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Volume 2009, Article ID 192197,13pages doi:10.1155/2009/192197

Research Article

On a Multiple Hilbert-Type Integral

Operator and Applications

Qiliang Huang and Bicheng Yang

Department of Mathematics, Guangdong Institute of Education, Guangzhou, Guangdong 510303, China

Correspondence should be addressed to Qiliang Huang,qlhuang@yeah.net

Received 19 August 2009; Accepted 9 December 2009

Academic Editor: Wing-Sum Cheung

Copyrightq2009 Q. Huang and B. Yang. 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.

By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral operator with the homogeneous kernel of−λ-degreeλ∈Rand its norm are considered. As for applications, two equivalent inequalities with the best constant factors, the reverses, and some particular norms are obtained.

1. Introduction

Ifp > 1,1/p1/q 1,f≥ 0 ∈ Lp0,, g 0 Lq0,,f p

0 fpxdx

1/p

>

0,gq > 0,then we have the following famous Hardy-Hilbert’s integral inequality and its equivalent formcf.1:

0

0

fxgy

xy dx dy < π

sinπ/pfpgq, 1.1

0

0 fx xydx

p

dy

1/p

< π

sinπ/pfp, 1.2

where the constant factorπ/sinπ/pis the best possible. Define the Hardy-Hilbert’s integral operator T : Lp0, Lp0, as follows: for f Lp0,, Tfy :

01/x yfxdxy ∈0,.Then in view of1.2, it follows thatTfp < π/sinπ/pfpand

T ≤π/sinπ/p.Since the constant factor in1.2is the best possible, we find thatcf.2

(2)

Inequalities1.1and1.2are important in analysis and its applicationscf.3. In 2002, reference4considered the property of Hardy-Hilbert’s integral operator and gave an improvement of 1.1 for p q 2. In 2004-2005, introducing another pair of conjugate exponentsr, sr >1,1/r1/s 1and an independent parameterλ >0,5,6gave two best extensions of1.1as follows:

0

0

fxgy

yλ dx dy <

π

λsinπ/rfp,φgq,ψ, 1.3

0

0

fxgy

xyλ dx dy < B

λ r,

λ s

fp,φgq,ψ, 1.4

where Bu, v is the Beta function φx xp1−λ/r−1, ψx xq1−λ/s−1,f

p,φ :

0 φxfpxdx

1/p

> 0,gq,ψ > 0. In 2009, 7, Theorem 9.1.1 gave the following multiple Hilbert-type integral inequality: suppose that nN\ {1}, pi > 1,

n

i11/pi

1, λ > 0,thenkλx1, . . . , xn ≥ 0 is a measurable function of−λ-degree in Rn and for any

r1, . . . , rnri>1satisfiesni11/ri 1 and

Rn−1

kλu1, . . . , un−1,1

n−1

j1

ujλ/rj−1du1· · ·dun−1>0. 1.5

Ifφix xpi1−λ/ri−1, fi≥0∈Lφpii0,∞,fpi,φi >0i1, . . . , n,then we have the following

inequality:

Rn

kλx1, . . . , xn n

i1

fixidx1· · ·dxn< kλ n

i1

fi

pi,φi, 1.6

where the constant factor is the best possible. For n 2, kλx, y 1/xλ , and

1/xyλin1.6, we obtain1.3and1.4. Inequality1.6is some extensions of the results in6,8–11. In 2006, reference12also considered a multiple Hilbert-type integral operator with the homogeneous kernel of−n1-degree and its inequality with the norm, which is the best extension of1.2.

In this paper, by using the way of weight functions and the technic of real analysis, a new multiple Hilbert-type integral operator with the norm is considered, which is an extension of the result in12. As for applications, an extended multiple Hilbert-type integral inequality and the equivalent form, the reverses, and some particular norms are obtained.

2. Some Lemmas

Lemma 2.1. IfnN\ {1}, piR\ {0,1}, λiRi1, . . . , n,ni11/pi1,then

A:

n

i1

xλi−11−pi

i

n

j1j /i

xλj−1

j

⎤ ⎦

1/pi

(3)

Proof. We find that

A:

n

i1

xiλi−11−pi1−λi

n

j1 xλj−1

j

⎤ ⎦

1/pi

n

i1

⎡ ⎣x1−λipi

i n

j1 xλj−1

j

⎤ ⎦

1/pi

n

i1 x1−λi

i

n

j1 xλj−1

j

⎞ ⎠

n i11/pi

,

2.2

and then2.1is valid.

Definition 2.2. IfnN,Rn

:{x1, . . . , xnxi >0 i 1, . . . , n}, λ∈R,andkλx1, . . . , xnis a

measurable function inRn such that for anyu >0 andx1, . . . , xnRn, kλux1, . . . , uxn

uλk

λx1, . . . , xn,then callkλx1, . . . , xnthe homogeneous function of−λ-degree inRn.

Lemma 2.3. As for the assumption of Lemma2.1, ifni1λi λ, kλx1, . . . , xn≥0is a homogeneous

function of−λ-degree inRn

,

Hi:

Rn−1

kλu1, . . . , ui−1,1, ui1, . . . , un n

j1j /i

uλj−1

j du1· · ·dui−1dui1· · ·dun 2.3

i 1, . . . , n, and Hn R,then each Hi Hn kλi 1, . . . , n,and for any

i1, . . . , n, it follows that

ωixi:xλii

Rn−1

kλx1, . . . , xn n

j1j /i

xλj−1

j dx1· · ·dxi−1dxi1· · ·dxnkλ. 2.4

Proof. Settinguj unvj j /i, nin the integralHi,we find that

Hi

Rn−1

v1, . . . , vi−1, un1, vi1, . . . , vn−1,1

n−1

j1j /i

vλj−1

j u

1−λi

n dv1· · ·dvi−1dvi1· · ·dvn−1dun.

2.5

Settingvi un1in the above integral, we obtain Hi Hn.Settinguj xj/xi j /iin

2.4, we find thatωixi Hi Hn kλ.

Lemma 2.4. As for the assumption of Lemma2.3, setting

1, . . . ,λn−1

:

Rn−1

kλu1, . . . , un−1,1

n−1

j1 uλj−1

(4)

then there existδ0 > 0 andI {λ1, . . . ,λn−1 | λi λiδi,i| ≤ δ0 i 1, . . . , n−1},such that for anyλ1, . . . ,λn−1 ∈ I, kλ1, . . . ,λn−1 ∈R,if and only ifkλ1, . . . ,λn−1is continuous at

λ1, . . . , λn−1.

Proof. The sufficiency property is obvious. We prove the necessary property of the condition by mathematical induction. Forn2, since

1δ1

1

0

kλu1,1u1λ1δ1−1du1

1

kλu1,1u1λ1δ1−1du1,

kλu1,1u1λ1δ1−1≤kλu1,111−δ0−1du1, u1∈0,1,

kλu1,1u1λ1δ1−1≤kλu1,111δ0−1du1, u1∈1,,

2.7

and1−δ0 1δ0<∞,then by Lebesgue control convergence theoremcf.13, it

follows that1 δ1 1 o1δ1 → 0.Assuming that forn≥2, kλ1, . . . ,λn−1is

continuous atλ1, . . . , λn−1,then forn1,in view of the result forn2,we have that

lim

δn→0

1δ1, . . . , λnδn

lim

δn→0

0

Rn−1

kλu1, . . . , un,1 n−1

j1 uλjδj−1

j du1· · ·dun−1

⎞ ⎠uλnδn−1

n dun

0

Rn−1

kλu1, . . . , un,1 n−1

j1 uλjδj−1

j du1· · ·dun−1

⎞ ⎠uλn−1

n dun

Rn−1

0

kλu1, . . . , un,1uλnn−1dun

n−1

j1

jjδj−1du1· · ·dun−1,

2.8

then by the assumption forn,it follows that

lim

δn→0

1δ1, . . . , λnδn 1, . . . , λn o1 δi−→0, i1, . . . , n−1. 2.9

By mathematical induction, we prove that fornN\ {1}, kλ1, . . . ,λn−1is continuous at

λ1, . . . , λn−1.

Lemma 2.5. As for the assumption of Lemma2.4, if0< ε <min1≤in{|pi|}δ0, then forε → 0,

:ε

1 · · ·

1

kλx1, . . . , xn n

j1

xλjε/pj−1

(5)

Proof. Settinguj xj/xn j 1, . . . , n−1,we find that

Iεε

1 xn−1−ε

⎡ ⎣∞

x−1 n

· · ·

x−1 n

kλu1, . . . , un−1,1

n−1

j1

uλjε/pj−1

j du1· · ·dun−1

dxn. 2.11

SettingDj:{u1, . . . , un−1|uj ∈0, xn1, uk∈0,k /j}and

Ajxn:

· · ·

Dj

kλu1, . . . , un−1,1

n−1

j1

jjε/pj−1du1· · ·dun−1, 2.12

then by2.11, it follows that

Rn−1

kλu1, . . . , un−1,1

n−1

j1

uλjε/pj−1

j du1· · ·dun−1−ε n−1

j1

1

xn−1Ajxndxn. 2.13

Without loses of generality, we estimate that∞1 x−1

n An−1xndxn O1.In fact, settingα >0

such that|ε/pn−1 α|< δ0,since−uαn1lnun−1 → 0un−1 → 0,there existsM >0,such

that−uαn1lnun−1≤Mun−1∈0,1,and then by Fubini theorem, it follows that

0≤

1

xn1An−1xndxn

1 xn1

Rn−2

x−1 n

0

kλu1, . . . , un−1,1

n−1

j1

uλjε/pj−1

j dun−1du1· · ·dun−2

⎤ ⎦dxn

1

0

Rn−2

kλu1, . . . , un−1,1

n−1

j1

uλjε/pj−1

j

u−1 n−1

1

xn1dxn

du1· · ·dun−1

1

0

Rn−2

kλu1, . . . , un−1,1

n−1

j1

jjε/pj−1−lnun−1du1· · ·dun−1

M

1

0

Rn−2

kλu1, . . . , un−1,1

n−2

j1

uλjε/pj−1

j u

λn−1−ε/pn−1α−1

n−1 du1· · ·dun−1

M

Rn−1

kλu1, . . . , un−1,1

n−2

j1

uλjε/pj−1

j u

λn−1−ε/pn−1α−1

n−1 du1· · ·dun−1

M·k

λ1− ε p1

, . . . , λn−2− ε pn−2

, λn−1−

ε pn−1

α

<∞.

(6)

Hence by2.13, we have that

Rn−1

kλu1, . . . , un−1,1

n−1

j1

uλjε/pj−1

j du1· · ·dun−1−o11. 2.15

By Lemma2.4, we find that

ε

1 x−1−ε

n

⎡ ⎣∞

0 · · ·

0

kλu1, . . . , un−1,1

×n−1

j1

uλjε/pj−1

j du1· · ·dun−1

⎤ ⎦dxn

0 · · ·

0

kλu1, . . . , un−1,1

n−1

j1

uλjε/pj−1

j du1· · ·dun−1

k

λ1− ε p1

, . . . , λn−1− ε pn−1

kλo21,

2.16

Then by combination with2.15, we have2.10.

Lemma 2.6. Suppose that nN\ {1}, p1 ∈ R\ {1},

n

i11/pi 1,1/qn 1−1/pn,λ1, . . . , λnRn,ni1λi λ,thenkλx1, . . . , xn≥0is a measurable function of−λ-degree inRnsuch that

Rn−1

kλu1, . . . , un−1,1

n−1

j1 uλj−1

j du1· · ·dun−1∈R. 2.17

Iffi≥0are measurable functions inRi1, . . . , n−1,then (1) forpi >1i1, . . . , n,

J:

0 xqnλn−1

n

Rn−1

kλx1, . . . , xn n−1

i1

fixidx1· · ·dxn−1

qn dxn

1/qn

n−1

i1

0

xpi1−λi−1fpixdx

1/pi ,

2.18

(7)

Proof. 1Forpi>1 i1, . . . , n,by H ¨older’s inequalitycf.14 and2.4, it follows that

Rn−1

kλx1, . . . , xn n−1

i1

fixidx1· · ·dxn−1

qn ⎧ ⎪ ⎨ ⎪ ⎩

Rn−1

kλx1, . . . , xn n−1

i1

xiλi−11−pi

n

j1j /i

xλj−1

j

⎤ ⎦

1/pi fixi

×

xnλn−11−pn n−1

j1 xλj−1

j

⎤ ⎦

1/pn

dx1· · ·dxn−1

⎫ ⎪ ⎬ ⎪ ⎭ qn

Rn−1

kλx1, . . . , xn n−1

i1

xiλi−11−pi

n

j1j /i

jj−1

⎤ ⎦

qn/pi

×fqn

i xidx1· · ·dxn−1

×

⎧ ⎨ ⎩

Rn−1

kλx1, . . . , xnx

λn−11−pn

n

n−1

j1 xλj−1

j dx1· · ·dxn−1

⎫ ⎬ ⎭

qn−1

kλqn−1x1nqnλn

Rn−1

kλx1, . . . , xn

×n−1

i1

xiλi−11−pi

n

j1j /i

jj−1

⎤ ⎦

qn/pi fqn

i xidx1· · ·dxn−1,

2.19

J1/pn

⎧ ⎪ ⎨ ⎪ ⎩ 0

Rn−1

kλx1, . . . , xn

×n−1

i1

xiλi−11−pi

n

j1j /i

xλj−1

j

⎤ ⎦

qn/pi fqn

i xidx1· · ·dxn−1dxn

⎫ ⎪ ⎬ ⎪ ⎭

1/qn

1/pn

⎧ ⎪ ⎨ ⎪ ⎩

Rn−1

0

kλx1, . . . , xnxλnn−1dxn

×n−1

i1

xiλi−11−pi

n−1

j1j /i

xλj−1

j

⎤ ⎦

qn/pi fqn

i xidx1· · ·dxn−1

⎫ ⎪ ⎬ ⎪ ⎭

1/qn .

(8)

Forn≥3,by H ¨older’s inequality again, it follows that

J1/pn

⎧ ⎪ ⎨ ⎪ ⎩

n−1

i1

Rn−1

0

kλx1, . . . , xnxnλn−1dxn

×xiλi−11−pi

n−1

j1j /i

xλj−1

j f pi

i xidx1· · ·dxn−1

⎤ ⎦

qn/pi⎫⎪

⎪ ⎭

1/qn

1/pn n−1

i1

⎧ ⎨ ⎩

0

Rn−1

kλx1, . . . , xn

×xλi

i n

j1j /i

xλj−1

j dx1· · ·dxi−1dxi1· · ·dxn

xpi1−λi−1

i f pi

i xidxi

⎫ ⎬ ⎭

1/pi

1/pn n−1

i1

0

ωixixipi1−λi−1fipixidxi

1/pi .

2.21

Then by2.4, we have2.18 note that forn2,we do not use H ¨older’s inequality again.

2For 0< p1 <1, pi <0i2, . . . , n,by the reverse H ¨older’s inequality and the same way,

we obtain the reverses of2.18.

3. Main Results and Applications

As for the assumption of Lemma2.6, settingφix: xpi1−λi−1x ∈0,∞;i 1, . . . , n,then

we find thatφ1/1−pn

n x xqnλn−1.Ifpi >1i1, . . . , n,then define the following real function

spaces:

Lpi

φi0,∞:

f;fp i,φi

0

φix((fx((pidx

1/pi

<i1, . . . , n,

n−1

i1 Lpi

φi0,∞:

)

f1, . . . , fn−1

;fiLpφii0,, i1, . . . , n−1

*

,

3.1

and a multiple Hilbert-type integral operatorT : +ni11Lpi

φi0,∞ → L

qn

φ1/1npn

as follows: for

f f1, . . . , fn−1∈

+n1

i1L

pi

φi0,,

Tfxn:

Rn−1

kλx1, . . . , xn n−1

i1

(9)

Then by2.18, it follows thatTfLqn

φ1/1npn

,T is bounded,Tfq

n,φ1/1npn

+n−1

i1fipii,

andT ≤kλ,where

T: sup

f∈+n−1

i1Lpiφi0,fi/θ,i1,...,n−1

Tf

qn,φ1/n1−pn

+n1

i1fipi,φi

. 3.3

Define the formal inner product ofTf1, . . . , fn−1andfnas

Tf1, . . . , fn−1

, fn

:

Rn

kλx1, . . . , xn n

i1

fixidx1· · ·dxn. 3.4

Theorem 3.1. Suppose that nN\ {1}, p1 ∈ R \ {1},ni11/pi 1,1/qn 1−1/pn,

thenkλx1, . . . , xn≥0is a measurable function of−λ-degree inRn,and for anyλ1, . . . , λnRn,

it satisfiesni1λi λand

Rn−1

kλu1, . . . , un−1,1

n−1

j1 uλj−1

j du1· · ·dun−1>0. 3.5

Iffi≥0∈Lφpii0,,fpi,φi >0i1, . . . , n,then (1) forpi >1 i1, . . . , n,Tkλand the

following equivalent inequalities are obtained:

T

f1, . . . , fn−1q n,φ

1/1−pn n < kλ

n−1

i1

fi

pi,φi, 3.6

Tf1, . . . , fn−1

, fn

< kλ n

i1

fi

pi,φi, 3.7

where the constant factorkλis the best possible; (2) for0 < p1 <1, pi < 0 i2, . . . , n,using the

formal symbols of the case inpi>1 i1, . . . , n,the equivalent reverses of 3.6and3.7with the

best constant factor are given.

Proof. 1Forpi>1i1, . . . , n,if2.18takes the form of equality, then forn≥3 in2.21,

there exist constantsCiandCk i /ksuch that they are not all zero and

Cix

λi−11−pi

i

n−1

j1j /i

xλj−1

j f pi

i xi

Ckx

λk−11−pk

k

n−1

j1j /k

xλj−1

j f pk

k xka.e.in R n

,

3.8

viz. Cixpii1−λifipixi Ckxkpk1−λkfkpkxk C a.e. in Rn. Assuming that Ci > 0, then

xpi1−λi−1

i f pi

(10)

consider2.19forfpk

k xk 1 in the above. Hence we have3.6. By H ¨older’s inequality, it

follows that

Tf, fn

0

xλn−1/qn

n

Rn−1

kλx1, . . . , xn n−1

i1

fixidx1· · ·dxn−1

×x1/qnλn

n fnxn

dxnT

f1, . . . , fn−1qn1/1−pn n

fnpnn,

3.9

and then by3.6, we have3.7. Assuming that3.7is valid, setting

fnxn:xqnnλn−1

Rn−1

kλx1, . . . , xn n−1

i1

fixidx1· · ·dxn−1

qn−1

, 3.10

thenJ )∞0 xpn1−λn−1

n fnpnxndxn

*1/qn

.By2.18, it follows thatJ <∞.IfJ 0,then3.6is naturally valid. Assuming that 0< J <∞,by3.7, it follows that

0

xpn1−λn−1

n f

pn

n xndxnJqn

Tf, fn

< kλ n

i1

fi

pi,φi,

0

xpn1−λn−1

n fnpnxndxn

1/qn

J < kλ n−1

i1

fipii,

3.11

and then3.6is valid, which is equivalent to3.7.

For ε > 0 small enough, setting fixas follows: fix 0, x ∈ 0,1; fix

xλiε/pi−1, x1,i 1, . . . , n, if there existskk

λ, such that3.7is still valid as we

replacebyk,then in particular, by Lemma2.5, we have that

kλo1 Iεε

Tf1, . . . ,fn−1

,fn

< εk

n

i1

f

i pi,φi

k, 3.12

and → 0.Hencek is the best value of3.7. We conform that the constant

factorin3.6is the best possible; otherwise, we can get a contradiction by3.9that the

constant factor in3.7is not the best possible. ThereforeTkλ.

2For 0 < p1 < 1, pi <0 i 2, . . . , n,by using the reverse H ¨older’s inequality and

(11)

Example 3.2. Forλ >0, λi λ/ri i1, . . . , n,ni11/ri 1, kλx1, . . . , xn 1/ni1xi

λ

,

we obtain 1/Γλ

+n

iλ/ri cf.7,9.1.19. By Theorem3.1, it follows thatT

1/Γλ

+n

iλ/ri, and then by3.7, we find that

Rn

1

n

i1xi

λ n

i1

Fixidx1· · ·dxn

< 1

Γλ

n

i1

Γ

λ ri

0

xpi1−λ/ri−1

i Fipixidxi

1/pi .

3.13

SettingFixi xβ/ni fixiandλiλ/ri−β/ni1, . . . , nin3.13, we obtainni1λi

λβ, min1≤ini}>−β/nand

Rn

n

,+n

i1xi

β

n

i1xi

λ n

i1

fixidx1· · ·dxn

< 1

Γλ

n

i1

Γ

λi

β n

0

xpi1−λi−1

i fi pix

idxi

1/pi .

3.14

It is obvious that3.13and3.14are equivalent in which the constant factors are all the best possible. Hence forβx1, . . . , xn

n

,+n

i1xi

β

/ni1xi

λ

λ > 0,min1≤ini} > −β/n,

we can show thatTβ 1/Γλ+niλiβ/n.

Example 3.3. For λ > 0, λi λ/rii 1, . . . , n,

n

i11/ri 1, kλx1, . . . , xn

1/max1≤in{xi}λ,we obtain 1/λn−1+ni1ricf.7,9.1.24. By Theorem3.1, it follows

thatT 1/λn−1+ni1ri, and then by3.7, we find that

Rn

1

max1≤in{xi}λ n

i1

Fixidx1· · ·dxn

< 1 λn−1

n

i1 ri

0

xpi1−λ/ri−1

i Fi pix

idxi

1/pi .

3.15

SettingFixi xβ/ni fixiandλiλ/ri−β/ni1, . . . , nin3.15, we obtain

n

i1λi

λβ, min1≤ini}>−β/nand

Rn

n

,+n

i1xi

β

max1≤in{xi}λ n

i1

fixidx1· · ·dxn

< λ

n

i1

1

λiβ/n

0

xpi1−λi−1

i fi pix

idxi

1/pi .

(12)

Hence forβx1, . . . , xn

n

,+n

i1xi

β

/max1≤in{xi}λ λ > 0,min1≤ini} > −β/n, we

can show thatTβλ+ni11/λiβ/n.

Example 3.4. For λ > 0, λi −λ/ri,

n

i11/ri 1 i 1, . . . , n, kλx1, . . . , xn

min1≤in{xi}λ,by mathematical induction, we can show that

kλ

Rn−1

min{u1, . . . , un−1,1}λ

n−1

j1

ujλ/rj−1du1· · ·dun−1

+n i1ri

λn−1 . 3.17

In fact, forn2,we obtain

kλ

1

0 uλ/r2−1

1 du1

1

uλ/r1−1

1 du1

1

λr1r2. 3.18

Assuming that forn≥2 3.17is valid, then forn1, it follows that

kλ

Rn−1

n

j2 uλ/rj−1

j

-∞

0

min{u1, . . . , un,1}λu−1λ/r1−1du1

.

du2· · ·dun

Rn−1

n

j2 uλ/rj−1

j

min{u2,...,un,1}

0

1uλ/r1−1 1 du1

min{u2,...,un,1}

min{u2, . . . , un,1}λu1λ/r1−1du1

du2· · ·dun

r12

λr1−1

Rn−1

min{u2, . . . , un,1}λ1−1/r1 n

j2

uλ1−1/r1/1−1/r1rj−1

j du2· · ·dun

r12

λr1−1

1

λ1−1/r1n−1

n1

i2

1− 1

r1

ri 1

λn n1

i1 ri.

3.19

Then by mathematical induction,3.17is valid fornN\ {1}. By Theorem3.1, it follows thatTkλ 1/λn−1

+n

i1ri, and by3.7, we find that

Rn

min

1≤in{xi}

λ n

i1

Fixidx1· · ·dxn

< 1 λn−1

n

i1 ri

0

xpi1λ/ri−1

i Fipixidxi

1/pi .

(13)

SettingFixi xiβ/nfixiand λi −λ/riβ/ni 1, . . . , nin 3.20, we obtain

n

i1λiβλ,max1≤ini}< β/nand

Rn

min1≤in{xi}λ

n

,+n

i1xi

β n

i1

fixidx1· · ·dxn

< λ

n

i1

1

β/nλi

0

xpi1−λi−1

i fi pix

idxi

1/pi .

3.21

Hence forλx1, . . . , xn min1≤in{xi}λ/

n

,+n

i1xi

β

λ >0,max1≤ini}< β/n, we can

show thatTλλ

+n

i11/β/nλi.

Acknowledgments

This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and Universityno. 05Z026, and Guangdong Natural Science Foundationno. 7004344.

References

1 G. H. Hardy, J. E. Littlewood, and G. P ´olya,Inequalities, Cambridge University Press, Cambridge, UK, 1934.

2 T. Carleman, Sur les Equations Integrals Singulieres a Noyau Reed et Symetrique, vol. 923, Uppsala universitet, Uppsala, Sweden.

3 D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink,Inequalities Involving Functions and Their Integrals and Derivatives, vol. 53 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.

4 K. Zhang, “A bilinear inequality,”Journal of Mathematical Analysis and Applications, vol. 271, no. 1, pp. 288–296, 2002.

5 B. Yang, “On a extansion of Hilbert’s integral inequality with some parameters,”The Australian Journal of Mathematical Analysis and Applications, vol. 1, no. 1, article 11, pp. 1–8, 2004.

6 B. Yang, I. Brneti´c, M. Krni´c, and J. Peˇcari´c, “Generalization of Hilbert and Hardy-Hilbert integral inequalities,”Mathematical Inequalities & Applications, vol. 8, no. 2, pp. 259–272, 2005.

7 B. Yang,The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijin, China, 2009.

8 Y. Hong, “All-sided generalization about Hardy-Hilbert integral inequalities,” Acta Mathematica Sinica, vol. 44, no. 4, pp. 619–625, 2001.

9 L. He, J. Yu, and M. Gao, “An extension of the Hilbert’s integral inequality,”Journal of Shaoguan University ( Natural Science), vol. 23, no. 3, pp. 25–30, 2002.

10 B. Yang, “A multiple Hardy-Hilbert integral inequality,”Chinese Annals of Mathematics. Series A, vol. 24, no. 6, pp. 25–30, 2003.

11 B. Yang and Th. 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.

12 A. B´enyi and O. Choonghong, “Best constants for certain multilinear integral operators,”´ Journal of Inequalities and Applications, vol. 2006, Article ID 28582, 12 pages, 2006.

References

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