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 haswhere 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 thenwhere the constant factoris 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 haswhere 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 thatthen one haswhere the constant factoris 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).