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A New Method to Study Analytic Inequalities
Journal of Inequalities and Applications volume 2010, Article number: 698012 (2010)
Abstract
We present a new method to study analytic inequalities involving variables. Regarding its applications, we proved some well-known inequalities and improved Carleman's inequality.
1. Monotonicity Theorems
Throughout this paper, we denote the set of real numbers and the set of strictly positive real numbers, , .
In this section, we present the main results of this paper.
Theorem 1.1.
Suppose that with and , has continuous partial derivatives and
If for all , then
for all .
Proof.
Without loss of generality, since we assume that and .
For , we clearly see that , then
From the continuity of the partial derivatives of and
we know that there exists such that and
for any . Hence, since is strictly monotone increasing, then we have
Next, for , then and
Hence, we get
If , then Theorem 1.1 is true. Otherwise, we repeat the above process and we clearly see that the first and second variables in are decreasing and no less than . Let be their limit values, respectively, then and . If , then Theorem 1.1 is also true; otherwise, we repeat the above process again and denote and the greatest lower bounds for the first and the second variables , respectively. We clearly see that ; therefore, and Theorem 1.1 is true.
Similarly, we have the following theorem.
Theorem 1.2.
Suppose that with and , has continuous partial derivatives and
If for all , then
for all .
It follows from Theorems 1.1 and 1.2 that we get the following Corollaries 1.3–1.6.
Corollary 1.3.
Suppose that with , has continuous partial derivatives and
If for all and , then
for all with .
Corollary 1.4.
Suppose with , then
and is symmetric with continuous partial derivatives. If for all , then
where . Equality holds if and only if .
Corollary 1.5.
Suppose with , has continuous partial derivatives and
If for all and , then
where . Equality holds if and only if .
Corollary 1.6.
Suppose with , then
and is symmetric with continuous partial derivatives . If for all , then
where . Equality holds if and only if .
2. Unifying Proof of Some Well-Known Inequality
In this section, we denote , , and
Proposition 2.1 (Power Mean Inequality).
If the power mean of order is defined by for and , then for ; equality holds if and only if .
Proof.
It is well known that is symmetric with respect to and is continuous. Without loss of generality, we assume that . Then
If , then . It follows from Corollary 1.4 that we get
Equality holds if and only if .
Proposition 2.2 (Holder Inequality).
Suppose that , . If , then
Proof.
Let and
If , then
Similarly, if , then . From Theorem 1.1, we get
Therefore, Proposition 2.2 follows from and .
Proposition 2.3 (Minkowski Inequality).
Suppose that ,. If , then
Proof.
Let and
If , then
Similarly, If , then . It follows from Theorem 1.1 that we get
Therefore, Proposition 2.3 follows from and .
3. A Brief Proof for Hardy's Inequality
If with , then the well-known Hardy's inequality (see [1,Theorem ]) is
In this section, we establish the following result involving Hardy's inequality.
Theorem 3.1.
Let , , and . If
then
Proof.
Let , then inequality (3.3) is equivalent to
and . Let
If , then
Making use of the well-known Hadamard's inequality of convex functions, we get
Then Theorem 1.1 leads to
and we clearly see that inequalities (3.4) and (3.3) are true.
Corollary 3.2.
Let , , and . If
then
Proof.
From inequality (3.3), we clearly see that
Remark 3.3.
If , then inequality (3.1) follows from inequality (3.10).
4. A Refinement of Carleman's Inequality
If with , then the well-known Carleman's inequality is
with the best possible constant factor (see [2]).
Recently, Yang and Debnath [3] gave a strengthened version of (4.1) as follows:
Some other strengthened versions of (4.1) were given in [4–9]. In this section, we give a refinement for Carleman's inequality (see Corollary 4.4).
Lemma 4.1.
If and , then
Proof.
Let , then inequality is equivalent to inequality
If , then simple computation leads to inequality (4.5).
If , then it is not difficult to verify that and
If , then ; this implies that
From inequalities (4.6) and (4.7), we get
From the well-known Stirling Formula (), we get
Therefore, inequality (4.5) follows from inequalities (4.8) and (4.9).
From the monotonicity of sequence and , we get ; therefore, inequality (4.3) is proved.
Meanwhile, we have
Therefore, inequality (4.4) follows from inequalities (4.10) and
Theorem 4.2.
Let , , and . If , then
Proof.
Let , , and ,
Then inequality (4.12) is equivalent to the following inequality:
where .
If , then
From inequality (4.3) and together with Theorem 1.1, we clearly see that
Therefore, inequality (4.14) is proved.
Corollary 4.3.
Let , , and . If , then
Proof.
Let (), then inequality (4.4) implies that is a strictly increasing sequence. Then from inequality (4.12) we get
Let ; thus, we know that Corollary 4.4 is true.
Corollary 4.4.
If with , then
Remark 4.5.
Many other applications for Theorem 1.1 appeared in [10].
References
Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1952:xii+324.
Carleman T: Sur les fonctions quasi-analytiques. Comptes rendus du Ve Congres des Mathematiciens, Scandinaves, 1922, Helsinki, Finland 181–196.
Yang B, Debnath L: Some inequalities involving the constant , and an application to Carleman's inequality. Journal of Mathematical Analysis and Applications 1998, 223(1):347–353. 10.1006/jmaa.1997.5617
Alzer H: On Carleman's inequality. Portugaliae Mathematica 1993, 50(3):331–334.
Alzer H: A refinement of Carleman's inequality. Journal of Approximation Theory 1998, 95(3):497–499. 10.1006/jath.1996.3256
Johansson M, Persson L-E, Wedestig A: Carleman's inequality-history, proofs and some new generalizations. JIPAM Journal of Inequalities in Pure and Applied Mathematics 2003, 4(3, article 53):-19.
Liu HP, Zhu L: New strengthened Carleman's inequality and Hardy's inequality. Journal of Inequalities and Applications 2007, 2007:-7.
Pečarić J, Stolarsky KB: Carleman's inequality: history and new generalizations. Aequationes Mathematicae 2001, 61(1–2):49–62. 10.1007/s000100050160
Sunouchi GI, Takagi N: A generalization of the Carleman's inequality theorem. Proceedings of the Physico-Mathematical Society of Japan 1934, 16: 164–166.
Zhang XM, Chu YM: New Discussion to Analytic Inequality. Harbin Institute of Technology Press, Harbin, China; 2009.
Acknowledgments
The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This work was partly supported by the National Nature Science Foundation of China under Grant no. 60850005, the Nature Science Foundation of Zhejiang Province under Grant no. Y607128, the Nature Science Foundation of China Central Radio & TV University under Grant no. GEQ1633, and the Nature Science Foundation of Zhejiang Broadcast & TV University under Grant no. XKT-07G19.
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Zhang, XM., Chu, YM. A New Method to Study Analytic Inequalities. J Inequal Appl 2010, 698012 (2010). https://doi.org/10.1155/2010/698012
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DOI: https://doi.org/10.1155/2010/698012