Journal of Inequalities in Pure and
Applied Mathematics
OSTROWSKI TYPE INEQUALITIES FOR ISOTONIC
LINEAR FUNCTIONALS
volume 3, issue 5, article 68,
2002.
S.S. DRAGOMIR
School of Communications and Informatics
Victoria University of Technology
PO Box 14428
Melbourne City MC
8001 Victoria, Australia
EMail: 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
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
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.
2000 Mathematics Subject Classification: Primary 26D15, 26D10.
Key words: Ostrowski Type Inequalities, Isotonic Linear Functionals.
Ostrowski Type Inequalities for
Isotonic Linear Functionals
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Ostrowski Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
The Case where |f 0 | is Convex . . . . . . . . . . . . . . . . . . . . . . . . .
5
Some Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Some Discrete Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
S.S. Dragomir
3
8
13
19
21
24
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1.
Introduction
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
1
2
f (x) −
(1.1)
+
(b − a) M
f (t) dt ≤
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]).
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−a2
(b − a) kf 0 k∞
if f 0 ∈ L∞ [a, b] ;

4



1 h
i1 0
(b−a) p
x−a p+1
b−x p+1 p
≤
+
kf kq if f 0 ∈ Lq [a, b] , p1 + 1q = 1, p > 1;
1

b−a
b−a
p

(p+1)

i
h

 1 x− a+b
2 
+
kf 0 k1 ;
2
b−a Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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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:
Z b
1
(1.4) f (x) −
f (t) dt
b−a a
"
r+1 r+1 #
H
b−x
x−a
≤
+
(b − a)r .
r+1
b−a
b−a
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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J. Ineq. Pure and Appl. Math. 3(5) Art. 68, 2002
The constant
1
r+1
is also sharp in the above sense.
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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
x− 2
1
f (x) − 1
 (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]).
Theorem
1.4. Assume that f : [a, b] → R is of bounded variation and denote
W
by ba (f ) its total variation. Then
#
"
Z b
b
a+b _
1
1 x − 2 (1.6)
f (t) dt ≤
+
(f )
f (x) − b − a
2 b−a a
a
for all x ∈ [a, b].
The constant 12 is the best possible.
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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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]).
Close
Theorem 1.5. Let f : [a, b] → R be monotonic nondecreasing. Then for all
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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
1
2
S.S. Dragomir
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
(1.8)
Ostrowski Type Inequalities for
Isotonic Linear Functionals
1
(b − x)2 f+0 (x) − (x − a)2 f_0 (x)
2
Z b
≤
f (t) dt − (b − a) f (x)
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a
≤
1
(b − x)2 f−0 (b) − (x − a)2 f+0 (a) .
2
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J. Ineq. Pure and Appl. Math. 3(5) Art. 68, 2002
In both parts of the inequality (1.8) the constant
1
2
is sharp.
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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.
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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2.
Preliminaries
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;
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
(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
or
Z
1
Aw (f ) := R
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.
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In particular,
for x̄ = (x1 , . . . , xn ) , w̄ := (w1 , . . . , wn ) ∈ Rn with wi ≥ 0,
Pn
Wn := i=1 wi > 0 we have
n
1X
A (x̄) :=
xi
n i=1
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 be an absolutely
R 1 0 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
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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giving the equality
f (x) = f (t) + (x − t) g (t, x)
(2.2)
for any t, x ∈ [a, b] .
Applying the functional A, we get
A (f (x) · 1) = A (f + (x − e) g (·, x)) ,
Ostrowski Type Inequalities for
Isotonic Linear Functionals
for any x ∈ [a, b] .
Since
A (f (x) · 1) = f (x) A (1) = f (x)
S.S. Dragomir
A (f + (x − e) · g (·, x)) = A (f ) + A ((x − e) · g (·, x)) ,
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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:
(2.3) f (x) = R b
a
w (t) f (t) dt
w (t) dt
a
1
Z
+ Rb
a
w (t) dt
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Z
w (t) (x − t)
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Z
1
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0
1
f [(1 − λ) x + λt] dλ dt
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for any x ∈ [a, b] , where p : [a, b] → R is a Lebesgue integrable function with
Rb
w (t) dt 6= 0.
a
In particular, we have
Z b
1
(2.4) 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
Ostrowski Type Inequalities for
Isotonic Linear Functionals
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
JJ
J
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:
1
Wn
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defined on
(2.5) f (x) =
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wi f (xi )
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i=1
+
1
Wn
n
X
i=1
Z
wi (x − xi )
0
1
f 0 [(1 − λ) x + λxi ] dλ
J. Ineq. Pure and Appl. Math. 3(5) Art. 68, 2002
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for
Pnany x ∈ [a, b] , where xi ∈ [a, b] , wi ∈ R (i = {1, . . . , n}) with Wn :=
i=1 wi 6= 0.
In particular, we have
n
n
1X
1X
(2.6) f (x) =
f (xi ) +
(x − xi )
n i=1
n i=1
Z
1
f [(1 − λ) x + λxi ] dλ
0
0
for any x ∈ [a, b] .
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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3.
Ostrowski Type Inequalities
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
|f (x) − A (f )|
 0

A |x − e| kf k[x,·],∞






 1
0
q
A |x − e| kf k[x,·],p
≤







 A kf 0 k
[x,·],1
(3.1)
if |x − e| kf 0 k[x,·],∞ ∈ L, f 0 ∈ L∞ [a, b] ;
Ostrowski Type Inequalities for
Isotonic Linear Functionals
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,
S.S. Dragomir
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where
khk[m,n],∞ := ess sup |h (t)| and
t∈[m,n]
(t∈[n,m])
khk[m,n],p
Z
:= n
m
1
p
|h (t)| dt , p ≥ 1.
p
If we denote
M∞ (x) := A |x − e| kf 0 k[x,·],∞ ,
1
Mp (x) := A |x − e| q kf 0 k[x,·],p , and M1 (x) := A kf 0 k[x,·],1 ,
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then we have the inequalities:
(3.2)










≤









M∞ (x)
kf 0 k[a,b],∞ A (|x − e|) if |x − e| ∈ L, f 0 ∈ L∞ [a, b] ;
i β1
h 1
[A (|x − e|α )] α
A kf 0 kβ[x,·],∞
if kf 0 kβ[x,·],∞ , |x − e|α ∈ L, f 0 ∈ L∞ [a, b] , α > 1, α1 + β1 = 1;
1
x − a+b A kf 0 k
(b
−
a)
+
if kf 0 k[x,·],∞ ∈ L, f 0 ∈ L∞ [a, b] .
[x,·],∞
2
2
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
(3.3)











≤










Mp (x)
n
o 1
1
0
0
q
max kf k[a,x],p , kf k[x,b],p A |x − e|
if |x − e| q ∈ L, f 0 ∈ Lp [a, b] ;
i α1
h i β1 h α
A |x − e| q
A kf 0 kβ[x,·],p
α
q
if kf 0 kβ[x,·],p , |x − e| ∈ L, f 0 ∈ Lp [a, b] , α > 1, α1 + β1 = 1;
1 1
x − a+b q A kf 0 k
(b
−
a)
+
if kf 0 k[x,·],p ∈ L, f 0 ∈ Lp [a, b]
[x,·],p
2
2
and
(3.4)
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M1 (x) ≤




1
2
kf k[a,b],1 + kf 0 k[a,x],1 − kf 0 k[x,b],1 ,
0
1
2
h i 1


 A kf 0 kβ[x,·],1 β ,
β > 1.
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Proof. Using (2.1) and taking the modulus, we have
|f (x) − A (f )| = |A ((x − e) · g (·, x))|
≤ A (|(x − e) · g (·, x)|)
= A (|x − e| |g (·, x)|) .
(3.5)
For t 6= x (t, x ∈ [a, b]) we may state
Z 1
|g (t, x)| ≤
|f 0 ((1 − λ) x + λt)| dλ
0
≤ ess sup |f 0 ((1 − λ) x + λt)|
Ostrowski Type Inequalities for
Isotonic Linear Functionals
λ∈[0,1]
S.S. Dragomir
0
= kf k[x,t],∞ .
Hölder’s inequality will produce
Z 1
|g (t, x)| ≤
|f 0 ((1 − λ) x + λt)| dλ
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0
Z
≤
1
p1
|f ((1 − λ) x + λt)| dλ
p
0
0
=
1
x−t
− p1
= |x − t|
x
Z
0
p
p1
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and finally
Z
|g (t, x)| ≤
0
1
|f 0 ((1 − λ) x + λt)| dλ =
1
kf 0 k[x,t],1 .
t−x
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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
0
M∞ (x) ≤ sup kf k[x,t],∞ A (|x − e|)
t∈[a,b]
o
n
= 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
α α
β
(3.7)
|A (hg)| ≤ [A (|h| )] A |g|
, 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,·],∞
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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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,·],∞
a
+
b
1
A kf 0 k
=
(b − a) + x −
[x,·],∞
2
2 and the inequality (3.2) is completely proved.
We also have
n
o 1
Mp (x) ≤ sup kf 0 k[x,t],p A |x − e| q
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
t∈[a,b]
n
o 1
= max kf 0 k[a,x],p , kf 0 k[x,b],p A |x − e| q .
Using Hölder’s inequality (3.7) one has
h i α1 h i β1
α
1
1
0 β
q
Mp (x) ≤ A |x − e|
A kf k[x,·],p
, α > 1, + = 1
α β
and
n
o 1
Mp (x) ≤ sup |x − t| q A kf 0 k[x,·],p
t∈[a,b]
n
o 1
1
= max (x − a) q , (b − x) q A kf 0 k[x,·],p
1 q
1
a
+
b
A kf 0 k
=
(b − a) + x −
[x,·],p ,
2
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proving the inequality (3.3).
Finally,
n
o
A kf 0 k[x,·],1 ≤ sup kf 0 k[x,t],1 A (1)
t∈[a,b]
o
n
= max kf 0 k[a,x],1 , kf 0 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
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
A kf 0 k[x,·],1
i β1
h 0 β
,
≤ A kf k[x,·],1
β > 1,
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and the last part of (3.4) is also proved.
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The Case where |f 0| is Convex
4.
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
(4.1)
≤



















1 0
[|f (x)| A (|x − e|) + A (|x − e| |f 0 |)]
2
h
i
1
0
0
kf
k
+
|f
(x)|
A (|x − e|) , if f 0 ∈ L∞ [a, b] ;
[a,b],∞
2
h i β1 1
α
β
1
|f 0 (x)| A (|x − e|) + [A (|x − e| )] α A |f 0 |
2
|f (x) − A (f )| ≤
if |x − e|α , |f 0 |β ∈ L, α > 1, α1 + β1 = 1;
0
1
1
x − a+b A (|f 0 |) if |f 0 | ∈ L.
|f
(x)|
A
(|x
−
e|)
+
(b
−
a)
+
2
2
2
Proof. Since |f 0 | is convex, we have
Z 1
|g (t, x)| ≤
|f 0 ((1 − λ) x + λt)| dλ
0
Z 1
Z
0
0
= |f (x)|
(1 − λ) dλ + |f (t)|
0
=
|f 0 (x)| + |f 0 (t)|
.
2
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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1
λdλ
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Thus,
|f 0 (x)| + |f 0 (t)|
|f (x) − A (f )| ≤ A |x − e| ·
2
1 0
= [|f (x)| A (|x − e|) + A (|x − e| |f 0 |)]
2
and the first part of (4.1) is proved.
We have
0
0
A (|x − e| |f |) ≤ ess sup {|f (t)|} · A (|x − e|)
t∈[a,b]
0
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
= kf k[a,b],∞ A (|x − e|) .
By Hölder’s inequality for isotonic linear functionals, we have
h i β1
1
1
0 β
A (|x − e| |f |) ≤ [A (|x − e| )] A |f |
, α > 1, + = 1
α β
α
0
1
α
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.
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5.
Some Integral Inequalities
Rb
1
f , then
If we consider the normalised isotonic linear functional A (f ) = b−a
a
by Theorem 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

a+b 2 
x−
1
0

(b − a) (Ostrowski’s inequality)
kf k[a,b],∞ 4 + b−a2






provided f 0 ∈ L∞ [a, b] ;






h
i β1 h
i1

R

(b−x)α+1 +(x−a)α+1 α
 1 b kf 0 kβ
dt
[x,t],∞
b−a a
(α+1)(b−a)
≤



if f 0 ∈ L∞ [a, b] , kf 0 k[x,·],∞ ∈ Lβ [a, b] , α > 1, α1 + β1 = 1;






Rb 0

|x− a+b
1
2 |


kf k[x,t],∞ dt
+

b−a
2
a




if f 0 ∈ L [a, b] , and if kf 0 k
∈ L [a, b] ,
∞
[x,·],∞
1
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S.S. Dragomir
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for each x ∈ [a, b] ;
(5.2)
Z b
Z b
1
1
f (x) − 1
f (t) dt ≤
|x − t| q kf 0 k[x,t],p dt
b−a a
b−a a
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
n
o
1 +1
1 +1
(b−x) q +(x−a) q

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 
α +1
α +1

(b−x) q +(x−a) q
 q α1 1 R b kf 0 kβ
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 |

kf k[x,t],p dt
+

2
b−a
b−a a




if f 0 ∈ Lp [a, b] , and kf 0 k[x,·],p ∈ L1 [a, b] ,
for each x ∈ [a, b] and
Z b
1
(5.3)
f
(t)
dt
f
(x)
−
b−a a
Z b
1
≤
kf 0 k[x,t],1 dt
b−a a

1
1 0
0
0

kf k[a,b],1 + 2 kf k[a,x],1 − kf k[x,b],1 if f 0 ∈ L1 [a, b] ;

2


 β1
Rb 0 β
≤
1
kf 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] .
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
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
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inequalities established in [1]
f (x) −
(5.4)


a+b
2
1
b−a
!2 
Z
a
b
f (t) dt
b

x−
1
1 0
 (b − a) + 1
|f (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)
|f (x)| 4 + b−a2

2




i β1 i1 h
h

Rb 0
β
(b−x)α+1 +(x−a)α+1 α
1
|f (t)| dt
+
≤
b−a a
(α+1)(b−a)



if f 0 ∈ Lβ [a, b] , α > 1, α1 + β1 = 1;





a+b 2 
Rb 0

x− a+b
x− 2
|
|

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] ,
≤
Z
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S.S. Dragomir
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for each x ∈ [a, b] .
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6.
Some Discrete Inequalities
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
A (f ; In , ξ) :=
1 X
f (ξi ) hi
b − a i=0
f (xi ) + f (xi+1 )
· hi
2
i=0
n−1 xi + xi+1
1 X
f
· hi
AM (f ; In ) :=
b − a i=0
2
AT (f ; In ) :=
1
b−a
n−1
X
1
2
AS (f ; In ) := AT (f ; In ) + AM (f ; In ) .
3
3
(Riemann Rule)
S.S. Dragomir
(Trapezoid Rule)
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(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)
Ostrowski Type Inequalities for
Isotonic Linear Functionals
|f (x) − A (f )| ≤ kf 0 k[a,b] A (|x − e|) , x ∈ [a, b] ,
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provided f : [a, b] → R is absolutely continuous and f 0 ∈ L∞ [a, b] , then we
get the inequalities
n−1
n−1
1 X
1 X
0
(6.2)
f (ξi ) hi ≤ kf k[a,b],∞
|x − ξi | hi ,
f (x) −
b − a i=0
b − a i=0
(6.3)
n−1
1 X f (xi ) + f (xi+1 )
· hi f (x) −
b − a i=0
2
n−1
1 X |x − xi | + |x − xi+1 |
hi ,
≤ kf k[a,b],∞ ·
b − a i=0
2
0
Ostrowski Type Inequalities for
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S.S. Dragomir
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(6.4)
Contents
n−1 xi + xi+1
1 X
f
· hi f (x) −
b − a i=0
2
0
≤ kf k[a,b],∞
n−1 1 X xi + xi+1 x−
hi ,
b − a i=0 2
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for each x ∈ [a, b] .
Similar results may be stated if one uses for example Theorem 4.1. We omit
the details.
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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, 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), 3337.
Ostrowski Type Inequalities for
Isotonic Linear Functionals
S.S. Dragomir
Title Page
[4] S.S. DRAGOMIR, On the Ostrowski’s inequality for mappings of bounded
variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33–40.
[5] S.S. DRAGOMIR, Ostrowski’s inequality for monotonous mappings and
applications, J. KSIAM, 3(1) (1999), 127–135.
[6] S.S. DRAGOMIR, An Ostrowski type inequality for convex
functions, Res. Rep. Coll., 5(1) (2002), Article 5. [ONLINE:
http://rgmia.vu.edu.au/v5n1.html]
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[7] S.S. DRAGOMIR, P. CERONE, J. ROUMELIOTIS AND S. WANG, A
weighted version of Ostrowski inequality for mappings of Hölder type and
applications in numerical analysis, Bull. Math. Soc. Sci. Math. Roumanie,
42(90)(4) (1992), 301–314.
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[8] S.S. DRAGOMIR AND Th.M. RASSIAS (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht/Boston/London, 2002.
[9] S.S. DRAGOMIR AND S. WANG, A new inequality of Ostrowski’s type in
L1 −norm and applications to some special means and to some numerical
quadrature rules, Tamkang J. of Math., 28 (1997), 239–244.
[10] S.S. DRAGOMIR AND S. WANG, Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and some
numerical quadrature rules, Appl. Math. Lett., 11 (1998), 105–109.
[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) (1998), 245–304.
[12] A.M. FINK, Bounds on the deviation of a function from its averages,
Czech. Math. J., 42(117) (1992), 289–310.
[13] A. OSTROWSKI, Über die Absolutabweichung einer differentienbaren
Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938),
226–227.
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Isotonic Linear Functionals
S.S. Dragomir
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