Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 494257, 18 pages
doi:10.1155/2009/494257
Research Article
A Hilbert-Type Linear Operator with the Norm and
Its Applications
Wuyi Zhong
Department of Mathematics, Guangdong Institute of Education, Guangzhou, Guangdong 510303, China
Correspondence should be addressed to Wuyi Zhong,
Received 9 February 2009; Accepted 9 March 2009
Recommended by Nikolaos Papageorgiou
A Hilbert-type linear operator T :
p
φ
→
p
ψ
is defined. As for applications, a more precise operator
inequality with the norm and its equivalent forms are deduced. Moreover, three equivalent
reverses from them are given as well. The constant factors in these inequalities are proved to be
the best possible.
Copyright q 2009 Wuyi Zhong. 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.
1. Introduction
In 1925, Hardy 1 extended Hilbert inequality as follows.
If p>1, 1/p 1/q 1, a
n
,b
n
π/p
∞
n1
a
p
n
1/p
∞
n1
b
q
n
1/q
,
1.1
∞
n1
∞
m1
a
ln
m/n
a
m
b
n
m − n
<
π
sin
π/p
2
∞
n1
a
p
n
1/p
∞
n1
a
p
n
, 1.4
where the constant factors π/sinπ/p
2
and π/sinπ/p
2p
are also the best possible. The
expression 1.3 is well known as a Hilbert-type inequality.
By setting a real space of sequences:
p
: {a; a {a
n
}
∞
n0
, a
p
{
∞
n1
|a
n
|
p
}
q
,
1.5
Ta
p
<
T
a
p
, 1.6
respectively, where T π/sinπ/p
2
, b ∈
q
. Ta,b is the formal inner product of Ta
and b.
The inequalities 1.1–1.4 play important roles in theoretical analysis and appli-
cations 3. These inequalities and their integral forms have been recently extended
or strengthened in 4–8. Zhao and Debnath 9 obtained a Hilbert-Pachpatte’s reverse
inequality. Zhong and Yang 10, 11 have given some reverses concerning some extensions
of 1.1. Papers in 12–15 studied some multiple Hardy-Hilbert-type or Hilbert-type
inequalities. Articles in 16, 17 got some Hilbert-type linear operator inequalities. In 2006,
Yang 18 deduced a new Hilbert-type inequality as follows.
Set p, q as a pair of conjugate exponents, and p>1, 1/2 ≤ α ≤ 1,a
n α
a
m
b
n
m − n
<
π
sin
π/p
2
∞
n0
a
p
n
1/p
∞
n0
sin
π/p
2p
∞
n0
a
p
n
. 1.8
It has been proved that 1.7 and 1.8 are two equivalent inequalities and their constant
factors π/sinπ/p
2
and π/sinπ/p
2p
are the best possible. When α 1, the expressions
1.7 and 1.8 can be reduced to 1.3 and 1.4, respectively.
This paper reports the studies on a Hilbert-type linear operator T :
p
φ
→
p
ψ
.As
for the applications, a more precise linear operator’s general form of Hilbert-type inequality
1.3 incorporating the norm and its equivalent form are deduced. Moreover, three equivalent
reverses of the new general forms are deduced as well. The constant factors in these
2
B
1
r
,
1
s
2
. 1.9
2Euler-Maclaurin’s summation formula.Setf ∈ C
3
0, ∞,if−1
i
f
i
x >
0,f
i
∞0 i 0, 1, 2, 3, then cf. 19, Lemma 1
∞
n0
f
n
n
>
∞
0
f
x
dx
1
2
f
0
. 1.11
2. Lemmas
Lemma 2.1. Set r, s as a pair of conjugate exponents, s>1, α>0, 0 <λ≤ 1, and define
g
u
:
⎧
⎪
⎨
⎪
⎩
λ
x α
m α
λ
1/s−1/λ
,x∈
−α, ∞
,m∈ N
0
. 2.2
Then, one has the following:
1 the function f
s
x satisfies the conditions of 1.10 and 1.11. This means
−1
i
f
i
m α
∞
−α
f
s
x
dx
B
1/s, 1/r
λ
2
π
λ sin
π/s
2
. 2.4
g
u
λ
m α
x α
m α
λ−1
< 0,
z
x
g
u
λ
m α
x α
m α
u
λ
m α
x α
m α
λ−1
3
3g
u
λ
2
λ − 1
m α
3
x α
m α
m α
x α
m α
λ/s−2
< 0,
t
x
λ/s − 1
λ/s − 2
m α
2
x α
m α
λ/s−3
> 0,
t
x
t
x
> 0,f
s
∞
0,
f
s
x
z
x
t
x
z
x
x
t
x
z
x
t
x
> 0,f
s
∞
0,
f
s
x
x
t
x
< 0,f
s
∞
0.
2.7
Then inequality 2.3 holds.
2 For x>−α, m ∈ N
0
and λ>0, s>1, set u x α/m α
λ
, then one has
1
λ
m α
∞
−α
x α
m α
λ
1/s−1/λ
d
x α
m α
1
λ
2
∞
0
ln u
u − 1
u
1/s−1
du.
2.8
By 1.9, then 2.4 holds. Lemma 2.1 is proved.
Journal of Inequalities and Applications 5
Lemma 2.2. Set r, s as a pair of conjugate exponents, s>1, α ≥ 1/2, 0 <λ≤ 1 and define
ω
λ
n α
1−λ/s
m ∈ N
0
, 2.9
then, one has
0 <ω
λ
m, s
<k
λ
s
, 2.10
0 <ω
λ
n, r
<k
λ
r
/
m α
λ
n α
/
m α
λ
− 1
n α
m α
λ
1/s−1/λ
1
λ
m α
∞
<
1
λ
m α
∞
0
f
s
x
dx
1
2
f
s
0
−
1
12
f
s
1
2
f
s
0
1
12
f
s
0
k
λ
s
−
1
λ
m α
R
0
g
α
m α
λ
α
m α
λ/s−1
, 2.14
f
s
0
z
0
t
0
α
m α
λ
α
m α
λ/sλ−1
.
2.15
6 Journal of Inequalities and Applications
Set u x α/m α
λ
, with the partial integration, by the strictly monotonic increase of
g
ug
u > 0 and s r/r − 1,itgives
0
−α
f
x
m α
λ
1/s−1/λ
d
x α
m α
m α
λ
α/
mα
λ
0
g
u
u
1/s−1
du
s
m α
α
m α
λ/s
−
s
m α
λ
α/
mα
λ
0
u
1/s
g
u
du
>
sα
λ
α/
mα
λ
0
u
1−1/r
du
sα
λ
g
α
m α
λ
α
m α
λ/s−1
−
r
g
α
m α
λ
α
m α
λ/s−1
−
r
2
α
λ
r − 1
2r − 1
g
α
m α
λ
α
m α
λ/s−1
−
r
2
α
λ
r − 1
2r − 1
−
λ
12α
g
α
m α
6sα
2sα − λ
− λ
s − λ
12sαλ
≥
6sα
s − λ
− λ
s − λ
12sαλ
6sα − λ
s − λ
12sαλ
> 0,
r
2
2
6α
2
− λ
2
λ
2
3r − 1
12λα
r − 1
2r − 1
> 0.
2.18
Journal of Inequalities and Applications 7
This means that Rs, m > 0. By 2.13 and 2.4, the inequalities 2.10 and 2.11 hold.
Lemma 2.2 is proved.
Lemma 2.3. Set r, s as a pair of conjugate exponents, s>1, α ≥ 1/2, 0 <λ≤ 1, and ω
λ
m, s,
k
λ
s are defined by 2.9, 2.4, respectively, then,
r
< 1
θ
λ
r
:
1
k
λ
s
λ
2
1
0
ln u
u − 1
u
−1/r
du
, 2.20
s
0, f
s
x is defined by 2.2.
Proof. By 2.12, 1.11,and2.4,
ω
λ
m, s
1
λ
m α
∞
n0
f
s
n
>
1
λ
m α
x
dx −
0
−α
f
s
x
dx
1
2
f
s
0
k
λ
s
1 −
1
k
s
1 − η
λ
m
.
2.22
This implies that 2.19 holds.
From the monotonic decrease of the function f
s
xsee 2.3, f
s
0 > 0andα ≥ 1/2,
one has η
λ
m > 1/k
λ
sλm ααf
s
0 − 1/2f
s
0 ≥ 0. On the other hand, if f
s
0 > 0
and by the computation as in 2.16,
η
0
<
1
k
λ
s
λ
m α
0
−α
f
s
x
dx
1
k
λ
s
λ
|≤L u ∈ 0, α/m α
λ
. Then,
0 <η
λ
m
<
L
k
λ
s
λ
2
α/
mα
λ
0
u
1/2s−1
du
2sL
k
s is defined by 2.4. Defining
I
1
: ε
∞
m0
m α
p
1−λ/r
−1
a
p
m
1/p
∞
n0
n α
q
y α
x α
λ
−
y α
λ
dx dy,
2.25
then
1
0 <I
1
<
ε
α
1ε
1
α
ε
, 2.26
1/p
∞
n0
n α
−1−ε
1/q
ε
1
α
1ε
∞
n1
1
n α
1ε
<ε
1
α
,
2.28
which implies that inequality 2.26 holds.
Journal of Inequalities and Applications 9
2 By y ≥ 1 − α, letting 0 <ε<pλ/2r, one has y α
−1−ε
≤ y α
−1
. And setting
u x α/y α
λ
, with lim
u → 0
ln u/u − 1u
1/2r
0 r>1, |ln u/u − 1u
1/2r
|≤
L
1
u ∈ 0, 1,L
1
> 0, one has
I
2
ε
λ
1−α
y α
−1−ε
⎡
⎣
∞
0
ln u
u − 1
u
1/r−ε/pλ−1
du −
1/
yα
λ
0
ln u
u − 1
u
1/r−ε/pλ−1
du
⎤
⎦
ln u
u − 1
u
1/r−ε/pλ−1
du
⎤
⎦
dy
≥
B
2
1/r − ε/pλ, 1/s ε/pλ
λ
2
−
εL
1
λ
2
∞
1−α
y α
−1
⎡
⎣
∞
1−α
y α
−λ
1/2r−ε/pλ
−1
dy
B
1/r − ε/pλ, 1/s ε/pλ
λ
2
εL
1
λ
3
1/2r − ε/pλ
x α
q
1−λ/s
−1
,
ψ
x
:
ϕ
x
1−p
x α
pλ/s−1
,
x ∈
n
|
a
n
|
p
1/p
< ∞
⎫
⎬
⎭
. 3.2
10 Journal of Inequalities and Applications
It is a real space of sequences, where
a
p,φ
∞
n0
φ
n
Ta
n
: C
n
:
∞
m0
ln
m α
/
n α
m α
λ
−
n α
λ
a
m
a
m
m α
λ
−
n α
λ
b
n
∞
n0
∞
m0
ln
m α
/
n α
a
p
ψ
. 3.6
It means that T :
p
φ
→
p
ψ
.
2 T is a bounded linear operator and
T
p,ψ
: sup
a∈
p
φ
a
/
θ
Ta
p,ψ
a
p,φ
∞
m0
lnm α/n α
m α
λ
−
n α
λ
m α
1−λ/r/q
n α
1−λ/s/p
a
m
n α
1−λ/s/p
m α
1−λ/r/q
1−λ/r
n α
1−λ/s
a
p
m
×
∞
m0
ln
m α
/
n α
m α
λ
−
n α
λ
−
n α
λ
m α
p−1
1−λ/r
n α
1−λ/s
a
p
m
ω
λ
n, r
ϕ
n
m α
p−1
1−λ/r
n α
1−λ/s
a
p
m
ϕ
p−1
n
.
3.8
And if ψnϕ
1−p
n,by2.9 and 2.10, it follows that
Ta
/
n α
m α
λ
−
n α
λ
m α
p−1
1−λ/r
n α
1−λ/s
a
p
m
k
p−1
< ∞.
3.9
This means that C {C
n
}
∞
n0
∈
p
ψ
and T
p,ψ
≤ k
λ
s.
If there exists a constant K<k
λ
s, such that T
p,ψ
≤ K, then for 0 <ε<pλ/2r,by
the definition 3.5, and by using H
¨
older’s inequality and the result 2.26, one has
ε
T a,
b
≤ ε
where a {a
m
}
∞
m0
∈
p
φ
,
b {
b
n
}
∞
n0
∈
q
ϕ
and a
m
,
b
n
are defined as in Lemma 2.4.
On the other hand, from the strictly monotonic decrease of the function gu
ln u/u − 1 and the exponents λ/r − ε/p − 1 < 0, λ/s − ε/q − 1 < 0and1− α ≥ 0, and
by α>0, λ>0, in view of 2.27, one has
λ
−
y α
λ
dx dy
I
2
≥ k
λ
s
o
1
ε −→ 0
.
3.11
12 Journal of Inequalities and Applications
In view of 3.10 and 3.11, one has k
λ
so1 <Kε/α
1ε
1/α
ε
> 0,then
Ta,b
<k
λ
s
a
p,φ
b
q,ϕ
. 3.12
2 If a ∈
p
φ
and a
p,φ
> 0,then
Ta
p,ψ
<k
λ
s
a
p
p,φ
. 3.14
And by p>1, 3.13 holds.
By using H
¨
older’s inequality and 3.13, one has
Ta,b
∞
n0
∞
m0
ln
m α
/
n α
1−λ/s/p
m α
1−λ/r/q
b
n
≤
Ta
p,ψ
b
q,ϕ
<k
λ
s
a
p,φ
b
q,ϕ
0
.Bya
combination as in 3.15 and by 1/2 ≤ α ≤ 1, 0 <λ≤ 1, and with 2.10 and 2.11, then,
0 <
K
n0
ϕ
n
b
q
n
K
K
n0
ψ
n
K
m0
ln
/
n α
a
m
b
n
K
m α
λ
−
n α
λ
<k
λ
s
K
n0
n0
ϕ
n
b
q
n
K
<k
p
λ
s
∞
n0
φ
n
a
p
n
< ∞. 3.17
Journal of Inequalities and Applications 13
p,ψ
k
λ
s, it is obvious that the constant factor
k
λ
sk
λ
r is the best possible. This completes the proof of Theorem 3.2.
Theorem 3.3. Set p, q and r, s as two pairs of conjugate exponents, 0 <p<1q<0, r>1,
1/2 ≤ α ≤ 1, 0 <λ≤ 1, a
n
,b
n
≥ 0.Let
H
a, b
∞
n0
∞
m0
ln
m α
s
∞
m0
1 − η
λ
m
φ
m
a
p
m
1/p
b
q,ϕ
. 3.19
2 If 0 < a
p,φ
λ
p
>k
p
λ
s
∞
m0
1 − η
λ
m
φ
m
a
p
m
. 3.20
3 If 0 < b
q,ϕ
< ∞,then
m α
λ
−
n α
λ
q
<k
q
λ
s
b
q
q,ϕ
, 3.21
where the marks a
p,φ
and b
q,ϕ
0 <p<1 as two formal norms are still defined like in 3.3
and the factor η
λ
m
a
p
m
1/p
∞
n0
ω
λ
n, r
ϕ
n
b
q
n
1/q
. 3.22
If 1/2 ≤ α ≤ 1, 0 <λ≤ 1, the expressions 2.19 and 2.11 are established for ω
λ
m, s and
m
φ
m
a
p
m
1/p
b
q,ϕ
K
∞
m0
1 − η
λ
m
1 −
∞
m0
O
1/
m α
1ελ/2s
∞
m0
1/
m α
1ε
⎫
⎬
⎭
1/p
,
3.23
where a {a
< ∞.
On the other hand, by 3.18, 2.2, 2.12,and2.10,
H
a,
b
∞
m0
∞
n0
ln
n α
/
m α
m α
λ/r−ε/p−1
n α
λ/s−ε/q−1
/
m α
λ
n α
/
m α
λ/s−ε/q−1
n α
/
m α
λ
− 1
∞
m0
m α
n0
1
n α
1ε
.
3.24
In view of 3.23 and 3.24, one has
K
⎧
⎨
⎩
1 −
∞
m0
O
1/
m α
1ελ/2s
λ
s. The constant factor k
λ
s in 3.19
is the best possible.
Journal of Inequalities and Applications 15
2 By 0 < a
p,φ
< ∞, one has
∞
n0
ψn
∞
m0
lnm α/n αa
m
/m α
λ
−
n α
λ
p
> 0. Setting b
n
: ψn
∞
n
∞
n0
ψ
n
∞
m0
ln
m α
/
n α
a
m
m α
λ
−
n α
m
1/p
b
q,ϕ
,
3.26
one has
∞
n0
ψ
n
∞
m0
ln
m α
/
n α
a
m
a
p
m
. 3.27
Therefore, 3.20 holds.
On the other hand, if 3.20 is valid, by 0 <p<1 q<0 and by using the reverse
H
¨
older’s inequality, it has
H
a, b
∞
n0
n α
λ/s−1/p
∞
m0
ln
n0
ψ
n
∞
m0
ln
m α
/
n α
a
m
m α
λ
−
n α
λ
p
λ
m
φ
m
a
p
m
1/p
b
q,ϕ
.
3.28
Then 3.19 holds. It means that 3.20 is equivalent to 3.19.
3 By 0 < b
q,ϕ
< ∞, it is obvious that there exist n
0
∈ N, such that
K
m0
m α
λ
−
n α
λ
⎤
⎥
⎦
q
> 0,
K
n0
ϕ
n
b
q
n
> 0 when K>n
0
.
3.29
m0
1 − η
λ
m
φ
m
a
p
m
K
K
m0
φ
−1
m
1 − η
λ
K
m0
K
n0
ln
m α
/
n α
b
n
a
m
K
m α
λ
−
n α
K
n0
ϕ
n
b
q
n
1/q
.
3.30
Further one has
K
m0
1 − η
λ
m
φ
m
m0
1 − η
λ
m
φ
m
a
p
m
K
<k
q
λ
s
K
n0
ϕ
a
p
m
∞
<
1
1 − θ
λ
r
∞
m0
1 − η
λ
m
φ
m
a
p
λ
m
φ
−1
m
1/p
a
m
⎧
⎨
⎩
φ
−1
m
1 − η
λ
m
1/p
1− η
λ
m
φ
m
a
p
m
1/p
⎧
⎨
⎩
∞
m0
φ
−1
m
1− η
λ
⎫
⎬
⎭
1/q
>k
λ
s
∞
m0
1 − η
λ
m
φ
m
a
p
m
1/p
3 D. S. Mitrinovi
´
c, J. Pe
ˇ
cari
´
c,andA.M.Fink,Inequalities Involving Functions and Their Integrals and
Derivatives, vol. 53 of Mathematics and Its Applications (East European Series), Kluwer Academic
Publishers, Dordrecht, The Netherlands, 1991.
4 M. Gao and B. Yang, “On the extended Hilbert’s inequality,” Proceedings of the American Mathematical
Society, vol. 126, no. 3, pp. 751–759, 1998.
5 B. G. Pachpatte, “On some new inequalities similar to Hilbert’s inequality,” Journal of Mathematical
Analysis and Applications, vol. 226, no. 1, pp. 166–179, 1998.
6 B. Yang, I. Brneti
´
c, M. Krni
´
c, and J. Pe
ˇ
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