- Research Article
- Open Access

# On an Extension of Shapiro's Cyclic Inequality

- Nguyen Minh Tuan
^{1}Email author and - Le Quy Thuong
^{2}

**2009**:491576

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

© N. M. Tuan and L. Q. Thuong. 2009

**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.

## Keywords

- Vector Space
- Quadratic Form
- Direct Calculation
- Basic Vector
- Simple Calculation

## 1. Introduction

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

Theorem 2.1.

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

Proof.

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.

Remark 2.2.

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 .

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 .

- (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.

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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## Copyright

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.