# An Optimal Double Inequality between Power-Type Heron and Seiffert Means

- Yu-Ming Chu
^{1}Email author, - Miao-Kun Wang
^{2}and - Ye-Fang Qiu
^{2}

**2010**:146945

https://doi.org/10.1155/2010/146945

© Yu-Ming Chu et al. 2010

**Received: **29 August 2010

**Accepted: **31 October 2010

**Published: **16 November 2010

## Abstract

## 1. Introduction

respectively.

Recently, the means of two variables have been the subject of intensive research [1–15]. In particular, many remarkable inequalities for and can be found in the literature [16–20].

The main properties for power mean are given in [21].

The purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean . Our main result is the following Theorem 1.1.

Theorem 1.1.

and and are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean , respectively.

## 2. Lemmas

In order to prove our main result, Theorem 1.1, we need two lemmas which we present in this section.

Lemma 2.1.

Proof.

and is strictly decreasing in because of for .

Therefore, Lemma 2.1 follows from (2.2)–(2.4) together with the monotonicity of .

Lemma 2.2.

If , , and , then there exists such that for and for .

Proof.

From (2.24), we know that is strictly decreasing in . Then (2.22) implies that is strictly decreasing in .

From (2.20) and (2.21) together with the monotonicity of , we clearly see that there exists such that is strictly increasing in and strictly decreasing in .

Inequality (2.17) and (2.18) together with the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in .

The piecewise monotonicity of together with (2.14) and (2.15) leads to the fact that there exists such that is strictly increasing in and strictly decreasing in .

From (2.11) and (2.12) together with the piecewise monotonicity of , we conclude that there exists such that is strictly increasing in and strictly decreasing in .

Equations (2.8) and (2.9) together with the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in .

Therefore, Lemma 2.2 follows from (2.5) and (2.6) together with the piecewise monotonicity of .

## 3. Proof of Theorem 1.1

Proof of Theorem 1.1.

Therefore, follows from (3.1)–(3.5).

From (3.12) and (3.13) together with Lemma 2.2, we clearly see that there exists such that is strictly increasing in and strictly decreasing in .

Therefore, follows from (3.6) and (3.7) together with (3.14).

At last, we prove that and are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean , respectively.

Equations (3.15) and (3.17) together with inequality (3.16) imply that for any , there exist and such that for and for .

## Declarations

### Acknowledgments

This work was supported by the Natural Science Foundation of China under Grant no. 11071069, the Natural Science Foundation of Zhejiang Province under Grant no. Y7080106, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no. T200924.

## Authors’ Affiliations

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