# On a New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane

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

**2010**:256796

https://doi.org/10.1155/2010/256796

© Zheng Zeng and Zitian Xie. 2010

**Received: **5 May 2010

**Accepted: **14 July 2010

**Published: **2 August 2010

## Abstract

By introducing some parameters and estimating the weight functions, we build a new Hilbert's inequality with the homogeneous kernel of 0 order and the integral in whole plane. The equivalent inequality and the reverse forms are considered. The best constant factor is calculated using Complex Analysis.

## 1. Introduction

where the constant factor is the best possible. Inequality (1.1) is well known as Hilbert's integral inequality, which has been extended by Hardy-Riesz as [2].

where the constant factor also is the best possible.

Both of them are important in Mathematical Analysis and its applications [3]. It attracts some attention in recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variations. Equation (1.1) has been strengthened by Yang and others (including double series inequalities) [4–21].

In 2008, Xie and Zeng gave a new Hilbert-type Inequality [4] as follows.

where the constant factor is the best possible.

The main purpose of this paper is to build a new Hilbert-type inequality with homogeneous kernel of degree 0, by estimating the weight function. The equivalent inequality is considered.

## 2. Some Lemmas

## 3. Main Results

Theorem 3.1.

Inequalities (3.1) and (3.2) are equivalent, and where the constant factors and are the best possibles.

Proof.

We claim that . In fact, if , then a.e. in which contradicts the fact that . In the same way, we claim that This is too a contradiction and hence by (2.13), we have (3.2).

Using (3.2), we have (3.1).

Inequalities (3.1) and (3.2) are equivalent.

Hence, we find For , it follows that , which contradicts the fact that . Hence the constant in (3.1) is the best possible.

Thus we complete the proof of the theorem.

Remark 3.2.

## Authors’ Affiliations

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