International Journal of Mathematics and Mathematical Sciences

Volume 2008, Article ID 918534, 12 pages

http://dx.doi.org/10.1155/2008/918534

## The Generalizations of Hilbert's Inequality

^{1}Guangzhou Sontan Polytechnic College, Guangzhou 511370, China^{2}Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

Received 24 June 2008; Accepted 2 October 2008

Academic Editor: Feng Qi

Copyright © 2008 Liubin Hua et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By introducing some parameters and estimating the weight functions, we establish a new Hilbert-type inequalities with best constant factors. The equivalent inequalities are also considered.

#### 1. Introduction

If are real functions such that and , then we have (see [1])where the constant factor is the best possible. Inequality (1.1) is the well-known Hilbert's inequality. Inequality (1.1) had been generalized by Hardy-Riesz (see [2]) in 1925 as if are real functions such that and , thenwhere the constant factor is the best possible. When , (1.2) reduces to (1.1). Inequality (1.2) is named after Hardy-Hilbert's integral inequality, which is important in the analysis and its applications (see [3]), it has been studied and generalized in many directions by a number of mathematicians (see [4–8]).

Under the same condition of (1.2), we have Hardy-Hilbert's type inequality (see [1, Theorems 341 and 342]):where the constant factors 4 and are both the best possible.

Recently, Li et al. [9], by introducing the function , establish new inequalities similar to Hilbert-type inequality for integrals.

Theorem 1.1. *If .
Then, one has**where the constant factor 4 is
the best possible.*

In this paper, we give further analogs of Hilbert-type inequality and its applications. The main result unifies and generalizes the classical results as follows.

Assume that , and If such that thenwhere the constant factoris the best possible.

#### 2. Some Lemmas

Our results will be based on the following results. In the following lemmas, assume that , , , and

Lemma 2.1. *Define the
following weight functions:**Then, where is defined by (1.1).*

*Proof. *For let and we have

For first let and then let Thus, we haveHence, the lemma is proved.

Lemma 2.2. *Assume that , then*

*Proof. *Sinceit follows that there exists ,
such that if then .
Moreover, there exists ,
such that if then .
Using the expression of ,
if ,
thenTherefore, Lemma 2.2 is proved
for .
If ,
then we can replace the right-hand side of the first strict inequality above
withBy the same way, we can show
that the lemma is valid for .
Hence, the lemma is proved.

Lemma 2.3. *Assume that , then*

*Proof. *Using introduced in the proof of Lemma 2.2, if ,
thenTherefore, Lemma 2.3 is proved
for .
If ,
then we can replace the right-hand side of the second strict inequality
mentioned above withBy the same way, we can show that
the lemma is valid for .
Therefore, the lemma is proved.

#### 3. Integral Case

In this section, we will state our main results.

Proposition 3.1. *Assume that ,
and If such that then**where the constant factor**is the best possible.*

*Proof. *Using
Hölder's inequality, we have

If (3.3) takes the form of equality, then there are
constants and ,
such that they are not all zero, andHence,Therefore, there is 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 a contradiction. Hence, by (3.3), we have (3.1).

If the constant factor in (3.1) is not the best possible factor, then
there exists a positive constant (with ), such that (3.1) is still valid if we
replace with .
For sufficiently small, construct the following functions:Thus, we obtainSetting ,
we haveSinceit follows thatHence,This contradicts the fact that .
So the constant factor in (3.1) is the best possible.
Then Proposition 3.1 is proved.

Proposition 3.2. *Under the same assumptions of Proposition 3.1, one has**where the constant factor is the best possible. Inequalities
(3.1) and (3.13) are equivalent.*

*Proof. *Settingby (3.1), we haveAs a result,Hence,Then, we have (3.13).

Conversely, by Hölder's inequality, we
haveThen, by (3.13), we have (3.1).
Hence inequalities (3.1) and (3.13) are equivalent.

If the constant factor in (3.13) is not the best possible, then by
(3.18) we can get a contradiction that the constant factor in (3.1) is not the best possible. Hence Proposition 3.2 is proved.

*Remark 3.3. *In (3.1), let ,
and we have Hilbert's integral inequalityLet ;
we have Hardy-Hilbert's classical inequalityLet ,
and we can combine the above two inequalities as
follows:Let ,
and we can get the following inequality:

#### 4. Discrete Case

We also give results for the discrete case.

Proposition 4.1. *Assume that and .
If ,
such that**then one has**where the constant factor**is the best possible, and inequalities (4.2) and (4.3) are
equivalent.*

*Proof. *Define
the following weight functions:then,where are defined in Lemma 2.1. By Hölder's inequality, we haveThe last strict inequality holds
because both series and have positive terms. Thus, we have (4.2).

If the constant factor in (4.2) is not the best possible, then there
exists a positive constant (with ), such that (4.2) is still valid if we
replace by .
For small enough, construct seriesThen, we havewhere and are defined in the proof of
Proposition 3.1. Sinceit follows thatHence,This contradicts with the fact
that .
So the constant factor in (4.2) is the best possible one.

Settingwe findBy the same argument used in the
proof of Proposition 3.2, we can show that (4.3)
is valid, the constant factor in (4.3) is the best possible one and
inequalities (4.2) and (4.3) are equivalent.

*Remark 4.2. *Proposition 4.1 is the corresponding series form
of Propositions 3.1 and 3.2 for ,
and it is also a generalization of Hilbert's inequality. Here, we restrict the
constants so that we can use the monotony of functions
to obtain (4.6) and (4.9).

#### Acknowledgment

This work was partially supported by the National Natural Science Foundation of China (Grant no. 10871213).

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