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

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

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)

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Correspondence to Zitian Xie.

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Zeng, Z., Xie, Z. On a New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane. J Inequal Appl 2010, 256796 (2010). https://doi.org/10.1155/2010/256796

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