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volume 1, issue 2, article 12, 2000.

Received 5 February, 2000;

accepted 29 February, 2000.

Communicated by:B. Mond

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Journal of Inequalities in Pure and Applied Mathematics

A GRUSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN INNER PRODUCT SPACES AND APPLICATIONS

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

c

2000Victoria University ISSN (electronic): 1443-5756 002-00

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A Grüss Type Inequality for Sequences of Vectors in Inner

Product Spaces and Applications S.S. Dragomir

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J. Ineq. Pure and Appl. Math. 1(2) Art. 12, 2000

http://jipam.vu.edu.au

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.

1. Introduction

In 1935, G. Grüss proved the following integral inequality (see [11] or [12]) (1.1)

1 b−a

Z b a

f(x)g(x)dx− 1 b−a

Z b a

f(x)dx· 1 b−a

Z b a

g(x)dx

≤ 1

4(Φ−φ) (Γ−γ), provided thatf andg 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]. The constant ¹₄ is the best possible and is achieved for

f(x) =g(x) = sgn

x− a+b 2

.

The discrete version of (1.1) states that:

Ifa≤ ai ≤A, b ≤bi ≤B (i = 1, ..., n) wherea, A, ai, b, B, bi are real numbers, then

(1.3)

1 n

n

X

i=1

a_ib_i− 1 n

n

X

i=1

a_i 1 n

n

X

i=1

b_i

≤ 1

4(A−a) (B −b)

and the constant ¹₄ 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 overK,K =C,R,and e ∈X,kek= 1. Ifφ,Φ, γ,Γ∈Kandx, y ∈X such that

(1.4) RehΦe−x, x−φei ≥0 and RehΓe−y, y−γei ≥0 holds, then we have the inequality

(1.5) |hx, yi − hx, ei he, yi| ≤ 1

4|Φ−φ| |Γ−γ|.

The constant ¹₄ 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, if ρ : (−∞,+∞) → (−∞,+∞) is a probability density func- tion, i.e., R∞

−∞ρ(t)dt = 1, thenρ¹² ∈ L²(−∞,∞)and obviouslykρ¹²k₂ = 1.

Consequently, if we assume thatf, g ∈L²(−∞,∞)and

(1.6) αρ¹² ≤f ≤ψρ¹², βρ¹² ≤g ≤θρ¹² a.e. on (0,∞), then we have the inequality

(1.7)

Z ∞

−∞

f(t)g(t)dt− Z ∞

−∞

f(t)ρ¹² (t)dt Z ∞

−∞

g(t)ρ¹² (t)dt

≤ 1

4(ψ−α) (θ−β).

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In a similar way, if e = (e_i)_i∈

N ∈ l²(R) with P

i∈N

|e_i|² = 1 and x = (x_i)_i∈

N, y = (y_i)_i∈

N ∈l²(R)are such that

(1.8) αe_i ≤x_i ≤ψe_i, βe_i ≤y_i ≤θe_i for alli∈N,then we have

(1.9)

X

i∈N

x_iy_i−X

i∈N

x_ie_i X

i∈N

y_ie_i

≤ 1

4(ψ−α) (θ−β).

In the recent paper [6], the author also proved the following discrete inequality in inner product spaces:

(1.10)

n

X

i=1

p_ia_ix_i−

n

X

i=1

p_ia_i

n

X

i=1

p_ix_i

≤ 1

4|A−a| kX−xk

providedx_i ∈H,a_i ∈K(K=C,R)anda, A∈K,x, X ∈Hare such that (1.11)

Re [(A−a_i) (¯a_i−¯a)]≥0 and RehX−x_i, x_i−xi ≥0 for alli∈ {1, ..., n}. The constant ¹₄ is sharp.

For other recent developments of the Grüss inequality, see the papers [1]-[6], [10] and the websitehttp://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).

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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 number fieldK,xi ∈Handpi ≥0(i= 1, ..., n) such that

n

P

i=1

pi = 1(n ≥2).

Ifx, X ∈Hare such that

(2.1) RehX−x_i, x_i−xi ≥0 for alli∈ {1, ..., n}, then we have the inequality

(2.2) 0≤

n

X

i=1

pikxik²−

n

X

i=1

pixi

2

≤ 1

4kX−xk².

The constant ¹₄ is sharp.

Proof. Define

I₁ :=

* X−

n

X

i=1

p_ix_i,

n

X

i=1

p_ix_i−x +

and

I₂ :=

n

X

i=1

p_ihX−x_i, x_i−xi.

Then

I₁ =

n

X

i=1

p_ihX, x_ii − hX, xi −

n

X

i=1

p_ix_i

2

+

n

X

i=1

p_ihx_i, xi

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and

I₂ =

n

X

i=1

p_ihX, x_ii − hX, xi −

n

X

i=1

p_ikx_ik²+

n

X

i=1

p_ihx_i, xi.

Consequently

(2.3) I₁−I₂ =

n

X

i=1

p_ikx_ik²−

n

X

i=1

p_ix_i

2

.

Taking the real value in (2.3) we can state

(2.4)

n

X

i=1

p_ikx_ik² −

n

X

i=1

p_ix_i

2

= Re

* X−

n

X

i=1

p_ix_i,

n

X

i=1

p_ix_i−x +

−

n

X

i=1

p_iRehX−x_i, x_i−xi,

which is an identity of interest in itself.

Using the assumption (2.1), we can conclude, by (2.4), that (2.5)

n

X

i=1

pikxik²−

n

X

i=1

pixi

2

≤Re

* X−

n

X

i=1

pixi,

n

X

i=1

pixi−x +

.

It is known that ify, z ∈H, then

(2.6) 4 Rehz, yi ≤ kz+yk²,

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with equality iffz =y.

Now, by (2.6), we can state that Re

* X−

n

X

i=1

p_ix_i,

n

X

i=1

p_ix_i−x +

≤ 1 4

X−

n

X

i=1

pixi+

n

X

i=1

pixi−x

2

= 1

4kX−xk². 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 constantc >0, i.e.,

(2.7) 0≤

n

X

i=1

p_ikx_ik²−

n

X

i=1

p_ix_i

2

≤ckX−xk²

for allp_i, x_iandx,Xas in the hypothesis of Lemma2.1.

Assume thatn = 2, p₁ = p₂ = ¹₂, x₁ =xandx₂ =X withx, X ∈ H and x6=X. Then, obviously,

hX−x₁, x₁ −xi=hX−x₂, x₂−xi= 0, which shows that the condition (2.1) holds.

If we replacen,p₁,p₂,x₁,x₂ in (2.7), we obtain

2

X

i=1

pikxik²−

2

X

i=1

pixi

2

= 1

2 kxk²+kXk²−

x+X 2

2!

= kX−xk²

4 ≤ckX−xk²,

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from where we deducec≥ 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 condi- tion

(2.8)

n

X

i=1

p_iRehX−x_i, x_i−xi ≥0

and the conclusion (2.2) will still remain valid.

The following corollary is natural.

Corollary 2.2. Let a_i ∈ K, p_i ≥ 0 (i= 1, ..., n) (n ≥2)with

n

P

i=1

p_i = 1. If a, A ∈Kare such that

(2.9) Re [(A−a_i) (¯a_i−a)]¯ ≥0 for alli∈ {1, ..., n}, then we have the inequality

(2.10) 0≤

n

X

i=1

p_i|a_i|² −

n

X

i=1

p_ia_i

2

≤ 1

4|A−a|².

The proof follows by the above Lemma2.1by choosingH = K, hx, yi :=

xy, x¯ i =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 assump- tion

(2.11)

n

X

i=1

piRe [(A−ai) (¯ai−a)]¯ ≥0.

Remark 2.3. If we assume thatK=R, then (2.8) is equivalent with (2.12) a≤a_i ≤A for alli∈ {1, ..., n}

and then, with the assumption (2.12), we get the discrete Grüss type inequality

(2.13) 0≤

n

X

i=1

pia²_i −

n

X

i=1

piai

!2

≤ 1

4(A−a)²

and the constant ¹₄ 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, xi, yi ∈H,pi ≥0 (i= 0, ..., n) (n ≥2)with

n

P

i=1

pi = 1. Ifx, X, y, Y ∈Hare such that

(3.1)

RehX−x_i, x_i−xi ≥0and RehY −y_i, y_i−yi ≥0 for alli∈ {1, ..., n}, then we have the inequality

(3.2)

n

X

i=1

p_ihx_i, y_ii −

* _n X

i=1

p_ix_i,

n

X

i=1

p_iy_i +

≤ 1

4kX−xk kY −yk.

Proof. A simple calculation shows that

(3.3)

n

X

i=1

p_ihx_i, y_ii −

* _n X

i=1

p_ix_i,

n

X

i=1

p_iy_i +

= 1 2

n

X

i,j=1

p_ip_jhx_i−x_j, y_i−y_ji.

Taking the modulus in both parts of (3.3), and using the generalized triangle inequality, we obtain

(3.4)

n

X

i=1

p_ihx_i, y_ii −

* _n X

i=1

p_ix_i,

n

X

i=1

p_iy_i +

≤ 1 2

n

X

i,j=1

p_ip_j|hx_i−x_j, y_i−y_ji|.

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By Schwartz’s inequality in inner product spaces we have (3.5) |hx_i−x_j, y_i−y_ji| ≤ kx_i−x_jk ky_i−y_jk for alli, j ∈ {1, ..., n},and therefore

(3.6)

n

X

i=1

p_ihx_i, y_ii −

* _n X

i=1

p_ix_i,

n

X

i=1

p_iy_i +

≤ 1 2

n

X

i,j=1

p_ip_jkx_i−x_jk ky_i−y_jk.

Using the Cauchy-Buniakowsky-Schwartz inequality for double sums, we can state that

(3.7) 1 2

n

X

i,j=1

p_ip_jkx_i−x_jk ky_i−y_jk

≤ 1 2

n

X

i,j=1

pipjkxi −xjk²

!¹₂

× 1 2

n

X

i,j=1

pipjkyi−yjk²

!¹₂

and, a simple calculation shows that, 1

2

n

X

i,j=1

p_ip_jkx_i−x_jk² =

n

X

i=1

p_ikx_ik²−

n

X

i=1

p_ix_i

2

and

1 2

n

X

i,j=1

p_ip_jky_i−y_jk² =

n

X

i=1

p_iky_ik²−

n

X

i=1

p_iy_i

2

.

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We obtain (3.8)

n

X

i=1

p_ihx_i, y_ii −

* _n X

i=1

p_ix_i,

n

X

i=1

p_iy_i +

≤





n

X

i=1

pikxik²−

n

X

i=1

pixi

2



1 2

×





n

X

i=1

pikyik²−

n

X

i=1

piyi

2



1 2

.

Using Lemma2.1, we know that





n

X

i=1

p_ikx_ik²−

n

X

i=1

p_ix_i

2



1 2

≤ 1

2kX−xk

and





n

X

i=1

p_iky_ik²−

n

X

i=1

p_iy_i

2



1 2

≤ 1

2kY −yk.

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 constantc >0, i.e.,

(3.9)

n

X

i=1

p_ihx_i, y_ii −

* _n X

i=1

p_ix_i,

n

X

i=1

p_iy_i +

≤ckX−xk kY −yk

under the above assumptions forpi, xi, yi, x, X, y, Y andn ≥2.

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If we choosen = 2, x₁ = x, x₂ = X, y₁ = y, y₂ = Y (x6=X, y 6=Y) andp1 =p2 = ¹₂, then

2

X

i=1

p_ihx_i, y_ii −

* ₂ X

i=1

p_ix_i,

2

X

i=1

p_iy_i +

= 1 2

2

X

i,j=1

p_ip_jhx_i−x_j, y_i−y_ji

= X

1≤i<j≤2

p_ip_jhx_i−x_j, y_i−y_ji

= 1

4hx−X, y−Yi and then

2

X

i=1

pihxi, yii −

* ₂ X

i=1

pixi,

2

X

i=1

piyi

+

= 1

4|hx−X, y−Yi|.

ChooseX−x=z, Y −y=z, z 6= 0. Then using (3.9), we derive 1

4kzk² ≤ckzk², z6= 0 which implies thatc≥ ¹₄, and the theorem is proved.

Remark 3.1. The condition (3.1) can be replaced by the more general assump- tion

(3.10)

n

X

i=1

p_iRehX−x_i, x_i−xi ≥0,

n

X

i=1

p_iRehY −y_i, y_i−yi ≥0

and the conclusion (3.2) still remains valid.

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The following corollary for real or complex numbers holds.

Corollary 3.2. Letai, bi ∈K(K=C,R), pi ≥0 (i= 1, ..., n)with

n

P

i=1

pi = 1.

Ifa, A, b, B ∈Kare such that

(3.11) Re [(A−ai) (¯ai−¯a)]≥0 and Re

(B−bi) ¯bi−¯b

≥0, then we have the inequality

(3.12)

n

X

i=1

p_ia_i¯b_i−

n

X

i=1

p_ia_i

n

X

i=1

p_i¯b_i

≤ 1

4|A−a| |B−b|

and the constant ¹₄ is sharp.

The proof is obvious by Theorem 3.1 applied for the inner product space (C,h·,·i)wherehx, yi=x·y. We omit the details.¯

Remark 3.2. The condition (3.11) can be replaced by the more general condi- tion

(3.13)

n

X

i=1

p_iRe [(A−a_i) (¯a_i−¯a)]≥0,

n

X

i=1

p_iRe

(B −b_i) ¯b_i−¯b

≥0

and the conclusion of the above corollary will still remain valid.

Remark 3.3. If we assume thata_i, b_i,a, b, A,B are real numbers, then (3.11) is equivalent to

(3.14) a≤ai ≤A, b ≤bi ≤B for alli∈ {1, ..., n}

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and (3.12) becomes

(3.15) 0≤

n

X

i=1

p_ia_ib_i−

n

X

i=1

p_ia_i

n

X

i=1

p_ib_i

≤ 1

4(A−a) (B−b),

which is the classical Grüss inequality for sequences of real numbers.

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4. Applications for Convex Functions

Let(H;h·,·i)be a real inner product space andF :H →Ra Fréchet differentiable convex mapping onH. Then we have the “gradient inequality”

(4.1) F(x)−F (y)≥ h∇F(y), x−yi

for allx, y ∈H, where∇F :H →His the gradient operator associated to the differentiable convex functionF.

The following theorem holds.

Theorem 4.1. LetF :H →Rbe as above andx_i ∈H (i= 1, ..., n). Suppose that there exists the vectors γ, φ ∈ H such that hx_i−γ, φ−x_ii ≥ 0 for all i ∈ {1, ..., m} andm, M ∈ H such that h∇F(x_i)−m, M − ∇F (x_i)i ≥ 0 for all i∈ {1, ..., m}. Then for all p_i ≥0 (i= 1, ..., m)withP_m :=

m

P

i=1

p_i >0, we have the inequality

(4.2) 0≤ 1 Pm

m

X

i=1

p_iF (x_i)−F 1 Pm

m

X

i=1

p_ix_i

!

≤ 1

4kφ−γk kM −mk.

Proof. Choose in (4.1),x= _P¹

M

m

P

i=1

p_ix_iandy=x_j to obtain

(4.3) F 1

P_m

m

X

i=1

pixi

!

−F (xj)≥

*

∇F (xj), 1 P_m

m

X

i=1

pixi−xj

+

for allj ∈ {1, ..., n}.

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If we multiply (4.3) byp_j ≥0and sum overj from1tom, we have P_mF 1

P_m

m

X

i=1

p_ix_i

!

−

m

X

j=1

p_jF (x_j)

≥ 1 P_m

* _m X

j=1

p_j∇F (x_j),

m

X

i=1

p_ix_i +

−

m

X

i=1

h∇F (x_i), x_ii.

Dividing byP_m >0,we obtain the inequality

0 ≤ 1

P_m

m

X

i=1

p_iF (x_i)−F 1 P_m

m

X

i=1

p_ix_i

! (4.4)

≤ 1 P_m

m

X

i=1

pih∇F (xi), xii

−

* 1 P_m

m

X

i=1

p_i∇F (x_i), 1 P_m

m

X

i=1

p_ix_i +

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 onRⁿ[9].

Applying Theorem3.1 for real inner product spaces, X = φ, x = γ, y_i =

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∇F (x_i),y=m,Y =M andn =m, we easily deduce (4.5) 1

P_m

m

X

i=1

pihxi,∇F (xi)i −

* 1 P_m

m

X

i=1

pixi, 1 P_m

m

X

i=1

pi∇F (xi) +

≤ 1

4kΦ−φk kM −mk 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) hx_i−γ, φ−x_ii ≥0, h∇F (x_i)−m, M − ∇F (x_i)i ≥0, for alli∈ {1, ..., m}can be replaced by the more general conditions (4.7)

m

X

i=1

pihxi−γ, φ−xii ≥0 and

m

X

i=1

pih∇F (xi)−m, M − ∇F(xi)i ≥0

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 x_i (i= 1, ..., n) are known. In other words, it provides the opportunity to estimate the difference

∆ (F, x, p) := 1 P_m

m

X

i=1

p_iF (x_i)−F 1 P_m

m

X

i=1

p_ix_i

! ,

where the differenceskφ−γkandkM −mkare known.

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Remark 4.3. For example, if we know thath∇F (x_i)−m, M − ∇F (x_i)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 hx_i−γ, φ−x_ii ≥ 0 for all i ∈ {1, ..., m} and kφ−γk ≤ _kM−mk^4ε (ε >0), then by (4.2), we can conclude that

0≤∆ (F, x, p)≤ε.

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5. Applications for Some Discrete Transforms

Let(H;h·,·i)be an inner product space overK,K=C,Randx¯= (x₁, ..., x_n) be a sequence of vectors inH.

For a givenm∈K, define the discrete Fourier transform (5.1) F_w(¯x) (m) =

n

X

k=1

exp (2wimk)×x_k, m= 1, ..., n.

The complex number

n

P

k=1

exp (2wimk)hx_k, y_ki is actually the usual Fourier transform of the vector (hx₁, y₁i, ...,hx_n, y_ni) ∈ Kⁿ and will be denoted by

(5.2) F_w(¯x·y) (m) =¯

n

X

k=1

exp (2wimk)hx_k, y_ki, m = 1, ..., n.

The following result holds.

Theorem 5.1. Let x,¯ y¯∈ Hⁿbe sequences of vectors such that there exists the vectorsc, C, y, Y ∈Hwith the properties

(5.3) RehC−exp (2wimk)x_k,exp (2wimk)x_k−ci ≥0, k, m= 1, ..., n and

(5.4) RehY −y_k, y_k−yi ≥0, k = 1, ..., n.

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Then we have the inequality

(5.5)

F_w(¯x·y) (m)¯ −

*

F_w(¯x) (m), 1 n

n

X

k=1

y_k +

≤ n

4 kC−ck kY −yk,

for allm ∈ {1, ..., n}.

The proof follows by Theorem3.1applied forp_k= _n¹ and for the sequences x_k →c_k= exp (2wimk)x_kandy_k (k= 1, ..., n). We omit the details.

We can also consider the Mellin transform

(5.6) M(¯x) (m) :=

n

X

k=1

k^m−1x_k, m= 1, ..., n,

of the sequencex¯= (x₁, ..., x_n)∈Hⁿ. We remark that the complex number

n

P

k=1

k^m−1hx_k, y_kiis actually the Mellin transform of the vector(hx₁, y₁i, ...,hx_n, y_ni)∈Kⁿand will be denoted by (5.7) M(¯x·y) (m) :=¯

n

X

k=1

k^m−1hx_k, y_ki.

The following theorem holds.

Theorem 5.2. Let x,¯ y¯∈ Hⁿbe sequences of vectors such that there exist the vectorsd, D, y, Y ∈H with the properties

(5.8) Re

D−k^m−1xk, k^m−1xk−d

≥0

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for allk, m∈ {1, ..., n},and (5.4) is fulfilled.

Then we have the inequality

(5.9)

M(¯x·y) (m)¯ −

*

M(¯x) (m), 1 n

n

X

k=1

y_k +

≤ n

4 kD−dk kY −yk

for allm ∈ {1, ..., n}.

The proof follows by Theorem3.1applied forp_k= _n¹ and for the sequences x_k →d_k =kx_kandy_k(k = 1, ..., n). We omit the details.

Another result which connects the Fourier transforms for different parame- terswalso holds.

Theorem 5.3. Let x,¯ y¯ ∈ Hⁿ and w, z ∈ K. If there exists the vectors e, E, f, F ∈Hsuch that

RehE−exp (2wimk)x_k,exp (2wimk)x_k−ei ≥0, k, m= 1, ..., n and

RehF −exp (2zimk)y_k,exp (2zimk)y_k−fi ≥0, k, m= 1, ..., n then we have the inequality:

1

nFw+z(¯x·y) (m)¯ − 1

nFw(¯x) (m),1

nFz(¯y) (m)

≤ 1

4kE−ek kF −fk, for allm ∈ {1, ..., n}.

The proof follows by Theorem3.1for the sequencesexp (2wimk)x_k, exp (2zimk)y_k (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 r- Hölder’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 applica- tions 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 ANDG.L. BOOTH, On a Grüss-Lupa¸s type inequality and its applications for the estimation ofp-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.

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[9] S.S. DRAGOMIR ANDC.J. GOH, A counterpart of Jensen’s discrete inequality for differentiable convex mappings and applications in informa- tion theory, Mathl. Comput. Modelling, 24(2) (1996), 1–11.

[10] A.M. FINK, A treatise on Grüss’ inequality, submitted.

[11] G. GRÜSS, Über das Maximum des absoluten Betrages von

1 b−a

Rb

a f(x)g(x)dx − _(b−a)¹ 2

Rb

a f(x)dxRb

ag(x)dx, Math. Z., 39 (1935), 215–226.

[12] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.