Open Access

On an Extension of Shapiro's Cyclic Inequality

Journal of Inequalities and Applications20092009:491576

https://doi.org/10.1155/2009/491576

Received: 21 August 2009

Accepted: 13 October 2009

Published: 15 October 2009

Abstract

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.

1. Introduction

In 1954, Harold Seymour Shapiro proposed the inequality for a cyclic sum in variables as follows:
(1.1)
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 [18]). Motivated by the directions of generalizations and proofs of (1.1), we consider the following inequality:
(1.2)
where and . It is clear that (1.2) is true for . Indeed, by the Cauchy inequality, we have
(1.3)
It follows that
(1.4)

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
(2.1)

Theorem 2.1.

It holds that (2.1) is true for , and it is false for .

Proof.

By the Cauchy inequality, we have
(2.2)
Hence
(2.3)
It is an equality if and only if
(2.4)
Consider the following quadratic form:
(2.5)
By a simple calculation we obtain the canonical quadratic form as follows:
(2.6)
where
(2.7)

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
(2.8)

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
(2.9)
It is obvious that
(2.10)

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,
(2.11)
Let , , and be the principal minors of orders , , and respectively, of . By direct calculation we obtain
(2.12)

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
(2.13)
The matrix of in an appropriate system of basic vectors is of the form
(2.14)
which has the principal minors
(2.15)
This implies that the necessary and sufficient condition for the positivity of the quadratic form is
(2.16)

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 counter-example, we conclude .

Open Question 2 s.
  1. (a)

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

     
  2. (b)

    For certain , which is sufficient condition of the pair , so that (1.2) is true.

     

Declarations

Acknowledgment

This work is supported partially by Vietnam National Foundation for Science and Technology Development.

Authors’ Affiliations

(1)
Department of Mathematical Analysis, University of Hanoi
(2)
Department of Mathematics, University of Hanoi

References

  1. Troesch BA: The validity of Shapiro's cyclic inequality. Mathematics of Computation 1989,53(188):657–664.MathSciNetMATHGoogle Scholar
  2. Bushell PJ: Shapiro's cyclic sum. The Bulletin of the London Mathematical Society 1994,26(6):564–574. 10.1112/blms/26.6.564MathSciNetView ArticleMATHGoogle Scholar
  3. Bushell PJ, McLeod JB: Shapiro's cyclic inequality for even . Journal of Inequalities and Applications 2002,7(3):331–348. 10.1155/S1025583402000164MathSciNetMATHGoogle Scholar
  4. Diananda PH: On a cyclic sum. Proceedings of the Glasgow Mathematical Association 1963, 6: 11–13. 10.1017/S2040618500034626MathSciNetView ArticleMATHGoogle Scholar
  5. Drinfeld VG: A certain cyclic inequality. Mathematical Notes 1971, 9: 68–71. 10.1007/BF01316982MathSciNetView ArticleGoogle Scholar
  6. 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.Google Scholar
  7. Mordell LJ: On the inequality and some others Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1958, 22: 229–241. 10.1007/BF02941955MathSciNetView ArticleMATHGoogle Scholar
  8. Mordell LJ: Note on the inequality and some others Journal of the London Mathematical Society 1962, 37: 176–178. 10.1112/jlms/s1-37.1.176MathSciNetView ArticleMATHGoogle Scholar

Copyright

© N. M. Tuan and L. Q. Thuong. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.