On a Multiple Hilbert's Inequality with Parameters

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Volume 2010, Article ID 309319,12pages doi:10.1155/2010/309319

Research Article

On a Multiple Hilbert’s Inequality with Parameters

Qiliang Huang

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

Correspondence should be addressed to Qiliang Huang,[email protected]

Received 12 May 2010; Accepted 31 August 2010

Academic Editor: Wing-Sum Cheung

Copyrightq2010 Qiliang Huang. 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 introducing multiparameters and conjugate exponents and using Hadamard’s inequality and the way of real analysis, we estimate the weight coeﬃcients and give a multiple more accurate Hilbert’s inequality, which is an extension of some published results. We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.

1. Introduction

In 1908, Weyl published the following famous Hilbert’s inequalitycf.1. Ifan, bn ≥0, 0<

_∞

n1a2n<∞and 0<

_∞

n1b2n<∞,then

∞

n1

∞

m1

ambn

mn < π _∞

m1

a2m

∞

n1

b2n

1/2

, 1.1

where the constant factorπ is the best possible. In 1934, Hardy proved the following more accurate Hilbert’s inequalitycf.2:

∞

n1

∞

m1

ambn

mn−1 < π _∞

m1

a2m

∞

n1

bn2

1/2

, 1.2

where the constant factorπis the best possible. For 0<∞n1a2n<∞,the equivalent forms of

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∞

n1

_∞

m1

am

mn 2

< π2

∞

m1

a2m, 1.3

∞

n1

_∞

m1

am

mn−1 2

< π2

∞

m1

a2m, 1.4

where the constant factorπ2 _{is the best possible. Inequalities} _1.1_–_1.4 _{are important in}

analysis and their applicationscf.3. In near one century, there are many improvements,

generalizations and, applications of1.1–1.4in numerous literatures and monographs of mathematicscf.2–18. Yang and Huang also considered the multiple Hilbert-type integral

inequalitycf.19,20. Recently, Yang summarized the methods of introducing parameters

and estimating the weight coeﬃcients to extend Hilbert-type inequalities for the past 100 years. Some representative results are as followscf.21,22:

iifp, r >1, 1/p1/q1/r1/s1, 0< α≤1, 0< λ≤min{r, s},then

∞

n1

∞

m1

ambn

mn−1λ < B

λ r,

λ s

×

_∞

m1

m−1

2

p1−λ/r−1

apm

1/p_∞

n1

n−1

2

q1−λ/s−1

bqn

1/q

,

1.5

∞

n1

n−1

2

pλ/s−1∞

m1

am

mn−1λ p

<

B

λ r,

λ s

p∞

m1

m−1

2

p1−λ/r−1

apm, 1.6

∞

n1

∞

m1

ambn

m−1/2α_n₋_1/2α <

π αsinπ/r×

_∞

m1

m−1

2

p1−α/r−1

apm

1/p

×

_∞

n1

n−1

2

q1−α/s−1

bqn

1/q

,

1.7

∞

n1

n−1

2

pα/s−1∞

m1

am

m−1/2α_n₋_1/2α

p

<

π αsinπ/r

p∞

m1

m−1

2

p1−α/r−1

apm,

1.8

iiifpi, ri>1,

_n

i11/pi

_n

i11/ri 1, 0< α≤1, 0< λα≤min1≤i≤n{ri}, then

∞

mn1

· · · ∞

m11

1 n

i1miα

λ n

i1

amii< α1−n

Γλ

n

i1

Γ

λ ri

∞

mi1

mipi1−λα/ri−1

amii

pi

1/pi

. 1.9

The constant factors in the above five inequalities are all the best possible. Inequalities 1.5 and 1.7 are generalizations of inequality 1.2, and inequality 1.9 is a multiple extension of 1.1. Inequalities1.6 and 1.8 are the equivalent forms of 1.5 and 1.7,

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In this paper, by introducing multi-parameters and conjugate exponents and using Hadamard’s inequality, we estimate the weight coeﬃcients and give a multiple more accurate Hilbert ’s inequality, which is an extension of inequalities1.5,1.7, and1.9. We also prove

that the constant factor in the new inequality is the best possible and consider its equivalent form.

2. Some Lemmas

Lemma 2.1. Ifn∈N\ {1}, pi, ri >1i1, . . . , n,ni11/pi ni11/ri 1,λ >0< α <2,

β≥ −1/2,λαmax{1/2−α,1} ≤min1≤i≤n{ri}, then

A:

n

i1

⎡

⎣_m_i_βλα/ri−11−pi

n

j1j /i

mjβ

λα/rj−1 ⎤ ⎦

1/pi

1. 2.1

Proof. We find the following:

A

n

i1

⎡

⎣_m_i_βλα/ri−11−pi1−λα/ri

n

j1

mjβ

λα/rj−1 ⎤ ⎦

1/pi

n

i1

⎡

⎣_m_i_βpi1−λα/ri

n

j1

mjβ

λα/rj−1 ⎤ ⎦

1/pi

n

i1

miβ

1−λα/ri ⎡ ⎣n

j1

mjβ

λα/rj−1 ⎤ ⎦

n i11/pi

1,

2.2

and then2.1is valid.

Lemma 2.2. Ifλ, y >0,r >1,1/r1/s 1,0 < α <2,β≥ −1/2,λαmax{1/2−α,1} ≤r, then

Γλ/rΓλ/s αΓλ

1−O

1 yλ/r

<

∞

m1

yλ/s_m_βλα/r−1

ymβαλ <

Γλ/rΓλ/s

αΓλ . 2.3

Proof. For fixed y >0, we set

fx: yλ/s

xβλα/r−1

yxβαλ , x∈

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In virtue ofαλα/r−2 ≤ 0 andλα/r−1 ≤ 0,we find −1ifi_x _> _0,_i _1,_2._Putting

u xβα/y,we have the following:

_∞

−β

fxdx 1 α

_∞

0

uλ/r−1

1uλdu

Γλ/rΓλ/s

αΓλ . 2.5

Since−β≤1/2,by the following Hadamard’s inequalitycf.5:

fm< _m₁_/₂

m−1/2

fxdxm∈N, 2.6

it follows that

∞

m1

ymβαλ

∞

m1

fm<

∞

m1

m1/2

m−1/2

fxdx

_∞

1/2

fxdx≤

_∞

−β

fxdx Γλ/rΓλ/s αΓλ ,

2.7

and then we have the right-hand side of2.3. Since

1

−β

fxdx

xβα_/y

0

uλ/r−1

α1uλdu

< 1 α

xβα_/y

0

uλ/r−1du r

1βλα/r λαyλ/r ,

2.8

andfxis strictly decreasing in−β,∞, we get

∞

m1

fm> _∞

1

fxdx _∞

−β

fxdx−

1

−β

fxdx

> Γλ/rΓλ/s αΓλ −

r1βλα/r λαyλ/r .

2.9

Hence, we prove that the left-hand side of2.3is valid.

Lemma 2.3. As the assumption of Lemma 2.1, define the weight coeﬃcients ωimi ωmi;r1,

. . . , rnas

ωimi:

miβ

λα/ri

∞

mn1

· · · ∞

mi11 ∞

mi−11

· · ·∞

m11

n j1j /i

mjβ

λα/rj−1 n

i1

miβ

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i1, . . . ,n, then there existsδn>0, such that

α1−n

Γλ

n

j1

Γ

λ rj

1−O

1

mnβ

δn

< ωnmn

mnβ

λα/rn ∞

mn−11

· · · ∞

m11

n−1

j1

mjβ

λα/rj−1 n

i1

miβ

αλ <

α1−n

Γλ

n

j1

Γ

λ rj

.

2.11

Moreover, for anyi∈ {1, . . . , n},it follows that

ωimi< α

1−n

Γλ

n

j1

Γ

λ rj

. 2.12

Proof. We prove 2.11 by mathematical induction. For n 2, we set r r1 and s r2

satisfying 1/r1/s1.Puttingmm1,y m2βα,δ2λα/r >0,we have the following:

ω2m2

∞

m11

m1β

λα/r1−1

m2β

λα/r2

m1β

α

m2β

αλ

∞

m1

ymβαλ , 2.13

and then2.11is valid by using inequality2.3.

Assuming that forn≥2,2.11is valid, then forn1,settingy n_i₂1miβα>

mn1βα,s1 1−1/r1 −1,by2.3, we have the following:

Γλ/r1Γλ/s1

αΓλ

1−O1

1 yλ/r1

<

∞

m11

yλ/s1_m

1β

λα/r1−1

ym1β

αλ <

Γλ/r1Γλ/s1

αΓλ . 2.14

Settingλλ/s1,rjrj1/s1,mjmj1 j1, . . . , n,we findnj11/rj 1,αλmax{1/2−

α,1} ≤min1≤i≤n{ri}.By the assumption of induction, it follows that

ωn1mn1

mnβ

λα/ rn _×

∞

mn−11

· · ·∞

m11

_n₋₁

j1

mjβ

λα/rj−1

n i1

miβ

αλ

×

⎧ ⎨ ⎩

∞

m11

yλ/s1_m₁_βλα/r1−1

ym1β

αλ

⎫ ⎬ ⎭

<mnβ

λα/ rn ∞

mn−11

· · · ∞

m11

n−1

j1

mjβ

λα/ rj−1

_n

i1

miβ

αλ ·

Γλ/r1Γλ/s1

αΓλ

< α

1−n

Γλ

n

i1

Γ

λ ri

·Γλ/r1Γ

λ αΓλ

α1−n1 Γλ

n1

i1

Γri

λ

,

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ωn1mn1>

mnβ

λα/ rn _× ∞

mn−11

· · ·∞

m11

n−1

j1

mjβ

λα/rj−1

n i1

miβ

αλ ·

Γλ/r1Γλ/s1

αΓλ

×

1−O1

1 yλ/r1

> Γλ/r1Γλ/s1 αΓλ

⎡ ⎢

⎣mnβ

λα/ rn

∞

mn−11

· · · ∞

m11

_n₋₁

j1

mjβ

λα/ rj−1

n i1

miβ

αλ −γ

⎤ ⎥ ⎦

> α

1−n1 Γλ

n1

i1

Γ

λ ri

×

⎡ ⎣1−O2

⎛

⎝ 1

mnβ

δn ⎞ ⎠ ⎤

⎦₋ Γλ/r1Γλ/s1

αΓλ γ,

2.16

whereδn>0 and

0< γ:mnβ

λα/ rn ∞

mn−11

· · · ∞

m11

n−1

j1

mjβ

λα/ rj−1

n i1

miβ

αλ O1

1

mn1β

αλ/r1

< α

1−n

Γλ/s1

n1

i2

Γri

λ

×O1

1

mn1β

αλ/r1

.

2.17

Settingδn1min{δn, αλ/r1}>0,by2.16, we have the following:

ωn1mn1> α 1−n1

Γλ

n1

i1

Γ

λ ri

×

1−O

1

mn1β

δn1

, 2.18

and then by2.15,2.18, and mathematical induction,2.11is valid. Settingmj mj,rj

rj j 1, . . . , i−1,mj mj1,rj rj1 j i, . . . , n−1,mn mi,rn ri,then we have the

following:

ωimi ωmn;r1, . . . ,rn< α

1−n

Γλ

n

j1

Γ

λ rj

α1−n

Γλ

n

j1

Γ

λ rj

. 2.19

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3. Main Results

Theorem 3.1. Suppose thatn ∈ N\ {1},pi,ri > 1i 1, . . . , n, ni11/pi ni11/ri

1,1/qn 1−1/pn,λ > 0,0 < α < 2,β ≥ −1/2,λαmax{1/2−α,1} ≤ min1≤i≤n{ri},amii ≥ 0mi∈N,such that

0<

∞

mi1

miβ

pi1−λα/ri−1 amii

pi

<∞ i1, . . . , n, 3.1

then one has the following equivalent inequalities:

I:

∞

mn1

· · · ∞

m11

1 n

i1

miβ

αλ n

i1

amii

< α

1−n

Γλ

n

i1

Γ

λ ri

∞

mi1

miβ

pi

1/pi ,

3.2

J: ⎧ ⎨ ⎩

∞

mn1

mnβ

λαqn/rn−1 ⎡ ⎣ ∞

mn−11

· · · ∞

m11

n−1

i1a

i

mi n

i1

miβ

αλ

⎤ ⎦

qn⎫_⎬

⎭

1/qn

< Γλ/rn αn−1Γλ

n−1

i1

Γ

λ ri

∞

mi1

miβ

pi

1/pi .

3.3

Proof. Since 1/pn1/qn1,by2.1and H ¨older’s inequalitycf.5, we find that

⎡ ⎣ ∞

mn−11

· · · ∞

m11

n−1

i1a

i

mi n

i1

miβ

αλ

⎤ ⎦

qn

⎧ ⎪ ⎨ ⎪ ⎩

∞

mn−11

· · · ∞

m11

1 n

i1

miβ

αλ

⎡

⎣_m_n_βλα/rn−11−pn

n−1

j1

mjβ

λα/rj−1 ⎤ ⎦

1/pn

×n−1

i1

⎡

n

j1j /i

mjβ

λα/rj−1 ⎤ ⎦

1/pi

amii ⎫ ⎪ ⎬ ⎪ ⎭

qn

≤)ωnmn

mnβ

pn1−λα/rn−1*qn/pn

∞

mn−11

· · · ∞

m11

1 n

i1

miβ

αλ

×n−1

i1

⎡

n

j1j /i

mjβ

λα/rj−1 ⎤ ⎦

qn/pi

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≤

n

i1Γλ/ri

αn−1_Γλ

qn/pn mnβ

1−λαqn/rn

∞

mn−11

· · · ∞

m11

1 n

i1

miβ

αλ

×n−1

i1

⎡

n

j1j /i

mjβ

λα/rj−1 ⎤ ⎦

qn/pi

amii qn

,

3.4

J ≤

n

i1Γλ/ri

αn−1Γλ

1/pn

× ⎧ ⎪ ⎨ ⎪ ⎩ ∞

mn1

∞

mn−11

· · · ∞

m11

1 n

i1

miβ

αλ × n−1

i1

⎡

⎣_m_i_βλα/ri−11−pi

n

j1j /i

mjβ

λα/rj−1 ⎤ ⎦

qn/pi

×amii qn ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

1/qn

n

i1Γλ/ri

αn−1_Γλ

1/pn ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∞

mn−11

· · · ∞

m11

⎡ ⎣∞

mn1

mnβ

λα/rn−1 _n

i1

miβ

αλ

⎤ ⎦

×n−1

i1

⎡ ⎢ ⎢ ⎣miβ

pi1−λα/ri−1 miβ

λα/ri

n−1

j1

j /i

mjβ

λα/rj−1 ⎤ ⎥ ⎥ ⎦

qn/pi

amii qn ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

1/qn

.

3.5

For n ≥ 3, since _in−₁1qn/pi 1, by H ¨older’s inequality again in 3.5, we have the

following:

J≤

n

i1Γλ/ri

αn−1Γλ

1/pnn−1

i1

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∞

mn−11

· · · ∞

m11 ∞

mn1

mnβ

λα/rn−1 n

i1

miβ

αλ

×

⎡ ⎢ ⎢ ⎣miβ

pi1−λα/ri−1 miβ

λα/ri

n−1

j1

j /i

mjβ

λα/rj−1 ⎤ ⎥ ⎥ ⎦

amii pi ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

1/pi

n

i1Γλ/ri

αn−1_Γλ

1/pnn−1

i1

_∞

mi1 ωimi

miβ

pi

1/pi .

3.6

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Since 1/qn1/pn1,by H ¨older’s inequality once again, it follows that

I

∞

mn1 ⎡

⎣_m_n_βλα/rn−1/qn

∞

mn−11

· · · ∞

m11

n−1

i1a

i

mi n

i1

miβ

αλ

⎤

⎦_×+_m_n_β1/qn−λα/rn amnn

,

≤J

_∞

mn1

mnβ

pn1−λα/rn−1 amnn

pn

1/pn .

3.7

By3.3, we have3.2. On the other hand, assuming that3.2is valid, setting

amnn:

mnβ

λαqn/rn−1 ⎡ ⎣ ∞

mn−11

· · · ∞

m11

_n₋₁

i1a

i mi n

i1

miβ

αλ

⎤ ⎦

qn−1

, 3.8

then we find that

J _∞

mn1

mnβ

pn

1/qn

I1/qn_. 3.9

By3.2, it follows thatJ <∞.IfJ 0,then3.3is naturally valid. Suppose thatJ >0,by 3.2, we find that

0<

∞

mn1

mnβ

pn

Jqn _I

< n

i1Γλ/ri

αn−1_Γλ

n

i1

_∞

mi1

miβ

pi

1/pi <∞.

3.10

Dividing outJqn/pn _{into two sides of}_3.10, we have the following:

_∞

mn1

mnβ

pn

1/qn J

< _n

i1Γλ/ri

αn−1_Γλ

n−1

i1

_∞

mi1

miβ

pi

1/pi .

3.11

Then3.3is valid, which is equivalent to3.2.

Theorem 3.2. Let the assumptions of Theorem 3.1 be fulfilled, then the same constant factor α1−n_/Γλn

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Proof. By2.11and

lim

N→ ∞

mnβ

λα/rn

N

mn−11

· · · N

m11

_n₋₁

j1

mjβ

λα/rj−1 n

i1

miβ

αλ ωnmn, 3.12

there existsN0∈N,such that forN > N0,

mnβ

λα/rn

N

mn−11

· · ·N

m11

_n₋₁

j1

mjβ

λα/rj−1 n

i1

miβ

αλ >

α1−n

Γλ

n

j1

Γ

λ rj

1−O

1

mnβ

δn

,

3.13

whereδn>0.Setting

amii :

⎧ ⎨ ⎩

miβ

λα/ri−1

, mi≤N,

0, mi> N ,

i1, . . . , n 3.14

we find that

I:

∞

mn1

· · · ∞

m11

1 n

i1

miβ

αλ n

i1

amii

N

mn1

mnβ

λα/rn

mnβ N

mn−11

· · · N

m11

n−1

i1

miβ

λα/ri−1 n

i1

miβ

αλ

>

N

mn1 1 mnβ ·

α1−n Γλ

n

j1

Γ

λ rj

1−O

1

mnβ

δn

α1−n Γλ

n

j1

Γ

λ rj

_N

mn1 1 mnβ

×

⎧ ⎨ ⎩1−

_N

mn1 1 mnβ

−1_N

mn1 O

1

mnβ

δn1 ⎫_⎬

⎭.

3.15

If there exists a constantk≤α1−n_/Γλn

i1Γλ/ri,such that3.2is still valid as we replace

α1−n_/Γλn

i1Γλ/ribyk,then in particular, we have the following:

I < k

n

i1

_∞

mi1

miβ

pi

1/pi k

N

mn1 1

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In virtue of3.15and3.16, it follows that

α1−n

Γλ

n

j1

Γ

λ rj

⎧_⎨

⎩1− _N

mn1 1 mnβ

−1_N

mn1 O

1

mnβ

δn1 ⎫_⎬

⎭< k. 3.17

ForN → ∞, we haveα1−n_/Γλn

i1Γλ/ri ≤ k. Hence,k α1−n/Γλni1Γλ/riis

the best value of3.2.

We conform that the constant factor α1−n_/Γλn

i1Γλ/ri in 3.3 is the best

possible, otherwise we can get a contradiction by 3.7 that the constant factor in 3.2 is not the best possible.

Remarks 3.3. iWhen 0 < α ≤ 1, the assumption λαmax{1/2 −α,1} ≤ min1≤i≤n{ri} of

two theorems becomes λα ≤ min1≤i≤n{ri}. ii When 0 < α ≤ 1,β 0, 3.2 reduces to

1.9.iiiFor n 2, r1 r, r2 s, p1 p, p2 q,settingα 1, β −1/2 in 3.2, then

Γλ/r1Γλ/r2/Γλ Bλ/r, λ/s, we obtain1.5. Settingβ−1/2,λ1 in3.2, we get

1.7.

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.

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