- Open Access
The connection between Hilbert and Hardy inequalities
© Azar; licensee Springer. 2013
- Received: 10 April 2013
- Accepted: 12 September 2013
- Published: 7 November 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.
- Hilbert’s inequality
- Hölder’s inequality
- Hardy’s inequality
here as in (1.2).
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 .
where and or . If , then the reverse form of (1.9) holds. The constant is best possible.
In this paper, we assume that u and v are defined as in inequality (1.3) from the introduction.
Substituting the last inequality in (2.2), we get (2.1). □
Substituting the last inequality in (2.4), we get (2.3).
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.
where the constant is best possible.
It is obvious that when from (3.3) and (3.4), we obtain a contradiction. Thus, the proof of the theorem is completed. □
where C is as in Theorem 3.1.
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). □
- 1.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 .
- 2.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.
- 3.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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