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The connection between Hilbert and Hardy inequalities
Journal of Inequalities and Applications volume 2013, Article number: 452 (2013)
In this paper we introduce some new forms of the Hilbert integral inequality, and we study the connection between the obtained inequalities with Hardy inequalities. The reverse form and some applications are also given.
The famous Hardy-Hilbert inequality for positive functions f, g and two conjugate parameters p and q such that , is given as
provided that the integrals on the right-hand side are convergent. The constant is best possible . In the last years, inequality (1.1) has been extended in different ways. In  the authors obtained the following extension of (1.1):
where is the best possible constant ( is the beta function), , , and . For , , , , and , the reverse form of (1.2) is also valid with the same constant factor. In  the following extension was given:
where is best possible ( is the hypergeometric function), , , and , , , , u and v are differentiable nonnegative strictly increasing functions on (), and they satisfy the following conditions: and . In particular, if we let , and consider , instead of and respectively in (1.3), we get
here as in (1.2).
The following inequalities are special cases of (1.4):
Refinements of some Hilbert-type inequalities by virtue of various methods were obtained in [4, 5] and . A survey of some recent results concerning Hilbert and Hilbert-type inequalities can be found in  and .
If , , and , then the well-known Hardy inequality  is given as
the constant is best possible. A weighted form of (1.7) was given also by Hardy  as
where , or , and the constant is best possible. For (), inequality (1.8) holds in the reverse direction. Inequality (1.7) was discovered by Hardy while he was trying to introduce a simple proof of Hilbert’s inequality. In the book , the following Hardy-type inequality is given:
where and or . If , then the reverse form of (1.9) holds. The constant is best possible.
Recently, in , for , , and , , the following form of (1.1) was obtained:
In this paper, by estimating the double integral , we introduce an extension of (1.11) with the best constant factor. The reverse form is also obtained. Some applications are given. The connection between Hilbert and Hardy inequalities is also considered. As a consequence of Theorem 3.1, we obtain the following interesting inequality:
2 Preliminaries and lemmas
Recall that the gamma function and the beta function are defined respectively by
In this paper, we assume that u and v are defined as in inequality (1.3) from the introduction.
Lemma 2.1 Let , , , , , and let h be a differentiable nonnegative strictly increasing function on such that , . Then, for , we have
Proof Using integration by parts, we get
Applying Hölder’s inequality, we obtain
Substituting the last inequality in (2.2), we get (2.1). □
Lemma 2.2 Let , , , , , and let h be as in Lemma 2.1. Then, for and (), we have
Proof Integration by parts yields
Using the reverse Hölder inequality, we obtain
Substituting the last inequality in (2.4), we get (2.3).
By the definition of the gamma function above, we may write
3 Main results
In this section, we introduce two main results in this paper. Theorem 3.1 gives an extended form of inequality (1.11) and it is connected to the famous Hardy inequality. In Theorem 3.2, we introduce the reverse form obtained in Theorem 3.1.
Theorem 3.1 Let , , , , , , define and . If and , then
where the constant is best possible.
Proof By using (2.5) and applying Hölder’s inequality, we have
By Lemma 2.1, for , , and then for , , , we obtain, respectively,
Substituting these two inequalities in (3.2), we have
Now, since and , we get
Inequality (3.1) is proved. We need to show that the constant factor C in (3.1) is best possible. For , we define the functions , for and for and for and for , where and are such that . Then we get for and for , for , , respectively. Suppose that the constant is not best possible, then there exists such that
On the other hand, we have
It is obvious that when from (3.3) and (3.4), we obtain a contradiction. Thus, the proof of the theorem is completed. □
Theorem 3.2 Let , , , , , , define and . If and , then we obtain the reverse form of (3.1) as
where C is as in Theorem 3.1.
Proof If we use (2.5) and apply the reverse Hölder inequality, we have
By Lemma 2.2, for , , and then for , , , we obtain, respectively,
If we substitute these two inequalities in (3.6) and make some computations as we did in Theorem 3.1, we get inequality (3.5). □
In this section, we give some applications of Theorem 3.1 and Theorem 3.2. We consider some specific functions which satisfy the conditions of the functions u and v, and we see the connection between Hilbert and Hilbert-type inequalities with Hardy and Hardy-type inequalities from the introduction.
Let , , , then we find by (3.1)(4.1)
here and . If we put in (4.1), we get (1.11). If we let , , we obtain the following form:(4.2)
Applying Hardy’s inequality (1.7) to the right-hand side of (4.2), we get Hilbert’s inequality (1.1). If we apply the weighted Hardy inequality (1.8) to (4.1), we get(4.3)
where . Inequality (4.3) is equivalent to inequality (1.2) if we set () under the condition . By Theorem 3.2, we have the reverse form of (4.1)(4.4)
If we apply the reverse inequality of (1.8) to the first integral on the right-hand side in (4.4) and inequality (1.8) to the second integral (), we get(4.5)
Inequality (4.5) is equivalent to the reverse form of (1.2) if we set under the condition .
If , , , we obtain by (3.1)(4.6)
here and . If we apply (1.9) to the integrals on the right-hand side of (4.6) and set , we obtain (1.5). The reverse form of (4.6) is also valid, and we may obtain a reverse inequality of (1.5) if we use (1.9) and its reverse form.
If , , , then we have(4.7)
here and . In particular, for , , we get
if we apply (1.10), we get Hilbert-type inequality (1.6).
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The author declares that he has no competing interests.
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Azar, L.E. The connection between Hilbert and Hardy inequalities. J Inequal Appl 2013, 452 (2013). https://doi.org/10.1186/1029-242X-2013-452
- Hilbert’s inequality
- Hölder’s inequality
- Hardy’s inequality