Journal of Inequalities in Pure and
Applied Mathematics
A DISCRETE EULER IDENTITY
A. AGLIĆ ALJINOVIĆ AND J. PEČARIĆ
Department of Applied Mathematics
Faculty of Electrical Engineering and Computing
Unska 3, 10 000 Zagreb, Croatia.
EMail: andrea@zpm.fer.hr
Faculty of Textile Technology
University of Zagreb
Pierottijeva 6, 10000 Zagreb
Croatia.
EMail: pecaric@mahazu.hazu.hr
volume 5, issue 3, article 58,
2004.
Received 09 January, 2004;
accepted 22 April, 2004.
Communicated by: P.S. Bullen
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
086-04
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Abstract
A discrete analogue of the weighted Montgomery identity (i.e. Euler identity)
for finite sequences of vectors in normed linear space is given as well as a
discrete analogue of Ostrowski type inequalities and estimates of difference of
two arithmetic means.
2000 Mathematics Subject Classification: 26D15
Key words: Discrete Montgomery identity, Discrete Ostrowski inequality.
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
Contents
1
2
3
4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Discrete Weighted Euler Identity . . . . . . . . . . . . . . . . . . . . . . . . 5
Discrete Ostrowski Type Inequalities . . . . . . . . . . . . . . . . . . . . 13
Estimates of the Differences Between Two Weighted Arithmetic Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References
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1.
Introduction
The following Ostrowski inequality is well known [10]:
#
"
Z b
a+b 2
x
−
1
1
2
f (x) −
f (t) dt ≤
+
(b − a) kf 0 k∞ .
2
b−a a
4
(b − a)
It holds for every x ∈ [a, b] whenever f : [a, b] → R is continuous on [a, b] and
differentiable on (a, b) with derivative f 0 : (a, b) → R bounded on (a, b) i.e.
kf 0 k∞ = sup |f 0 (t)| < +∞.
t∈(a,b)
Let f : [a, b] → R be differentiable on [a, b], f 0 : [a, b] → R integrable on
[a, b] and w : [a, b] → [0, ∞) some probability density function, i.e. integrable
Rb
Rt
function satisfying a w (t) dt = 1; define W (t) = a w (x) dx for t ∈ [a, b],
W (t) = 0 for t < a and W (t) = 1 for t > b. The following identity, given by
Pečarić in [11], is the weighted Montgomery identity
Z b
Z b
f (x) =
w (t) f (t) dt +
Pw (x, t) f 0 (t) dt,
a
a
where the weighted Peano kernel is
(
W (t) ,
a ≤ t ≤ x,
Pw (x, t) =
W (t) − 1 x < t ≤ b.
All results in this paper are discrete analogues of results from [1]. The aim
of this paper is to prove the discrete analogue of the weighted Euler identity for
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
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finite sequences of vectors in normed linear spaces and to use it to obtain some
new discrete Ostrowski type inequalities as well as estimates of differences between two (weighted) arithmetic means. In Section 2, a discrete weighted Montgomery (i.e. Euler) identity is presented. In Section 3, Ostrowski’s inequality
and its generalization are proved. These are the discrete analogues of some results from [6]. In Section 4, estimates of differences between two (weighted)
arithmetic means are given and these are the discrete analogues of some results
from [2], [3], [4], [5] and [12].
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
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2.
Discrete Weighted Euler Identity
Let x1 , x2 , . . . , xn be a finite sequence of vectors in the normed linear space
(X, k·k) and w1 , w2 , . . . , wn finite sequence of positive real numbers. If, for
1 ≤ k ≤ n,
n
k
X
X
wi = Wn − Wk ,
Wk =
wi , Wk =
i=1
i=k+1
then we have, see [9],
(2.1)
n
X
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
wi xi
i=1
= xk Wn +
k−1
X
Wi (xi − xi+1 ) +
i=1
n−1
X
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Wi (xi+1 − xi ) , 1 ≤ k ≤ n.
The difference operator ∆ is defined by
∆xi = xi+1 − xi .
(2.2)
So using formula (2.1), we get the discrete analogue of weighted Montgomery identity
(2.3)
xk =
Contents
i=k
n
n−1
X
1 X
wi xi +
Dw (k, i) ∆xi ,
Wn i=1
i=1
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where the discrete Peano kernel is defined by
(
Wi ,
1 ≤ i ≤ k − 1,
1
(2.4)
Dw (k, i) =
·
Wn
−Wi , k ≤ i ≤ n.
If we take wi = 1, i = 1, . . . , n, then Wi = i and Wi = n − i, and (2.3)
reduces to the discrete Montgomery identity
n
(2.5)
n−1
X
1X
xi +
Dn (k, i) ∆xi ,
xk =
n i=1
i=1
where
(
Dn (k, i) =
i
,
n
i
n
1 ≤ i ≤ k − 1,
− 1, k ≤ i ≤ n.
If n ∈ N, ∆n is inductively defined by
∆n xi = ∆n−1 (∆xi ) .
It is then easy to prove, by induction or directly using the elementary theory of
operators,see [8], that
n X
n
n
∆ xi =
(−1)n−k xi+k .
k
k=0
In the next theorem we give the generalization of the identity (2.3).
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
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Theorem 2.1. Let (X, k·k) be a normed linear space, x1 , x2 , . . . , xn a finite sequence of vectors in X, w1 , w2 , . . . , wn finite sequence of positive real numbers.
Then for all m ∈ {2, 3, . . . , n − 1} and k ∈ {1, 2, . . . , n} the following identity
is valid:
!
n
m−1
n−r
X 1
X
1 X
r
(2.6) xk =
wi xi +
∆ xi
Wn i=1
n − r i=1
r=1
!
n−1 X
n−2
n−r
X
X
×
···
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
i1 =1 i2 =1
+
n−1 X
n−2
X
i1 =1 i2 =1
···
n−m
X
ir =1
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im ) ∆m xim .
im =1
Title Page
Proof. We prove our assertion by induction with respect to m. For m = 2 we
have to prove the identity
! n−1
!
n
n−1
X
X
1 X
1
wi xi +
∆xi
Dw (k, i)
xk =
Wn i=1
n − 1 i=1
i=1
+
n−1 X
n−2
X
Dw (k, i) Dn−1 (i, j) ∆2 xj .
i=1 j=1
Applying the identity (2.5) for the finite sequence of vectors ∆xi , i = 1, 2, . . . , n−
1, we obtain
n−1
∆xi =
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n−2
X
1 X
∆xi +
Dn−1 (i, j) ∆2 xj
n − 1 i=1
j=1
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so, again using (2.3), we have
n
n−1
X
1 X
xk =
wi xi +
Dw (k, i)
Wn i=1
i=1
n
1 X
1
=
wi xi +
Wn i=1
n−1
+
n−1
n−2
X
1 X
∆xi +
Dn−1 (i, j) ∆2 xj
n − 1 i=1
j=1
! n−1
!
n−1
X
X
∆xi
Dw (k, i)
i=1
n−1
n−2
XX
!
i=1
Dw (k, i) Dn−1 (i, j) ∆2 xj .
A Discrete Euler Identity
i=1 j=1
A. Aglić Aljinović and J. Pečarić
Hence the identity (2.6) holds for m = 2.
Now, we assume that it holds for a natural number m ∈ {2, 3, . . . , n − 2}.
Applying the identity (2.5) for the ∆m xim
m
∆ xim
n−m
n−m−1
X
1 X m
∆ xi +
=
Dn−m (im , im+1 ) ∆m+1 xim+1
n − m i=1
i
=1
m+1
×
n−1 X
n−2
X
i1 =1 i2 =1
···
n−r
X
ir =1
n−r
X
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and using the induction hypothesis, we get
n
m−1
X 1
1 X
wi xi +
xk =
Wn i=1
n−r
r=1
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!
∆r xi
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i=1
!
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
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n−1 X
n−2
X
+
···
i1 =1 i2 =1
1
=
Wn
×
n−m
X
1
n−m
n
X
i=1
n−1 X
n−2
X
n−m−1
X

Dn−m (im , im+1 ) ∆m+1 xim+1 
im+1 =1
m
X
1
n−r
r=1
n−r
X
···
i1 =1 i2 =1
+
∆m xi +
i=1
wi xi +
n−1 X
n−2
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im )
im =1

×
n−m
X
n−r
X
!
∆r xi
i=1
A Discrete Euler Identity
!
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
ir =1
n−(m+1)
···
i1 =1 i2 =1
A. Aglić Aljinović and J. Pečarić
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m (im , im+1 ) ∆m+1 xim+1 .
im+1 =1
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We see that (2.6) is valid for m + 1 and our assertion is proved.
Remark 2.1. For m = n − 1 (2.6) becomes
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xk =
1
Wn
×
n
X
wi xi +
i=1
n−1
n−2
XX
i1 =1 i2 =1
···
n−2
X
r=1
n−r
X
1
n−r
n−r
X
!
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∆r xi
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i=1
!
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
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ir =1
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+
n−1 X
n−2
X
1
X
···
i1 =1 i2 =1
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · D2 (in−2 , in−1 ) ∆n−1 xin−1 .
in−1 =1
Corollary 2.2. Let (X, k·k) be a normed linear space, x1 , x2 , . . . , xn a finite sequence of vectors in X. Then for all m ∈ {2, 3, . . . , n − 1} and k ∈ {1, 2, . . . , n}
the following identity is valid:
n
m−1
X 1
1X
xk =
xi +
n i=1
n−r
r=1
n−1 X
n−2
X
×
···
i1 =1 i2 =1
+
n−1 X
n−2
X
···
i1 =1 i2 =1
n−r
X
n−r
X
!
∆r xi
A Discrete Euler Identity
i=1
A. Aglić Aljinović and J. Pečarić
!
Dn (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
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ir =1
n−m
X
Dn (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im ) ∆m xim .
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im =1
Proof. Apply Theorem 2.1 with wi = 1, i = 1, . . . , n.
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Remark 2.2. If we apply (2.6) with n = 2l − 1 and k = l we get
Close
xl =
1
W2l−1
×
2l−1
X
wi xi +
i=1
2l−2
X 2l−3
X
i1 =1 i2 =1
m−1
X
r=1
···
2l−1−r
X
ir =1
1
2l − 1 − r
2l−1−r
X
!
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∆r xi
i=1
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!
Dw (l, i1 ) D2l−2 (i1 , i2 ) · · · D2l−r (ir−1 , ir )
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+
2l−2
X 2l−3
X
···
2l−1−m
X
i1 =1 i2 =1
Dw (l, i1 ) D2l−2 (i1 , i2 ) · · · D2l−m (im−1 , im ) ∆m xim .
im =1
We may regard this identity as a generalized midpoint identity since for m = 1
it reduces to
xl =
(2.7)
1
2l−1
X
W2l−1
i=1
wi xi +
2l−2
X
Dw (l, i) ∆xi
i=1
A Discrete Euler Identity
and further for wi = 1, i = 1, 2, . . . , 2l − 1 to
2l−1
A. Aglić Aljinović and J. Pečarić
l−1
1 X
1 X
xl =
xi +
i (∆xi − ∆x2l−1−i ) .
2l − 1 i=1
2l − 1 i=1
(2.8)
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Similarly, if we apply (2.6) with k = 1 and then with k = n, then sum these two
equalities and divide them by 2, we get
n
m−1
X 1
x1 + xn
1 X
(2.9)
=
wi xi +
2
Wn i=1
n−r
r=1
×
n−1
X
···
i1 =1
+
n−1
X
i1 =1
···
n−r
X
ir
n−r
X
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∆r xi
Close
i=1
!
Dw (1, i1 ) + Dw (n, i1 )
Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
2
=1
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n−m
X
im
Dw (1, i1 ) + Dw (n, i1 )
Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im ) ∆m xim .
2
=1
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We may regard this identity as a generalized trapezoid identity since for m = 1
it reduces to
(2.10)
n
n−1
X
x1 + xn
1 X
Dw (1, i) + Dw (n, i)
=
wi xi +
∆xi ,
2
Wn i=1
2
i=1
and further for wi = 1, i = 1, 2, . . . , n to
(2.11)
n
n−1
1X
1 X
n
x1 + xn
i−
=
xi +
∆xi .
2
n i=1
n i=1
2
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
(2.8) and (2.11) were obtained by Dragomir in [7].
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3.
Discrete Ostrowski Type Inequalities
The Bernoulli numbers Bi , i ≥ 0, are defined by the implicit recurrence relation
(
m X
1, if m = 0,
m+1
Bi =
i
0, if m 6= 0.
i=0
If, for n ∈ N and m ∈ R ,we write
Sm (n) = 1m + 2m + 3m + · · · + (n − 1)m ,
it is well known, see [8], that if m ∈ N
m 1 X m+1
Sm (n) =
Bi nm+1−i .
m + 1 i=0
i
Theorem 3.1. Let (X, k·k) be a normed linear space, x1 , x2 , . . . , xn a finite sequence of vectors in X, w1 , w2 , . . . , wn finite sequence of positive real numbers.
Let also (p, q) be a pair of conjugate exponents1 , m ∈ {2, 3, . . . , n − 1} and
k ∈ {1, 2, . . . , n} the following inequality holds:
!
n
m−1
n−r
X 1
X
1 X
(3.1) xk −
wi xi −
∆r xi
Wn i=1
n
−
r
r=1
i=1
!
n−1
n−2
n−r
XX
X
×
···
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir ) i1 =1 i2 =1
ir =1
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
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1
That is: 1 < p, q < ∞ ,
1
p
+
1
q
=1
n−1 n−2
n−m+1
X
X X
k∆m xk ,
≤
·
·
·
D
(k,
i
)
D
(i
,
i
)
·
·
·
D
(i
,
·)
w
1
n−1
1
2
n−m+1
m−1
p
i1 =1 i2 =1
im−1 =1
q
where
 p1

n−m

 P k∆m x kp
, if 1 ≤ p < ∞,
i
k∆m xkp =
i=1


 max k∆m xi k
if p = ∞.
1≤i≤n−m
Proof. By using the (2.6) and the Hölder inequality.
Corollary 3.2. Let (X, k·k) be a normed linear space, x1 , x2 , . . . , xn a finite
sequence of vectors in X, w1 , w2 , . . . , wn a finite sequence of positive real
numbers. Let also (p, q) be a pair of conjugate exponents. Then for all k ∈
{1, 2, . . . , n} the following inequalities hold:

n
P
1


|k − i| wi · k∆xk∞ ,



Wn i=1


!q
!q ! 1q
n


1 X
k−1
i
n−1
n
P
P
P
P
1
wi xi ≤
xk −
wj +
wi
· k∆xkp ,

Wn i=1

W

n
i=1
j=1
j=i+1
i=k




1


max {Wk−1 , Wn − Wk } · k∆xk1 .
Wn
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
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Proof. By using the discrete analogue of the weighted Montgomery identity
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(2.3) and applying the Hölder inequality we get
n
X
1
wi xi ≤ kDw (k, ·)kq k∆xkp .
xk −
Wn
i=1
We have
1
kDw (k, ·)k1 =
Wn
1
=
Wn
k−1
X
n−1
X
−Wi |Wi | +
i=1
i=k
k−1
X
(k − i) wi +
n−k
X
i=1
!
!
iwk+i
A. Aglić Aljinović and J. Pečarić
i=1
n
1 X
=
|k − i| wi
Wn i=1
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and the first inequality is proved.
Since
kDw (k, ·)kq =
=
1
Wn
1
Wn
k−1
X
|Wi |q +
n−1
X
i=1
i=k
k−1
i
X
X
!q
i=1
j=1
wj
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! 1q
−Wi q
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+
n−1
X
n
X
i=k
j=i+1
!q ! 1q
wi
the second inequality is proved.
Finally, for the third
kDw (k, ·)k∞
A Discrete Euler Identity
1
=
max {Wk−1 , Wn − Wk } ,
Wn
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which completes the proof.
The first and the third inequality from Corollary 3.2 and also the following
corollary was proved by Dragomir in [7].
Corollary 3.3. Let (X, k·k) be a normed linear space, x1 , x2 , . . . , xn a finite
sequence of vectors in X, w1 , w2 , . . . , wn finite sequence of positive real numbers, and also let (p, q) be a pair of conjugate exponents. Then for all k ∈
{1, 2, . . . , n} the following inequalities hold:
 1
n2 −1
n+1 2


· k∆xk∞ ,
+ k− 2

n
4

n

X
1
1
1
(3.2)
xi ≤
q · k∆xk ,
xk −
(S
(k)
+
S
(n
−
k
+
1))
q
q
p
n
n i=1 



 1 max {k − 1, n − k} · k∆xk .
1
n
Proof. If we apply Corollary 3.2 with wi = 1, i = 1, 2, . . . , n, or use the discrete
Montgomery identity (2.5), we have
n
n−1
X
X
1
xi = Dn (k, i) ∆xi xk −
n i=1 i=1
! 1q n−1
! p1
n−1
X
X
≤
|Dn (k, i)|q
k∆xi kp
.
i=1
i=1
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i=1
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Since for q = 1
n−1
X
A Discrete Euler Identity
1
|Dn (k, i)| =
n
2 !
n2 − 1
n+1
+ k−
,
4
2
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the first inequality follows.
For the second let 1 < q < ∞
n−1
X
i=1
1
|Dn (k, i)|q = q
n
k−1
X
i=1
iq +
n−1
X
!
(n − i)q
i=k
1
= q (Sq (k) + Sq (n − k + 1))
n
the second inequality follows.
Finally for q = ∞ and
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
1
max {|D (k, i)|} = max {k − 1, n − k}
1≤i≤n−1
n
Title Page
implies the last inequality.
Contents
Corollary 3.4. Assume that all assumptions from Theorem 3.1 hold. Then the
following inequality holds
!
2l−1
m−1
2l−1−r
X
X
1 X
1
r
wi xi −
∆ xi
xl −
W2l−1 i=1
2l − 1 − r
r=1
i=1
!
2l−2
2l−1−r
X 2l−3
X
X
×
···
Dw (l, i1 ) D2l−2 (i1 , i2 ) · · · D2l−r (ir−1 , ir ) i1 =1 i2 =1
ir =1
2l−2
2l−m
X
X
X 2l−3
m
≤
···
Dw (l, i1 ) D2l−2 (i1 , i2 ) · · · D2l−m (im−1 , ·)
k∆ xkp ;
i1 =1 i2 =1
im−1 =1
q
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it may be regarded as a generalized midpoint inequality since for m = 1 it
reduces to
2l−1
X
1
wi xi xl −
W2l−1 i=1

2l−1
P
1


|l − i| wi · k∆xk∞ ,



W2l−1 i=1



!q
!q ! 1q

l−1
i
2l−2
n
P
P
P
P
1
≤
wj +
wi
· k∆xkp ,


W2l−1 i=1 j=1

j=i+1
i=l




1


max {Wl−1 , W2l−1 − Wl } · k∆xk1 ;
W2l−1
if in addition wi = 1, i = 1, 2, . . . , 2l − 1 it further reduces to

l (l − 1)


· k∆xk∞ ,

2l − 1



2l−1

1
1 X 1
xi ≤
(3.3)
xl −
(2Sq (l)) q · k∆xkp ,
2l − 1 i=1 
2l − 1


 l−1



· k∆xk1 .
2l − 1
Proof. Apply (3.1) with n = 2l − 1 and k = l to get the first inequality.
For the second, taking m = 1, or applying Hölder’s inequality to (2.7), gives
2l−2
2l−1
X
X
1
wi xi = Dw (l, i) ∆xi ≤ kDw (l, ·)kq k∆xkp .
xl −
W2l−1
i=1
i=1
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Now
kDw (l, ·)k1 =
1
l−1
X
W2l−1
i=1
kDw (l, ·)kq =
=
|Wi | +
2l−1
X
!
−Wi i=l
1
l−1
X
W2l−1
i=1
i=l
l−1
i
X
X
!q
1
W2l−1
=
i=1
kDw (l, ·)k∞ =
|Wi |q +
2l−1
X
wj
1
2l−1
X
W2l−1
i=1
!
|l − i| wi ,
! 1q
−Wi q
+
j=1
2l−2
X
n
X
i=l
j=i+1
!q ! 1q
wi
A Discrete Euler Identity
,
1
max {Wl−1 , W2l−1 − Wl }
W2l−1
and the second inequality is proved.
Now if we take wi = 1, i = 1, 2, . . . , 2l − 1, or apply inequality (3.2) with
n = 2l − 1 and k = l,
2l−1
1 X
l (l − 1)
|l − i| =
,
kD2l−1 (l, ·)k1 =
2l − 1 i=1
2l − 1
kD2l−1 (l, ·)kq =
1
2l − 1
2l−1
X
i=1
! 1q
|l − i|q
=
1
1
(2Sq (l)) q ,
2l − 1
1
l−1
max {l − 1, 2l − 1 − l} =
,
2l − 1
2l − 1
and thus the third inequality is proved.
kD2l−1 (l, ·)k∞ =
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Corollary 3.5. Let all the assumptions from Theorem 3.1 hold. Then the following inequality holds:
!
n
m−1
n−r
x + x
X
X
X
1
1
1
n
∆r xi
−
wi xi −
2
Wn i=1
n − r i=1
r=1
!
n−1
n−r
X
X
Dw (1, i1 ) + Dw (n, i1 )
Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir ) ×
···
2
i =1
ir =1
1
n−1
n−m+1
X Dw (1, i1 ) + Dw (n, i1 )
X
≤
·
·
·
D
(i
,
i
)
·
·
·
D
(i
,
·)
n−1
1
2
n−m+1
m−1
2
i =1
i
=1
1
m−1
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
q
× k∆m xkp ;
this may regarded as a generalized trapezoid inequality since for m = 1 it
reduces to
 P
n−1 Wi
1

−
 i=1 Wn 2 · k∆xk∞ ,




n
x + x
 q 1
X
1
Pn−1 Wi
1
n
q
1
wi xi ≤
−
−
· k∆xkp ,
i=1
W
2
n
2

Wn i=1


o

n


 max w1 − 1 , wn − 1 · k∆xk .
1
Wn
2
Wn
2
and if in addition, wi = 1, i = 1, 2, . . . , n it further reduces to
n
x + x
X
1
1
n
(3.4) −
xi 2
n
i=1
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≤












1
n
n−1−
n n 



1
n
n
2
2Sq
2
1q
2
· k∆xk∞ ,
· k∆xkp ,


Sq (n−1)

1



− 2Sq

n
2q−1






 n−2 · k∆xk .
1
2n
n−1
2
1q
if n is even,
· k∆xkp , if n is odd,
Proof. To obtain the first inequality take (2.9) and apply Hölder’s inequality.
For the second we take m = 1 or apply Hölder’s inequality to (2.10),
n
n−1
X
x + x
X
1
D
(1,
i)
+
D
(n,
i)
1
n
w
w
−
wi xi = ∆xi 2
Wn i=1
2
i=1
Dw (1, ·) + Dw (n, ·) k∆xk .
≤
p
2
q
Now
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n−1 n−1 X
Dw (1, ·) + Dw (n, ·) Wi − Wi X
Wi
1
=
=
−
2Wn Wn 2 ,
2
1
i=1
i=1
Dw (1, ·) + Dw (n, ·) =
2
q
q ! 1q
n−1 X
Wi
1
−
,
Wn 2 i=1
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Wi
Dw (1, ·) + Dw (n, ·) 1 = max
− 1≤i≤n−1 Wn
2
2
∞
W1
1 Wn−1 1 = max − ,
− Wn 2 Wn
2
w1
1 wn
1
= max − , − Wn 2 Wn 2
and the second inequality is proved.
Now if we take wi = 1, i = 1, 2, . . . , n, or use (2.11) and apply Hölder’s inequality, we get
n
i
x + x
X
1
1
1
n
xi ≤
−
− k∆xkp .
2
n i=1 n 2 q
For q = 1
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A. Aglić Aljinović and J. Pečarić
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Contents
n−1 j n k j n k
X
i
1
n 1 − = 1
n−1−
;
i − =
n 2
n i=1
2
n
2
2
1
for 1 < q < ∞
i
−
n
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! 1q
n−1 X
1
n q
= 1
i
−
2 q n i=1
2

1q
1
n


,
 n 2Sq 2
=


 1 Sq (n−1)
− 2Sq
q−1
n
2
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if n is even,
n−1
2
1q
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J. Ineq. Pure and Appl. Math. 5(3) Art. 58, 2004
, if n is odd;
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and for q = ∞
i
i
1
n−2
− 1 = max
−
=
.
n 2
1≤i≤n−1
n 2
2n
∞
Remark 3.1. The first inequality from (3.3) was obtained by Dragomir in [7]
and also an incorrect version of the first inequality from (3.4), viz.:
 k−1
n
if n = 2k,
 2 k∆xk∞ ,
x + x
X
1
1
n
xi ≤
−
2
 2k2 +2k+1 k∆xk , if n = 2k + 1.
n
i=1
The second coefficient
1
2k + 1
2k2 +2k+1
2(2k+1)
should be
2k + 1
(2k + 1) − 1 −
2
A. Aglić Aljinović and J. Pečarić
∞
2(2k+1)
k2
2k+1
A Discrete Euler Identity
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since
2k + 1
2
Contents
=
k2
.
2k + 1
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4.
Estimates of the Differences Between Two
Weighted Arithmetic Means
In this section we will give the estimates of the differences between two weighted
arithmetic means using the discrete weighted Montgomery (Euler) identity. We
suppose l, m, n ∈ N. The first method is by subtracting two weighted Montgomery identities. The second is by summing the discrete weighted Montgomery identity. Both methods are possible for both the case 1 ≤ l ≤ m ≤ n,
i.e. [l, m] ⊆ [1, n] and the case 1 ≤ l ≤ n ≤ m, i.e. [1, n] ∩ [l, m] = [l, n].
Theorem 4.1. Let (X, k·k) be a normed linear space, x1 , x2 , . . . , xmax{m,n} a finite sequence of vectors in X, l, m, n ∈ N, w1 , w2 , . . . , wn and ulP
, ul+1 , . . . , um ,
n
two
finite
sequences
of
positive
real
numbers.
Let
also
W
=
i=1 wi , U =
Pm
i=l ui and for k ∈ N

k
 P
wi , 1 ≤ k ≤ n,
Wk =
 i=1
W,
k > n,
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(4.1)

0,
k < l,



 k
P
Uk =
ui l ≤ k ≤ m,


i=l


U,
k > m.
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If [1, n]∩[l, m] 6= ∅, then, for both cases [l, m] ⊆ [1, n] and [1, n]∩[l, m] = [l, n],
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the next formula is valid
(4.2)
max{m,n}
n
m
X
1 X
1 X
ui x i =
wi xi −
K (i) ∆xi ,
W i=1
U i=l
i=1
where
K (i) =
Ui W i
−
, 1 ≤ i ≤ max {m, n} .
U
W
Proof. For k ∈ ([1, n] ∩ [l, m]) ∩ N, we subtract the identities
n
n−1
X
1 X
wi xi +
Dw (k, i) ∆xi ,
xk =
W i=1
i=1
and
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
Title Page
Contents
m
m−1
X
1 X
ui x i +
Du (k, i) ∆xi .
xk =
U i=l
i=l
Then put
K (k, i) = Du (k, i) − Dw (k, i) .
As K (k, i) does not depend on k we write simply K (i):


− Wi ,
1 ≤ i ≤ l − 1,

 W
Ui
i
(4.3)
K (i) =
if [l, m] ⊆ [1, n] ,
−W
, l ≤ i ≤ m,
U
W


 1 − Wi , m + 1 ≤ i ≤ n,
W
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(4.4)


− Wi ,
1 ≤ i ≤ l − 1,

 W
Ui
i
K (i) =
−W
, l ≤ i ≤ n,
U
W


 Ui − 1, n + 1 ≤ i ≤ m,
U
if [1, n] ∩ [l, m] = [l, n] .
Theorem 4.2. Let all assumptions from Theorem 4.1 hold and (p, q) be a pair
of conjugate exponents. Then we have
n
m
1 X
1 X
wi xi −
ui xi ≤ kKkq k∆xkp .
W
U
i=1
i=l
The constant kKkq is sharp for 1 ≤ p ≤ ∞.
Proof. We use the identity (4.2) and apply the Hölder inequality to obtain
max{m,n}
n
m
1 X
X
X
1
wi xi −
ui xi = K (i) ∆xi ≤ kKkq k∆xkp .
W
U i=l
i=1
i=1
For the proof of the sharpness of the constant kKkq , we will find x, a finite
sequence of vectors in X such that

 1q
max{m,n}
max{m,n}
X
X
K (i) ∆xi = 
|K (i)|q  k∆xkp .
i=1
i=1
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For 1 < p < ∞ take x to be such that
1
∆xi = sgn K (i) · |K (i)| p−1 .
For p = ∞ take
∆xi = sgn K (i) .
For p = 1 we will find a finite sequence of vectors x such that


max{m,n}
max{m,n}
X
X
|∆xi | .
K (i) ∆xi =
max
|K (i)| 
i=1
1≤i≤max{m,n}
i=1
Suppose that |K (i)| attains its maximum at i0 ∈ ([1, n] ∪ [l, m]) ∩ N. First we
assume that K (i0 ) > 0. Define x such that ∆xi0 = 1 and ∆xi = 0, i 6= i0 , i.e.
(
0, 1 ≤ i ≤ i0 ,
xi =
1, i0 + 1 < i ≤ max {m, n} .
Then,


max{m,n}
max{m,n}
X
X
K (i) ∆xi = |K (i0 )| =
max
|K (i)| 
|∆xi | ,
1≤i≤max{m,n}
i=1
i=1
and the statement follows. In the case K (i0 ) < 0, we take x such that ∆xi0 =
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−1 and ∆xi = 0, i 6= i0 , i.e.
(
1, 1 ≤ i ≤ i0 ,
xi =
0, i0 + 1 ≤ i ≤ max {m, n} ,
and the rest of proof is the same as above.
Corollary 4.3. Assume all assumptions from the Theorem 4.2 hold and additionally assume 1 ≤ l < m ≤ n. Then we have
n
m
1 X
X
1
wi xi −
ui x i W
U
i=1
i=l
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A. Aglić Aljinović and J. Pečarić
Title Page
 "
#
m n
l−1 X
X

U i Wi Wi X
W

i

+
1 −
k∆xk ,
+
−

∞

U
W 
W
W

i=1
i=m+1
i=l






 " l−1 q
q # 1q
m n
X
X
X Wi U i W i q
W
≤
i
+
1 −
+
−
k∆xkp ,

U
W 
W
W


i=1
i=m+1
i=l







U i Wi Wl−1
Wm+1


k∆xk ,
,1 −
, max
−
 max
1
W
W l≤i≤m U
W
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and for 1 ≤ l < n ≤ m
n
m
1 X
1 X
wi xi −
ui xi W
U i=l
i=1
 "
#
n m l−1 X
X

U i Wi Ui
Wi X


+
− 1 k∆xk ,
+
−

∞

U
W 
W i=n+1 U

i=1
i=l






 " l−1 q
q
q # 1q
n m X
X
X
≤
Wi +
U i − Wi +
Ui − 1
k∆xkp ,

W U
U

W


i=1
i=n+1
i=l







U i Wi W
U

l−1
n+1

k∆xk .
,1 −
, max −
 max
1
W
U l≤i≤n U
W
Proof. Directly from the Theorem 4.2.
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Remark 4.1. If we suppose n = m in both of the cases 1 ≤ l < m ≤ n and
1 ≤ l < n ≤ m,then the analogous results coincides.
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Remark 4.2. By setting l = m = k and uk = 1 in the first inequality from
Corollary 4.3 we get the weighted Ostrowski inequality from Corollary 3.2.
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Corollary 4.4. If all assumptions from Theorem 4.2 hold and, in addition, asJ. Ineq. Pure and Appl. Math. 5(3) Art. 58, 2004
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sume 1 ≤ k ≤ m, then we have
k
m
1 X
1 X
ui xi wi xi −
W
U i=k
i=1
 "
#
m k−1 X
X

Ui
Wi Uk Wk 

+
− 1 k∆xk ,
+
−

∞

W U

W i=k+1 U

i=1






 " k−1 q q
q # 1q
m X
X
≤
Wi + Uk − Wk +
Ui − 1
k∆xkp ,

W U
U

W


i=1
i=k+1







Uk Wk W
U

k−1
k+1

k∆xk .
,1 −
,
−
 max
1
W
U U
W
Proof. By setting n = l = k in the second inequality from the Corollary 4.3.
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
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The second method of giving an estimate of the difference between the two
weighted arithmetic means is by summing the weighted Montgomery identity.
In this way we also get formula (4.2). For the case 1 ≤ l ≤ m ≤ n let
j ∈ {l, l + 1, . . . , m} , from (2.3) we have
uj xj = uj
n
n−1
X
1 X
wi xi + uj
Dw (j, i) ∆xi ,
W i=1
i=1
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so
m
1 X
uj xj =
U j=l
m
1 X
uj
U j=l
!
n
m
n−1
1 X
1 X X
wi xi +
uj
Dw (j, i) ∆xi .
W i=1
U j=l
i=1
By interchange of the order of summation we get
m
n
1 X
1 X
uj x j −
wi xi
U j=l
W i=1
j−1
m
m
n−1 1 X X Wi
1 X X Wi
uj
uj
=
∆xi +
− 1 ∆xi
U j=l
W
U j=l
W
i=1
i=j
l−1 m
m−1 m
1 X X Wi
1 X X Wi
=
uj ∆xi +
uj ∆xi
U i=1 j=l W
U i=l j=i+1 W
m−1 i
n−1 m
1 XX
Wi
1 XX
Wi
uj
uj
+
− 1 ∆xi +
− 1 ∆xi
U i=l j=l
W
U i=m j=l
W
l−1
m−1
X
X
Wi
Ui W i
=
∆xi +
∆xi
1−
W
U W
i=1
i=l
m−1
n−1 X Ui Wi
X
Wi
+
− 1 ∆xi +
− 1 ∆xi
U W
W
i=m
i=l
l−1
m n−1 X
X
X
Wi
Wi Ui
Wi
=
∆xi +
−
∆xi +
− 1 ∆xi .
W
W
U
W
i=1
i=m+1
i=l
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This identity is equivalent to (4.2) with (4.3).
For case 1 ≤ l ≤ n ≤ m, let j ∈ {l, l + 1, . . . , n} , and again from (2.3) we
have
n
n−1
X
1 X
uj xj = uj
wi xi + uj
Dw (j, i) ∆xi ,
W i=1
i=1
so
n
n
n
n
n−1
X
X
X
X
1 X
uj xj =
uj
wi xi +
uj
Dw (j, i) ∆xi
W i=1
i=1
j=l
j=l
j=l
and
A Discrete Euler Identity
A. Aglić Aljinović and J. Pečarić
m
m
1 X
1 X
uj xj −
uj x j
U j=l
U j=n+1
!
m
n
1 X
1 X
uj
wi xi −
=
U j=l
W i=1
Title Page
m
1 X
uj
U j=n+1
+
!
n
1 X
wi xi
W i=1
! n−1
n
X
1 X
uj
Dw (j, i) ∆xi .
U j=l
i=1
Thus
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m
n
1 X
1 X
uj xj −
wi xi
U j=l
W i=1
m
1 X
uj
=
U j=n+1
n
1 X
xj −
wi xi
W i=1
Quit
Page 32 of 38
!
n
n−1
1 X X
+
uj
Dw (j, i) ∆xi .
U j=l
i=1
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Since for j ∈ {n + 1, n + 2, . . . , m}
j−1
n
n
X
1 X
1 X
xj −
wi xi =
∆xi + xn −
wi xi
W i=1
W
i=n
i=1
=
j−1
X
∆xi +
i=n
n−1
X
Dw (n, i) ∆xi ,
i=1
we have
A Discrete Euler Identity
m
n
n
n−1
1 X
1 X
1 X X
uj xj −
wi xi =
uj
Dw (j, i) ∆xi
U j=l
W i=1
U j=l
i=1
m
1 X
+
uj
U j=n+1
n−1
X
Dw (n, i) ∆xi +
i=1
A. Aglić Aljinović and J. Pečarić
j−1
X
!
∆xi .
i=n
By interchange of the order of summation we get
m
n
1 X
1 X
uj xj −
wi xi
U j=l
W i=1
j−1
=
n
n
1 X X Wi
1 X
uj
∆xi +
uj
U j=l
W
U
i=1
j=l
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n−1 X
i=j
Wi
− 1 ∆xi
W
j−1
m
n−1
m
X
X
1 X
1 X
uj
Dw (n, i) ∆xi +
uj
+
∆xi
U j=n+1 i=1
U j=n+1 i=n
Quit
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n
l−1 n
n−1
n−1 i
Wi
1 X X Wi
1 X X Wi
1 XX
uj
=
uj ∆xi +
uj ∆xi +
− 1 ∆xi
U i=1 j=l W
U i=l j=i+1 W
U i=l j=l
W
n−1
m
m−1
m
1 X X
1 X X
+
uj Dw (n, i) ∆xi +
uj ∆xi
U i=1 j=n+1
U i=n+1 j=i+1
l−1
n−1 n−1
X
X
Ui Wi
Un Ui Wi
Un X Wi
∆xi +
−
∆xi +
− 1 ∆xi
=
U i=1 W
U
U
W
U
W
i=l
i=l
X
n−1
m−1
X
Wi
Ui
Un
+ 1−
∆xi +
1−
∆xi
U i=1 W
U
i=n+1
n m−1
l−1
X
X
X Wi
W i Ui
Ui
=
∆xi +
−
∆xi +
1−
∆xi .
W
W
U
U
i=1
i=n+1
i=l
This identity is equivalent to (4.2) with (4.4)
The next theorem is the generalization of Theorem 4.2.
Theorem 4.5. Let (X, k·k) be a normed linear space, x1 , x2 , . . . , xmax{m,n} a
finite sequence of vectors in X, w1 , w2 , . . . , wn and ul , ul+1 , . . . , um finite sequences of positive real numbers and (p, q) a pair of conjugate exponents. Then
for all s ∈ {2, 3, . . . , n − 1} and k ∈ ([1, n] ∩ [l, m]) ∩ N the following inequalityis valid
n
m
1 X
1 X
wi xi −
wi xi
W
U
i=1
i=l
!
Pn−r r n−1
s−1
n−r
X
X
X
∆
x
i
i=1
···
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
+
n
−
r
r=1
i =1
i =1
1
r
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−
s−1
X
Pm−r
r
i=l ∆ xi
m−r
r=1
m−1
X
×
i1 =l
···
m−r
X
ir =l
!
Du (k, i1 ) Dm−l (i1 , i2 ) · · · Dm−l−r+2 (ir−1 , ir ) ≤ kK (k, ·)kq k∆s xkp ,
where
K (k, im ) =
n−1 X
n−2
X
···
i1 =1 i2 =1
−
m−1
X m−2
X
i1 =l i2 =l
···
n−s+1
X
A Discrete Euler Identity
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−s+1 (is−1 , is )
A. Aglić Aljinović and J. Pečarić
Du (k, i1 ) Dm−l (i1 , i2 ) · · · Dm−l−s+2 (is−1 , is )
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is−1 =1
m−s+1
X
Contents
is−1 =l
and we suppose that
n−1 X
n−2
X
···
i1 =1 i2 =1
n−s+1
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−s+1 (is−1 , is ) = 0,
and
i1 =l i2 =l
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is−1 =1
for is ∈
/ [1, n] ∩ N
m−1
X m−2
X
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···
m−s+1
X
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Du (k, i1 ) Dm−l (i1 , i2 ) · · · Dm−l−s+2 (is−1 , is ) = 0,
is−1 =l
J. Ineq. Pure and Appl. Math. 5(3) Art. 58, 2004
for is ∈
/ [l, m] ∩ N.
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The constant kK (k, ·)kq is sharp for 1 ≤ p ≤ ∞
Proof. As in Theorem 4.2, we subtract two weighted Montgomery identities,
one for the interval [1, n] ∩ N and the other for [l, m] ∩ N. After that, our inequality follows by applying Hölder’s inequality. The proof for the sharpness
of the constant kK (k, ·)kq is similar to the proof of Theorem 4.2 (with K (k, ·)
instead of K and ∆s x instead of ∆x).
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References
[1] A. AGLIĆ ALJINOVIĆ AND J. PEČARIĆ, The weighted Euler identity,
Math. Inequal. & Appl., (accepted).
[2] A. AGLIĆ ALJINOVIĆ, J. PEČARIĆ AND I. PERIĆ, Estimations of the
difference of two weighted integral means via weighted Montgomery identity, Math. Inequal. & Appl., (accepted).
[3] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR AND A.M. FINK, Comparing two integral means for absolutely continuous mappings whose
derivatives are in L∞ [a, b] and applications, Computers and Math. with
Appl., 44 (2002), 241–251.
[4] P. CERONE AND S.S. DRAGOMIR, Diferences between means with
bounds from a Riemann-Stieltjes integral, RGMIA Res. Rep. Coll., 4(2)
(2001).
[5] P. CERONE AND S.S. DRAGOMIR, On some inequalities arising from
Montgomery’s identity, RGMIA Res. Rep. Coll., 3(2) (2000).
[6] Lj. DEDIĆ, M. MATIĆ AND J. PEČARIĆ, On generalizations of Ostrowski inequality via some Euler-type identities, Math. Inequal. & Appl.,
3(3) (2000), 337–353.
[7] S.S. DRAGOMIR, The discrete version of Ostrowski’s inequality in
normed linear spaces, J. Inequal. in Pure and Appl. Math., 3(1) (2002),
Art. 2.
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[8] R.L. GRAHAM, D.E. KNUTH AND O. PATASHNIK, Concrete Mathematics: A Foundation for Computer Science, Second edition, AddisonWesley 1994.
[9] D.S. MITRINOVIĆ AND J.E. PEČARIĆ, Monotone funkcije i nijhove nejednakosti, Naučna kniga, Beograd 1990.
[10] A. OSTROWSKI, Über die Absolutabweichung einer differentiebaren
Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938),
226–227.
A Discrete Euler Identity
[11] J. PEČARIĆ, On the Čebišev inequality, Bul. Inst. Politehn. Timisoara,
25(39) (1980), 10–11.
A. Aglić Aljinović and J. Pečarić
[12] J. PEČARIĆ, I. PERIĆ AND A. VUKELIĆ, Estimations of the difference
of two integral means via Euler-type identities, Math. Inequal. & Appl.,
7(6) (2002), 787–805.
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