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On an Extension of Shapiro's Cyclic Inequality
Journal of Inequalities and Applications volume 2009, Article number: 491576 (2009)
Abstract
We prove an interesting extension of the Shapiro's cyclic inequality for four and five variables and formulate a generalization of the wellknown Shapiro's cyclic inequality. The method used in the proofs of the theorems in the paper concerns the positive quadratic forms.
1. Introduction
In 1954, Harold Seymour Shapiro proposed the inequality for a cyclic sum in variables as follows:
where , and for . Although (1.1) was settled in 1989 by Troesch [1], the history of long year proofs of this inequality was interesting, and the certain problems remain (see [1–8]). Motivated by the directions of generalizations and proofs of (1.1), we consider the following inequality:
where and . It is clear that (1.2) is true for . Indeed, by the Cauchy inequality, we have
It follows that
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.
2. Main Result
Without loss generality of (1.2), we assume that . However, (1.2) for now is of the form
Theorem 2.1.
It holds that (2.1) is true for , and it is false for .
Proof.
By the Cauchy inequality, we have
Hence
It is an equality if and only if
Consider the following quadratic form:
By a simple calculation we obtain the canonical quadratic form as follows:
where
It is easily seen that if , that is, , then for all . This implies that is positive. We thus have .
Now let us consider the cases when vanishes. This depends considerably on the comparison of with . If , that is, , then the quadratic form attains at and . By (2.4) we assert that whenever and . Also, if , then vanishes if and only if
Combining these facts with (2.4) we conclude that when .
Now we give a counterexample to (2.1) in the case , that is, . Let , , and . We will prove that
It is obvious that
The last inequality is evident as and , so (2.9) follows.
The theorem is proved.
Remark 2.2.
Let denote the matrix of the quadratic form in the canonical base of the real vector space . Namely,
Let , , and be the principal minors of orders , , and respectively, of . By direct calculation we obtain
Then is positive if and only if for every . We find the first part of Theorem 2.1.
Thanks to the idea of using positive quadratic form we now study (1.2) in the case . It is sufficient to consider the case . By the Cauchy inequality, we reduce our work to the following inequality
The matrix of in an appropriate system of basic vectors is of the form
which has the principal minors
This implies that the necessary and sufficient condition for the positivity of the quadratic form is
We thus obtain a sufficient condition under which (1.2) holds for .
Theorem 2.3.
If , then (1.2) is true for .
Remark 2.4.
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 counterexample, we conclude .
Open Question 2 s.

(a)
Find pairs of nonnegative numbers , so that (1.2) is true for every .

(b)
For certain , which is sufficient condition of the pair , so that (1.2) is true.
References
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Drinfeld VG: A certain cyclic inequality. Mathematical Notes 1971, 9: 68–71. 10.1007/BF01316982
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Acknowledgment
This work is supported partially by Vietnam National Foundation for Science and Technology Development.
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Tuan, N.M., Thuong, L.Q. On an Extension of Shapiro's Cyclic Inequality. J Inequal Appl 2009, 491576 (2009). https://doi.org/10.1155/2009/491576
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DOI: https://doi.org/10.1155/2009/491576
Keywords
 Vector Space
 Quadratic Form
 Direct Calculation
 Basic Vector
 Simple Calculation