32
f (a) + f (b)
≤
√ C2 ,
(2.24)
(b
−
a)
−
f
(t)dt
2
2 3
a
Contents
JJ
J
II
I
Go Back
where
(
(2.25)
Title Page
C2 =
2
2
kf 0 k2 −
[f (b) − f (a)]
− [P (f ; a, b)]2
b−a
) 12
Close
Quit
Page 11 of 20
and P (f ; a, b) is defined by (2.11).
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
Proof. Let p(t) be defined by (2.15). We calculate
b
Z
p(t)χ(t)dt
(2.26)
a
Z
a+b
2
=
a
=
5a + b
(t − a) t −
6
Z
b
a + 5b
dt +
(t − b) t −
a+b
6
2
dt
(b − a)3
.
24
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Integrating by parts, we have
Z b
(2.27)
f 0 (t)χ(t)dt|
Nenad Ujević
a
a+b
2
Z b a + 5b
5a + b
0
f 0 (t)dt
t−
f (t)dt +
t−
=
a+b
6
6
a
2
Z b
f (a) + 4f a+b
+
f
(b)
2
=
(b − a) −
f (t)dt.
6
a
Z
Using (2.16), (2.19), (2.26), (2.27) and (2.8) we get
Title Page
Contents
JJ
J
II
I
Go Back
(2.28) Sχ (f 0 , p) =
Z
a
b
p(t)f 0 (t)dt −
1
b−a
Z
b
f 0 (t)dt
Z
b
p(t)dt
a
a
Z b
Z b
−
p(t)χ0 (t)dt
f 0 (t)χ0 (t)dt
a
Close
Quit
Page 12 of 20
a
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
a+b
2
Z
b
(b − a) −
f (t)dt
a
"
#
Z b
)
+
f
(b)
3 f (a) + 4f ( a+b
2
(b − a) −
−
f (t)dt
2
6
a
Z
1
f (a) + f (b) 1 b
= − (b − a)
+
f (t)dt.
2
2
2 a
=f
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
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
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
(2.31)
Title Page
Contents
JJ
J
II
I
Go Back
C22 = Sχ (f 0 , f 0 ).
From (2.29)-(2.31) we easily get (2.24).
Remark 2.3. We see that (2.24) is better than the first estimation in (1.6).
Close
Quit
Page 13 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
We now consider a simple quadrature rule of the form
(2.32)
f (a) + 2f
a+b
2
+ f (b)
Z
b
(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 2.5. Under the assumptions of Theorem 2.3 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
(2.34) C3 = kf 0 k2 −
[f (b) − f (a)]2
b−a
2 # 12
1
a+b
−
f (a) − 2f
+ f (b)
.
b−a
2
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
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
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
We easily find that
Z
(2.37)
b
Nenad Ujević
η(t)dt = 0, kηk22 = b − a.
a
Title Page
Let p(t) be defined by (2.15). Then we have
b
Z
Z
p(t)η(t)dt =
(2.38)
a
a+b
2
Z
(t − a)dt −
(t − b)dt
a+b
2
a
=
Contents
b
(b − a)2
.
4
JJ
J
II
I
Go Back
Close
We also have
Z
b
0
f (t)η(t)dt = −f (a) + 2f
(2.39)
a
a+b
2
Quit
− f (b).
Page 15 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
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
=f
(b − a) −
f (t)dt
2
a
a+b
b−a
−
−f (a) + 2f
− f (b)
4
2
Z b
+ f (b)
f (a) + 2f a+b
2
(b − a) −
f (t)dt.
=
4
a
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
From (2.3) and (2.40) it follows that
Z b
f (a) + 2f a+b + f (b)
2
(2.41) (b − a) −
f (t)dt
4
a
Title Page
Contents
1
1
≤ Sη (f 0 , f 0 ) 2 Sη (p, p) 2 .
II
I
Go Back
We now calculate
(2.42)
JJ
J
1
Sη (p, p) =
−
b−a
3
(b − a)
=
.
48
kpk22
Z
a
b
2 Z b
2
p(t)dt −
p(t)η0 (t)dt
Close
Quit
a
Page 16 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
We also have
(2.43)
C32 = Sη (f 0 , f 0 ).
From (2.41)-(2.43) we easily get (2.33).
Remark 2.4. It is not difficult to see that (2.33) is better than the first estimation
in (1.7).
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) −
f (t)dt ≤
(Γ − γ),
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).
(2.45) (b − a) −
f (t)dt ≤
4
8 3
a
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Inequality (2.45) follows from (2.33), since
(2.46)
Sη (f 0 , f 0 ) ≤ (b − a)
Quit
Γ−γ
2
2
Page 17 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
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 2.5. 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 Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
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.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
References
[1] G.A. ANASTASSIOU, Ostrowski type inequalities, Proc. Amer. Math.
Soc., 123(12) (1995), 3775–3781.
[2] N.S. BARNETT, S.S. DRAGOMIR AND A. SOFO, Better bounds for an
inequality of the Ostrowski type with applications, RGMIA Research Report Collection, 3(1) (2000), Article 11.
[3] P. CERONE AND S.S. DRAGOMIR, Midpoint-type rules from an inequalities point of view, Handbook of Analytic-Computational Methods in Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York, (2000),
135–200.
[4] P. CERONE AND S.S. DRAGOMIR, Trapezoidal-type rules from an inequalities point of view, Handbook of Analytic-Computational Methods
in Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York,
(2000), 65–134.
[5] S.S. DRAGOMIR, P. CERONE AND J. ROUMELIOTIS, A new generalization of Ostrowski integral inequality for mappings whose derivatives are
bounded and applications in numerical integration and for special means,
Appl. Math. Lett., 13 (2000), 19–25.
[6] S.S. DRAGOMIR AND T.C. PEACHY, New estimation of the remainder in
the trapezoidal formula with applications, Stud. Univ. Babeş-Bolyai Math.,
XLV (4), (2000), 31–42.
[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
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au
means and some numerical quadrature rules, Comput. Math. Appl., 33
(1997), 15–20.
[8] A. GHIZZETTI AND A. OSSICINI, Quadrature Formulae, Birkhaüser
Verlag, Basel/Stuttgart, 1970.
[9] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Improvement and further generalization of some inequalities of Ostrowski-Grüss type, Comput. Math.
Appl., 39 (2000), 161–179.
[10] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Acad. Publ.,
Dordrecht/Boston/Lancaster/Tokyo, 1991.
A Generalization of the
Pre-Grüss Inequality and
Applications to some
Quadrature Formulae
Nenad Ujević
[11] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Acad. Publ., Dordrecht/Boston/Lancaster/Tokyo, 1993.
[12] C.E.M. PEARCE, J. PEČARIĆ, N. UJEVIĆ AND S. VAROŠANEC, Generalizations of some inequalities of Ostrowski-Grüss type, Math. Inequal.
Appl., 3(1) (2000), 25–34.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 20
J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002
http://jipam.vu.edu.au