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
volume 6, issue 2, article 55,
2005.
Received 01 April, 2005;
accepted 23 April, 2005.
Communicated by: J.M. Rassias
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
100-05
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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 inequalities, 2-Inner Products, 2-Norms.
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
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.
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
Title Page
Contents
1
2
3
4
5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bessel’s Inequality in 2-Inner Product Spaces . . . . . . . . . . . . .
Some Inequalities for 2-Norms . . . . . . . . . . . . . . . . . . . . . . . . .
Some Inequalities for Fourier Coefficients . . . . . . . . . . . . . . . .
Some Boas-Bellman Type Inequalities in 2-Inner Product
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Applications for Determinantal Integral Inequalities . . . . . . .
References
Contents
3
5
10
17
19
25
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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
|(x, ei )|2 ≤ kxk2
i=1
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. If x, y1 , . . . , yn are elements of an inner product space (H; (·, ·)) ,
then the following inequality:

! 12 
n
X
X
|(x, yi )|2 ≤ kxk2  max kyi k2 +
|(yi , yj )|2 
i=1
1≤i≤n
1≤i6=j≤n
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 generalizations and related results are also pointed out. Applications for determinantal
integral inequalities are provided.
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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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 references are given.
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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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 intensively studied by many authors in the last three decades. A systematic presentation of the recent results related to the theory of 2-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 of 2-inner product
spaces.
Let X be a linear space of dimension greater than 1 over the field K = R of
real numbers or the field K = C of complex numbers. Suppose that (·, ·|·) is a
K-valued function defined on X × X × X satisfying the following conditions:
(2I1 ) (x, x|z) ≥ 0 and (x, x|z) = 0 if and only if x and z are linearly dependent;
(2I2 ) (x, x|z) = (z, z|x),
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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(2I3 ) (y, x|z) = (x, y|z),
Contents
(2I4 ) (αx, y|z) = α(x, y|z) for any scalar α ∈ K,
0
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
0
(2I5 ) (x + x , y|z) = (x, y|z) + (x , y|z).
(·, ·|·) is called a 2-inner product on X and (X, (·, ·|·)) is called a 2-inner
product space (or 2-pre-Hilbert space). Some basic properties of 2-inner products (·, ·|·) can be immediately obtained as follows [4]:
(1) If K = R, then (2I3 ) reduces to
(y, x|z) = (x, y|z).
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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) = ᾱ(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)
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
and
(2.2)
1
Re(x, y|z) = [(z, z|x + y) − (z, z|x − y)].
4
In the real case, (2.2) reduces to
(2.3)
1
(x, y|z) = [(z, z|x + y) − (z, z|x − y)]
4
and, using this formula, it is easy to see that, for any α ∈ R,
(2.4)
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
(x, y|αz) = α2 (x, y|z).
In the complex case, using (2.1) and (2.2), we have
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1
Im(x, y|z) = Re[−i(x, y|z)] = [(z, z|x + iy) − (z, z|x − iy)],
4
J. Ineq. Pure and Appl. Math. 6(2) Art. 55, 2005
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which, in combination with (2.2), yields
1
i
(2.5) (x, y|z) = [(z, z|x+y)−(z, z|x−y)]+ [(z, z|x+iy)−(z, z|x−iy)].
4
4
Using the above formula and (2.1), we have, for any α ∈ C,
(x, y|αz) = |α|2 (x, y|z).
(2.6)
However, for α ∈ R, (2.6) reduces to (2.4). Also, from (2.6) it follows that
(x, y|0) = 0.
(4) For any three given vectors x, y, z ∈ X, consider the vector u = (y, y|z)x −
(x, y|z)y. By (2I1 ), we know that (u, u|z) ≥ 0 with the equality if and only if
u and z are linearly dependent. The inequality (u, u|z) ≥ 0 can be rewritten as
(2.7)
(y, y|z)[(x, x|z)(y, y|z) − |(x, y|z)|2 ] ≥ 0.
For x = z, (2.7) becomes
−(y, y|z)|(z, y|z)|2 ≥ 0,
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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which implies that
(2.8)
Close
(z, y|z) = (y, z|z) = 0,
provided y and z are linearly independent. Obviously, when y and z are linearly
dependent, (2.8) holds too. Thus (2.8) is true for any two vectors y, z ∈ X.
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Now, if y and z are linearly independent, then (y, y|z) > 0 and, 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 y and z are
linearly dependent. Therefore, the inequality (2.9) holds for any three vectors
x, y, z ∈ X and is strict unless the vectors u = (y, y|z)x − (x, y|z)y and z are
linearly dependent. In fact, we have the equality in (2.9) if and only if the three
vectors x, y and z are linearly dependent.
In any given 2-inner product space (X, (·, · | ·)), we can define a function
k · | · k on X × X by
p
(2.10)
kx|zk = (x, x|z)
for all x, z ∈ X.
It is easy to see that this function satisfies the following conditions:
(2N1 ) kx|zk ≥ 0 and kx|zk = 0 if and only if x and z are linearly dependent,
(2N2 ) kz|xk = kx|zk,
(2N3 ) kαx|zk = |α|kx|zk for any scalar α ∈ K,
(2N4 ) kx + x0 |zk ≤ kx|zk + kx0 |zk.
Any function k · | · k defined on X × X and satisfying the conditions (2N1 )
– (2N4 ) is called a 2-norm on X and (X, k · | · k) is called a linear 2-normed
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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space [8]. Whenever a 2-inner product space (X, (·, ·|·)) is given, we consider
it as a linear 2-normed space (X, k · | · k) with the 2-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
space X, and, for a given z ∈ X, (ei , ej |z) = δij for all i, j ∈ {1, . . . , n} where
δij is the Kronecker delta (we say that the family (ei )1≤i≤n is z−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
|(x, ei |z)|2 ≤ kx|zk2
i=1
for any x ∈ X. For more details about this inequality, see the recent paper [4]
and the references therein.
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
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
2
X
µi zi |z (3.1) i=1

P

max |µi |2 ni=1 kzi |zk2 ;


1≤i≤n


1


 Pn |µ |2α α1 Pn kz |zk2β β ,
i
i
i=1
i=1
≤

where
α
>
1, α1 + β1 = 1;




P


 ni=1 |µi |2 max kzi |zk2 ,
1≤i≤n

P
max
{|µ
µ
|}

i
j
1≤i6=j≤n |(zi , zj |z)| ;

 1≤i6=j≤n


! 1δ


1
h
i

P

 (Pn |µi |γ )2 − Pn |µi |2γ γ
,
|(zi , zj |z)|δ
i=1
i=1
+
1≤i6=j≤n



where γ > 1, γ1 + 1δ = 1;



hP
i

P


 ( n |µ |)2 − n |µ |2 max |(z , z |z)| .
i=1
i
i=1
i
1≤i6=j≤n
i
j
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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Proof. We observe that
n
2
X
(3.2)
µi zi |z =
i=1
=
n
X
µi zi ,
i=1
n
X
!
µj zj |z
j=1
n X
n
X
µi µj (zi , zj |z)
i=1 j=1
n X
n
X
=
µi µj (zi , zj |z)
≤
=
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
i=1 j=1
n X
n
X
|µi | |µj | |(zi , zj |z)|
i=1 j=1
n
X
X
|µi |2 kzi |zk2 +
i=1
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
|µi | |µj | |(zi , zj |z)| .
Title Page
1≤i6=j≤n
Using Hölder’s inequality, we may write that

n
2P


max
|µ
|
kzi |zk2 ;
i


1≤i≤n
i=1






n
α1 n
β1


n
P
 P
2α
2β
X
,
|µi |
kzi |zk
(3.3)
|µi |2 kzi |zk2 ≤
i=1
i=1


i=1

where α > 1, α1 + β1 = 1;







n
P



|µ |2 max kz |zk2 .
i
i=1
1≤i≤n
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By Hölder’s inequality for double sums, we also have
X
(3.4)
|µi | |µj | |(zi , zj |z)|
1≤i6=j≤n
≤











1≤i6=j≤n
P
|(zi , zj |z)| ;
1≤i6=j≤n
! γ1
|µi |γ |µj |γ
P
! 1δ
P
1≤i6=j≤n










=
max |µi µj |
|(zi , zj |z)|δ
where γ > 1,
P
1≤i6=j≤n
,
1≤i6=j≤n
1
γ
+
1
δ
|µi | |µj | max |(zi , zj |z)| ,
1≤i6=j≤n
P

max
{|µ
µ
|}
|(zi , zj |z)| ;
i
j


1≤i6=j≤n

1≤i6=j≤n


"

2 n
# γ1


n
P
P
P



|µi |γ −
|µi |2γ

i=1
i=1
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
! 1δ
|(zi , zj |z)|δ
Title Page
,
1≤i6=j≤n

where γ > 1,




"
#

2


n
n
P
P



|µi | −
|µi |2 max |(zi , zj |z)| .

1≤i6=j≤n
i=1
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
= 1;
1
γ
+
1
δ
= 1;
i=1
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 Lemma 3.1, we have
2
n
X
(3.5)
µi zi |z i=1
(
n
X
2
≤
|µi |
max kzi |zk2
1≤i≤n
i=1
h P
n
2 2
Pn
4
|µi | − i=1 |µi |
Pn
2
i=1 |µi |

n

X
2
≤
|µi |
max kzi |zk2 +
1≤i≤n
i=1
+
i=1
i 12
X
|(zi , zj |z)|2

! 12 



1≤i6=j≤n
X
2
|(zi , zj |z)|
! 12 

Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
.

1≤i6=j≤n
Title Page
The first inequality follows by taking the third branch in the first curly bracket
with the second branch in the second curly bracket for γ = δ = 2.
The second inequality in (3.5) follows by the fact that

 12
!2
n
n
n
X
X
X
2
4

|µi |
−
|µi |
≤
|µi |2 .
i=1
i=1
i=1
Applying the following Cauchy-Bunyakovsky-Schwarz inequality
!2
n
n
X
X
(3.6)
ai
≤n
a2i , ai ∈ R+ , 1 ≤ i ≤ n,
i=1
i=1
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we may write that
(3.7)
n
X
!2
γ
|µi |
−
i=1
n
X
2γ
|µi |
≤ (n − 1)
i=1
n
X
|µi |2γ
(n ≥ 1)
i=1
and
(3.8)
n
X
!2
|µi |
−
i=1
n
X
|µi |2 ≤ (n − 1)
i=1
n
X
|µi |2
(n ≥ 1) .
i=1
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
Also, it is obvious that:
(3.9)
max {|µi µj |} ≤ max |µi |2 .
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
1≤i≤n
1≤i6=j≤n
Consequently, we may state the following coarser upper bounds for k
that may be useful in applications.
Pn
i=1
Corollary 3.3. With the assumptions in Lemma 3.1, we have the inequalities:
2
n
X
(3.10)
µi zi |z i=1

2 Pn
2
max
|µ
|

i
i=1 kzi |zk ;

1≤i≤n



1

1 
Pn

2α α Pn
2β β

,
i=1 |µi |
i=1 kzi |zk
≤



where α > 1, α1 + β1 = 1,




 Pn |µ |2 max kz |zk2 ,

i=1
i
1≤i≤n
2
µi zi |zk
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+

2P
max
|µ
|
i

1≤i6=j≤n |(zi , zj |z)| ;

1≤i≤n




1

1 
δ δ
 (n − 1) γ1 Pn |µi |2γ γ P
,
i=1
1≤i6=j≤n |z (i , zj |z)|



where γ > 1,




P

 (n − 1) n |µi |2 max |(zi , zj |z)| .
i=1
1
γ
+
1
δ
= 1;
1≤i6=j≤n
The proof is obvious by Lemma 3.1 on applying the inequalities (3.7) – (3.9).
Remark 2. The following inequalities which are incorporated in (3.10) are of
special interest:
2
" n
#
n
X
X
X
2
2
|(zi , zj |z)| ;
(3.11)
µi zi |z ≤ max |µi |
kzi |zk +
1≤i≤n
i=1
i=1
1≤i6=j≤n
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
Title Page
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(3.12)
JJ
J
n
2
X
µi zi |z i=1
≤
n
X
! p1 
2p
|µi |
n
X

i=1
! 1q
Go Back
2q
kzi |zk
Close
i=1
! 1q 
+ (n − 1)
II
I
1
p
X
|(zi , zj |z)|q
,
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where p > 1,
(3.13)
1
p
+
1
q
= 1; and
2
n
X
µ
z
|z
i i i=1
n
X
2
2
≤
|µi | max kzi |zk + (n − 1) max |(zi , zj |z)| .
i=1
1≤i≤n
1≤i6=j≤n
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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4.
Some Inequalities for Fourier Coefficients
The following results holds
Theorem 4.1. Let x, y1 , . . . , yn , z be vectors of a 2-inner product space (X; (·, ·|·))
and c1 , . . . , cn ∈ K (K = C, R) . Then one has the inequalities:
n
2
X
(4.1) ci (x, yi |z)
i=1

P

max |ci |2 ni=1 kyi |zk2 ;

 1≤i≤n


1
1 
Pn

2α α Pn
2β β

,
i=1 |ci |
i=1 kyi |zk
≤ kx|zk2 ×
1
1
where α > 1, α + β = 1;




n

P


|ci |2 max kyi |zk2 ;

1≤i≤n
i=1
P

max {|ci cj |}
|(yi , yj |z)| ;


1≤i6=j≤n

1≤i6=j≤n


hP
i1

Pn

n
γ 2
2γ γ

 ( i=1 |ci | ) −
i=1 |ci |
! 1δ
+ kx|zk2 ×

P

δ

 ×
|(yi , yj |z)|
, where γ > 1, γ1 + 1δ = 1;



1≤i6
=
j≤n

i
 h Pn
P
2
( i=1 |ci |) − ni=1 |ci |2 max |(yi , yj |z)| .
1≤i6=j≤n
Proof. We note that
n
X
i=1
ci (x, yi |z) =
x,
n
X
i=1
!
ci yi |z
Some Boas-Bellman Type
Inequalities in 2-Inner Product
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.
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Using Schwarz’s inequality
in 2-inner
wehave
n
2 productspaces,
n
X
2
X
2
ci yi |z .
ci (x, yi |z) ≤ kx|zk i=1
i=1
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 Remark 2, hold.
Corollary 4.2. With the assumptions in Theorem 4.1, one has the inequalities:
2
n
X
(4.2) ci (x, yi |z)
i=1


! 21 



Pn
P

2
2
2


;
|(yi , yj |z)|
max kyi |zk +

i=1 |ci |


1≤i≤n

1≤i6=j≤n


(
)



P
P

n
2


max |ci |2
|(yi , yj |z)| ;

i=1 kyi |zk +

1≤i≤n

1≤i6
=
j≤n


1 n P
1
 Pn
2p p
n
2q q
2
|c
|
ky
|zk
i
i
i=1
i=1
≤ kx|zk ×

! 1q 




1
P

q

p

+
(n
−
1)
|(y
,
y
|z)|
,
i j




1≤i6
=
j≤n





where p > 1, p1 + 1q = 1;




Pn
2
2

 i=1 |ci | max kyi |zk + (n − 1) max |(yi , yj |z)| .
1≤i≤n
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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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 obtain 9 different inequalities between the Fourier coefficients (x, yi |z) and the
2-norms and 2-inner products of the vectors yi (i = 1, . . . , n) . We restrict ourselves only to those inequalities that may be obtained from (4.2).
From the first inequality in (4.2) for ci = (x, yi |z), we get
n
X
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
!2
|(x, yi |z)|2
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
i=1
2
≤ kx|zk
n
X
|(x, yi |z)|2
i=1


max kyi |zk2 +
1≤i≤n
X
|(yi , yj |z)|2
! 12 

,

1≤i6=j≤n
Title Page
Contents
which is clearly equivalent to the following Boas-Bellman type inequality for
2-inner products:
(5.1)
n
X
|(x, yi |z)|2
≤ kx|zk2
max ky |zk2 +
1≤i≤n i
II
I
Go Back
i=1


JJ
J
X
1≤i6=j≤n
|(yi , yj |z)|2
! 12 


Close
.
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From the second inequality in (4.2) for ci = (x, yi |z) , we get
n
X
!2
|(x, yi |z)|2
i=1
( n
X
≤ kx|zk2 max |(x, yi |z)|2
kyi |zk2 +
1≤i≤n
i=1
)
X
|(yi , yj |z)| .
1≤i6=j≤n
Taking the square root in this inequality, we obtain
(5.2)
n
X
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
|(x, yi |z)|2
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
i=1
≤ kx|zk max |(x, yi |z)|
1≤i≤n
( n
X
) 12
kyi |zk2 +
i=1
X
|(yi , yj |z)|
1≤i6=j≤n
for any x, y1 , . . . , yn , z vectors in the 2-inner product space (X; (·, ·|·)) .
If we assume that (ei )1≤i≤n is an orthonormal family in X with 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 ≤
√
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J
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n kx|zk max |(x, ei |z)| , x ∈ X.
1≤i≤n
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From the third inequality in (4.2) for ci = (x, yi |z) , we deduce
n
X
!2
|(x, yi |z)|2
i=1
n
X
≤ kx|zk2
! p1
|(x, yi |z)|2p
i=1
×

n
 X

for p > 1 with
(5.4)
n
X
! 1q
1
kyi |zk2q
X
+ (n − 1) p
i=1
1
p
+
|(yi , yj |z)|q
! 1q 


1≤i6=j≤n
1
q
= 1. Taking the square root in this inequality, we get
Contents
≤ kx|zk
n
X
! 2p1
JJ
J
|(x, yi |z)|2p
i=1

n
 X

S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
Title Page
|(x, yi |z)|2
i=1
×
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
! 1q
kyi |zk
2q
+ (n − 1)
1
p
i=1
X
1≤i6=j≤n
1
p
1
q
q
|(yi , yj |z)|
! 1q  12

II
I
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Close

for any x, y1 , . . . , yn , z ∈ X and p > 1 with + = 1.
The above inequality (5.4) becomes, for an orthornormal family (ei )1≤i≤n
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with respect of the vector z,
n
X
(5.5)
1
q
2
|(x, ei |z)| ≤ n kx|zk
i=1
n
X
! 2p1
2p
|(x, ei |z)|
, x ∈ X.
i=1
Finally, the choice ci = (x, yi |z) (i = 1, . . . , n) will produce in the last inequality in (4.2)
!2
n
X
|(x, yi |z)|2
i=1
2
≤ kx|zk
n
X
i=1
2
|(x, yi |z)|
max kyi |zk + (n − 1) max |(yi , yj |z)| ,
2
1≤i≤n
1≤i≤n
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
1≤i6=j≤n
which gives the following inequality
n
X
2
2
2
(5.6)
|(x, yi |z)| ≤ kx|zk max kyi |zk + (n − 1) max |(yi , yj |z)|
i=1
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
1≤i6=j≤n
for any x, y1 , . . . , yn , z ∈ X.
It is obvious that (5.6) will give for z−orthonormal families, the Bessel inequality 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 of z−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
! 12
An :=
X
|(yi , yj |z)|2
1≤i6=j≤n
and
Bn := (n − 1) max |(yi , yj |z)| ,
1≤i6=j≤n
where n ≥ 1, and y1 , . . . , yn , z ∈ X.
If we choose n = 3, we have
√
1
A3 = 2 (y1 , y2 |z)2 + (y2 , y3 |z)2 + (y3, y1 |z)2 2
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
and
B3 = 2 max {|(y1 , y2 |z)| , |(y2 , y3 |z)| , |(y3, y1 |z)|} ,
Title Page
where y1 , 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
√
1
A3 := 2 a2 + b2 + c2 2
Contents
with
B3 = 2 max {a, b, c} .
√
1
If we assume that b = c = 1, then
A
:=
2 (a2 + 2) 2 , B3 = 2 max {a, 1} .
3
√
Finally, for a = 1, we√get A3 = 6, B3 = 2 showing that A3 > B3 , while for
a = 2 we have A3 = 12, B3 = 4 showing that B3 > A3 .
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In conclusion, we may state that the bounds

! 21 


X
M1 := kx|zk2 max kyi |zk2 +
|(yi , yj |z)|2

1≤i≤n
1≤i6=j≤n
and
2
M2 := kx|zk
Pn
max kyi |zk + (n − 1) max |(yi , yj |z)|
2
1≤i≤n
1≤i6=j≤n
for the Bessel’s sum i=1 |(x, yi |z)|2 cannot be compared in general, meaning
that sometimes one is better than the other.
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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6.
Applications for Determinantal Integral
Inequalities
Let (Ω, Σ, µ) be a measure space consisting of a set Ω, a σ−algebra Σ of subsets of Ω and a countably additive and positive measure µ on Σ with values in
R ∪ {∞}.
Denote by L2ρ (Ω) the Hilbert space of all real-valued functions f defined
R
on Ω that are 2 − ρ−integrable on Ω, i.e., Ω ρ (s) |f (s)|2 dµ (s) < ∞, where
ρ : Ω → [0, ∞) is a measurable function on Ω.
We can introduce the following 2-inner product on L2ρ (Ω) by the formula
(6.1)
(f, g|h)ρ
Z Z
f (s) f (t)
1
:=
ρ (s) ρ (t) h (s) h (t)
2 Ω Ω
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
g (s) g (t)
h (s) h (t)
dµ (s) dµ (t) ,
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where, by
f (s) f (t)
h (s) h (t)
,
II
I
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we denote the determinant of the matrix
"
#
f (s) f (t)
Close
,
h (s) h (t)
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generating the 2-norm on L2ρ (Ω) expressed by

(6.2)
kf |hkρ := 
1
2
f (s) f (t)
ρ (s) ρ (t) h (s) h (t)
Ω
Z Z
Ω
 12
2
dµ (s) dµ (t) .
A simple calculation with integrals reveals that
R
R
Ω ρf gdµ Ω ρf hdµ
(6.3)
(f, g|h)ρ = R
R
ρghdµ
ρh2 dµ
Ω
Ω
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
and
R
R
1
Ω ρf 2 dµ Ω ρf hdµ 2
(6.4)
kf |hkρ = R
,
R
2
ρf
hdµ
ρh
dµ
Ω
Ω
R
Rwhere, for simplicity, instead of Ω ρ (s) f (s) g (s) dµ (s) , we have written
ρf gdµ.
Ω
Using the representations (6.2), (6.3) and the inequalities for 2-inner products and 2-norms established in the previous sections, one may state some interesting determinantal integral inequalities, as follows.
Proposition 6.1. Let f, g1 , . . . , gn , h ∈
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
L2ρ
(Ω) , where ρ : Ω → [0, ∞) is a
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measurable function on Ω. Then we have the inequality
R
R
2
n ρf
g
dµ
ρf
hdµ
X
i
Ω
Ω
R
R
2
ρg
hdµ
ρh
dµ
i=1
i
Ω
Ω
R
R
R
(
R
Ω ρf 2 dµ Ω ρf hdµ Ω ρgi2 dµ
ρgi hdµ
Ω
≤ R
×
max
R
R
R
1≤i≤n ρf hdµ Ω ρh2 dµ ρgi hdµ Ω ρh2 dµ
Ω
Ω


R
R
2  12 
n

Ω ρgj gi dµ Ω ρgj hdµ X

.
+
R
R


ρg hdµ
ρh2 dµ 1≤i6=j≤n
Ω
i
Ω
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 on Ω. Then we have the inequality
R
R
2
n ρf
g
dµ
ρf
hdµ
X
i
Ω
Ω
R
R
2
ρg
hdµ
ρh
dµ
i=1
i
Ω
Ω
R
R
R
(
R
Ω ρf 2 dµ Ω ρf hdµ Ω ρgi2 dµ
ρgi hdµ Ω
≤ R
× max R
R
R
1≤i≤n 2
2
ρf
hdµ
ρh
dµ
ρg
hdµ
ρh
dµ
i
Ω
Ω
Ω
Ω
R
R
)
Ω ρgj gi dµ Ω ρgj hdµ + (n − 1) max R
.
R
1≤i6=j≤n ρg hdµ
ρh2 dµ Ω
i
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
S.S. Dragomir, Y.J. Cho, S.S. Kim
and A. Sofo
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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. KIM AND A. MISIAK, Theory of 2-Inner Product Spaces, Nova Science Publishers, Inc., New York, 2001.
[4] Y.J. CHO, M. MATIĆ AND J.E. PEČARIĆ, On Gram’s determinant in 2inner product spaces, J. Korean Math. Soc., 38(6) (2001), 1125–1156.
[5] S.S. DRAGOMIR AND J. SÁNDOR, On Bessel’s and Gram’s inequality in
prehilbertian spaces, Periodica Math. Hung., 29(3) (1994), 197–205.
[6] S.S. DRAGOMIR AND B. 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. PEČARIĆ, Some remarks on
Bessel’s inequality in inner product spaces, Studia Univ. Babeş-Bolyai,
Mathematica, 37(4) (1992), 77–86.
[8] R.W. FREESE AND Y.J. CHO, Geometry of Linear 2-Normed Spaces, Nova
Science Publishers, Inc., New York, 2001.
[9] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
Some Boas-Bellman Type
Inequalities in 2-Inner Product
Spaces
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and A. Sofo
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