• Research Article
• Open access
• Published:

# On a New Hilbert-Hardy-Type Integral Operator and Applications

## Abstract

By applying the way of weight functions and a Hardy's integral inequality, a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to Hilbert-type integral inequality is given, and two equivalent inequalities with the best constant factors as well as some particular examples are considered.

## 1. Introduction

In 1934, Hardy published the following theorem (cf. [1, Theorem 319]).

Theorem A.

If is a homogeneous function of degree in , , , and , then for , , and , one has

(1.1)

where the constant factor is the best possible.

Hardy [2] also published the following Hardy's integral inequality.

Theorem B.

If , , , and ; , , then one has

(1.2)

where the constant factor is the best possible (cf. [1, Theoremâ€‰â€‰330]).

In 2009, Yang [3] published the following theorem.

Theorem C.

If , , , is a homogeneous function of degree in , and for any , , then for , , , , , and , we have

(1.3)

where the constant factor is the best possible.

For, , (1.3) reduces to (1.1). We name of (1.1) and (1.3) Hilbert-type integral inequalities. Inequalities (1.1), (1.2) and (1.3) are important in analysis and its applications (cf. [4â€“6]).

Setting , , , , by applying (1.2) (for ), Das and Sahoo gave a new integral inequality similar to Pachpatte's inequality (cf. [7, 8]) as follows:

(1.4)

where the constant factor is the best possible (cf. [9]). Sulaiman [10] also considered a Hilbert-Hardy-type integral inequality similar to (1.4) with the kernel , , . But he cannot show that the constant factor in the new inequality is the best possible.

In this paper, by applying the way of weight functions and inequality (1.2) for , a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to (1.3) is given, and two equivalent inequalities with a best constant factor as well as some particular examples are considered.

## 2. A Lemma and Two Equivalent Inequalities

Lemma 2.1.

If , is a nonnegative homogeneous function of degree in with , and for any , , then and

(2.1)

Proof.

Setting , we find

(2.2)

There exists , satisfying and . Since we find

(2.3)

there exists , such that , and then

(2.4)

The lemma is proved.

Theorem 2.2.

If , , , is a homogeneous function of degree in , and for any , , then for , , ,

(2.5)

, and , one has the following equivalent inequalities:

(2.6)
(2.7)

Proof.

Setting the weight functions and as follows:

(2.8)

then by Lemma 2.1, we find

(2.9)

By HÃ¶lder's inequality (cf. [11]) and (2.8), (2.9), we obtain

(2.10)

Then by Fubini theorem (cf. [12]), it follows:

(2.11)

Since , , then by (1.2) (for , we have

(2.12)

Hence by (2.11), we have (2.7). Still by HÃ¶lder's inequality, we find

(2.13)

Then by (2.7), we have (2.6).

On the other-hand, supposing that (2.6) is valid, by (2.11) and (1.2) (for , it follows . If , then (2.7) is naturally valid; if , setting

(2.14)

then by (2.6), we find

(2.15)

Hence, we have (2.7), which is equivalent to (2.6).

## 3. A Hilbert-Hardy-Type Integral Operator and Applications

Setting a real function space as follows:

(3.1)

for , , define an integral operator as follows:

(3.2)

Then, by (2.7), , and is bounded with

(3.3)

Theorem 3.1.

Let the assumptions of Theorem 2.2 be fulfilled, and additionally setting . Then one has

(3.4)

where the constant factor is the best possible. Moreover the constant factor in (2.6) and (2.7) is the best possible and then

(3.5)

Proof.

Since , by (1.2), for , it follows:

(3.6)

Then, by (2.6), we have (3.4).

For , setting as follows:

(3.7)

then for , , we find

(3.8)

where , , and are indicated as follows;

(3.9)

If there exists a positive constant , such that (3.4) is still valid as we replace by , then in particular, we find

(3.10)

By (3.8) and (3.10), we find

(3.11)

Since by Fubini theorem, we obtain

(3.12)

then for in (3.10), by Lemma 2.1, we obtain . Hence , and then is the best value of (3.4).

We conclude that the constant factor in (2.6) is the best possible, otherwise we can get a contradiction by (1.2) that the constant factor in (3.4) is not the best possible. By the same way, if the constant factor in (2.7) is not the best possible, then by (2.13), we can get a contradiction that the constant factor in (2.6) is not the best possible. Therefore in view of (3.3), we have (3.5).

Corollary 3.2.

For , , , , , in (2.6), (2.7) and (3.4), one has the following basic Hilbert-Hardy-type integral inequalities with the best constant factors:

(3.13)
(3.14)
(3.15)

where , and (3.13) is equivalent to (3.14).

Example 3.3.

For , , , , and in(3.4),

(a)if , , and , then we obtain the following integral inequalities:

(3.16)

(b)if, , then we have

(3.17)

(c)if , then we find

(3.18)

where the constant factors in the above inequalities are the best possible.

## References

1. Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.

2. Hardy GH: Note on some points in the integral calculus (LXIV). Messenger of Math 1928, 57: 12â€“16.

3. Yang B: A survey of the study of Hilbert-type inequalities with parameters. Advances in Math 2009, 38(3):257â€“268.

4. Mintrinovic DS, Pecaric JE, Kink AM: Inequalities Involving Functions and their Integrals and Derivertives. Kluwer Academic Publishers, Boston, Mass, USA; 1991.

5. Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science, Beijin, China; 2009.

6. Yang B: On the norm of a Hilbert's type linear operator and applications. Journal of Mathematical Analysis and Applications 2007, 325(1):529â€“541. 10.1016/j.jmaa.2006.02.006

7. Pachpatte BG: On some new inequalities similar to Hilbert's inequality. Journal of Mathematical Analysis and Applications 1998, 226(3):166â€“179.

8. Pachpatte BG: Inequalities similar to certain extensions of Hilbert's inequality. Journal of Mathematical Analysis and Applications 2000, 243(2):217â€“227. 10.1006/jmaa.1999.6646

9. Das N, Sahoo S: New inequalities similar to Hardy-Hilbert's inequality. Turkish Journal of Mathematics 2009, 33: 1â€“13.

10. Sulaiman WT: On three inequalities similar to Hardy-Hilbert's integral inequality. Acta Mathematica Universitatis Comenianae 2007, 76(2):273â€“278.

11. Kuang J: Applied Inequalities. Shangdong Science Technic, Jinan, China; 2004.

12. Kuang J: Introduction to Real Analysis. Hunan Education, Changsha, China; 1996.

## Acknowledgments

This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026) and the Guangdong Natural Science Foundation (no. 7004344).

## Author information

Authors

### Corresponding author

Correspondence to Bicheng Yang.

## Rights and permissions

Reprints and permissions

Liu, X., Yang, B. On a New Hilbert-Hardy-Type Integral Operator and Applications. J Inequal Appl 2010, 812636 (2010). https://doi.org/10.1155/2010/812636