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volume 6, issue 2, article 55, 2005.

Received 01 April, 2005;

accepted 23 April, 2005.

Communicated by:J.M. Rassias

Abstract Contents

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Journal of Inequalities in Pure and Applied Mathematics

SOME BOAS-BELLMAN TYPE INEQUALITIES IN 2-INNER PRODUCT SPACES

S.S. DRAGOMIR, Y.J. CHO, S.S. KIM, AND A. SOFO

School of Computer Science and Mathematics Victoria University of Technology

PO Box 14428, MCMC Victoria 8001, Australia.

EMail:sever.dragomir@vu.edu.au URL:http://rgmia.vu.edu.au/dragomir Department of Mathematics College of Education

Gyeongsang National University Chinju 660-701, Korea.

EMail:yjcho@gsnu.ac.kr Department of Mathematics Dongeui University Pusan 614-714, Korea.

EMail:sskim@deu.ac.kr

School of Computer Science and Mathematics Victoria University of Technology

PO Box 14428, MCMC Victoria 8001, Australia.

EMail:sofo@matilda.vu.edu.au URL:http://rgmia.vu.edu.au/sofo

c

2000Victoria University ISSN (electronic): 1443-5756 100-05

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Some Boas-Bellman Type Inequalities in 2-Inner Product

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S.S. Dragomir, Y.J. Cho, S.S. Kim and A. Sofo

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J. Ineq. Pure and Appl. Math. 6(2) Art. 55, 2005

Abstract

Some inequalities in 2-inner product spaces generalizing Bessel’s result that are similar to the Boas-Bellman inequality from inner product spaces, are given.

Applications for determinantal integral inequalities are also provided.

2000 Mathematics Subject Classification:26D15, 26D10, 46C05, 46C99.

Key words: Bessel’s inequality in 2-Inner Product Spaces, Boas-Bellman type in- equalities, 2-Inner Products, 2-Norms.

S.S. Dragomir and Y.J. Cho greatly acknowledge the financial support from the Brain Pool Program (2002) of the Korean Federation of Science and Technology Societies.

The research was performed under the "Memorandum of Understanding" between Victoria University and Gyeongsang National University.

Contents

1 Introduction. . . 3

2 Bessel’s Inequality in 2-Inner Product Spaces. . . 5

3 Some Inequalities for 2-Norms . . . 10

4 Some Inequalities for Fourier Coefficients. . . 17

5 Some Boas-Bellman Type Inequalities in 2-Inner Product Spaces. . . 19

6 Applications for Determinantal Integral Inequalities . . . 25 References

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S.S. Dragomir, Y.J. Cho, S.S. Kim and A. Sofo

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1. Introduction

Let(H; (·,·))be an inner product space over the real or complex number field K. If (ei)1≤i≤n are orthonormal vectors in the inner product space H, i.e., (ei, ej) = δij for all i, j ∈ {1, . . . , n} where δij is the Kronecker delta, then the following inequality is well known in the literature as Bessel’s inequality (see for example [9, p. 391]):

n

X

i=1

|(x, ei)|2 ≤ kxk2

for any x∈H.

For other results related to Bessel’s inequality, see [5] – [7] and Chapter XV in the book [9].

In 1941, R.P. Boas [2] and in 1944, independently, R. Bellman [1] proved the following generalization of Bessel’s inequality (see also [9, p. 392]).

Theorem 1.1. Ifx, y1, . . . , ynare elements of an inner product space(H; (·,·)), then the following inequality:

n

X

i=1

|(x, yi)|2 ≤ kxk2

max

1≤i≤nkyik2+ X

1≤i6=j≤n

|(yi, yj)|2

!12

holds.

It is the main aim of the present paper to point out the corresponding version of Boas-Bellman inequality in 2-inner product spaces. Some natural general- izations and related results are also pointed out. Applications for determinantal integral inequalities are provided.

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Some Boas-Bellman Type Inequalities in 2-Inner Product

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For a comprehensive list of fundamental results on 2-inner product spaces and linear 2-normed spaces, see the recent books [3] and [8] where further ref- erences are given.

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2. Bessel’s Inequality in 2-Inner Product Spaces

The concepts of 2-inner products and 2-inner product spaces have been inten- sively studied by many authors in the last three decades. A systematic presenta- tion of the recent results related to the theory of2-inner product spaces as well as an extensive list of the related references can be found in the book [3]. Here we give the basic definitions and the elementary properties of2-inner product spaces.

LetX be a linear space of dimension greater than1over the fieldK=Rof real numbers or the fieldK =Cof complex numbers. Suppose that(·,·|·)is a K-valued function defined onX×X×Xsatisfying the following conditions:

(2I1) (x, x|z)≥0and(x, x|z) = 0if and only ifxandzare linearly dependent;

(2I2) (x, x|z) = (z, z|x), (2I3) (y, x|z) = (x, y|z),

(2I4) (αx, y|z) =α(x, y|z)for any scalarα∈K, (2I5) (x+x0, y|z) = (x, y|z) + (x0, y|z).

(·,·|·) is called a 2-inner product on X and (X,(·,·|·)) is called a 2-inner product space (or2-pre-Hilbert space). Some basic properties of2-inner prod- ucts(·,·|·)can be immediately obtained as follows [4]:

(1) IfK=R, then(2I3)reduces to

(y, x|z) = (x, y|z).

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Some Boas-Bellman Type Inequalities in 2-Inner Product

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(2) From(2I3)and(2I4), we have

(0, y|z) = 0, (x,0|z) = 0 and

(2.1) (x, αy|z) = ¯α(x, y|z).

(3) Using(2I2)–(2I5), we have

(z, z|x±y) = (x±y, x±y|z) = (x, x|z) + (y, y|z)±2 Re(x, y|z) and

(2.2) Re(x, y|z) = 1

4[(z, z|x+y)−(z, z|x−y)].

In the real case, (2.2) reduces to

(2.3) (x, y|z) = 1

4[(z, z|x+y)−(z, z|x−y)]

and, using this formula, it is easy to see that, for anyα∈R,

(2.4) (x, y|αz) = α2(x, y|z).

In the complex case, using (2.1) and (2.2), we have Im(x, y|z) = Re[−i(x, y|z)] = 1

4[(z, z|x+iy)−(z, z|x−iy)],

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which, in combination with (2.2), yields (2.5) (x, y|z) = 1

4[(z, z|x+y)−(z, z|x−y)] +i

4[(z, z|x+iy)−(z, z|x−iy)].

Using the above formula and (2.1), we have, for anyα∈C,

(2.6) (x, y|αz) = |α|2(x, y|z).

However, forα∈R, (2.6) reduces to (2.4). Also, from (2.6) it follows that (x, y|0) = 0.

(4) For any three given vectorsx, y, z ∈X, consider the vectoru= (y, y|z)x− (x, y|z)y.By(2I1), we know that(u, u|z) ≥ 0with the equality if and only if uandz are linearly dependent. The inequality(u, u|z)≥0can be rewritten as (2.7) (y, y|z)[(x, x|z)(y, y|z)− |(x, y|z)|2]≥0.

Forx=z, (2.7) becomes

−(y, y|z)|(z, y|z)|2 ≥0, which implies that

(2.8) (z, y|z) = (y, z|z) = 0,

providedyandzare linearly independent. Obviously, whenyandzare linearly dependent, (2.8) holds too. Thus (2.8) is true for any two vectors y, z ∈ X.

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Some Boas-Bellman Type Inequalities in 2-Inner Product

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S.S. Dragomir, Y.J. Cho, S.S. Kim and A. Sofo

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Now, if y andz are linearly independent, then (y, y|z) > 0and, from (2.7), it follows that

(2.9) |(x, y|z)|2 ≤(x, x|z)(y, y|z).

Using (2.8), it is easy to check that (2.9) is trivially fulfilled when yandz are linearly dependent. Therefore, the inequality (2.9) holds for any three vectors x, y, z ∈ X and is strict unless the vectorsu = (y, y|z)x−(x, y|z)yandz are linearly dependent. In fact, we have the equality in (2.9) if and only if the three vectorsx, y andz are linearly dependent.

In any given 2-inner product space (X,(·,· | ·)), we can define a function k · | · konX×Xby

(2.10) kx|zk=p

(x, x|z) for allx, z ∈X.

It is easy to see that this function satisfies the following conditions:

(2N1) kx|zk ≥0andkx|zk= 0if and only ifxandz are linearly dependent, (2N2) kz|xk=kx|zk,

(2N3) kαx|zk=|α|kx|zkfor any scalarα∈K, (2N4) kx+x0|zk ≤ kx|zk+kx0|zk.

Any functionk · | · kdefined onX×X and satisfying the conditions(2N1) – (2N4)is called a2-norm onX and (X,k · | · k)is called a linear 2-normed

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space [8]. Whenever a2-inner product space(X,(·,·|·))is given, we consider it as a linear2-normed space(X,k · | · k)with the2-norm defined by (2.10).

Let(X; (·,·|·))be a 2-inner product space over the real or complex number field K. If (ei)1≤i≤n are linearly independent vectors in the 2-inner product spaceX,and, for a givenz ∈X,(ei, ej|z) =δij for alli, j ∈ {1, . . . , n}where δij is the Kronecker delta (we say that the family(ei)1≤i≤n isz−orthonormal), then the following inequality is the corresponding Bessel’s inequality (see for example [4]) for the z−orthonormal family (ei)1≤i≤n in the 2-inner product space(X; (·,·|·)):

(2.11)

n

X

i=1

|(x, ei|z)|2 ≤ kx|zk2

for any x∈ X.For more details about this inequality, see the recent paper [4]

and the references therein.

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S.S. Dragomir, Y.J. Cho, S.S. Kim and A. Sofo

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3. Some Inequalities for 2-Norms

We start with the following lemma which is also interesting in itself.

Lemma 3.1. Let z1, . . . , zn, z ∈ X and µ1, . . . , µn ∈ K. Then one has the inequality:

n

X

i=1

µizi|z

2

(3.1)

















1≤i≤nmax|µi|2Pn

i=1kzi|zk2; Pn

i=1i|α1 Pn

i=1kzi|zkβ1 , where α >1,α1 + 1β = 1;

Pn

i=1i|2 max

1≤i≤nkzi|zk2,

+





















1≤i6=j≤nmax {|µiµj|}P

1≤i6=j≤n|(zi, zj|z)|; h

(Pn

i=1i|γ)2 − Pn

i=1i|iγ1

P

1≤i6=j≤n

|(zi, zj|z)|δ

!1δ ,

where γ >1, 1γ +1δ = 1;

h (Pn

i=1i|)2 −Pn

i=1i|2i

1≤i6=j≤nmax |(zi, zj|z)|.

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Proof. We observe that

n

X

i=1

µizi|z

2

=

n

X

i=1

µizi,

n

X

j=1

µjzj|z

! (3.2)

=

n

X

i=1 n

X

j=1

µiµj(zi, zj|z)

=

n

X

i=1 n

X

j=1

µiµj(zi, zj|z)

n

X

i=1 n

X

j=1

i| |µj| |(zi, zj|z)|

=

n

X

i=1

i|2kzi|zk2+ X

1≤i6=j≤n

i| |µj| |(zi, zj|z)|. Using Hölder’s inequality, we may write that

(3.3)

n

X

i=1

i|2kzi|zk2

























1≤i≤nmax|µi|2

n

P

i=1

kzi|zk2;

n P

i=1

i|

α1 n P

i=1

kzi|zk β1

, where α >1,α1 + 1β = 1;

n

P

i=1

i|2 max

1≤i≤nkzi|zk2.

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By Hölder’s inequality for double sums, we also have X

1≤i6=j≤n

i| |µj| |(zi, zj|z)|

(3.4)





















1≤i6=j≤nmax |µiµj| P

1≤i6=j≤n

|(zi, zj|z)|;

P

1≤i6=j≤n

i|γj|γ

!1γ

P

1≤i6=j≤n

|(zi, zj|z)|δ

!1δ , where γ >1, 1γ +1δ = 1;

P

1≤i6=j≤n

i| |µj| max

1≤i6=j≤n|(zi, zj|z)|,

=

























1≤i6=j≤nmax {|µiµj|} P

1≤i6=j≤n

|(zi, zj|z)|;

"

n P

i=1

i|γ 2

n

P

i=1

i| #1γ

P

1≤i6=j≤n

|(zi, zj|z)|δ

!1δ , where γ >1, γ1 + 1δ = 1;

"

n P

i=1

i| 2

n

P

i=1

i|2

#

1≤i6=j≤nmax |(zi, zj|z)|.

Utilizing (3.3) and (3.4) in (3.2), we may deduce the desired result (3.1).

Remark 1. Inequality (3.1) contains in fact 9 different inequalities which may be obtained combining the first 3 ones with the last 3 ones.

A particular result of interest is embodied in the following inequality.

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Corollary 3.2. With the assumptions in Lemma3.1, we have

n

X

i=1

µizi|z

2

(3.5)

n

X

i=1

i|2 (

1≤i≤nmax kzi|zk2

+ h Pn

i=1i|22

−Pn

i=1i|4i12 Pn

i=1i|2

X

1≤i6=j≤n

|(zi, zj|z)|2

!12





n

X

i=1

i|2

1≤i≤nmaxkzi|zk2 + X

1≤i6=j≤n

|(zi, zj|z)|2

!12

 .

The first inequality follows by taking the third branch in the first curly bracket with the second branch in the second curly bracket forγ =δ = 2.

The second inequality in (3.5) follows by the fact that

n

X

i=1

i|2

!2

n

X

i=1

i|4

1 2

n

X

i=1

i|2.

Applying the following Cauchy-Bunyakovsky-Schwarz inequality (3.6)

n

X

i=1

ai

!2

≤n

n

X

i=1

a2i, ai ∈R+, 1≤i≤n,

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we may write that (3.7)

n

X

i=1

i|γ

!2

n

X

i=1

i| ≤(n−1)

n

X

i=1

i| (n≥1)

and (3.8)

n

X

i=1

i|

!2

n

X

i=1

i|2 ≤(n−1)

n

X

i=1

i|2 (n≥1).

Also, it is obvious that:

(3.9) max

1≤i6=j≤n{|µiµj|} ≤ max

1≤i≤ni|2.

Consequently, we may state the following coarser upper bounds forkPn

i=1µizi|zk2 that may be useful in applications.

Corollary 3.3. With the assumptions in Lemma3.1, we have the inequalities:

n

X

i=1

µizi|z

2

(3.10)

















1≤i≤nmax|µi|2Pn

i=1kzi|zk2; Pn

i=1i|α1 Pn

i=1kzi|zkβ1 ,

where α >1,α1 + 1β = 1, Pn

i=1i|2 max

1≤i≤nkzi|zk2,

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+

















1≤i≤nmax |µi|2P

1≤i6=j≤n|(zi, zj|z)|; (n−1)γ1 Pn

i=1i|1γ P

1≤i6=j≤n|z(i, zj|z)|δ1δ ,

where γ >1, γ1 + 1δ = 1;

(n−1)Pn

i=1i|2 max

1≤i6=j≤n|(zi, zj|z)|.

The proof is obvious by Lemma3.1on applying the inequalities (3.7) – (3.9).

Remark 2. The following inequalities which are incorporated in (3.10) are of special interest:

(3.11)

n

X

i=1

µizi|z

2

≤ max

1≤i≤ni|2

" n X

i=1

kzi|zk2+ X

1≤i6=j≤n

|(zi, zj|z)|

#

;

n

X

i=1

µizi|z

2

(3.12)

n

X

i=1

i|2p

!1p

n

X

i=1

kzi|zk2q

!1q

+ (n−1)1p X

1≤i6=j≤n

|(zi, zj|z)|q

!1q

,

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wherep >1, 1p +1q = 1;and

n

X

i=1

µizi|z

2

(3.13)

n

X

i=1

i|2

1≤i≤nmax kzi|zk2+ (n−1) max

1≤i6=j≤n|(zi, zj|z)|

.

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4. Some Inequalities for Fourier Coefficients

The following results holds

Theorem 4.1. Letx, y1, . . . , yn, zbe vectors of a 2-inner product space(X; (·,·|·)) andc1, . . . , cn ∈K(K=C,R).Then one has the inequalities:

n

X

i=1

ci(x, yi|z)

2

(4.1)

≤ kx|zk2×

















1≤i≤nmax |ci|2Pn

i=1kyi|zk2; Pn

i=1|ci|α1 Pn

i=1kyi|zkβ1 , where α >1,α1 + 1β = 1;

n

P

i=1

|ci|2 max

1≤i≤nkyi|zk2;

+kx|zk2×

















1≤i6=j≤nmax {|cicj|} P

1≤i6=j≤n

|(yi, yj|z)|; h

(Pn

i=1|ci|γ)2− Pn

i=1|ci|iγ1

× P

1≤i6=j≤n

|(yi, yj|z)|δ

!1δ

, where γ >1, γ1+1δ= 1;

h (Pn

i=1|ci|)2−Pn

i=1|ci|2i

1≤i6=j≤nmax |(yi, yj|z)|. Proof. We note that

n

X

i=1

ci(x, yi|z) = x,

n

X

i=1

ciyi|z

! .

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Using Schwarz’s inequality in 2-inner product spaces, we have

n

X

i=1

ci(x, yi|z)

2

≤ kx|zk2

n

X

i=1

ciyi|z

2

.

Now using Lemma 3.1 with µi = ci, zi = yi (i= 1, . . . , n), we deduce the desired inequality (4.1).

The following particular inequalities that may be obtained by the Corollaries 3.2,3.3, and Remark2, hold.

Corollary 4.2. With the assumptions in Theorem4.1, one has the inequalities:

(4.2)

n

X

i=1

ci(x, yi|z)

2

≤ kx|zk2×









































 Pn

i=1|ci|2

1≤i≤nmaxkyi|zk2+ P

1≤i6=j≤n

|(yi, yj|z)|2

!12

;

1≤i≤nmax|ci|2 (

Pn

i=1kyi|zk2+ P

1≤i6=j≤n

|(yi, yj|z)|

)

;

Pn

i=1|ci|2p1pn Pn

i=1kyi|zk2q1q

+ (n−1)1p P

1≤i6=j≤n

|(yi, yj|z)|q

!1q

 ,

where p > 1,1p + 1q = 1;

Pn i=1|ci|2

1≤i≤nmax kyi|zk2+ (n−1) max

1≤i6=j≤n|(yi, yj|z)|

.

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5. Some Boas-Bellman Type Inequalities in 2-Inner Product Spaces

If one chooses ci = (x, yi|z) (i= 1, . . . , n)in (4.1), then it is possible to ob- tain 9 different inequalities between the Fourier coefficients (x, yi|z) and the 2-norms and 2-inner products of the vectorsyi (i= 1, . . . , n).We restrict our- selves only to those inequalities that may be obtained from (4.2).

From the first inequality in (4.2) forci = (x, yi|z),we get

n

X

i=1

|(x, yi|z)|2

!2

≤ kx|zk2

n

X

i=1

|(x, yi|z)|2

1≤i≤nmaxkyi|zk2+ X

1≤i6=j≤n

|(yi, yj|z)|2

!12

 ,

which is clearly equivalent to the following Boas-Bellman type inequality for 2-inner products:

n

X

i=1

|(x, yi|z)|2 (5.1)

≤ kx|zk2

1≤i≤nmaxkyi|zk2+ X

1≤i6=j≤n

|(yi, yj|z)|2

!12

 .

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From the second inequality in (4.2) forci = (x, yi|z),we get

n

X

i=1

|(x, yi|z)|2

!2

≤ kx|zk2 max

1≤i≤n|(x, yi|z)|2 ( n

X

i=1

kyi|zk2+ X

1≤i6=j≤n

|(yi, yj|z)|

) .

Taking the square root in this inequality, we obtain

n

X

i=1

|(x, yi|z)|2 (5.2)

≤ kx|zk max

1≤i≤n|(x, yi|z)|

( n X

i=1

kyi|zk2 + X

1≤i6=j≤n

|(yi, yj|z)|

)12

for anyx, y1, . . . , yn, zvectors in the 2-inner product space(X; (·,·|·)).

If we assume that(ei)1≤i≤nis an orthonormal family inXwith respect with the vector z, i.e., (ei, ej|z) = δij for all i, j ∈ {1, . . . , n}, then by (5.1) we deduce Bessel’s inequality (2.11), while from (5.2) we have

(5.3)

n

X

i=1

|(x, ei|z)|2 ≤√

nkx|zk max

1≤i≤n|(x, ei|z)|, x∈X.

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From the third inequality in (4.2) forci = (x, yi|z),we deduce

n

X

i=1

|(x, yi|z)|2

!2

≤ kx|zk2

n

X

i=1

|(x, yi|z)|2p

!1p

×

n

X

i=1

kyi|zk2q

!1q

+ (n−1)1p X

1≤i6=j≤n

|(yi, yj|z)|q

!1q

 forp >1with 1p +1q = 1.Taking the square root in this inequality, we get

n

X

i=1

|(x, yi|z)|2 (5.4)

≤ kx|zk

n

X

i=1

|(x, yi|z)|2p

!2p1

×

n

X

i=1

kyi|zk2q

!1q

+ (n−1)p1 X

1≤i6=j≤n

|(yi, yj|z)|q

!1q

1 2

for anyx, y1, . . . , yn, z∈Xandp >1with 1p +1q = 1.

The above inequality (5.4) becomes, for an orthornormal family (ei)1≤i≤n

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with respect of the vectorz,

(5.5)

n

X

i=1

|(x, ei|z)|2 ≤n1q kx|zk

n

X

i=1

|(x, ei|z)|2p

!2p1

, x∈X.

Finally, the choiceci = (x, yi|z) (i= 1, . . . , n)will produce in the last inequal- ity in (4.2)

n

X

i=1

|(x, yi|z)|2

!2

≤ kx|zk2

n

X

i=1

|(x, yi|z)|2

1≤i≤nmaxkyi|zk2 + (n−1) max

1≤i6=j≤n|(yi, yj|z)|

,

which gives the following inequality (5.6)

n

X

i=1

|(x, yi|z)|2 ≤ kx|zk2

1≤i≤nmax kyi|zk2+ (n−1) max

1≤i6=j≤n|(yi, yj|z)|

for anyx, y1, . . . , yn, z ∈X.

It is obvious that (5.6) will give forz−orthonormal families, the Bessel in- equality mentioned in(2.11)from the Introduction.

Remark 3. Observe that, both the Boas-Bellman type inequality for 2-inner products incorporated in (5.1) and the inequality (5.6) become in the particular case ofz−orthonormal families, the regular Bessel’s inequality. Consequently, a comparison of the upper bounds is necessary.

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It suffices to consider the quantities

An:= X

1≤i6=j≤n

|(yi, yj|z)|2

!12

and

Bn:= (n−1) max

1≤i6=j≤n|(yi, yj|z)|, wheren ≥1,andy1, . . . , yn, z ∈X.

If we choosen= 3,we have A3 =√

2 (y1, y2|z)2+ (y2, y3|z)2+ (y3,y1|z)212 and

B3 = 2 max{|(y1, y2|z)|,|(y2, y3|z)|,|(y3,y1|z)|}, wherey1, y2, y3, z ∈X.

If we consider a := |(y1, y2|z)| ≥ 0, b := |(y2, y3|z)| ≥ 0 and c :=

|(y3,y1|z)| ≥0, then we have to compare

A3 :=√

2 a2+b2+c212 with

B3 = 2 max{a, b, c}. If we assume that b = c = 1,thenA3 := √

2 (a2+ 2)12 , B3 = 2 max{a,1}. Finally, fora = 1,we getA3 =√

6, B3 = 2 showing thatA3 > B3,while for a = 2we haveA3 =√

12, B3 = 4showing thatB3 > A3.

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In conclusion, we may state that the bounds

M1 :=kx|zk2

1≤i≤nmaxkyi|zk2+ X

1≤i6=j≤n

|(yi, yj|z)|2

!12

and

M2 :=kx|zk2

1≤i≤nmaxkyi|zk2+ (n−1) max

1≤i6=j≤n|(yi, yj|z)|

for the Bessel’s sumPn

i=1|(x, yi|z)|2 cannot be compared in general, meaning that sometimes one is better than the other.

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6. Applications for Determinantal Integral Inequalities

Let(Ω,Σ, µ)be a measure space consisting of a setΩ,aσ−algebraΣof sub- sets ofΩand a countably additive and positive measureµonΣwith values in R∪ {∞}.

Denote by L2ρ(Ω) the Hilbert space of all real-valued functions f defined on Ωthat are2−ρ−integrable onΩ, i.e.,R

ρ(s)|f(s)|2dµ(s) < ∞, where ρ: Ω→[0,∞)is a measurable function onΩ.

We can introduce the following 2-inner product onL2ρ(Ω)by the formula (f, g|h)ρ

(6.1)

:= 1 2

Z

Z

ρ(s)ρ(t)

f(s) f(t) h(s) h(t)

g(s) g(t) h(s) h(t)

dµ(s)dµ(t),

where, by

f(s) f(t) h(s) h(t)

,

we denote the determinant of the matrix

" f(s) f(t) h(s) h(t)

# ,

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generating the 2-norm onL2ρ(Ω)expressed by

(6.2) kf|hkρ :=

 1 2

Z

Z

ρ(s)ρ(t)

f(s) f(t) h(s) h(t)

2

dµ(s)dµ(t)

1 2

.

A simple calculation with integrals reveals that

(6.3) (f, g|h)ρ=

R

ρf gdµ R

ρf hdµ R

ρghdµ R

ρh2dµ and

(6.4) kf|hkρ=

R

ρf2dµ R

ρf hdµ R

ρf hdµ R

ρh2

1 2

,

where, for simplicity, instead of R

ρ(s)f(s)g(s)dµ(s), we have written R

ρf gdµ.

Using the representations (6.2), (6.3) and the inequalities for 2-inner prod- ucts and 2-norms established in the previous sections, one may state some in- teresting determinantal integral inequalities, as follows.

Proposition 6.1. Let f, g1, . . . , gn, h ∈ L2ρ(Ω), where ρ : Ω → [0,∞) is a

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measurable function onΩ.Then we have the inequality

n

X

i=1

R

ρf gidµ R

ρf hdµ R

ρgihdµ R

ρh2

2

R

ρf2dµ R

ρf hdµ R

ρf hdµ R

ρh2

× (

1≤i≤nmax

R

ρg2idµ R

ρgihdµ R

ρgihdµ R

ρh2

+

n

X

1≤i6=j≤n

R

ρgjgidµ R

ρgjhdµ R

ρgihdµ R

ρh2

2

1 2



 .

The proof follows by the inequality(5.1)applied for the 2-inner product and 2-norm defined in(??)and(6.1),and utilizing the identities(6.2)and(6.3).

If one uses the inequality (5.6), then the following result may also be stated.

Proposition 6.2. Let f, g1, . . . , gn, h ∈ L2ρ(Ω), where ρ : Ω → [0,∞) is a measurable function onΩ.Then we have the inequality

n

X

i=1

R

ρf gidµ R

ρf hdµ R

ρgihdµ R

ρh2

2

R

ρf2dµ R

ρf hdµ R

ρf hdµ R

ρh2

× (

1≤i≤nmax

R

ρg2idµ R

ρgihdµ R

ρgihdµ R

ρh2

+ (n−1) max

1≤i6=j≤n

R

ρgjgidµ R

ρgjhdµ R

ρgihdµ R

ρh2

) .

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References

[1] R. BELLMAN, Almost orthogonal series, Bull. Amer. Math. Soc., 50 (1944), 517–519.

[2] R.P. BOAS, A general moment problem, Amer. J. Math., 63 (1941), 361–

370.

[3] Y.J. CHO, P.C.S. LIN, S.S. KIMANDA. MISIAK, Theory of 2-Inner Prod- uct Spaces, Nova Science Publishers, Inc., New York, 2001.

[4] Y.J. CHO, M. MATI ´C AND J.E. PE ˇCARI ´C, On Gram’s determinant in 2- inner product spaces, J. Korean Math. Soc., 38(6) (2001), 1125–1156.

[5] S.S. DRAGOMIRANDJ. SÁNDOR, On Bessel’s and Gram’s inequality in prehilbertian spaces, Periodica Math. Hung., 29(3) (1994), 197–205.

[6] S.S. DRAGOMIRANDB. MOND, On the Boas-Bellman generalisation of Bessel’s inequality in inner product spaces, Italian J. of Pure & Appl. Math., 3 (1998), 29–35.

[7] S.S. DRAGOMIR, B. MOND AND J.E. PE ˇCARI ´C, Some remarks on Bessel’s inequality in inner product spaces, Studia Univ. Babe¸s-Bolyai, Mathematica, 37(4) (1992), 77–86.

[8] R.W. FREESEANDY.J. CHO, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, Inc., New York, 2001.

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

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