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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 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.
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 left-hand 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 non-zero 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 left-hand 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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DOI: https://doi.org/10.1155/2009/158690
Keywords
- Positive Integer
- Nonnegative Integer
- Real Function
- Constant Factor
- Equivalent Form