Journal of Inequalities in Pure and
Applied Mathematics
A GRUSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS
IN INNER PRODUCT SPACES AND APPLICATIONS
volume 1, issue 2, article 12,
2000.
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 5 February, 2000;
accepted 29 February, 2000.
Communicated by: B. Mond
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
002-00
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Abstract
A Grüss type inequality for sequences of vectors in inner product spaces which
complement a recent result from [6] and applications for differentiable convex
functions defined on inner product spaces and applications for Fourier and
Mellin transforms, are given.
2000 Mathematics Subject Classification: 26D15, 26D99, 46Cxx
Key words: Grüss’ Inequality, Inner Product Spaces.
Contents
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
A Discrete Inequality of Grüss Type . . . . . . . . . . . . . . . . . . . . . 11
4
Applications for Convex Functions . . . . . . . . . . . . . . . . . . . . . . 17
5
Applications for Some Discrete Transforms . . . . . . . . . . . . . . 21
References
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1.
Introduction
In 1935, G. Grüss proved the following integral inequality (see [11] or [12])
Z b
Z b
Z b
1
1
1
(1.1) f (x) g (x) dx −
f (x) dx ·
g (x) dx
b−a a
b−a a
b−a a
1
≤ (Φ − φ) (Γ − γ) ,
4
provided that f and g are two integrable functions on [a, b] and satisfy the condition
(1.2)
φ ≤ f (x) ≤ Φ and γ ≤ g (x) ≤ Γ for all x ∈ [a, b] .
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
The constant
1
4
is the best possible and is achieved for
a+b
f (x) = g (x) = sgn x −
.
2
The discrete version of (1.1) states that:
If a ≤ ai ≤ A, b ≤ bi ≤ B (i = 1, ..., n) where a, A, ai , b, B, bi are real
numbers, then
n
n
n
1 X
1 X 1 X 1
(1.3)
ai b i −
ai
bi ≤ (A − a) (B − b)
n
4
n
n
i=1
i=1
i=1
and the constant 41 is the best possible.
In the recent paper [2], the author proved the following generalisation in
inner product spaces.
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Theorem 1.1. Let (X; h·, ·i) be an inner product space over K, K = C, R, and
e ∈ X, kek = 1. If φ, Φ, γ, Γ ∈ K and x, y ∈ X such that
Re hΦe − x, x − φei ≥ 0 and Re hΓe − y, y − γei ≥ 0
(1.4)
holds, then we have the inequality
|hx, yi − hx, ei he, yi| ≤
(1.5)
The constant
1
4
1
|Φ − φ| |Γ − γ| .
4
is the best possible.
It has been shown in [1] that the above theorem, for the real case, contains
the usual integral and discrete Grüss inequality and also some Grüss type inequalities for mappings defined on infinite intervals.
Namely,
+∞) is a probability density funcR ∞if ρ : (−∞, +∞) →1 (−∞,
1
2
tion, i.e., −∞ ρ (t) dt = 1, then ρ 2 ∈ L (−∞, ∞) and obviously kρ 2 k2 = 1.
Consequently, if we assume that f, g ∈ L2 (−∞, ∞) and
1
2
1
2
1
2
αρ ≤ f ≤ ψρ , βρ ≤ g ≤ θρ
(1.6)
1
2
a.e. on (0, ∞) ,
then we have the inequality
(1.7)
Z
∞
−∞
A Grüss Type Inequality for
Sequences of Vectors in Inner
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S.S. Dragomir
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Z
∞
f (t) g (t) dt −
1
2
Z
∞
f (t) ρ (t) dt
−∞
−∞
g (t) ρ (t) dt
1
2
≤
1
(ψ − α) (θ − β) .
4
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In a similar way, if e = (ei )i∈N ∈ l2 (R) with
P
i∈N
|ei |2 = 1 and x = (xi )i∈N ,
y = (yi )i∈N ∈ l2 (R) are such that
(1.8)
αei ≤ xi ≤ ψei , βei ≤ yi ≤ θei
for all i ∈ N, then we have
X
1
X
X
(1.9)
xi yi −
xi ei
yi ei ≤ (ψ − α) (θ − β) .
4
i∈N
i∈N
i∈N
In the recent paper [6], the author also proved the following discrete inequality in inner product spaces:
n
n
n
1
X
X
X
p
a
x
−
p
a
p
x
|A − a| kX − xk
(1.10)
i i i
i i
i i ≤
4
i=1
i=1
i=1
provided xi ∈ H, ai ∈ K (K = C, R) and a, A ∈ K, x, X ∈ H are such that
(1.11)
Re [(A − ai ) (āi − ā)] ≥ 0 and Re hX − xi , xi − xi ≥ 0 for all i ∈ {1, ..., n} .
The constant 41 is sharp.
For other recent developments of the Grüss inequality, see the papers [1]-[6],
[10] and the website http://rgmia.vu.edu.au/Gruss.html
In this paper we point out some other Grüss type inequalities in inner product
spaces which will complement the above result (1.10).
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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2.
Preliminary Results
The following lemma is of interest in itself (see also [6]).
Lemma 2.1. Let (H; h·, ·i) be an inner product space over the real or complex
n
P
number field K, xi ∈ H and pi ≥ 0 (i = 1, ..., n) such that
pi = 1 (n ≥ 2).
i=1
If x, X ∈ H are such that
Re hX − xi , xi − xi ≥ 0 for all i ∈ {1, ..., n} ,
(2.1)
then we have the inequality
0≤
(2.2)
n
X
i=1
The constant
1
4
2
n
X
1
pi kxi k − pi xi ≤ kX − xk2 .
4
i=1
2
is sharp.
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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Proof. Define
Contents
*
I1 :=
X−
n
X
i=1
and
I2 :=
n
X
pi xi ,
n
X
+
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i=1
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pi hX − xi , xi − xi .
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i=1
Then
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2
n
n
n
X
X
X
I1 =
pi hX, xi i − hX, xi − pi xi +
pi hxi , xi
i=1
i=1
i=1
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and
I2 =
n
X
pi hX, xi i − hX, xi −
i=1
n
X
2
pi kxi k +
i=1
n
X
pi hxi , xi .
i=1
Consequently
I1 − I2 =
(2.3)
n
X
i=1
n
2
X
pi kxi k2 − pi xi .
i=1
Taking the real value in (2.3) we can state
(2.4)
2
n
X
pi kxi k − pi xi i=1
i=1
*
+
n
n
n
X
X
X
= Re X −
pi xi ,
pi xi − x −
pi Re hX − xi , xi − xi ,
n
X
2
i=1
i=1
i=1
i=1
S.S. Dragomir
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i=1
which is an identity of interest in itself.
Using the assumption (2.1), we can conclude, by (2.4), that
2
*
+
n
n
n
n
X
X
X
X
(2.5)
pi kxi k2 − pi xi ≤ Re X −
pi xi ,
pi xi − x .
i=1
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
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It is known that if y, z ∈ H, then
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(2.6)
2
4 Re hz, yi ≤ kz + yk ,
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with equality iff z = y.
Now, by (2.6), we can state that
*
+
n
n
X
X
pi xi ,
pi xi − x
Re X −
i=1
i=1
2
n
n
X
X
1
1
pi xi +
pi xi − x = kX − xk2 .
≤ X −
4
4
i=1
i=1
Using (2.5), we can easily deduce (2.2).
To prove the sharpness of the constant 14 , let us assume that the inequality
(2.2) holds with a constant c > 0, i.e.,
2
n
n
X
X
2
(2.7)
0≤
pi kxi k − pi xi ≤ c kX − xk2
i=1
i=1
for all pi , xi and x, X as in the hypothesis of Lemma 2.1.
Assume that n = 2, p1 = p2 = 12 , x1 = x and x2 = X with x, X ∈ H and
x 6= X. Then, obviously,
hX − x1 , x1 − xi = hX − x2 , x2 − xi = 0,
which shows that the condition (2.1) holds.
If we replace n, p1 , p2 , x1 , x2 in (2.7), we obtain
2
!
2
2
X
X
x + X 2
1
2
2
2
pi kxi k − pi xi =
kxk + kXk − 2 2
i=1
i=1
kX − xk2
=
≤ c kX − xk2 ,
4
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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from where we deduce c ≥ 14 , which proves the sharpness of the constant factor
1
.
4
Remark 2.1. The assumption (2.1) can be replaced by the more general condition
n
X
(2.8)
pi Re hX − xi , xi − xi ≥ 0
i=1
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
and the conclusion (2.2) will still remain valid.
The following corollary is natural.
Corollary 2.2. Let ai ∈ K, pi ≥ 0 (i = 1, ..., n) (n ≥ 2) with
n
P
pi = 1. If
i=1
a, A ∈ K are such that
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Re [(A − ai ) (āi − ā)] ≥ 0 for all i ∈ {1, ..., n} ,
(2.9)
then we have the inequality
0≤
(2.10)
n
X
i=1
The constant
1
4
S.S. Dragomir
2
n
X
1
pi |ai | − pi ai ≤ |A − a|2 .
4
i=1
2
is sharp.
The proof follows by the above Lemma 2.1 by choosing H = K, hx, yi :=
xȳ, xi = ai , x = a, X = A. We omit the details.
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Remark 2.2. The condition (2.9) can be replaced by the more general assumption
n
X
(2.11)
pi Re [(A − ai ) (āi − ā)] ≥ 0.
i=1
Remark 2.3. If we assume that K = R, then (2.8) is equivalent with
a ≤ ai ≤ A for all i ∈ {1, ..., n}
(2.12)
and then, with the assumption (2.12), we get the discrete Grüss type inequality
!2
n
n
X
X
1
2
(2.13)
0≤
p i ai −
p i ai
≤ (A − a)2
4
i=1
i=1
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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and the constant
1
4
is sharp.
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3.
A Discrete Inequality of Grüss Type
The following Grüss type inequality holds.
Theorem 3.1. Let (H; h·, ·i) be an inner product space over K; K = C, R,
n
P
xi , yi ∈ H, pi ≥ 0 (i = 0, ..., n) (n ≥ 2) with
pi = 1. If x, X, y, Y ∈ H are
i=1
such that
(3.1)
Re hX − xi , xi − xi ≥ 0 and Re hY − yi , yi − yi ≥ 0 for all i ∈ {1, ..., n} ,
then we have the inequality
+
* n
n
n
1
X
X
X
pi yi ≤ kX − xk kY − yk .
pi xi ,
(3.2)
pi hxi , yi i −
4
i=1
i=1
i=1
The constant
1
4
S.S. Dragomir
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is sharp.
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Proof. A simple calculation shows that
* n
+
n
n
n
X
X
X
1X
pi hxi , yi i −
pi xi ,
pi yi =
(3.3)
pi pj hxi − xj , yi − yj i .
2
i=1
i=1
i=1
i,j=1
Taking the modulus in both parts of (3.3), and using the generalized triangle
inequality, we obtain
(3.4)
* n
+
n
n
n
X
1X
X
X
pi hxi , yi i −
pi xi ,
pi yi ≤
pi pj |hxi − xj , yi − yj i| .
2
i=1
A Grüss Type Inequality for
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i=1
i=1
i,j=1
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By Schwartz’s inequality in inner product spaces we have
(3.5)
|hxi − xj , yi − yj i| ≤ kxi − xj k kyi − yj k
for all i, j ∈ {1, ..., n} , and therefore
(3.6)
* n
+
n
n
n
X
1X
X
X
pi pj kxi − xj k kyi − yj k .
pi hxi , yi i −
pi xi ,
pi yi ≤
2
i,j=1
i=1
i=1
i=1
Using the Cauchy-Buniakowsky-Schwartz inequality for double sums, we can
state that
(3.7)
n
1X
pi pj kxi − xj k kyi − yj k
2 i,j=1
≤
n
1X
pi pj kxi − xj k2
2 i,j=1
S.S. Dragomir
! 12
×
n
1X
pi pj kyi − yj k2
2 i,j=1
and, a simple calculation shows that,
2
n
n
n
X
X
1X
pi pj kxi − xj k2 =
pi kxi k2 − pi xi 2 i,j=1
i=1
i=1
and
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
! 12
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2
n
n
n
X
X
1X
pi pj kyi − yj k2 =
pi kyi k2 − pi yi .
2 i,j=1
i=1
i=1
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We obtain
n
+
* n
n
X
X
X
(3.8) pi hxi , yi i −
pi xi ,
pi yi i=1
i=1
i=1

2  12
2  21 
n
n
n
n
X
X
X
X

2
2


×
pi kyi k − pi yi  .
≤
pi kxi k − pi xi i=1
i=1
i=1
i=1
Using Lemma 2.1, we know that

n
X

i=1
2  12
n
X
1
pi kxi k2 − pi xi  ≤ kX − xk
2
i=1
A Grüss Type Inequality for
Sequences of Vectors in Inner
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Applications
S.S. Dragomir
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and
2  21
n
n
X
X
1

pi kyi k2 − pi yi  ≤ kY − yk .
2
i=1
i=1

Therefore, by (3.8) we may deduce the desired inequality (3.3).
To prove the sharpness of the constant 14 , let us assume that (3.2) holds with
a constant c > 0, i.e.,
+
* n
n
n
X
X
X
(3.9)
p
hx
,
y
i
−
p
x
,
p
y
i
i i
i i
i i ≤ c kX − xk kY − yk
i=1
i=1
i=1
under the above assumptions for pi , xi , yi , x, X, y, Y and n ≥ 2.
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If we choose n = 2, x1 = x, x2 = X, y1 = y, y2 = Y (x 6= X, y 6= Y )
and p1 = p2 = 21 , then
* 2
+
2
2
2
X
X
X
1X
pi hxi , yi i −
pi xi ,
pi yi
=
pi pj hxi − xj , yi − yj i
2
i=1
i=1
i=1
i,j=1
X
=
pi pj hxi − xj , yi − yj i
1≤i<j≤2
=
1
hx − X, y − Y i
4
and then
* 2
+
2
2
X
1
X
X
pi hxi , yi i −
pi xi ,
pi yi = |hx − X, y − Y i| .
4
i=1
i=1
i=1
Choose X − x = z, Y − y = z, z 6= 0. Then using (3.9), we derive
1
kzk2 ≤ c kzk2 , z 6= 0
4
which implies that c ≥ 14 , and the theorem is proved.
Remark 3.1. The condition (3.1) can be replaced by the more general assumption
(3.10)
n
X
pi Re hX − xi , xi − xi ≥ 0,
i=1
n
X
i=1
pi Re hY − yi , yi − yi ≥ 0
A Grüss Type Inequality for
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S.S. Dragomir
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and the conclusion (3.2) still remains valid.
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The following corollary for real or complex numbers holds.
Corollary 3.2. Let ai , bi ∈ K (K = C, R) , pi ≥ 0 (i = 1, ..., n) with
Re [(A − ai ) (āi − ā)] ≥ 0 and
pi = 1.
i=1
If a, A, b, B ∈ K are such that
(3.11)
n
P
Re (B − bi ) b̄i − b̄ ≥ 0,
then we have the inequality
n
n
n
X
1
X
X
(3.12)
pi ai b̄i −
p i ai
pi b̄i ≤ |A − a| |B − b|
4
i=1
i=1
i=1
and the constant
1
4
S.S. Dragomir
is sharp.
The proof is obvious by Theorem 3.1 applied for the inner product space
(C, h·, ·i) where hx, yi = x · ȳ. We omit the details.
Remark 3.2. The condition (3.11) can be replaced by the more general condition
(3.13)
n
X
i=1
pi Re [(A − ai ) (āi − ā)] ≥ 0,
n
X
pi Re (B − bi ) b̄i − b̄ ≥ 0
i=1
and the conclusion of the above corollary will still remain valid.
Remark 3.3. If we assume that ai , bi , a, b, A, B are real numbers, then (3.11)
is equivalent to
(3.14)
A Grüss Type Inequality for
Sequences of Vectors in Inner
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a ≤ ai ≤ A, b ≤ bi ≤ B for all i ∈ {1, ..., n}
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and (3.12) becomes
n
n
n
X
1
X
X
(3.15)
0≤
p i ai b i −
p i ai
pi bi ≤ (A − a) (B − b) ,
4
i=1
i=1
i=1
which is the classical Grüss inequality for sequences of real numbers.
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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4.
Applications for Convex Functions
Let (H; h·, ·i) be a real inner product space and F : H → R a Fréchet differentiable convex mapping on H. Then we have the “gradient inequality”
F (x) − F (y) ≥ h∇F (y) , x − yi
(4.1)
for all x, y ∈ H, where ∇F : H → H is the gradient operator associated to the
differentiable convex function F .
The following theorem holds.
Theorem 4.1. Let F : H → R be as above and xi ∈ H (i = 1, ..., n). Suppose
that there exists the vectors γ, φ ∈ H such that hxi − γ, φ − xi i ≥ 0 for all
i ∈ {1, ..., m} and m, M ∈ H such that h∇F (xi ) − m, M − ∇F (xi )i ≥ 0
m
P
for all i ∈ {1, ..., m}. Then for all pi ≥ 0 (i = 1, ..., m) with Pm :=
pi > 0,
i=1
A Grüss Type Inequality for
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S.S. Dragomir
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we have the inequality
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(4.2)
m
1 X
0≤
pi F (xi ) − F
Pm i=1
Proof. Choose in (4.1), x =
(4.3)
F
m
1 X
pi xi
Pm i=1
1
PM
m
P
m
1 X
pi xi
Pm i=1
!
≤
1
kφ − γk kM − mk .
4
Close
*
− F (xj ) ≥
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pi xi and y = xj to obtain
i=1
!
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m
1 X
∇F (xj ) ,
pi xi − xj
Pm i=1
+
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for all j ∈ {1, ..., n}.
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If we multiply (4.3) by pj ≥ 0 and sum over j from 1 to m, we have
Pm F
m
1 X
pi xi
Pm i=1
!
−
m
X
pj F (xj )
j=1
+
* m
m
m
X
X
1 X
≥
pj ∇F (xj ) ,
pi xi −
h∇F (xi ) , xi i .
Pm j=1
i=1
i=1
Dividing by Pm > 0, we obtain the inequality
(4.4)
m
1 X
0 ≤
pi F (xi ) − F
Pm i=1
m
1 X
pi xi
Pm i=1
!
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
m
1 X
≤
pi h∇F (xi ) , xi i
Pm i=1
+
*
m
m
1 X
1 X
−
pi ∇F (xi ) ,
pi xi
Pm i=1
Pm i=1
which is a generalisation for the case of inner product spaces of the result by
Dragomir and Goh established in 1996 for the case of differentiable mappings
defined on Rn [9].
Applying Theorem 3.1 for real inner product spaces, X = φ, x = γ, yi =
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∇F (xi ), y = m, Y = M and n = m, we easily deduce
*
+
m
m
m
1 X
1 X
1 X
(4.5)
pi hxi , ∇F (xi )i −
pi xi ,
pi ∇F (xi )
Pm i=1
Pm i=1
Pm i=1
1
kΦ − φk kM − mk
4
and then, by (4.4) and (4.5) we can conclude that the desired inequality (4.2)
holds.
≤
Remark 4.1. The conditions
(4.6)
hxi − γ, φ − xi i ≥ 0, h∇F (xi ) − m, M − ∇F (xi )i ≥ 0,
for all i ∈ {1, ..., m} can be replaced by the more general conditions
(4.7)
m
m
X
X
pi hxi − γ, φ − xi i ≥ 0 and
pi h∇F (xi ) − m, M − ∇F (xi )i ≥ 0
i=1
i=1
and the conclusion (4.2) will still be valid.
Remark 4.2. Even if the inequality (4.2) is not as sharp as (4.4), it can be
more useful in practice when only some bounds of the gradient operator ∇F
and of the vectors xi (i = 1, ..., n) are known. In other words, it provides the
opportunity to estimate the difference
!
m
m
1 X
1 X
∆ (F, x, p) :=
pi F (xi ) − F
pi xi ,
Pm i=1
Pm i=1
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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where the differences kφ − γk and kM − mk are known.
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Remark 4.3. For example, if we know that h∇F (xi ) − m, M − ∇F (xi )i ≥ 0
for all i ∈ {1, ..., m} and the vectors xi (i = 1, ..., n) are not too far from
each other in the sense that hxi − γ, φ − xi i ≥ 0 for all i ∈ {1, ..., m} and
(ε > 0), then by (4.2), we can conclude that
kφ − γk ≤ kM4ε
−mk
0 ≤ ∆ (F, x, p) ≤ ε.
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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5.
Applications for Some Discrete Transforms
Let (H; h·, ·i) be an inner product space over K, K = C, R and x̄ = (x1 , ..., xn )
be a sequence of vectors in H.
For a given m ∈ K, define the discrete Fourier transform
(5.1)
Fw (x̄) (m) =
n
X
exp (2wimk) × xk , m = 1, ..., n.
k=1
The complex number
n
P
exp (2wimk) hxk , yk i is actually the usual Fourier
k=1
transform of the vector (hx1 , y1 i , ..., hxn , yn i) ∈ Kn and will be denoted by
(5.2)
Fw (x̄ · ȳ) (m) =
n
X
exp (2wimk) hxk , yk i , m = 1, ..., n.
k=1
The following result holds.
Theorem 5.1. Let x̄, ȳ ∈ H n be sequences of vectors such that there exists the
vectors c, C, y, Y ∈ H with the properties
(5.3) Re hC − exp (2wimk) xk , exp (2wimk) xk − ci ≥ 0, k, m = 1, ..., n
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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(5.4)
Re hY − yk , yk − yi ≥ 0, k = 1, ..., n.
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Then we have the inequality
+
*
n
n
1X
(5.5) Fw (x̄ · ȳ) (m) − Fw (x̄) (m) ,
yk ≤ kC − ck kY − yk ,
4
n k=1
for all m ∈ {1, ..., n}.
The proof follows by Theorem 3.1 applied for pk = n1 and for the sequences
xk → ck = exp (2wimk) xk and yk (k = 1, ..., n). We omit the details.
We can also consider the Mellin transform
(5.6)
M (x̄) (m) :=
n
X
k m−1 xk , m = 1, ..., n,
k=1
of the sequence x̄ = (x1 , ..., xn ) ∈ H n .
We remark that the complex number
k m−1 hxk , yk i is actually the Mellin
k=1
M (x̄ · ȳ) (m) :=
Contents
n
transform of the vector (hx1 , y1 i , ..., hxn , yn i) ∈ K and will be denoted by
(5.7)
S.S. Dragomir
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P
n
X
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
k m−1 hxk , yk i .
k=1
The following theorem holds.
Theorem 5.2. Let x̄, ȳ ∈ H n be sequences of vectors such that there exist the
vectors d, D, y, Y ∈ H with the properties
(5.8)
Re D − k m−1 xk , k m−1 xk − d ≥ 0
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for all k, m ∈ {1, ..., n} , and (5.4) is fulfilled.
Then we have the inequality
+
*
n
n
1X
yk ≤ kD − dk kY − yk
(5.9) M (x̄ · ȳ) (m) − M (x̄) (m) ,
4
n k=1
for all m ∈ {1, ..., n}.
The proof follows by Theorem 3.1 applied for pk = n1 and for the sequences
xk → dk = kxk and yk (k = 1, ..., n). We omit the details.
Another result which connects the Fourier transforms for different parameters w also holds.
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
Theorem 5.3. Let x̄, ȳ ∈ H n and w, z ∈ K. If there exists the vectors
e, E, f, F ∈ H such that
S.S. Dragomir
Re hE − exp (2wimk) xk , exp (2wimk) xk − ei ≥ 0, k, m = 1, ..., n
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and
Re hF − exp (2zimk) yk , exp (2zimk) yk − f i ≥ 0, k, m = 1, ..., n
then we have the inequality:
1
1
1
1
Fw+z (x̄ · ȳ) (m) −
Fw (x̄) (m) , Fz (ȳ) (m) ≤ kE − ek kF − f k ,
n
n
n
4
for all m ∈ {1, ..., n}.
The proof follows by Theorem 3.1 for the sequences exp (2wimk) xk ,
exp (2zimk) yk (k = 1, ..., n). We omit the details.
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References
[1] S.S. DRAGOMIR, Grüss inequality in inner product spaces, Austral.
Math. Soc. Gazette, 26(2) (1999), 66–70.
[2] S.S. DRAGOMIR, A generalization of Grüss’ inequality in inner product
spaces and applications, J. Math. Anal. Appl., 237 (1999), 74–82.
[3] S.S. DRAGOMIR, A Grüss type integral inequality for mappings of rHölder’s type and applications for trapezoid formula, Tamkang J. of Math.,
31(1) (2000), 43–47.
[4] S.S. DRAGOMIR, Some discrete inequalities of Grüss type and applications in guessing theory, Honam Math. J., 21(1) (1999), 115–126.
[5] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. of
Pure and Appl. Math., 31(4) (2000), 397–415.
[6] S.S. DRAGOMIR, A Grüss type discrete inequality in inner product
spaces and applications, J. Math. Anal. Appl., (in press).
[7] S.S. DRAGOMIR AND G.L. BOOTH, On a Grüss-Lupaş type inequality
and its applications for the estimation of p-moments of guessing mappings,
Math. Comm., (in press).
[8] S.S. DRAGOMIR AND I. FEDOTOV, An inequality of Grüss’ type for
Riemann-Stieltjes integral and applications for special means, Tamkang J.
of Math., 29(4) (1998), 286–292.
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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[9] S.S. DRAGOMIR AND C.J. GOH, A counterpart of Jensen’s discrete inequality for differentiable convex mappings and applications in information theory, Mathl. Comput. Modelling, 24(2) (1996), 1–11.
[10] A.M. FINK, A treatise on Grüss’ inequality, submitted.
[11] G. RGRÜSS, Über das Maximum
absoluten Betrages von
Rb
R des
b
b
1
1
f (x)g(x)dx − (b−a)2 a f (x)dx a g(x)dx, Math. Z., 39 (1935),
b−a a
215–226.
[12] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
A Grüss Type Inequality for
Sequences of Vectors in Inner
Product Spaces and
Applications
S.S. Dragomir
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