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

Journal of Inequalities and Applications20102010:256796

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

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.

Keywords

Weight FunctionMeasurable FunctionMathematical AnalysisComplex AnalysisConstant Factor

1. Introduction

If , and satisfy that and then we have [1]
(1.1)

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].

If , , such that and then we have the following Hardy-Hilbert's integral inequality:
(1.2)

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) [421].

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

If , , such that and , then
(1.3)

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.

In the following, we always suppose that: ,

2. Some Lemmas

We start by introducing some lemmas.

Lemma 2.1.

If , then
(2.1)

Proof.

We have
(2.2)
Setting , then
(2.3)
we find that , then
(2.4)

The lemma is proved.

Lemma 2.2.

Define the weight functions as follow:
(2.5)

then .

Proof.

We only prove that for .

Using Lemma 2.1, setting and ,
(2.6)

and the lemma is proved.

Lemma 2.3.

For and define both functions as follows:
(2.7)
then
(2.8)

Proof.

Easily, we get the following:
(2.9)
Let , using and
(2.10)
we have that is an even function on , then
(2.11)
Setting then
(2.12)

where and we have

Similarly, The lemma is proved.

Lemma 2.4.

If is a nonnegative measurable function and , then
(2.13)

Proof.

By Lemma 2.2, we find that
(2.14)

3. Main Results

Theorem 3.1.

If both functions, and , are nonnegative measurable functions and satisfy and , then
(3.1)
(3.2)

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

Proof.

If (2.13) takes the form of equality for some , then there exists constants and , such that they are not all zero, and
(3.3)
Hence, there exists a constant , such that
(3.4)

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).

By Hölder's inequality with weight [22] and (3.2), we have the following:
(3.5)

Using (3.2), we have (3.1).

Setting , then by (2.13), we have . If then (3.2) is proved. If by (3.1), we obtain
(3.6)

Inequalities (3.1) and (3.2) are equivalent.

If the constant factor in (3.1) is not the best possible, then there exists a positive (with ), such that
(3.7)
For , by (3.7), using Lemma 2.3, we have
(3.8)

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.

For in (3.1), we have the following particular result:
(3.9)

Authors’ Affiliations

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

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Copyright

© Zheng Zeng and Zitian Xie. 2010

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.

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