- Research Article
- Open Access
On an Extension of Shapiro's Cyclic Inequality
© N. M. Tuan and L. Q. Thuong. 2009
- Received: 21 August 2009
- Accepted: 13 October 2009
- Published: 15 October 2009
We prove an interesting extension of the Shapiro's cyclic inequality for four and five variables and formulate a generalization of the well-known Shapiro's cyclic inequality. The method used in the proofs of the theorems in the paper concerns the positive quadratic forms.
- Vector Space
- Quadratic Form
- Direct Calculation
- Basic Vector
- Simple Calculation
Obviously, (1.2) is true for every if or .
In this note, by studying (1.2) in the case , we show that it is true when , and false when . Moreover, we give a sufficient condition of , under which (1.2) is true in the case . It is worth saying that if , then (1.2) is false for every even . Two open questions are discussed at the end of this paper.
It holds that (2.1) is true for , and it is false for .
It is easily seen that if , that is, , then for all . This implies that is positive. We thus have .
Combining these facts with (2.4) we conclude that when .
The last inequality is evident as and , so (2.9) follows.
The theorem is proved.
Then is positive if and only if for every . We find the first part of Theorem 2.1.
We thus obtain a sufficient condition under which (1.2) holds for .
If , then (1.2) is true for .
Consider (1.2) in the case , is even, and . According to the proof of the second part of Theorem 2.1, this inequality is false. Indeed, we choose , . By the above counter-example, we conclude .
Find pairs of nonnegative numbers , so that (1.2) is true for every .
For certain , which is sufficient condition of the pair , so that (1.2) is true.
This work is supported partially by Vietnam National Foundation for Science and Technology Development.
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