# A Hilbert's Inequality with a Best Constant Factor

## Abstract

We give a new Hilbert's inequality with a best constant factor and some parameters.

## 1. Introduction

If , , such that and , then the well-known Hardy-Hilbert's inequality and its equivalent form are given by (1.1) (1.2)

where the constant factors are all the best possible . It attracted some attention in the recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variants. Equation (1.1) has been strengthened by Yang and others ( including integral inequalities ) .

In 2006, Yang gave an extension of  as follows.

If , , such that , , then (1.3) is the Beta function.

In 2007 Xie gave a new Hilbert-type Inequality  as follows.

If , and the right of the following inequalities converges to some positive numbers, then (1.4)

The main objective of this paper is to build a new Hilbert's inequality with a best constant factor and some parameters.

In the following, we always suppose that

(1)  , , ,

(2)both functions and are differentiable and strict increasing in and respectively,

(3) are strictly increasing in and respectively. is strict decreasing on and ,

(4) ## 2. Some Lemmas

Lemma 2.1.

Define the weight coefficients as follows: (2.1) (2.2) (2.3) (2.4)

then (2.5)

where (2.6)

Proof.

Let then if then  (2.7)

if ), then  (2.8)

On the other hand, . Setting , then Similarly, .

Lemma 2.2.

For one has (2.9)

Proof. (2.10)

The lemma is proved.

Lemma 2.3.

Setting (or and (or , resp.), then is strictly decreasing, then (2.11)

There for any ).

Proof.

We have (2.12)

Easily, had up bounded when ## 3. Main Results

Theorem 3.1.

If , , , then (3.1) (3.2) is defined by Lemma 2.1.

Proof.

By Hölder's inequality  and (2.5), (3.3)

setting By(3.1) we have (3.4) By and (3.4) taking the form of strict inequality, we have (3.1). By Hölder's inequality, we have (3.5)

as . By (3.2), (3.5) taking the form of strict inequality, we have (3.1).

Theorem 3.2.

If , then both constant factors, and of (3.1) and (3.2), are the best possible.

Proof.

We only prove that is the best possible. If the constant factor in (3.1) is not the best possible, then there exists a positive (with ), such that (3.6)

For , setting , then (3.7)

On the other hand and ), (3.8)

By (3.6), (3.7), (3.8), and Lemma 2.3, we have (3.9) (3.10)

We have , . This contracts the fact that .

## References

1. 1.

Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1952:xii+324.

2. 2.

Yang BC: On Hilbert's inequality with some parameters. Acta Mathematica Sinica. Chinese Series 2006,49(5):1121–1126.

3. 3.

Xie Z: A new Hilbert-type inequality with the kernel of --homogeneous. Journal of Jilin University. Science Edition 2007,45(3):369–373.

4. 4.

Xie Z, Yang B: A new Hilbert-type integral inequality with some parameters and its reverse. Kyungpook Mathematical Journal 2008,48(1):93–100.

5. 5.

Yang B: A Hilbert-type inequality with a mixed kernel and extensions. Journal of Sichuan Normal University. Natural Science 2008,31(3):281–284.

6. 6.

Xie Z, Zeng Z: A Hilbert-type integral with parameters. Journal of Xiangtan University. Natural Science 2007,29(3):24–28.

7. 7.

Wenjie W, Leping H, Tieling C: On an improvenment of Hardy-Hilbert's type inequality with some parameters. Journal of Xiangtan University. Natural Science 2008,30(2):12–14.

8. 8.

Xie Z: A new reverse Hilbert-type inequality with a best constant factor. Journal of Mathematical Analysis and Applications 2008,343(2):1154–1160. 10.1016/j.jmaa.2008.02.007

9. 9.

Yang B: On an extended Hardy-Hilbert's inequality and some reversed form. International Mathematical Forum 2006,1(37–40):1905–1912.

10. 10.

Xie Z: A Hilbert-type inequality with the kernel of irrational expression. Mathematics in Practice and Theory 2008,38(16):128–133.

11. 11.

Xie Z, Rong JM: A new Hilbert-type inequality with some parameters. Journal of South China Normal University. Natural Science Edition 2008,120(2):38–42.

12. 12.

Kang J: Applied Inequalities. Shangdong Science and Technology Press, Jinan, China; 2004.

## Author information

Authors

### Corresponding author

Correspondence to Zi-tian Xie.

## Rights and permissions

Reprints and Permissions

Zeng, Z., Xie, Z. A Hilbert's Inequality with a Best Constant Factor. J Inequal Appl 2009, 820176 (2009). https://doi.org/10.1155/2009/820176

• Revised:

• Accepted:

• Published:

### Keywords

• Constant Factor
• Weight Coefficient
• Equivalent Form
• Strict Inequality
• Integral Inequality 