• Research Article
• Open Access

# A Hilbert's Inequality with a Best Constant Factor

Journal of Inequalities and Applications20092009:820176

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

• Received: 6 February 2009
• Accepted: 23 July 2009
• Published:

## 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

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

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

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

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.

Let then if then if ), then On the other hand, . Setting , then Similarly, .

Lemma 2.2.

For one has

Proof.

The lemma is proved.

Lemma 2.3.

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

There for any ).

Proof.

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

Theorem 3.1.

If , , , then

Proof.

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

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
For , setting , then
On the other hand and ),

We have , . This contracts the fact that .

## Authors’ Affiliations

(1)
Department of Mathematics, Shaoguan University, Shaoguan, Guangdong, 512005, China
(2)
Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong, 526061, China

## References 