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.
If , and satisfy that and then we have 
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 .
If , , such that and then we have the following Hardy-Hilbert's integral inequality:
where the constant factor also is the best possible.
Both of them are important in Mathematical Analysis and its applications . 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  as follows.
If,, such that and , then
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.
Setting , then
we find that , then
The lemma is proved.
Define the weight functions as follow:
We only prove that for .
Using Lemma 2.1, setting and,
and the lemma is proved.
For and define both functions as follows:
Easily, we get the following:
Let , using and
we have that is an even function on , then
where and we have
Similarly, The lemma is proved.
If is a nonnegative measurable function and , then
By Lemma 2.2, we find that
3. Main Results
If both functions,and, are nonnegative measurable functions and satisfy and , then
Inequalities (3.1) and (3.2) are equivalent, and where the constant factors and are the best possibles.
If (2.13) takes the form of equality for some , then there exists constants and , such that they are not all zero, and
Hence, there exists a constant , such that
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  and (3.2), we have the following:
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
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
For , by (3.7), using Lemma 2.3, we have
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.
For in (3.1), we have the following particular result:
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Xie, Z: A new reverse Hilbert-type inequality with a best constant factor. Journal of Mathematical Analysis and Applications. 343(2), 1154–1160 (2008). Publisher Full Text
Brnetić, I, Krnić, M, Pečarić, J: Multiple Hilbert and Hardy-Hilbert inequalities with non-conjugate parameters. Bulletin of the Australian Mathematical Society. 71(3), 447–457 (2005). Publisher Full Text