Journal of Inequalities in Pure and
Applied Mathematics
AN INEQUALITY OF OSTROWSKI TYPE VIA POMPEIU’S MEAN
VALUE THEOREM
volume 6, issue 3, article 83,
2005.
S.S. DRAGOMIR
School of Computer Science and Mathematics
Victoria University
PO Box 14428, MCMC 8001
VIC, Australia.
EMail: sever@csm.vu.edu.au
URL: http://rgmia.vu.edu.au/dragomir
Received 08 February, 2005;
accepted 12 June, 2005.
Communicated by: B.G. Pachpatte
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
032-05
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Abstract
An inequality providing some bounds for the integral mean via Pompeiu’s mean
value theorem and applications for quadrature rules and special means are
given.
2000 Mathematics Subject Classification: Primary 26D15, 26D10; Secondary
41A55.
Key words: Ostrowski’s inequality, Pompeiu mean value theorem, quadrature rules,
Special means.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Pompeiu’s Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Evaluating the Integral Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
The Case of Weighted Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 12
5
A Quadrature Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6
Applications for Special Means . . . . . . . . . . . . . . . . . . . . . . . . . 17
References
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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1.
Introduction
The following result is known in the literature as Ostrowski’s inequality [1].
Theorem 1.1. Let f : [a, b] → R be a differentiable mapping on (a, b) with the
property that |f 0 (t)| ≤ M for all t ∈ (a, b) . Then

!2 
Z b
a+b
x− 2
1
f (x) − 1
 (b − a) M,
(1.1)
f (t) dt ≤  +
b−a a
4
b−a
for all x ∈ [a, b] . The constant
replaced by a smaller constant.
1
4
is best possible in the sense that it cannot be
In [2], the author has proved the following Ostrowski type inequality.
Theorem 1.2. Let f : [a, b] → R be continuous on [a, b] with a > 0 and
differentiable on (a, b) . Let p ∈ R\ {0} and assume that
Kp (f 0 ) := sup u1−p |f 0 (u)| < ∞.
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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u∈(a,b)
Then we have the inequality
Z b
1
Kp (f 0 )
(1.2) f (x) −
f (t) dt ≤
b−a a
|p| (b − a)
 p
p
2x (x − A) + (b − x) Lp (b, x) − (x − a) Lpp (x, a)




if p ∈ (0, ∞) ;


p
p
p
(x − a) Lp (x, a) − (b − x) Lp (b, x) − 2x (x − A)
×


if p ∈ (−∞, −1) ∪ (−1, 0)




(x − a) L−1 (x, a) − (b − x) L−1 (b, x) − x2 (x − A) if p = −1,
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for any x ∈ (a, b) , where for a 6= b,
A = A (a, b) :=
a+b
, is the arithmetic mean,
2
Lp = Lp (a, b)
p+1
1
b
− ap+1 p
=
, is the p − logarithmic mean p ∈ R\ {−1, 0} ,
(p + 1) (b − a)
and
L = L (a, b) :=
b−a
is the logarithmic mean.
ln b − ln a
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
Another result of this type obtained in the same paper is:
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Theorem 1.3. Let f : [a, b] → R be continuous on [a, b] (with a > 0) and
differentiable on (a, b) . If
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P (f 0 ) := sup |uf 0 (x)| < ∞,
u∈(a,b)
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then we have the inequality
(1.3)
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Z b
1
f (x) −
f (t) dt
b−a a
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" "
≤
#
P (f 0 )
[I (x, b)]b−x
ln
+ 2 (x − A) ln x
b−a
[I (a, x)]x−a
#
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for any x ∈ (a, b) , where for a 6= b
1
I = I (a, b) :=
e
bb
aa
1
b−a
, is the identric mean.
If some local information around the point x ∈ (a, b) is available, then we
may state the following result as well [2].
Theorem 1.4. Let f : [a, b] → R be continuous on [a, b] and differentiable on
(a, b) . Let p ∈ (0, ∞) and assume, for a given x ∈ (a, b) , we have that
Mp (x) := sup |x − u|1−p |f 0 (u)| < ∞.
u∈(a,b)
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
Then we have the inequality
Title Page
(1.4)
Z b
1
f (x) −
f
(t)
dt
b−a a
1
≤
(x − a)p+1 + (b − x)p+1 Mp (x) .
p (p + 1) (b − a)
For recent results in connection to Ostrowski’s inequality see the papers
[3],[4] and the monograph [5].
The main aim of this paper is to provide some complementary results, where
instead of using Cauchy mean value theorem, we use Pompeiu mean Value
Theorem to evaluate the integral mean of an absolutely continuous function.
Applications for quadrature rules and particular instances of functions are given
as well.
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2.
Pompeiu’s Mean Value Theorem
In 1946, Pompeiu [6] derived a variant of Lagrange’s mean value theorem, now
known as Pompeiu’s mean value theorem (see also [7, p. 83]).
Theorem 2.1. For every real valued function f differentiable on an interval
[a, b] not containing 0 and for all pairs x1 6= x2 in [a, b] , there exists a point ξ
in (x1 , x2 ) such that
x1 f (x2 ) − x2 f (x1 )
= f (ξ) − ξf 0 (ξ) .
x1 − x2
Proof. Define a real valued function F on the interval 1b , a1 by
1
(2.2)
F (t) = tf
.
t
Since f is differentiable on 1b , a1 and
1
1 0 1
0
− f
,
(2.3)
F (t) = f
t
t
t
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
(2.1)
S.S. Dragomir
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then applying the mean value theorem to F on the interval [x, y] ⊂
get
(2.4)
for some η ∈ (x, y) .
F (x) − F (y)
= F 0 (η)
x−y
1
1
,
we
b a
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Let x2 = x1 , x1 =
1
y
and ξ = η1 . Then, since η ∈ (x, y) , we have
x1 < ξ < x2 .
Now, using (2.2) and (2.3) on (2.4), we have
1
xf x − yf y1
1
1 0 1
=f
− f
,
x−y
η
η
η
that is
x1 f (x2 ) − x2 f (x1 )
= f (ξ) − ξf 0 (ξ) .
x1 − x2
This completes the proof of the theorem.
Remark 1. Following [7, p. 84 – 85], we will mention here a geometrical
interpretation of Pompeiu’s theorem. The equation of the secant line joining the
points (x1 , f (x1 )) and (x2 , f (x2 )) is given by
y = f (x1 ) +
f (x2 ) − f (x1 )
(x − x1 ) .
x2 − x1
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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This line intersects the y−axis at the point (0, y) , where y is
f (x2 ) − f (x1 )
(0 − x1 )
x2 − x1
x1 f (x2 ) − x2 f (x1 )
=
.
x1 − x2
y = f (x1 ) +
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The equation of the tangent line at the point (ξ, f (ξ)) is
y = (x − ξ) f 0 (ξ) + f (ξ) .
The tangent line intersects the y−axis at the point (0, y) , where
y = −ξf 0 (ξ) + f (ξ) .
Hence, the geometric meaning of Pompeiu’s mean value theorem is that the
tangent of the point (ξ, f (ξ)) intersects on the y−axis at the same point as the
secant line connecting the points (x1 , f (x1 )) and (x2 , f (x2 )) .
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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3.
Evaluating the Integral Mean
The following result holds.
Theorem 3.1. Let f : [a, b] → R be continuous on [a, b] and differentiable on
(a, b) with [a, b] not containing 0. Then for any x ∈ [a, b] , we have the inequality
(3.1)
Z b
a + b f (x)
1
·
−
f
(t)
dt
2
x
b−a a

b − a 1
≤
+
|x|
4
x − a+b
2
b−a
!2 
 kf − `f 0 k ,
∞
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
where ` (t) = t, t ∈ [a, b] .
The constant 41 is sharp in the sense that it cannot be replaced by a smaller
constant.
Proof. Applying Pompeiu’s mean value theorem, for any x, t ∈ [a, b] , there is
a ξ between x and t such that
tf (x) − xf (t) = [f (ξ) − ξf 0 (ξ)] (t − x)
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giving
(3.2)
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|tf (x) − xf (t)| ≤ sup |f (ξ) − ξf 0 (ξ)| |x − t|
ξ∈[a,b]
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0
= kf − `f k∞ |x − t|
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for any t, x ∈ [a, b] .
Integrating over t ∈ [a, b] , we get
Z b
Z b
f (x)
(3.3)
tdt
−
x
f
(t)
dt
a
a
Z b
≤ kf − `f 0 k∞
|x − t| dt
a
#
"
2
2
(x
−
a)
+
(b
−
x)
= kf − `f 0 k∞
2
"
2 #
1
a
+
b
2
= kf − `f 0 k∞
(b − a) + x −
4
2
Rb
2
2
and since a tdt = b −a
, we deduce from (3.3) the desired result (3.1).
2
Now, assume that (3.2) holds with a constant k > 0, i.e.,
(3.4)
Z b
a + b f (x)
1
·
−
f
(t)
dt
2
x
b−a a

b−a
≤
k+
|x|
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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a+b
2
x−
b−a
!2 
 kf − `f 0 k ,
∞
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for any x ∈ [a, b] .
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Consider f : [a, b] → R, f (t) = αt + β; α, β 6= 0. Then
kf − `f 0 k∞ = |β| ,
Z b
1
a+b
f (t) dt =
· α + β,
b−a a
2
and by (3.4) we deduce

a + b
β
a+b
b−a
α+
−
α + β ≤
k+
2
x
2
|x|
x − a+b
2
b−a
!2 
 |β|
S.S. Dragomir
giving
(3.5)
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
a + b
≤ (b − a) k +
−
x
2
x − a+b
2
b−a
!2
for any x ∈ [a, b] .
If in (3.5) we let x = a or x = b, we deduce k ≥ 14 , and the sharpness of the
constant is proved.
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The following interesting particular case holds.
Corollary 3.2. With the assumptions in Theorem 3.1, we have
Z b
a
+
b
1
f
≤ (b − a) kf − `f 0 k .
(3.6)
−
f
(t)
dt
∞
2 |a + b|
2
b−a a
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4.
The Case of Weighted Integrals
We will consider now the weighted integral case.
Theorem 4.1. Let f : [a, b] → R be continuous on [a, b] and differentiable on
(a, b) with [a, b] not containing 0. If w : [a, b] → R is nonnegative integrable on
[a, b] , then for each x ∈ [a, b] , we have the inequality:
(4.1)
Z b
Z b
f
(x)
f (t) w (t) dt −
tw (t) dt
x
a
a
Z x
Z b
0
≤ kf − `f k∞ sgn (x)
w (t) dt −
w (t) dt
a
x
Z b
Z x
1
tw (t) dt −
tw (t) dt .
+
|x|
x
a
Proof. Using the inequality (3.2), we have
Z b
Z b
(4.2) f (x)
tw (t) dt − x
f (t) w (t) dt
a
a
Z b
0
≤ kf − `f k∞
w (t) |x − t| dt
a
Z x
Z b
0
= kf − `f k∞
w (t) (x − t) dt +
w (t) (t − x) dt
a
x
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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Z x
Z x
Z b
Z b
= kf − `f k∞ x
w (t) dt −
tw (t) dt +
tw (t) dt − x
w (t) dt
a
a
x
x
Z x
Z b
Z b
Z x
0
= kf − `f k∞ x
w (t) dt −
w (t) dt +
tw (t) dt −
tw (t) dt
0
a
x
x
a
from where we get the desired inequality (4.1).
Now, if we assume that 0 < a < b, then
Rb
tw (t) dt
(4.3)
a ≤ Rab
≤ b,
w (t) dt
a
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
Rb
provided a w (t) dt > 0.
With this extra hypothesis, we may state the following corollary.
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Corollary 4.2. With the above assumptions, we have
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(4.4)
f
− Rb
f (t) w (t) dt
Rab
w (t) dt
w (t) dt a
a
a
"R x
Rb
w (t) dt − x w (t) dt
≤ kf − `f 0 k∞ a
Rb
w (t) dt
a
#
Rb
Rx
w
(t)
tdt
−
tw
(t)
dt
a
+ x
.
Rb
tw
(t)
dt
a
Rb
tw (t) dt
!
1
Z
b
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5.
A Quadrature Formula
We assume in the following that 0 < a < b.
Consider the division of the interval [a, b] given by
In : a = x0 < x1 < · · · < xn−1 < xn = b,
and ξi ∈ [xi , xi+1 ], i = 0, . . . , n − 1 a sequence of intermediate points. Define
the quadrature
(5.1)
Sn (f, In , ξ) :=
=
n−1
X
i=0
n−1
X
i=0
f (ξi ) x2i+1 − x2i
·
ξi
2
f (ξi ) xi + xi+1
·
· hi ,
ξi
2
where hi := xi+1 − xi , i = 0, . . . , n − 1.
The following result
R b concerning the estimate of the remainder in approximating the integral a f (t) dt by the use of Sn (f, In , ξ) holds.
Theorem 5.1. Assume that f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) . Then we have the representation
Z
b
f (t) dt = Sn (f, In , ξ) + Rn (f, In , ξ) ,
(5.2)
a
where Sn (f, In , ξ) is as defined in (5.1), and the remainder Rn (f, In , ξ) satis-
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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fies the estimate
(5.3)
|Rn (f, In , ξ)| ≤ kf − `f 0 k∞
i=0


xi +xi+1 2
ξ
−
i

i 1
2
+
ξi 4 hi
n−1 2
X
h
n−1 2
n−1
X
kf − `f 0 k∞ X 2
hi
1
0
≤ kf − `f k∞
≤
hi .
2
ξ
2a
i=0 i
i=0
Proof. Apply Theorem 3.1 on the interval [xi , xi+1 ] for the intermediate points
ξi to obtain
Z xi+1
f (ξi ) xi + xi+1
(5.4)
·
·
h
f
(t)
dt
−
i
ξ
2
i
xi

!2 
xi +xi+1
ξi − 2
1
1
 kf − `f 0 k
≤ h2i  +
∞
ξi
4
hi
1 2
1
≤
hi kf − `f 0 k∞ ≤ h2i kf − `f 0 k∞
2ξi
2a
for each i ∈ {0, . . . , n − 1} .
Summing over i from 1 to n−1 and using the generalised triangle inequality,
we deduce the desired estimate (5.3).
Now, if we consider the mid-point rule (i.e., we choose ξi =
i ∈ {0, . . . , n − 1})
n−1 X
xi + xi+1
Mn (f, In ) :=
f
hi ,
2
i=0
xi +xi+1
2
above,
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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then, by Corollary 3.2, we may state the following result as well.
Corollary 5.2. With the assumptions of Theorem 5.1, we have
Z b
(5.5)
f (t) dt = Mn (f, In ) + Rn (f, In ) ,
a
where the remainder satisfies the estimate:
(5.6)
|Rn (f, In )| ≤
≤
n−1
kf − `f 0 k∞ X
h2i
2
x + xi+1
i=0 i
n−1
kf − `f 0 k∞ X 2
hi .
4a
i=0
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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6.
Applications for Special Means
For 0 < a < b, let us consider the means
a+b
,
2
√
G = G (a, b) := a · b,
2
H = H (a, b) := 1 1 ,
+b
a
b−a
L = L (a, b) :=
(logarithmic mean),
ln b − ln a
1
1 bb b−a
I = I (a, b) :=
(identric mean)
e aa
A = A (a, b) :=
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
Title Page
and the p−logarithmic mean
bp+1 − ap+1
Lp = Lp (a, b) =
(p + 1) (b − a)
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p1
, p ∈ R\ {−1, 0} .
It is well known that
H ≤ G ≤ L ≤ I ≤ A,
and, denoting L0 := I and L−1 = L, the function R 3 p 7→ Lp ∈ R is
monotonic increasing.
In the following we will use the following inequality obtained in Corollary
3.2,
Z b
a
+
b
1
f
≤ (b − a) kf − `f 0 k ,
(6.1)
−
f
(t)
dt
∞
2 (a + b)
2
b−a a
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provided 0 < a < b.
1. Consider the function f : [a, b] ⊂ (0, ∞) → R, f (t) = tp , p ∈ R\ {−1, 0} .
Then
a+b
f
= [A (a, b)]p ,
2
Z b
1
f (t) dt = Lpp (a, b) ,
b−a a
(
(1 − p) ap if p ∈ (−∞, 0) \ {−1} ,
0
kf − `f k[a,b],∞ =
|1 − p| bp if p ∈ (0, 1) ∪ (1, ∞) .
Consequently, by (6.1) we deduce
(6.2) Ap (a, b) − Lpp (a, b)

(1 − p) ap (b − a)


if p ∈ (−∞, 0) \ {−1} ,
A (a, b)
1 
≤ ×
4 
|1 − p| bp (b − a)


if p ∈ (0, 1) ∪ (1, ∞) .
A (a, b)
2. Consider the function f : [a, b] ⊂ (0, ∞) → R, f (t) =
a+b
1
f
=
,
2
A (a, b)
Z b
1
1
f (t) dt =
,
b−a a
L (a, b)
1
.
t
Then
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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2
kf − `f 0 k[a,b],∞ = .
a
Consequently, by (6.1) we deduce
(6.3)
0 ≤ A (a, b) − Lp (a, b) ≤
b−a
L (a, b) .
2a
3. Consider the function f : [a, b] ⊂ (0, ∞) → R, f (t) = ln t. Then
a+b
f
= ln [A (a, b)] ,
2
Z b
1
f (t) dt = ln [I (a, b)] ,
b−a a
a b 0
kf − `f k[a,b],∞ = max ln
.
, ln
e
e Consequently, by (6.1) we deduce
A (a, b)
b−a
a b (6.4) 1 ≤
≤ exp
max ln
.
, ln
I (a, b)
4A (a, b)
e
e An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
S.S. Dragomir
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References
[1] A. OSTROWSKI, Über die Absolutabweichung einer differentienbaren
Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938),
226–227.
[2] S.S. DRAGOMIR, Some new inequalities of Ostrowski type, RGMIA
Res. Rep. Coll., 5(2002), Supplement, Article 11, [ON LINE: http:
//rgmia.vu.edu.au/v5(E).html]. Gazette, Austral. Math. Soc.,
30(1) (2003), 25–30.
[3] G.A. ANASTASSIOU, Multidimensional Ostrowski inequalities, revisited.
Acta Math. Hungar., 97(4) (2002), 339–353.
[4] G.A. ANASTASSIOU, Univariate Ostrowski inequalities, revisited.
Monatsh. Math., 135(3) (2002), 175–189.
[5] S.S. DRAGOMIR AND Th.M. RASSIAS (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht/Boston/London, 2002.
[6] D. POMPEIU, Sur une proposition analogue au théorème des accroissements finis, Mathematica (Cluj, Romania), 22 (1946), 143–146.
[7] P.K. SAHOO AND T. RIEDEL, Mean Value Theorems and Functional
Equations, World Scientific, Singapore, New Jersey, London, Hong Kong,
2000.
An Inequality of Ostrowski Type
via Pompeiu’s Mean Value
Theorem
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