Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 393025, 18 pages
doi:10.1155/2010/393025
Research Article
A New Hilbert-Type Linear Operator with
a Composite Kernel and Its Applications
Wuyi Zhong
Department of Mathematics, Guangdong Institute of Education, Guangzhou, Guangdong 510303, China
Correspondence should be addressed to Wuyi Zhong, [email protected]
Received 20 April 2010; Accepted 31 October 2010
Academic Editor: Ond
ˇ
rej Do
ˇ
sl
´
y
Copyright q 2010 Wuyi Zhong. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any m edium, provided the original work is properly cited.
A new Hilbert-type linear operator with a composite kernel function is built. As the applications,
two new more accurate operator inequalities and their equivalent forms are deduced. The constant
factors in these inequalities are proved to be the best possible.
1. Introduction
In 1908, Weyl 1 published the well-known Hilbert’s inequality as follows:
if a
n
, b
n
≥ 0 are real sequences, 0 <
a
2
n
∞
n1
b
2
n
1/2
,
1.1
where the constant factor π is the best possible.
Under the same conditions, there are the classical inequalities 2
∞
n0
∞
m0
a
m
b
n
m n 1
<π
∞
2
∞
n1
a
2
n
∞
n1
b
2
n
1/2
,
1.3
2 Journal of Inequalities and Applications
where the constant factors π and π
2
are the best possible also. Expression 1.2 is called a
more accurate form of 1.1. Some more accurate inequalities were considered by 3–5. In 2009,
Zhong 5 gave a more accurate form of 1.3.
Set p, q, s, r as two pairs of conjugate exponents, and p>1, s>1, α ≥ 1/2, and a
n
,
b
n
≥ 0, such that 0 <
n α
a
m
b
n
m α
λ
−
n α
λ
<
∞
n0
n α
p1−λ/r−1
a
p
n
1/p
{
∞
n0
φn|a
n
|
p
}
1/p
< ∞},
q
ϕ
: {b; b {b
n
}
∞
n0
,and b
q,ϕ
: {
∞
n0
ϕn|b
n
|
q
}
1/q
n α
λ
<k
λ
s
a
p,φ
b
q,ϕ
,
1.5
where T :
p
φ
→
p
ψ
is a linear operator, k
λ
sT. a
p,φ
is the norm of the sequence a with
a weight function φ. Ta,b is a formal inner product of the sequences Tan:
n
<
∞
m
f
x
dx
1
2
f
m
−
1
12
f
m
,
∞
nm
1
ln
λ
αm ln
λ
βy
ln
λ/r
αm
ln
1−λ/s
βy
y
,y∈
1, ∞
,m∈ N,
2.1
g
y
:
1
ln
λ
αy ln
du −
1
2
f
1
1
12
f
1
, 2.3
R
λ
n, r
:
ln α/ ln βn
0
u
λ/r−1
1 u
λ
1
2
f
1
. 2.5
Then, it has the following.
1 The functions fy, gy satisfy the conditions of 1.6.Itmeansthat
−1
i
F
i
y
> 0
F f, g, y ∈
1, ∞
,
F
i
∞
1
ln αm
ρ
ρ>0,m−→ ∞
.
2.8
Proof. 1 For β ≥ α ≥ e
7/12
, y ≥ 1, m ∈ N,0<λ≤ 1, and s>1, set
f
1
y
:
1
ln
λ
αm ln
λ
βy
,f
2
y
y
f
3
y
2.10
when y ≥ 1. It is easy to find that
−1
i
f
i
j
y
> 0,f
i
j
∞
0
y ∈
1, ∞
i
g
i
y > 0, g
i
∞0 y ≥ 1, i 0, 1, 2, 3, 4.These
tell us that 2.6 holds and the functions fy, gy satisfy the conditions of 1.6.
2 Set t u
λ
. With the partial integration, it has
ln β/ln αm
0
u
λ/s−1
1 u
λ
du
1
λ
ln β/ ln αm
λ
0
t
1/s−1
1 t
dt
s
λ
λ
ln
λ
β
ln
λ
αm ln
λ
β
ln αm
ln β
λ/r
s
2
λ
1 s
ln β/ ln αm
λ
0
dt
1/s1
1 t
αm ln
λ
β
2
ln αm
ln β
λ/r
.
2.12
By 2.1,ithas
f
1
ln αm
ln β
λ/r
ln
λ−1
β
ln
λ
αm ln
λ
ln
λ−2
β
ln
λ
αm ln
λ
β
−
ln
λ−1
β
ln
λ
αm ln
λ
β
.
2.14
In view of 2.12∼2.14,ithas
R
λ
m, s
≥
ln αm
λ
αm ln
λ
β
2
s
2
λ
1 s
−
λ
12ln
2
β
.
2.15
As ln β ≥ 7/12, s>1r>1,and0<λ≤ 1, it has
−
1 − λ/s
12 ln β
−
7
12
s
1 −
λ
s
7
12
−
1
12 ln β
> 0,
s
2
λ
1 s
−
λ
12ln
2
β
>
s
4
−
s
12ln
2
≥
s
λ
ln
λ
β
ln
λ
αm ln
λ
β
ln αm
ln β
λ/r
−
1
2
ln
λ−1
β
ln
λ
αm ln
λ
β
ln αm
ln β
λ−1
β
ln
λ
αm ln
λ
β
ln β −
1
2
> 0,
η
λ
m, s
<
1
λ
ln β/ ln αm
λ
0
u
1/s−1
du
s
λ
λ/s−1
,y∈
0, ∞
,m∈ N
0
,
g
1
y
:
1
λ
n β
ln y α/n β
λ
y α/n β
λ
− 1
y α
n β
λ/r−1
,y∈
1
12
f
1
0
,
R
λ
n, r
:
1
λ
2
α/nβ
λ
0
ln u
u − 1
u
1/r−1
du −
ln u
u − 1
u
1/s−1
du −
1
2
f
1
0
.
2.18
Then, it has
1 The functions f
1
y, g
1
y satisfy the conditions of 1.6.Itmeansthat
−1
i
F
i
y
> 0
m, s
> 0,
R
λ
n, r
> 0,
2.20
6 Journal of Inequalities and Applications
3
0 <η
λ
m, s
O
1
m α
ρ
ρ>0,m−→ ∞
.
2.21
λ
2
β/mα
λ
0
ln u
u − 1
u
1/s−1
du
1
λ
2
u
0
0
h
u
u
1/s−1
du
s
λ
2
u
du
≥
s
λ
2
h
u
0
u
1/s
0
− h
u
0
u
0
0
u
2.22
f
1
0
1
λ
m α
ln β/m α
λ
β/m α
λ
− 1
β
m α
λ/s−1
1
λβ
h
u
0
s
−
1
λ
h
u
0
. 2.24
With 2.22∼2.24,ithas
R
λ
m, s
≥ h
u
0
u
1/s
0
s
λ
2
.
2.25
By hu
0
> 0, h
u
0
< 0, and β ≥ 1/2, s>1, 0 <λ≤ 1, it has
s
λ
2
−
1
2λβ
1
12β
2
1
s
−
1
λ
6βs
4β
2
s − s
8β
2
s − 1
12βλ
1 s
> 0.
2.26
So R
λ
m, s > 0 holds. Similarly, it can be shown that
R
λ
n, r > 0.
Journal of Inequalities and Applications 7
3 In view of 2.22, 2.23,byhu > 0, h
u < 0, it has
η
λ
β
> 0,
2.27
and by lim
u → 0
ln u/u − 1u
1/2s
0, so there exists a constant L>0, such that |ln u/u −
1u
1/2s
| <Lu ∈ 0, β/m α
λ
.Thenithas
η
λ
m, s
<
1
λ
2
β/mα
λ
0
ln u
u − 1
u
> 0,U∞∞U u, v.Give
some notations as follows:
1
φ
x
:
u
x
p1−λ/r−1
u
x
1−p
,
ϕ
x
:
v
pλ/s−1
v
x
x ∈
n
0
, ∞
,
3.1
2 set
p
φ
:
⎧
⎨
⎩
a; a
{
a
n
}
∞
nn
a real space of sequences,where
a
p,φ
∞
nn
0
φ
n
|
a
n
|
p
1/p
3.3
is called the norm of the sequence with a weight function φ. Similarly, the real spaces of sequences
q
ϕ
,
p
ψ
and the norm b
,v
n
a
m
n ≥ n
0
,
3.4
4 for all a ∈
p
φ
, b ∈
q
ϕ
,define the formal inner product of Ta and b as
Ta,b
:
∞
nn
0
∞
m
,v
n
a
m
b
n
,
3.5
5 define two weight coefficients ωm, s and ϑn, r as
ω
λ
m, s
:
∞
nn
0
K
u
m
mn
0
K
u
m
,v
n
v
n
λ/s
u
m
1−λ/r
u
m
,m,n≥ n
0 <ω
λ
m, s
<k
λ
, 0 <ϑ
λ
n, r
<k
λ
m, n ≥ n
0
, 3.7
k
λ
1 − O
1
u
m
,
3.9
It means that T :
p
φ
→
p
ψ
,
Journal of Inequalities and Applications 9
2 T is a bounded linear operator and
T
p,ψ
: sup
a∈
p
φ
a
/
θ
Ta
p,ψ
a
p,φ
,v
n
u
m
1−λ/r/q
v
n
1/p
v
n
1−λ/s/p
u
m
1/p
p
≤
∞
mn
0
K
u
m
,v
n
u
m
p−11−λ/r
v
n
,v
n
v
n
q−11−λ/s
u
m
u
m
1−λ/r
v
n
q−1
p−1
∞
mn
a
p
m
ϑ
λ
n, r
ϕ
n
p−1
<k
p−1
λ
∞
mn
0
K
u
m
m
ϕ
p−1
n
.
3.11
And by ψnϕ
1−p
n, it follows that
Ta
p
p,ψ
∞
nn
0
ψ
n
C
p
n
v
n
1−λ/s
u
m
p−1
a
p
m
k
p−1
λ
∞
mn
0
ω
λ
≤ k
λ
a
p,φ
,andT
p,ψ
≤ k
λ
. T is a bounded linear
operator.
If there exists a constant K<k
λ
,suchthatT
p,ψ
≤ K,thenforε>0, setting a
m
:
um
λ/r−ε/p−1
u
m,
b
n
:vn
λ/s−ε/q−1
v
n,ithasa {a
T
p,ψ
a
p,φ
b
q,ϕ
≤ K
∞
mn
0
u
m
u
m
u
m
1ε
.
3.13
10 Journal of Inequalities and Applications
But on the other side, by 3.8,ithas
T a,
b
∞
mn
0
∞
nn
0
K
u
∞
mn
0
u
m
u
m
1ε
∞
nn
0
K
u
m
,v
n
nn
0
K
u
m
,v
n
u
m
λ/rε/q
v
n
1−λ/sε/q
v
n
n
v
n
1−λ/sε/q
n
1
−1
nn
0
K
u
m
,v
n
u
m
,v
n
u
m
λ/r
v
n
v
n
1−λ/s
n
1
−1
nn
0
K
m
ε/q
ω
λ
m, s
−
n
1
−1
nn
0
K
u
m
,v
n
u
u
m
λ/r
v
n
v
n
1−λ/sε/q
.
3.15
The series is uniformly convergent for ε ≥ 0, so it has
lim
ε → 0
∞
nn
0
K
3.16
and for m>n
0
,thereexistsε
0
> 0, when 0 <ε<ε
0
,ithas
∞
nn
0
K
u
m
,v
n
u
m
λ/rε/q
b
≥
∞
mn
0
u
m
u
m
1ε
ω
λ
m, s
−
1
u
m
1
k
λ
u
m
k
λ
∞
mn
0
u
m
u
m
1ε
⎧
⎨
⎩
1 −
m
1ε
O
1
u
m
ρ
1
k
λ
u
m
⎫
⎬
⎭
.
3.18
λ/r
m
v
1−λ/s
y
v
y
,
g
y
: K
λ
u
y
,v
n
1,u
u
λ/s−1
du −
1
2
f
n
0
1
12
f
n
0
,
3.20
R
n, r
:
m, s
:
vn
0
/um
0
K
λ
1,u
u
λ/s−1
du −
1
2
f
n
0
.
3.22
If (a) K
λ
x, y ≥ 0 is a homogeneous measurable kernel function of “λ”degreeinR
2
> 0
y>n
0
,F
i
∞
0
F f, g, i 0, 1, 2, 3, 4
,
3.24
12 Journal of Inequalities and Applications
(c) there exists ρ>0,suchthat
R
m, s
> 0,
R
n, r
> 0, 0 <η
∞
nn
0
∞
mn
0
K
λ
u
m
,v
n
a
m
b
n
<k
λ
s
pλ/s−1
v
n
∞
mn
0
K
u
m
,v
n
a
m
p
1/p
<k
λ
dy −
1
2
f
n
0
<ω
λ
m, s
∞
nn
0
K
λ
u
m
,v
n
0
f
y
dy −
1
2
f
n
0
1
12
f
n
0
,
3.28
0 <ϑ
λ
n, r
mn
0
g
m
<
∞
n
0
g
y
dy −
1
2
g
n
0
1
12
g
y
u
λ/r
m
v
1−λ/s
y
v
y
dy
∞
vn
0
/um
K
λ
1,ν
ν
λ/s−1
−
vn
0
/um
0
K
λ
1,ν
ν
λ/s−1
dν,
3.30
∞
n
0
g
y
dy
∞
n
0
K
λ
K
λ
μ, 1
μ
λ/r−1
dμ
∞
0
K
λ
μ, 1
μ
λ/r−1
dμ −
un
0
/vn
0
K
λ
μ, 1
K
λ
u, 1u
λ/r−1
du
∞
0
K
λ
1,tt
λ/s−1
dt k
λ
s.In
view of 3.28, 3.30, 3.20, 3.22,andwith3.25,ithas
0 <ω
λ
m, s
<k
λ
s
− R
λ
m, s
1
u
ρ
m
ρ>0,m−→ ∞
.
3.32
Similarly, with 3.29, 3.31, 3.21,and3.25,ithas
0 <ϑ
λ
n, r
<k
λ
s
3.33
also. By Theorem 3.1,ithas
Ta
p,ψ
1
>n
0
,suchthat
H
mn
0
φma
p
m
> 0andb
n
H : ψn
H
mn
0
K
λ
um,vna
m
p−1
> 0whenH>n
1
.For
b : {b
n
H
mn
0
K
λ
u
m
,v
n
a
m
p
H
nn
0
H
mn
0
K
λ
a
p
n
1/p
H
nn
0
ϕ
n
b
q
n
H
1/q
< ∞.
3.36
By p>1andq>1, it follows that
0 <
H
nn
0
∞
nn
0
ϕnb
q
n
∞ < ∞, and it means that b
{b
n
∞}
∞
nn
0
∈
q
ϕ
and b
q,ϕ
> 0. Therefore, the inequality 3.36 keeps the form of the strict
inequality when H →∞.Inviewof
∞
nn
0
ϕnb
q
n
∞Ta
p
∞
n1
ln βn
q1−λ/s−1
b
q
n
/n
q−1
<
∞,then
∞
m1
∞
n1
a
m
b
n
ln
λ
αm ln
λ
βn
<
π
λ sin
1/q
.
4.1
2 If 0 <
∞
n1
ln αn
p1−λ/r−1
a
p
n
/n
p−1
< ∞,then
∞
n1
ln βn
pλ/s−1
n
∞
m1
a
m
λ
r : π/λsinπ/s and π/λsinπ/s
p
are both the
best possible. Inequality 4.2 is equivalent to 4.1.
Journal of Inequalities and Applications 15
Proof. Setting K
λ
x, y1/x
λ
y
λ
, x, y ∈ R
2
, it is a homogeneous m easurable kernel
function of “λ” degree. Letting t u
λ
,ithas
0 <k
λ
s
:
∞
0
K
λ
,
1
r
k
λ
r
< ∞.
4.3
Setting uxln αx, vxln βx,thenbothux and vx are strictly monotonic increasing
differentiable functions in 1, ∞ and satisfy
U
1
> 0,U
∞
∞
U u, v
,
0 <
∞
n1
u
n
1ε
∞
n1
1
n
ln αn
1ε
< ∞
4.4
for ε>0. As β ≥ α ≥ e
7/12
,0<λ≤ 1, s>1, and n
0
1, letting
f
y
K
λ
λ/r
αm
ln
1−λ/s
βy
y
,
g
y
: K
λ
u
y
,v
n
v
λ/s
n
,n,m∈ N,
4.5
with 2.1∼2.8,ithas
R
λ
m, s
v1/um
0
K
λ
1,u
u
λ/s−1
du −
1
2
f
1
1
12
:
u1/vn
0
K
λ
μ, 1
μ
λ/r−1
dμ −
1
2
g
1
1
12
g
1
> 0.
4.6
When 0 <
> 0, b
q,ϕ
> 0, by Theorem 3.2, inequality 4.1 holds, so does
4.2.And4.2 is equivalent to 4.1, and the constant factors k
λ
sk
λ
r : π/λsinπ/s
and π/λsinπ/s
p
are both the best possible.
Example 4.2. Set p, q, r, s be two pairs of conjugate exponents and p>1, s>1, β ≥ α ≥ 1 /2,
0 <λ≤ 1. Then it has the following.
16 Journal of Inequalities and Applications
1 If 0 <
∞
n0
n α
p1−λ/r−1
a
p
n
< ∞ and 0 <
∞
n0
n β
q1−λ/s−1
b
λ
<
π
λ sinπ/s
2
∞
n0
n α
p1−λ/r−1
a
p
n
1/p
∞
n0
n β
q1−λ/s−1
b
q
λ
−
n β
λ
p
<
π
λ sinπ/s
2p
∞
n0
n α
p1−λ/r−1
a
p
n
,
4.8
where inequality 4.8 is equivalent to 4.7 and the constant factors k
λ
sk
λ
1,u
u
λ/s−1
du
1
λ
∞
0
− ln u
λ
1 − u
λ
u
λ/s−1
du
1
λ
2
∞
0
ln t
t − 1
t
1/s−1
0
> 0,U
∞
∞
U u, v
,
0 <
∞
n0
v
n
v
n
1ε
∞
n0
4.10
Journal of Inequalities and Applications 17
for ε>0. As β ≥ α ≥ 1/2, 0 <λ≤ 1, s>1, and n
0
0, letting
f
1
y
K
λ
u
m
,v
y
u
λ/r
m
v
1−λ/s
y
1−λ/s
1
λ
m α
ln y β/m α
λ
y β/m α
λ
− 1
y β
m α
λ/s−1
,
g
1
y
: K
λ
u
y
y α
λ
−
n β
λ
n β
λ/s
y α
1−λ/r
1
λ
n β
ln y α/n β
λ
y α/n β
λ
− 1
y α
n β
λ/r−1
,y∈
1
12
f
1
0
1
λ
2
β/mα
λ
0
ln u
u − 1
u
1/s−1
du −
1
2
f
1
0
1
0
1
12
g
1
0
> 0,
0 <η
λ
m, s
O
1
u
m
ρ
ρ>0,m−→ ∞
> 0, b
q,ϕ
> 0, by Theorem 3.2, inequality 4.7 holds, so does 4.8.
And 4.8 is equivalent to 4.7, and the constant factors k
λ
sk
λ
r :π/λsinπ/s
2
and
π/λsinπ/s
2p
are both the best possible.
Remark 4.3. It can be proved similarly that, if the conditions “β ≥ α ≥ e
7/12
”inLemma 2.1 and
“β ≥ α ≥ 1/2” in Lemma 2.2 are changed into “α ≥ β ≥ e
7/12
”and“α ≥ β ≥ 1/2”, respectively,
Lemmas 2.1 and 2.2 are also valid. So the conditions “β ≥ α ≥ e
7/12
”inExample 4.1 and
“β ≥ α ≥ 1/2” in Example 4.2 can be replaced by “β ≥ e
7/12
, α ≥ e
7/12
”and“β ≥ 1/2,
α ≥ 1/2”, respectively.
Acknowledgment
This paper is supported by the National Natural Science Foundation of China (no. 10871073).The
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