On a New Hilbert-Type Intergral Inequality with the Intergral in Whole Plane
© Zheng Zeng and Zitian Xie. 2010
Received: 5 May 2010
Accepted: 14 July 2010
Published: 2 August 2010
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.
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 .
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.
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.
2. Some Lemmas
We start by introducing some lemmas.
The lemma is proved.
and the lemma is proved.
3. Main Results
Using (3.2), we have (3.1).
Inequalities (3.1) and (3.2) are equivalent.
Thus we complete the proof of the theorem.
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