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

- Xingdong Liu
^{1}and - Bicheng Yang
^{2}Email author

**2010**:812636

https://doi.org/10.1155/2010/812636

© Xingdong Liu and Bicheng Yang. 2010

**Received: **7 September 2010

**Accepted: **26 October 2010

**Published: **27 October 2010

## 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.

## Keywords

## 1. Introduction

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

Theorem A.

where the constant factor is the best possible.

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

Theorem B.

where the constant factor is the best possible (cf. [1, Theorem 330]).

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

Theorem C.

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. [4–6]).

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.

Proof.

The lemma is proved.

Theorem 2.2.

Proof.

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

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

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

Theorem 3.1.

Proof.

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

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.

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

Example 3.3.

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

## Declarations

### 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).

## Authors’ Affiliations

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## Copyright

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