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, wp@bao.ac.cn
Received 9 February 2009; Accepted 9 March 2009
Recommended by Nikolaos Papageorgiou
p
p
A Hilbert-type linear operator T : φ → 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.
∞ q
p
If p > 1, 1/p 1/q 1, an , bn ≥ 0, 0 < ∞
n1 an < ∞, and 0 <
n1 bn < ∞, then
∞
∞ π
am bn
<
m
n
sin π/p
n1 m1
∞
n1
p
an
1/p 1/q
∞
q
bn
,
1.1
n1
p p ∞
∞ ∞
am
π
p
<
an ,
m
n
sinπ/p
n1 m1
n1
1.2
where p, q is a pair of conjugate exponents. The constant factors π/sinπ/p and
π/sinπ/pp are the best possible. The expression 1.1 is the famous Hardy-Hilbert’s
inequality.
2
Journal of Inequalities and Applications
Under the same conditions, there are the classic inequalities 2:
∞
∞ lnm/nam bn
m−n
n1 m1
π
<
sin π/p
p
∞ ∞
lnm/nam
n1 m1
m−n
2 ∞
p
an
1/p 1/q
∞
q
bn
,
n1
π
<
sin π/p
1.3
n1
2p
∞
p
1.4
an ,
n1
where the constant factors π/sinπ/p2 and π/sinπ/p2p are also the best possible. The
expression 1.3 is well known as a Hilbert-type inequality.
∞
p 1/p
< ∞}
By setting a real space of sequences: p : {a; a {an }∞
n0 , ap { n1 |an | }
∞
p
p
and defining a linear operator T : → , T an Cn n1 lnm/nam /m − n
n ∈ N0 , the expressions 1.3 and 1.4 can be rewritten as
T a, b < T ap bq ,
1.5
T ap < T ap ,
1.6
respectively, where T π/sinπ/p2 , b ∈ q . T a, b is the formal inner product of T a
and b.
The inequalities 1.1–1.4 play important roles in theoretical analysis and applications 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, an , bn ≥ 0, such that
∞ q
p
0< ∞
n1 an < ∞, 0 <
n1 bn < ∞, then one has
∞ ∞
lnm α/n αam bn
m−n
n0 m0
π
<
sin π/p
p
∞ ∞
lnm α/n αam
n0 m0
m−n
1/p 1/q
2 ∞
∞
p
q
an
bn
,
n0
π
<
sin π/p
1.7
n0
2p
∞
p
an .
1.8
n0
It has been proved that 1.7 and 1.8 are two equivalent inequalities and their constant
factors π/sinπ/p2 and π/sinπ/p2p are the best possible. When α 1, the expressions
1.7 and 1.8 can be reduced to 1.3 and 1.4, respectively.
p
p
This paper reports the studies on a Hilbert-type linear operator T : φ → 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
inequalities are all the best possible.
At first, two known results are introduced.
Journal of Inequalities and Applications
3
1 If s > 1, r, s is a pair of conjugate exponents, then the Beta function is defined as
follows cf. 2, Theorem 342,
∞
0
2 ln u 1/s−1
π
1 1 2
1 1 2
u
,
,
du B
B
.
u−1
sinπ/s
s r
r s
1.9
2 Euler-Maclaurin’s summation formula. Set f ∈ C3 0, ∞, if −1i f i x >
0, f ∞ 0 i 0, 1, 2, 3, then cf. 19, Lemma 1
i
∞
∞
1
1
fn <
fxdx f0 − f 0,
2
12
0
n0
1.10
∞
∞
1
fn >
fxdx f0.
2
0
n0
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
⎧
⎪
⎨ ln u ,
gu : u − 1
⎪
⎩1,
u ∈ 0, 1 ∪ 1, ∞,
2.1
u 1,
1/s−1/λ
1
xα λ
xα λ
fs x : hm,λ x,
: g
,
s
mα
mα
x ∈ −α, ∞, m ∈ N0 . 2.2
Then, one has the following:
1 the function fs x satisfies the conditions of 1.10 and 1.11. This means
i
i
−1i fs x > 0 x > −α,
fs ∞ 0
i 0, 1, 2, 3,
2.3
2
1
kλ s :
λm α
∞
B1/s, 1/r
fs xdx λ
−α
2
π
λ sinπ/s
2
.
2.4
Proof. 1 For α > 0, x > −α, m ∈ N0 , 0 < λ ≤ 1 and s > 1, set zx gx α/m αλ ,
1/s−1/λ
tx x α/m αλ x α/m αλ/s−1 and u x α/m αλ . These
show that zx gu and fs x zxtx gutx when u > 0. With the settings,
4
Journal of Inequalities and Applications
−1i g i u > 0, g i ∞ 0 u > 0, i 0, 1, 2, 3 cf. 16, Lemma 2.2, one has zx > 0,
tx > 0,
λ
x α λ−1
< 0,
mα mα
2
λ
λλ − 1 x α λ−2
x α λ−1
z x g u
g u
> 0,
mα mα
m α2 m α
z x g u
3
λ
λ2 λ − 1 x α 2λ−3
x α λ−1
z x g u
3g u
mα mα
m α3 m α
g u
λλ − 1λ − 2
xα
mα
λ−3
2.5
< 0,
m α3
λ/s − 1 x α λ/s−2
< 0,
t x mα mα
λ/s − 1λ/s − 2 x α λ/s−3
> 0,
t x mα
m α2
t x λ/s − 1λ/s − 2λ/s − 3
m α3
xα
mα
λ/s−4
< 0.
These are followed by
fs x zxtx > 0,
fs ∞ 0,
fs x z xtx zxt x < 0,
2.6
fs ∞ 0,
fs x z xtx 2z xt x zxt x > 0,
fs ∞ 0,
fs x z xtx 3zxt x 3z xt x zxt x < 0,
fs ∞ 0.
2.7
Then inequality 2.3 holds.
2 For x > −α, m ∈ N0 and λ > 0, s > 1, set u x α/m αλ , then one has
1
λm α
∞
1
fs xdx λ
−α
∞
1
λ2
ln x α/m αλ
−α x
∞
0
α/m αλ − 1
ln u 1/s−1
u
du.
u−1
By 1.9, then 2.4 holds. Lemma 2.1 is proved.
xα
mα
λ 1/s−1/λ xα
d
mα
2.8
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
ωλ m, s :
∞
lnn α/m α
n0
n αλ − m αλ
·
m αλ/r
n α1−λ/s
m ∈ N0 ,
2.9
then, one has
0 < ωλ m, s < kλ s,
2.10
0 < ωλ n, r < kλ r kλ s n ∈ N0 ,
2.11
where kλ s is defined by 2.4.
Proof. By 2.9 and 2.2, it is evident that
1/s−1/λ
∞
ln n α/m αλ
1
nα λ
0 < ωλ m, s λm α n0 n α/m αλ − 1
mα
∞
∞
1
1
1
hm,λ n,
fs n.
λm α n0
s
λm α n0
2.12
In view of 2.3, 1.10, and 2.4, one has
∞
1
1
fs xdx fs 0 − fs 0
2
12
0
∞
0
1
1
1 fs xdx −
fs xdx − fs 0 fs 0
λm α
2
12
−α
−α
ωλ m, s <
1
λm α
kλ s −
where Rs, m :
0
1
Rs, m,
λm α
f xdx
−α s
fs 0 z0t0 g
2.13
− 1/2fs 0 1/12fs 0 m ∈ N0 . With 2.6, it follows that
α
mα
λ α
mα
λ/s−1
,
fs 0 z 0t0 z0t 0
λ λ/s−1
λ λ/sλ−1
λ−s
λ α
α
α
α
g
g
.
sα
mα
mα
α
mα
mα
2.14
2.15
6
Journal of Inequalities and Applications
Set u x α/m αλ , with the partial integration, by the strictly monotonic increase of
g ug u > 0 and s r/r − 1, it gives
0
−α
fxdx
m α
mα
λ
0
ln x α/m αλ
−α x
λ
α/m α − 1
α/mαλ
xα
mα
λ 1/s−1/λ
d
xα
mα
guu1/s−1 du
0
sm α
λ
α/mαλ
gudu1/s
0
λ
λ λ/s
sm α
sm α α/mα 1/s α
α
g
−
u g udu
λ
mα
mα
λ
0
λ λ/s−1
λ α/mαλ
rm α sα
α
α
α
g
g
−
u1−1/r du
>
λ
mα
mα
λr − 1
mα
0
sα
g
λ
sα
g
λ
α
mα
α
mα
λ λ α
mα
α
mα
λ/s−1
λ/s−1
r 2 m α
g
−
λr − 12r − 1
r2α
g
−
λr − 12r − 1
α
mα
λ α
mα
λ α
mα
α
mα
λ 2−1/r
λ/sλ−1
.
2.16
In view of 2.13–2.16, one has
λ λ/s−1
sα 1 λ − s
α
α
Rs, m >
− g
λ
2 12sα
mα
mα
λ λ/sλ−1
λ
r2α
α
α
−
g
−
.
λr − 12r − 1 12α
mα
mα
2.17
If α ≥ 1/2, s > 1r > 1, 0 < λ ≤ 1, gu > 0, −g u > 0, one has
sα 1 λ − s 12s2 α2 − 6sαλ λλ − s 6sα2sα − λ − λs − λ
− λ
2 12sα
12sαλ
12sαλ
≥
6sαs − λ − λs − λ 6sα − λs − λ
> 0,
12sαλ
12sαλ
r2α
λ
12r 2 α2 − λ2 r − 12r − 1
−
λr − 12r − 1 12α
12λαr − 12r − 1
2r 2 6α2 − λ2 λ2 3r − 1
> 0.
12λαr − 12r − 1
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 ωλ m, s,
kλ s are defined by 2.9, 2.4, respectively, then,
1
2
3
ωλ m, s > kλ s 1 − ηλ m ,
2.19
1
ln u −1/r
1
u
0 < ηλ m < θλ r < 1 θλ r :
du ,
kλ sλ2 0 u − 1
λ/2s 1
ηλ m O
m −→ ∞,
mα
where ηλ m : 1/kλ sλm α
0
f xdx
−α s
2.20
2.21
− 1/2fs 0, fs x is defined by 2.2.
Proof. By 2.12, 1.11, and 2.4,
∞
∞
1
1
1
fs n >
fs xdx fs 0
λm α n0
λm α 0
2
∞
0
1
1
fs xdx −
fs xdx fs 0
λm α −α
2
−α
0
1
1
kλ s 1 −
fs xdx − fs 0
kλ sλm α −α
2
ωλ m, s 2.22
kλ s 1 − ηλ m .
This implies that 2.19 holds.
From the monotonic decrease of the function fs x see 2.3, fs 0 > 0 and α ≥ 1/2,
one has ηλ m > 1/kλ sλm ααfs 0 − 1/2fs 0 ≥ 0. On the other hand, if fs 0 > 0
and by the computation as in 2.16,
1
ηλ m kλ sλm α
<
1
kλ sλm α
Equation 2.20 is valid.
0
0
α/mαλ
1
fs xdx − fs 0
2
−α
−α
fs xdx 1
kλ sλ2
0
2.23
ln u 1/s−1
u
du ≤ θλ r < 1.
u−1
8
Journal of Inequalities and Applications
Since limu → 0 ln u/u − 1u1/2s 0 s > 1, there exists a constant L > 0, such that
|ln u/u − 1u1/2s | ≤ L u ∈ 0, α/m αλ . Then,
L
kλ sλ2
0 < ηλ m <
α/mαλ
u1/2s−1 du 0
2sL
kλ sλ2
α
mα
λ/2s
.
2.24
This means that ηλ m O1/m αλ/2s m → ∞, the proof is finished.
Lemma 2.4. Set p, q and r, s as two pairs of conjugate exponents, p > 1, r > 1, α > 0, λ > 0,
0 < ε < pλ/2r, am : m αλ/r−ε/p−1 , bn : n αλ/s−ε/q−1 , and kλ s is defined by 2.4. Defining
I1 : ε
∞
m p
αp1−λ/r−1 am
1/p 1/q
∞
q1−λ/s−1 q
,
bn
n α
m0
∞
I2 : ε
n0
x α
λ/s−ε/q−1 ln x α/ y α
yα
dx dy,
λ
x αλ − y α
λ/r−ε/p−1 1−α
2.25
then
ε
1
,
α1ε αε
1
0 < I1 <
2
I2 ≥ kλ s o1
2.26
ε −→ 0 .
2.27
Proof. 1 By α > 0 and ε > 0, one has
0 < I1 ε
∞
1/p m α
−1−ε
m0
ε
α1ε
ε
1
1
α1ε
∞
n α−1−ε
1/q
n0
∞
n1 n
1
α
1ε
∞ 1
− x α−ε ,
ε
0
which implies that inequality 2.26 holds.
<ε
1
α1ε
∞
0
1
x α
1ε
dx
2.28
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 αλ , with limu → 0 ln u/u − 1u1/2r 0 r > 1, |ln u/u − 1u1/2r | ≤
L1 u ∈ 0, 1, L1 > 0, one has
ε
I2 2
λ
ε
λ2
∞
−1−ε
yα
∞
1−α
ln u 1/r−ε/pλ−1
u
du dy
λ
1/yα u − 1
∞
1−α
⎡
⎤
1/yαλ
−1−ε ∞ ln u 1/r−ε/pλ−1
ln u 1/r−ε/pλ−1 ⎦
⎣
u
u
du −
du dy
yα
u
−
1
u
−1
0
0
⎡
⎤
λ
B2 1/r − ε/pλ, 1/s ε/pλ
−1−ε 1/yα ln u 1/r−ε/pλ−1
ε ∞ ⎣
u
− 2
du⎦dy
yα
u−1
λ2
λ 1−α
0
⎡
⎤
λ
B2 1/r − ε/pλ, 1/s ε/pλ
−1 1/yα 1/2r−ε/pλ−1
εL1 ∞ − 2
u
du⎦dy
yα ⎣
≥
λ2
λ 1−α
0
2
∞
B 1/r − ε/pλ, 1/s ε/pλ
−λ 1/2r−ε/pλ−1
εL1
− 2
dy
yα λ
λ 1/2r − ε/pλ 1−α
2
B 1/r − ε/pλ, 1/s ε/pλ
εL1
2 .
3
λ
λ 1/2r − ε/pλ
2.29
Set ε → 0 , then the inequality 2.27 holds. Lemma 2.4 is proved.
3. Main Results
Firstly, the following notations are given.
1 Set p > 0, p /
1, r > 1, p, q and r, s are two pairs of conjugate exponents. Let
φx : x αp1−λ/r−1 ,
ϕx : x αq1−λ/s−1 ,
3.1
1−p
ψx : ϕx
x αpλ/s−1 ,
x ∈ 0, ∞.
2 Set p > 1, p, q is a pair of conjugate exponents. Let
p
φ :
⎧
⎨
∞
⎩
n0
a; a {an }∞
n0 , ap,φ :
1/p
φn|an |p
⎫
⎬
<∞ .
⎭
3.2
10
Journal of Inequalities and Applications
It is a real space of sequences, where
ap,φ ∞
φn|an |p
1/p
3.3
n0
is a norm of sequence a with the weight function φ. Similarly, it can define the real spaces
q
p
of sequences: ϕ , ψ and the norm of sequence b with the weight function ϕ: bq,ϕ as well.
For 0 < p < 1 or q < 0, the marks ap,φ and bq,ϕ as two formal norms are still used in
Theorem 3.3.
3 Set p > 1, p, q is a pair of conjugate exponents. Define a Hilbert-type linear
p
operator T , for all a ∈ φ , one has
T an : Cn :
∞
lnm α/n α
m0
p
m αλ − n αλ
am
n ∈ N0 .
3.4
q
4 For a ∈ φ , b ∈ ϕ , define the formal inner product of T a and b as
T a, b :
∞
∞
lnm α/n αam
n0
m0
m αλ − n αλ
bn ∞ ∞
lnm α/n αam bn
n0 m0
m αλ − n αλ
,
3.5
Then one will have some results in the following theorem.
Theorem 3.1. Suppose that p, q and r, s are two pairs of conjugate exponents and p > 1, r > 1,
p
1/2 ≤ α ≤ 1, 0 < λ ≤ 1, an ≥ 0. Then for ∀a ∈ φ , one has
1
T a C {Cn }∞
n0 ∈ ψ .
p
p
3.6
p
It means that T : φ → ψ .
2 T is a bounded linear operator and
T p,ψ :
sup
p
a∈φ a /
θ
T ap,ψ
ap,φ
kλ s,
3.7
where Cn , T are defined by 3.4, T ap,ψ Cp,ψ is defined as by 3.3, and kλ s is a constant
defined by 2.4.
Journal of Inequalities and Applications
11
Proof. If p > 1, by using Hölder’s inequality cf. 20 and the result 2.11, for n ∈ N0 , it is
obvious that Cn ≥ 0 and
p
Cn
∞
lnm α/n α m α1−λ/r/q
m0
≤
m αλ − n αλ
∞
lnm α/n α m αp−11−λ/r
m0
×
m αλ − n αλ
n α1−λ/s
am
m αλ − n αλ
n α1−λ/s/p
p
m α1−λ/r/q
p
am
∞
lnm α/n α n αq−11−λ/s
m0
n α1−λ/s/p
p−1
3.8
m α1−λ/r
∞
lnm α/n α m αp−11−λ/r
p
am
"p−1
!
ωλ n, rϕn
m αλ − n αλ
n α1−λ/s
∞
lnm α/n α m αp−11−λ/r p # p−1 $
p−1
≤ kλ s
am ϕ n .
λ
λ
n α1−λ/s
m0 m α − n α
m0
And if ψn ϕ1−p n, by 2.9 and 2.10, it follows that
p
T ap,ψ ∞
p
ψnCn
n0
≤
p−1
kλ s
p−1
kλ s
∞ ∞
lnm α/n α m αp−11−λ/r
m0 n0
∞
m αλ − n αλ
p
p
n α1−λ/s
p
am
3.9
p
ωλ m, sφmam ≤ kλ sap,φ < ∞.
m0
This means that C {Cn }∞
n0 ∈ ψ and T p,ψ ≤ kλ s.
If there exists a constant K < kλ s, such that T p,ψ ≤ K, then for 0 < ε < pλ/2r, by
the definition 3.5, and by using Hölder’s inequality and the result 2.26, one has
p
' '
%
&
' '
ε T a, b ≤ εT p,ψ ap,φ 'b'
q,ϕ
≤ KI1 < K
ε
α1ε
1
,
αε
3.10
p q
∞
m , bn are defined as in Lemma 2.4.
where a {
am }∞
m0 ∈ φ , b {bn }n0 ∈ ϕ and a
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 < 0 and 1 − α ≥ 0, and
by α > 0, λ > 0, in view of 2.27, one has
%
&
ε T a, b ≥ ε
λ/s−ε/q−1 ln x α/ y α
x αλ/r−ε/p−1 y α
dx dy
λ
1−α
x αλ − y α
∞
I2 ≥ kλ s o1
ε −→ 0 .
3.11
12
Journal of Inequalities and Applications
In view of 3.10 and 3.11, one has kλ s o1 < Kε/α1ε 1/αε . Setting ε → 0 , one has
kλ s ≤ K. This means that K kλ s, that is, T p,ψ kλ s. Theorem 3.1 is proved.
Theorem 3.2. Suppose that p, q and r, s are two pairs of conjugate exponents, r > 1, p > 1,
1/2 ≤ α ≤ 1, 0 < λ ≤ 1, an , bn ≥ 0 n ∈ N0 . Then one has the following.
p
q
1 If a ∈ φ , b ∈ ϕ , and ap,φ > 0, bq,ϕ > 0, then
3.12
T a, b < kλ sap,φ bq,ϕ .
p
2 If a ∈ φ and ap,φ > 0, then
3.13
T ap,ψ < kλ sap,φ ,
where the mark T ap,ψ is defind as in Theorem 3.1. The inequality 3.13 is equivalent to 3.12, and
the constant factor kλ s 1/λB1/s, 1/r2 kλ r is the best possible.
Proof. In view of 3.9 and 0 < ap,φ < ∞, one has
p
p
p
3.14
T ap,ψ < kλ sap,φ .
And by p > 1, 3.13 holds.
By using Hölder’s inequality and 3.13, one has
T a, b ∞ ∞
lnm α/n α m α1−λ/r/q
n0 m0
m αλ − n αλ
n α1−λ/s/p
am
n α1−λ/s/p
m α1−λ/r/q
bn
3.15
≤ T ap,ψ bq,ϕ < kλ sap,φ bq,ϕ .
The inequality 3.12 is obtained.
p
From 3.12 and ap,φ > 0, there exists k0 ∈ N, such that K
m0 φmam > 0 and
K
p−1
> 0 when K > k0 . By a
bn K ψn m0 lnm α/n αam /m αλ − n αλ combination as in 3.15 and by 1/2 ≤ α ≤ 1, 0 < λ ≤ 1, and with 2.10 and 2.11, then,
p
K
K
K
lnm α/n αam
q
ϕnbn K ψn
0<
λ
λ
n0
n0
m0 m α − n α
1/q
1/p K K
K
K
lnm α/n αam bn K
p
q
< kλ s
φnan
ϕnbn K
< ∞.
m αλ − n αλ
n0 m0
n0
n0
3.16
By p > 1 and q > 1, it follows that
0<
K
n0
q
p
ϕnbn K < kλ s
∞
p
φnan < ∞.
n0
3.17
Journal of Inequalities and Applications
13
q
q
∞
Letting K → ∞ in 3.17, this means 0 < ∞
n0 ϕnbn ∞ < ∞, that is, b {bn ∞}n0 ∈ ϕ
and bq,ϕ > 0. Therefore the inequality 3.16 keeps the form of the strict inequality when
q
p
K → ∞. So does 3.17. In view of ∞
n0 ϕnbn ∞ T ap,ψ , the inequality 3.13 holds,
and 3.12 is equivalent to 3.13. By T p,ψ kλ s, it is obvious that the constant factor
kλ s kλ 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, an , bn ≥ 0. Let
Ha, b ∞
∞ lnm α/n αam bn
n0 m0
m αλ − n αλ
3.18
.
Then the reverse inequalities can be established as follows.
1 If 0 < ap,φ < ∞ and 0 < bq,ϕ < ∞, then
Ha, b > kλ s
∞
p
1 − ηλ m φmam
1/p
3.19
bq,ϕ .
m0
2 If 0 < ap,φ < ∞, then
p
∞
∞
∞
lnm α/n αam
p
p
ψn
>
k
1 − ηλ m φmam .
s
λ
λ
λ
−
α
α
n
m
n0
m0
m0
3.20
3 If 0 < bq,ϕ < ∞, then
∞
m0
φ−1 m
1 − ηλ m
q
q−1 ∞
lnm α/n αbn
n0
λ
m α − n α
λ
q
q
< kλ sbq,ϕ ,
3.21
where the marks ap,φ and bq,ϕ 0 < p < 1 as two formal norms are still defined like in 3.3
and the factor ηλ m in 3.19–3.21 is defined in Lemma 2.3. The inequalities 3.20 and 3.21 are
p
q
equivalent to 3.19. The constant factors kλ s, kλ s and kλ s in 3.19, 3.20, and 3.21 are all
the best possible.
Proof. By 0 < p < 1 q < 0, with the reverse Hölder’s inequality, one has the following.
1 By the combination as in 3.15 for 3.18, then one has
Ha, b ≥
∞
p
ωλ m, sφmam
m0
1/p ∞
1/q
q
ωλ n, rϕnbn
.
3.22
n0
If 1/2 ≤ α ≤ 1, 0 < λ ≤ 1, the expressions 2.19 and 2.11 are established for ωλ m, s and
ωλ n, r, respectively. And by q < 0, 3.19 holds.
14
Journal of Inequalities and Applications
then for
≥ kλ r, 3.19 is still valid if we replace kλ r by K,
Setting a constant K
0 < ε < −qλ/r, by 3.19 and 2.21, one will have
1/p
∞
&
p
am
H a, b > K
1 − ηλ m φm
bq,ϕ
%
m0
∞
1 − ηλ m
K
1ε
m0 m α
K
∞
1
n0 n
1/p ∞
n0 n
(
⎧
⎨
1−
∞
m0 O
α1ε ⎩
1/q
1
3.23
α1ε
%
1/m α1ελ/2s
(
)
∞
1ε
m0 1/m α
&) ⎫1/p
⎬
⎭
,
∞ m m αλ/r−ε/p−1 , bn n αλ/s−ε/q−1 and it is apparent
where a {
am }∞
m0 , b {bn }n0 , a
that 0 < ap,φ < ∞, 0 < b
q,ϕ < ∞.
On the other hand, by 3.18, 2.2, 2.12, and 2.10,
∞ ∞
%
& lnn α/m αm αλ/r−ε/p−1 n αλ/s−ε/q−1
H a, b n αλ − m αλ
m0 n0
∞
m α
−1−ε
m0
∞
m α
m0
−1−ε
∞
1
ln n α/m αλ n α/m αλ/s−ε/q−1
λm α n0
n α/m αλ − 1
∞
1
ε
1
hm n, −
λm α n0
s qλ
∞
B2 1/s − ε/qλ, 1/r ε/qλ 1
<
.
1ε
λ2
n0 n α
3.24
In view of 3.23 and 3.24, one has
K
(
⎧
⎨
⎩
1−
∞
m0 O
%
1/m α1ελ/2s
(
)
∞
1ε
1/m
α
m0
&) ⎫1/p
⎬
⎭
<
B2 1/s − ε/qλ, 1/r ε/qλ
.
λ2
3.25
≤ kλ s, which means K
kλ s. The constant factor kλ s in 3.19
Setting ε → 0 , one has K
is the best possible.
Journal of Inequalities and Applications
15
∞
λ
2 By 0 < ap,φ < ∞, one has ∞
n0 ψn m0 lnm α/n αam /m α −
p
λ
λ p−1
,
n αλ > 0. Setting bn : ψn ∞
m0 lnm α/n αam /m α − n α q
making the calculations as in 3.8 and 3.9, one has 0 < bq,ϕ < ∞. By using 3.19 in the
following:
q
bq,ϕ
∞
q
ϕnbn
n0
∞
ψn
n0
m0
Ha, b > kλ s
∞
lnm α/n αam
p
m αλ − n αλ
∞
p
1 − ηλ m φmam
3.26
1/p
bq,ϕ ,
m0
one has
p
∞
∞
∞
lnm α/n αam
p
p
ψn
> kλ s
1 − ηλ m φmam .
λ
λ
n0
m0 m α − n α
m0
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
Hölder’s inequality, it has
H a, b ∞
n α
n0
≥
λ/s−1/p
∞
)
lnm α/n αam (
1/p−λ/s
b
α
n
n
λ
λ
m0 m α − n α
1/q
p 1/p ∞
∞
∞
lnm α/n αam
q1−λ/s−1 q
ψn
bn
n α
λ
λ
n0
m0 m α − n α
n0
∞
p
> kλ s
1 − ηλ m φmam
3.28
1/p
bq,ϕ .
m0
Then 3.19 holds. It means that 3.20 is equivalent to 3.19.
3 By 0 < bq,ϕ < ∞, it is obvious that there exist n0 ∈ N, such that
K
m0
φ−1 m
1 − ηλ m
q−1
⎡
⎤q
K
lnm
α/n
αb
⎢
n⎥
& ⎦ > 0,
⎣ %
λ
λ
n0 m α − n α
K
q
ϕnbn
> 0 when K > n0 .
n0
3.29
16
Journal of Inequalities and Applications
λ
λ q−1
>
Setting am K φ−1 m/1− ηλ m K
n0 lnm α/n αbn /m α −n α K
p
0, one has 0 < m0 φmam K < ∞. By 3.19,
K
p
1 − ηλ m φmam K
m0
K
m0
φ−1 m
1 − ηλ m
q−1 K
lnm α/n αbn
n0
m αλ − n αλ
3.30
K K
lnm α/n αbn am K
m0 n0
m αλ − n αλ
> kλ s
q
K
p
1 − ηλ m φmam K
1/p K
m0
1/q
q
ϕnbn
.
n0
Further one has
K
p
1 − ηλ m φmam K
1/q
m0
K
q
> kλ s
ϕnbn
1/q
> 0.
3.31
n0
By q < 0, one has
0<
K
K
∞
p
q
q
q
q
1 − ηλ m φmam K < kλ s ϕnbn < kλ s ϕnbn < ∞.
m0
n0
3.32
n0
Setting K → ∞ in 3.32, via 2.20, one has
0<
∞
p
φmam ∞ <
m0
∞
1
p
1 − ηλ m φmam ∞ < ∞.
1 − θλ r m0
3.33
It means that 0 < ap,φ < ∞a : {am ∞}∞
m0 . The conditions for 3.19 are satisfied.
Equation 3.30 keeps a strict form when K → ∞. So does 3.31. By q < 0, the inequality
3.21 holds.
Journal of Inequalities and Applications
17
Also, from 3.21, by q < 0 and by using the reverse Hölder’s inequality,
⎫
⎧
∞
∞
⎨ φ−1 m 1/p 1 − ηλ m 1/p
lnm α/n αbn ⎬
a
Ha, b m
λ
λ ⎭
⎩ 1 − ηλ m
φ−1 m
m0
n0 m α − n α
∞
p
≥
1− ηλ m φmam
m0
1/p⎧ q ⎫1/q
q−1 ∞
∞
⎨
φ−1 m
lnm α/n αbn ⎬
λ
λ
⎩m0 1− ηλ m
⎭
n0 m α −n α
∞
p
> kλ s
1 − ηλ m φmam
m0
1/p
bq,ϕ .
3.34
Therefore, 3.19 holds. Equation 3.21 is equivalent to 3.19.
p
q
If the constant factor kλ s or kλ s in 3.20 or in 3.21 is not the best possible,
by 3.28 or by 3.34, then it leads to a contradiction in which the constant factor kλ s in
3.19 is not the best possible. Theorem 3.3 is proved.
Remark 3.4. Set r q, s p, λ 1, the inequalities 3.12 and 3.13 can be reduced to 1.7
and 1.8, respectively. So 3.12 or 3.13 is an extension of 1.7 or 1.8.
Acknowledgments
The work is supported by the National Natural Science Foundation of China no.10871073.
The author would like to thank the anonymous referee for his or her suggestions and
corrections.
References
1 G. Hardy, “Not on a theorem of Hilbert concering series of positive terms,” Proceedings of the London
Mathematical Society, vol. 23, no. 2, pp. 415–416, 1925.
2 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK,
2nd edition, 1952.
3 D. S. Mitrinović, J. Pečarić, and A. 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. Brnetić, M. Krnić, and J. Pečarić, “Generalization of Hilbert and Hardy-Hilbert integral
inequalities,” Mathematical Inequalities & Applications, vol. 8, no. 2, pp. 259–272, 2005.
7 W. Zhong and B. Yang, “A best extension of Hilbert inequality involving seveial parameters,” Journal
of Jinan University (Natural Science), vol. 28, no. 1, pp. 20–23, 2007 Chinese.
8 H. Leping, G. Xuemei, and G. Mingzhe, “On a new weighted Hilbert inequality,” Journal of Inequalities
and Applications, vol. 2008, Article ID 637397, 10 pages, 2008.
9 C.-J. Zhao and L. Debnath, “Some new inverse type Hilbert integral inequalities,” Journal of
Mathematical Analysis and Applications, vol. 262, no. 1, pp. 411–418, 2001.
10 W. Zhong and B. Yang, “On the extended form on the reverse Hardy-Hilbert’s integral inequalities,”
Journal of Southwest China Normal University (Natural Science), vol. 29, no. 4, pp. 44–48, 2007 Chinese.
18
Journal of Inequalities and Applications
11 W. Zhong, “A reverse Hilbert’s type integral inequality,” International Journal of Pure and Applied
Mathematics, vol. 36, no. 3, pp. 353–360, 2007.
12 I. Brnetić and J. Pečarić, “Generalization of Hilbert’s integral inequality,” Mathematical Inequalities &
Applications, vol. 7, no. 2, pp. 199–205, 2004.
13 B. Yang and T. M. Rassias, “On the way of weight coefficient and research for the Hilbert-type
inequalities,” Mathematical Inequalities & Applications, vol. 6, no. 4, pp. 625–658, 2003.
14 H. Yong, “On multiple Hardy-Hilbert integral inequalities with some parameters,” Journal of
Inequalities and Applications, vol. 2006, Article ID 94960, 11 pages, 2006.
15 W. Zhong and B. Yang, “On a multiple Hilbert-type integral inequality with the symmetric kernel,”
Journal of Inequalities and Applications, vol. 2007, Article ID 27962, 17 pages, 2007.
16 Y. Li, Z. Wang, and B. He, “Hilbert’s type linear operator and some extensions of Hilbert’s inequality,”
Journal of Inequalities and Applications, vol. 2007, Article ID 82138, 10 pages, 2007.
17 W. Zhong, “The Hilbert-type integral inequalities with a homogeneous kernel of −λ-degree,” Journal
of Inequalities and Applications, vol. 2008, Article ID 917392, 12 pages, 2008.
18 B. C. Yang, “A more accurate Hardy-Hilbert-type inequality and its applications,” Acta Mathematica
Sinica, vol. 49, no. 2, pp. 363–368, 2006 Chinese.
19 B. C. Yang and Y. H. Zhu, “Inequalities for the Hurwitz zeta-function on the real axis,” Acta
Scientiarum Naturalium Universitatis Sunyatseni, vol. 36, no. 3, pp. 30–35, 1997.
20 J. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan, China, 2004.