# 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.

### Keywords

Weight Function Measurable Function Mathematical Analysis Complex Analysis Constant Factor## 1. Introduction

where the constant factor Open image in new window 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 Open image in new window 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, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

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

where the constant factor Open image in new window 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: Open image in new window Open image in new window , Open image in new window

## 2. Some Lemmas

We start by introducing some lemmas.

Lemma 2.1.

Proof.

The lemma is proved.

Lemma 2.2.

then Open image in new window .

Proof.

We only prove that Open image in new window for Open image in new window .

and the lemma is proved.

Lemma 2.3.

Proof.

where Open image in new window and we have Open image in new window

Similarly, Open image in new window The lemma is proved.

Lemma 2.4.

Proof.

## 3. Main Results

Theorem 3.1.

Inequalities (3.1) and (3.2) are equivalent, and where the constant factors Open image in new window and Open image in new window are the best possibles.

Proof.

We claim that Open image in new window . In fact, if Open image in new window , then Open image in new window a.e. in Open image in new window which contradicts the fact that Open image in new window . In the same way, we claim that Open image in new window 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 Open image in new window For Open image in new window , it follows that Open image in new window , which contradicts the fact that Open image in new window . Hence the constant Open image in new window in (3.1) is the best possible.

Thus we complete the proof of the theorem.

Remark 3.2.

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