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A note on a class of Hardy-Rellich type inequalities
Journal of Inequalities and Applications volume 2013, Article number: 84 (2013)
In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results.
It is well known that Hardy’s inequality and its generalizations play important roles in many areas of mathematics. The classical Hardy inequality is given by, for ,
where , the constant is optimal and not attained.
Recently there has been a considerable interest in studying the Hardy-type and Rellich-type inequalities. See, for example, [1–7]. In  Caffarelli, Kohn and Nirenberg proved a rather general interpolation inequality with weights. That is the following so-called Caffarelli-Kohn-Nirenberg inequality. For any , there exists such that
In  Costa proved the following -case version for a class of Caffarelli-Kohn-Nirenberg inequalities with a sharp constant by an elementary method. For all and ,
where the constant is sharp.
On the other hand, the Rellich inequality is a generalization of the Hardy inequality to second-order derivatives, and the classical Rellich inequality in states that for and ,
The constant is sharp and never achieved. In  Tetikas and Zographopoulos obtained a corresponding stronger versions of the Rellich inequality which reads
for all and . In  Costa obtained a new class of Hardy-Rellich type inequalities which contain (1.5) as a special case. If , then
where the constant is sharp.
The goal of this paper is to extend the above (1.3) and (1.6) to the general case for by a different and direct approach.
2 Main results
In this section, we will give the proof of the main theorems.
Theorem 1 For all and , one has
where and the constant is sharp.
Proof Let , and . By integration by parts and the Hölder inequality, one has
It remains to show the sharpness of the constant. By the condition with equality in the Hölder inequality, we consider the following family of functions:
where is a positive number sequence converging to as . By direct computation and the limit process, we know the constant is sharp. □
Remark 1 When , the inequality (2.1) covers the inequality (2.4) in .
Remark 2 When , , the inequality (2.1) is the classical Hardy inequality:
When we take special values for a, b, the following corollary holds.
Corollary 1 (i) When , the inequality (2.1) is just the weighted Hardy inequality:
When , according to the inequality (2.1), we have(2.5)
When and , we obtain the inequality(2.6)
By a similar method, we can prove the following case Hardy-Rellich type inequality.
Theorem 2 Let , . Then, for any , the following holds:
where , and is the p-Laplacian operator.
Proof Set , it is easy to see
On the other hand,
Then, we can deduce from (2.8) and (2.9)
By the Hölder inequality,
note that . Thus
We mention that we do not know whether the constant in (2.7) is optimal or not. □
Corollary 2 When , we have the following inequalities:
when , , the inequality (2.7) is equivalent to the inequality(2.14)
When , , we obtain the inequality(2.15)
When , , we get(2.16)
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This work is supported by NNSF of China (11001240), ZJNSF (LQ12A01023) and the foundation of the Zhejiang University of the Technology (20100229).
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the final manuscript.
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Di, Y., Jiang, L., Shen, S. et al. A note on a class of Hardy-Rellich type inequalities. J Inequal Appl 2013, 84 (2013). https://doi.org/10.1186/1029-242X-2013-84
- Hardy inequality
- Hardy-Rellich inequality
- Caffarelli-Kohn-Nirenberg inequality