Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 851360, 10 pages
doi:10.1155/2009/851360
Research Article
Some New Hilbert’s Type Inequalities
Chang-Jian Zhao1 and Wing-Sum Cheung2
1
Department of Information and Mathematics Sciences, College of Science,
China Jiliang University, Hangzhou 310018, China
2
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Correspondence should be addressed to Chang-Jian Zhao, chjzhao@163.com
Received 25 December 2008; Accepted 24 April 2009
Recommended by Peter Pang
Some new inequalities similar to Hilbert’s type inequality involving series of nonnegative terms
are established.
Copyright q 2009 C.-J. Zhao and W.-S. Cheung. 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 recent years, several authors 1–10 have given considerable attention to Hilbert’s type
inequalities and their various generalizations. In particular, in 1, Pachpatte proved some
new inequalities similar to Hilbert’s inequality 11, page 226 involving series of nonnegative
terms. The main purpose of this paper is to establish their general forms.
2. Main Results
In 1, Pachpatte established the following inequality involving series of nonnegative terms.
Theorem A. Let p ≥ 1, q ≥ 1, and let {am } and {bn } be two nonnegative sequences of real numbers
defined for m 1, . . . , k, and n 1, . . . , r, where k, r are natural numbers. Let Am m
s1 as and
n
Bn t1 bt . Then
p q
r
k Am B n
m1 n1
mn
≤ C p, q, k, r
k
k − m 1
m1
p−1 2
am Am
1/2
r
1/2
q−1 2
×
,
r − n 1 bn Bn
n1
2.1
2
Journal of Inequalities and Applications
where
1
C p, q, k, r pqkr1/2 .
2
2.2
We first establish the following general form of inequality 2.1.
Theorem 2.1. Let p ≥ 1, q ≥ 1, t > 0, and 1/α 1/β 1, α > 1. Let {am1 ,...,mn }, and {bn1 ,...,nn }
be positive sequences of real numbers defined for mi 1, 2, . . . , ki , and ni 1, 2, . . . , ri , where
m1
mn
ki , ri i 1, . . . , n are natural numbers. Let Am1 ,...,mn s1 1 · · ·
sn 1 as1 ,...,sn , and Bn1 ,...,nn n1
nn
·
·
·
b
.
Then
t
,...,t
n
t1 1
tn 1 1
k1
···
m1 1
kn r1
···
mn 1n1 1
p
q
rn
αβt1/β Am1 ,...,mn Bn1 ,...,nn
m · · · mn β n1 · · · nn αt
nn 1 1
≤ L k1 , . . . , kn , r1 , . . . , rn , p, q, α, β
×
k1
m1 1
⎛
×⎝
r1
···
···
n1 1
β
p−1
kj − mj 1 am1 ,...,mn Am1 ,...,mn
kn n mn 1j1
1/β
2.3
⎞1/α
α
q−1
rj − nj 1 bn1 ,...,nn Bn1 ,...,nn ⎠ ,
rn n
nn 1 j1
where
L k1 , . . . , kn , r1 , . . . , rn , p, q, α, β pqk1 · · · kn 1/α r1 · · · rn 1/β .
2.4
Proof. By using the following inequality see 12:
n1
···
m1 1
nn
p
zm1 ,...,mn
≤p
mn 1
n1
,...,
m1 1
nn
zm1 ,...,mn
mn 1
m
1
···
k1 1
mn
p−1
zk1 ,...,kn
,
2.5
kn 1
where p ≥ 1 is a constant, and zm1 ,...,mn ≥ 0, mi 1, 2, . . . , ki , i 1, 2, . . . , n, we obtain
m1
p
Am1 ,...,mn ≤ p
···
s1 1
mn
p−1
as1 ,...,sn As1 ,...,sn .
2.6
sn 1
Similarly, we have
q
Bn1 ,...,nn ≤ q
n1
t1 1
···
nn
tn 1
q−1
bt1 ,...,tn Bt1 ,...,tn .
2.7
Journal of Inequalities and Applications
3
From 2.6 and 2.7, using Hölder’s inequality 13 and the elementary inequality:
a
t1/α b
,
β
αt1/β
a1/α b1/β ≤
2.8
where 1/α 1/β 1, α > 1, b > 0, a > 0, and t > 0, we have
p
q
Ami ,...,mn Bni ,...,nn
≤ pq
m
1
mn
p−1
···
as1 ,...,sn As1 ,...,sn
s1 1
sn 1
1/α
≤ pqm1 , . . . , mn m
1
···
s1 1
n
1
nn
q−1
···
bt1 ,...,tn Bt1 ,...,tn
t1 1
tn 1
mn sn 1
β
p−1
as1 ,...,sn As1 ,...,sn
1/β
n
nn α 1/α
1
q−1
1/β
bt1 ,...,tn Bt1 ,...,tn
× n1 , . . . , nn ···
t1 1
≤ pq
×
tn 1
m1 · · · mn n1 · · · nn t1/α
β
αt1/β
n
1
nn ...
t1 1
m
1
α
q−1
bt1 ,...,tn Bt1 ,...,tn
tn 1
···
s1 1
mn sn 1
β
p−1
as1 ,...,sn As1 ,...,sn
1/β
1/α
.
2.9
Dividing both sides of 2.9 by m1 · · · mn β n1 · · · nn αt/αβt1/β , summing up over ni from
1 to ri i 1, 2, . . . , n first, then summing up over mi from 1 to ki i 1, 2, . . . , n, using again
Hölder’s inequality, then interchanging the order of summation, we obtain
k1
kn r1
···
m1 1
···
mn 1 n1 1
≤ pq
×
⎧
k1
⎨
⎩m
⎧
r1
⎨
⎩n
1 1
kn
···
1 1
p
q
rn
αβt1/β Am1 ,...,mn Bn1 ,...,nn
m · · · mn β n1 · · · nn αt
nn 1 1
m
1
mn 1
rn
···
n
1
nn 1
1/α
k1
1/β
× r1 · · · rn k1
n1 1
nn tn 1
···
m1 1
···
β
p−1
as1 ,...,sn As1 ,...,sn
sn 1
···
t1 1
≤ pqk1 · · · kn ···
s1 1
mn kn
α
q−1
bt1 ,...,tn Bt1 ,...,tn
m
1
mn 1
rn
n
1
nn 1
···
s1 1
t1 1
···
1/β ⎫
⎬
⎭
1/α ⎫
⎬
⎭
mn sn 1
nn tn 1
β
p−1
as1 ,...,sn As1 ,...,sn
α
q−1
bt1 ,...,tn Bt1 ,...,tn
1/β
1/α
4
Journal of Inequalities and Applications
L k1 , . . . , kn , r1 , . . . , rn , p, q, α, β
×
×
k1
kn ···
s1 1
sn 1
r1
rn ···
t1 1
tn 1
β
p−1
as1 ,...,sn As1 ,...,sn
α
q−1
bt1 ,...,tn Bt1 ,...,tn
k1
m1 s1
r1
n1 t1
···
···
kn
1/β
1
mn sn
rn
1/α
1
nn tn
L k1 , . . . , kn , r1 , . . . , rn , p, q, α, β
×
×
⎧
k1
⎨
⎩s
···
1 1
⎧
r1
⎨
⎩t
⎫1/β
β ⎬
p−1
kj − sj 1 as1 ,...,sn As1 ,...,sn
⎭
kn n
sn 1 j1
···
1 1
⎫1/α
⎬
α
q−1
rj − tj 1 bt1 ,...,tn Bt1 ,...,tn
⎭
rn n
tn 1 j1
L k1 , . . . , kn , r1 , . . . , rn , p, q, α, β
×
⎧
k1
⎨
⎩m 1
×
⎧
r1
⎨
⎩n
⎫1/β
β ⎬
p−1
kj − mj 1 am1 ,...,mn Am1 ,...,mn
⎭
kn n
···
mn 1 j1
1
⎫1/α
⎬
α
q−1
.
rj − nj 1 bn1 ,...,nn Bn1 ,...,nn
⎭
rn n
···
1 1
nn 1 j1
2.10
This completes the proof.
Remark 2.2. Taking α β n j 2, 2.3 becomes
k1 k2
p
r1 r2
q
Am1 ,m2 Bn1 ,n2
−1/2 n n t1/2
m
m
1 2
n1 1 n2 1 1 2 t
m1 m2 1
1 ≤ pq k1 k2 r1 r2
2
×
r r
1 2
k2
k1 k1 − m1 1k2 − m2 1
m1 m2 1
r1 − n1 1r2 − n2 1
n1 n2 1
p−1 2
bn1 ,n2 Bn1 ,n2
2
p−1
am1 ,m2 Am1 ,m2
1/2
2.11
1/2
.
Taking t 1, and changing {am1 ,m2 }, {bn1 ,n2 }, {Am1 ,m2 }, and {Bn1 ,n2 } into {am }, {bn }, {Am }, and
{Bn }, respectively, and with suitable changes, 2.11 reduces to Pachpatte 1, inequality 1.
In 1, Pachpatte also established the following inequality involving series of nonnegative terms.
Journal of Inequalities and Applications
5
Theorem B. Let {am }, {bn }, Am , Bn be as defined in Theorem A. Let {pm } and {qn } be positive
sequences for m 1, . . . , k, and n 1, . . . , r, where k, r are natural numbers. Define Pm m
s1 ps ,
n
q
.
Let
φ
and
ψ
be
real-valued,
nonnegative,
convex,
submultiplicative
functions
and Qn t1 t
defined on R 0, ∞. Then
r
k φAm ψBn m1 n1
mn
k
am
≤ Mk, r
k − m 1 pm φ
pm
m1
2 1/2
2.12
2 1/2
r
bn
×
,
r − n 1 qn φ
qn
n1
where
1
Mk, r 2
k φPm 2
1/2 1/2
r
φQn 2
Pm
m1
n1
Qn
.
2.13
Inequality 2.12 can also be generalized to the following general form.
Theorem 2.3. Let {am1 ,...,mn }, {bn1 ,...,nn }, α, β, t, Am1 ,...,mn , and Bn1 ,...,nn be as defined in Theorem 2.1.
Let {pm1 ,...,mn } and {qn1 ,...,nn } be positive sequences for mi 1, 2, . . . , ki , and ni 1, 2, . . . , ri i 1
m n
n1
nn
1, 2, . . . , n. Define Pm1 ,...,mn m
s1 1 · · ·
sn 1 ps1 ,...,sn , and Qn1 ,...,nn t1 1 · · ·
tn 1 qt1 ,...,tn . Let φ
and ψ be real-valued, nonnegative, convex, submultiplicative functions defined on R 0, ∞.
Then
k1
···
m1 1
kn r1
rn
αβt1/β φAm1 ,...,mn ψBn1 ,...,nn ···
mn 1n1 1
nn 1
m1 · · · mn β n1 · · · nn αt
≤ M k1 , . . . , kn , r1 , . . . , rn , α, β
×
×
⎧
k1
⎨
⎩m
···
1 1
⎧
r1
⎨
⎩n 1
1
⎫
β ⎬1/β
am1 ,...,mn
kj − mj 1 pm1 ,...,mn φ
⎭
pm1 ,...,mn
kn n
mn 1 j1
···
2.14
⎫
α ⎬1/α
bn1 ,...,nn
,
rj − nj 1 qn1 ,...,nn ψa
⎭
qn1 ,...,mn
rn n
nn 1 j1
where
M k1 , . . . , kn , r1 , . . . , rn , α, β
k1
m1 1
···
kn φPm1 ,...,mn α
mn 1
Pm1 ,...,mn
1/α r1
n1 1
···
rn ψQn1 ,...,nn β
nn 1
Qn1 ,...,nn
1/β
.
2.15
6
Journal of Inequalities and Applications
Proof. By the hypotheses, Jensen’s inequality, and Hölder’s inequality, we obtain
φAm1 ,...,mn φ
Pm1 ,...,mn
m1
ps1 ,...,sn as1 ,...,sn /ps1 ,...,sn
m n
s1 1 · · ·
sn 1 ps1 ,...,sn
...
m1
s1 1
m1
≤ φPm1 ,...,mn φ
s1 1
m n
sn 1
n
··· m
sn 1 ps1 ,··· ,sn as1 ,...,sn /ps1 ,...,sn
m n
m1
s1 1 · · ·
sn 1 ps1 ,...,sn
2.16
mn
m1
φPm1 ,...,mn as1 ,...,sn
···
ps ,...,s φ
≤
Pm1 ,...,mn s1 1 sn 1 1 n
ps1 ,...,sn
φPm1 ,...,mn ≤
m1 · · · mn 1/α
Pm1 ,...,mn
m
1
···
s1 1
mn sn 1
as1 ,...,sn
ps1 ,...,sn φ
ps1 ,...,sn
β 1/β
.
Similarly,
φQn1 ,...,nn φBn1 ,...,nn ≤
n1 · · · nn 1/β
Qn1 ,...,nn
n1
nn ···
n1 1
nn 1
bt1 ,...,tn
qt1 ,...,tn ψ
qt1 ,...,tn
α 1/α
.
2.17
By 2.16 and 2.17, and using the elementary inequality:
a1/α b1/β ≤
a
t1/α b
,
1/β
β
αt
2.18
where 1/α 1/β 1, α > 1, b > 0, a > 0, and t > 0, we have
φAm1 ,...,mn φBn1 ,...,nn ≤
m1 · · · mn n1 · · · nn t1/α
β
αt1/β
φPm1 ,...,mn ×
Pm1 ,...,mn
φQn1 ,...,nn ×
Qn1 ,...,nn
m
1
···
s1 1
n
1
t1 1
...
mn sn 1
nn tn 1
as1 ,...,sn
ps1 ,...,sn φ
ps1 ,...,sn
bt1 ,...,tn
qt1 ,...,tn ψ
qt1 ,...,tn
β 1/β
α 1/α
2.19
.
Journal of Inequalities and Applications
7
Dividing both sides of 2.19 by m1 · · · mn β n1 · · · nn αt/αβt1/β , and summing up over ni
from 1 to ri i 1, 2, . . . , n first, then summing up over mi from 1 to ki i 1, 2, . . . , n, using
again inverse Hölder’s inequality, and then interchanging the order of summation, we obtain
k1
···
m1 1
kn r1
rn
αβt1/β φAm1 ,...,mn ψBn1 ,...,nn ···
mn 1 n1 1
m1 . . . mn β n1 . . . nn αt
nn 1
k1
kn
⎛
⎝ φPm1 ,...,mn ≤
...
Pm1 ,...,mn
m1 1
mn 1
r1
m
1
⎛
rn
β
mn as1 ,...,sn
···
ps1 ,...,sn φ
ps1 ,...,sn
s1 1
sn 1
⎝ φQn1 ,...,nn ×
···
Qn1 ,...,nn
n1 1
nn 1
≤
k1
···
m1 1
×
kn φPm1 ,...,mn α
kn
···
m1 1
×
r1
×
r1
β
mn as1 ,...,sn
···
ps1 ,...,sn φ
ps1 ,...,sn
s1 1
sn 1
rn ψQn1 ,...,nn β
···
···
n1 1
n
1
nn 1
1/β
1/β
Qn1 ,...,nn
nn 1
rn
1/α ⎞
⎠
1/α
m
1
mn 1
n1 1
α
nn bt1 ,...,tn
···
qt1 ,...,tn ψ
qt1 ,...,tn
n1 1
nn 1
n1
Pm1 ,...,mn
mn 1
k1
1/β ⎞
⎠
···
t1 1
nn tn 1
bt1 ,...,tn
qt1 ,...,tn ψ
qt1 ,...,tn
α 1/α
M k1 , . . . , kn , r1 , . . . , rn , α, β
×
×
⎧
k1
⎨
⎩m
···
1 1
⎧
r1
⎨
⎩n 1
1
···
⎫
β ⎬1/β
am1 ,...,mn
kj − mj 1 pm1 ,...,mn φ
⎭
pm1 ,...,mn
kn n
mn 1 j1
⎫
α ⎬1/α
bn1 ,...,nn
.
rj − nj 1 qn1 ,...,nn ψ
⎭
qn1 ,...,mn
rn n
nn 1 j1
2.20
The proof is complete.
8
Journal of Inequalities and Applications
Remark 2.4. Taking α β n j 2, 2.14 becomes
k2
k1 r2
r1 φAm1 ,m2 ψBn1 ,n2 −1/2 n n t1/2
1 2
n1 1 n2 1 m1 m2 t
m1 m2 1
≤ Mk1 , k2 , r1 , r2 am1 ,...,mn
k1 − m1 1k2 − m2 1 pm1 ,m2 φ
pm1 ,...,mn
m1 m2 1
×
×
k1 k2
r r
1 2
2 1/2
2.21
2 1/2
bn1 ,...,nn
r1 − n1 1r2 − n2 1 qn1 ,n2 ψ
qn1 ,...,nn
n1 n2 1
,
where
k1 k2 φPm1 ,m2 2
Mk1 , k2 , r1 , r2 m1 1 m2 1
1/2 Pm1 ,m2
r1 r2 ψQn1 ,n2 2
n1 1 n2 1
1/2
Qn1 ,n2
.
2.22
Taking t 1, and changing {am1 ,m2 }, {bn1 ,n2 }, {Am1 ,m2 }, and {Bn1 ,n2 } into {am }, {bn }, {Am }, and
{Bn }, respectively, and with suitable changes, 2.21 reduces to Pachpatte 1, Inequality 7.
Theorem 2.5. Let {am1 ,...,mn }, {bn1 ,...,nn }, {pm1 ,...,mn }, {qn1 ,...,nn }, Pm1 ,...,mn , and Qn1 ,...,nn , α, β, t, be as
defined in Theorem 2.3. Define
Am1 ,...,mn Bn1,...,n2 1
m1
Pm1 ,...,mn s1 1
1
n1
Qn1 ,...,nn t1 1
···
mn
ps1 ,...,sn as1 ,...,sn ,
sn 1
···
nn
2.23
qt1 ,...,tn bt1 ,...,tn ,
tn 1
for mi 1, 2, . . . , ki , and ni 1, 2, . . . , ri i 1, 2, . . . , n, where ki , ri i 1, . . . , n are natural
numbers. Let φ and ψ be real-valued, nonnegative, convex functions defined on R 0, ∞. Then
k1
m1 1
···
kn r1
rn
αβt1/β Pm1 ,...,mn Qn1 ,...,nn φAm1 ,...,mn ψBn1 ,...,nn ···
mn 1n1 1
m1 · · · mn β n1 · · · nn αt
nn 1
k1 · · · kn 1/α r1 · · · rn 1/β
×
×
⎧
k1
⎨
⎩m
1 1
⎧
r1
⎨
⎩n
···
1 1
···
⎫1/β
β ⎬
kj − mj 1 pm1 ,...,mn φam1 ,...,mn ⎭
kn n
mn 1 j1
⎫1/α
α ⎬
.
rj − nj 1 qn1 ,...,nn ψbn1 ,...,nn ⎭
rn n
nn 1 j1
2.24
Journal of Inequalities and Applications
9
Proof. By the hypotheses, Jensen’s inequality, and Hölder’s inequality, it is easy to observe
that
φAm1 ,...,mn φ
≤
≤
Pm1 ,...,mn s1 1
Pm1 ,...,mn s1 1
1
···
n1
Qn1 ,...,nn t1 1
Qn1 ,...,nn
ps1 ,...,sn as1 ,...,sn
sn 1
mn
ps1 ,...,sn φas1 ,...,sn 2.25
sn 1
n1
Qn1 ,...,nn t1 1
1
···
1/α
m
1
···
s1 1
1
1
mn
m1 · · · mn Pm1 ,...,mn
ψBn1 ,...,nn ψ
≤
m1
1
≤
m1
1
···
···
nn
mn
sn 1
ps1 ,...,sn φas1 ,...,sn β
1/β
,
qt1 ,...,tn bt1 ,...,tn
tn 1
nn
qt1 ,...,tn ψbt1 ,...,tn 2.26
tn 1
n1 · · · nn 1/β
n
1
···
t1 1
nn
qt1 ,...,tn ψbt1 ,...,tn tn 1
α
1/α
.
Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifications, it is not hard to arrive at the desired inequality. The details are omitted here.
Remark 2.6. In the special case where j 1, t 1, α β 2, and n 1, Theorem 2.5 reduces to
the following result.
Theorem C. Let {am }, {bn }, {pm }, {qn }, P m , Qn be as defined in Theorem B. Define Am n
1/Pm m
s1 ps as , and Bn 1/Qn t1 qt bt for m 1, . . . , k, and n 1, . . . , r, where k, r are
natural numbers. Let φ and ψ be real-valued, nonnegative, convex functions defined on R 0, ∞.
Then
r
k Pm Qn φAm ψBn m1 n1
mn
1
≤ kr1/2
2
×
r
k
k − m 1 pm φam m1
1/2
2.27
1/2
r − n 1qn ψbn 2
2
.
n1
This is the new inequality of Pachpatte in [1, Theorem 4].
Remark 2.7. Taking j 1, t 1, pm 1, qn 1, α β 2, and n 1 in Theorem 2.5, and in view
of Pm m, Qn n, we obtain the following theorem.
10
Journal of Inequalities and Applications
Theorem D. Let {am }, {bn } be as defined in Theorem A. Define Am 1/m m
s1 as , and Bn n
1/n t1 bt , for m 1, . . . , k, and n 1, . . . , r, where k, r are natural numbers. Let φ and ψ be
real-valued, nonnegative, convex functions defined on R 0, ∞. Then
r
k mn
1
φAm ψBn ≤ kr1/2
m
n
2
m1 n1
×
r
k
k − m 1 φam m1
2
r − n 1 ψbn 2
1/2
2.28
1/2
.
n1
This is the new inequality of Pachpatte in [1, Theorem 3].
Acknowledgments
Research is supported by Zhejiang Provincial Natural Science Foundation of ChinaY605065,
Foundation of the Education Department of Zhejiang Province of China 20050392. Research
is partially supported by the Research Grants Council of the Hong Kong SAR, China Project
no. HKU7016/07P and a HKU Seed Grant forBasic Research.
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