Abstract
The purpose of this paper is to investigate a sharpened version of Hardy’s inequality for parameter . By evaluating the weight coefficient , sharpened Hardy’s inequality that contains the best coefficient is established.
MSC:26D15, 26D20, 26D07.
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1 Introduction
Let , , (), . Then
where is the best coefficient. Inequality (1) is called Hardy’s inequality which is of great use in the field of modern mathematics (see [1, 2]).
A special case of (1) yields the following inequalities:
In 1998, Yang and Zhu [3] evaluated the weight coefficient ,
and established an improved version of inequality (2) as follows:
With the same approach, that is, evaluating the weight coefficient , Huang [4–7] gave some improvements on Hardy’s inequality for and , i.e.,
Some further extensions of Hardy’s inequality related to the range of parameter p were given in Huang [7, 8].
In 2005, Yang [9] proved an inequality for the weight coefficient
and established the following inequality:
where is the best coefficient under the weight coefficient .
In 2009, Zhang and Xu made use of the monotonicity theorem [10–13] and obtained an improvement of inequality (1):
where
By evaluating the weight coefficient , and with the help of an inequality-proving package called BOTTEMA [14, 15], He [16] investigated a sharpened version of Hardy’s inequality for and obtained the following improved version of inequality (3):
where is the best coefficient under the weight coefficient .
In addition, in [16] the author wrote the computer program HDISCOVER to accomplish the automated verification of the following inequality for (N is the set of natural numbers):
where is the best coefficient of (11) under the weight coefficient .
Recently, based on the program HDISCOVER 2012 written by Deng, He and Wu [17], an automated verification of inequality (11) is achieved for (Q is the set of rational numbers).
For more detailed information of Hardy’s inequality, we refer the interested readers to relevant research papers [10, 12, 18–23].
In this paper, by evaluating the weight coefficient , we establish an improvement of Hardy’s inequality for parameter as follows:
where is the best coefficient under the weight coefficient .
2 Lemmas
To prove the main results in Section 3, we will use the following lemmas.
Lemma 1 (see[22])
If , then for all integers , it holds that
Lemma 2 (see[3])
If , then for all integers , it holds that
Lemma 3 Let , , and let , be the functions defined by
Then , .
Proof Since , , hence for .
Further, we have
and consequently, is strictly increasing on .
Now, from and , it follows that and .
Similarly, from
we deduce that .
Lemma 3 is proved. □
Lemma 4 Let . If , then
If , then
Proof When . By using the Maclaurin formula
and noticing , we find
Thus
When . We have
Thus
The proof of Lemma 4 is complete. □
Lemma 5 Let , , , and let denote the greatest integer less than or equal to the real number x. Then we have
Proof By Lemma 1 and the identity , , it follows that
Combining Lemmas 3 and 4, we obtain
This completes the proof of Lemma 5. □
Lemma 6 Let , . If , then
If , then
Proof Since , , using Lemma 1 gives
When . From Lemmas 3 and 4, we have
When . Using Lemmas 3 and 4, we obtain
Lemma 6 is proved. □
Lemma 7 (see[3])
Let , (), . Then
3 Main results
Theorem 1 For an arbitrary natural number k, the following inequality holds true:
where
Proof
Using Lemma 5 gives
where
Hence
Using Lemma 2 and taking in the right-hand side of inequality (13), respectively, we get
Adding up the above inequalities, we obtain
Theorem 1 is proved. □
Theorem 2 For an arbitrary natural number k, the following inequality holds true:
where
Proof
Utilizing Lemma 6 gives
where
Hence
Using Lemma 2 and taking in the left-hand side of inequality (13), respectively, we get
Adding up the above inequalities, we obtain
Theorem 2 is proved. □
Theorem 3 Let (), . Then
where is the best possible under the weight coefficient .
Proof
By Lemma 7, we have
Therefore, to prove inequality (14), it suffices to show that
Obviously, inequality (15) becomes an equality for . In what follows, we will assume that .
By Theorem 1 , we need only to prove that
Note that
it suffices to show
Substituting in (16), inequality (16) becomes
which is equivalent to the following inequality:
where
From the hypothesis , we have
Further, we have
Consequently, inequality (17) holds true, and inequality (14) is proved.
Let us now show that is the best possible under the weight coefficient .
Consider inequality (14) in a general form as
Putting in (18) yields
Thus the best possible value for in (18) should be .
This completes the proof of Theorem 3. □
Remark 1 From the definition of and in the same way as in [17], we can establish the following accurate estimates of :
Further, the approximation of can be derived as follows:
Remark 2 For , inequality (11) can be written as
It is easy to observe that
and
hence
This implies that inequality (14) is stronger than inequality (11).
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Acknowledgements
The present investigation was supported, in part, by the Natural Science Foundation of Fujian province of China under Grant 2012J01014 and, in part, by the Foundation of Scientific Research Project of Fujian Province Education Department of China under Grant JK2012049.
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YD finished the proof and the writing work. SW gave YD some advice on the proof and writing. DH gave YD lots of help in revising the paper. All authors read and approved the final manuscript.
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Deng, Y., Wu, S. & He, D. A sharpened version of Hardy’s inequality for parameter . J Inequal Appl 2013, 63 (2013). https://doi.org/10.1186/1029-242X-2013-63
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DOI: https://doi.org/10.1186/1029-242X-2013-63