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On strengthened form of Copson’s inequality
Journal of Inequalities and Applications volume 2012, Article number: 305 (2012)
In this paper, the famous Copson inequality has been improved. We obtain some new results by a different method.
Suppose that , , then we obtain the following Hardy inequality:
Hardy’s inequality plays an important role in the field of analysis; see [1–4]. In recent decades, some generalizations and strengthening of Hardy’s inequality have been obtained in [1–5]. We list some previous results as follows.
Suppose that (). If , then
If , then
In this paper, inequalities (1.2) and (1.3) were strengthened by using a new method.
2 Relational lemmas and definitions
In this section, some relational lemmas and definitions will be introduced.
Theorem A [, Th. 1.1]
Suppose that , , , and has continuous partial derivatives, and
If holds for all (), then
Theorem B [, Cor. 1.3]
Suppose that , , and has continuous partial derivatives and
If holds for all , where , then
where (), .
Definition 1 
Let be a convex set, be a continuous function. If
holds for all , , then the function φ is convex (concave).
Lemma 1 (Hermite-Hadamard’s inequality)
Let be a convex (concave) function. Then
and the equality holds if and only if ϕ is linear.
Lemma 2 Suppose that .
If , or , then(2.2)
If , then(2.3)
If , then(2.4)
Proof Set . Then we have
Obviously, is monotone decreasing for , is monotone increasing for , and , , then (2.2) and (2.3) hold. Let
Then h is concave for . Because and , then and hold for . From , we have for . Inequality (2.4) is proved. □
If , then the equation(2.5)
has only a positive root for .
If , then the equation(2.6)
has only a positive root for .
Let . Then is monotone decreasing. According to inequality (2.2), we have
and . So, equation (2.5) has only a positive root for .
Let . Thus,
By inequality (2.4), is monotone increasing. According to inequality (2.3), we get
Therefore, equation (2.6) has only a positive root for .
Lemma 4 If , , and is the only one positive root of equation (2.5), then
Proof (1) If , by Lemma 1, we get
If , by Lemma 1, we get
holds for every and . Since inequalities (2.9), (2.10) and
inequality (2.7) holds.
Let and . Using Hölder’s inequality, we have
inequality (2.8) holds. □
Lemma 5 If , , and is the only one positive root of equation (2.6), then
By Hölder’s inequality, we have
And by using inequality (2.12), we obtain
From inequality (2.12) and inequality (2.13), we get
Then inequality (2.11) holds. □
3 Strengthened Copson’s inequality ()
Theorem 1 Assume that , , , (), is the only one positive root of equation (2.5) and . Then
Proof Set (). Then inequality (3.1) is equivalent to
where . Let
If , then
By inequality (2.7), we know . By Theorem B, inequality (3.2) holds, the proof is completed. □
Corollary 1 If , , , (), is the only one positive root of equation (2.5), and , then
Proof By (3.1) and (2.8), we can obtain
Corollary 2 If , (), and is the only one positive root of equation (2.5), then
Proof Because of , the infimum of is zero. Then there exists a sequence such that decrease to zero. Since (3.3), we have
Let in inequality (3.5), we have and
Then by (3.5), we can obtain
Therefore, inequality (3.4) holds. □
Remark Obviously, inequality (3.4) strengthens inequality (1.2).
4 Strengthened Copson’s inequality ()
Theorem 2 If , , , (), is the only one positive root of equation (2.6) and . Then
Proof Let (). Then inequality (4.1) is equivalent to
where . Set
and . If , then
By Lemma 1, we have
As , by the definition of d, we have
As , because , and is concave, we have
Thus, for every , . By Theorem B, inequality (4.2) holds. □
Corollary 3 If , , , (), is the only one positive root of equation (2.6) and . Then
Proof From Theorem 2 and Lemma 5, we have
Then inequality (4.3) holds. □
Corollary 4 If , (), is the only one positive root of equation (2.6) and series . Then
Proof According to inequality (4.3), we obtain
The following proof is the same as the relevant proof for Corollary 2, omitted here. □
Remark For , there is no doubt that inequality (4.4) strengthens inequality (1.3).
Hardy GH, Littlewood JE, Poly AG: Inequalities. Cambridge University Press, Cambridge; 1952.
Yang B-C: Arithmetic Operators and Hilbert Type Inequality. Science Press, Beijing; 2009. in Chinese
Kuang J-C: Inequality of Regular. Shandong Science and Technology Press, Jinan; 2004. in Chinese
Pachpatte BG: Mathematical Inequalities. Elsevier, Amsterdam; 2005:118–127.
Bullen PS: A Dictionary of Inequalities. Chapman & Hall, London; 1998:65.
Gao P: On weighted remainder form of Hadry-type inequalities. RGMIA 2009, 12(3):17–32. http://www.staff.vu.edu.au/RGMIA/v12n3.asp
Zhang Xiao-Ming, Chu Yu-Ming: A new method to study analytic inequalities. J. Inequal. Appl. 2010., 2010: Article ID 698012
The research is supported by the Nature Science Foundation of China (No. 110771069) and the NS Foundation of the Educational Committee of Zhejiang Province under Grant Y201223283.
The author declares that they have no competing interests.
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Cite this article
Xu, Q. On strengthened form of Copson’s inequality. J Inequal Appl 2012, 305 (2012). https://doi.org/10.1186/1029-242X-2012-305
- Copson’s inequality
- analytic inequalities