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An Extension of the Hilbert's Integral Inequality
Journal of Inequalities and Applications volume 2009, Article number: 158690 (2009)
Abstract
It is shown that an extension of the Hilbert's integral inequality can be established by introducing two parameters and . The constant factors expressed by the Euler number and as well as by the Bernoulli number and , respectively, are proved to be the best possible. Some important and especial results are enumerated. As applications, some equivalent forms are given.
1. Introduction and Lemmas
Let . Define , when . If , then
where the constant factor is the best possible. This is the famous Hilbert's integral inequality (see [1, 2]). Owing to the importance of the Hilbert's inequality and the Hilberttype inequality in analysis and applications, some mathematicians have been studying them. Recently, various improvements and extensions of (1.1) appear in a great deal of papers (see [3–11], etc.). Specially, Gao and Hsu enumerated the research articles more than 40 in the paper [6]. The purpose of the present paper is to establish the Hilberttype inequality of the form
where is a nonnegative integer and is a positive number. We will give the constant factor and the expression of the weigh function , prove the constant factor to be the best possible, and then give some especial results and discuss some equivalent forms of them. Evidently inequality (1.2) is an extension of (1.1). The new inequality established is significant in theory and applications. We will discover that the constant factor in (1.2) is very interesting. It can be expressed by and the Bernoulli number, when is an odd number, and it can be expressed by and the Euler number, when is an even number, and that seems to play a bridge role between two cases.
In order to prove our main results, we need the following lemmas.
Lemma 1.1.
Let be a positive number and . Then
Proof.
According to the definition of function, (1.3) easily follows. This result can be also found in the paper [12, page 226, formula 1053].
Lemma 1.2.
Let be a positive integer. Then
where the are the Bernoulli numbers, namely, and so forth.
Proof.
It is known from the paper [13, page 231] that
where the are the Bernoulli numbers, namely, and so forth. It is easy to deduce that
Notice that . Equality (1.4) follows.
Lemma 1.3.
Let be a positive number.
(i)If is a positive integer, then
where the are the Bernoulli numbers.

(ii)
If is a nonnegative integer, then
(1.8)
where the are the Euler numbers, namely, and so forth.
Proof.
We prove firstly equality (1.7). Expanding the hyperbolic cosecant function , and then using Lemma 1.1 and noticing that , we have
By Lemma 1.2, we obtain (1.7) at once.
Next we consider (1.8). Similarly by expanding the hyperbolic secant function and then using Lemma 1.1, we have
It is known from the paper [13, pp. 231] that
where the are Euler numbers, namely, and so forth. In particular, when , we have , hence we can define . It follows from (1.10) and (1.11) that the equality (1.8) is true.
By the way, there is an error in the paper [12, page 260, formula 1566], namely, the integral in the paper [12] is wrong. It should be .
By applying this correct result, it is easy to verify the formulas 1562–1565 in the paper [12, pp. 259]. These are omitted here.
2. Main Results
In this section, we will prove our assertions by using the above lemmas.
Theorem 2.1.
Let and be two real functions, and let be a positive integer, . If and , then
where the constant factor is defined by
andthe are the Bernoulli numbers, namely, and so forth. And the constant factor in (2.1) is the best possible.
Proof.
We may apply the Cauchy inequality to estimate the lefthand side of (2.1) as follows:
where .
By using Lemma 1.3, it is easy to deduce that
where the constant factor is defined by (2.2).
It follows from (2.3) and (2.4) that
If (2.5) takes the form of the equality, then there exists a pair of nonzero constants and such that
Then we have
Without losing the generality, we suppose that , then
This contradicts that . Hence it is impossible to take the equality in (2.5). So the inequality (2.1) is valid.
It remains only to show that in (2.1) is the best possible, for all . Define two functions by
It is easy to deduce that
If in (2.1) is not the best possible, then there exists , such that
On the other hand, we have
When is sufficiently small, we obtain from (2.12) that
Noticing the proof of (2.4), we have
Evidently, inequality (2.14) is in contradiction with that in (2.11). Therefore, the constant factor in (2.1) is the best possible. Thus the proof of the theorem is completed.
Based on Theorem 2.1, we may build some important and interesting inequalities.
In particular, when , we have , the inequality (2.1) can be reduced to (1.1).
It shows that Theorem 2.1 is an extension of (1.1).
Corollary 2.2.
If and , then
where the constant factor is the best possible.
Corollary 2.3.
If and , then
where the constant factor is the best possible.
Corollary 2.4.
If and , then
where the constant factor is the best possible.
Corollary 2.5.
Let be a positive integer. If and , then
where the constant factor is defined by
and the are the Bernoulli numbers. And the constant factor in (2.18) is the best possible.
Similarly, we can establish also a great deal of new inequalities. They are omitted here.
Theorem 2.6.
Let and be two real functions, and let be a nonnegative integer and . If and , then
where the constant factor is defined by
where and the are the Euler numbers,namely, and so forth. And the constant factor in (2.20) is the best possible.
Proof.
By applying Cauchy's inequality to estimate the lefthand side of (2.20), we have
where .
By proper substitution of variable, and then by Lemma 1.3, it is easy to deduce that
where the constant factor is defined by (2.21).
It follows from (2.22) and (2.23) that
The proof of the rest is similar to that of Theorem 2.1, it is omitted here.
In particular, when and , we have , inequality (2.20) can be reduced to (1.1). It shows that Theorem 2.6 is also an extension of (1.1).
Corollary 2.7.
If and , then
where the constant factor is the best possible.
Corollary 2.8.
If and , then
where the constant factor is the best possible.
Corollary 2.9.
If and , then
where the constant factor is the best possible.
Corollary 2.10.
Let be a nonnegative integer. If and , then
where and the are the Euler numbers. And the constant factor in (2.28) is the best possible.
Similarly, we can establish also a great deal of new inequalities. They are omitted here.
3. Some Equivalent Forms
As applications, we will build some new inequalities.
Theorem 3.1.
Let be a real function, and let be a positive integer, let .
If , then
where is defined by (2.2) and the constant factor in (3.1) is the best possible. And the inequality (3.1) is equivalent to (2.1).
Proof.
First, we assume that inequality (2.1) is valid. Set a real function as
By using (2.1), we have
It follows from (3.3) that inequality (3.1) is valid after some simplifications.
On the other hand, assume that inequality (3.1) keeps valid, by applying in turn Cauchy's inequality and (3.1), we have
Therefore the inequality (3.1) is equivalent to (2.1).
If the constant factor in (3.1) is not the best possible, then it is known from (3.4) that the constant factor in (2.1) is also not the best possible. This is a contradiction. The theorem is proved.
Corollary 3.2.
Let be a real function. If , then
where the constant factor is the best possible. And the inequality (3.5) is equivalent to (2.15).
Its proof is similar to the one of Theorem 3.1. Hence it is omitted.
Similarly, we can establish also some new inequalities which are, respectively, equivalent to inequalities (2.16), (2.17), and (2.18). They are omitted here.
Theorem 3.3.
Let be a real function,and let be a nonnegative integer, .
If , then
where is defined by (2.19) and the constant factor in (3.6) is the best possible. Inequality (3.6) is equivalent to (2.20).
Corollary 3.4.
If , then
where the constant factor in (3.7) is the best possible. And inequality (3.7) is equivalent to (2.25).
Similarly, we can establish also some new inequalities which are, respectively, equivalent to inequalities (2.26), (2.27), and (2.28). These are omitted here.
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Yu, Z., Xuemei, G. & Mingzhe, G. An Extension of the Hilbert's Integral Inequality. J Inequal Appl 2009, 158690 (2009). https://doi.org/10.1155/2009/158690
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Keywords
 Positive Integer
 Nonnegative Integer
 Real Function
 Constant Factor
 Equivalent Form