ON THE WEIGHTED OSTROWSKI INEQUALITY
N.S. BARNETT AND S.S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University, PO Box 14428
Melbourne City, VIC 8001, Australia.
EMail: {neil.barnett, sever.dragomir}@vu.edu.au
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
Title Page
Received:
14 May, 2007
Accepted:
30 September, 2007
Communicated by:
B.G. Pachpatte
2000 AMS Sub. Class.:
26D15, 26D10.
Key words:
Ostrowski inequality, Integral inequalities, Absolutely continuous functions.
Abstract:
On utilising an identity from [5], some weighted Ostrowski type inequalities
are established.
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Contents
1
Introduction
3
2
Ostrowski Type Inequalities
6
3
Some Examples
17
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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1.
Introduction
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
a
where the weighted Peano kernel Pw is
( Rt
w (s) ds,
a ≤ t ≤ x,
a
(1.4)
Pw (x, t) :=
Rb
− t w (s) ds, x < t ≤ b.
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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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,
Weighted Ostrowski Inequality
if x < t ≤ b,
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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
a
a
b
0
a
Z
x
w (x) ϕ
w (t) dt
a
a
Z b
Z b
0
0
×
Pw,ϕ (x, t) f (t) dt
Pw,ϕ (x, t) g (t) dt dx
Z
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the authors obtained the representation:
1
(1.7) Tϕ (w, f, g) = 2
ϕ (1)
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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
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a
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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.
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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2.
Ostrowski Type Inequalities
In order to state some Ostrowski type inequalities, we consider the Lebesgue norms
kgk[α,β],∞ := ess sup |g (t)|
t∈[α,β]
and
Z
β
1`
|g (t)| dt ,
Weighted Ostrowski Inequality
`
kgk[α,β],` :=
` ∈ [1, ∞);
α
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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
Z t
Z b
1
0
(2.1) f (x) −
w (t) ϕ
w (s) ds f (t) dt
ϕ (1) a
a
Z x Z t
Z b Z t
0
0
ϕ
|f (t)| dt +
ϕ
|f (t)| dt
≤
w
(s)
ds
w
(s)
ds
−
ϕ
(1)
a
a
x
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for any x ∈ [a, b] .
If
Close
Z
H1 (x) :=
a
and
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x
Z t
ϕ
w (s) ds |f 0 (t)| dt
a
Z b Z t
0
ϕ
|f (t)| dt,
H2 (x) :=
w
(s)
ds
−
ϕ
(1)
x
a
then
(2.2)
H1 (x) ≤









R·
ϕ
w (s) ds [a,x],∞ kf 0 k[a,x],1 ;
a
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] ;
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
and
(2.3)
Weighted Ostrowski Inequality
H2 (x)
 R·
ϕ
w
(s)
ds
−
ϕ
(1)
kf 0 k[x,b],1 ;


a
[x,b],∞




R

 ϕ · w (s) ds − ϕ (1)
kf 0 k[x,b],t
if r > 1, 1r + 1t = 1
a
[x,b],r
≤

and f 0 ∈ Lt [x, b] ;





R

 ϕ · w (s) ds − ϕ (1)
kf 0 k[x,b],∞ if f 0 ∈ L∞ [x, b]
a
[x,b],1
for any x ∈ [a, b] .
Proof. Follows from the identity (1.1) on observing that
Z t
Z b
1
0
w (t) ϕ
w (s) ds f (t) dt
(2.4)
f (x) − ϕ (1)
a
a
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Z x Z t
Z b Z t
w (s) ds f 0 (t) dt +
ϕ
w (s) ds − ϕ (1) f 0 (t) dt
= ϕ
Za x Za t
xZ b aZ 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
ϕ
|f (t)| dt +
ϕ
|f (t)| dt
≤
w
(s)
ds
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.
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
Remark 1. 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
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although they are not stated explicitly.
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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:
Page 8 of 21
Theorem 2.2. 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
f (x) −
(2.5)
w (t) ϕ
w (s) ds f (t) dt
ϕ (1) a
a
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1
≤
ϕ (1)
"
Z t
_
x
w (s) ds · (f )
sup ϕ
t∈[a,x]
a
a
#
Z t
_
b
+ sup ϕ
w (s) ds − ϕ (1) · (f )
t∈[x,b]
a
x
(
Z t
1
· max sup ϕ
w (s) ds ,
≤
ϕ (1)
t∈[a,x]
a
Z t
) b
_
sup ϕ
w (s) ds − ϕ (1) · (f ) ,
t∈[x,b]
where
a
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
a
Title Page
Wb
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
Rβ
of bounded variation, then the Riemann-Stieltjes integral α p (t) dv (t) exists and
Z β
β
_
(2.6)
p (t) dv (t) ≤ sup |p (t)| (v) .
t∈[α,β]
α
a
a
x
a
exist and
Z
a
x
Z
ϕ
a
t
Z t
_
x
w (s) ds df (t) ≤ sup ϕ
w (s) ds · (f ) ,
t∈[a,x]
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α
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)
(2.7)
Weighted Ostrowski Inequality
a
a
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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
= f (t) ϕ
w (s) ds −
f (t) d ϕ
w (s) ds
a
a
a
a
Z x
Z x
Z t
0
= f (x) ϕ
w (s) ds −
w (t) ϕ
w (s) ds f (t) dt
a
a
a
and
(2.10)
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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Z b Z t
ϕ
w (s) ds − ϕ (1) df (t)
x
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a
b
Z t
w (s) ds − ϕ (1) f (t)
= ϕ
a
x
Z t
Z b
−
f (t) d ϕ
w (s) ds − ϕ (1)
x
a
Z t
=− ϕ
w (s) ds − ϕ (1) f (x)
a
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Z
−
b
w (t) ϕ
0
x
Z
t
w (s) ds f (t) dt.
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:
Z t
Z b
1
0
(2.11) f (x) =
w (t) ϕ
w (s) ds f (t) dt
ϕ (1) a
a
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
for any x ∈ [a, b] .
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.
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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The following particular case is of interest for applications.
Corollary 2.3. Assume that f, ϕ, w are as in Theorem 2.2. In addition, if ϕ is monotonic nondecreasing on [0, 1] , then
Z t
Z b
1
0
f (x) −
(2.12)
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 _
1 ϕ a w (s) ds
+
− (f ) .
≤
2 ϕ (1)
2
a
Proof. Follows by Theorem 2.2 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
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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Page 12 of 21
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
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Corollary 2.4. With the assumptions of Theorem 2.2 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
x
Z b
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. If w (s) ≥ 0 for s ∈ [a, b] , then from (2.13) we get
Z t
Z b
1
0
f (x) −
(2.14)
w (t) ϕ
w (s) ds f (t) dt
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
_
K
1
1 ≤
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 2.5. 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
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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[x, b] , with x ∈ [a, b] , then
Z t
Z b
1
0
f (x) −
(2.15)
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
Z x Z t
1
ϕ
L1 ·
≤
w (s) ds dt
ϕ (1)
a
a
Z b Z t
ϕ
dt
+ L2 ·
w
(s)
ds
−
ϕ
(1)
x
a
Z x Z t
1
ϕ
dt
w
(s)
ds
≤ max {L1 , L2 } ·
ϕ (1) a a
Z b Z t
ϕ
dt .
+
w
(s)
ds
−
ϕ
(1)
x
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
α
a
then we can state that
Z x Z t
Z
(2.17)
ϕ
w (s) ds df (t) ≤ L1 ·
a
a
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
Title Page
Contents
a
Proof. We recall that, if p : [α, β] → R is L−Lipschitzian and v is Riemann inteRβ
grable, then 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.
a
Weighted Ostrowski Inequality
a
x
Z t
ϕ
w (s) ds dt
a
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and
(2.18)
Z b Z t
ϕ
w
(s)
ds
−
ϕ
(1)
df
(t)
x
a
Z b Z t
≤ L2 ·
w (s) ds − ϕ (1) dt.
ϕ
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.
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
The following particular case is of interest as well.
Corollary 2.6. With the assumptions of Theorem 2.5 and if K := supt∈(0,1) |ϕ0 (t)| <
∞, then
Z t
Z b
1
0
f (x) −
(2.19)
w
(t)
ϕ
w
(s)
ds
f
(t)
dt
ϕ (1) a
a
Z x Z t
Z b Z b
K
≤
L1 ·
w (s) ds dt + L2 ·
w (s) ds dt
ϕ (1)
a
a
x Zt b Z b
Z x Z t
K
w (s) ds dt +
dt .
≤
max {L1 , L2 }
w
(s)
ds
ϕ (1)
a
a
x
t
Rt
Rb
Remark 3. 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
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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
x
b
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
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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3.
Some Examples
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
(3.1)
w (t)
w (s) ds
f (t) dt
f (x) − α
a
a
Z x
α _
Z x
α _
x
b
≤
w (s) ds · (f ) + 1 −
w (s) ds
· (f )
a
a
Z x
α
1 ≤
+
w (s) ds −
2 a
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"
w
(t)
1
(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
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
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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
1
f (x) −
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
# b
"
Rx
1 exp a w (s) ds − 1 1 _
+
− · (f ) ,
≤
2 e−1
2
Weighted Ostrowski Inequality
N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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a
for any x ∈ [a, b] , provided f is of bounded variation on [a, b] and the involved
integrals exist.
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N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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Weighted Ostrowski Inequality
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