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Volume 2008, Article ID 693248, 6 pagesdoi:10.1155/2008/693248 Research Article On Inverse Hilbert-Type Inequalities Zhao Changjian 1 and Wing-Sum Cheung 2 1 Department of Information an

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Volume 2008, Article ID 693248, 6 pages

doi:10.1155/2008/693248

Research Article

On Inverse Hilbert-Type Inequalities

Zhao Changjian 1 and Wing-Sum Cheung 2

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 Zhao Changjian, chjzhao@163.com

Received 14 November 2007; Revised 1 December 2007; Accepted 4 December 2007

Recommended by Martin J Bohner

This paper deals with new inverse-type Hilbert inequalities Our results in special cases yield some

of the recent results and provide some new estimates on such types of inequalities.

Copyright q 2008 Z Changjian 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

Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities and their various generalizations by several authors including Handley et al.1, Minzhe and Bicheng2, Minzhe 3, Hu 4, Jichang 5, Bicheng 6, and Zhao 7,8 In 1998, Pachpatte

9 gave some new integral inequalities similar to Hilbert inequality see 10, page 226 In

2000, Zhao and Debnath11 established some inverse-type inequalities of the above integral inequalities This paper deals with some new inverse-type Hilbert inequalities which provide some new estimates on such types of inequalities

2 Main results

Theorem 2.1 Let 0 < p i ≤ 1 i 1, , n and r ≤ 0 Let {a i,m i } be n positive sequences of real numbers defined for m i 1, 2, , k i , where k i i 1, , n are natural numbers, define A i,m i

m i

s i1a i,s i , and define A i,0 0 Then for p−1 q−1 1, p < 0 or 0 < p < 1, one has

k1

m1 1

· · · k n

m n1

n

i1A p i

i,m i

1/nn

i1m r in/ pr ≥

n

i1

p i k 1/p i

k

i

m i1

k i − m i 1a i,m i A p i−1

i,m i

q

1/q

. 2.1

Trang 2

Proof By using the following inequalitysee 10, page 39:

h i a h i−1

i,m i

a i,m i − b i,m i

≤ a h i

i,m i − b h i

i,m i ≤ h i b h i−1

i,m i

a i,m i − b i,m i

where a i,m i > 0, b i,m i > 0, and 0 ≤ h i ≤ 1 i 1, 2, , n, we obtain that

A p i

i,m i1− A p i

i,m i ≥ p i A p i−1

i,m i1

A i,m i1− A i,m i

 p i a i,m i1A p i−1

i,m i1,

ki−1

m i0

A p i

i,m i1− A p i

i,m i A p i

i,k i≥ ki−1

m i0

p i a i,m i1A p i−1

i,m i1 p i

k i

m i1

a i,m i A p i−1

i,m i ,

2.3

thus

A p i

i,m i ≥ p i

m i

s i1

a i,s i A p i−1

From inequality2.4 and in view of the following mean inequality and inverse H¨older’s in-equality10, page 24, we have

n

i1

m 1/n i ≥

1

n

i1

m r i

1/r

n

i1A p i

i,m i

1/nn

i1m r in/ pr ≥

n

i1

p i

m

i

s i1

a i,s i A p i−1

i,s i

q

1/q

Taking the sum of both sides of2.6 over m i from 1 to k i 1, 2, , n first and then using again

inverse H ¨older’s inequality, we obtain that

k1

m1 1

· · · k n

m n1

n

i1A p i

i,m i

1/nn

i1m r in/ pr ≥

n

i1

p i

k

i

m i1

m

i

s i1

a i,s i A p i−1

i,s i

q 1/q

≥n

i1

p i k 1/p i

k

i

m i1

m i

s i1

a i,s i A p i−1

i,s i

q 1/q

n

i1

p i k 1/p i

k

i

s i1

k i − s i 1a i,s i A p i−1

i,s i

q 1/q

n

i1

p i k 1/p i

k

i

m i1

k i − m i 1a i,m i A p i−1

i,m i

q

1/q

.

2.7

This completes the proof

Remark 2.2 Taking n 2, q −2, r −1 to 2.1, 2.1 becomes

k1

m1 1

k2

m2 1

A p1

1,m1A p2

2,m2

m−11  m−1

2

−3 ≥ 8p1p2

k1k2

3/2

k

1

m1 1

k1− m1 1a 1,m1A p1 −1

1,m1

−2 −1/2

×

k

2

m1

k2− m2 1a 2,m2A p2 −1

2,m2

−2 −1/2

.

2.8

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This is just an inverse form of the following inequality which was proven by Pachpatte

9:

k

m1

r

n1

A p m B q n

m n ≤

1

2pqkr 1/2

k

m1

k − m 1a m A p m−12 1/2r

n1

r − n 1b n B n q−12 1/2

.

2.9

Theorem 2.3 Let {a i,m i }, A i,m i , k i , p, and q be as defined in Theorem 2.1 Let {p i,m i } be n positive sequences for m i 1, 2, , k i i 1, 2, , n Set P i,m i m i

s i1p i,s i i 1, 2, , n Let φ i i

1, 2, , n be n real-valued nonnegative, concave, and supermultiplicative functions defined on R

0, ∞ Then,

k1

m1 1

· · ·k n

m n1

n

i1φ i

A i,m i

1/nn

i1m r in/ pr ≥Mk1, k2, , k nn

i1

k

i

m i1

k i − m i 1 p i,m i φ i a i,m i

p i,m i

q 1/q

,

2.10

where

M

k1, k2, , k n

n

i1

k

i

m i1

φ i

P i,m i

p 1/p

Proof From the hypotheses and by Jensen’s inequality, the means inequality, and inverse

H ¨older’s inequality, we obtain that

n

i1

φ i

A i,m i

n

i1

φ i

P i,m i

m i

s i1p i,s i

a i,s i /p i,s i

m i

s i1p i,s i

≥n

i1

φ i

P i,m i

φ i

m i

s i1p i,s i

a i,s i /p i,s i

m i

s i1p i,s i

≥n

i1

φ i

P i,m i

m i

s i1

p i,s i φ i

a i,s i

p i,s i

≥n

i1

φ i

P i,m i

m 1/p i

m

i

s i1

p i,s i φ i

a i,s i

p i,s i

q 1/q

≥

1

n

i1

m r i

n/ pr n

i1

φ i P i,m i

P i,m i

m

i

s i1

p i,s i φ i a i,s i

p i,s i

q 1/q

.

2.12 Dividing both sides of 2.12 by 1/nn

i1m r

in/ pr and then taking the sum over m i i 

1, 2, , n from 1 to k iand in view of inverse H¨older’s inequality, we have

k1

m1 1

· · · k n

m n1

n

i1φ i

A i,m i

1/nn

i1m r in/ pr ≥n

i1

k

i

m i1

φ i

P i,m i

m

i

s i1

p i,s i φ i a i,s i

p i,s i

q 1/q

≥n

i1

k

i

m i1

φ i

P i,m i

p 1/p k

i

m i1

m i

s i1

p i,s i φ i

a i,s i

p i,s i

q 1/q

 Mk1, k2, , k n

n

i1

k

i

m i1

m i

s i1

p i,s i φ i

a i,s i

p i,s i

q 1/q

 Mk1, k2, , k n

n

i1

k

i

m i1

k i −m i1 p i,m i φ i

a i,m i

p i,m i

q 1/q

.

2.13 The proof is complete

Trang 4

k1

m1 1

k2

m2 1

φ1

A 1,m1

φ2

A 2,m2

m−11  m−1

2

−3 ≥ Mk1, k2

k1

m1 1

k1− m1 1 p 1,m1φ1

a 1,m1

p 1,m1

−2 −1/2

×

k

2

m2 1

k2− m2 1 p 2,m2φ2

a 2,m2

p 2,m2

−2 −1/2

,

2.14

where

M

k1, k2

8

k

1

m1 1

φ1

P 1,m1

2/3 3/2 k

2

m2 1

φ2

P 2,m2

2/3 3/2

This is just an inverse of the following inequality which was proven by Pachpatte9:

k

m1

r

n1

φ

A m

ψ

B 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 ψ b n

q n

2 1/2

,

2.16

where

Mk, r  1

2

k

m1

φ

P m

2 1/2r

n1

ψ

Q n

2 1/2

Similarly, the following theorem also can be established

Theorem 2.5 Let P i,m i , {a i,m i }, {p i,m i }, k i , p, and q be as in Theorem 2.3 and define A i,m i

1/P i,m im i

s i1p i,s i a i,s i , for m i 1, 2, , k i Let φ i i 1, 2, , n be n real-valued, nonnegative, and concave functions defined on R.Then,

k1

m1 1

· · · k n

m n1

n

i1P i,m i φ i

A i,m i

1/nn

i1m r i

n/ pr ≥

n

i1

k i 1/p

k

i

m i1

k i − m i 1p i,m i φ i

a i,m i

q

1/q

. 2.18

The proof ofTheorem 2.5can be completed by following the same steps as in the proof

ofTheorem 2.3with suitable changes Here, we omit the details

k1

m1 1

k2

m2 1

P 1,m1P 2,m2φ1

A 1,m1

φ2

A 2,m2

m−11  m−1

2

−3

≥ 8k1k2

3/2k1

m1 1

k1−m11p 1,m1φ1

a 1,m1

−2 −1/2k2

m2 1

k2−m21p 2,m2φ2

a 2,m2

−2 −1/2

.

2.19

Trang 5

This is just an inverse of the following inequality which was proven by Pachpatte9:

k

m1

r

n1

P m Q n φ

A m

ψ

B n

m n

≤ 1

2kr 1/2

k

m1

k − m 1p m φ

a m

2

1/2r

n1

r − n 1q n ψ

b n

2

1/2

.

2.20

Remark 2.7 In view of L’H ˆopital law, we have the following fact:

lim

r→0

1

n

i1

m r i

n/ pr

exp n

plimr→0

ln

1/nn

i1m r i

r

exp n

plimr→0

n

i1m r i ln m i

n

i1m r i

m1·m2· · · · ·m n

1/p

.

2.21

Accordingly, in the special case when n 2, p 0.1, and p i,m i 1, let r → 0, then the

inequality2.18 reduces to the following inequality:

k1

m1 1

k2

m2 1

φ1

A 1,m1

φ2

A 2,m2

m1m2

−2

≥k1k2

−1k1

m1 1

k1− m11φ1

a 1,m1

1/2 2k2

m2 1

k2−m21φ2

a 2,m2

1/2 2

.

2.22

This is just a discrete form of the following inequality which was proven by Zhao and Debnath

11:

x

0

y

0

φ

F sψ

Gt

st−2 ds dt ≥ xy

−1 x 0

x − sφ

fs1/2

ds

2 y

0

y − tφ

gt1/2

dt

2

.

2.23

Acknowledgments

The authors cordially thank the anonymous referee for his/her valuable comments which lead

to the improvement of this paper Research is supported by Zhejiang Provincial Natural Science Foundation of China, Grant no Y605065, Foundation of the Education Department of Zhejiang Province of China, Grant no 20050392, partially supported by the Research Grants Council of the Hong Kong SAR, China, Project no HKU7016/07P

References

1 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.

2 G Minzhe and Y Bicheng, “On the extended Hilbert’s inequality,” Proceedings of the American

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

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3 G Minzhe, “On Hilbert’s inequality and its applications,” Journal of Mathematical Analysis and

Applica-tions, vol 212, no 1, pp 316–323, 1997.

4 K Hu, “On Hilbert inequality and its application,” Advances in Mathematics, vol 22, no 2, pp 160–163,

1993.

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

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

6 Y Bicheng, “On new generalizations of Hilbert’s inequality,” Journal of Mathematical Analysis and

Ap-plications, vol 248, no 1, pp 29–40, 2000.

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 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.

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

2nd edition, 1934.

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

Mathemat-ical Analysis and Applications, vol 262, no 1, pp 411–418, 2001.

3 G Minzhe, ? ?On Hilbert’s inequality and its applications,” Journal of Mathematical Analysis and. .. 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, ... edition, 1934.

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

Mathemat-ical Analysis and

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