Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 385824, 22 pages
doi:10.1155/2010/385824
Research Article
On Reverses of Some Inequalities in
n-Inner Product Spaces
Renu Chugh and Sushma Lather
Department of Mathematics, Maharshi D University, Rohtak 124001, India
Correspondence should be addressed to Renu Chugh, chughrenu@yahoo.com
Received 1 February 2010; Revised 2 June 2010; Accepted 31 July 2010
Academic Editor: Irena Lasiecka
Copyright q 2010 R. Chugh and S. Lather. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We present some new reverses of Cauchy-Bunyakovsky-Schwarz inequality, and Triangle and
Boas-Bellman Type inequalities in n-inner product spaces. The results obtained generalize the
results of Dragomir2003–2005 in n-inner product spaces. Also we provide some applications for
determinantal integral inequalities.
1. Introduction
In 1964, Gähler 1 introduced the concept of 2-norm and 2-inner product spaces as
generalization of norm and inner product spaces. A systematic presentation of the results
related to the theory of 2-inner product spaces can be found in the book in 2, 3 and in list of
references in it. Generalization of 2-inner product space for n ≥ 2 was developed by Misiak
4 in 1989. Gunawan and Mashadi 5 in 2001 introduced the concept of n-normed linear
spaces. Gunawan 6 obtained Cauchy-Bunyakovsky inequality in these spaces. Cho et al.
7 extended parallelogram law and proved Hlwaka’s type inequality in n-inner product
spaces. In this paper by, using same ideas of 8–13 we establish some results regarding
new reverses of Cauchy-Bunyakovsky-Schwarz inequality, and Triangle and Boas-Bellman
Type inequalities in n-inner product spaces and we also provide some applications for
determinantal integral inequalities.
2. Preliminaries
Definition 2.1 see 4. Assume that n is a positive integer and X is a vector space over the
field K R of real numbers or the field K C of complex numbers, such that dim X ≥ n and
X × X × ··· × X
·, · | ·, . . . , · is a K valued function defined on such that:
n−1
n1
2
International Journal of Mathematics and Mathematical Sciences
nI1 x1 , x1 | x2 , . . . , xn ≥ 0, for any x1 , x2 , . . . , xn ∈ X and x1 , x1 | x2 , . . . , xn 0 if
and only if x1 , x2 , . . . , xn are linearly dependent vectors;
nI2 a, b | x1 , . . . , xn−1 a, b | πx1 , . . . , πxn−1 , for any a, b, x1 , x2 , . . . , xn−1 ∈ X and
for any bijections π : {x1 , x2 , . . . , xn−1 } → {x1 , x2 , . . . , xn−1 };
nI3 If n > 1, then x1 , x1 | x2 , . . . , xn x2 , x2 | x1 , x3 , . . . , xn , for any x1 , x2 , . . . , xn ∈ X;
nI4 a, b | x1 , . . . , xn−1 b, a | x1 , . . . , xn−1 nI5 αa, b | x1 , . . . , xn−1 αa, b | x1 , . . . , xn−1 , for any a, b, x1 , . . . , xn−1 ∈ X and any
scalar α ∈ R;
nI6 a a1 , b | x1 , . . . , xn−1 a, b | x1 , . . . , xn−1 a1 , b | x1 , . . . , xn−1 , for any
a, b, a1 , x1 , . . . , xn−1 ∈ X.
Then ·, · | ·, . . . , · is called the n-inner product and X, ·, · | ·, . . . , · is called the n-prehilbert
n−1
n−1
space. If n 1, then Definition 2.1 reduces to the ordinary inner product. In any given n-inner
× ··· × X
product space X · ·, · | ·, ·, . . . , ·, we can define a function ·, ·, . . . , · on X n X
× X n-times
as
x1 , x2 , . . . , xn x1 , x1 | x2 , . . . , xn 2.1
for any x1 , x2 , . . . , xn ∈ X. It is easy to see that this function satisfies the following conditions:
nN1 x1 , x2 , . . . , xn ≥ 0 and x1 , x2 , . . . , xn 0 if and only if x1 , x2 , . . . , xn are linearly
dependent,
nN2 x1 , x2 , . . . , xn is invariant under any permutation,
nN3 x1 , x2 , . . . , axn |a|x1 , x2 , . . . , xn , for any a ∈ K,
nN4 x1 , x2 , . . . , xn − 1, y z ≤ x1 , x2 , . . . , xn−1 , y x1 , x2 , . . . , xn−1 , z.
A function ·, ·, . . . , · defined on X n and satisfying the above conditions is called an nnorm on X and the pair X, ·, . . . , · is called n-normed linear space. Whenever an n-inner
product space X, ·, · | ·, ·, . . . , · is given, we consider it as an n-normed space X, ·, ·, . . . , ·
with the n-norm defined by 2.1.
Let H; ·, · be an inner product space over the real or complex number fieldK. The
following inequality is known as Cauchy-Schwarz’s inequality:
x, y 2 ≤ x2 y
2 ,
x, y ∈ H,
2.2
where z2 z, z, z ∈ H. The equality occurs in 2.2 if and only if x and y are linearly
dependent.
In 8, Dragomir obtained the following reverse of Cauchy-Schwarz’s inequality:
2 4
2 1
0 ≤ x2 y
− x, y ≤ |A − a|2 y
,
4
2.3
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3
provided x, y ∈ H and a, A ∈ K are so that either
Re Ay − x, x − ay ≥ 0,
2.4
x − a A , y
≤ 1 |A − a|
y
,
2
2
2.5
or, equivalently
holds. The constant 1/4 is best possible in 2.3 in the sense that it cannot be replaced by a
smaller quantity.
If x, y, A, a satisfy either 2.4 or 2.5, then the following reverse of CauchySchwarz’s inequality also holds:
1 Re A x, y a x, y
1
|A| |a| ≤ ·
x, y x
y
≤ ·
1/2
1/2
2
2 ReaA
ReaA
2.6
provided that, the complex numbers a and A satisfy the condition ReaA > 0. In both
inequalities in 2.6, the constant 1/2 is best possible.
The following reverse of the triangle inequality in inner product space was also
obtained by Dragomir 9
Let H; ·, · be an inner product space overK and x, y ∈ H, M ≥ m > 0 such that
either ReAy − x, x − ay ≥ 0 or, equivalently x − a A/2, y ≤ 1/2|A − a|y, holds.
0 ≤ x y
− x y
≤
√
√
M − m Re x, y
√
mM
2.7
holds.
Gunawan 6 generalized the Cauchy-Bunyakovsky-Schwarz inequality shortly, the
CBS inequality for inner product space to n-inner product space and obtained the following:
x, y | z2 , . . . , zn 2 ≤ x, x | z2 , . . . , zn y, y | z2 , . . . , zn .
2.8
Moreover, the equality holds if and only if x, y, z2 , . . . , zn are linearly dependent. In terms
of the n-norms, the CBS-inequality 2.8 can be written as
x, y | z2 , . . . , zn 2 ≤ x, z2 , . . . , zn 2 y, z2 , . . . , zn 2 .
2.9
The equality holds if and only if x, y, z2 , . . . , zn are linearly dependent.
3. Main Results
The aim of the present paper is to generalize the above mentioned results of Dragomir 8, 9,
that is, reverses of CBS inequality, Triangle inequality and Boas-bellman type inequalities in
inner product space to n-inner product spaces.
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3.1. Reverses of the CBS Inequality
Theorem 3.1. Let A, a ∈ KK C, R and x, y, z2 , . . . , zn ∈ X, where X, ·, · | ·, . . . , · is
n-inner product space over K. If
Re Ay − x, x − ay | z2 , . . . , zn ≥ 0.
3.1
x − a A y, z2 , . . . , zn ≤ 1 |A − a|
y, z2 , . . . , zn .
2
2
3.2
or, equivalently,
holds, then one has
2 4
2 1
0 ≤ x, z2 , . . . , zn 2 y, z2 , . . . , zn − x, y | z2 , . . . , zn ≤ |A − a|2 y, z2 , . . . , zn .
4
3.3
The constant 1/4 in 3.3 cannot be replaced by a smaller constant.
Proof. Using nI2 –nI6 , we get
z2 , z2 | x ± y, . . . , zn x ± y, x ± y | z2 , . . . , zn
3.4
x, x | z2 , . . . , zn y, y | z2 , . . . , zn ± 2 Re x, y | z2 , . . . , zn ,
Re x, y | z2 , . . . , zn
3.5
1 z2 , z2 | x y, . . . , zn − z2 , z2 | x − y, . . . , zn .
4
Considering the vectors x, u, U, z2 , . . . , zn ∈ X and using 3.5 and nN1–nN6, we have
ReU − x, x − u | z2 , . . . , zn 1
z2 , z2 | U − u, . . . , zn − z2 , z2 | −2x U u, . . . , zn 4
uU
1
uU
−
u,
U
−
u
|
z
,
x
−
|
z
,
.
.
.
,
z
,
.
.
.
,
z
−
4
x
−
U
2
n
2
n
4
2
2
2
uU
1
2
, z2 , . . . , zn U − u, z2 , . . . , zn − x −
.
4
2
3.6
Therefore, ReU − x, x − u | z2 , . . . , zn ≥ 0 if and only if
x − u U , z2 , . . . , zn ≤ 1 U − u, z2 , . . . , zn .
2
2
3.7
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Applying this to the vectors U Ay and u ay, we obtain that the 3.1 and 3.2 are
equivalent. If we consider the real numbers
I1 Re
2 A
y, z2 , . . . , zn − x, y | z2 , . . . , zn
×
2 x, y | z2 , . . . , zn − a
y, z2 , . . . , zn ,
3.8
2 I2 y, z2 , . . . , zn Re Ay − x, x − ay | z2 , . . . , zn .
Then we obtain, by using properties of n-inner product space,
2 I1 y, z2 , . . . , zn Re A x, y | z2 , . . . , zn a x, y | z2 , . . . , zn
I2
4
2 − x, y | z2 , . . . , zn − y, z2 , . . . , zn ReAa.
2 y, z2 , . . . , zn Re A x, y | z2 , . . . , zn a x, y | z2 , . . . , zn
3.9
2 4
− x, z2 , . . . , zn 2 y, z2 , . . . , zn − y, z2 , . . . , zn ReAa
which gives
2 2
I1 − I2 x, z2 , . . . , zn 2 y, z2 , . . . , zn − x, y | z2 , . . . , zn ,
3.10
for any x, y, z2 , . . . , zn ∈ X and a, A ∈ K. If 3.1 holds, then I2 ≥ 0 and thus I1 − I2 ≤ I1
2 2
⇒ x, z2 , . . . , zn 2 y, z2 , . . . , zn − x, y | z2 , . . . , zn 2 ≤ Re A
y, z2 , . . . , zn − x, y | z2 , . . . , zn
×
3.11
2 x, y | z2 , . . . , zn − a
y, z2 , . . . , zn .
Now using the elementary inequality Reαβ ≤ 1/4|α β|2 for any α, β ∈ K K R, C, one
yields
Re
2 2 x, y | z2 , . . . , zn − a
y, z2 , . . . , zn A
y, z2 , . . . , zn − x, y | z2 , . . . , zn
≤
4
1
|A − a|2 y, z2 , . . . , zn .
4
If we combine 3.11 and 3.12, we get the required inequality.
3.12
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To prove the sharpness of the constant 1/4, assume that 3.3 holds with a constant
C > 0, that is,
2 2
x, z2 , . . . , zn 2 y, z2 , . . . , zn − x, y | z2 , . . . , zn 4
≤ C|A − a|2 y, z2 , . . . , zn ,
3.13
where x, y, z2 , . . . , zn , A, and a satisfy the hypothesis of the theorem.
A, m ∈ X with m, z2 , . . . , zn 1 and
Consider y ∈ X with y, z2 , . . . , zn 1, a /
y, m | z2 , . . . , zn 0 and define
x
A−a
Aa
y
m
2
2
3.14
then we have
|A − a|2 Re Ay − x, x − ay | z2 , . . . , zn Re y − m, y m | z2 , . . . , zn 0
4
3.15
and then the condition 3.1 is fulfilled. From 3.13 we deduce
2
A a
Aa
A−a
A−a
−
≤ C|A − a|2
y
m,
z
y
m,
y
|
z
,
.
.
.
,
z
,
.
.
.
,
z
2
n
2
n
2
2
2
2
3.16
and, since
2 2
2 A a
A−a
A a A − a ,
y
m,
z
,
.
.
.
,
z
2
n
2
2
2
2 2 A a 2
Aa
A−a
y
m,
y
|
z
,
.
.
.
,
z
2
n 2 .
2
2
3.17
By 3.16, we have
A − a 2
2
2 ≤ C|A − a| ,
for any A, a ∈ K with a /
A, which implies C ≥ 1/4. This completes the proof.
3.18
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Theorem 3.2. Assume that x, y, z2 , . . . , zn , a and A are the same as in above Theorem 3.1. If
ReaA > 0, then one has
A
a
,
.
.
.
,
z
Re
x,
y
|
z
2
n
1
x, z2 , . . . , zn y, z2 , . . . , zn ≤
2
ReaA1/2
1
|A a| ≤
x, y | z2 , . . . , zn .
1/2
2 ReaA
3.19
The constant 1/2 is best possible in both inequalities in the sense that it cannot be replaced by a smaller
constant.
Proof. Define
I Re Ay − x, x − ay | z2 , . . . , zn
Re A x, y | z2 , . . . , zn a x, y | z2 , . . . , zn
3.20
2
− x, z2 , . . . , zn 2 − y, z2 , . . . , zn ReAa.
We know that, for a complex number α ∈ C, Reα Reα and thus
Re A x, y | z2 , . . . , zn Re A x, y | z2 , . . . , zn
3.21
which implies
I Re
2
A a x, y | z2 , . . . , zn − x, z2 , . . . , zn 2 − y, z2 , . . . , zn ReAa.
3.22
Since x, y, z2 , . . . , zn , a and A are assumed to satisfy the condition 3.1, by 3.22, we deduce
2
x, z2 , . . . , zn 2 y, z2 , . . . , zn ReAa ≤ Re A a x, y | z2 , . . . , zn ,
3.23
which gives
1
1/2 2
y, z2 , . . . , zn 2
,
.
.
.
,
z
z
ReaA
x,
2
n
1/2
ReaA
Re A a x, y | z2 , . . . , zn
≤
ReaA1/2
since ReAa > 0.
3.24
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On the other hand, by the elementary inequality
αp2 1 2
q ≥ 2pq
α
3.25
for p, q ≥ 0 and α > 0, we have
2x, z2 , . . . , zn y, z2 , . . . , zn ≤
1
ReaA
1/2 2
y, z2 , . . . , zn 2 .
,
.
.
.
,
z
z
ReaA
x,
2
n
1/2
3.26
Using 3.24 and 3.26, we deduce the first inequality in 3.22. The last part is obvious by
the fact, for z ∈ C, | Rez| ≤ |z|.
To prove the sharpness of the constant 1/2 in the first inequality in 3.22, we assume
that 3.22 holds with a constant c > 0, that is,
Re
x, z2 , . . . , zn y, z2 , . . . , zn ≤ c
A a x, y | z2 , . . . , zn
ReaA1/2
3.27
provided x, y, z2 , . . . , zn , a and A satisfy 2.1. If we take a A 1, y x /
0, then
obviously 2.1 holds and from 3.27, we obtain
x, z2 , . . . , zn 2 ≤ 2cx, z2 , . . . , zn 2
3.28
for any linearly independent vectors x, z2 , . . . , zn ∈ X, which implies c ≥ 1/2. This completes
the proof.
When the constants involved are assumed to be positive, then we may state the
following result.
Corollary 3.3. Let M ≥ m > 0 and assume that, for x, y, z2 , . . . , zn ∈ X, one has
Re My − x, x − my | z2 , . . . , zn ≥ 0
3.29
x − M m , z2 , . . . , zn ≤ 1 M − m
y, z2 , . . . , zn .
2
2
3.30
or, equivalently,
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Then one has the following reverse of CBS inequality:
1Mm Re x, y | z2 , . . . , zn
x, z2 , . . . , zn y, z2 , . . . , zn ≤ √
2 mM
1 M m ≤ √
x, y | z2 , . . . , zn .
2 mM
3.31
The constant 1/2 is sharp in 3.31.
Corollary 3.4. With the assumptions of the Theorem 3.2, one has
2 2
0 < x, z2 , . . . , zn 2 y, z2 , . . . , zn − x, y | z2 , . . . , zn ≤
2
1 |A − a|2 x, y | z2 , . . . , zn .
4 ReaA
3.32
The constant 1/4 is best possible in 3.32.
Corollary 3.5. With the assumption of Corollary 3.3, one has
0 ≤ x, z2 , . . . , zn y, z2 , . . . , zn − x, y | z2 , . . . , zn ≤ x, z2 , . . . , zn y, z2 , . . . , zn − Re x, y | z2 , . . . , zn
√
√ 2
√ 2
M
−
m
M
−
m 1
1
x, y | z2 , . . . , zn ,
≤
Re x, y | z2 , . . . , zn ≤
√
√
2
2
mM
mM
2
0 ≤ x, z2 , . . . , zn 2 y, z2 , . . . , zn − x, y | z2 , . . . , zn 2 2
≤ x, z2 , . . . , zn 2 y, z2 , . . . , zn − Re x, y | z2 , . . . , zn
√
≤
3.33
3.34
2 1 M − m2 1 M − m2 x, y | z2 , . . . , zn 2 .
≤
Re x, y | z2 , . . . , zn
4 mM
4 mM
The constant 1/2 in 3.33 and the constant 1/4 in 3.34 are best possible.
3.2. Reverse of the Triangle Inequality
Corollary 3.6. Assume x, y, z2 , . . . , zn , m, M are the same as in Corollary 3.3 and Rex, y |
z2 , . . . , zn ≥ 0. Then one has the following reverse of the triangle inequality:
0 ≤ x, z2 , . . . , zn y, z2 , . . . , zn − x y, z2 , . . . , zn √
√
M − m ≤
Re x, y | z2 , . . . , zn .
1/4
mM
3.35
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International Journal of Mathematics and Mathematical Sciences
Proof. It is easy to see that
2 2
0 ≤ x, z2 , . . . , zn y, z2 , . . . , zn − x y, z2 , . . . , zn 2 x, z2 , . . . , zn y, z2 , . . . , zn − Re x, y | z2 , . . . , zn
3.36
for any x, y, z2 , . . . , zn ∈ X. If the assumption of Corollary 3.3 holds, then 3.33 is valid and,
by 3.36, we deduce
2 2
0 ≤ x, z2 , . . . , zn y, z2 , . . . , zn − x y, z2 , . . . , zn √
≤
√ 2
M− m
Re x, y | z2 , . . . , zn
√
mM
3.37
which gives
2
x, z2 , . . . , zn y, z2 , . . . , zn 2
≤ x y, z2 , . . . , zn √
√ 2
M− m
Re x, y | z2 , . . . , zn .
√
mM
3.38
Taking square root in 3.38, we have
x, z2 , . . . , zn y, z2 , . . . , zn √
√ 2
M
−
m
2
≤ x y, z2 , . . . , zn Re x, y | z2 , . . . , zn
√
mM
√
√ M− m
Re x, y | z2 , . . . , zn .
≤ x y, z2 , . . . , zn 1/4
mM
3.39
From where we deduce the desire inequality 3.35. This completes the proof.
Let X, ·, · | ·, . . . be a n-inner product space over real or complex number field K.
If ei 1<i<m are linearly independent vector in the n-inner product space X, and, for given
z2 , . . . , zn ∈ X, ei , ej | z2 , . . . , zn δij for all i, j ∈ {1, 2, . . . , m} where δij is the Kronecker
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delta,then the following inequality is the Bessel’s inequality for z2 , . . . , zn orthonormal family
ei 1<i<m in n-inner product space X, ·, · | ·, . . .:
m
|x, ei | z2 , . . . , zn |2 ≤ x, z2 , . . . , zn 2
for any x ∈ X.
3.40
i1
Theorem 3.7. Let x1 , x2 , . . . , xm , z2 , . . . , zn ∈ X and α1 , . . . , αm ∈ K. then one has
2
m
αi xi , z2 , . . . , zn i1
⎧
m
⎪
2
⎪
max
|α
|
xi , z2 , . . . , zn 2 ,
⎪
i
⎪
1≤i≤m
⎪
⎪
i1
⎪
⎪
⎪
⎨ m
m
2a 1/a ≤
|αi |
xi , z2 , . . . , zn 2b
⎪
⎪
⎪
i1
i1
⎪
⎪
m
⎪
⎪
⎪
2
2
⎪
⎩ |αi | max xi , z2 , . . . , zn i1
1/b
,
where a > 1,
1 1
1,
a b
1≤i≤m
3.41
⎧
" !
xi , xj | z2 , . . . , zn ,
αi αj ⎪
max
⎪
⎪
⎪
1≤i
j≤m
/
⎪
1≤i / j≤m
⎪
⎪
⎪
⎡
⎤1/c
⎪
⎪
2
⎪
m
m
⎪
⎪⎣ c
2c ⎦
⎪
⎪
−
|αi |
|αi |
⎨
i1
i1
⎪
1/d
1 1
⎪
⎪
xi , xj | z2 , . . . , zn d
⎪
×
, where c > 1, 1,
⎪
1≤i
j≤m
⎪
/
c d
⎪
⎡
⎤
⎪
⎪
2
⎪
m
m
⎪
⎪
⎪
⎣
⎪
− |αi |2 ⎦ max xi , xj | z2 , . . . , zn .
|αi |
⎪
⎩
1≤i /
j≤m
i1
i1
Proof. We observe that
2 '
(
m
m
m
αi xi ,
αj xj , | z2 , . . . , zn
αi xi , z2 , . . . , zn i1
i1
j1
m
m αi αj xi , xj | z2 , . . . , zn
i1 j1
m
m αi αj xi , xj | z2 , . . . , zn i1 j1
≤
m
m 3.42
|αi |αj xi , xj | z2 , . . . , zn i1 j1
m
i1
|αi |2 xi , z2 , . . . , zn 2 |αi |αj xi , xj | z2 , . . . , zn .
1≤i /
j≤m
12
International Journal of Mathematics and Mathematical Sciences
Using Holder’s inequality, we may write that
m
|αi |2 xi , z2 , . . . , zn 2
i1
⎧
m
⎪
⎪
⎪
max |αi |2 xi , z2 , . . . , zn 2 ,
⎪
⎪
1≤i≤m
⎪
i1
⎪
⎪
⎪
⎪
⎪
1/a m
⎨ m
2a
≤
|αi |
xi , z2 , . . . , zn 2b
⎪
⎪
⎪
i1
i1
⎪
⎪
⎪
⎪
m
⎪
⎪
2
2
⎪
⎪
⎩ |αi | max xi , z2 , . . . , zn .
3.43
1/b
,
where a > 1,
1 1
1,
a b
1≤i≤m
i1
By Holder’s inequality for double sum, we also have
|αi |αj xi , xj | z2 , . . . , zn 1≤i /
j≤m
≤
⎧
xi , xj | z2 , . . . , zn ,
⎪
max αi αj ⎪
⎪
⎪1≤i / j≤m
⎪
1≤i /
j≤m
⎪
⎪
⎪
⎪
⎛
⎞1/c
⎪
⎪
⎪
⎪
⎪⎝
c
⎪
⎪
|α |c αj ⎠
⎪
⎪ 1≤i j≤m i
⎨
/
⎛
⎞1/d
⎪
⎪
⎪
⎪
⎪
xi , xj | z2 , . . . , zn d ⎠ , where c > 1, 1 1 1,
⎪
×⎝
⎪
⎪
c d
⎪
1≤i /
j≤m
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
|αi |αj max xi , xj | z2 , . . . , zn ⎪
⎩
1≤i /
j≤m
1≤i /
j≤m
⎧
" !
xi , xj | z2 , . . . , zn ,
⎪
max αi αj ⎪
⎪
1≤i /
j≤m
⎪
⎪
1≤i /
j≤m
⎪
⎪
⎪
⎪
⎪
⎡
⎤1/c
⎪
2
⎪
m
m
⎪
⎪
⎪
⎪
⎣
−
|αi |c
|αi |2c ⎦
⎪
⎪
⎪
⎪
i1
i1
⎪
⎨
⎛
⎞1/d
⎪
⎪
⎪
⎪
xi , xj | z2 , . . . , zn d ⎠ , where c > 1, 1 1 1,
⎪
×⎝
⎪
⎪
c d
⎪
⎪
1≤i /
j≤m
⎪
⎪
⎪
⎪
⎡
⎤
⎪
⎪
2
⎪
m
m
⎪
⎪
⎪⎣
⎪
− |αi |2 ⎦ max xi , xj | z2 , . . . , zn .
|αi |
⎪
⎩
1≤i j≤m
/
i1
i1
Using 3.43 and 3.44 in 3.42, we may deduce the desired result.
3.44
International Journal of Mathematics and Mathematical Sciences
13
Corollary 3.8. With the assumption in above Theorem 3.7, one has
2
m
αi xi , z2 , . . . , zn i1
≤
n
|αi |
i1
2
⎧
⎪
⎪
⎪
⎪
⎪
⎨
-.
n
/
|αi |
i1
2
maxxi , z2 , . . . , zn ⎪
1≤i≤n
⎪
⎪
⎪
⎪
⎩
2
02
−
/n
i1
11/2
/n
i1
|αi |
4
|αi |2
⎫
⎪
⎪
⎛
⎞1/2 ⎪
⎪
⎪
⎬
2
⎝
⎠
×
xi , xj | z2 , . . . , zn
⎪
⎪
1≤i /
j≤n
⎪
⎪
⎪
⎭
⎞1/2 ⎫
⎪
⎬
2
2
2
⎝
⎠
.
xi , xj | z2 , . . . , zn
≤
|αi | max xi , z2 , . . . , zn ⎪
⎪
⎩1≤i≤m
⎭
i1
1≤i / j≤m
m
⎧
⎪
⎨
3.45
⎛
The first inequality follows by taking the third branch in the first curly bracket with the second branch
in the second curly bracket for c d 2. The second inequality in 3.45 follows by the fact that
⎡
m
⎣
|αi |2
2
−
i1
m
⎤1/2
|αi |4 ⎦
≤
i1
m
|αi |2 .
3.46
i1
By applying the following Cauchy-Bunyakovsky-Schwarz inequality:
m
2
m
≤m
ai
i1
ai 2 ,
ai ∈ R , 1 ≤ i ≤ m.
3.47
i1
one may write that
m
2
|αi |c
−
i1
m
i1
m
|αi |2c ≤ m − 1
i1
2
|αi |
−
m
m
|αi |2c ,
m ≥ 1,
3.48
i1
|αi |2 ≤ m − 1
i1
m
|αi |2 ,
m ≥ 1.
3.49
i1
It is obvious that
max
"
!
αi αj ≤ max |αi |2 .
1≤i /
j≤m
1≤i≤m
3.50
14
International Journal of Mathematics and Mathematical Sciences
Corollary 3.9. With the assumption in above Theorem 3.7, one has
2
m
αi xi , z2 , . . . , zn i1
⎧
m
⎪
⎪
max |αi |2 xi , z2 , . . . , zn 2 ,
⎪
⎪
⎪
1≤i≤m
⎪
i1
⎪
⎪
⎪
⎪
1/a m
⎨ m
2a
≤
|α
|
xi , z2 , . . . , zn 2b
i
⎪
⎪
⎪
i1
i1
⎪
⎪
⎪
m
⎪
⎪
2
2
⎪
⎪
⎩ |αi | max xi , z2 , . . . , zn i1
1/b
,
1 1
1,
a b
where a > 1,
1≤i≤m
⎧
⎪
max |αi |2
|xi , xi | z2 , . . . , zn |,
⎪
⎪1≤i≤m
⎪
1≤i≤m
⎪
⎪
⎛
⎞1/d
⎪
⎨
1/c
m
m−11/c
xi , xj | z2 , . . . , zn d⎠ ,
⎝
|αi |2c
⎪
⎪
⎪
⎪
i1
1≤i /
j≤m
⎪
⎪
m
⎪
⎩
2
m − 1 |αi | max xi , z2 , . . . , zn 2 .
i1
where c > 1,
1≤i≤m
1 1
1,
c d
3.51
The proof is obvious by Theorem 3.7 on applying 3.48–3.50.
Theorem 3.10. Let x, y1 , . . . , ym , z2 , . . . , zn be vectors of an n-inner product space (X, ·, · | ·, . . .)
and α1 , α2 , . . . , αn ∈ KK R, C. Then
2
m
αi x, yi | z2 , . . . , zn i1
⎧
m ⎪
2
yi , z2 , . . . , zn 2 ,
⎪
max
|α
|
⎪
i
⎪
⎪
1≤i≤m
⎪
i1
⎪
⎪
⎪
⎪
1/a m
⎨ m
2a
yi , z2 , . . . , zn 2b
≤ x, z2 , . . . , zn 2
|α
|
i
⎪
⎪
⎪
i1
i1
⎪
⎪
⎪
m
⎪
2
⎪
2
⎪
⎪
⎩ |αi | max yi , z2 , . . . , zn i1
1/b
,
where a > 1,
1 1
1,
a b
1≤i≤m
⎧
" !
yi , yj | z2 , . . . , zn ,
⎪
max αi αj ⎪
⎪
1≤i /
j≤m
⎪
⎪
1≤i /
j≤m
⎪
⎪
⎡
⎞1/d
⎤1/c ⎛
⎪
⎪
2
⎪
m
m
⎪
⎪
d
c
2c
⎨⎣
yi , yj | z2 , . . . , zn ⎠ ,
⎦ ⎝
−
|αi |
|αi |
x, z2 , . . . , zn 2
i1
i1
1≤i
j≤m
/
⎪
⎪
⎪
⎪
⎡
⎤
⎪
⎪
2
⎪
m
m
m
⎪
⎪
2 ⎦
⎪
⎣
−
|α
|
|α
|
|αi |2 max xi , z2 , . . . , zn 2 .
⎪
i
i
⎪
⎩
1≤i≤m
i1
i1
i1
3.52
International Journal of Mathematics and Mathematical Sciences
15
Proof. Since
m
6
5 αi x, yi | z2 , . . . , zn x,
αi yi | z2 , . . . , zn
3.53
i1
and by Schwarz’s inequality in n-inner product spaces,
2
2
m
m
2
αi x, yi | z2 , . . . , zn ≤ x, z2 , . . . , zn αi yi , z2 , . . . , zn .
i1
i1
3.54
Using αi αi , zi yi i 1, 2, . . . , n in the Theorem 3.7, we get the desired inequality
3.52.
Corollary 3.11. With the assumptions in Theorem 3.10, the following holds:
2
m
αi x, yi | z2 , . . . , zn i1
≤ x, z2 , . . . , zn 2
⎧
⎧
⎞1/2 ⎫
⎛
⎪
⎪
⎪
m
⎨
⎬
⎪
2
⎪
⎪
yi , z2 , . . . , zn 2 ⎝
yi , yj | z2 , . . . , zn 2 ⎠
⎪
max
,
|α
|
⎪
i
⎪
⎪
⎪
⎪
⎩1≤i≤m
⎭
⎪
i1
1≤i /
j≤m
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎫
⎧
⎪
⎪
⎪
m ⎬
⎨
⎪
⎪
2
⎪
yi , z2 , . . . , zn 2 yi , yj | z2 , . . . , zn ,
max
|α
|
⎪
i
⎪
⎪
⎭
⎩ i1
1≤i≤m
⎪
1≤i /
j≤m
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
1/p
m
2p
×
|αi |
⎪
⎪
⎪
i1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎧
⎪
⎛
⎞1/q ⎫
⎪
⎪
1/q
⎪
⎪
⎪
m ⎨
⎬
⎪
2q
q
⎪
⎪
⎝
⎠
⎪
y
,
×
,
z
,
.
.
.
,
z
−
1
,
y
|
z
,
.
.
.
,
z
y
m
i
2
n
i
j
2
n
⎪
⎪
⎪
⎪
⎪
⎩ i1
⎭
⎪
1≤i / j≤m
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
8
7
⎪
m
⎪
2
⎪
2
⎪
⎪
.
⎩ |αi | max yi , z2 , . . . , zn m − 1 max yi , yj | z2 , . . . , zn
i1
1≤i≤m
1≤i /
j≤m
3.55
16
International Journal of Mathematics and Mathematical Sciences
3.3. Some Boas-Bellman Type Inequalities in n-Inner Product Spaces
If we put αi x, yi | z2 , . . . , zn i 1, 2, . . . , m, in the first inequality of 3.55
m x, yi | z2 , . . . , zn 2
2
i1
≤ x, z2 , . . . , zn 2
m x, yi | z2 , . . . , zn 2
i1
⎧
⎪
⎨
3.56
⎞1/2 ⎫
⎪
⎬
2
2
⎠
⎝
yi , yj | z2 , . . . , zn
× max yi , z2 , . . . , zn ⎪
⎪
⎩1≤i≤m
⎭
1≤i /
j≤m
⎛
which is equivalent to the following Boass-Bellman type inequality for n-inner products:
m x, yi | z2 , . . . , zn 2
i1
⎧
⎪
⎨
⎞1/2 ⎫ 3.57
⎪
⎬
2
2
2
⎝
⎠
.
≤ x, z2 , . . . , zn max yi , z2 , . . . , zn yi , yj | z2 , . . . , zn
⎪
⎪
⎩1≤i≤m
⎭
1≤i /
j≤m
⎛
Now, if we take αi x, yi | z2 , . . . , zn i 1, 2, . . . , m, in second inequality of 3.55, we have
m x, yi | z2 , . . . , zn 2
2
i1
2
≤ x, z2 , . . . , zn 2 max x, yi | z2 , . . . , zn 1≤i≤m
3.58
⎫
⎧
m ⎨
2
⎬
y , z , . . . , zn yi , yj | z2 , . . . , zn .
×
⎭
⎩ i1 i 2
1≤i j≤m
/
By taking the square root in
m x, yi | z2 , . . . , zn 2
i1
≤ x, z2 , . . . , zn max x, yi | z2 , . . . , zn 1≤i≤m
⎧
⎫1/2
m ⎨
2
⎬
y , z , . . . , zn yi , yj | z2 , . . . , zn ×
⎩ i1 i 2
⎭
1≤i j≤m
/
3.59
International Journal of Mathematics and Mathematical Sciences
17
for any x, y1 , . . . , ym , z2 , . . . , zn be vectors of an n-inner product space X, ·, · | ·, . . ..
If we assume that ei 1≤i≤m is an orthonormal family in X with respect to the vector
z2 , . . . , zn , ei , ej | z2 , . . . , zn δij for all i, j ∈ {1, . . . , m} then by 3.57 we deduce Bessel’s
/
2
2
inequality m
i1 |x, ei | z2 , . . . , zn | ≤ x, z2 , . . . , zn , and 3.59 implies
m
|x, ei | z2 , . . . , zn |2 ≤
√
mx, z2 , . . . , zn max |x, ei | z2 , . . . , zn |2 .
3.60
1≤i≤m
i1
For the third inequality in 3.55 αi x, yi | z2 , . . . , zn i 1, 2, . . . , m, we have
m x, yi | z2 , . . . , zn 2
2
i1
m x, yi | z2 , . . . , zn 2p
≤ x, z2 , . . . , zn 1/p
2
i1
⎧
⎪
m ⎨ yi , z2 , . . . , zn 2q
×
⎪
⎩ i1
1/q
3.61
⎞1/q ⎫
⎪
⎬
yi , yj | z2 , . . . , zn q ⎠
m − 1⎝
⎪
⎭
1≤i /
j≤m
⎛
for p > 1, 1/p 1/q 1. Taking the square root in this inequality, we get
m x, yi | z2 , . . . , zn 2
i1
m x, yi | z2 , . . . , zn 2p
≤ x, z2 , . . . , zn 1/2p
i1
⎧
⎪
m ⎨ yi , z2 , . . . , zn 2q
×
⎪
⎩ i1
1/q
⎞1/q ⎫1/2
⎪
⎬
q
yi , yj | z2 , . . . , zn ⎠
m − 1⎝
.
⎪
⎭
1≤i /
j≤m
⎛
3.62
For any x, y1 , . . . , ym , z2 , . . . , zn ∈ X, and p > 1, with 1/p 1/q 1, then the above
inequality 3.62 becomes, for an orthonormal family with respect to the vector z2 , . . . , zn ,
m x, yi | z2 , . . . , zn 2
i1
≤m
1/q
x, z2 , . . . , zn m
i1
1/2p
|x, ei | z2 , . . . , zn |
2p
.
3.63
18
International Journal of Mathematics and Mathematical Sciences
We take αi x, yi | z2 , . . . , zn i 1, 2, . . . , m, in the last inequality of 3.55
m x, yi | z2 , . . . , zn 2
2
i1
≤ x, z2 , . . . , zn 2
7
×
m x, yi | z2 , . . . , zn 2
3.64
i1
8
2
max yi , z2 , . . . , zn m − 1 max yi , yj | z2 , . . . , zn 1≤i /
j≤m
1≤i≤m
which implies
m x, yi | z2 , . . . , zn 2
i1
7
2
≤ x, z2 , . . . , zn 8
2
max yi , z2 , . . . , zn m − 1 max yi , yj | z2 , . . . , zn 3.65
1<i /
j<m
1<i<m
for any x, y1 , . . . , ym , z2 , . . . , zn ∈ X.
4. Applications for 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 measure μ on Σ with values in R ∪ {∞}. Denote by L2 ρΩ the Hilbert
space
of all real-valued functions f defined on Ω that are n-ρ-integrable on Ω, that is,
9
ρs|fs|n dμs < ∞ where ρ : Ω → 0, ∞ is a measurable function on Ω.
Ω
We can introduce the following n-inner product on L2 ρΩ:
f, g | h2 , . . . , hn
ρ
1
n!
: :
Ω
:
· · · ρs1 ρs2 · · · ρsn Ω
Ω
n-times
fs1 h s 2 1
× .
..
hn s1 gs1 h s 2 1
× .
..
hn s1 fs2 . . . fsn h2 s2 . . . h2 sn .. ...
. hn s2 . . . hn sn gs2 . . . gsn h2 s2 . . . h2 sn .. dμs1 dμs2 · · · dμsn ,
...
. hn s2 . . . hn sn 4.1
International Journal of Mathematics and Mathematical Sciences
where by
fs1 h2 s1 ..
.
h s n 1
fs2 . . .
h2 s2 . . .
...
hn s2 . . .
19
hn sn fsn h2 sn ..
.
4.2
we denote the determinant of matrix
⎡
fs1 fs2 · · ·
⎢ h2 s1 h2 s2 · · ·
⎢
⎢ .
..
⎣ ..
.
···
hn s1 hn s2 · · ·
⎤
· · · fsn · · · h2 sn ⎥
⎥
.. ⎥.
···
. ⎦
· · · hn sn 4.3
Generating the n-norm on L2 ρΩ is expressed by
f, h2 , . . . , hn p
⎛
:
⎜1: :
⎜
⎜
· · · ρs1 ρs2 · · · ρsn ⎝ n! Ω Ω
Ω
n−times
fs1 fs2 · · ·
h2 s1 h2 s2 · · ·
× ..
..
.
.
···
h s h s · · ·
n 1
n 2
···
···
···
···
⎞1/2
2
fsn ⎟
h2 sn ⎟
.. dμs1 dμs2 · · · dμsn ⎟
⎟ .
. ⎠
hn sn 4.4
A simple calculation with integrals shows that
:
:
ρfh2 dμ · · ·
Ω ρfgdμ
Ω
:
:
ρfh
dμ
ρh2 2 dμ · · ·
2
f, g | h2 , . . . , hn ρ Ω
Ω
..
..
.
.
···
:
:
ρgh dμ
ρh2 hn dμ · · ·
n
Ω
Ω
:
:
ρf 2 dμ
ρfh2 dμ · · ·
Ω
Ω
:
:
ρfh2 dμ
ρh2 2 dμ · · ·
f, h2 , . . . , hn Ω
Ω
p
..
..
.
.
···
:
:
ρgh dμ
ρh2 hn dμ · · ·
n
Ω
where, for simplicity, instead of
9
Ω
Ω
:
ρfhn dμ Ω
:
···
ρh2 hn dμ
Ω
,
..
···
.
:
2
···
ρhn dμ ···
4.5
Ω
:
ρfhn dμ Ω
:
···
ρh2 hn dμ
Ω
,
..
··· :
.
2
···
ρhn dμ ···
Ω
ρsfsgsdμs, we have written
9
Ω
ρfgdμ.
4.6
20
International Journal of Mathematics and Mathematical Sciences
Proposition 4.1. Let f, g1 , . . . , gm , h2 , . . . , hn ∈ L2 ρΩ, where ρ : Ω → 0, ∞ a measure function
is on Ω. Then one has
:
2
:
:
ρfg
dμ
ρfh
dμ
·
·
·
·
·
·
ρfh
dμ
i
2
n
: Ω
Ω
Ω
:
:
2
m ρgi h2 dμ
ρh2 dμ · · · · · ·
ρh2 hn dμ
Ω
Ω
Ω
..
..
..
i1 .
.
·
·
·
·
·
·
.
:
:
:
2
ρgi hn dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
2
n
n
Ω
Ω
Ω
:
:
:
2
ρf dμ
ρfh2 dμ · · · · · ·
ρfhn dμ : Ω
:Ω
:Ω
2
ρh2 dμ · · · · · ·
ρh2 hn dμ
ρfh2 dμ
≤ Ω
Ω
Ω
..
..
..
.
.
·
·
·
·
·
·
.
:
:
:
2
ρfhn dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
2
n
n
Ω
Ω
Ω
:
:
:
⎧
⎫
⎪
⎪
2
⎪
⎪
ρg
dμ
ρg
h
dμ
·
·
·
·
·
·
ρg
h
dμ
⎪
⎪
i
i
2
i
n
⎪
⎪
⎪
⎪
Ω
Ω
Ω
⎪
⎪
:
:
:
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
ρg
h
dμ
ρh
dμ
·
·
·
·
·
·
ρh
h
dμ
⎪
⎪
i 2
2
2 n
⎪
⎪
⎪
⎪
max
Ω
Ω
Ω
⎪
⎪
⎪
⎪
1≤i≤m
⎪
⎪
.
.
.
⎪
⎪
.
.
.
⎪
⎪
⎪
⎪
.
.
·
·
·
·
·
·
.
⎪
⎪
:
:
:
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
ρg
h
dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
i
n
2
n
n
⎪
⎪
⎪
⎪
⎨
⎬
Ω
Ω
Ω
×
:
:
:
⎛
⎞1/2 ⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
ρgj gi dμ
⎪
⎪
ρg
h
dμ
·
·
·
·
·
·
ρg
h
dμ
j
2
j
n
⎪
⎪
⎜
⎟
⎪
⎪
⎪
⎪
⎜
⎟
Ω
Ω
Ω
⎪
⎪
:
:
:
⎪
⎪
⎜
⎟
⎪
⎪
⎪
⎪
⎜
⎟
2
⎪
⎪
⎪
⎪
ρg
h
dμ
ρh
dμ
·
·
·
·
·
·
ρh
h
dμ
⎜
⎟
i 2
2
2 n
⎪
⎪
⎪
⎪
⎜
⎟
⎪
⎪
.
Ω
Ω
Ω
⎪
⎪
⎜
⎟
⎪
⎪
.
.
.
⎪
⎪
⎜
⎟
1≤i
j≤m
⎪
⎪
/
.
.
.
⎪
⎪
⎜
⎟
.
.
·
·
·
·
·
·
.
⎪
⎪
⎪
⎪
:
:
:
⎜
⎟
⎪
⎪
⎪
⎪
⎝
⎠
⎪
⎪
2
⎪
⎪
ρg
h
dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
⎩
⎭
i
n
2
n
n
Ω
Ω
4.7
Ω
Proof. Applying the n-inner product and n-norm defined in 4.1 and 4.4, and using 4.5
and 4.6 in 3.57
m x, yi | z2 , . . . , zn 2
i1
≤ x, z2 , . . . , zn 2
⎧
⎞1/2 ⎫
⎛
⎪
⎪
⎬
⎨
2
yi , yj | z2 , . . . , zn 2 ⎠
,
× max yi , z2 , . . . , zn ⎝
⎪
⎪1≤i≤m
⎭
⎩
1≤i j≤m
/
we get the required proof of the proposition.
4.8
International Journal of Mathematics and Mathematical Sciences
21
Proposition 4.2. Let f, g1 , . . . , gm , h2 , . . . , hn ∈ L2 ρΩ, where ρ:Ω → 0, ∞ a measure function
is on Ω. Then one has
:
2
:
:
ρfg
dμ
ρfh
dμ
·
·
·
·
·
·
ρfh
dμ
i
2
n
: Ω
Ω
Ω
:
:
2
m ρgi h2 dμ
ρh2 dμ · · · · · ·
ρh2 hn dμ
Ω
Ω
Ω
..
..
..
i1 .
.
··· ··· :
.
:
:
2
ρgi hn dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
2
n
n
Ω
Ω
Ω
:
:
:
2
ρf
dμ
ρfh
dμ
·
·
·
·
·
·
ρfh
dμ
2
n
: Ω
Ω
Ω
:
:
2
ρh2 dμ · · · · · ·
ρh2 hn dμ
ρfh2 dμ
≤ Ω
Ω
Ω
..
..
..
.
.
··· ··· :
.
:
:
2
ρfhn dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
2
n
n
Ω
Ω
Ω
⎧
⎫
:
:
:
⎪
⎪
2
⎪
⎪
ρg
dμ
ρg
h
dμ
·
·
·
·
·
·
ρg
h
dμ
⎪
⎪
i
i
2
i
n
⎪
⎪
⎪
⎪
⎪
⎪
Ω
Ω
Ω
⎪
⎪
:
:
:
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
ρg
h
dμ
ρh
dμ
·
·
·
·
·
·
ρh
h
dμ
⎪
⎪
i 2
2
2 n
⎪
⎪
⎪
⎪
max Ω
⎪
⎪
Ω
Ω
⎪
⎪
⎪
⎪
1≤i≤m
..
..
..
⎪
⎪
⎪
⎪
⎪
⎪
.
.
·
·
·
·
·
·
.
⎪
⎪
:
:
⎪
⎪
:
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
ρg
h
dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
⎨
⎬
i
n
2
n
n
Ω
Ω
Ω
:
:
:
×
⎪.
⎪
ρgj gi dμ
⎪
⎪
ρg
h
dμ
·
·
·
·
·
·
ρg
h
dμ
⎪
j
2
j
n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
Ω
Ω
Ω
⎪
:
:
:
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
ρg
h
dμ
ρh
dμ
·
·
·
·
·
·
ρh
h
dμ
⎪
⎪
i 2
2
2 n
⎪
⎪
⎪ m − 1 max Ω
⎪
⎪
⎪
Ω
Ω
⎪
⎪
⎪
1≤i
j≤m
/
.
.
.
⎪
⎪
⎪
⎪
⎪
.
.
.
⎪
⎪
.
.
·
·
·
·
·
·
.
⎪
:
:
:
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
ρg
h
dμ
ρh
h
dμ
·
·
·
·
·
·
ρh
dμ
⎩
i
n
2
n
n
⎭
Ω
Ω
4.9
Ω
Proof. By 3.65,
x, yi | z2 , . . . , zn 2
≤ x, z2 , . . . , zn 2
8
7
2
× max yi , z2 , . . . , zn m − 1 max yi , yj | z2 , . . . , zn ,
1≤i≤m
1≤i /
j≤m
and using 4.1 and 4.4, we get the proof of the proposition.
4.10
22
International Journal of Mathematics and Mathematical Sciences
Acknowledgment
The authors would like to thank the Editor-in-Chief and referees for their valuable
suggestions.
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