- Research Article
- Open Access

# An Extension of the Hilbert's Integral Inequality

- Zhou Yu
^{1}, - Gao Xuemei
^{1}and - Gao Mingzhe
^{1}Email author

**2009**:158690

https://doi.org/10.1155/2009/158690

© Zhou Yu et al. 2009

**Received: **30 May 2009

**Accepted: **20 August 2009

**Published: **27 September 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.

## Keywords

- Positive Integer
- Nonnegative Integer
- Real Function
- Constant Factor
- Equivalent Form

## 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 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 [3–11], 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 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.

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.

where the are the Bernoulli numbers, namely, and so forth.

Proof.

Notice that . Equality (1.4) follows.

Lemma 1.3.

where the are the Euler numbers, namely, and so forth.

Proof.

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

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.

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

Proof.

where the constant factor is defined by (2.2).

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

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.

where the constant factor is the best possible.

Corollary 2.3.

where the constant factor is the best possible.

Corollary 2.4.

where the constant factor is the best possible.

Corollary 2.5.

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.

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

Proof.

where the constant factor is defined by (2.21).

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.

where the constant factor is the best possible.

Corollary 2.8.

where the constant factor is the best possible.

Corollary 2.9.

where the constant factor is the best possible.

Corollary 2.10.

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 .

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.

It follows from (3.3) that inequality (3.1) is valid after some simplifications.

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.

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, .

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.

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.

## Authors’ Affiliations

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