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.
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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)
is a nonnegative integer, then 
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