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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 918534, 12 pages
http://dx.doi.org/10.1155/2008/918534
Research Article

The Generalizations of Hilbert's Inequality

1Guangzhou Sontan Polytechnic College, Guangzhou 511370, China
2Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

Received 24 June 2008; Accepted 2 October 2008

Academic Editor: Feng Qi

Copyright © 2008 Liubin Hua et al. 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.

Abstract

By introducing some parameters and estimating the weight functions, we establish a new Hilbert-type inequalities with best constant factors. The equivalent inequalities are also considered.

1. Introduction

If are real functions such that and , then we have (see [1])where the constant factor is the best possible. Inequality (1.1) is the well-known Hilbert's inequality. Inequality (1.1) had been generalized by Hardy-Riesz (see [2]) in 1925 as if are real functions such that and , thenwhere the constant factor is the best possible. When , (1.2) reduces to (1.1). Inequality (1.2) is named after Hardy-Hilbert's integral inequality, which is important in the analysis and its applications (see [3]), it has been studied and generalized in many directions by a number of mathematicians (see [48]).

Under the same condition of (1.2), we have Hardy-Hilbert's type inequality (see [1, Theorems 341 and 342]):where the constant factors 4 and are both the best possible.

Recently, Li et al. [9], by introducing the function , establish new inequalities similar to Hilbert-type inequality for integrals.

Theorem 1.1. If . Then, one haswhere the constant factor 4 is the best possible.

In this paper, we give further analogs of Hilbert-type inequality and its applications. The main result unifies and generalizes the classical results as follows.

Assume that , and If such that thenwhere the constant factoris the best possible.

2. Some Lemmas

Our results will be based on the following results. In the following lemmas, assume that , , , and

Lemma 2.1. Define the following weight functions:Then, where is defined by (1.1).

Proof. For let and we have
For first let and then let Thus, we haveHence, the lemma is proved.

Lemma 2.2. Assume that , then

Proof. Sinceit follows that there exists , such that if then . Moreover, there exists , such that if then . Using the expression of , if , thenTherefore, Lemma 2.2 is proved for . If , then we can replace the right-hand side of the first strict inequality above withBy the same way, we can show that the lemma is valid for . Hence, the lemma is proved.

Lemma 2.3. Assume that , then

Proof. Using introduced in the proof of Lemma 2.2, if , thenTherefore, Lemma 2.3 is proved for . If , then we can replace the right-hand side of the second strict inequality mentioned above withBy the same way, we can show that the lemma is valid for . Therefore, the lemma is proved.

3. Integral Case

In this section, we will state our main results.

Proposition 3.1. Assume that , and If such that thenwhere the constant factoris the best possible.

Proof. Using Hölder's inequality, we have
If (3.3) takes the form of equality, then there are constants and , such that they are not all zero, andHence,Therefore, there is a constant , such that We claim that . In fact, if , then , a.e., in , which contradicts the fact that . In the same way, we claim that . This is a contradiction. Hence, by (3.3), we have (3.1).
If the constant factor in (3.1) is not the best possible factor, then there exists a positive constant (with ), such that (3.1) is still valid if we replace with . For sufficiently small, construct the following functions:Thus, we obtainSetting , we haveSinceit follows thatHence,This contradicts the fact that . So the constant factor in (3.1) is the best possible. Then Proposition 3.1 is proved.

Proposition 3.2. Under the same assumptions of Proposition 3.1, one haswhere the constant factor is the best possible. Inequalities (3.1) and (3.13) are equivalent.

Proof. Settingby (3.1), we haveAs a result,Hence,Then, we have (3.13).
Conversely, by Hölder's inequality, we haveThen, by (3.13), we have (3.1). Hence inequalities (3.1) and (3.13) are equivalent.
If the constant factor in (3.13) is not the best possible, then by (3.18) we can get a contradiction that the constant factor in (3.1) is not the best possible. Hence Proposition 3.2 is proved.

Remark 3.3. In (3.1), let , and we have Hilbert's integral inequalityLet ; we have Hardy-Hilbert's classical inequalityLet , and we can combine the above two inequalities as follows:Let , and we can get the following inequality:

4. Discrete Case

We also give results for the discrete case.

Proposition 4.1. Assume that and . If , such thatthen one haswhere the constant factoris the best possible, and inequalities (4.2) and (4.3) are equivalent.

Proof. Define the following weight functions:then,where are defined in Lemma 2.1. By Hölder's inequality, we haveThe last strict inequality holds because both series and have positive terms. Thus, we have (4.2).
If the constant factor in (4.2) is not the best possible, then there exists a positive constant (with ), such that (4.2) is still valid if we replace by . For small enough, construct seriesThen, we havewhere and are defined in the proof of Proposition 3.1. Sinceit follows thatHence,This contradicts with the fact that . So the constant factor in (4.2) is the best possible one.
Settingwe findBy the same argument used in the proof of Proposition 3.2, we can show that (4.3) is valid, the constant factor in (4.3) is the best possible one and inequalities (4.2) and (4.3) are equivalent.

Remark 4.2. Proposition 4.1 is the corresponding series form of Propositions 3.1 and 3.2 for , and it is also a generalization of Hilbert's inequality. Here, we restrict the constants so that we can use the monotony of functions to obtain (4.6) and (4.9).

Acknowledgment

This work was partially supported by the National Natural Science Foundation of China (Grant no. 10871213).

References

  1. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 1934. View at Zentralblatt MATH
  2. G. H. Hardy, “Note on a theorem of Hilbert,” Mathematische Zeitschrift, vol. 6, no. 3-4, pp. 314–317, 1920. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, vol. 53 of Mathematics and Its Applications (East European Series), Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1991. View at Zentralblatt MATH · View at MathSciNet
  4. Y. C. Chow, “On inequalities of Hilbert and Widder,” Journal of the London Mathematical Society, vol. 14, no. 2, pp. 151–154, 1939. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. Gao, “On Hilbert's inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 212, no. 1, pp. 316–323, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. K. Jichang, “On new extensions of Hilbert's integral inequality,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 608–614, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. G. Pachpatte, “Inequalities similar to the integral analogue of Hilbert's inequality,” Tamkang Journal of Mathematics, vol. 30, no. 2, pp. 139–146, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. B. C. Yang, “An extension of Hardy-Hilbert's inequality,” Chinese Annals of Mathematics. Series A, vol. 23, no. 2, pp. 247–254, 2002 (Chinese). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. Li, Y. Qian, and B. He, “On further analogs of Hilbert's inequality,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 76329, 6 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet