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Hindawi Publishing CorporationJournal of Inequalities and Applications Volume 2009, Article ID 851360, 10 pages doi:10.1155/2009/851360 Research Article Some New Hilbert’s Type Inequalit

Trang 1

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

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

Copyrightq 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, in1, Pachpatte proved some new inequalities similar to Hilbert’s inequality11, page 226 involving series of nonnegative terms The main purpose of this paper is to establish their general forms

2 Main Results

In1, Pachpatte established the following inequality involving series of nonnegative terms

Theorem A Let p ≥ 1, q ≥ 1, and let {am } and {b n } be two nonnegative sequences of real numbers

defined for m 1, , k, and n 1, , r, where k, r are natural numbers Let A m m s1 a s and

B nn

t1 b t Then

k

m1

r

n1

A p m B q n

m n ≤ C

p, q, k, r

k

m1

k − m 1a m A p−1 m 2

1/2

×

r

n1

r − n 1b n B q−1 n 2

1/2

,

2.1

Trang 2

Cp, q, k, r 1

We first establish the following general form of inequality2.1

Theorem 2.1 Let p ≥ 1, q ≥ 1, t > 0, and 1/α 1/β 1, α > 1 Let {a m1, ,m n }, and {b n1, ,n n}

be positive sequences of real numbers defined for m i 1, 2, , k i , and n i 1, 2, , r i , where

k i , r i i 1, , n are natural numbers Let A m1, ,m n m1

s1 1· · ·m n

s n1a s1, ,s n , and B n1, ,n n

n1

t1 1· · ·n n

t n1b t1, ,t n Then

k1

m1 1

· · · k n

m n1

r1

n1 1

· · ·r n

n n1

αβt1/β A p m1, ,m n B n q1, ,n n

m1· · · m n β n1· · · n n αt

≤ Lk1, , k n , r1, , r n , p, q, α, β

×

k1

m1 1· · · k n

m n1

n j1

k j − m j 1a m1, ,m n A p−1 m1, ,m nβ

1/β

×

⎛

⎝r1

n1 1

· · ·r n

n n1

n

j1

r j − n j 1b n1, ,n n B n q−11, ,n nα

⎞

⎠

1/α

,

2.3

where

Lk1, , k n , r1, , r n , p, q, α, β pqk1· · · k n1/α r1· · · r n1/β 2.4

Proof By using the following inequalitysee 12:

n

1

m1 1

· · · n n

m n1

z m1, ,m n

p

≤ pn1

m1 1

, ,n n

m n1

z m1, ,m n

m

1

k1 1

· · ·m n

k n1

z k1, ,k n

p−1

, 2.5

wherep ≥ 1 is a constant, and z m1, ,m n ≥ 0, m i 1, 2, , k i,i 1, 2, , n, we obtain

A p m1, ,m n ≤ pm1

s1 1

· · ·m n

s n1

a s1, ,s n A p−1 s1, ,s n 2.6

Similarly, we have

B n q1, ,n n ≤ qn1

t1 1

· · ·n n

t n1

b t1, ,t n B q−1 t1, ,t n 2.7

Trang 3

Journal of Inequalities and Applications 3 From2.6 and 2.7, using H¨older’s inequality 13 and the elementary inequality:

a1/α b1/β≤ a

αt1/β t1/α β b , 2.8 where 1/α 1/β 1, α > 1, b > 0, a > 0, and t > 0, we have

A p m i , ,m n B q n i , ,n n ≤ pq

m

1

s1 1

· · ·m n

s n1

a s1, ,s n A p−1 s1, ,s n

n

1

t1 1

· · ·n n

t n1

b t1, ,t n B t q−11, ,t n

≤ pqm1, , m n1/α

m

1

s1 1

· · ·m n

s n1

a s1, ,s n A p−1 s1, ,s nβ

1/β

× n1, , n n1/β

n

1

t1 1

· · ·n n

t n1

b t1, ,t n B q−1 t1, ,t nα

1/α

≤ pq

m1· · · m n

αt1/β n1· · · n n t1/α

β

m

1

s1 1

· · ·m n

s n1

a s1, ,s n A p−1 s1, ,s n

β 1/β

×

n

1

t1 1

.n n

t n1

b t1, ,t n B q−1 t1, ,t nα

1/α

.

2.9

Dividing both sides of2.9 by m1· · · m n β n1· · · n n αt/αβt1/β , summing up over n ifrom

1 tor i i 1, 2, , n first, then summing up over m ifrom 1 tok i i 1, 2, , n, using again

H ¨older’s inequality, then interchanging the order of summation, we obtain

k1

m1 1

· · · k n

m n1

r1

n1 1

· · ·r n

n n1

αβt1/β A p m1, ,m n B n q1, ,n n

m1· · · m n β n1· · · n n αt

≤ pq

⎧

⎨

⎩

k1

m1 1

· · · k n

m n1

m

1

s1 1

· · ·m n

s n1

a s1, ,s n A p−1 s1, ,s n

β 1/β⎫

⎬

⎭

×

⎧

⎨

⎩

r1

n1 1

· · ·r n

n n1

n

1

t1 1

· · ·n n

t n1

b t1, ,t n B q−1 t1, ,t nα

1/α⎫⎬

⎭

≤ pqk1· · · k n1/α

k

1

m1 1

· · · k n

m n1

m

1

s1 1

· · ·m n

s n1

a s1, ,s n A p−1 s1, ,s nβ

1/β

× r1· · · r n1/β

k

1

n1

· · ·r n

n 1

n

1

t1

· · ·n n

t1

b t1, ,t n B t q−11, ,t nα

1/α

Trang 4

Lk1, , k n , r1, , r n , p, q, α, β

×

k

1

s1 1

· · ·k n

s n1

a s1, ,s n A p−1 s1, ,s nβ

k

1

m1s1

· · · k n

m n s n

1 1/β

×

r

1

t1 1

· · ·r n

t n1

b t1, ,t n B q−1 t1, ,t nα

r

1

n1t1

· · · r n

n n t n

1 1/α

Lk1, , k n , r1, , r n , p, q, α, β

×

⎧

⎨

⎩

k1

s1 1

· · ·k n

s n1

n

j1

k j − s j 1a s1, ,s n A p−1 s1, ,s n

β⎫⎬

⎭

1/β

×

⎧

⎨

⎩

r1

t1 1

· · ·r n

t n1

n j1

r j − t j 1b t1, ,t n B t q−11, ,t nα

⎫

⎬

⎭

1/α

Lk1, , k n , r1, , r n , p, q, α, β

×

⎧

⎨

⎩

k1

m1 1

· · · k n

m n1

n j1

k j − m j 1a m1, ,m n A p−1 m1, ,m nβ

⎫

⎬

⎭

1/β

×

⎧

⎨

⎩

r1

n1 1

· · ·r n

n n1

n

j1

r j − n j 1b n1, ,n n B n q−11, ,n nα

⎫

⎬

⎭

1/α

.

2.10 This completes the proof

Remark 2.2 Taking α β n j 2, 2.3 becomes

k1

m1

k2

m2 1

r

1

n1 1

r2

n2 1

A p m1,m2B q n1,n2

m1m2t −1/2 n1n2t1/2

≤ 1

2pqk1k2r1r2

k

1

m1

k2

m2 1

k1− m1 1k2− m2 1a m1,m2A p−1 m1,m22

1/2

×

r

1

n1

r2

n2 1

r1− n1 1r2− n2 1b n1,n2B n p−11,n22

1/2

.

2.11

Takingt 1, and changing {a m1,m2}, {b n1,n2}, {A m1,m2}, and {B n1,n2} into {a m }, {b n }, {A m }, and {B n }, respectively, and with suitable changes, 2.11 reduces to Pachpatte 1, inequality1

In1, Pachpatte also established the following inequality involving series of nonneg-ative terms

Trang 5

Journal of Inequalities and Applications 5

Theorem B Let {am }, {b n }, A m , B n be as defined in Theorem A Let {p m } and {q n } be positive

sequences for m 1, , k, and n 1, , r, where k, r are natural numbers Define P m m

s1 p s , and Q n n t1 q t Let φ and ψ be real-valued, nonnegative, convex, submultiplicative functions defined onR 0, ∞ Then

k

m1

r

n1

φA m ψB n

m n ≤ Mk, r

k

m1

k − m 1

p m φ

a

m

p m

2 1/2

×

r

n1

r − n 1

q n φ

b

n

q n

2 1/2

,

2.12

where

Mk, r 1

2

k

m1

φP m

P m

2 1/2 r

n1

φQ n

Q n

2 1/2

Inequality2.12 can also be generalized to the following general form

Theorem 2.3 Let {a m1, ,m n }, {b n1, ,n n }, α, β, t, A m1, ,m n , and B n1, ,n n be as defined in Theorem 2.1 Let {p m1, ,m n } and {q n1, ,n n } be positive sequences for m i 1, 2, , k i , and n i 1, 2, , r i i

1, 2, , n Define P m1, ,m n m1

s1 1· · ·m n

s n1p s1, ,s n , and Q n1, ,n n n1

t1 1· · ·n n

t n1q t1, ,t n Let φ and ψ be real-valued, nonnegative, convex, submultiplicative functions defined on R 0, ∞.

Then

k1

m1 1

· · · k n

m n1

r1

n1 1

· · ·r n

n n1

αβt1/β φA m1, ,m n ψB n1, ,n n

m1· · · m n β n1· · · n n αt

≤ Mk1, , k n , r1, , r n , α, β

×

⎧

⎨

⎩

k1

m1 1

· · · k n

m n1

n j1

k j − m j 1p m1, ,m n φ a m1, ,m n

p m1, ,m n

β⎫⎬

⎭

1/β

×

⎧

⎨

⎩

r1

n1 1

· · ·r n

n n1

n

j1

r j − n j 1q n1, ,n n ψa b n1, ,n n

q n1, ,m n

α⎫

⎬

⎭

1/α

,

2.14

where

Mk1, , k n , r1, , r n , α, β

k

1

m1 1

· · · k n

m n1

φP m1, ,m n

P m1, ,m n

α 1/α r

1

n1 1

· · ·r n

n n1

ψQ n1, ,n n

Q n1, ,n n

β 1/β

. 2.15

Trang 6

Proof By the hypotheses, Jensen’s inequality, and H ¨older’s inequality, we obtain

φA m1, ,m n φ

P m1, ,m n

m1

s1 1 .m n

s n1p s1, ,s n

a s1, ,s n /p s1, ,s n

m1

s1 1· · ·m n

s n1p s1, ,s n

≤ φP m1, ,m n φ

m1

s1 1· · ·m n

s n1p s1,··· ,s na s1, ,s n /p s1, ,s n

m1

s1 1· · ·m n

s n1p s1, ,s n

≤ φP m1, ,m n

P m1, ,m n

m1

s1 1

· · ·m n

s n1

p s1, ,s n φ a s1, ,s n

p s1, ,s n

≤ φP m1, ,m n

P m1, ,m n m1· · · m n1/α

m

1

s1 1

· · ·m n

s n1

p s1, ,s n φ a s1, ,s n

p s1, ,s n

β 1/β

.

2.16

Similarly,

φB n1, ,n n ≤ φQ n1, ,n n

Q n1, ,n n n1· · · n n1/β

n

1

n1 1

· · ·n n

n n1

q t1, ,t n ψ b t1, ,t n

q t1, ,t n

α 1/α

. 2.17

By2.16 and 2.17, and using the elementary inequality:

a1/α b1/β≤ a

αt1/β t1/α β b , 2.18

where 1/α 1/β 1, α > 1, b > 0, a > 0, and t > 0, we have

φA m1, ,m n φB n1, ,n n ≤

m1· · · m n

αt1/β n1· · · n n t1/α

β

× φP m1, ,m n

P m1, ,m n

m

1

s1 1

· · ·m n

s n1

p s1, ,s n φ a s1, ,s n

p s1, ,s n

β 1/β

× φQ n1, ,n n

Q n1, ,n n

n

1

t1 1

.n n

t n1

q t1, ,t n ψ b t1, ,t n

q t1, ,t n

α 1/α

.

2.19

Trang 7

Journal of Inequalities and Applications 7 Dividing both sides of2.19 by m1· · · m n β n1· · · n n αt/αβt1/β , and summing up over n i

from 1 tor i i 1, 2, , n first, then summing up over m ifrom 1 tok i i 1, 2, , n, using

again inverse H ¨older’s inequality, and then interchanging the order of summation, we obtain

k1

m1 1

· · · k n

m n1

r1

n1 1

· · ·r n

n n1

αβt1/β φA m1, ,m n ψB n1, ,n n

m1 m n β n1 n n αt

≤ k1

m1 1

. k n

m n1

⎛

⎝φP m1, ,m n

P m1, ,m n

m

1

s1 1

· · ·m n

s n1

p s1, ,s n φ a s1, ,s n

p s1, ,s n

β 1/β⎞

⎠

×r1

n1 1

· · ·r n

n n1

⎛

⎝φQ n1, ,n n

Q n1, ,n n

n

1

n1 1

· · ·n n

n n1

q t1, ,t n ψ b t1, ,t n

q t1, ,t n

α 1/α⎞

⎠

≤

k

1

m1 1

· · · k n

m n1

φP m1, ,m n

P m1, ,m n

α 1/α

×

k

1

m1 1

· · · k n

m n1

m

1

s1 1

· · ·m n

s n1

p s1, ,s n φ a s1, ,s n

p s1, ,s n

β 1/β

×

r

1

n1 1

· · ·r n

n n1

ψQ n1, ,n n

Q n1, ,n n

β 1/β

×

r

1

n1 1

· · ·r n

n n1

n

1

t1 1

· · ·n n

t n1

q t1, ,t n ψ b t1, ,t n

q t1, ,t n

α 1/α

Mk1, , k n , r1, , r n , α, β

×

⎧

⎨

⎩

k1

m1 1

· · · k n

m n1

n

j1

k j − m j 1p m1, ,m n φ a m1, ,m n

p m1, ,m n

β⎫⎬

⎭

1/β

×

⎧

⎨

⎩

r1

n1 1

· · ·r n

n n1

n j1

r j − n j 1q n1, ,n n ψ b n1, ,n n

q n1, ,m n

α⎫⎬

⎭

1/α

.

2.20 The proof is complete

Trang 8

Remark 2.4 Taking α β n j 2, 2.14 becomes

k1

m1

k2

m2 1

r

1

n1 1

r2

n2 1

φA m1,m2ψB n1,n2

m1m2t −1/2 n1n2t1/2

≤ Mk1, k2, r1, r2

×

k

1

m1

k2

m2 1

k1− m1 1k2− m2 1

p m1,m2φ a m1, ,m n

p m1, ,m n

2 1/2

×

r

1

n1

r2

n2 1

r1− n1 1r2− n2 1

q n1,n2ψ b n1, ,n n

q n1, ,n n

2 1/2

,

2.21

where

Mk1, k2, r1, r2

k

1

m1 1

k2

m2 1

φP m1,m2

P m1,m2

2 1/2 r

1

n1 1

r2

n2 1

ψQ n1,n2

Q n1,n2

2 1/2

. 2.22

Takingt 1, and changing {a m1,m2}, {b n1,n2}, {A m1,m2}, and {B n1,n2} into {a m }, {b n }, {A m }, and {B n }, respectively, and with suitable changes, 2.21 reduces to Pachpatte 1, Inequality 7

Theorem 2.5 Let {am1, ,m n }, {b n1, ,n n }, {p m1, ,m n }, {q n1, ,n n }, P m1, ,m n , and Q n1, ,n n , α, β, t, be as defined in Theorem 2.3 Define

A m1, ,m n P 1

m1, ,m n

m1

s1 1

· · ·m n

s n1

p s1, ,s n a s1, ,s n ,

B n1, ,n2 Q 1

n1, ,n n

n1

t1 1

· · ·n n

t n1

q t1, ,t n b t1, ,t n ,

2.23

for m i 1, 2, , k i , and n i 1, 2, , r i i 1, 2, , n, where k i , r i i 1, , n are natural

numbers Let φ and ψ be real-valued, nonnegative, convex functions defined on R 0, ∞ Then

k1

m1 1

· · · k n

m n1

r1

n1 1

· · ·r n

n n1

αβt1/β P m1, ,m n Q n1, ,n n φA m1, ,m n ψB n1, ,n n

m1· · · m n β n1· · · n n αt

k1· · · k n1/α r1· · · r n1/β

×

⎧

⎨

⎩

k1

m1 1

· · · k n

m n1

n

j1

k j − m j 1p m1, ,m n φa m1, ,m nβ

⎫

⎬

⎭

1/β

×

⎧

⎨

⎩

r1

n1 1

· · ·r n

n n1

n

j1

r j − n j 1q n1, ,n n ψb n1, ,n nα

⎫

⎬

⎭

1/α

.

2.24

Trang 9

Journal of Inequalities and Applications 9

Proof By the hypotheses, Jensen’s inequality, and H ¨older’s inequality, it is easy to observe

that

φA m1, ,m n φ

1

P m1, ,m n

m1

s1 1

· · ·m n

s n1

p s1, ,s n a s1, ,s n

≤ P 1

m1, ,m n

m1

s1 1

· · ·m n

s n1

p s1, ,s n φa s1, ,s n

≤ P 1

m1, ,m n

m1· · · m n1/α

m

1

s1 1

· · ·m n

s n1

p s1, ,s n φa s1, ,s nβ

1/β

,

2.25

ψB n1, ,n n ψ

1

Q n1, ,n n

n1

t1 1

· · ·n n

t n1

q t1, ,t n b t1, ,t n

≤ Q 1

n1, ,n n

n1

t1 1

· · ·n n

t n1

q t1, ,t n ψb t1, ,t n

≤ Q 1

n1, ,n n

n1· · · n n1/β

n

1

t1 1

· · ·n n

t n1

q t1, ,t n ψb t1, ,t nα

1/α

.

2.26

Proceeding now much as in the proof of Theorems2.1and2.3, and with suitable modifica-tions, 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 2.5reduces to the following result

Theorem C Let {a m }, {b n }, {p m }, {q n }, P m , Q n be as defined in Theorem B Define A m

1/P mm

s1 p s a s , and B n 1/Q nn

t1 q t b t 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

k

m1

r

n1

P m Q n φA m ψB n

1

2kr1/2

k

m1

k − m 1p m φa m2

1/2

×

r

n1

r − n 1q n ψb n2

1/2

.

2.27

This is the new inequality of Pachpatte in [ 1 , Theorem 4].

Remark 2.7 Taking j 1, t 1, p m 1, q n 1, α β 2, and n 1 inTheorem 2.5, and in view

ofP m m, Q n n, we obtain the following theorem.

Trang 10

Theorem D Let {am }, {b n } be as defined in Theorem A Define A m 1/mm s1 a s , and B n

1/nn

t1 b t , 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

k

m1

r

n1

mn

m n φA m ψB n ≤ 1

2kr1/2

k

m1

k − m 1φa m2

1/2

×

r

n1

r − n 1ψb n2

1/2

.

2.28

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 China20050392 Research

is partially supported by the Research Grants Council of the Hong Kong SAR, ChinaProject

no HKU7016/07P and a HKU Seed Grant forBasic Research

References

1 B G Pachpatte, “On some new inequalities similar to Hilbert’s inequality,” Journal of Mathematical

Analysis and Applications, vol 226, no 1, pp 166–179, 1998.

2 G D Handley, J J Koliha, and J E Peˇcari´c, “New Hilbert-Pachpatte type integral inequalities,”

Journal of Mathematical Analysis and Applications, vol 257, no 1, pp 238–250, 2001.

3 M Gao and B Yang, “On the extended Hilbert’s inequality,” Proceedings of the American Mathematical

Society, vol 126, no 3, pp 751–759, 1998.

4 K Jichang, “On new extensions of Hilbert’s integral inequality,” Journal of Mathematical Analysis and

Applications, vol 235, no 2, pp 608–614, 1999.

5 B Yang, “On new generalizations of Hilbert’s inequality,” Journal of Mathematical Analysis and

Applications, vol 248, no 1, pp 29–40, 2000.

6 W Zhong and B Yang, “On a multiple Hilbert-type integral inequality with the symmetric kernel,”

Journal of Inequalities and Applications, vol 2007, Article ID 27962, 17 pages, 2007.

7 C.-J Zhao, “Inverses of disperse and continuous Pachpatte’s inequalities,” Acta Mathematica Sinica,

vol 46, no 6, pp 1111–1116, 2003

8 C.-J Zhao, “Generalization on two new Hilbert type inequalities,” Journal of Mathematics, vol 20, no.

4, pp 413–416, 2000

9 C.-J Zhao and L Debnath, “Some new inverse type Hilbert integral inequalities,” Journal of

Mathematical Analysis and Applications, vol 262, no 1, pp 411–418, 2001.

10 G D Handley, J J Koliha, and J Peˇcari´c, “A Hilbert type inequality,” Tamkang Journal of Mathematics,

vol 31, no 4, pp 311–315, 2000

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

1934

12 J N´emeth, “Generalizations of the Hardy-Littlewood inequality,” Acta Scientiarum Mathematicarum,

vol 32, pp 295–299, 1971

13 E F Beckenbach and R Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N.

F., Band 30, Springer, Berlin, Germany, 1961

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