Volume 8 (2007), Issue 4, Article 96, 10 pp.
ON THE WEIGHTED OSTROWSKI INEQUALITY
N.S. BARNETT AND S.S. DRAGOMIR
S CHOOL OF C OMPUTER S CIENCE AND M ATHEMATICS
V ICTORIA U NIVERSITY, PO B OX 14428
M ELBOURNE C ITY, VIC 8001, AUSTRALIA .
neil.barnett@vu.edu.au
sever.dragomir@vu.edu.au
Received 14 May, 2007; accepted 30 September, 2007
Communicated by B.G. Pachpatte
A BSTRACT. On utilising an identity from [5], some weighted Ostrowski type inequalities are
established.
Key words and phrases: Ostrowski inequality, Integral inequalities, Absolutely continuous functions.
2000 Mathematics Subject Classification. 26D15, 26D10.
1. I NTRODUCTION
In [5], the authors obtained the following generalisation of the weighted Montgomery identity:
Z t
Z b
1
0
(1.1)
f (x) =
w (t) ϕ
w (s) ds f (t) dt
ϕ (1) a
a
Z b
1
+
Pw,ϕ (x, t) f 0 (t) dt,
ϕ (1) a
where f : [a, b] → R is an absolutely continuous function, ϕ : [0, 1] → R is a differentiable
function with ϕ (0) = 0, ϕ (1) 6= 0 and w : [a, b] → [0, ∞) is a probability density function
such that the weighed Peano kernel
 R
 ϕ t w (s) ds ,
a ≤ t ≤ x,
Ra
(1.2)
Pw,ϕ (x, t) :=
 ϕ t w (s) ds − ϕ (1) , x < t ≤ b,
a
is integrable for any x ∈ [a, b] .
If ϕ (t) = t, then (1.1) reduces to the weighted Montgomery identity obtained by Pečarić in
[21]:
Z b
Z b
(1.3)
f (x) =
w (t) f (t) dt +
Pw (x, t) f 0 (t) dt,
a
160-07
a
2
N.S. BARNETT AND S.S. D RAGOMIR
where the weighted Peano kernel Pw is
( Rt
w (s) ds,
a ≤ t ≤ x,
Rb
− t w (s) ds, x < t ≤ b.
a
Pw (x, t) :=
(1.4)
Finally, the uniform distribution is used to provide the Montgomery identity [17, p. 565]:
Z b
Z b
1
(1.5)
f (x) =
f (t) dt +
P (x, t) f 0 (t) dt,
b−a a
a
with
(
P (x, t) :=
t−a
b−a
t−b
b−a
if a ≤ t ≤ x,
if x < t ≤ b,
that has been extensively used to obtain Ostrowski type results, see for instance the research
papers [3] – [6], [7] – [16], [19] – [20], [22] and the book [15].
In the same paper [5], on introducing the generalised Čebyšev functional,
b
0
Z
x
(1.6) Tϕ (w, f, g) :=
w (x) ϕ
w (t) dt f (x) g (x) dx
a
a
Z b
Z x
1
0
−
w (x) ϕ
w (t) dt f (x) dx
ϕ (1) a
a
Z b
Z x
0
×
w (x) ϕ
w (t) dt g (x) dx ,
Z
a
a
the authors obtained the representation:
1
(1.7) Tϕ (w, f, g) = 2
ϕ (1)
Z
a
b
0
Z
x
w (t) dt
w (x) ϕ
a
Z b
Z b
0
0
×
Pw,ϕ (x, t) f (t) dt
Pw,ϕ (x, t) g (t) dt dx
a
a
and used it to obtain an upper bound for the absolute value of the Čebyšev functional in the case
where f 0 , g 0 , ϕ0 ∈ L∞ [a, b] . This bound can be stated as:
Z b
1
0
0
0
(1.8)
|Tϕ (w, f, g)| ≤ 2
kf k∞ kg k∞ kϕ k∞
w (x) H 2 (x) dx,
ϕ (1)
a
Rb
where H (x) := a |Pw,ϕ (x, t)| dt. The inequality (1.8) provides a generalisation of a result
obtained by Pachpatte in [18].
The main aim of this paper is to obtain some weighted inequalities of the Ostrowski type by
providing various upper bounds for the deviation of f (x) , x ∈ [a, b] , from the integral mean
Z t
Z b
1
0
w (t) ϕ
w (s) ds f (t) dt,
ϕ (1) a
a
when f is absolutely continuous, of bounded variation or Lipschitzian on the interval [a, b].
Some particular cases of interest are also given.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
http://jipam.vu.edu.au/
W EIGHTED O STROWSKI I NEQUALITY
3
2. O STROWSKI T YPE I NEQUALITIES
In order to state some Ostrowski type inequalities, we consider the Lebesgue norms
kgk[α,β],∞ := ess sup |g (t)|
t∈[α,β]
and
β
Z
1`
|g (t)| dt ,
`
kgk[α,β],` :=
` ∈ [1, ∞);
α
provided that the integral and the supremum are finite.
Theorem 2.1. Let ϕ : [0, 1] → R be continuous on [0, 1] , differentiable on (0, 1) with the
property that ϕ (0) = 0 and ϕ (1) 6= 0. If w : [a, b] → R+ is a probability density function, then
for any f : [a, b] → R an absolutely continuous function, we have
(2.1)
Z t
Z b
1
0
f (x) −
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
Z x Z t
Z b Z t
0
≤
w (s) ds |f (t)| dt +
w (s) ds − ϕ (1) |f 0 (t)| dt
ϕ
ϕ
a
a
x
a
for any x ∈ [a, b] .
If
Z
H1 (x) :=
a
x
Z t
0
ϕ
|f (t)| dt
w
(s)
ds
a
and
Z b Z t
H2 (x) :=
w (s) ds − ϕ (1) |f 0 (t)| dt,
ϕ
x
a
then
(2.2)
H1 (x) ≤









R·
ϕ
w
(s)
ds
kf 0 k[a,x],1 ;
a
[a,x],∞
R·
ϕ
w
(s)
ds
kf 0 k[a,x],q
a
[a,x],p






R

 ϕ · w (s) ds kf 0 k[a,x],∞
a
[a,x],1
if p > 1, p1 + 1q = 1
and f 0 ∈ Lq [a, x] ;
if f 0 ∈ L∞ [a, x] ;
and
(2.3)
H2 (x) ≤









R·
ϕ
w (s) ds − ϕ (1)[x,b],∞ kf 0 k[x,b],1 ;
a
R·
ϕ
w
(s)
ds
−
ϕ
(1)
kf 0 k[x,b],t
a
[x,b],r






R

 ϕ · w (s) ds − ϕ (1)
kf 0 k[x,b],∞
a
[x,b],1
if r > 1, 1r + 1t = 1
and f 0 ∈ Lt [x, b] ;
if f 0 ∈ L∞ [x, b]
for any x ∈ [a, b] .
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
http://jipam.vu.edu.au/
4
N.S. BARNETT AND S.S. D RAGOMIR
Proof. Follows from the identity (1.1) on observing that
Z t
Z b
1
0
(2.4)
w (t) ϕ
w (s) ds f (t) dt
f (x) − ϕ (1)
a
a
Z x Z t
Z b Z t
0
0
= ϕ
w (s) ds f (t) dt +
ϕ
w (s) ds − ϕ (1) f (t) dt
a
a
x
a
Z b Z t
Z x Z t
0
0
≤ ϕ
w (s) ds f (t) dt + ϕ
w (s) ds − ϕ (1) f (t) dt
a
a
x
a
Z x Z t
Z b Z t
0
0
|f (t)| dt
≤
ϕ
w
(s)
ds
|f
(t)|
dt
+
ϕ
w
(s)
ds
−
ϕ
(1)
a
a
x
a
for any x ∈ [a, b] , and the first part of (2.1) is proved.
The bounds from (2.2) and (2.3) follow by the Hölder inequality.
Remark 2.2. It is obvious that, the above theorem provides 9 possible upper bounds for the
absolute value of the deviation of f (x) from the integral mean,
Z t
Z b
1
0
w (t) ϕ
w (s) ds f (t) dt
ϕ (1) a
a
although they are not stated explicitly.
The above result, which provides an Ostrowski type inequality for the absolutely continuous
function f, can be extended to the larger class of functions of bounded variation as follows:
Theorem 2.3. Let ϕ and w be as in Theorem 2.1. If w is continuous on [a, b] and f : [a, b] → R
is a function of bounded variation on [a, b] , then:
Z t
Z b
1
0
(2.5)
f
(x)
−
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
"
Z t
_
x
1
≤
sup ϕ
w (s) ds · (f )
ϕ (1) t∈[a,x] a
a
#
Z t
_
b
+ sup ϕ
w (s) ds − ϕ (1) · (f )
t∈[x,b]
≤
1
· max
ϕ (1)
(
a
x
Z t
sup ϕ
w (s) ds ,
t∈[a,x]
a
Z t
) b
_
sup ϕ
w (s) ds − ϕ (1) · (f ) ,
t∈[x,b]
where
Wb
a
a
a
(f ) denotes the total variation of f on [a, b] .
Proof. We recall that, if p : [α, β] → R is continuous on [α, β] and v : [α, β] → R is of bounded
Rβ
variation, then the Riemann-Stieltjes integral α p (t) dv (t) exists and
(2.6)
Z
β
α
β
_
p (t) dv (t) ≤ sup |p (t)| (v) .
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
t∈[α,β]
α
http://jipam.vu.edu.au/
W EIGHTED O STROWSKI I NEQUALITY
5
R·
R·
Since the functions ϕ a w (s) ds and ϕ a w (s) ds − ϕ (1) are continuous on [a, x] and
[x, b], respectively, the Riemann-Stieltjes integrals
Z x Z t
Z b Z t
ϕ
w (s) ds df (t) and
ϕ
w (s) ds − ϕ (1) df (t)
a
a
x
a
exist and
Z
(2.7)
x
Z
t
ϕ
a
Z t
_
x
w (s) ds df (t) ≤ sup ϕ
w (s) ds · (f ) ,
t∈[a,x]
a
a
a
while
(2.8)
Z b Z t
ϕ
w (s) ds − ϕ (1) df (t)
x
a
Z t
b
_
≤ sup ϕ
w (s) ds − ϕ (1) · (f ) .
t∈[x,b]
a
x
Integrating by parts in the Riemann-Stieltjes integral, we have
Z x Z t
(2.9)
ϕ
w (s) ds df (t)
a
a
Z t
x Z x
Z t
w (s) ds
f (t) d ϕ
w (s) ds −
= f (t) ϕ
a
a
a
a
Z t
Z x
Z x
0
w (s) ds f (t) dt
w (t) ϕ
w (s) ds −
= f (x) ϕ
a
a
a
and
(2.10)
Z b Z t
w (s) ds − ϕ (1) df (t)
ϕ
x
a
b Z b
Z t
Z t
= ϕ
w (s) ds − ϕ (1) f (t) −
f (t) d ϕ
w (s) ds − ϕ (1)
a
x
a
x
Z t
Z t
Z b
0
w (s) ds − ϕ (1) f (x) −
w (t) ϕ
w (s) ds f (t) dt.
=− ϕ
a
x
a
If we add (2.9) and (2.10) we deduce the following identity of the Montgomery type for the
Riemann-Stieltjes integral which is of interest in itself:
1
(2.11) f (x) =
ϕ (1)
Z
b
w (t) ϕ
a
0
Z
t
w (s) ds f (t) dt
Z x Z t
1
+
ϕ
w (s) ds df (t)
ϕ (1) a
a
Z b Z t
1
+
ϕ
w (s) ds − ϕ (1) df (t) ,
ϕ (1) x
a
a
for any x ∈ [a, b] .
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
http://jipam.vu.edu.au/
6
N.S. BARNETT AND S.S. D RAGOMIR
Now, by (2.11) and (2.7) – (2.8) we obtain the estimate:
Z t
Z b
1
0
f (x) −
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
Z x Z t
Z b Z t
1
1 ≤
ϕ
w (s) ds df (t) +
ϕ
w (s) ds − ϕ (1) df (t)
ϕ (1) a
ϕ (1) x
a
a
Z t
_
x
1
· sup ϕ
w (s) ds · (f )
≤
ϕ (1) t∈[a,x]
a
a
Z t
b
_
1
+
· sup ϕ
w (s) ds − ϕ (1) · (f ) ,
x ∈ [a, b]
ϕ (1) t∈[x,b]
a
x
which provides the first inequality in (2.5).
The last part of (2.5) is obvious.
The following particular case is of interest for applications.
Corollary 2.4. Assume that f, ϕ, w are as in Theorem 2.3. In addition, if ϕ is monotonic
nondecreasing on [0, 1] , then
Z t
Z b
1
0
f (x) −
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
"
x
# b
Rx
Rx
_
ϕ a w (s) ds _
ϕ a w (s) ds
· (f ) + 1 −
· (f )
≤
ϕ (1)
ϕ (1)
a
x
Rx
"
#
b
1 ϕ a w (s) ds
1 _
≤
+
− (f ) .
2 ϕ (1)
2 a
(2.12)
Proof. Follows by Theorem 2.3 on observing that, if ϕ is monotonic nondecreasing on [a, b] ,
then:
Z t
Z t
Z x
sup ϕ
w (s) ds = sup ϕ
w (s) ds = ϕ
w (s) ds
t∈[a,x]
a
t∈[a,x]
a
a
and
Z t
Z t
sup ϕ
w (s) ds − ϕ (1) = sup ϕ (1) − ϕ
w (s) ds
t∈[x,b]
t∈[x,b]
a
a
Z t
= ϕ (1) − inf ϕ
w (s) ds
t∈[x,b]
a
Z x
= ϕ (1) − ϕ
w (s) ds .
a
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
http://jipam.vu.edu.au/
W EIGHTED O STROWSKI I NEQUALITY
7
Corollary 2.5. With the assumptions of Theorem 2.3 and if K := supt∈(0,1) |ϕ0 (t)| < ∞, then
we have the bounds:
Z t
Z b
1
0
(2.13)
f
(x)
−
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
"
#
Z t
_
Z b
_
x
b
1
≤
· K sup w (s) ds · (f ) + sup w (s) ds · (f )
ϕ (1)
t∈[a,x]
t∈[x,b]
a
t
a
x
(
Z t
Z b
) b
_
K
max sup w (s) ds , sup w (s) ds
(f ) .
≤
ϕ (1)
t∈[a,x]
t∈[x,b]
a
t
a
Remark 2.6. If w (s) ≥ 0 for s ∈ [a, b] , then from (2.13) we get
Z t
Z b
1
0
(2.14)
w (t) ϕ
w (s) ds f (t) dt
f (x) − ϕ (1)
a
a
"Z
#
Z b
x
b
x
_
_
K
≤
w (s) ds · (f ) +
w (s) ds · (f )
ϕ (1) a
x
a
x
Z x
b
Z b
Z b
_
1
1 K
w (s) ds + w (s) ds −
≤
w (s) ds · (f ) .
ϕ (1) 2 a
2 a
x
a
The following result, that provides an Ostrowski type inequality for L−Lipschitzian functions, can be stated as well.
Theorem 2.7. Let ϕ and w be as in Theorem 2.1. If w is continuous on [a, b] and f : [a, b] → R
is an L1 −Lipschitzian function on [a, x] and L2 −Lipschitzian on [x, b] , with x ∈ [a, b] , then
Z t
Z b
1
0
(2.15)
w (t) ϕ
w (s) ds f (t) dt
f (x) − ϕ (1)
a
a
Z x Z t
1
dt
ϕ
≤
L1 ·
w
(s)
ds
ϕ (1)
a
a
Z b Z t
ϕ
+ L2 ·
w (s) ds − ϕ (1) dt
x
a
Z x Z t
1
dt
≤ max {L1 , L2 } ·
ϕ
w
(s)
ds
ϕ (1) a a
Z b Z t
ϕ
+
w (s) ds − ϕ (1) dt .
x
a
Proof. We recall that, if p : [α, β] → R is L−Lipschitzian and v is Riemann integrable, then
Rβ
the Riemann-Stieltjes integral α f (t) du (t) exists and
Z β
Z β
(2.16)
p (t) dv (t) ≤ L
|p (t)| dt.
α
α
Now, if we apply the above property to the integrals
Z x Z t
Z b Z t
ϕ
w (s) ds df (t) and
ϕ
w (t) ds − ϕ (1) df (t) ,
a
a
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
α
a
http://jipam.vu.edu.au/
8
N.S. BARNETT AND S.S. D RAGOMIR
then we can state that
Z x Z t
Z
(2.17)
ϕ
w (s) ds df (t) ≤ L1 ·
a
a
x
a
Z t
ϕ
w (s) ds dt
a
and
Z b Z t
ϕ
w
(s)
ds
−
ϕ
(1)
df
(t)
(2.18)
x
a
Z b Z t
dt.
ϕ
w
(s)
ds
−
ϕ
(1)
≤ L2 ·
x
a
By making use of the identity (2.11), by (2.17) and (2.18) we deduce the first part of (2.15).
The last part is obvious.
The following particular case is of interest as well.
Corollary 2.8. With the assumptions of Theorem 2.7 and if K := supt∈(0,1) |ϕ0 (t)| < ∞, then
Z t
Z b
1
0
w (t) ϕ
(2.19)
w (s) ds f (t) dt
f (x) − ϕ (1)
a
a
Z x Z t
Z b Z b
K
w (s) ds dt + L2 ·
dt
L1 ·
≤
w
(s)
ds
ϕ (1)
a
a
x
t
Z x Z t
Z b Z b
K
w (s) ds dt +
≤
max {L1 , L2 }
w (s) ds dt .
ϕ (1)
a
a
x
t
Rt
Rb
Remark 2.9. If w : [a, b] → R is a nonnegative weight, then a w (s) ds, t w (s) ds ≥ 0 for
each t ∈ [a, b] and since
Z t
x Z x
Z x Z t
w (s) ds dt =
w (s) ds · t −
w (t) dt
a
a
a
a
a
Z x
Z x
Z x
=x
w (t) dt −
tw (t) dt =
(x − t) w (t) dt
a
a
a
and
Z b Z
x
t
b
b Z b
w (s) ds dt = t ·
w (s) ds +
w (t) dt
t
x
x
Z b
Z b
Z b
= −x
w (t) dt +
tw (t) dt =
(t − x) w (t) dt,
Z
b
x
x
x
then we get, from (2.19), the following result:
Z t
Z b
1
0
f (x) −
(2.20)
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
Z x
Z b
K
≤
L1 ·
(x − t) w (t) dt + L2 ·
(t − x) w (t) dt
ϕ (1)
a
x
Z b
K
≤
max {L1 , L2 }
|t − x| w (t) dt.
ϕ (1)
a
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
http://jipam.vu.edu.au/
W EIGHTED O STROWSKI I NEQUALITY
9
3. S OME E XAMPLES
The inequality (2.12) is a source of numerous particular inequalities that can be obtained
by specifying the function ϕ : [0, 1] → R which is continuous, differentiable and monotonic
nondecreasing with ϕ (0) = 0.
For instance, if we choose ϕ (t) = tα , α > 0, then we get the inequality:
Z t
α−1
Z b
w (t)
w (s) ds
f (t) dt
(3.1)
f (x) − α
a
a
Z x
α _
Z x
α _
x
b
≤
w (s) ds · (f ) + 1 −
w (s) ds
· (f )
a
a
a
Z x
α
1 ≤
+
w (s) ds −
2 a
x
b
1 _
(f ) ,
2 a
for any x ∈ [a, b] provided that f is of bounded variation on [a, b] , w (s) ≥ 0 for any s ∈ [a, b]
and the involved integrals exist.
Another simple example can be given by choosing ϕ (t) = ln (t + 1) . In this situation, we
obtain the inequality:
#
Z b"
1
w (t)
(3.2)
f (t) dt
Rt
f (x) −
ln 2 a
w (s) ds + 1
a
"
x
# b
Rx
Rx
_
ln a w (s) ds + 1 _
ln a w (s) ds + 1
≤
· (f ) + 1 −
· (f )
ln 2
ln 2
a
x
"
#
R
x
b
1 ln a w (s) ds + 1
1 _
≤
+
− · (f ) ,
2 ln 2
2
a
for any x ∈ [a, b] provided that f is of bounded variation on [a, b] , w (s) ≥ 0 for any s ∈ [a, b]
and the involved integrals exist.
Finally, by choosing the function ϕ (t) = exp(t) − 1, we obtain, from the inequality (2.12),
the following result as well:
Z t
Z b
f (x) − 1
w (t) exp
w (s) ds f (t) dt
e−1 a
a
b
Rx
Rx
x
exp a w (s) ds − 1 _
e − exp a w (s) ds _
≤
· (f ) +
· (f )
e−1
e−1
a
x
#
"
Rx
b
1 exp a w (s) ds − 1 1 _
+
− · (f ) ,
≤
2 e−1
2
a
for any x ∈ [a, b] , provided f is of bounded variation on [a, b] and the involved integrals exist.
R EFERENCES
[1] A. AGLIĆ ALJINOVIĆ AND J. PEČARIĆ, On some Ostrowski type inequalities via Montgomery
identity and Taylor’s formula, Tamkang J. Math., 36(3) (2005), 199–218.
[2] A. AGLIĆ ALJINOVIĆ, J. PEČARIĆ AND A. VUKELIĆ, On some Ostrowski type inequalities
via Montgomery identity and Taylor’s formula. II, Tamkang J. Math., 36(4) (2005), 279–301.
[3] G.A. ANASTASSIOU, Ostrowski type inequalities, Proc. Amer. Math. Soc., 123(12) (1995), 3775–
3781.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
http://jipam.vu.edu.au/
10
N.S. BARNETT AND S.S. D RAGOMIR
[4] N. S. BARNETT, C. BUŞE, P. CERONE AND S.S. DRAGOMIR, Ostrowski’s inequality for vectorvalued functions and applications, Comput. Math. Appl., 44(5-6) (2002), 559–572.
[5] K. BOUKERRIOUA AND A. GUEZANE-LAKOUD, On generalisation of Čebyšev type inequality, J. Ineq. Pure and Appl. Math., 8(2) (2007), Art. 55. [ONLINE http://jipam.vu.edu.
au/article.php?sid=865].
[6] C. BUŞE, S.S. DRAGOMIR AND A. SOFO, Ostrowski’s inequality for vector-valued functions of
bounded semivariation and applications, New Zealand J. Math., 31(2) (2002), 137–152.
[7] J. de la CAL AND J. CÁRCAMO, A general Ostrowski-type inequality, Statist. Probab. Lett., 72(2)
(2005), 145–152.
[8] P. CERONE, On relationships between Ostrowski, trapezoidal and Chebychev identities and inequalities, Soochow J. Math., 28(3) (2002), 311–328.
[9] P. CERONE, A new Ostrowski type inequality involving integral means over end intervals,
Tamkang J. Math., 33(2) (2002), 109–118.
[10] P. CERONE, Approximate multidimensional integration through dimension reduction via the Ostrowski functional, Nonlinear Funct. Anal. Appl., 8(3) (2003), 313–333.
[11] S.S. DRAGOMIR, A refinement of Ostrowski’s inequality for absolutely continuous functions and
applications, Acta Math. Vietnam., 27(2) (2002), 203–217.
[12] S.S. DRAGOMIR, Ostrowski type inequalities for isotonic linear functionals. J. Inequal. Pure Appl.
Math., 3(5) (2002), Art. 68. [ONLINE http://jipam.vu.edu.au/article.php?sid=
220].
[13] S.S. DRAGOMIR, A weighted Ostrowski type inequality for functions with values in Hilbert spaces
and applications, J. Korean Math. Soc., 40(2) (2003), 207–224.
[14] S.S. DRAGOMIR, An Ostrowski like inequality for convex functions and applications, Rev. Mat.
Complut., 16(2) (2003), 373–382.
[15] S.S. DRAGOMIR AND Th. M. RASSIAS (Eds.), Ostrowski Type Inequalities and Applications in
Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002.
[16] A. FLOREA AND P.C. NICULESCU, A note on Ostrowski’s inequality, J. Inequal. Appl., 8(5)
(2005), 459–468.
[17] D.S. MITRINOVIĆ, J. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their
Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
[18] B.G. PACHPATTE, On Čebyšev-Grüss type inequalities via Pečarić extensions of the Montgomery
identity, J. Ineq. Pure and Appl. Math., 7(1) (2007), Art. 11. [ONLINE http://jipam.vu.
edu.au/article.php?sid=624].
[19] B.G. PACHPATTE, New Ostrowski type inequalities for mappings whose derivatives belong to
Lp − spaces. Demonstratio Math., 37(2) (2004), 293–298.
[20] B.G. PACHPATTE, On a new generalisation of Ostrowski’s inequality. J. Inequal. Pure Appl.
Math., 5(2) (2004), Art. 36. [ONLINE http://jipam.vu.edu.au/article.php?sid=
378].
[21] J. PEČARIĆ, On the Čebyšev inequality, Bull. Şti. Tehn. Inst. Politech. “Traian Vuia”, Timişoara
(Romania), (25)(39)(1) (1980), 5–9.
[22] J. ROUMELIOTIS, Improved weighted Ostrowski-Grüss type inequalities. Inequality Theory and
Applications. Vol. 3, 153–160, Nova Sci. Publ., Hauppauge, NY, 2003.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 96, 10 pp.
http://jipam.vu.edu.au/