# On a New Weighted Hilbert Inequality

- He Leping
^{1}, - Gao Xuemei
^{2, 3}and - Gao Mingzhe
^{2}Email author

**2008**:637397

**DOI: **10.1155/2008/637397

© He Leping et al. 2008

**Received: **13 January 2008

**Accepted: **18 May 2008

**Published: **20 May 2008

## Abstract

It is shown that a weighted Hilbert inequality for double series can be established by introducing a proper weight function. Thus, a quite sharp result of the classical Hilbert inequality for double series is obtained. And a similar result for the Hilbert integral inequality is also proved. Some applications are considered.

## 1. Introduction

where , and the constant factor in (1.2) is the best possible value. Recently, various improvements and extensions of (1.1) and (1.2) appear in a great deal of papers (see [2–6], etc.). The aim of the present paper is to build some new inequalities by using the weight function method and the technique of analysis, and then to study some applications of them.

First we give some lemmas.

Lemma 1.1.

Proof.

where is an arbitrary constant. This result is given in the paper (see [7]). Based on this indefinite integral it is easy to deduce that the equality (1.3) holds.

Lemma 1.2.

If and , where , then

- (2)(1.5)

- (1)
At first, notice that , hence we can write in the form of: , where and . It is obvious that the functions and are monotonously decreasing in . So, is also monotonously decreasing in . In the next place, notice that , therefore we can write in the form of: , where and . It is clear that the functions and are monotonously decreasing in . So, is also monotonously decreasing in .

- (2)Below, we need only to compute the first integral,(1.7)

By Lemma 1.1, we obtain the first integral of (1.5) at once after some simple computations and simplifications.

Similarly, the second integral of (1.5) can be gotten.

Lemma 1.3.

Proof.

Lemma 1.4.

Its proof is similar to that of Lemma 1.3. Hence, it is omitted.

## 2. Main Results

Theorem 2.1.

where the weight function is defined by (1.6).

Proof.

where the weight function is defined by (1.6).

And then by using the inequality (2.8), the inequality (2.1) follows from (2.9) at once. It is obvious that the inequality (2.1) is a refinement of (1.1). Below, we give an extension of (1.2).

Theorem 2.2.

Specially, when , it is a refinement of (1.2).

Proof.

Let be a real function, and .

We need only to compute the weight function .

where the weight function is defined by (2.11).

It follows from (2.19) and (2.20) that the inequality (2.10) is valid.

## 3. Applications

As applications, we will give some new refinements of Hardy-Littlewood's theorem and Widder's theorem below.

where is the best constant that the inequality (3.2) keeps valid.

Theorem 3.1.

where is defined by (1.6).

Proof.

By our assumptions, we may write in the form of:

It is known from (2.8) and (3.4) that the inequality (3.3) is valid. Theorem is therefore proved.

This is the famous Widder theorem (see [1]).

Theorem 3.2.

where is defined by (2.11).

Proof.

where . By using (2.19), the inequality (3.6) follows from (3.8) at once.

## Declarations

### Acknowledgment

This work is a project supported by Scientific Research Fund of Hunan Provincial Education Department (07C520 and 06C657).

## Authors’ Affiliations

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## Copyright

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