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On a New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane
Journal of Inequalities and Applications volume 2010, Article number: 256796 (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
If ,
and satisfy that
and
then we have [1]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ1_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ2_HTML.gif)
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.
If,
,
such that
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ4_HTML.gif)
Proof.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ5_HTML.gif)
Setting ,
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ6_HTML.gif)
we find that , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ7_HTML.gif)
The lemma is proved.
Lemma 2.2.
Define the weight functions as follow:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ8_HTML.gif)
then .
Proof.
We only prove that for
.
Using Lemma 2.1, setting and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ9_HTML.gif)
and the lemma is proved.
Lemma 2.3.
For and
define both functions
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ10_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ11_HTML.gif)
Proof.
Easily, we get the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ12_HTML.gif)
Let , using
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ13_HTML.gif)
we have that is an even function on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ14_HTML.gif)
Setting then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ15_HTML.gif)
where and we have
Similarly, The lemma is proved.
Lemma 2.4.
If is a nonnegative measurable function and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ16_HTML.gif)
Proof.
By Lemma 2.2, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ17_HTML.gif)
3. Main Results
Theorem 3.1.
If both functions,and
, are nonnegative measurable functions and satisfy
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ20_HTML.gif)
Hence, there exists a constant , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ21_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ24_HTML.gif)
For , by (3.7), using Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ25_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F256796/MediaObjects/13660_2010_Article_2102_Equ26_HTML.gif)
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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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DOI: https://doi.org/10.1155/2010/256796