Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 3, Article 58, 2004
A DISCRETE EULER IDENTITY
A. AGLIĆ ALJINOVIĆ AND J. PEČARIĆ
D EPARTMENT OF A PPLIED M ATHEMATICS
FACULTY OF E LECTRICAL E NGINEERING AND C OMPUTING
U NSKA 3, 10 000 Z AGREB , C ROATIA .
andrea@zpm.fer.hr
FACULTY OF T EXTILE T ECHNOLOGY
U NIVERSITY OF Z AGREB
P IEROTTIJEVA 6, 10000 Z AGREB
C ROATIA .
pecaric@mahazu.hazu.hr
Received 09 January, 2004; accepted 22 April, 2004
Communicated by P.S. Bullen
A BSTRACT. 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.
Key words and phrases: Discrete Montgomery identity, Discrete Ostrowski inequality.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
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] →
Rb
[0, ∞) some probability density function, i.e. integrable function satisfying a w (t) dt = 1;
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
086-04
2
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
Rt
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
(
Pw (x, t) =
a ≤ t ≤ x,
W (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 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].
2. D ISCRETE W EIGHTED E ULER I DENTITY
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,
Wk =
k
X
wi , Wk =
i=1
n
X
wi = Wn − Wk ,
i=k+1
then we have, see [9],
(2.1)
n
X
i=1
wi xi = xk Wn +
k−1
X
Wi (xi − xi+1 ) +
i=1
n−1
X
Wi (xi+1 − xi ) , 1 ≤ k ≤ n.
i=k
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)
n
n−1
X
1 X
xk =
wi xi +
Dw (k, i) ∆xi ,
Wn i=1
i=1
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
xk =
xi +
Dn (k, i) ∆xi ,
n i=1
i=1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
where
(
Dn (k, i) =
i
,
n
i
n
3
1 ≤ i ≤ k − 1,
− 1, k ≤ i ≤ n.
n
If 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).
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
X
X
1
1
(2.6) xk =
wi xi +
∆r 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
ir =1
···
i1 =1 i2 =1
n−m
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im ) ∆m xim .
im =1
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
1 X
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
n−2
X
1 X
∆xi =
∆xi +
Dn−1 (i, j) ∆2 xj
n − 1 i=1
j=1
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 X
n−2
X
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
!
i=1
Dw (k, i) Dn−1 (i, j) ∆2 xj .
i=1 j=1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
4
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
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
and using the induction hypothesis, we get
n
m−1
X 1
1 X
xk =
wi xi +
Wn i=1
n−r
r=1
n−1 X
n−2
X
×
n−r
X
+
···
i1 =1 i2 =1
n−m
X
···
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
ir =1
n−m
X
1
n−m
×
∆m xi +
i=1
n
m
X
1
1 X
wi xi +
Wn i=1
n−r
r=1
n−1 X
n−2
X
n−m−1
X

Dn−m (im , im+1 ) ∆m+1 xim+1 
im+1 =1
n−r
X
!
∆r xi
i=1
···
i1 =1 i2 =1
+
!
im =1
×
n−2
n−1 X
X
n−r
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im )

=
∆r xi
i=1
i1 =1 i2 =1
n−1 X
n−2
X
!
n−r
X
!
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
ir =1
n−(m+1)
X
···
i1 =1 i2 =1
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m (im , im+1 ) ∆m+1 xim+1 .
im+1 =1
We see that (2.6) is valid for m + 1 and our assertion is proved.
Remark 2.2. For m = n − 1 (2.6) becomes
n
n−2
X
1 X
1
xk =
wi xi +
Wn 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
!
∆r xi
i=1
n−r
X
!
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
ir =1
···
1
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · D2 (in−2 , in−1 ) ∆n−1 xin−1 .
in−1 =1
Corollary 2.3. 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
xi +
xk =
n i=1
n−r
r=1
n−r
X
!
∆r xi
i=1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
×
n−1 X
n−2
X
···
i1 =1 i2 =1
+
5
!
n−r
X
Dn (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir )
ir =1
n−1 X
n−2
X
···
i1 =1 i2 =1
n−m
X
Dn (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im ) ∆m xim .
im =1
Proof. Apply Theorem 2.1 with wi = 1, i = 1, . . . , n.
Remark 2.4. If we apply (2.6) with n = 2l − 1 and k = l we get
!
2l−1
m−1
2l−1−r
X
X
1
1 X
∆r xi
xl =
wi xi +
W2l−1 i=1
2l
−
1
−
r
r=1
i=1
×
2l−2
X 2l−3
X
···
2l−1−r
X
i1 =1 i2 =1
+
2l−2
X 2l−3
X
!
Dw (l, i1 ) D2l−2 (i1 , i2 ) · · · D2l−r (ir−1 , ir )
ir =1
···
i1 =1 i2 =1
2l−1−m
X
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
and further for wi = 1, i = 1, 2, . . . , 2l − 1 to
2l−1
l−1
1 X
1 X
xl =
xi +
i (∆xi − ∆x2l−1−i ) .
2l − 1 i=1
2l − 1 i=1
(2.8)
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
n−r
X 1
X
x1 + xn
1 X
(2.9)
=
wi xi +
∆r xi
2
Wn i=1
n
−
r
r=1
i=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
i =1
r
1
+
n−1
X
i1 =1
···
n−m
X
im
Dw (1, i1 ) + Dw (n, i1 )
Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , im ) ∆m xim .
2
=1
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
x1 + xn
1X
1 X
n
=
xi +
i−
∆xi .
2
n i=1
n i=1
2
(2.8) and (2.11) were obtained by Dragomir in [7].
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
6
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
3. D ISCRETE O STROWSKI T YPE I NEQUALITIES
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 ) i =1 i =1
ir =1
1 2
n−1 n−2
n−m+1
X
X X
m
≤
···
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−m+1 (im−1 , ·)
k∆ xkp ,
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
1
That is: 1 < p, q < ∞ ,
1
p
+
1
q
=1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
7
Proof. By using the discrete analogue of the weighted Montgomery identity (2.3) and applying
the Hölder inequality we get
n
1 X
wi xi ≤ kDw (k, ·)kq k∆xkp .
xk −
Wn
i=1
We have
1
kDw (k, ·)k1 =
Wn
k−1
X
n−1
X
−Wi |Wi | +
i=1
i=k
k−1
X
1
=
Wn
(k − i) wi +
n−k
X
i=1
!
!
iwk+i
i=1
n
1 X
|k − i| wi
=
Wn i=1
and the first inequality is proved.
Since
kDw (k, ·)kq =
1
Wn
1
=
Wn
k−1
X
n−1
X
q
−Wi q
|Wi | +
i=1
i=k
k−1
i
X
X
!q
i=1
j=1
wj
+
! 1q
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∞ =
1
max {Wk−1 , Wn − Wk } ,
Wn
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:
 2
1
n −1
n+1 2

+
k
−
· k∆xk∞ ,

n
4
2

n

X
1
1
1
(3.2)
xi ≤
xk −
(Sq (k) + Sq (n − k + 1)) q · k∆xkp ,
n
n i=1 


 1
max {k − 1, n − k} · k∆xk1 .
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−1
n
1X X
xi = Dn (k, i) ∆xi xk −
n i=1 i=1
! p1
! 1q n−1
n−1
X
X
p
q
k∆xi k
.
≤
|Dn (k, i)|
i=1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
i=1
http://jipam.vu.edu.au/
8
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
Since for q = 1
n−1
X
i=1
1
|Dn (k, i)| =
n
2 !
n2 − 1
n+1
+ k−
,
4
2
the first inequality follows.
For the second let 1 < q < ∞
n−1
X
i=1
1
|Dn (k, i)| = q
n
q
=
k−1
X
i=1
iq +
n−1
X
!
q
(n − i)
i=k
1
(Sq (k) + Sq (n − k + 1))
nq
the second inequality follows.
Finally for q = ∞ and
max {|D (k, i)|} =
1≤i≤n−1
1
max {k − 1, n − k}
n
implies the last inequality.
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
X
1
1
wi xi −
∆r xi
xl −
W2l−1 i=1
2l − 1 − r
r=1
i=1
!
2l−2
2l−3
2l−1−r
XX
X
×
···
Dw (l, i1 ) D2l−2 (i1 , i2 ) · · · D2l−r (ir−1 , ir ) i1 =1 i2 =1
ir =1
2l−2
2l−3
2l−m
X
X
X
m
≤
···
Dw (l, i1 ) D2l−2 (i1 , i2 ) · · · D2l−m (im−1 , ·)
k∆ xkp ;
i1 =1 i2 =1
im−1 =1
q
it may be regarded as a generalized midpoint inequality since for m = 1 it reduces to

2l−1
P
1


|l − i| wi · k∆xk∞ ,



W2l−1 i=1


!q
!q ! 1q
2l−1


X
1
l−1
i
2l−2
n
P
P
P
P
1
wi xi ≤
xl −
wj +
wi
· k∆xkp ,

W2l−1 i=1

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
(3.3)
xi ≤
xl −
(2Sq (l)) q · k∆xkp ,
2l − 1 i=1 
2l − 1




l−1


· k∆xk1 .
2l − 1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
9
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−1
2l−2
X
X
1
wi xi = Dw (l, i) ∆xi ≤ kDw (l, ·)kq k∆xkp .
xl −
W2l−1
i=1
i=1
Now
kDw (l, ·)k1 =
1
l−1
X
W2l−1
i=1
kDw (l, ·)kq =
=
|Wi | +
2l−1
X
!
−Wi 1
l−1
X
W2l−1
i=1
i=l
l−1
i
X
X
!q
1
W2l−1
kDw (l, ·)k∞ =
=
i=l
|Wi |q +
i=1
2l−1
X
wj
W2l−1
2l−1
X
W2l−1
i=1
!
|l − i| wi ,
! 1q
−Wi q
+
j=1
1
1
2l−2
X
n
X
i=l
j=i+1
!q ! 1q
wi
,
max {Wl−1 , W2l−1 − Wl }
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
kD2l−1 (l, ·)k1 =
kD2l−1 (l, ·)kq =
1
2l − 1
kD2l−1 (l, ·)k∞ =
1 X
l (l − 1)
|l − i| =
,
2l − 1 i=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.
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
−
wi xi −
∆r xi
2
Wn i=1
n
−
r
r=1
i=1
!
n−1
n−r
X
X
Dw (1, i1 ) + Dw (n, i1 )
×
···
Dn−1 (i1 , i2 ) · · · Dn−r+1 (ir−1 , ir ) 2
i1 =1
ir =1
n−1
n−m+1
X Dw (1, i1 ) + Dw (n, i1 )
X
k∆m xk ;
D
(i
,
i
)
·
·
·
D
(i
,
·)
≤
·
·
·
n−1
1
2
n−m+1
m−1
p
2
i =1
i
=1
1
m−1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
q
http://jipam.vu.edu.au/
10
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
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  1
n − 1 − n2
· k∆xk∞ ,

n
2




  1
1




2Sq n2 q · k∆xkp ,
if n is even,

n

x + x

n
1X 1
n
(3.4) −
xi ≤
1q
2
 
n i=1 
Sq (n−1)

1
n−1


− 2Sq 2
· k∆xkp , if n is odd,

n
2q−1





 n−2
· k∆xk1 .
2n
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
n−1 n−1 X
Dw (1, ·) + Dw (n, ·) Wi − Wi X
Wi
1
=
=
− ,
2
2Wn
Wn 2
1
i=1
i=1
Dw (1, ·) + Dw (n, ·) =
2
q
q ! 1q
n−1 X
Wi
1
,
Wn − 2 i=1
Dw (1, ·) + Dw (n, ·) Wi
1 = max
− 1≤i≤n−1 Wn
2
2
∞
Wn−1 1 W1
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
x + x
i
X
1
1
1
n
−
xi ≤
− k∆xkp .
2
n i=1 n 2 q
For q = 1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
11
n−1 j n k j n k
X
i
n 1 − 1 = 1
i
−
=
n
−
1
−
;
n 2
n i=1
2
n
2
2
1
for 1 < q < ∞
!1
n−1 q q
X
i
1
1
n
− =
i − n 2
n i=1
2
q

1q
n
1


,
 n 2Sq 2
=


 1 Sq (n−1)
− 2Sq
q−1
n
2
if n is even,
n−1
2
1q
, if n is odd;
and for q = ∞
i
i
1
n−2
− 1 = max
−
=
.
n 2
1≤i≤n−1
n 2
2n
∞
Remark 3.6. 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,
x + x
 2 k∆xk∞ ,
1X 1
n
−
xi ≤
2
 2k2 +2k+1 k∆xk , if n = 2k + 1.
n
i=1
The second coefficient
2k2 +2k+1
2(2k+1)
∞
k2
should be 2k+1 since
2k + 1
2k + 1
k2
1
(2k + 1) − 1 −
=
.
2k + 1
2
2
2k + 1
2(2k+1)
4. E STIMATES OF THE D IFFERENCES B ETWEEN T WO W EIGHTED A RITHMETIC
M EANS
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, w1P
, w2 , . . . , wn and
l , ul+1 , . . . , um , two finite sequences of positive
Pu
n
m
real numbers. Let also W = i=1 wi , U = i=l ui and for k ∈ N

k
 P
wi , 1 ≤ k ≤ n,
Wk =
i=1

W,
k > n,
(4.1)

0,
k < l,



 k
P
Uk =
ui l ≤ k ≤ m,


i=l


U,
k > m.
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
12
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
If [1, n] ∩ [l, m] 6= ∅, then, for both cases [l, m] ⊆ [1, n] and [1, n] ∩ [l, m] = [l, n], the next
formula is valid
max{m,n}
n
m
X
1 X
1 X
wi xi −
ui xi =
K (i) ∆xi ,
W i=1
U i=l
i=1
(4.2)
where
K (i) =
Ui Wi
−
, 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
xk =
wi xi +
Dw (k, i) ∆xi ,
W i=1
i=1
and
m
m−1
X
1 X
xk =
ui x i +
Du (k, i) ∆xi .
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) =
−W
, l ≤ i ≤ m,
U
W


 1 − Wi , m + 1 ≤ i ≤ n,
if [l, m] ⊆ [1, n] ,
W
(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=l
i=1
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=1
i=l
i=1
For the proof of the sharpness of the constant kKkq , we will find x, a finite sequence of vectors
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
13
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
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
K (i) ∆xi =
max
|K (i)| 
|∆xi | .
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 = −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 xi W
U
i=1
 " i=l #
l−1
m n
X
X

Wi X
U i Wi W


+
+
1 − i k∆xk ,

−

∞

W
U
W
W

i=1
i=m+1

i=l





 " l−1 q
q # 1q
m n
X Wi X
X
U i W i q
W
≤
i
+
+
1 −
−
k∆xkp ,

W U

W
W


i=1
i=m+1
i=l






U i Wi 
W
W
l−1
m+1

k∆xk ,

,1 −
, max −
 max
1
W
W l≤i≤m U
W
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
14
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
and for 1 ≤ l < n ≤ m
n
m
1 X
1 X
ui xi wi xi −
W
U
i=1
 "i=l #
n m l−1
X
X

U i Wi Ui
Wi X


+
− 1 k∆xk ,
+

−

∞

W
U
W
U

i=n+1
i=1

i=l





 " l−1 q
q
q # 1q
n m X
X
X
W
U
i
U
W
≤
i
i +
i − 1
−
+
k∆xkp ,


W
U
W
U


i=1
i=n+1
i=l






U i Wi 
Wl−1
Un+1

k∆xk .

,1 −
, max
−
 max
1
W
U l≤i≤n U
W
Proof. Directly from the Theorem 4.2.
Remark 4.4. If we suppose n = m in both of the cases 1 ≤ l < m ≤ n and 1 ≤ l < n ≤
m,then the analogous results coincides.
Remark 4.5. 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.
Corollary 4.6. If all assumptions from Theorem 4.2 hold and, in addition, assume 1 ≤ k ≤ m,
then we have
k
m
1 X
X
1
w
x
−
u
x
i i
i i
W
U
i=1
i=k

" k−1 #
m X Wi Uk Wk X

Ui


+
+
− 1 k∆xk ,

−

∞
W U

W i=k+1 U

i=1






 " k−1 q q
q # 1q
m X
X
≤
W i + Uk − W k +
Ui − 1
k∆xkp ,

W U
U

W


i=1
i=k+1






Uk W k 
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.
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 x j = uj
n
n−1
X
1 X
wi xi + uj
Dw (j, i) ∆xi ,
W i=1
i=1
so
m
1 X
uj xj =
U j=l
m
1 X
uj
U j=l
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
!
n
m
n−1
1 X
1 X X
wi xi +
uj
Dw (j, i) ∆xi .
W i=1
U j=l
i=1
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
15
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
m
m
n−1 1 X X Wi
1 X X Wi
=
uj
∆xi +
uj
− 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
Wi
1 XX
1 XX
Wi
uj
− 1 ∆xi +
uj
− 1 ∆xi
+
U i=l j=l
W
U i=m j=l
W
l−1
X
Wi
m−1
X
Ui W i
=
∆xi +
1−
∆xi
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
m
n−1 l−1
X Wi
X Wi Ui
X
Wi
=
∆xi +
−
∆xi +
− 1 ∆xi .
W
W
U
W
i=1
i=m+1
i=l
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
wi xi + uj
Dw (j, i) ∆xi ,
uj x j = uj
W i=1
i=1
so
n
X
uj xj =
j=l
n
X
j=l
n
n
n−1
X
X
1 X
uj
wi xi +
uj
Dw (j, i) ∆xi
W i=1
i=1
j=l
and
m
m
1 X
1 X
uj xj −
uj xj
U j=l
U j=n+1
=
m
1 X
uj
U j=l
!
n
1 X
wi xi −
W i=1
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
m
n
1 X
1 X
uj x j −
wi xi
U j=l
W i=1
m
1 X
=
uj
U j=n+1
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
n
1 X
xj −
wi xi
W i=1
!
+
n
n−1
1 X X
uj
Dw (j, i) ∆xi .
U j=l
i=1
http://jipam.vu.edu.au/
16
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
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=1
i=n
=
j−1
X
∆xi +
i=n
n−1
X
Dw (n, i) ∆xi ,
i=1
we have
m
n
1 X
1 X
uj xj −
wi xi
U j=l
W i=1
n
n−1
m
1 X X
1 X
=
uj
Dw (j, i) ∆xi +
uj
U j=l
U j=n+1
i=1
n−1
X
Dw (n, i) ∆xi +
i=1
j−1
X
!
∆xi .
i=n
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
n
n
n−1 1 X X Wi
1 X X Wi
=
uj
∆xi +
uj
− 1 ∆xi
U j=l
W
U
W
i=1
i=j
j=l
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
l−1 n
n−1
n
n−1 i
1 X X Wi
1 X X Wi
1 XX
Wi
uj ∆xi +
uj ∆xi +
uj
− 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
Un X Wi
Un Ui Wi
Ui Wi
=
∆xi +
−
∆xi +
− 1 ∆xi
U i=1 W
U
U W
U W
i=l
i=l
n−1
m−1
X Un X Wi
Ui
+ 1−
∆xi +
1−
∆xi
U i=1 W
U
i=n+1
l−1
n
m−1
X Wi
X W i Ui
X 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 4.7. 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 ∈
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
A D ISCRETE E ULER IDENTITY
17
([1, n] ∩ [l, m]) ∩ N the following inequality is valid
n
m
1 X
1 X
wi xi −
wi xi
W
U i=l
i=1
!
n−1
n−r
s−1 Pn−r
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
i1 =1
ir =1
!
Pm−r r m−1
m−r
s−1
X
X
X
∆
x
i
i=l
−
···
Du (k, i1 ) Dm−l (i1 , i2 ) · · · Dm−l−r+2 (ir−1 , ir ) m−r
r=1
i =l
i =l
r
1
≤ kK (k, ·)kq k∆s xkp ,
where
n−1 X
n−2
X
K (k, im ) =
i1 =1 i2 =1
···
n−s+1
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−s+1 (is−1 , is )
is−1 =1
−
m−1
X m−2
X
i1 =l i2 =l
···
m−s+1
X
Du (k, i1 ) Dm−l (i1 , i2 ) · · · Dm−l−s+2 (is−1 , is )
is−1 =l
and we suppose that
n−1 X
n−2
X
···
i1 =1 i2 =1
and
m−1
X
X m−2
i1 =l i2 =l
···
n−s+1
X
Dw (k, i1 ) Dn−1 (i1 , i2 ) · · · Dn−s+1 (is−1 , is ) = 0,
for is ∈
/ [1, n] ∩ N
is−1 =1
m−s+1
X
Du (k, i1 ) Dm−l (i1 , i2 ) · · · Dm−l−s+2 (is−1 , is ) = 0,
for is ∈
/ [l, m] ∩ N.
is−1 =l
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).
R EFERENCES
[1] A. AGLIĆ ALJINOVIĆ
(accepted).
AND
J. PEČARIĆ, The weighted Euler identity, Math. Inequal. & Appl.,
[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 RiemannStieltjes 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).
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/
18
A. AGLI Ć A LJINOVI Ć AND J. P E ČARI Ć
[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.
[8] R.L. GRAHAM, D.E. KNUTH AND O. PATASHNIK, Concrete Mathematics: A Foundation for
Computer Science, Second edition, Addison-Wesley 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.
[11] J. PEČARIĆ, On the Čebišev inequality, Bul. Inst. Politehn. Timisoara, 25(39) (1980), 10–11.
[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.
J. Inequal. Pure and Appl. Math., 5(3) Art. 58, 2004
http://jipam.vu.edu.au/