- Research Article
- Open Access
On an Extension of Shapiro's Cyclic Inequality
Journal of Inequalities and Applications volume 2009, Article number: 491576 (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.
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 , 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
It holds that (2.1) is true for , and it is false for .
By the Cauchy inequality, we have
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:
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 counter-example 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.
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 .
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 .
Open Question 2 s.
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.
Troesch BA: The validity of Shapiro's cyclic inequality. Mathematics of Computation 1989,53(188):657–664.
Bushell PJ: Shapiro's cyclic sum. The Bulletin of the London Mathematical Society 1994,26(6):564–574. 10.1112/blms/26.6.564
Bushell PJ, McLeod JB: Shapiro's cyclic inequality for even . Journal of Inequalities and Applications 2002,7(3):331–348. 10.1155/S1025583402000164
Diananda PH: On a cyclic sum. Proceedings of the Glasgow Mathematical Association 1963, 6: 11–13. 10.1017/S2040618500034626
Drinfeld VG: A certain cyclic inequality. Mathematical Notes 1971, 9: 68–71. 10.1007/BF01316982
Fink AM: Shapiro's inequality. In Recent Progress in Inequalities, Mathematics and Its Applications, Part 13. Volume 430. 1st edition. Edited by: Milovanovic GV. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:241–248.
Mordell LJ: On the inequality and some others Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1958, 22: 229–241. 10.1007/BF02941955
Mordell LJ: Note on the inequality and some others Journal of the London Mathematical Society 1962, 37: 176–178. 10.1112/jlms/s1-37.1.176
This work is supported partially by Vietnam National Foundation for Science and Technology Development.
About this article
Cite this article
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
- Vector Space
- Quadratic Form
- Direct Calculation
- Basic Vector
- Simple Calculation