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A note on Hardy’s inequality
Journal of Inequalities and Applications volume 2014, Article number: 271 (2014)
Abstract
In this paper, we prove that the inequality holds for and if (), and .
MSC:26D15.
1 Introduction
Let and () with , then Hardy’s well-known inequality [1] is given by
Recently, the refinement, improvement, generalization, extension, and application for Hardy’s inequality have attracted the attention of many researchers [2–10].
Yang and Zhu [11] presented an improvement of Hardy’s inequality (1.1) for as follows:
For , Huang [12] proved that
In [13], Wen and Zhang proved that the inequality
holds for if (), with , where for and for .
Xu et al. [14] gave a further improvement of the inequality (1.2):
where for and for .
For the special parameter , Deng et al. [15] established
where .
In [16], Long and Linh discussed Hardy’s inequality with the parameter , and proved that
for and
for if () with .
It is the aim of this paper to present an improvement of inequality (1.3) for the parameter . Our main result is Theorem 1.1.
Theorem 1.1 Let , and () with , then
2 Lemmas
In order to establish our main result we need several lemmas, which we present in this section.
Lemma 2.1 (see [[17], Corollary 1.3])
Suppose that with , has continuous partial derivatives and
If holds for all and , then
for all (), where .
Lemma 2.2 Let be a positive natural number and with . Then
Proof We use mathematical induction to prove inequality (2.1). We clearly see that inequality (2.1) becomes equality for . We assume that inequality (2.1) holds for (, ), namely
Then for we have
Note that () is concave on , therefore Hermite-Hadamard’s inequality implies that
From (2.2) and (2.3) we know that inequality (2.1) holds for . □
Remark 2.1 The inequality
holds for all with equality if and only if .
Proof We clearly see that inequality (2.4) becomes equality for .
If , then it is well known that the function is strictly increasing on , so we get
Therefore, inequality (2.4) follows from (2.5). □
Lemma 2.3 The inequality
holds for all .
Proof Let , then we clearly see that
Inequality (2.7) leads to
It follows from the well-known inequality () that
From (2.8) and (2.9) we have
Therefore, inequality (2.6) follows easily from (2.10). □
Lemma 2.4 Let and
Then f is convex on .
Proof From (2.11) we have
It follows from Lemma 2.3 and (2.12) that
for all .
Therefore, Lemma 2.4 follows from inequality (2.13). □
Lemma 2.5 Let , and , then
Proof If , then we clearly see that inequality (2.14) becomes equality. Next, we assume that . Let
Then simple computations lead to
Note that
It follows from Remark 2.1 and (2.17)-(2.19) that
Let
Then from and together with the fact that is a concave parabola we know that
for .
Therefore, Lemma 2.5 follows easily from (2.15) and (2.16) together with (2.20)-(2.22). □
Lemma 2.6 Let , , N is a positive natural number, () and , then
Proof Let (), then and inequality (2.23) becomes
Let (), and
Then for any () we have
From Lemma 2.2 and (2.26) one has
It clearly follows from Lemma 2.4 and the Hermite-Hadamard inequality that
and
Note that
From Lemma 2.5 and (2.30) one has
Inequalities (2.27), (2.28), and (2.31) together with (2.29) lead to the conclusion that
for any and .
It follows from Lemma 2.1 and (2.32) that
Therefore, inequality (2.24) follows from (2.25) and (2.33). □
Lemma 2.7 Let , , then
Proof We clearly see that inequality (2.34) holds for . Next, we assume that , let , then and Lemma 2.5 leads to
Note that
for all . In fact, let and
Then
It follows from (2.38) and (2.39) that
Equation (2.37) and inequality (2.40) lead to the conclusion that
From (2.35) and (2.36) together with the fact that we have
Therefore, inequality (2.34) follows from (2.42). □
Lemma 2.8 Let , , N is a positive natural number, () and , then
Proof Let , and
Then
It follows from Lemma 2.2 and (2.46) together with Remark 2.1 that
Let
Then
for .
From (2.47)-(2.50) we get
Therefore, Lemma 2.8 follows easily from Lemma 2.6, (2.44), (2.45), and (2.51). □
3 Proof of Theorem 1.1
Let , and (), then , , and .
It follows from Lemmas 2.7 and 2.8 that one has
Letting , (3.1) leads to
Therefore, Theorem 1.1 follows immediately from (3.2) and together with .
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Acknowledgements
The authors would like to express their deep gratitude to the referees for giving many valuable suggestions. This research was supported by the Natural Science Foundation of China under Grants 11171307 and 61374086, and the Natural Science Foundation of Zhejiang Province under Grant Y13A01000.
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Authors’ contributions
Y-MC provided the main idea and carried out the proof of Lemmas 2.1 and 2.2. QX carried out the proof of Lemmas 2.3-2.5 and Theorem 1.1. X-MZ carried out the proof of Lemmas 2.6-2.8. All authors read and approved the final manuscript.
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Chu, YM., Xu, Q. & Zhang, XM. A note on Hardy’s inequality. J Inequal Appl 2014, 271 (2014). https://doi.org/10.1186/1029-242X-2014-271
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DOI: https://doi.org/10.1186/1029-242X-2014-271