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# A Hilbert's Inequality with a Best Constant Factor

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 820176 (2009)

## Abstract

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

## 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 [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.

If , , such that , , then

is the Beta function.

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.

Define the weight coefficients as follows:

then

where

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.

We have

Easily, had up bounded when

## 3. Main Results

Theorem 3.1.

If , , , then

is defined by Lemma 2.1.

Proof.

By Hölder's inequality [12] and (2.5),

setting By(3.1) we have

By and (3.4) taking the form of strict inequality, we have (3.1). By Hölder's inequality[12], 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 ),

By (3.6), (3.7), (3.8), and Lemma 2.3, we have

We have , . This contracts the fact that .

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**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Cite this article

Zeng, Z., Xie, Z. A Hilbert's Inequality with a Best Constant Factor.
*J Inequal Appl* **2009, **820176 (2009). https://doi.org/10.1155/2009/820176

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### Keywords

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