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# The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

*Journal of Inequalities and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 436457 (2010)

## Abstract

We find the greatest value and least value such that the double inequality holds for all with . Here , , and denote the arithmetic, harmonic, and Seiffert's means of two positive numbers and , respectively.

## 1. Introduction

For with the Seiffert's mean was introduced by Seiffert [1] as follows:

Recently, the inequalities for means have been the subject of intensive research [2â€“11]. In particular, many remarkable inequalities for the Seiffert's mean can be found in the literature [12â€“17]. The Seiffert's mean can be rewritten as (see [14, ()])

Let , , , , and be the arithmetic, geometric, harmonic, identric, and logarithmic means of two positive real numbers and with . Then

In [1], Seiffert proved that

for all with .

Later, Seiffert [18] established that

for all with .

In [19], SÃ¡ndor proved that

for all with .

The following bounds for the Seiffert's mean in terms of the power mean were presented by Jagers in [17]:

for all with .

HÃ¤stÃ¶ [13] found the sharp lower power bound for the Seiffert's mean as follows:

for all with .

The purpose of this paper is to find the greatest value and the least value such that the double inequality holds for all with .

## 2. Main Result

Theorem 2.1.

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

Proof.

Firstly, we prove that

for all with .

Without loss of generality, we assume . Let and . Then (1.1) leads to

Let

Then simple computations lead to

where

We divide the proof into two cases.

Case 1.

If , then it follows from (2.8) that

for .

Therefore, inequality (2.1) follows from (2.3)â€“(2.5) and (2.7) together with (2.9).

Case 2.

If , then from (2.8) we have

From (2.24) and (2.25) we clearly see that for , hence is strictly decreasing in . It follows from (2.22) and (2.23) together with the monotonicity of that there exists such that for and for , hence is strictly increasing in and strictly decreasing in .

From (2.19) and (2.20) together with the monotonicity of we know that there exists such that for ) and for , hence, is strictly increasing in and strictly decreasing in .

From (2.16) and (2.17) together with the monotonicity of we clearly see that there exists such that is strictly increasing in and strictly decreasing in . It follows from (2.13) and (2.14) together with the monotonicity of that there exists such that is strictly increasing in and strictly decreasing in . Then (2.7), (2.10) and (2.11) imply that there exists such that is strictly increasing in and strictly decreasing in .

Note that (2.6) becomes

for .

It follows from (2.5) and (2.26) together with the monotonicity of that

for .

Therefore, inequality (2.2) follows from (2.3) and (2.4) together with (2.27).

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

For any and , from (1.1) we have

where

It follows from (2.29) that

If , then (2.31) leads to

From (2.32) and the continuity of we clearly see that there exists such that

for . Then (2.30) and (2.33) imply that

for .

Therefore, for follows from (2.28) and (2.34).

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

For , then from (1.1) one has

Inequality (2.35) implies that for any there exists such that for .

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## Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China (Grant 60850005), N S Foundation of Zhejiang Province (Grants Y7080106 and Y607128), and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant T200924).

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Chu, YM., Qiu, YF., Wang, MK. *et al.* The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean.
*J Inequal Appl* **2010**, 436457 (2010). https://doi.org/10.1155/2010/436457

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DOI: https://doi.org/10.1155/2010/436457