An Extension of the Hilbert's Integral Inequality

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Abstract

It is shown that an extension of the Hilbert's integral inequality can be established by introducing two parameters Open image in new window and Open image in new window . The constant factors expressed by the Euler number and Open image in new window as well as by the Bernoulli number and Open image in new window , respectively, are proved to be the best possible. Some important and especial results are enumerated. As applications, some equivalent forms are given.

Keywords

Positive Integer Nonnegative Integer Real Function Constant Factor Equivalent Form 

1. Introduction and Lemmas

Let Open image in new window . Define Open image in new window , when Open image in new window . If Open image in new window , then

where the constant factor Open image in new window 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 Hilbert-type 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, 4, 5, 6, 7, 8, 9, 10, 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 Hilbert-type inequality of the form

where Open image in new window is a nonnegative integer and Open image in new window is a positive number. We will give the constant factor Open image in new window and the expression of the weigh function Open image in new window , prove the constant factor Open image in new window 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 Open image in new window in (1.2) is very interesting. It can be expressed by Open image in new window and the Bernoulli number, when Open image in new window is an odd number, and it can be expressed by Open image in new window and the Euler number, when Open image in new window is an even number, and that Open image in new window seems to play a bridge role between two cases.

In order to prove our main results, we need the following lemmas.

Lemma 1.1.

Proof.

According to the definition of Open image in new window -function, (1.3) easily follows. This result can be also found in the paper [12, page 226, formula 1053].

Lemma 1.2.

Let Open image in new window be a positive integer. Then

where the Open image in new window are the Bernoulli numbers, namely, Open image in new window and so forth.

Proof.

It is known from the paper [13, page 231] that
where the Open image in new window are the Bernoulli numbers, namely, Open image in new window and so forth.  It is easy to deduce that

Notice that Open image in new window . Equality (1.4) follows.

Lemma 1.3.

Let Open image in new window be a positive number.

(i)If Open image in new window is a positive integer, then
where the Open image in new window are the Bernoulli numbers.
  1. (ii)
    If Open image in new window is a nonnegative integer, then
     

where the Open image in new window are the Euler numbers, namely, Open image in new window and so forth.

Proof.

We prove firstly equality (1.7). Expanding the hyperbolic cosecant function Open image in new window , and then using Lemma 1.1 and noticing that Open image in new window , we have

By Lemma 1.2, we obtain (1.7) at once.

Next we consider (1.8). Similarly by expanding the hyperbolic secant function Open image in new window and then using Lemma 1.1, we have

It is known from the paper [13, pp. 231] that

where the Open image in new window are Euler numbers, namely, Open image in new window and so forth. In particular, when Open image in new window , we have Open image in new window , hence we can define Open image in new window . 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] Open image in new window is wrong. It should be Open image in new window .

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.

where the constant factor Open image in new window is defined by

andthe Open image in new window are the Bernoulli numbers, namely, Open image in new window Open image in new window and so forth. And the constant factor Open image in new window in (2.1) is the best possible.

Proof.

We may apply the Cauchy inequality to estimate the left-hand side of (2.1) as follows:

where Open image in new window .

By using Lemma 1.3, it is easy to deduce that

where the constant factor Open image in new window 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 non-zero constants Open image in new window and Open image in new window such that
Then we have
Without losing the generality, we suppose that Open image in new window , then

This contradicts that Open image in new window . Hence it is impossible to take the equality in (2.5). So the inequality (2.1) is valid.

It remains only to show that Open image in new window in (2.1) is the best possible, for all Open image in new window . Define two functions by
It is easy to deduce that
If Open image in new window in (2.1) is not the best possible, then there exists Open image in new window , such that
On the other hand, we have
When Open image in new window 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 Open image in new window 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 Open image in new window , we have Open image in new window , 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.

where the constant factor Open image in new window is the best possible.

Corollary 2.3.

where the constant factor Open image in new window is the best possible.

Corollary 2.4.

where the constant factor Open image in new window is the best possible.

Corollary 2.5.

where the constant factor Open image in new window is defined by

and the Open image in new window are the Bernoulli numbers. And the constant factor Open image in new window 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.

where the constant factor Open image in new window is defined by

where Open image in new window and the Open image in new window are the Euler numbers,namely, Open image in new window and so forth. And the constant factor Open image in new window in (2.20) is the best possible.

Proof.

By applying Cauchy's inequality to estimate the left-hand side of (2.20), we have

where Open image in new window .

By proper substitution of variable, and then by Lemma 1.3, it is easy to deduce that

where the constant factor Open image in new window 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 Open image in new window and Open image in new window , we have Open image in new window , 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.

where the constant factor Open image in new window is the best possible.

Corollary 2.8.

where the constant factor Open image in new window is the best possible.

Corollary 2.9.

where the constant factor Open image in new window is the best possible.

Corollary 2.10.

where Open image in new window and the Open image in new window are the Euler numbers. And the constant factor Open image in new window 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 Open image in new window be a real function, and let Open image in new window be a positive integer, let Open image in new window .

where Open image in new window is defined by (2.2) and the constant factor Open image in new window 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 Open image in new window 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 Open image in new window in (3.1) is not the best possible, then it is known from (3.4) that the constant factor Open image in new window in (2.1) is also not the best possible. This is a contradiction. The theorem is proved.

Corollary 3.2.

where the constant factor Open image in new window 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 Open image in new window be a real function,and let Open image in new window be a nonnegative integer, Open image in new window .

where Open image in new window is defined by (2.19) and the constant factor Open image in new window in (3.6) is the best possible. Inequality (3.6) is equivalent to (2.20).

Corollary 3.4.

where the constant factor Open image in new window 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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Copyright information

© Zhou Yu et al. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceNormal College of Jishou UniversityHunan JishouChina

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