A New Method to Study Analytic Inequalities
© X.-M. Zhang and Y.-M. Chu. 2010
Received: 16 October 2009
Accepted: 24 December 2009
Published: 5 January 2010
1. Monotonicity Theorems
In this section, we present the main results of this paper.
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.
It follows from Theorems 1.1 and 1.2 that we get the following Corollaries 1.3–1.6.
2. Unifying Proof of Some Well-Known Inequality
Proposition 2.1 (Power Mean Inequality).
Proposition 2.2 (Holder Inequality).
Proposition 2.3 (Minkowski Inequality).
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.
and we clearly see that inequalities (3.4) and (3.3) are true.
4. A Refinement of Carleman's Inequality
with the best possible constant factor (see ).
Recently, Yang and Debnath  gave a strengthened version of (4.1) as follows:
Therefore, inequality (4.5) follows from inequalities (4.8) and (4.9).
Meanwhile, we have
Therefore, inequality (4.14) is proved.
Many other applications for Theorem 1.1 appeared in .
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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