General Mathematics Vol. 13, No. 2 (2005), 51–64
Grüss Type Inequalities for Forward
Difference of Vectors in Inner Product
Spaces
Sever S. Dragomir
Dedicated to Professor Dumitru Acu on the occasion of his 60th birthday
Abstract
Some Grüss type inequalities for two sequences of vectors in terms
of the forward difference are given. An application for the Jensen
inequality for convex functions defined on inner product spaces is
also pointed out.
2000 Mathematics Subject Classification: 26D15, 47A05, 46A05.
Keywords and Phrases: Grüss type inequalities, Inner product spaces.
1
Introduction
In [1], we have proved the following generalisation of the Grüss inequality.
51
52
Sever S. Dragomir
Theorem 1. Let (H, h·, ·i) be an inner product space over K, K = C, R and
e ∈ H, kek = 1. If φ, Φ, γ, Γ ∈ K and x, y ∈ H are such that
Re hΦe − x, x − φei ≥ 0 and
Re hΓe − y, y − γei ≥ 0,
hold, then we have the inequality
|hx, yi − hx, ei he, yi| ≤
The constant
1
4
1
|Φ − φ| |Γ − γ| .
4
is the best possible.
A Grüss type inequality for sequences of vectors in inner product spaces
was pointed out in [2].
Theorem 2. Let H and K be as in Theorem 1 and xi ∈ H, ai ∈ K, pi ≥ 0
n
P
(i = 1, . . . , n) (n ≥ 2) with
pi = 1. If a, A ∈ K and x, X ∈ H are such
i=1
that:
Re [(A − ai ) (ai − a)] ≥ 0,
Re hX − xi , xi − xi ≥ 0
for any i ∈ {1, . . . , n} , then we have the inequality
n
n
n
X
1
X
X
0≤
pi ai xi −
pi ai ·
pi xi ≤ |A − a| kX − xk .
4
i=1
The constant
1
4
i=1
i=1
is best possible.
A complementary result for two sequences of vectors in inner product
spaces is the following result that has been obtained in [3].
Theorem 3. Let H and K be as above, xi , yi ∈ H, pi ≥ 0 (i = 1, . . . , n)
n
P
(n ≥ 2) with
pi = 1. If x, X, y, Y ∈ H are such that:
i=1
Re hX − xi , xi − xi ≥ 0
and
Re hY − yi , yi − yi ≥ 0
Grüss Type Inequalities for Forward Difference of Vectors...
53
for all i ∈ {1, . . . , n} , then we have the inequality
n
+
* n
n
1
X
X
X
0≤
pi hxi , yi i −
p i xi ,
pi yi ≤ kX − xk kY − yk .
4
i=1
i=1
The constant
1
4
i=1
is best possible.
In the general case of normed linear spaces, the following Grüss type
inequality in terms of the forward difference is known, see [4].
Theorem 4. Let (E, k·k) be a normed linear space over K = C, R, xi ∈ E,
n
P
αi ∈ K and pi ≥ 0 (i = 1, . . . , n) such that
pi = 1. Then we have the
i=1
inequality
(1)
n
n
n
X
X
X
0≤
pi αi xi −
pi αi ·
pi xi ≤
i=1
i=1
i=1

!2 
n
n
X
X
≤ max |∆αj | max k∆xj k 
i2 pi −
ipi  ,
1≤j≤n−1
1≤j≤n−1
i=1
i=1
where ∆αj = αj+1 − αj and ∆xj = xj+1 − xj (j = 1, . . . , n − 1) are the forward differences of the vectors having the components αj and xj (j = 1, . . . , n − 1) ,
respectively.
The inequality (1) is sharp in the sense that the multiplicative constant
C = 1 in the right hand side cannot be replaced by a smaller one.
An important particular case is the one where all the weights are equal,
giving the following corollary [4].
Corollary 1. Under the above assumptions for αi , xi (i = 1, . . . , n) we have
the inequality
(2)
n
n
n
1 X
1X
1X 0≤
α i xi −
αi ·
xi ≤
n
n i=1
n i=1 i=1
54
Sever S. Dragomir
≤
The constant
1
12
n2 − 1
max |∆αj | max k∆xj k .
1≤j≤n−1
12 1≤j≤n−1
is best possible.
Another result of this type was proved in [6].
Theorem 5. With the assumptions of Theorem 4, one has the inequality
n
n
n
X
X
X
(3)
pi αi xi −
pi αi ·
pi xi ≤
0≤
i=1
i=1
i=1
n−1
n−1
n
X
X
1X
≤
|∆αj |
k∆xj k
pi (1 − pi ) .
2 j=1
j=1
i=1
The constant
1
2
is best possible.
As a useful particular case, we have the following corollary [6].
Corollary 2. If αi , xi (i = 1, . . . , n) are as in Theorem 4, then
n
n
n
1 X
1X
1X 0≤
αi xi −
αi ·
xi ≤
n
n i=1
n i=1 i=1
1
≤
2
1
1−
n
X
n−1
|∆αi |
i=1
n−1
X
k∆xi k .
i=1
The constant 12 is the best possible.
Finally, the following result is also known [5].
Theorem 6. With the assumptions in Theorem 4, we have the inequality:
n
n
n
X
X
X
(4)
pi xi ≤
pi αi ·
pi αi xi −
0≤
i=1
i=1
i=1
Grüss Type Inequalities for Forward Difference of Vectors...
n−1
X
≤
! p1
|∆αj |p
n−1
X
j=1
where p > 1,
1
p
+
1
q
! 1q
X
k∆xj kq
j=1
55
(j − i) pi pj ,
1≤i<j≤n
= 1.
The constant c = 1 in the right hand side of (4) is sharp.
The case of equal weights is embodied in the following corollary [5].
Corollary 3. With the above assumptions for αi , xi (i = 1, . . . , n) one has
n
n
n
1 X
X
X
1
1
0≤
α i xi −
αi ·
xi ≤
n
n
n
i=1
i=1
i=1
n2 − 1
≤
6n
where p > 1,
1
p
+
The constant
n−1
X
! p1
|∆αj |p
j=1
n−1
X
! 1q
k∆xj kq
,
j=1
1
q
= 1.
1
6
is the best possible.
The main aim of this section is to establish some similar bounds for the
absolute value of the difference
n
X
*
pi hxi , yi i −
i=1
n
X
i=1
p i xi ,
n
X
+
pi yi
i=1
provided that xi , yi (i = 1, . . . , n) are vectors in an inner product space H,
n
P
pi = 1.
and pi ≥ 0 (i = 1, . . . , n) with
i=1
2
The Main Results
We assume that (H, h·, ·i) is an inner product space over K, K = C or
K = R. The following discrete inequality of Grüss type holds.
56
Sever S. Dragomir
Theorem 7. If xi , yi ∈ H, pi ≥ 0 (i = 1, . . . , n) with
n
P
pi = 1, then one
i=1
has the inequalities:
* n
+
n
n
X
X
X
(5)
pi hxi , yi i −
pi xi ,
pi yi ≤
i=1
i=1
i=1
#
 "
2
n
n

P
P

2

ipi
max k∆xk k max k∆yk k ;
i pi −


k=1,...,n−1
k=1,...,n−1

i=1
i=1







 "
#

p1 n−1
1q

n−1

P
P
P

p
q
pi pj (i − j)
k∆xk k
k∆yk k
≤
1≤j<i≤n
k=1
k=1





if p > 1, p1 + 1q = 1;








n−1
n

n−1

P
P
1 P


pi (1 − pi )
k∆xk k
k∆yk k .

2 i=1
k=1
k=1
All the inequalities in (5) are sharp.
The following particular case for equal vectors holds.
Corollary 4. With the assumptions of Theorem 7, one has the inequalities
2
n
n
X
X
2
0≤
pi kxi k − p i xi ≤
i=1
i=1
#
 "
n
2
n

P
P

2

ipi
i pi −
max k∆xk k2 ;


k=1,...,n−1

i=1
i=1









1q
n−1
p1 n−1
 P

P
P

q
p
k∆xk k
pi pj (i − j)
k∆xk k
≤
1≤j<i≤n
k=1
k=1




if p > 1, p1 + 1q = 1;









n−1
2

n

P
1P



pi (1 − pi )
k∆xk k .
2 i=1
k=1
Grüss Type Inequalities for Forward Difference of Vectors...
57
The following particular case for equal weights may be useful in practice.
Corollary 5. If xi , yi ∈ H (i = 1, . . . , n), then one has the inequalities:
* n
+
n
n
1 X
X
X
1
1
hxi , yi i −
xi ,
yi ≤
n
n
n
i=1
i=1
i=1
 2
n −1


max k∆xk k max k∆yk k ;


k=1,...,n−1
k=1,...,n−1
12








p1 n−1
n−1
1q

2

P
P
n
−
1

p
q

k∆xk k
k∆yk k
6n
≤
k=1
k=1



if p > 1, p1 + 1q = 1;










n−1
P
P

n − 1 n−1


k∆xk k
k∆yk k .
2n k=1
k=1
The constants
1
, 1
12 6
and
1
2
are best possible.
In particular, the following corollary holds.
Corollary 6. If xi ∈ H (i = 1, . . . , n) , then one has the inequality
2
n
n
1 X
1X
0≤
kxi k2 − xi ≤
n
n i=1
i=1
 2
n −1


max k∆xk k2 ;



12 k=1,n








p1 n−1
1q

 n2 − 1 n−1
P
P

p
q

k∆xk k
k∆xk k
6n
≤
k=1
k=1



if p > 1, p1 + 1q = 1;









2


P
n − 1 n−1


k∆xk k .

2n
k=1
58
Sever S. Dragomir
The constants
3
1
, 1
12 6
and
1
2
are best possible.
Proof of the Main Result
It is well known that, the following identity holds in inner product spaces:
* n
+
n
n
X
X
X
(6)
pi hxi , yi i −
p i xi ,
pi yi =
i=1
i=1
i=1
n
X
1X
=
pi pj hxi − xj , yi − yj i .
pi pj hxi − xj , yi − yj i =
2 i,j=1
1≤j<i≤n
We observe, for i > j, we can write that
(7)
xi − xj =
i−1
X
∆xk ,
yi − yj =
i−1
X
∆yk .
k=j
k=j
Taking the modulus in (6) and by the use of (7) and Schwarz0 s inequality
in inner product spaces, i.e., we recall that |hz, ui| ≤ kzk kuk , z, u ∈ H, we
have:
* n
+
n
n
X
X
X
pi hxi , yi i −
p i xi ,
pi yi ≤
i=1
≤
i=1
X
i=1
pi pj kxi − xj k kyi − yj k =
1≤j<i≤n
X
X
1≤j<i≤n
≤
X
1≤j<i≤n
pi pj
i−1
X
k∆xk k
k=j
pi pj |hxi − xj , yi − yj i| ≤
1≤j<i≤n
i−1
X
i−1
i−1
X
X
pi pj ∆xk ∆yl ≤
k=j
l=j
k∆yl k := M.
l=j
It is obvious that
i−1
X
k=j
k∆xk k ≤ (i − j) max k∆xk k ≤ (i − j) max k∆xk k
k=j,...,i−1
k=1,...,n
Grüss Type Inequalities for Forward Difference of Vectors...
59
and
i−1
X
k∆yk k ≤ (i − j) max k∆yk k ≤ (i − j) max k∆yk k ,
k=j,...,i−1
k=j
k=1,...,n
giving that
M≤
X
pi pj (i − j)2 · max k∆xk k max k∆yk k ,
k=1,...,n
1≤j<i≤n
k=1,...,n
and since
n
n
X
1X
pi pj (i − j)2 =
pi pj (i − j)2 =
pi i2 −
2 i,j=1
i=1
1≤j<i≤n
n
X
X
!2
ipi
,
i=1
the first inequality in (5) is proved.
Using Hölder0 s discrete inequality, we can state that
i−1
X
k∆xk k ≤ (i − j)
1
q
k=j
i−1
X
! p1
k∆xk k
p
≤ (i − j)
n−1
X
1
q
k=j
! p1
p
k∆xk k
k=1
and
i−1
X
k∆yk k ≤ (i − j)
1
p
k=j
i−1
X
! 1q
k∆yk kq
≤ (i − j)
1
p
k=j
for p > 1,
1
p
+
M≤
1
q
n−1
X
! 1q
k∆yk kq
k=1
= 1, giving that:
X
pi pj (i − j) ·
1≤j<i≤n
n−1
X
! p1
k∆xk kp
k=1
n−1
X
! 1q
k∆yk kq
k=1
and the second inequality in (5) is proved.
Also, observe that
i−1
X
k=j
k∆xk k ≤
n−1
X
k=1
k∆xk k and
i−1
X
k=j
k∆yk k ≤
n−1
X
k=1
k∆yk k
,
60
Sever S. Dragomir
and thus
X
M≤
n−1
X
pi pj (i − j)
1≤j<i≤n
k∆xk k
k=1
n−1
X
k=1
Since
" n
#
n
X
1 X
1
pi pj =
pi pj −
p2k =
2 i,j=1
2
1≤j<i≤n
k=1
X
k∆yk k .
1−
n
X
!
p2k
k=1
n
1X
=
pi (1 − pi ) ,
2 i=1
the last part of (5) is also proved.
Now, assume that the first inequality in (5) holds with a constant c > 0,
i.e.,
n
X
*
pi hxi , yi i −
i=1

n
X

i2 pi −
≤c
n
X
i=1
i=1
!2 
ipi

n
X
pi xi ,
n
X
i=1
max
k=1,...,n−1
+
pi yi
≤
i=1
k∆xk k
max
k=1,...,n−1
k∆yk k
and choose n = 2 to get
(8)
p1 p2 |hx2 − x1 , y2 − y1 i| ≤ cp1 p2 kx2 − x1 k ky2 − y1 k
for any p1 , p2 > 0 and x1 , x2 , y1 , y2 ∈ H.
If in (8) we choose y2 = x2 , y1 = x1 and x2 6= x1 , then we deduce c ≥ 1,
which proves the sharpness of the constant in the first inequality in (5).
In a similar way one may show that the other two inequalities are sharp,
and the theorem is completely proved.
4
A Reverse for Jensen0s Inequality
Let (H; h·, ·i) be a real inner product space and F : H → R a Fréchet
differentiable convex function on H. If OF : H → H denotes the gradient
Grüss Type Inequalities for Forward Difference of Vectors...
61
operator associated to F, then we have the inequality
F (x) − F (y) ≥ hOF (y) , x − yi
for each x, y ∈ H.
The following result holds.
Theorem 8. Let F : H → R be as above and zi ∈ H, i ∈ {1, . . . , n} . If
n
P
qi ≥ 0 (i ∈ {1, . . . , n}) with
qi = 1, then we have the following reverse
i=1
of Jensen0 s inequality
0≤
(9)
n
X
qi F (zi ) − F
i=1
n
X
!
qi z i
≤
i=1
 "
n
2 #
n

P
P


i2 qi −
iqi
max k∆ (OF (zi ))k max k∆zi k ;


k=1,...,n−1
k=1,...,n−1

i=1
i=1








"
#

p1 n−1
1q

n−1

P
P
P

p
q
k∆ (OF (zi ))k
qi qj (i − j)
k∆zi k
≤
i=1
i=1
1≤j<i≤n





if p > 1, p1 + 1q = 1;








n−1
n

n−1

P
P
1 P



qi (1 − qi )
k∆ (OF (zi ))k
k∆zi k .
2 i=1
i=1
i=1
Proof. We know, see for example [3, Eq. (4.4)], that the following reverse
of Jensen0 s inequality for Fréchet differentiable convex functions
!
n
n
X
X
0≤
qi F (zi ) − F
(10)
qi z i ≤
i=1
≤
n
X
i=1
i=1
*
qi hOF (zi ) , zi i −
n
X
i=1
qi OF (zi ) ,
n
X
i=1
+
qi zi
62
Sever S. Dragomir
holds.
Now, if we apply Theorem 7 for the choices xi = OF (zi ) , yi = zi and
pi = qi (i = 1, ..., n) , then we may state
(11)
n
* n
+
n
X
X
X
q
hOF
(z
)
,
z
i
−
q
OF
(z
)
,
q
z
i
i
i
i
i
i i ≤
i=1
i=1
i=1
 "
n
2 #
n

P
P


i2 qi −
iqi
max k∆ (OF (zk ))k max k∆zk k ;


k=1,...,n−1
k=1,...,n−1

i=1
i=1








"
#

p1 n−1
1q

n−1

P
P
P

p
q
qi qj (i − j)
k∆ (OF (zk ))k
k∆zk k
≤
1≤j<i≤n
k=1
k=1





if p > 1, p1 + 1q = 1;








n−1
n

n−1
 1 P
P
P


qi (1 − pi )
k∆ (OF (zk ))k
k∆zk k .

2 i=1
k=1
k=1
Finally, on making use of the inequalities (10) and (11), we deduce the
desired result (9).
The unweighted case may useful in application and is incorporated in
the following corollary.
Corollary 7. Let F : H → R be as above and zi ∈ H, i ∈ {1, . . . , n} . Then
we have the inequalities
n
1X
0≤
F (zi ) − F
n i=1
n
1X
zi
n i=1
!
≤
Grüss Type Inequalities for Forward Difference of Vectors...
63
 2
n −1


max k∆ (OF (zk ))k max k∆zk k ;


k=1,...,n−1
k=1,...,n−1
12








p1 n−1
1q
n−1

2

P

q
 n − 1 P k∆ (OF (zk ))kp
k∆zk k
6n
≤
k=1
k=1



if p > 1, p1 + 1q = 1;










n−1
P
P

n − 1 n−1


k∆zk k .
k∆ (OF (zk ))k
2n k=1
k=1
References
[1] S. S. Dragomir, A generalisation of Grüss inequality in inner product
space and applications, J. Math. Anal. Appl., 237 (1999), 74-82.
[2] S. S. Dragomir, A Grüss type discrete inequality in inner product spaces
and applications, J. Math. Anal. Appl., 250 (2000), 494-511.
[3] S. S. Dragomir, A Grüss type inequality for sequences of vectors in inner product spaces and applications, J. Ineq. Pure & Appl. Math., 1(2)
(2000), Article 12. [ON LINE: http://jipam.vu.edu.au].
[4] S. S. Dragomir, G.L. Booth, On a Grüss-Lupaş type inequality and
its applications for the estimation of p-moments of guessing mappings,
Math. Comm., 5 (2000), 117-126.
64
Sever S. Dragomir
[5] S. S. Dragomir, Another Grüss type inequality for sequences of vectors
in normed linear spaces and applications, J. Comp. Anal. & Appl., 4(2)
(2002), 155-172.
[6] S. S. Dragomir, A Grüss type inequality for sequences of vectors in
normed linear spaces and applications, RGMIA Res. Rep. Coll., 5(2002),
No.2, Article 9 [ON LINE: http://rgmia.vu.edu.au/v5n2.html]
[7] J. E. Pečarić, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, San Diego, 1992.
[8] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.
School of Computer Science and Mathematics
Victoria University of Technology
PO Box 14428, MCMC 8001
Victoria, Australia.
E-mail: sever@matilda.vu.edu.au
http://rgmia.vu.edu.au/SSDragomirWeb.html