- Research Article
- Open Access

# A Hilbert's Inequality with a Best Constant Factor

- Zheng Zeng
^{1}and - Zi-tian Xie
^{2}Email author

**2009**:820176

https://doi.org/10.1155/2009/820176

© Z. Zeng and Z.-t. Xie. 2009

**Received:**6 February 2009**Accepted:**23 July 2009**Published:**23 August 2009

## Abstract

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

## Keywords

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

## 1. Introduction

where the constant factors are all the best possible [1]. 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 ) [2–11].

In 2006, Yang gave an extension of [2] as follows.

In 2007 Xie gave a new Hilbert-type Inequality [3] as follows.

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

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.

Proof.

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

Lemma 2.2.

Proof.

The lemma is proved.

Lemma 2.3.

There for any ).

Proof.

Easily, had up bounded when

## 3. Main Results

Theorem 3.1.

Proof.

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 have , . This contracts the fact that .

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.