# 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.

## 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.

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 .

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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## Copyright

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