• Research Article
• Open Access

On an Extension of Shapiro's Cyclic Inequality

Journal of Inequalities and Applications20092009:491576

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

• Accepted: 13 October 2009
• Published:

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.

Keywords

• Vector Space
• Direct Calculation
• Basic Vector
• Simple Calculation

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 , the history of long year proofs of this inequality was interesting, and the certain problems remain (see ). 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

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 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

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
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 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, 334 Nguyen Trai Street, Hanoi, Vietnam
(2)
Department of Mathematics, University of Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam

References 