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A Hilbert's Inequality with a Best Constant Factor
Journal of Inequalities and Applications volume 2009, Article number: 820176 (2009)
We give a new Hilbert's inequality with a best constant factor and some parameters.
If , , such that and , then the well-known Hardy-Hilbert's inequality and its equivalent form are given by
where the constant factors are all the best possible . It attracted some attention in the recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variants. Equation (1.1) has been strengthened by Yang and others ( including integral inequalities ) [2–11].
In 2006, Yang gave an extension of  as follows.
If , , such that , , then
is the Beta function.
In 2007 Xie gave a new Hilbert-type Inequality  as follows.
If , and the right of the following inequalities converges to some positive numbers, then
The main objective of this paper is to build a new Hilbert's inequality with a best constant factor and some parameters.
In the following, we always suppose that
(1), , ,
(2)both functions and are differentiable and strict increasing in and respectively,
(3) are strictly increasing in and respectively. is strict decreasing on and ,
2. Some Lemmas
Define the weight coefficients as follows:
Let then if then
if ), then
On the other hand, . Setting , then Similarly, .
For one has
The lemma is proved.
Setting (or and (or , resp.), then is strictly decreasing, then
There for any ).
Easily, had up bounded when
3. Main Results
If , , , then
is defined by Lemma 2.1.
By Hölder's inequality  and (2.5),
setting By(3.1) we have
By and (3.4) taking the form of strict inequality, we have (3.1). By Hölder's inequality, we have
as . By (3.2), (3.5) taking the form of strict inequality, we have (3.1).
If , then both constant factors, and of (3.1) and (3.2), are the best possible.
We only prove that is the best possible. If the constant factor in (3.1) is not the best possible, then there exists a positive (with ), such that
For , setting , then
On the other hand and ),
By (3.6), (3.7), (3.8), and Lemma 2.3, we have
We have , . This contracts the fact that .
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Zeng, Z., Xie, Z. A Hilbert's Inequality with a Best Constant Factor. J Inequal Appl 2009, 820176 (2009). https://doi.org/10.1155/2009/820176
- Constant Factor
- Weight Coefficient
- Equivalent Form
- Strict Inequality
- Integral Inequality