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On a Multiple Hilbert's Inequality with Parameters
Journal of Inequalities and Applications volume 2010, Article number: 309319 (2010)
Abstract
By introducing multiparameters and conjugate exponents and using Hadamard's inequality and the way of real analysis, we estimate the weight coefficients and give a multiple more accurate Hilbert's inequality, which is an extension of some published results. We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.
1. Introduction
In 1908, Weyl published the following famous Hilbert's inequality (cf. [1]). If , and then
where the constant factor is the best possible. In 1934, Hardy proved the following more accurate Hilbert's inequality (cf. [2]):
where the constant factor is the best possible. For the equivalent forms of (1.1) and (1.2) are given as follows (cf. [2]):
where the constant factor is the best possible. Inequalities (1.1)–(1.4) are important in analysis and their applications (cf. [3]). In near one century, there are many improvements, generalizations and, applications of (1.1)–(1.4) in numerous literatures and monographs of mathematics (cf. [2–18]). Yang and Huang also considered the multiple Hilberttype integral inequality (cf. [19, 20]). Recently, Yang summarized the methods of introducing parameters and estimating the weight coefficients to extend Hilberttype inequalities for the past 100 years. Some representative results are as follows (cf. [21, 22]):
(i)if , , then

(ii)
if ,,, then
The constant factors in the above five inequalities are all the best possible. Inequalities (1.5) and (1.7) are generalizations of inequality (1.2), and inequality (1.9) is a multiple extension of (1.1). Inequalities (1.6) and (1.8) are the equivalent forms of (1.5) and (1.7), which are extensions of (1.4).
In this paper, by introducing multiparameters and conjugate exponents and using Hadamard's inequality, we estimate the weight coefficients and give a multiple more accurate Hilbert 's inequality, which is an extension of inequalities (1.5), (1.7), and (1.9). We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.
2. Some Lemmas
Lemma 2.1.
If , , , , , then
Proof.
We find the following:
and then (2.1) is valid.
Lemma 2.2.
If ,, ,,, then
Proof.
For fixedwe set
In virtue of and we find , Putting we have the following:
Since by the following Hadamard's inequality (cf. [5]):
it follows that
and then we have the righthand side of (2.3). Since
and is strictly decreasing in , we get
Hence, we prove that the lefthand side of (2.3) is valid.
Lemma 2.3.
As the assumption of Lemma 2.1, define the weight coefficients as
, then there exists such that
Moreover, for any it follows that
Proof.
We prove (2.11) by mathematical induction. For we set and satisfying Putting ,, we have the following:
and then (2.11) is valid by using inequality (2.3).
Assuming that for (2.11) is valid, then for setting , by (2.3), we have the following:
Setting , , we find , By the assumption of induction, it follows that
where and
Setting by (2.16), we have the following:
and then by (2.15), (2.18), and mathematical induction, (2.11) is valid. Setting ,,,,, then we have the following:
Hence, (2.12) is valid.
3. Main Results
Theorem 3.1.
Suppose that ,, ,, , , , , such that
then one has the following equivalent inequalities:
Proof.
Since by (2.1) and Hölder's inequality (cf. [5]), we find that
For since by Hölder's inequality again in (3.5), we have the following:
Note that for by (3.5), we directly get (3.6). Hence, (3.3) is valid by (3.6) and (2.12).
Since by Hölder's inequality once again, it follows that
By (3.3), we have (3.2). On the other hand, assuming that (3.2) is valid, setting
then we find that
By (3.2), it follows that If then (3.3) is naturally valid. Suppose that by (3.2), we find that
Dividing out into two sides of (3.10), we have the following:
Then (3.3) is valid, which is equivalent to (3.2).
Theorem 3.2.
Let the assumptions of Theorem 3.1 be fulfilled, then the same constant factor in (3.2) and (3.3) is the best possible.
Proof.
By (2.11) and
there exists such that for ,
where Setting
we find that
If there exists a constant such that (3.2) is still valid as we replace by then in particular, we have the following:
In virtue of (3.15) and (3.16), it follows that
For , we have Hence, is the best value of (3.2).
We conform that the constant factor in (3.3) is the best possible, otherwise we can get a contradiction by (3.7) that the constant factor in (3.2) is not the best possible.
Remarks 3.3 ..

(i)
When the assumption of two theorems becomes (ii) When , (3.2) reduces to (1.9). (iii) For , setting ,in (3.2), then we obtain (1.5). Setting , in (3.2), we get (1.7).
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Acknowledgments
This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026), and Guangdong Natural Science Foundation (no. 7004344).
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Huang, Q. On a Multiple Hilbert's Inequality with Parameters. J Inequal Appl 2010, 309319 (2010). https://doi.org/10.1155/2010/309319
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DOI: https://doi.org/10.1155/2010/309319
Keywords
 Constant Factor
 Weight Coefficient
 Equivalent Form
 Representative Result
 Integral Inequality