Journal of Inequalities in Pure and
Applied Mathematics
ON OSTROWSKI-GRÜSS-ČEBYŠEV TYPE INEQUALITIES FOR
FUNCTIONS WHOSE MODULUS OF DERIVATIVES ARE CONVEX
volume 6, issue 4, article 128,
2005.
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India.
Received 17 August, 2005;
accepted 30 August, 2005.
Communicated by: I. Gavrea
EMail: bgpachpatte@hotmail.com
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
245-05
Quit
Abstract
The aim of the present paper is to establish some new Ostrowski-Grüss-Čebyšev
type inequalities involving functions whose modulus of the derivatives are convex.
2000 Mathematics Subject Classification: 26D15, 26D20.
Key words: Ostrowski-Grüss-Čebyšev type inequalities, Modulus of derivatives,
Convex, Log-convex, Integral identities.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Proofs of Theorems 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 12
4
Proofs of Theorems 2.3 and 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 22
5
Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
References
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
1.
Introduction
In 1938, A.M. Ostrowski [5] proved the following classial inequality.
Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) whose
derivative f 0 : (a, b) → R is bounded on (a, b) i.e., |f 0 (x)| ≤ M < ∞. Then

!2 
Z b
a+b
x− 2
1
f (x) − 1
 (b − a) M,
f (t) dt ≤  +
(1.1)
b−a a
4
b−a
for all x ∈ [a, b] , where M is a constant.
For two absolutely continuous functions f, g : [a, b] → R, consider the
functional
Z b
1
(1.2) T (f, g) =
f (x) g (x) dx
b−a a
Z b
Z b
1
1
f (x) dx
g (x) dx ,
−
b−a a
b−a a
provided the involved integrals exist. In 1882, P.L.Čebyšev [6] proved that, if
f 0 , g 0 ∈ L∞ [a, b], then
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
(1.3)
1
|T (f, g)| ≤
(b − a)2 kf 0 k∞ kg 0 k∞ .
12
In 1934, G. Grüss [6] showed that
(1.4)
1
|T (f, g)| ≤ (M − m) (N − n) ,
4
Close
Quit
Page 3 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
provided m, M, n, N are real numbers satisfying the condition −∞ < m ≤
f (x) ≤ M < ∞, −∞ < n ≤ g (x) ≤ N < ∞, for all x ∈ [a, b].
During the past few years many researchers have given considerable attention to the above inequalities and various generalizations, extensions and variants of these inequalities have appeared in the literature, see [1] – [10] and the
references cited therein. Motivated by the recent results given in [1] – [3], in
the present paper, we establish some inequalities similar to those given by Ostrowski, Grüss and Čebyšev, involving functions whose modulus of derivatives
are convex. The analysis used in the proofs is elementary and based on the use
of integral identities proved in [1] and [2].
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
2.
Statement of Results
Let I be a suitable interval of the real line R. A function f : I → R is called
convex if
f (λx + (1 − λ) y) ≤ λf (x) + (1 − λ) f (y) ,
for all x, y ∈ I and λ ∈ [0, 1] . A function f : I → (0, ∞) is said to be logconvex, if
f (tx + (1 − t) y) ≤ [f (x)]t [f (y)]1−t ,
for all x, y ∈ I and t ∈ [0, 1] (see [10]). We need the following identities proved
in [1] and [2] respectively:
Z 1
Z b
Z b
1
1
0
f (x) =
f (t) dt +
(x − t)
f [(1 − λ) x + λt] dt dt,
b−a a
b−a a
0
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
f (x) =
1
b−a
Z
a
b
1
1
λf 0 [(1 − λ) a + λx] dλ
b−a 0
Z 1
1
2
− (b − x)
λf 0 [λx + (1 − λ) b] dλ,
b−a 0
f (t) dt + (x − a)2
Z
for x ∈ [a, b] where f : [a, b] → R is an absolutely continuous function on [a, b]
and λ ∈ [0, 1].
We use the following notation to simplify the details of presentation:
Z b
Z b
1
S (f, g) = f (x) g (x) −
g (x)
f (t) dt + f (x)
g (t) dt ,
2 (b − a)
a
a
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
and define k·k∞ as the usual Lebesgue norm on L∞ [a, b] i.e.,
khk∞ := ess supt∈[a,b] |h (t)|
for h ∈ L∞ [a, b] .
The following theorems deal with Ostrowski type inequalities involving two
functions.
Theorem 2.1. Let f, g : [a, b] → R be absolutely continuous functions on [a, b].
(a1 ) If |f 0 | , |g 0 | are convex on [a, b], then

!2 
a+b
x− 2
1 1
 (b − a)
(2.1) |S (f, g)| ≤  +
4 4
b−a
0
0
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
0
0
× {|g (x)| [|f (x)| + kf k∞ ] + |f (x)| [|g (x)| + kg k∞ ]} ,
Title Page
for x ∈ [a, b].
(a2 ) If |f 0 | , |g 0 | are log-convex on [a, b], then
Z b
1
A−1
0
(2.2) |S (f, g)| ≤
|g (x)| |f (x)|
|x − t|
dt
2 (b − a)
log A
a
Z b
B−1
0
+ |f (x)| |g (x)|
|x − t|
dt ,
log B
a
for x ∈ [a, b], where
(2.3)
Contents
JJ
J
II
I
Go Back
Close
Quit
A=
|f 0 (t)|
|g 0 (t)|
,
B
=
.
|f 0 (x)|
|g 0 (x)|
Page 6 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
Theorem 2.2. Let f, g : [a, b] → R be absolutely continuous functions on [a, b].
(b1 ) If |f 0 | , |g 0 | are convex on [a, x] and [x, b], then
(2.4)
|S (f, g)| ≤
1
{|g (x)| F (x) + |f (x)| G (x)} ,
2
for x ∈ [a, b], where
1h
(2.5) F (x) = |f 0 (a)|
6
2
2
x−a
b−x
0
+ |f (b)|
b−a
b−a


!2 


a+b
x− 2
+ 1+4
|f 0 (x)| (b − a) ,


b−a
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
1h
(2.6) G (x) = |g 0 (a)|
6
2
2
x−a
b−x
0
+ |g (b)|
b−a
b−a


!2 


a+b
x− 2
+ 1+4
|g 0 (x)| (b − a) ,


b−a
Contents
JJ
J
II
I
Go Back
Close
for x ∈ [a, b].
(b2 ) If |f 0 | , |g 0 | are log-convex on [a, x] and [x, b], then
Quit
Page 7 of 31
(2.7)
|S (f, g)| ≤ |g (x)| H (x) + |f (x)| L (x) ,
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
for x ∈ [a, b], where
"
2
1
x − a A1 log A1 + 1 − A1
0
(2.8) H (x) = (b − a) |f (a)|
2
b−a
(log A1 )2
#
2
b
−
x
B
log
B
+
1
−
B
1
1
1
,
+ |f 0 (b)|
b−a
(log B1 )2
"
(2.9) L (x) =
2
x − a A2 log A2 + 1 − A2
b−a
(log A2 )2
#
2
B
log
B
+
1
−
B
b
−
x
2
2
2
+ |g 0 (b)|
,
b−a
(log B2 )2
1
(b − a) |g 0 (a)|
2
and
(2.10)
(2.11)
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
|f 0 (x)|
,
|f 0 (a)|
|g 0 (x)|
A2 = 0
,
|g (a)|
A1 =
|f 0 (x)|
,
|f 0 (b)|
|g 0 (x)|
B2 = 0
,
|g (b)|
B1 =
for x ∈ [a, b].
JJ
J
II
I
Go Back
Close
Quit
The Grüss type inequalities are embodied in the following theorems.
Page 8 of 31
Theorem 2.3. Let f, g : [a, b] → R be absolutely continuous functions on [a, b].
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
(c1 ) If |f 0 | , |g 0 | are convex on [a, b], then
(2.12)
1
|T (f, g)| ≤
4 (b − a)2
Z bh
|g (x)| [|f 0 (x)| + kf 0 k∞ ]
a
i
+ |f (x)| [|g 0 (x)| + kg 0 k∞ ] E (x) dx,
where
E (x) =
(2.13)
(x − a)2 + (b − x)2
,
2
for x ∈ [a, b].
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
(c2 ) If |f 0 | , |g 0 | are log-convex on [a, b], then
Title Page
(2.14) |T (f, g)|
1
≤
2 (b − a)2
Z b
Z b
A−1
0
|g (x)|
|x − t| |f (x)|
dt
log A
a
a
Z b
B−1
0
+ |f (x)|
|x − t| |g (x)|
dt dx,
log B
a
where A, B are defined by (2.3).
Theorem 2.4. Let f, g : [a, b] → R be absolutely continuous functions on [a, b].
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
(d1 ) If |f 0 | , |g 0 | are convex on [a, b], then
2 Z "
1 b
1 0
x−a
1 0
(2.15) |T (f, g)| ≤
|g (x)|
|f (a)| + |f (x)|
2 a
b−a
6
3
1 0
1
+ |f (x)|
|g (a)| + |g 0 (x)|
6
3
2 b−x
1 0
1 0
+
|g (x)|
|f (x)| + |f (b)|
b−a
3
6
1 0
1 0
+ |f (x)|
|g (x)| + |g (b)|
dx,
3
6
(d2 ) If |f 0 | , |g 0 | are log-convex on [a, x] and [x, b], then
(2.16) |T (f, g)|
2
Z "
x−a
A1 log A1 + 1 − A1
1 b
≤
{|g (x)| |f 0 (a)|
2 a
b−a
(log A1 )2
A2 log A2 + 1 − A2
0
+ |f (x)| |g (a)|
(log A2 )2
2 b−x
B1 log B1 + 1 − B1
+
|g (x)| |f 0 (b)|
b−a
(log B1 )2
B2 log B2 + 1 − B2
0
+ |f (x)| |g (b)|
dx,
(log B2 )2
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 31
where A1 , B1 and A2 , B2 are defined by (2.10) and (2.11).
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
The next theorem contains Čebyšev type inequalities.
Theorem 2.5. Let f, g : [a, b] → R be absolutely continuous functions on [a, b].
(e1 ) If |f 0 | , |g 0 | are convex on [a, b], then
(2.17) |T (f, g)|
1
≤
4 (b − a)3
Z
b
[|f 0 (x)| + kf 0 k∞ ] [|g 0 (x)| + kg 0 k∞ ] E 2 (x) dx,
a
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
where E(x) is given by (2.13).
(e2 ) If |f 0 | , |g 0 | are log-convex on [a, b], then
Z b Z
B.G. Pachpatte
b
A−1
|x − t| |f (x)|
dt
log A
a
a
Z b
B−1
0
×
|x − t| |g (x)|
dt dx,
log B
a
1
(2.18) |T (f, g)| ≤
(b − a)3
where A, B are defined by (2.3).
0
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
3.
Proofs of Theorems 2.1 and 2.2
Proof of Theorem 2.1. From the hypotheses of Theorem 2.1, the following identities hold:
Z b
1
(3.1) f (x) =
f (t) dt
b−a a
Z 1
Z b
1
0
+
(x − t)
f [(1 − λ) x + λt]dλ dt,
b−a a
0
(3.2) g (x) =
1
b−a
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
b
Z
g (t) dt
a
1
+
b−a
b
Z
Z
(x − t)
a
1
g [(1 − λ) x + λt]dλ dt,
0
0
for x ∈ [a, b] . Multiplying both sides of (3.1) and (3.2) by g(x) and f (x) respectively, adding the resulting identities and rewriting we have
Z 1
Z b
1
0
(3.3) S (f, g) =
g (x)
(x − t)
f [(1 − λ) x + λt]dλ dt
2 (b − a)
a
0
Z 1
Z b
0
+f (x)
(x − t)
g [(1 − λ) x + λt]dλ dt .
a
Title Page
Contents
JJ
J
II
I
Go Back
0
(a1 ) Since |f 0 | , |g 0 | are convex on [a, b] , from (3.3) we observe that
Close
Quit
|S (f, g)|
1
≤
2 (b − a)
B.G. Pachpatte
Z
Z b
|g (x)|
|x − t|
a
0
1
|f [(1 − λ) x + λt]| dλ dt
Page 12 of 31
0
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
≤
=
=
≤
=
b
Z
1
+ |f (x)|
|x − t|
|g [(1 − λ) x + λt]| dλ dt
a
0
Z 1
Z b
1
0
0
|g (x)|
|x − t|
{(1 − λ) |f (x)| + λ |f (t)|} dλ dt
2 (b − a)
a
0
Z 1
Z b
0
0
+ |f (x)|
|x − t|
{(1 − λ) |g (x)| + λ |g (t)|} dλ dt
a
0
Z b
Z 1
Z 1
1
0
0
|g (x)|
|x − t| |f (x)|
(1 − λ) dλ + |f (t)|
λdλ dt
2 (b − a)
a
0
0
Z b
Z 1
Z 1
0
0
+ |f (x)|
|x − t| |g (x)|
(1 − λ) dλ + |g (t)|
λdλ dt
a
0
0
Z b
1
1
|g (x)|
|x − t| [|f 0 (x)| + |f 0 (t)|] dt
2 (b − a)
2
a
Z b
1 0
0
+ |f (x)|
|x − t| [|g (x)| + |g (t)|] dt
2
a
Z b
1
ess. sup
0
0
|g (x)|
[|f (x)| + |f (t)|]
|x − t| dt
t ∈ [a, b]
4 (b − a)
a
Z b
ess. sup
0
0
+ |f (x)|
[|g (x)| + |g (t)|]
|x − t| dt
t ∈ [a, b]
a
1
{|g (x)| [|f 0 (x)| + kf 0 k∞ ]
4 (b − a)
Z
0
b
0
+ |f (x)| |g (x)| + kg k∞
|x − t| dt
Z
0
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 31
a
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
"
#
1 (x − a)2 + (b − x)2
=
4
2 (b − a)

=
1 1
4 4
× {|g (x)| [|f 0 (x)| + kf 0 k∞ ] + |f (x)| |g 0 (x)| + kg 0 k∞
!2 
a+b
x− 2

+
b−a
× (b − a) {|g (x)| [|f 0 (x)| + kf 0 k∞ ] + |f (x)| |g 0 (x)| + kg 0 k∞ .
This is the required inequality in (2.1).
(a2 ) Since |f 0 | , |g 0 | are log-convex on [a, b] , from (3.3) we observe that
Z 1
Z b
1
0
|S (f, g)| ≤
|g (x)|
|x − t|
|f [(1 − λ) x + λt]| dλ dt
2 (b − a)
a
0
Z 1
Z b
0
+ |f (x)|
|x − t|
|g [(1 − λ) x + λt]| dλ dt
a
0
Z 1
Z b
1
1−λ
λ
0
0
|g (x)|
≤
|x − t|
[|f (x)|]
[|f (t)|] dλ dt
2 (b − a)
a
0
Z 1
Z b
1−λ
λ
0
0
+ |f (x)|
|x − t|
[|g (x)|]
[|g (t)|] dλ dt
a
0
(
"
λ #
Z b
Z 1 0
1
|f (t)|
0
=
|g (x)|
|x − t| |f (x)|
dλ dt
2 (b − a)
|f 0 (x)|
a
0
"
λ # )
Z b
Z 1 0
|g (t)|
0
+ |f (x)|
|x − t| |g (x)|
dλ dt
|g 0 (x)|
a
0
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
Z b
A−1
0
|g (x)| |f (x)|
dt
|x − t|
logA
a
Z b
B−1
0
+ |f (x)| |g (x)|
|x − t|
dt .
log B
a
1
=
2 (b − a)
This completes the proof of the inequality (2.2).
Proof of Theorem 2.2. From the hypotheses of Theorem 2.2, the following identities hold:
1
(3.4) f (x) =
b−a
b
Z
f (t) dt
1
(3.5) g (x) =
b−a
B.G. Pachpatte
a
+ (x − a)2
Z
1
b−a
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
Z
1
λf 0 [(1 − λ) a + λx] dλ
0
Z 1
1
2
− (b − x)
λf 0 [λx + (1 − λ) b] dλ,
b−a 0
b
g (t) dt
Title Page
Contents
JJ
J
II
I
Go Back
a
1
+ (x − a)
b−a
2
Z
1
λg 0 [(1 − λ) a + λx] dλ
0
Z 1
1
2
− (b − x)
λg 0 [λx + (1 − λ) b] dλ.
b−a 0
Close
Quit
Page 15 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
Multiplying both sides of (3.4) and (3.5) by g(x) and f (x) respectively, adding
the resulting identities and rewriting we have
Z 1
1
2
g (x) (x − a)
λf 0 [(1 − λ) a + λx] dλ
b−a 0
Z 1
1
2
0
− (b − x)
λf [λx + (1 − λ) b] dλ
b−a 0
Z 1
1
2
+ f (x) (x − a)
λg 0 [(1 − λ) a + λx] dλ
b−a 0
Z 1
1
2
0
λg [λx + (1 − λ) b] dλ .
− (b − x)
b−a 0
1
(3.6) S (f, g) =
2
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
(b1 ) Since |f 0 | , |g 0 | are convex on [a, x] and [x, b], from (3.6) we observe that
(3.7)
|S (f, g)| ≤
1
{|g (x)| M (x) + |f (x)| N (x)} ,
2
where
(3.8) M (x) = (x − a)2
1
b−a
1
Z
λ |f 0 [(1 − λ) a + λx]| dλ
0
Z 1
1
2
+ (b − x)
λ |f 0 [λx + (1 − λ) b]| dλ,
b−a 0
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
(x − a)2
(3.9) N (x) =
b−a
Z
Page 16 of 31
1
0
λ |g [(1 − λ) a + λx]| dλ
0
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
(b − x)2
+
b−a
Z
1
λ |g 0 [λx + (1 − λ) b]| dλ.
0
Next, we observe that
Z 1
(3.10)
λ |f 0 [(1 − λ) a + λx]| dλ
0
Z 1
Z
0
0
≤ |f (a)|
λ (1 − λ) dλ + |f (x)|
0
1
λ2 dλ
0
1
1
= |f 0 (a)| + |f 0 (x)|
6
3
and
B.G. Pachpatte
1
Z
0
(3.11)
0
λ |f [λx + (1 − λ) b]| dλ
Z 1
Z
0
2
0
≤ |f (x)|
λ dλ + |f (b)|
0
=
Title Page
1
λ (1 − λ) dλ
(3.13)
0
1
Contents
0
1 0
1
|f (x)| + |f 0 (b)| .
3
6
Similarly we have
Z 1
1
1
(3.12)
λ |g 0 [(1 − λ) a + λx]| dλ ≤ |g 0 (a)| + |g 0 (x)| ,
6
3
0
Z
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
1
1
λ |g [λx + (1 − λ) b]| dλ ≤ |g 0 (x)| + |g 0 (b)| .
3
6
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 31
0
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
From (3.8), (3.10) and (3.11) we observe that
"
2 Z 1
x−a
M (x) =
(3.14)
λ |f 0 [(1 − λ) a + λx]| dλ
b−a
0
#
2 Z 1
b−x
+
λ |f 0 [λx + (1 − λ) b]| dλ (b − a)
b−a
0
"
2 1 0
1 0
x−a
≤
|f (a)| + |f (x)|
b−a
6
3
2 #
b−x
1 0
1 0
+
|f (x)| + |f (b)| (b − a)
b−a
3
6
"
#
2
2
1
x−a
b
−
x
=
|f 0 (a)| +
|f 0 (b)| (b − a)
6
b−a
b−a
"
2 2 #
1
x−a
b−x
+
|f 0 (x)| (b − a)
+
3
b−a
b−a
"
2
2
1
x−a
b−x
0
=
|f (a)| +
|f 0 (b)|
6
b−a
b−a
"
#
2 2 #
x−a
b−x
+2
+
|f 0 (x)| (b − a) .
b−a
b−a
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
It is easy to observe that
"
#
"
2 2 #
b−x
4
x−a
(x − a)2 + (b − x)2
(3.15) 2
+
=
b−a
b−a
b−a
2 (b − a)

!2 
a+b
x− 2
4 1
 (b − a)
=
+
b−a 4
b−a

!2 
a+b
x− 2
.
= 1 + 4
b−a
Using (3.15) in (3.14) we get
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
M (x) ≤ F (x) .
(3.16)
Title Page
Similarly, from (3.9), (3.12), (3.13) we get
Contents
N (x) ≤ G (x) .
(3.17)
Using (3.16), (3.17) in (3.7) we get the required inequality in (2.4).
(b2 ) Since |f 0 | , |g 0 | are log-convex on [a, x] and [x, b] , from (3.6) we observe
that
(3.18)
|S (f, g)|
JJ
J
II
I
Go Back
Close
Quit
(
1
≤ (b − a) |g (x)|
2
"
x−a
b−a
2 Z
1
λ |f 0 [(1 − λ) a + λx]|dλ
Page 19 of 31
0
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
#
2 Z 1
b−x
+
λ |f 0 [λx + (1 − λ) b]|dλ
b−a
0
"
2 Z 1
x−a
+ |f (x)|
λ |g 0 [(1 − λ) a + λx]|dλ
b−a
0
#)
2 Z 1
b−x
0
λ |g [λx + (1 − λ) b]|dλ
+
b−a
0
(
"
2 Z 1
x−a
1
1−λ
λ
≤ (b − a) |g (x)|
λ [|f 0 (a)|]
[|f 0 (x)|] dλ
2
b−a
0
#
2 Z 1
b−x
λ
1−λ
0
0
+
λ [|f (x)|] [|f (b)|]
b−a
0
"
2 Z 1
x−a
1−λ
λ
+ |f (x)|
λ [|g 0 (a)|]
[|g 0 (x)|] dλ
b−a
0
#)
2 Z 1
b−x
λ
1−λ
+
λ [|g 0 (x)|] [|g 0 (b)|] dλ
b−a
0
(
"
2
Z 1
1
x−a
0
|f (a)|
λAλ1 dλ
= (b − a) |g (x)|
2
b−a
0
#
2
Z 1
b−x
+
|f 0 (b)|
λB1λ dλ
b−a
0
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
"
2
Z 1
x−a
0
+ |f (x)|
|g (a)|
λAλ2 dλ
b−a
0
#)
2
Z 1
b−x
+
|g 0 (b)|
λB2λ dλ
.
b−a
0
A simple calculation shows that for any C > 0 we have (see [2])
Z 1
C log C + 1 − C
(3.19)
λC λ dλ =
.
(log C)2
0
Using this fact in (3.18) we get the required inequality in (2.7).
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
4.
Proofs of Theorems 2.3 and 2.4
Proof of Theorem 2.3. From the hypotheses of Theorem 2.3 the identities (3.1)
– (3.3) hold. Integrating both sides of (3.3) with respect to x from a to b and
rewriting we have
(4.1) T (f, g)
1
=
2 (b − a)2
Z
Z b
Z b
g (x)
(x − t)
a
Z
a
b
Z
(x − t)
+f (x)
a
0
1
1
f [(1 − λ) x + λt] dλ dt
0
0
g [(1 − λ) x + λt] dλ dt dx.
0
(c1 ) Since |f 0 | , |g 0 | are convex on [a, b], from (4.1) we observe that
B.G. Pachpatte
Title Page
|T (f, g)|
1
≤
2 (b − a)2
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
Z
Z b
Z b
|g (x)|
|x − t|
a
a
b
[(1 − λ) |f (x)| + λ |f (t)|] dλ dt
0
0
0
1
0
0
+ |f (x)|
|x − t|
[(1 − λ) |g (x)| + λ |g (t)|] dλ dt dx
a
0
0
Z b
Z b
1
|f (x)| + |f 0 (t)|
dt
=
|g (x)|
|x − t|
2
2 (b − a)2 a
a
0
Z b
|g (x)| + |g 0 (t)|
+ |f (x)|
|x − t|
dt dx
2
a
Z
Z
1
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 22 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
Z b
Z b
ess sup |f 0 (x)| + |f 0 (t)|
|g (x)|
|x − t|
dt
t ∈ [a, b]
2
a
a
Z b
ess sup |g 0 (x)| + |g 0 (t)|
+ |f (x)|
|x − t|
dt dx
t ∈ [a, b]
2
a
Z b
1
=
[|g (x)| [|f 0 (x)| + kf 0 k∞ ]dt
2
4 (b − a) a
Z b
0
0
+ |f (x)| [|g (x)| + kg k∞ ]]
|x − t|dt dx
a
Z b
1
[|g (x)| [|f 0 (x)| + kf 0 k∞ ]dt
=
4 (b − a)2 a
+ |f (x)| [|g 0 (x)| + kg 0 k∞ ]] E (x) dx.
1
≤
2 (b − a)2
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
This completes the proof of the inequality (2.14).
Title Page
(c2 ) Since |f 0 | , |g 0 | are log-convex on [a, b] from (4.1) we observe that
Z 1
Z b
Z b
1
1−λ
λ
0
0
|T (f, g)| ≤
|g (x)|
|x − t|
[|f (x)|]
[|f (t)|] dλ dt
2 (b − a)2 a
a
0
Z 1
Z b
1−λ
λ
0
0
+ |f (x)|
|x − t|
[|g (x)|]
[|g (t)|] dλ dt dx
a
0
"
λ #
Z b"
Z b
Z 1 0
1
|f
(t)|
=
|g (x)|
|x − t| |f 0 (x)|
dλ dt
|f 0 (x)|
2 (b − a)2 a
a
0
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
λ # #
0
|g
(t)|
+ |f (x)|
|x − t| |g 0 (x)|
dλ dt dx
|g 0 (x)|
a
0
Z b
Z b
A−1
1
0
=
|g (x)|
|x − t| |f (x)|
dt
log A
2 (b − a)2 a
a
Z b
B−1
0
+ |f (x)|
|x − t| |g (x)|
dt dx,
log B
a
Z
b
"
1
Z
where A, B are defined by (2.3). This is the required inequality in (2.14).
Proof of Theorem 2.4. From the hypotheses of Theorem 2.4 the identities (3.4)
– (3.6) hold. Integrating both sides of (3.6) with respect to x from a to b and
rewriting we have
Z b "
2 Z 1
x−a
g (x)
λf 0 [(1 − λ) a + λx] dλ
b
−
a
a
0
Z 1
0
+f (x)
λg [(1 − λ) a + λx]dλ
0
2 Z 1
b−x
−
g (x)
λf 0 [λx + (1 − λ) b] dλ
b−a
0
Z 1
0
+ f (x)
λg [λx + (1 − λ) b]dλ dx.
1
(4.2) T (f, g) =
2
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
0
Page 24 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
(d1 ) Since |f 0 | , |g 0 | are convex on [a, x] and [x, b] from (4.2) we observe that
2 Z "
Z 1
1 b
x−a
|T (f, g)| ≤
|g (x)|
λ |f 0 [(1 − λ) a + λx]| dλ
2 a
b−a
0
Z 1
0
+ |f (x)|
λ |g [(1 − λ) a + λx]|dλ
0
2 Z 1
b−x
+
|g (x)|
λ |f 0 [λx + (1 − λ) b]| dλ
b−a
0
Z 1
0
+ |f (x)|
λ |g [λx + (1 − λ) b]|dλ dx
0
2 Z "
Z 1
x−a
1 b
|g (x)|
λ {(1 − λ) |f 0 (a)| + λ |f 0 (x)|} dλ
≤
2 a
b−a
0
Z 1
0
0
+ |f (x)|
λ {(1 − λ) |g (a)| + λ |g (x)|}dλ
0
2 Z 1
b−x
+
|g (x)|
λ {λ |f 0 (x)| + (1 − λ) |f 0 (b)|} dλ
b−a
0
Z 1
0
0
+ |f (x)|
λ {λ |g (x)| + (1 − λ) |g (b)|}dλ dx
0
2 Z "
1 b
x−a
1 0
1 0
=
|g (x)|
|f (a)| + |f (x)|
2 a
b−a
6
3
1 0
1
+ |f (x)|
|g (a)| + |g 0 (x)|
6
3
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
2 b−x
1 0
1 0
+
|g (x)|
|f (x)| + |f (b)|
b−a
3
6
1 0
1
+ |f (x)|
|g (x)| + |g 0 (b)|
dx.
3
6
This proves the inequality in (2.15).
(d2 ) Since |f 0 | , |g 0 | are log-convex on [a, x] and [x, b], from (4.2) and the fact
(3.19) we observe that
|T (f, g)|
2 Z b "
Z 1
1
x−a
1−λ
λ
≤
|g (x)|
λ [|f 0 (a)|]
[|f 0 (x)|] dλ
2 a
b−a
0
Z 1
1−λ
λ
0
0
+ |f (x)|
λ [|g (a)|]
[|g (x)|] dλ
0
2 Z 1
b−x
λ
1−λ
+
|g (x)|
λ [|f 0 (x)|] [|f 0 (b)|]
dλ
b−a
0
Z 1
λ
1−λ
0
0
+ |f (x)|
λ [|g (x)|] [|g (b)|]
dλ dx
0
2 Z "
Z 1
Z 1
1 b
x−a
0
λ
0
λ
=
|g (x)| |f (a)|
λA1 dλ + |f (x)| |g (a)|
λB1 dλ
2 a
b−a
0
0
2 #
Z 1
Z 1
b−x
|g (x)| |f 0 (b)|
λAλ2 dλ+ |f (x)| |g 0 (b)|
+
λB2λ dλ dx
b−a
0
0
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
1
=
2
Z b "
2 A1 log A1 + 1 − A1
|g (x)| |f 0 (a)|
(log A1 )2
a
B1 log B1 + 1 − B1
0
+ |f (x)| |g (a)|
(log B1 )2
2 b−x
A2 log A2 + 1 − A2
+
|g (x)| |f 0 (b)|
b−a
(log A2 )2
B2 log B2 + 1 − B2
0
+ |f (x)| |g (b)|
dx.
(log B2 )2
x−a
b−a
This is the desired inequality in (2.16).
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
5.
Proof of Theorem 2.5
From the hypotheses, the identities (3.1) and (3.2) hold. From (3.1) and (3.2)
we observe that
Z b
Z b
1
1
f (x) −
f (t) dt
g (x) −
g (t) dt
b−a a
b−a a
Z 1
Z b
1
0
=
(x − t)
f [(1 − λ) x + λt] dλ dt
b−a a
0
Z 1
Z b
1
0
×
(x − t)
g [(1 − λ) x + λt] dλ dt
b−a a
0
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
that is,
Z b
Z b
1
1
(5.1) f (x) g (x) − f (x)
g (t) dt − g (x)
f (t) dt
b−a a
b−a a
Z b
Z b
1
1
+
f (t) dt
g (t) dt
b−a a
b−a a
Z 1
Z b
1
0
=
(x − t)
f [(1 − λ) x + λt] dλ dt
b−a a
0
Z 1
Z b
1
0
×
(x − t)
g [(1 − λ) x + λt] dλ dt .
b−a a
0
Integrating both sides of (5.1) with respect to x from a to b and rewriting we
have
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 28 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
(5.2) T (f, g)
1
=
b−a
Z b
Z 1
Z b
1
0
(x − t)
f [(1 − λ) x + λt] dλ dt
b−a a
a
0
Z 1
Z b
1
0
×
(x − t)
g [(1 − λ) x + λt] dλ dt dx.
b−a a
0
(e1 ) Since |f 0 | , |g 0 | are convex on [a, b] , from (5.2) we observe that
Z 1
Z b
Z b
1
1
0
|T (f, g)| ≤
|x − t|
|f [(1 − λ) x + λt]| dλ dt
b−a a b−a a
0
Z 1
Z b
1
0
×
|x − t|
|g [(1 − λ) x + λt]| dλ dt dx
b−a a
0
Z 1
Z b Z b
1
0
0
|x − t|
[(1 − λ) |f (x)| + λ |f (t)|] dλ dt
≤
(b − a)3 a
a
0
Z b
Z 1
0
0
×
|x − t|
[(1 − λ) |g (x)| + λ |g (t)|] dλ dt dx
a
0
0
Z b Z b
1
|f (x)| + |f 0 (t)|
=
|x − t|
dt
2
(b − a)3 a
a
Z b
0
|g (x)| + |g 0 (t)|
×
|x − t|
dt dx.
2
a
The rest of the proof of inequality (2.17) can be completed by closely looking
at the proof of Theorem 2.3, part (c1 ).
(e2 ) The proof follows by closely looking at the proof of (e1 ) given above and
the proof of Theorem 2.3, part (c2 ) . We omit the details.
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 29 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
References
[1] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR, M.R. PINHEIRO AND
A. SOFO, Ostrowski type inequalities for functions whose modulus of
derivatives are convex and applications, RGMIA Res. Rep. Coll., 5(2)
(2002), 219–231. [ONLINE: http://rgmia.vu.edu.au/v5n2.
html]
[2] P.CERONE AND S.S.DRAGOMIR, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions, Demonstratio Math., 37(2) (2004), 299–308.
[3] S.S.DRAGOMIR AND A.SOFO, Ostrowski type inequalities for functions
whose derivatives are convex, Proceedings of the 4th International Conference on Modelling and Simulation, November 11-13, 2002. Victoria
University, Melbourne, Australia. RGMIA Res. Rep. Coll., 5(Supp) (2002),
Art. 30. [ONLINE: http://rgmia.vu.edu.au/v5(E).html]
[4] S.S.DRAGOMIR AND Th.M. RASSIAS (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002.
[5] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
[6] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 30 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au
[7] B.G. PACHPATTE, A note on integral inequalities involving two logconvex functions, Math. Inequal. Appl., 7(4) (2004), 511–515.
[8] B.G. PACHPATTE, A note on Hadamard type integral inequalities involving several log-convex functions, Tamkang J. Math., 36(1) (2005), 43–47.
[9] B.G. PACHPATTE, Mathematical Inequalities, North-Holland Mathematical Library, Vol. 67 Elsevier, 2005.
[10] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TANG, Convex functions, Partial Orderings and Statistcal Applications, Academic Press, New York,
1991.
On Ostrowski-Grüss-Čebyšev
Type Inequalities for Functions
Whose Modulus of Derivatives
are Convex
B.G. Pachpatte
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 31 of 31
J. Ineq. Pure and Appl. Math. 6(4) Art. 128, 2005
http://jipam.vu.edu.au