On a Multiple Hilbert’s Inequality with Parameters

(1)

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<

_∞

n1a²_n<∞and 0<_∞

n1b²_n<∞,then

∞ n1

∞ m1

ambn

mn < π _∞

m1

a²_m ∞ n1

b²_n _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

a²_m ∞ n1

b_n² _1/2

, 1.2

where the constant factorπis the best possible. For 0<_∞

n1a²_n<∞,the equivalent forms of 1.1and1.2are given as followscf.2:

(2)

∞ n1

_∞

m1

am

mn 2

< π² ∞ m1

a²_m, 1.3

∞ n1

_∞

m1

am

mn−1 ₂

< π² ∞ m1

a²_m, 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−1^λ < B λ

r,λ s

× _∞

m1

m−1

2

_{p1−λ/r−1} a^pm

1/p_∞

n1

n−1

2

_{q1−λ/s−1} b^qn

1/q

, 1.5 ∞

n1

n−1

2

_pλ/s−1_∞

m1

am

mn−1^λ _p

<

B

λ r,λ

s p∞

m1

m−1

2

_{p1−λ/r−1}

a^pm, 1.6

∞ n1

∞ m1

ambn

m−1/2^α n−1/2^α < π αsinπ/r×

_∞

m1

m−1

2

_{p1−α/r−1} a^pm

1/p

× _∞

n1

n−1

2

_{q1−α/s−1} b^qn

1/q

,

1.7

∞ n1

n−1

2

_pα/s−1_∞

m1

am

m−1/2^α n−1/2^α p

<

π αsinπ/r

_p_∞

m1

m−1

2

_{p1−α/r−1} a^pm,

1.8

iiifpi, ri>1,_n

i11/pi

_n

i11/ri 1, 0< α≤1, 0< λα≤min_1≤i≤n{ri}, then

∞ mn1

· · · ^∞

m11

_n 1

i1miα_λ n

i1

aⁱmi< α¹⁻ⁿ Γλ

n i1

Γ λ

ri

_∞

mi1

mipi1−λα/ri−1 aⁱmi

_p_i ^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, which are extensions of1.4.

(3)

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,_n

i11/pi _n

i11/ri 1,λ >0< α <2, β≥ −1/2,λαmax{1/2−α,1} ≤min_1≤i≤n{ri}, then

A:ⁿ

i1

⎡

⎣miβ_λα/r_i_−11−p_i ⁿ

j1j /i

mjβ_λα/r_j₋₁⎤

⎦

1/pi

1. 2.1

Proof. We find the following:

Aⁿ

i1

⎡

⎣miβ_λα/r_i_−11−p_i_1−λα/r_iⁿ

j1

⎦

1/pi

ⁿ

i1

⎡

⎣miβ_p_i_1−λα/r_iⁿ

j1

⎦

1/pi

ⁿ

i1

miβ_1−λα/r_i⎡

⎣ⁿ

j1

⎦

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 y^λ/r

<

∞ m1

y^λ/s

mβ_λα/r−1 y

mβ_αλ < Γλ/rΓλ/s

αΓλ . 2.3

Proof. For fixed y >0, we set

fx: y^λ/s

xβ_λα/r−1 y

xβ_αλ , x∈

−β,∞

. 2.4

(4)

In virtue ofαλα/r−2 ≤ 0 andλα/r−1 ≤ 0,we find −1ⁱfⁱx > 0,i 1,2.Putting u xβ^α/y,we have the following:

_∞

−βfxdx 1 α

_∞

0

u^λ/r−1

1u^λdu Γλ/rΓλ/s

αΓλ . 2.5

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

fm<

_m1/2

m−1/2fxdxm∈N, 2.6

it follows that ∞ m1

y^λ/s

mβ_λα/r−1 y

mβ_αλ ^∞

m1

fm<

∞ m1

_m1/2

m−1/2fxdx

_∞

1/2

fxdx≤ _∞

−βfxdx Γλ/rΓλ/s

αΓλ ,

2.7

and then we have the right-hand side of2.3. Since ₁

−βfxdx

_xβ^α_/y

0

u^λ/r−1 α1u^λdu

< 1 α

_xβ^α_/y

0

u^λ/r−1du r

1βλα/r

λαy^λ/r ,

2.8

andfxis strictly decreasing in−β,∞, we get ∞

m1

fm>

_∞

1

fxdx

_∞

−βfxdx− ₁

−βfxdx

> Γλ/rΓλ/s

αΓλ −r

1β_λα/r λαy^λ/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/r_i ^∞

mn1

· · · ^∞

mi11

∞ mi−11

· · ·^∞

m11

_n

j1j /i

mjβ_λα/r_j₋₁ _n

i1

miβ_αλ 2.10

(5)

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

α¹⁻ⁿ Γλ

n j1

Γ λ

rj

1−O

1 mnβ_δ_n

< ωnmn

mnβ_λα/r_n ^∞

mn−11

· · · ^∞

m11

_n−1

j1

mjβ_λα/r_j₋₁ _n

i1

miβαλ < α¹⁻ⁿ Γλ

n j1

Γ λ

rj

.

2.11

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

ωimi< α¹⁻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 m2β^α,δ2λα/r >0,we have the following:

ω2m2

∞ m11

m1βλα/r1−1

m2βλα/r2

m1β_α

m2β_αλ ^∞

m1

y^λ/s

mβ_λα/r−1 y

mβ_αλ , 2.13 and then2.11is valid by using inequality2.3.

Assuming that forn≥2,2.11is valid, then forn1,settingy _n1

i2 miβ^α>

mn1β^α,s1 1−1/r1 ⁻¹,by2.3, we have the following:

Γλ/r1Γλ/s1 αΓλ

1−O1

1 y^λ/r¹

<

∞ m11

y^λ/s¹

m1βλα/r1−1

y

m1β_αλ < Γλ/r1Γλ/s1

αΓλ . 2.14

Settingλλ/s1,rjrj1/s1,mjmj1 j1, . . . , n,we find_n

j11/rj 1,αλmax{1/2− α,1} ≤min_1≤i≤n{ri}.By the assumption of induction, it follows that

ωn1mn1

mnβλα/ rn × ^∞

mn−11

· · ·^∞

m11

_n−1

j1

mjβλα/rj−1

_n

i1

miβ_αλ

×

⎧⎨

⎩ ∞ m11

y^λ/s¹

m1β_λα/r₁₋₁ y

m1βαλ

⎫⎬

⎭

<

mnβλα/ rn ∞ mn−11

· · · ^∞

m11

_n−1

j1

mjβλα/ rj−1

_n

i1

miβ_αλ ·Γλ/r1Γλ/s1 αΓλ

< α¹⁻ⁿ Γ

λⁿ

i1

Γ λ

ri

·Γλ/r1Γ λ

αΓλ α¹⁻ⁿ¹

Γλ n1

i1

Γri

λ

,

2.15

(6)

ω_n1m_n1>

mnβλα/ rn × ^∞

mn−11

· · ·^∞

m11

_n−1

j1

mjβλα/rj−1

_n

i1

miβ_αλ ·Γλ/r1Γλ/s1 αΓλ

×

1−O1

1 y^λ/r¹

> Γλ/r1Γλ/s1 αΓλ

⎡

⎢⎣

mnβλα/ rn

∞ mn−11

· · · ^∞

m11

_n−1

j1

mjβλα/ rj−1

_n

i1

miβ_αλ −γ

⎤

⎥⎦

> α¹⁻ⁿ¹ Γλ

n1 i1

Γ λ

ri

×

⎡

⎣1−O2

⎛

⎝ 1 mnβδn

⎞

⎠

⎤

⎦− Γλ/r1Γλ/s1

αΓλ γ,

2.16

whereδn>0 and

0< γ:

mnβλα/ rn ∞ mn−11

· · · ^∞

m11

_n−1

j1

mjβλα/ rj−1

_n

i1

miβ_αλ O1

1 m_n1βαλ/r1

< α¹⁻ⁿ Γλ/s1

n1 i2

Γri

λ ×O1

1 mn1β_αλ/r₁

.

2.17

Settingδ_n1min{δn, αλ/r1}>0,by2.16, we have the following:

ωn1mn1> α¹⁻ⁿ¹ Γλ

n1 i1

Γ λ

ri

×

1−O

1 mn1β_δ_n1

, 2.18

and then by2.15,2.18, and mathematical induction,2.11is valid. Settingmj mj,rj rj j 1, . . . , i−1,mj m_j1,rj r_j1 j i, . . . , n−1,mn mi,rn ri,then we have the following:

ωimi ωmn;r1, . . . ,rn< α¹⁻ⁿ Γλ

n j1

Γ λ

rj

α¹⁻ⁿ

Γλ n

j1

Γ λ

rj

. 2.19

Hence,2.12is valid.

(7)

3. Main Results

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

i11/pi _n

i11/ri

1, 1/qn 1−1/pn,λ > 0, 0< α < 2,β ≥ −1/2,λαmax{1/2−α,1} ≤ min_1≤i≤n{ri},aⁱmi ≥ 0mi∈N,such that

0<

∞ mi1

miβ_p_i_1−λα/r_i₋₁ aⁱmi

pi

<∞ i1, . . . , n, 3.1

then one has the following equivalent inequalities:

I: ^∞

mn1

· · · ^∞

m11

_n 1

i1

miβαλ

n i1

aⁱmi

< α¹⁻ⁿ Γλ

n i1

Γ λ

ri

_∞

mi1

_p_i ^1/pⁱ ,

3.2

J:

⎧⎨

⎩ ∞ mn1

mnβ_λαq_n_/r_n₋₁⎡

⎣ ^∞

mn−11

· · · ^∞

m11

_n−1

i1aⁱmi

_n

i1

miβ_αλ

⎤

⎦

qn⎫

⎬

⎭

1/qn

< Γλ/rn αⁿ⁻¹Γλ

n−1 i1

Γ λ

ri

_∞

mi1

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

i1aⁱmi

_n

i1

miβ_αλ

⎤

⎦

qn

⎧⎪

⎨

⎪⎩ ∞ mn−11

· · · ^∞

m11

_n 1

i1

miβ_αλ

⎡

⎣mnβ_λα/r_n_−11−p_nⁿ⁻¹

j1

⎦

1/pn

×ⁿ⁻¹

i1

⎡

j1j /i

mjβλα/rj−1

⎤

⎦

1/pi

aⁱmi

⎫⎪

⎬

⎪⎭

qn

≤)

ωnmn

mnβ_p_n_1−λα/r_n₋₁*_q_n_/p_n ^∞

mn−11

· · · ^∞

m11

_n 1

i1

miβ_αλ

×ⁿ⁻¹

i1

⎡

j1j /i

⎦

qn/pi aⁱmi

_q_n

(8)

≤ n

i1Γλ/ri αⁿ⁻¹Γλ

qn/pn

mnβ_1−λαq_n_/r_n ^∞

mn−11

· · · ^∞

m11

_n 1

i1

miβ_αλ

×ⁿ⁻¹

i1

⎡

j1j /i

mjβλα/rj−1

⎤

⎦

qn/pi aⁱmi

_q_n ,

3.4 J ≤

n

1/pn

×

⎧⎪

⎨

⎪⎩ ∞ mn1

∞ mn−11

· · · ^∞

m11

_n 1

i1

miβαλ ×ⁿ⁻¹

i1

⎡

j1j /i

mjβλα/rj−1

⎤

⎦

qn/pi

× aⁱmi

_q_n

⎫⎪

⎪⎬

⎪⎪

⎭

1/qn

n

1/pn

⎧⎪

⎪⎨

⎪⎪

⎩ ∞ m_n−11

· · · ^∞

m11

⎡

⎣^∞

mn1

mnβ_λα/r_n₋₁ _n

i1

miβ_αλ

⎤

⎦

×ⁿ⁻¹

i1

⎡

⎢⎢

⎣

miβpi1−λα/ri−1

miβλα/ri n−1

j /j1i

mjβλα/rj−1

⎤

⎥⎥

⎦

qn/pi

aⁱmi

_q_n

⎫⎪

⎪⎬

⎪⎪

⎭

1/qn

.

3.5 For n ≥ 3, since _n−1

i1qn/pi 1, by H ¨older’s inequality again in 3.5, we have the following:

J≤ n

1/pnn−1 i1

⎧⎪

⎪⎨

⎪⎪

⎩ ∞ mn−11

· · · ^∞

m11

∞ mn1

mnβλα/rn−1

_n

i1

miβ_αλ

×

⎡

⎢⎢

⎣

miβ_p_i_1−λα/r_i₋₁

miβ_λα/r_iⁿ⁻¹

j /j1i

mjβ_λα/r_j₋₁

⎤

⎥⎥

⎦

aⁱmi

_p_i

⎫⎪

⎪⎬

⎪⎪

⎭

1/pi

n

1/pnn−1 i1

_∞

mi1

ωimi

miβpi1−λα/ri−1 aⁱmi

pi

1/pi

.

3.6

Note that forn2,by3.5, we directly get3.6. Hence,3.3is valid by3.6and2.12.

(9)

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

I ^∞

mn1

⎡

⎣mnβ_λα/r_n_−1/q_n ^∞

mn−11

· · · ^∞

m11

_n−1

i1aⁱmi

_n

i1

miβ_αλ

⎤

⎦×+

mnβ_1/q_n_−λα/r_n aⁿmn

,

≤J _∞

mn1

mnβ_p_n_1−λα/r_n₋₁ aⁿmn

_p_n ^1/pⁿ .

3.7

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

aⁿmn :

mnβλαqn/rn−1

⎡

⎣ ^∞

mn−11

· · · ^∞

m11

_n−1

i1aⁱmi

_n

i1

miβαλ

⎤

⎦

qn−1

, 3.8

then we find that

J _∞

mn1

mnβpn1−λα/rn−1 aⁿmn

_p_n ^1/qⁿ

I^1/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

mnβpn1−λα/rn−1 aⁿmn

_p_n

J^qⁿ I

<

_n

n i1

_∞

mi1

pi

1/pi

<∞.

3.10

Dividing outJ^qⁿ^/pⁿ into two sides of3.10, we have the following:

_∞

mn1

mnβ_p_n_1−λα/r_n₋₁ aⁿmn

_p_n ^1/qⁿ J

<

_n

n−1 i1

_∞

mi1

_p_i ^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/Γλ_n

i1Γλ/riin3.2and3.3is the best possible.

(10)

Proof. By2.11and

Nlim→ ∞

mnβ_λα/r_n ^N

m_n−11

· · · ^N

m11

_n−1

j1

mjβλα/rj−1

_n

i1

miβ_αλ ωnmn, 3.12

there existsN0∈N,such that forN > N0,

mnβ_λα/r_n ^N

m_n−11

· · ·^N

m11

_n−1

j1

mjβλα/rj−1

_n

i1

miβ_αλ > α¹⁻ⁿ Γλ

n j1

Γ λ

rj

1−O

1 mnβ_δ_n

, 3.13 whereδn>0.Setting

aⁱmi :

⎧⎨

⎩

miβλα/ri−1

, mi≤N,

0, mi> N , i1, . . . , n 3.14

we find that

I: ^∞

mn1

· · · ^∞

m11

_n 1

i1

miβ_αλ

n i1

aⁱmi

^N

mn1

mnβ_λα/r_n mnβ

N m_n−11

· · · ^N

m11

_n−1

i1

miβ_λα/r_i₋₁ _n

i1

miβ_αλ

>

N mn1

1

mnβ ·α¹⁻ⁿ Γλ

n j1

Γ λ

rj

1−O

1 mnβ_δ_n

α¹⁻ⁿ Γλ

n j1

Γ λ

rj

_N

mn1

1 mnβ

×

⎧⎨

⎩1− _N

mn1

1 mnβ

₋₁ _N

mn1

O

1 mnβ_δ_n₁

⎫⎬

⎭.

3.15

If there exists a constantk≤α¹⁻ⁿ/Γλ_n

i1Γλ/ri,such that3.2is still valid as we replace α¹⁻ⁿ/Γλ_n

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

I < k n

i1

_∞

mi1

miβpi1−λα/ri−1 aⁱmi

_p_i ^1/pⁱ k

N mn1

1

mnβ. 3.16

(11)

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_δ_n₁

⎫⎬

⎭< k. 3.17

ForN → ∞, we haveα¹⁻ⁿ/Γλ_n

i1Γλ/ri ≤ k. Hence,k α¹⁻ⁿ/Γλ_n

i1Γλ/riis the best value of3.2.

We conform that the constant factor α¹⁻ⁿ/Γλ_n

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} ≤ min_1≤i≤n{ri} of two theorems becomes λα ≤ min_1≤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.

References

1 H. Weyl, Singulare integral gleichungen mit besonderer berucksichtigung des fourierschen integral theorems, Inaugural-Dissertation, G ¨ottingen University, G ¨ottingen, Germany, 1908.

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

3 D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, vol. 53, Kluwer Academic, Boston, Mass, USA, 1991.

4 W. Zhong, “A Hilbert-type linear operator with the norm and its applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 494257, 18 pages, 2009.

5 J. C. Kuang, Applied Inequalitie, Shangdong Science Technic Press, Jinan, China, 2004.

6 K. Hu, Some Problems in Analysis Inequalities, Wuhan University Press, Wuhan, China, 2007.

7 W. Magnus, “On the spectrum of Hilbert’s matrix,” American Journal of Mathematics, vol. 72, pp. 699–

704, 1950.

8 B. C. Yang and M. Z. Gao, “On a best value of Hardy-Hilbert’s inequality,” Advances in Mathematics, vol. 26, no. 2, pp. 159–164, 1997Chinese.

9 M. Z. Gao and B. C. Yang, “On the extended Hilbert’s inequality,” Proceedings of the American Mathematical Society, vol. 126, no. 3, pp. 751–759, 1998.

10 K. Jichang, “On new extensions of Hilbert’s integral inequality,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 608–614, 1999.

11 B. C. Yang and L. Debnath, “On the extended Hardy-Hilbert’s inequality,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 187–199, 2002.

12 B. C. Yang, “An extension of Hardy-Hilbert’s inequality,” Chinese Annals of Mathematics, vol. 23, no. 2, pp. 247–254, 2002Chinese.

13 B. C. Yang and T. M. Rassias, “On a new extension of Hilbert’s inequality,” Mathematical Inequalities &

Applications, vol. 8, no. 4, pp. 575–582, 2005.

(12)

14 B. Yang, “On a new extension of Hilbert’s inequality with some parameters,” Acta Mathematica Hungarica, vol. 108, no. 4, pp. 337–350, 2005.

15 B. C. Yang, “Hilbert’s inequality with some parameters,” Acta Mathematica Sinica. Chinese Series, vol.

49, no. 5, pp. 1121–1126, 2006Chinese.

16 B. C. Yang, “A dual Hardy-Hilbert’s inequality and generalizations,” Advances in Mathematics, vol. 35, no. 1, pp. 102–108, 2006Chinese.

17 B. C. Yang, “On a Hilbert-type operator with a symmetric homogeneous kernel of -1-order and applications,” Journal of Inequalities and Applications, Article ID 47812, 9 pages, 2007.

18 B. C. Yang, “On the norm of a linear operator and its applications,” Indian Journal of Pure and Applied Mathematics, vol. 39, no. 3, pp. 237–250, 2008.

19 B. C. Yang, Hilbert-type Integral Inequalities, Bentham Science, Oak Park, Ill, USA, 2009.

20 Q. Huang and B. C. Yang, “On a multiple Hilbert-type integral operator and applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 192197, 13 pages, 2009.

21 B. C. Yang, The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijing, China, 2009.

22 B. C. Yang, “A survey of the study of Hilbert-type inequalities with parameters,” Advances in Mathematics, vol. 38, no. 3, pp. 257–268, 2009.