Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 820176, 8 pages
doi:10.1155/2009/820176
Research Article
A Hilbert’s Inequality with a Best Constant Factor
Zheng Zeng1 and Zi-tian Xie2
1
2
Department of Mathematics, Shaoguan University, Shaoguan, Guangdong 512005, China
Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China
Correspondence should be addressed to Zi-tian Xie, gdzqxzt@163.com
Received 6 February 2009; Revised 3 May 2009; Accepted 23 July 2009
Recommended by Yong Zhou
We give a new Hilbert’s inequality with a best constant factor and some parameters.
Copyright q 2009 Z. Zeng and Z.-t. Xie. 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
∞ q
p
If p > 1, 1/p 1/q 1, an , bn > 0 such that ∞ > ∞
n1 an > 0 and ∞ >
n1 bn > 0, then the
well-known Hardy-Hilbert’s inequality and its equivalent form are given by
1/p 1/q
∞
∞ ∞
∞
π
am bn
p
q
<
an
bn
,
m n sin π/p n1
n1 m1
n1
1.1
p p ∞
∞
∞
am
π
p
<
an ,
mn
sin π/p
n1 m1
n1
1.2
where the constant factors are all the best possible 1. It attracted some attention in the
recent years. Actually, inequalities 1.1 and 1.2 have many generalizations and variants.
Equation 1.1 has been strengthened by Yang and others including integral inequalities 2–11.
2
Journal of Inequalities and Applications
In 2006, Yang gave an extension of 2 as follows.
If p > 1, 1/p 1/q 1, r > 1, 1/r 1/s 1, t ∈ 0, 1, 2 − min{r, s}t min{r, s} ≥ λ >
p1−t2t−λ/r−1 p
q1−t2t−λ/s−1 q
an > 0, ∞ > ∞
bn > 0,
2 − min{r, s}t, such that ∞ > ∞
n1 n
n1 n
then
∞
∞ am bn
n1 m1 m
nλ
r − 2t λ s − 2t λ
,
<B
r
s
∞
1/p p
np1−t2t−λ/r−1 an
n1
∞
q
nq1−t2t−λ/s−1 bn
n1
1/q
.
1.3
Bu, v is the Beta function.
In 2007 Xie gave a new Hilbert-type Inequality 3 as follows.
If p > 1, 1/p 1/q 1, a, b, c > 0, 2/3 ≥ μ > 0, and the right of the following inequalities
converges to some positive numbers, then
∞
∞ m1 n1
<
nμ
a2 mμ nμ
am bn
b2 mμ nμ a2 mμ π
μa bb cc a
1/p 1/q
∞
∞
1−3μ/2p−1 p
1−3μ/2q−1 q
n
an
n
bn
.
n1
1.4
n1
The main objective of this paper is to build a new Hilbert’s inequality with a best constant
factor and some parameters.
In the following, we always suppose that
1 1/p 1/q 1, p > 1, a ≥ 0, −1 < α < 1,
2 both functions ux and vx are differentiable and strict increasing in n0 − 1, ∞
and m0 − 1, ∞, respectively,
3 u x/uα x, v x/vα x are strictly increasing in n0 − 1, ∞ and m0 − 1, ∞,
2
α
uαn vm
} is strict decreasing on n and m,
respectively. {u n v m /u2n 2aun vm vm
4 un un , un0 u0 , un0 − 1 vm0 − 1 0, u∞ ∞, v∞ ∞, u n .
u n , vm vm , vm0 v0 , v m vm
Journal of Inequalities and Applications
3
2. Some Lemmas
Lemma 2.1. Define the weight coefficients as follows:
∞
W p, m :
αp−1
1
v
· mα
2
2
un
nn0 un 2aun vm vm
ω p, m :
∞
p−1
vm
,
2.1
αp−1
αq−1
1
u
· nα
2
2
vm
u
2au
v
v
n m
m
mm0 n
ω
q, n :
un
1
u x
vm
·
·
dx,
2
2
p−1
uα x vm
no −1 u x 2auxvm vm
∞
q, n :
W
·
·
vm
un q−1
,
αq−1
v y
1
un
dy,
· α ·
2
2
v y un q−1
m0 −1 un 2aun v y v y
∞
2.2
2.3
2.4
then
pα−2α−1
Kvm
W p, m < ω p, m ,
p−1
vm
Kuqα−2α−1
n
W q, n < ω
q, n ,
q−1
un 2.5
where
K
∞
0
dσ
1 2aσ σ 2 σ α
⎧
√
⎪
1
π
α
⎪
⎪
a a2 − 1 −
√
⎪
√
α ,
⎪
⎪
2 a2 − 1 sin απ
⎪
a a2 − 1
⎪
⎪
⎪
⎪
⎪
⎪|απ|/ sin|απ|,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨πcscθcscαπ sinαθ,
⎪
√
⎪
1
⎪
⎪
ln a a2 − 1 ,
√
⎪
⎪
⎪
⎪
a2 − 1
⎪
⎪
⎪
⎪
⎪
⎪
θcscθ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1,
if α / 0, a > 1,
if α / 0, a 1,
if α /
0, a cos θ, 0 < θ < π,
if α 0, a > 1,
if α 0, a cos θ, 0 < θ <
π
,
2
if α 0, a 1,
2.6
4
Journal of Inequalities and Applications
z1 z − z2 zα then
Proof. Let fz 1/1 2az z2 zα 1/z − √
√ K 2πi/1 −
−2απi
Resf, z1 Resf, z2 if a > 1 then z1 −a − a2 − 1, z2 −a a2 − 1
e
⎡
−α −α ⎤
√
√
2
−a a2 − 1
2πi ⎢ −a − a − 1
⎥
K
√
√
⎣
⎦
1 − e−2απi
−2 a2 − 1
2 a2 − 1
π
√
2
2 a − 1 sin απ
2.7
⎤
⎡
α
⎢
⎣ a a2 − 1 − a
√
1
a2
⎥
α ⎦,
−1
if a cos θ 0 < θ < π/2, then z1 −eiθ , z2 −e−iθ
1
2πi
1
πcscθcscαπ sinαθ.
K
1 − e−2απi −2i sin θ−eiθ α 2i sin θ−e−iθ α
2.8
pα−2α−1
On the other hand, Wp, m < ωp, m. Setting ux vm σ, then ωp, m Kvm
qα−2α−1
p−1
n < ωq,
vm
. Similarly, Wq,
n Kun
/un q−1 .
/
Lemma 2.2. For 0 < ε < min{p, p1 − α} one has
∞
0
dσ
K o1
1 2aσ σ 2 σ αε/p
ε −→ 0 .
2.9
Proof.
∞
1
dσ
−
K
αε/p
2
1 2aσ σ σ
0
1 σ −α 1 − σ −ε/p ∞ σ −α 1 − σ −ε/p ≤
dσ dσ 0 1 2aσ σ 2
1 1 2aσ σ 2
2.10
1
∞
−α
−ε/p
≤ σ
1−σ
dσ σ −2−α 1 − σ −ε/p dσ 0
1
1
1
1
1
−→ 0
−
−
1 − α 1 − α − ε/p
1 α 1 α ε/p The lemma is proved.
for ε −→ 0 .
Journal of Inequalities and Applications
5
Lemma 2.3. Setting wn un (or vm and w0 n0 (or m0 , resp.), then k > 0. {τw /τwk } is strictly
decreasing, then
N
τw
k
ww0 τw
N
τ x
dx A.
k
w0 τ x
2.11
There A ∈ 0, τw 0 /τwk 0 , for any N).
Proof. We have
N
N N
N
τw 0
τw 0
τw
τw
τ x
τ x
<
dx
<
dx.
k x
k
k
k
k
k x
τ
τ
τ
τ
τ
τ
w0
w0
ww0 w
w0
w0
ww0 1 w
2.12
Easily, A had up bounded when N → ∞.
3. Main Results
Theorem 3.1. If an > 0, bn > 0, 0 <
q
un q−1 bn < ∞, then
∞ ∞
∞
am bn
<K
2
2
u
2au
n vm vm
nn0 mm0 n
∞
pαp−2α−1 un
un
nn0
pα−2α−1
n1
vm
1/p pα−2α−1
∞
vm
mm0
vm
∞
∞
p
p−1
/vm
an < ∞, 0 <
p
a
p−1 m
am
2
2
u
2au
n vm vm
mm0 n
< Kp
pα−2α−1
∞
vm
p−1
mm0 vm un
/
1/q
qα−2α−1
∞
un
q
b
q−1 n
u
nn0
n
p
qα−2α−1
nn0
p
am .
,
3.1
3.2
K is defined by Lemma 2.1.
Proof. By Hölder’s inequality 12 and 2.5,
J:
∞ ∞
2
nn0 mm0 un
am bn
2
2aun vm vm
∞ ∞
α/q
α/p
1/q
un 1/p
vm
1
vm
un
·
·
a
·
·
bn
m
2
2
α/p
1/q
α/q
vm un 1/p
nn0 mm0 un 2aun vm vm un
vm
∞
≤
1/p p
Wp, mam
mm0
<K
1/q
nbnq
Wq,
nn0
pα−2α−1
∞
vm
mm0
∞
vm
p
a
p−1 m
1/p qα−2α−1
∞
un
q
b
q−1 n
nn0 un 1/q
,
3.3
6
Journal of Inequalities and Applications
pα−2αp−1 ∞
un mm0
setting bn un
qα−2α−1
∞
un
q
bn
q−1
nn0 un p−1
2
am /u2n 2aun vm vm
∞
nn0
J≤K
∞
am
2
2
u
2au
n vm vm
mm0 n
pα−2αp−1 un
un
> 0. By3.1 we have
1/p pα−2α−1
∞
vm
mm0 vm p
a
p−1 m
p
qα−2α−1
∞
un
q−1
nn0 un 3.4
1/q
q
bn
.
qα−2α−1
q
By 0 < ∞
/un q−1 bn < ∞ and 3.4 taking the form of strict inequality, we have
nn0 un
3.1. By Hölder’s inequality12, we have
J
∞
−α2α/q1/q −11/q
un
un nn0
≤
∞
pα−2αp−1 un
un
nn0
α−2α/q−1/q am
1−1/q
un
bn un 2
2
u
2au
v
v
n m
m
mm0 n
∞
∞
am
2
2
mm0 un 2aun vm vm
p 1/p qα−2α−1
∞
un
q−1
nn0 un 3.5
1/q
q
bn
.
qα−2α−1
q 1/q
as 0 < { ∞
/un q−1 bn } < ∞. By 3.2, 3.5 taking the form of strict inequality,
nn0 un
we have 3.1.
Theorem 3.2. If α 0, then both constant factors, K and K p of 3.1 and 3.2, are the best possible.
Proof. We only prove that K is the best possible. If the constant factor K in 3.1 is not the best
possible, then there exists a positive H with H < K, such that
J<H
∞
−1
vm
mm0 vm 1/p −ε/p vm , bn
For 0 < ε < min{p, q}, setting am vm
∞
−1
vm
mm0 vm p
a
p−1 m
1/p ∞
∞
u−1
n
−ε/q un ,
then
p
a
p−1 m
u−1
n
q
b
q−1 n
nn0 un 1/q
q
b
q−1 n
nn0 un un
1/q
∞
vm
1ε
mm0 vm
3.6
.
1/p ∞
un
1ε
nn0 un
1/q
.
3.7
Journal of Inequalities and Applications
7
On the other hand ux σvy and vy τ,
−ε/p
−ε/q ∞ ∞
un un vm vm
2
mm0 nn0 un
2
2aun vm vm
u−ε/p xu xdx
>
vy−ε/q v y dy
2 x 2auxv y v 2 y
u
m0
n0
∞ ∞
σ −ε/p dσ
vy−1−ε v y dy
2
m0
u0 /vy σ 2aσ 1
∞ ∞
σ −ε/p dσ
τ −1−ε dτ
2 2aσ 1
σ
v0
0
∞ u0 /τ
σ −ε/p dσ
3.8
−
τ −1−ε dτ
σ 2 2aσ 1
v0
0
∞ ∞
∞
∞ u0 /τ −1−ε
≥ K o1 τ
σ −ε/p dσ dτ
dτ −
τ −1
v0
v0
0
∞
1−ε/p −1ε/p
v0
u
K o1 τ −1−ε dτ − 0
1 − ε/p2
v0
∞
K o1 τ −1−ε dτ − O1.
v0
By 3.6, 3.7, 3.8, and Lemma 2.3, we have
O1
<H
K o1 − ∞
τ −1−ε dτ
v0
∞
1ε
mm0 vm /vm
∞
τ −1−ε dτ
v0
1/p ∞ 1ε
nn0 un /un
∞
τ −1−ε dτ
v0
1/q
1/p 1/q
O1
O1
O1
.
1 ∞
< H 1 ∞
K o1 − ∞
τ −1−ε dτ
τ −1−ε dτ
τ −1−ε dτ
v0
v0
v0
,
3.9
3.10
We have K ≤ H, ε → 0 . This contracts the fact that H < K.
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