Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 5, Article 68, 2002
OSTROWSKI TYPE INEQUALITIES FOR ISOTONIC LINEAR FUNCTIONALS
S.S. DRAGOMIR
S CHOOL OF C OMMUNICATIONS AND I NFORMATICS
V ICTORIA U NIVERSITY OF T ECHNOLOGY
PO B OX 14428
M ELBOURNE C ITY MC
V ICTORIA 8001, AUSTRALIA .
sever@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 6 May, 2002; accepted 3 June, 2002
Communicated by P. Bullen
A BSTRACT. Some inequalities of Ostrowski type for isotonic linear functionals defined on a
linear class of function L := {f : [a, b] → R} are established. Applications for integral and
discrete inequalities are also given.
Key words and phrases: Ostrowski Type Inequalities, Isotonic Linear Functionals.
2000 Mathematics Subject Classification. Primary 26D15, 26D10.
1. I NTRODUCTION
The following result is known in the literature as Ostrowski’s inequality [13].
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
#
"
Z b
a+b 2
x
−
1
2
f (x) − 1
(1.1)
f (t) dt ≤
+
(b − a) M
b−a a
4
(b − a)2
for all x ∈ [a, b].
The constant 41 is the best possible in the sense that it cannot be replaced by a smaller constant.
The following Ostrowski type result for absolutely continuous functions whose derivatives
belong to the Lebesgue spaces Lp [a, b] also holds (see [9], [10] and [11]).
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
047-02
2
S.S. D RAGOMIR
Theorem 1.2. Let f : [a, b] → R be absolutely continuous on [a, b]. Then, for all x ∈ [a, b], we
have:
Z b
1
(1.2) f (x) −
f (t) dt
b−a a
 a+b 2 
x−
1

(b − a) kf 0 k∞
if f 0 ∈ L∞ [a, b] ;
 4 + b−a2





h

i1
1
b−x p+1 p
1
x−a p+1
(b − a) p kf 0 kq if f 0 ∈ Lq [a, b] ,
+
≤
1
b−a
b−a
(p+1) p


1


+ 1q = 1, p > 1;

p
h
i

a+b


 1 + x− 2 kf 0 k1 ;
2
b−a
where k·kr (r ∈ [1, ∞]) are the usual Lebesgue norms on Lr [a, b], i.e.,
kgk∞ := ess sup |g (t)|
t∈[a,b]
and
Z
kgkr :=
b
r1
|g (t)| dt , r ∈ [1, ∞).
r
a
The constants
1
,
4
1
1
(p+1) p
and
1
2
respectively are sharp in the sense presented in Theorem 1.1.
The above inequalities can also be obtained from Fink’s result in [12] on choosing n = 1 and
performing some appropriate computations.
If one drops the condition of absolute continuity and assumes that f is Hölder continuous,
then one may state the result (see [7]):
Theorem 1.3. Let f : [a, b] → R be of r − H−Hölder type, i.e.,
(1.3)
|f (x) − f (y)| ≤ H |x − y|r , for all x, y ∈ [a, b] ,
where r ∈ (0, 1] and H > 0 are fixed. Then for all x ∈ [a, b] we have the inequality:
"
r+1 r+1 #
Z b
1
H
b
−
x
x
−
a
f (x) −
(1.4)
f (t) dt ≤
+
(b − a)r .
b−a a
r+1
b−a
b−a
1
The constant r+1
is also sharp in the above sense.
Note that if r = 1, i.e., f is Lipschitz continuous, then we get the following version of
Ostrowski’s inequality for Lipschitzian functions (with L instead of H) (see [3])

!2 
Z b
a+b
f (x) − 1
≤ 1 + x − 2
 (b − a) L.
(1.5)
f
(t)
dt
b−a a
4
b−a
Here the constant 14 is also best.
Moreover, if one drops the continuity condition of the function, and assumes that it is of
bounded variation, then the following result may be stated (see [4]).
W
Theorem 1.4. Assume that f : [a, b] → R is of bounded variation and denote by ba (f ) its
total variation. Then
#
"
Z b
b
x − a+b _
1
1
2 (1.6)
f (t) dt ≤
+
(f )
f (x) − b − a
2 b−a a
a
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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O STROWSKI I NEQUALITY
3
for all x ∈ [a, b].
The constant 12 is the best possible.
If we assume more about f , i.e., f is monotonically increasing, then the inequality (1.6) may
be improved in the following manner [5] (see also [2]).
Theorem 1.5. Let f : [a, b] → R be monotonic nondecreasing. Then for all x ∈ [a, b], we have
the inequality:
Z b
1
f (x) −
(1.7)
f
(t)
dt
b−a a
Z b
1
≤
[2x − (a + b)] f (x) +
sgn (t − x) f (t) dt
b−a
a
1
≤
{(x − a) [f (x) − f (a)] + (b − x) [f (b) − f (x)]}
b
−
a
#
"
1 x − a+b
2 ≤
+
[f (b) − f (a)] .
2 b−a All the inequalities in (1.7) are sharp and the constant 21 is the best possible.
The version of Ostrowski’s inequality for convex functions was obtained in [6] and is incorporated in the following theorem:
Theorem 1.6. Let f : [a, b] → R be a convex function on [a, b]. Then for any x ∈ (a, b) we
have the inequality
1
(1.8)
(b − x)2 f+0 (x) − (x − a)2 f_0 (x)
2
Z b
≤
f (t) dt − (b − a) f (x)
a
1
≤
(b − x)2 f−0 (b) − (x − a)2 f+0 (a) .
2
In both parts of the inequality (1.8) the constant 12 is sharp.
For other Ostrowski type inequalities, see [8].
In this paper we extend Ostrowski’s inequality for arbitrary isotonic linear functionals A :
L → R, where L is a linear class of absolutely continuous functions defined on [a, b] . Some
applications for particular instances of linear functionals A are also provided.
2. P RELIMINARIES
Let L be a linear class of real-valued functions, g : E → R having the properties
(L1) f, g ∈ L imply (αf + βg) ∈ L for all α, β ∈ R;
(L2) 1 ∈ L, i.e., if f (t) = 1, t ∈ E, then f ∈ L.
An isotonic linear functional A : L → R is a functional satisfying
(A1) A (αf + βg) = αA (f ) + βA (g) for all f, g ∈ L and α, β ∈ R;
(A2) If f ∈ L and f ≥ 0, then A (f ) ≥ 0.
The mapping A is said to be normalised if
(A3) A (1) = 1.
Usual examples of isotonic linear functional that are normalised are the following ones
Z
1
A (f ) :=
f (x) dµ (x) , if µ (X) < ∞
µ (X) X
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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4
S.S. D RAGOMIR
or
Z
1
w (x) f (x) dµ (x) ,
w (x) dµ (x) X
X
R
where w (x) ≥ 0, X w (x) dµ (x) > 0, X is a measurable space and µ is a positive measure on
X.
P
In particular, for x̄ = (x1 , . . . , xn ) , w̄ := (w1 , . . . , wn ) ∈ Rn with wi ≥ 0, Wn := ni=1 wi >
0 we have
n
1X
A (x̄) :=
xi
n i=1
Aw (f ) := R
and
n
1 X
wi xi ,
Aw̄ (x̄) :=
Wn i=1
are normalised isotonic linear functionals on Rn .
The following representation result for absolutely continuous functions holds.
Lemma 2.1. Let f : [a, b] →
R 1 R0 be an absolutely continuous function on [a, b] and define
e (t) = t, t ∈ [a, b], g (t, x) = 0 f [(1 − λ) x + λt] dλ, t ∈ [a, b] and x ∈ [a, b] . If A : L → R
is a normalised linear functional on a linear class L of absolutely continuous functions defined
on [a, b] and (x − e) · g (·, x) ∈ L, then we have the representation
(2.1)
f (x) = A (f ) + A [(x − e) · g (·, x)] ,
for x ∈ [a, b] .
Proof. For any x, t ∈ [a, b] with t 6= x, one has
Rx 0
Z 1
f (u)
f (x) − f (t)
t
=
=
f 0 [(1 − λ) x + λt] dλ = g (t, x) ,
x−t
x−t
0
giving the equality
(2.2)
f (x) = f (t) + (x − t) g (t, x)
for any t, x ∈ [a, b] .
Applying the functional A, we get
A (f (x) · 1) = A (f + (x − e) g (·, x)) ,
for any x ∈ [a, b] .
Since
A (f (x) · 1) = f (x) A (1) = f (x)
and
A (f + (x − e) · g (·, x)) = A (f ) + A ((x − e) · g (·, x)) ,
the equality (2.1) is obtained.
The following particular cases are of interest:
Corollary 2.2. Let f : [a, b] → R be an absolutely continuous function on [a, b] . Then we have
the representation:
Z b
1
(2.3) f (x) = R b
w (t) f (t) dt
w
(t)
dt
a
a
Z 1
Z b
1
0
f [(1 − λ) x + λt] dλ dt
+ Rb
w (t) (x − t)
w
(t)
dt
a
0
a
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O STROWSKI I NEQUALITY
5
Rb
for any x ∈ [a, b] , where p : [a, b] → R is a Lebesgue integrable function with a w (t) dt 6= 0.
In particular, we have
Z 1
Z b
Z b
1
1
0
f (t) dt +
(x − t)
f [(1 − λ) x + λt] dλ dt
(2.4)
f (x) =
b−a a
b−a a
0
for each x ∈ [a, b] .
The proof is obvious by Lemma 2.1 applied for the normalised linear functionals
Z b
Z b
1
1
Aw (f ) := R b
w (t) f (t) dt, A (f ) :=
f (t) dt
b−a a
w (t) dt a
a
defined on
L := {f : [a, b] → R, f is absolutely continuous on [a, b]} .
The following discrete case also holds.
Corollary 2.3. Let f : [a, b] → R be an absolutely continuous function on [a, b] . Then we have
the representation:
Z 1
n
n
1 X
1 X
0
(2.5)
f (x) =
wi f (xi ) +
wi (x − xi )
f [(1 − λ) x + λxi ] dλ
Wn i=1
Wn i=1
0
P
for any x ∈ [a, b] , where xi ∈ [a, b] , wi ∈ R (i = {1, . . . , n}) with Wn := ni=1 wi 6= 0.
In particular, we have
Z 1
n
n
1X
1X
0
(2.6)
f (x) =
f (xi ) +
(x − xi )
f [(1 − λ) x + λxi ] dλ
n i=1
n i=1
0
for any x ∈ [a, b] .
3. O STROWSKI T YPE I NEQUALITIES
The following theorem holds.
Theorem 3.1. With the assumptions of Lemma 2.1, and assuming that A : L → R is isotonic,
then we have the inequalities
(3.1)
|f (x) − A (f )|
 0

A
|x
−
e|
kf
k

[x,·],∞





 1
A |x − e| q kf 0 k[x,·],p
≤







 A kf 0 k
[x,·],1
if |x − e| kf 0 k[x,·],∞ ∈ L, f 0 ∈ L∞ [a, b] ;
1
if |x − e| q kf 0 k[x,·],p ∈ L, f 0 ∈ Lp [a, b] ,
p > 1, p1 + 1q = 1;
if kf 0 k[x,·],1 ∈ L,
where
khk[m,n],∞ := ess sup |h (t)| and
t∈[m,n]
(t∈[n,m])
khk[m,n],p
Z
:= n
m
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
1
p
|h (t)| dt , p ≥ 1.
p
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6
S.S. D RAGOMIR
If we denote
0
M∞ (x) := A |x − e| kf k[x,·],∞ ,
1
Mp (x) := A |x − e| q kf 0 k[x,·],p ,
0
M1 (x) := A kf k[x,·],1 ,
then we have the inequalities:
M∞ (x)
(3.2)

kf 0 k[a,b],∞ A (|x − e|)







i β1
 h 1
0 β
A kf k[x,·],∞
[A (|x − e|α )] α
≤







 1 (b − a) + x − a+b A kf 0 k
2
if |x − e| ∈ L, f 0 ∈ L∞ [a, b] ;
if kf 0 kβ[x,·],∞ , |x − e|α ∈ L,
f 0 ∈ L∞ [a, b] , α > 1, α1 + β1 = 1;
[x,·],∞
2
if kf 0 k[x,·],∞ ∈ L, f 0 ∈ L∞ [a, b] .
Mp (x)
(3.3)

n
o 1
0
0

q

max
kf
k
,
kf
k
A
|x
−
e|

[a,x],p
[x,b],p





 h i β1 h i α1
α
β
0
q
≤
A kf k[x,·],p
A |x − e|






1 

 1 (b − a) + x − a+b q A kf 0 k
2
[x,·],p
2
1
if |x − e| q ∈ L, f 0 ∈ Lp [a, b] ;
α
if kf 0 kβ[x,·],p , |x − e| q ∈ L,
f 0 ∈ Lp [a, b] , α > 1, α1 + β1 = 1;
if kf 0 k[x,·],p ∈ L, f 0 ∈ Lp [a, b]
and
(3.4)
M1 (x) ≤




1
2
0
0
kf k[a,b],1 + kf k[a,x],1 − kf k[x,b],1 ,
0
1
2
h i β1


 A kf 0 kβ
,
[x,·],1
β > 1.
Proof. Using (2.1) and taking the modulus, we have
(3.5)
|f (x) − A (f )| = |A ((x − e) · g (·, x))|
≤ A (|(x − e) · g (·, x)|)
= A (|x − e| |g (·, x)|) .
For t 6= x (t, x ∈ [a, b]) we may state
Z
1
|f 0 ((1 − λ) x + λt)| dλ
|g (t, x)| ≤
0
≤ ess sup |f 0 ((1 − λ) x + λt)|
λ∈[0,1]
0
= kf k[x,t],∞ .
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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O STROWSKI I NEQUALITY
Hölder’s inequality will produce
Z
|g (t, x)| ≤
7
1
|f 0 ((1 − λ) x + λt)| dλ
0
Z
1
≤
p1
|f ((1 − λ) x + λt)| dλ
p
0
0
=
1
x−t
Z
x
0
p
p1
|f (u)| du
t
1
= |x − t|− p kf 0 k[x,t],p ,
p > 1,
1 1
+ = 1;
p q
and finally
1
Z
|f 0 ((1 − λ) x + λt)| dλ =
|g (t, x)| ≤
0
1
kf 0 k[x,t],1 .
t−x
Consequently
(3.6)


|x − e| kf 0 k[x,·],∞ if f 0 ∈ L∞ [a, b] ;





1
|(x − e)| |g (·, x)| ≤
|x − e| q kf 0 k[x,·],p if f 0 ∈ Lp [a, b] ,





 kf 0 k
[x,·],1
for any x ∈ [a, b] .
Applying the functional A to (3.6) and using (3.5) we deduce the inequality (3.1).
We have
n
o
M∞ (x) ≤ sup kf 0 k[x,t],∞ A (|x − e|)
t∈[a,b]
n
o
= max kf 0 k[a,x],∞ , kf 0 k[x,b],∞ A (|x − e|)
= kf 0 k[a,b],∞ A (|x − e|)
and the first inequality in (3.2) is proved.
Using Hölder’s inequality for the functional A, i.e.,
h i β1
1
1
1
(3.7)
|A (hg)| ≤ [A (|h|α )] α A |g|β
, α > 1, + = 1,
α β
where hg, |h|α , |g|β ∈ L, we have
h i β1
1
β
M∞ (x) ≤ [A (|x − e|α )] α A kf 0 k[x,·],∞
and the second part of (3.2) is proved.
In addition,
M∞ (x) ≤ sup |x − t| A kf 0 k[x,·],∞
t∈[a,b]
= max {x − a, b − x} A kf 0 k[x,·],∞
1
a + b 0
=
(b − a) + x −
A
kf
k
[x,·],∞
2
2 and the inequality (3.2) is completely proved.
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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8
S.S. D RAGOMIR
We also have
o n
1
0
q
Mp (x) ≤ sup kf k[x,t],p A |x − e|
t∈[a,b]
n
o 1
0
0
q
= max kf k[a,x],p , kf k[x,b],p A |x − e| .
Using Hölder’s inequality (3.7) one has
i α1 h i β1
h α
1
1
β
A kf 0 k[x,·],p
, α > 1, + = 1
Mp (x) ≤ A |x − e| q
α β
and
n
o 1
0
q
Mp (x) ≤ sup |x − t| A kf k[x,·],p
t∈[a,b]
n
o 1
1
0
q
q
= max (x − a) , (b − x) A kf k[x,·],p
1
1
a + b q 0
=
(b − a) + x −
A
kf
k
[x,·],p ,
2
2 proving the inequality (3.3).
Finally,
o
n
A kf 0 k[x,·],1 ≤ sup kf 0 k[x,t],1 A (1)
t∈[a,b]
n
o
0
0
= max kf k[a,x],1 , kf k[x,b],1
1 0
1 0
0
= kf k[a,b],1 + kf k[a,x],1 − kf k[x,b],1 .
2
2
By Hölder’s inequality, we have
h i β1
β
A kf 0 k[x,·],1 ≤ A kf 0 k[x,·],1
,
β > 1,
and the last part of (3.4) is also proved.
4. T HE C ASE
WHERE
|f 0 | IS C ONVEX
The following theorem also holds.
Theorem 4.1. Let f : [a, b] → R be an absolutely continuous function such that f 0 : (a, b) → R
is convex in absolute value, i.e., |f 0 | is convex on (a, b) . If A : L → R is a normalised isotonic
linear functional and |x − e|, |x − e| |f 0 | ∈ L, then
|f (x) − A (f )| ≤
(4.1)
≤

















1
2
h
1 0
[|f (x)| A (|x − e|) + A (|x − e| |f 0 |)]
2
i
kf k[a,b],∞ + |f (x)| A (|x − e|) ,
0
0
if f 0 ∈ L∞ [a, b] ;
1
2
h i β1 1
α α
0
0 β
|f (x)| A (|x − e|) + [A (|x − e| )] A |f |
1
2
0
|f (x)| A (|x − e|) + 12 (b − a) + x −
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
a+b 2
if |x − e|α , |f 0 |β ∈ L,
α > 1, α1 +
A (|f 0 |) if |f 0 | ∈ L.
1
β
= 1;
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O STROWSKI I NEQUALITY
9
Proof. Since |f 0 | is convex, we have
Z
1
|f 0 ((1 − λ) x + λt)| dλ
0
Z 1
Z
0
0
= |f (x)|
(1 − λ) dλ + |f (t)|
|g (t, x)| ≤
0
=
1
λdλ
0
|f 0 (x)| + |f 0 (t)|
.
2
Thus,
|f 0 (x)| + |f 0 (t)|
|f (x) − A (f )| ≤ A |x − e| ·
2
1
= [|f 0 (x)| A (|x − e|) + A (|x − e| |f 0 |)]
2
and the first part of (4.1) is proved.
We have
A (|x − e| |f 0 |) ≤ ess sup {|f 0 (t)|} · A (|x − e|)
t∈[a,b]
0
= kf k[a,b],∞ A (|x − e|) .
By Hölder’s inequality for isotonic linear functionals, we have
h i β1
1
1
1
β
, α > 1, + = 1
A (|x − e| |f 0 |) ≤ [A (|x − e|α )] α A |f 0 |
α β
and finally,
A (|x − e| |f 0 |) ≤ sup |x − t| · A (|f 0 |)
t∈[a,b]
= max (x − a, b − x) · A (|f 0 |)
1
a
+
b
A (|f 0 |) .
=
(b − a) + x −
2
2 The theorem is thus proved.
5. S OME I NTEGRAL I NEQUALITIES
Rb
1
If we consider the normalised isotonic linear functional A (f ) = b−a
f , then by Theorem
a
3.1 for f : [a, b] → R an absolutely continuous function, we may state the following integral
inequalities
Z b
1
(5.1) f (x) −
f (t) dt
b−a a
Z b
1
≤
|x − t| kf 0 k[x,t],∞ dt
b−a a
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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10
S.S. D RAGOMIR


!2 
a+b

x− 2

1

 (b − a) (Ostrowski’s inequality)

kf 0 k[a,b],∞  +


4
b
−
a






provided f 0 ∈ L∞ [a, b] ;





#1

β1 "

α+1
α+1 α
 1 Z b
(b − x)
+ (x − a)
β
kf 0 k[x,t],∞ dt
≤
b−a a
(α + 1) (b − a)




0
0

if f ∈ L∞ [a, b] , kf k[x,·],∞ ∈ Lβ [a, b] , α > 1, α1 + β1 = 1;




"
#


x − a+b Z b

1

2


kf 0 k[x,t],∞ dt
+

 2
b
−
a

a


if f 0 ∈ L∞ [a, b] , and if kf 0 k[x,·],∞ ∈ L1 [a, b] ,
for each x ∈ [a, b] ;
(5.2)
Z b
1
f (x) −
f
(t)
dt
b−a a
Z b
1
1
≤
|x − t| q kf 0 k[x,t],p dt
b−a a

n
o
1 +1
1 +1
q
q
+(x−a)
(b−x)

0
0

,
q max kf k[a,x],p , kf k[x,b],p

(b−a)(q+1)





p > 1, p1 + 1q = 1 and f 0 ∈ Lp [a, b] ;





α1

β1  1
α +1
α +1

 q α 1 R b kf 0 kβ
(b−x) q +(x−a) q
if f 0 ∈ Lp [a, b] ,
[x,t],p dt
b−a a
(b−a)(q+α)
≤


and kf 0 k[x,·],p ∈ Lβ [a, b] , where α > 1, α1 + β1 = 1





1q


Rb 0

x− a+b
|
|

1
1
2

+ b−a
kf k[x,t],p dt

2
b−a a




if f 0 ∈ Lp [a, b] , and kf 0 k[x,·],p ∈ L1 [a, b] ,
for each x ∈ [a, b] and
(5.3)
Z b
f (x) − 1
f (t) dt
b−a a
Z b
1
≤
kf 0 k[x,t],1 dt
b−a a

1 0
1 0
0

kf k[a,b],1 + kf k[a,x],1 − kf k[x,b],1 if f 0 ∈ L1 [a, b] ;



2
2

 β1
Z b
≤
1
β

kf 0 k[x,t],1 dt


b
−
a

a


if f 0 ∈ L1 [a, b] , kf 0 k[x,.],1 ∈ Lβ [a, b] , where β > 1,
for each x ∈ [a, b] .
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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O STROWSKI I NEQUALITY
11
If we assume now that f : [a, b] → R is absolutely continuous and such that |f 0 | is convex on
(a, b) , then by Theorem 4.1 we obtain the following integral inequalities established in [1]
Z b
1
(5.4)
f (t) dt
f (x) − b − a
a



!2 
Z b
a+b
x− 2
1
1
 (b − a) + 1
≤ |f 0 (x)|  +
|x − t| |f 0 (t)| dt
2
4
b−a
b−a a
 h
i
a+b 2 
x−
1
1
0
0


kf k[a,b],∞ + |f (x)| 4 + b−a2
(b − a) if f 0 ∈ L∞ [a, b] ;

2




a+b 2 
x−

1
1
0

(b − a)
 2 |f (x)| 4 + b−a2




h
i1 h
i β1 
R
b
β
(b−x)α+1 +(x−a)α+1 α
1
+
|f 0 (t)| dt
≤
(α+1)(b−a)
b−a a



if f 0 ∈ Lβ [a, b] , α > 1, α1 + β1 = 1;





a+b 2 
Rb 0

x− 2
|x− a+b

1
1
1
2 |
0

|f (x)| 4 + b−a
(b − a) + 2 + b−a
|f (t)| dt

2
a




if f 0 ∈ L1 [a, b] ,
for each x ∈ [a, b] .
6. S OME D ISCRETE I NEQUALITIES
For a given interval [a, b] , consider the division
In : a = x0 < x1 < · · · < xn−1 < xn = b
and the intermediate points ξi ∈ [xi , xi+1 ] , i = 0, n − 1. If hi := xi+1 − xi > 0 i = 0, n − 1
we may define the following functionals
n−1
1 X
A (f ; In , ξ) :=
f (ξi ) hi
b − a i=0
(Riemann Rule)
n−1
1 X f (xi ) + f (xi+1 )
· hi
b − a i=0
2
n−1 1 X
xi + xi+1
AM (f ; In ) :=
f
· hi
b − a i=0
2
AT (f ; In ) :=
1
2
AS (f ; In ) := AT (f ; In ) + AM (f ; In ) .
3
3
(Trapezoid Rule)
(Mid-point Rule)
(Simpson Rule)
We observe that, all the above functionals are obviously linear, isotonic and normalised.
Consequently, all the inequalities obtained in Sections 2 and 3 may be applied for these
functionals.
If, for example, we use the following inequality (see Theorem 3.1)
(6.1)
|f (x) − A (f )| ≤ kf 0 k[a,b] A (|x − e|) , x ∈ [a, b] ,
J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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12
S.S. D RAGOMIR
provided f : [a, b] → R is absolutely continuous and f 0 ∈ L∞ [a, b] , then we get the inequalities
(6.2)
(6.3)
n−1
n−1
X
1 X
1
0
f
(x)
−
f
(ξ
)
h
≤
kf
k
|x − ξi | hi ,
i
i
[a,b],∞
b − a i=0
b − a i=0
n−1
X
f
(x
)
+
f
(x
)
1
i
i+1
· hi f (x) −
b − a i=0
2
n−1
≤ kf 0 k[a,b],∞ ·
(6.4)
1 X |x − xi | + |x − xi+1 |
hi ,
b − a i=0
2
n−1 n−1 X
1
xi + xi+1
1 X xi + xi+1 0
f
· hi ≤ kf k[a,b],∞
x−
f (x) −
hi ,
b − a i=0
2
b − a i=0 2
for each x ∈ [a, b] .
Similar results may be stated if one uses for example Theorem 4.1. We omit the details.
R EFERENCES
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type inequalities for functions whose modulus of derivatives are convex and applications, Res. Rep.
Coll., 5(2) (2002), Article 1. [ONLINE: http://rgmia.vu.edu.au/v5n2.html]
[2] P. CERONE AND S.S. DRAGOMIR, Midpoint type rules from an inequalities point of view, in
Analytic-Computational Methods in Applied Mathematics, G.A. Anastassiou (Ed), CRC Press,
New York, 2000, 135-200.
[3] S.S. DRAGOMIR, The Ostrowski’s integral inequality for Lipschitzian mappings and applications,
Comp. and Math. with Appl., 38 (1999), 33-37.
[4] S.S. DRAGOMIR, On the Ostrowski’s inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33–40.
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(1997), 239–244.
[10] S.S. DRAGOMIR AND S. WANG, Applications of Ostrowski’s inequality to the estimation of error
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O STROWSKI I NEQUALITY
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[11] S.S. DRAGOMIR AND S. WANG, A new inequality of Ostrowski’s type in Lp −norm and applications to some special means and to some numerical quadrature rules, Indian J. of Math., 40(3)
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J. Inequal. Pure and Appl. Math., 3(5) Art. 68, 2002
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