- Research Article
- Open Access

# The Optimal Convex Combination Bounds for Seiffert's Mean

- Hong Liu
^{1}Email author and - Xiang-Ju Meng
^{2}

**2011**:686834

https://doi.org/10.1155/2011/686834

© H. Liu and X.-J. Meng. 2011

**Received:**28 November 2010**Accepted:**28 February 2011**Published:**13 March 2011

## Abstract

We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values , and the least values , such that the double inequalities and hold for all with . Here, , , , and denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers and , respectively.

## Keywords

- Real Number
- Simple Computation
- Positive Real Number
- Optimal Convex
- Convex Combination

## 1. Introduction

for all with .

holds for all with .

hold for all with .

## 2. Main Results

Firstly, we present the optimal convex combination bounds of contraharmonic and geometric means for Seiffert's mean as follows.

Theorem 2.1.

The double inequality holds for all with if and only if and .

Proof.

for all with .

We divide the proof into two cases.

Case 1 ( ).

Case 2 ( ).

Thus for all . Therefore, the first inequality in (2.1) follows from (2.2) and (2.3).

Secondly, we prove that is the best possible lower convex combination bound of the contraharmonic and geometric means for Seiffert's mean.

for .

Finally, we prove that is the best possible upper convex combination bound of the contraharmonic and geometric means for Seiffert's mean.

for .

Secondly, we present the optimal convex combination bounds of the contraharmonic and harmonic means for Seiffert's mean as follows.

Theorem 2.2.

The double inequality holds for all with if and only if and .

Proof.

for all with .

We divide the proof into two cases.

Case 1 ( ).

Case 2 ( ).

Thus for all . Therefore, the second inequality in (2.24) follows from (2.25) and (2.26).

Secondly, we prove that is the best possible upper convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.

for .

Finally, we prove that is the best possible lower convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.

for .

## Declarations

### Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. This research is partly supported by N S Foundation of Hebei Province (Grant A2011201011), and the Youth Foundation of Hebei University (Grant 2010Q24).

## Authors’ Affiliations

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## Copyright

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