Open Access

On Inverse Hilbert-Type Inequalities

Journal of Inequalities and Applications20072008:693248

https://doi.org/10.1155/2008/693248

Received: 14 November 2007

Accepted: 4 December 2007

Published: 10 December 2007

Abstract

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.

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 Bicheng [2], 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 Debnath [11] 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 and . Let be positive sequences of real numbers defined for , where are natural numbers, define , and define . Then for , or , one has

(2.1)

Proof.

By using the following inequality (see [10, page 39]):

(2.2)

where , , and , we obtain that

(2.3)

thus

(2.4)

From inequality (2.4) and in view of the following mean inequality and inverse Hölder's inequality [10, page 24], we have

(2.5)
(2.6)

Taking the sum of both sides of (2.6) over from 1 to first and then using again inverse Hölder's inequality, we obtain that

(2.7)

This completes the proof.

Remark 2.2.

Taking to (2.1), (2.1) becomes

(2.8)

This is just an inverse form of the following inequality which was proven by Pachpatte [9]:

(2.9)

Theorem 2.3.

Let , and be as defined in Theorem 2.1. Let be positive sequences for Set . Let be real-valued nonnegative, concave, and supermultiplicative functions defined on Then,

(2.10)

where

(2.11)

Proof.

From the hypotheses and by Jensen's inequality, the means inequality, and inverse Hölder's inequality, we obtain that

(2.12)

Dividing both sides of (2.12) by and then taking the sum over from 1 to (and in view of inverse Hölder's inequality), we have

(2.13)

The proof is complete.

Remark 2.4.

Taking to (2.10), (2.10) becomes

(2.14)

where

(2.15)

This is just an inverse of the following inequality which was proven by Pachpatte [9]:

(2.16)

where

(2.17)

Similarly, the following theorem also can be established.

Theorem 2.5.

Let , and be as in Theorem 2.3 and define for . Let be real-valued, nonnegative, and concave functions defined on Then,

(2.18)

The proof of Theorem 2.5 can be completed by following the same steps as in the proof of Theorem 2.3 with suitable changes. Here, we omit the details.

Remark 2.6.

Taking to (2.18), (2.18) becomes

(2.19)

This is just an inverse of the following inequality which was proven by Pachpatte [9]:

(2.20)

Remark 2.7.

In view of L'Hôpital law, we have the following fact:

(2.21)

Accordingly, in the special case when , , and , let , then the inequality (2.18) reduces to the following inequality:

(2.22)

This is just a discrete form of the following inequality which was proven by Zhao and Debnath [11]:

(2.23)

Declarations

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.

Authors’ Affiliations

(1)
Department of Information and Mathematics Sciences, College of Science, China Jiliang University
(2)
Department of Mathematics, The University of Hong Kong

References

  1. Handley GD, Koliha JJ, Pečarić JE: New Hilbert-Pachpatte type integral inequalities. Journal of Mathematical Analysis and Applications 2001,257(1):238–250. 10.1006/jmaa.2000.7350MATHMathSciNetView ArticleGoogle Scholar
  2. Minzhe G, Bicheng Y: On the extended Hilbert's inequality. Proceedings of the American Mathematical Society 1998,126(3):751–759. 10.1090/S0002-9939-98-04444-XMathSciNetView ArticleGoogle Scholar
  3. Minzhe G: On Hilbert's inequality and its applications. Journal of Mathematical Analysis and Applications 1997,212(1):316–323. 10.1006/jmaa.1997.5490MathSciNetView ArticleGoogle Scholar
  4. Hu K: On Hilbert inequality and its application. Advances in Mathematics 1993,22(2):160–163.MATHMathSciNetGoogle Scholar
  5. Jichang K: On new extensions of Hilbert's integral inequality. Journal of Mathematical Analysis and Applications 1999,235(2):608–614. 10.1006/jmaa.1999.6373MATHMathSciNetView ArticleGoogle Scholar
  6. Bicheng Y: On new generalizations of Hilbert's inequality. Journal of Mathematical Analysis and Applications 2000,248(1):29–40. 10.1006/jmaa.2000.6860MATHMathSciNetView ArticleGoogle Scholar
  7. Zhao C-J: Inverses of disperse and continuous Pachpatte's inequalities. Acta Mathematica Sinica 2003,46(6):1111–1116.MATHMathSciNetGoogle Scholar
  8. Zhao C-J: Generalization on two new Hilbert type inequalities. Journal of Mathematics 2000,20(4):413–416.MATHMathSciNetGoogle Scholar
  9. Pachpatte BG: On some new inequalities similar to Hilbert's inequality. Journal of Mathematical Analysis and Applications 1998,226(1):166–179. 10.1006/jmaa.1998.6043MATHMathSciNetView ArticleGoogle Scholar
  10. Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1934.Google Scholar
  11. Zhao C-J, Debnath L: Some new inverse type Hilbert integral inequalities. Journal of Mathematical Analysis and Applications 2001,262(1):411–418. 10.1006/jmaa.2001.7595MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Z. Changjian and W.-S. Cheung. 2008

This article is published under license to BioMed Central Ltd. 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.