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# Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 10 (2013)

## Abstract

In this paper, we prove three sharp inequalities as follows: P(a,b)>{L}_{2}(a,b), T(a,b)>{L}_{5}(a,b) and M(a,b)>{L}_{4}(a,b) for all a,b>0 with a\ne b. Here, {L}_{r}(a,b), M(a,b), P(a,b) and T(a,b) are the *r* th generalized logarithmic, Neuman-Sándor, first and second Seiffert means of *a* and *b*, respectively.

**MSC:**26E60.

## 1 Introduction

The Neuman-Sándor mean M(a,b) [1] and the first and second Seiffert means P(a,b) [2] and T(a,b) [3] of two positive numbers *a* and *b* are defined by

and

respectively.

Recently, these means, *M*, *P* and *T*, have been the subject of intensive research. In particular, many remarkable inequalities for *M*, *P* and *T* can be found in the literature [1, 4–8].

The power mean {M}_{p}(a,b) of order *r* of two positive numbers *a* and *b* is defined by

The main properties for {M}_{p}(a,b) are given in [9]. In particular, the function p\mapsto {M}_{p}(a,b) (a\ne b) is continuous and strictly increasing on ℝ.

The arithmetic-geometric mean \mathit{AG}(a,b) of two positive numbers *a* and *b* is defined as the common limit of sequences \{{a}_{n}\} and \{{b}_{n}\}, which are given by

Let H(a,b)=2ab/(a+b), G(a,b)=\sqrt{ab}, L(a,b)=(b-a)/(logb-loga), I(a,b)=1/e{({b}^{b}/{a}^{a})}^{1/(b-a)}, A(a,b)=(a+b)/2, and S(a,b)=\sqrt{({a}^{2}+{b}^{2})/2} be the harmonic, geometric, logarithmic, identric, arithmetic and root-square means of two positive numbers *a* and *b* with a\ne b, respectively. Then it is well known that the inequalities

hold for all a,b>0 with a\ne b.

For r>0 the *r* th generalized logarithmic mean {L}_{r}(a,b) of two positive numbers *a* and *b* is defined by

It is not difficult to verify that {L}_{r}(a,b) is continuous and strictly increasing with respect to r\in (0,+\mathrm{\infty}) for fixed a,b>0 with a\ne b.

In [2, 3] Seiffert proved that the double inequalities

and

hold for all a,b>0 with a\ne b.

The following bounds for the first Seiffert mean P(a,b) in terms of power mean were presented by Jagers in [10]:

for all a,b>0 with a\ne b.

Hästö [11, 12] proved that the function x\to T(1,x)/{M}_{p}(1,x) is increasing on (0,+\mathrm{\infty}) if p\le 1 and found the sharp lower power mean bound for the Seiffert mean P(a,b) as follows:

for all a,b>0 with a\ne b.

In [13] the authors presented the following best possible Lehmer mean bounds for the Seiffert means P(a,b) and T(a,b):

for all a,b>0 with a\ne b. Here, {\overline{L}}_{p}(a,b)=({a}^{p+1}+{b}^{p+1})/({a}^{p}+{b}^{p}) is the Lehmer mean of *a* and *b*.

In [14, 15] the authors proved that the inequalities

and

hold for all a,b>0 with a\ne b if and only if {\alpha}_{1}\le (4-\pi )/[(\sqrt{2}-1)\pi ], {\beta}_{1}\ge 2/3, {\alpha}_{2}\le 2/3, {\beta}_{2}\ge 4-2log\pi /log2, {\alpha}_{3}\le 3/5 and {\beta}_{3}\ge \pi /4.

For all a,b>0 with a\ne b, the following inequalities can be found in [16, 17]:

Neuman and Sándor [1] established that

and

for all a,b>0 with a\ne b. In particular, the Ky Fan inequalities

hold for all 0<a,b\le 1/2 with a\ne b, {a}^{\prime}=1-a and {b}^{\prime}=1-b.

It is the aim of this paper to find the best possible generalized logarithmic mean bounds for the Neuman-Sándor and Seiffert means.

## 2 Lemmas

In order to establish our main results, we need three lemmas, which we present in this section.

**Lemma 2.1**
*The inequality*

*holds for all* x>1.

*Proof* Let

Then f(x) can be rewritten as

where

Simple computations lead to

where

where

and

for x>1.

Inequality (2.11) implies that {f}_{3}^{\u2033}(x) is strictly decreasing in [1,+\mathrm{\infty}), then equation (2.10) leads to the conclusion that {f}_{3}^{\prime}(x) is strictly decreasing in [1,+\mathrm{\infty}).

From equations (2.4)-(2.9) and the monotonicity of {f}_{3}^{\prime}(x), we clearly see that

for x>1.

Therefore, inequality (2.1) follows from equations (2.2) and (2.3) together with inequality (2.12). □

**Lemma 2.2**
*The inequality*

*holds for all* x>1.

*Proof* Let

Then simple computations lead to

where

where

where

Let

Then

where

for x>1.

Equation (2.27) and inequality (2.28) lead to the conclusion that {g}_{4}(x) is strictly decreasing in [1,+\mathrm{\infty}). Then equation (2.26) implies that

for x>1.

Inequalities (2.25) and (2.29) imply that {g}_{3}^{\prime}(x)<0. Then equation (2.24) shows that

for x>1.

From equations (2.17)-(2.23) and inequality (2.30), we clearly see that

for x>1.

Therefore, inequality (2.13) follows easily from equations (2.14)-(2.16) and inequality (2.31). □

**Lemma 2.3**
*The inequality*

*holds for all* x>1.

*Proof* Let

Then simple computations lead to

where

where

where

where

where

for all x>1.

Equations (2.45) and (2.46) together with inequality (2.47) imply that {h}_{5}(x) is strictly decreasing in [1,+\mathrm{\infty}). Then equation (2.44) leads to

for all x>1.

From equations (2.36)-(2.43) and inequality (2.48), we clearly see that

for all x>1.

Therefore, inequality (2.32) follows from equations (2.33)-(2.35) and inequality (2.49). □

## 3 Main results

**Theorem 3.1**
*The inequality*

*holds for all* a,b>0 *with* a\ne b, *and* {L}_{2}(a,b) *is the best possible lower generalized logarithmic mean bound for the first Seiffert mean* P(a,b).

*Proof* From (1.2) and (1.4), we clearly see that both P(a,b) and {L}_{r}(a,b) are symmetric and homogenous of degree one. Without loss of generality, we assume that b=1 and a={x}^{2}>1. Then (1.2) and (1.4) lead to

Therefore, P({x}^{2},1)>{L}_{2}({x}^{2},1) follows from Lemma 2.1 and equation (3.1).

Next, we prove that {L}_{2}(a,b) is the best possible lower generalized logarithmic mean bound for the first Seiffert mean P(a,b).

For any \u03f5>0 and x>0, from (1.2) and (1.4), one has

Letting x\to 0 and making use of Taylor expansion, we get

Equations (3.2) and (3.3) imply that for any \u03f5>0, there exists {\delta}_{1}={\delta}_{1}(\u03f5)>0 such that {L}_{2+\u03f5}(1+x,1)>P(1+x,1) for x\in (0,{\delta}_{1}). □

**Remark 3.1** It follows from (1.2) and (1.4) that

for all \lambda >0.

Equation (3.4) implies that \lambda >0 such that {L}_{\lambda}(a,b)>P(a,b) for all a,b>0 does not exist.

**Theorem 3.2**
*The inequality*

*holds for all* a,b>0 *with* a\ne b, *and* {L}_{5}(a,b) *is the best possible lower generalized logarithmic mean bound for the second Seiffert mean* T(a,b).

*Proof* Without loss of generality, we assume that b=1 and a=x>1. Then (1.3) and (1.4) lead to

Therefore, T(x,1)>{L}_{5}(x,1) follows from Lemma 2.2 and equation (3.5).

Next, we prove that {L}_{5}(a,b) is the best possible lower generalized logarithmic mean bound for the second Seiffert mean T(a,b).

For any \u03f5>0 and x>0, from (1.3) and (1.4), one has

Letting x\to 0 and making use of Taylor expansion, we have

Equations (3.6) and (3.7) imply that for any \u03f5>0, there exists {\delta}_{2}={\delta}_{2}(\u03f5)>0 such that {L}_{5+\u03f5}(1+x,1)>T(1+x,1) for x\in (0,{\delta}_{2}). □

**Remark 3.2** It follows from (1.3) and (1.4) that

for all \mu >0.

Equation (3.8) implies that \mu >0 such that {L}_{\mu}(a,b)>T(a,b) for all a,b>0 does not exist.

**Theorem 3.3**
*The inequality*

*holds for all* a,b>0 *with* a\ne b, *and* {L}_{4}(a,b) *is the best possible lower generalized logarithmic mean bound for the Neuman*-*Sándor mean* M(a,b).

*Proof* Without loss of generality, we assume that b=1 and a=x>1. Then (1.1) and (1.4) lead to

Therefore, M(x,1)>{L}_{4}(x,1) follows from Lemma 2.3 and equation (3.9).

Next, we prove that {L}_{4}(a,b) is the best possible lower generalized logarithmic mean bound for the Neuman-Sándor mean M(a,b).

For any \u03f5>0 and x>0, from (1.1) and (1.4), one has

Letting x\to 0 and making use of Taylor expansion, we have

Equations (3.10) and (3.11) imply that for any \u03f5>0, there exists {\delta}_{3}={\delta}_{3}(\u03f5)>0 such that {L}_{4+\u03f5}(1+x,1)>M(1+x,1) for x\in (0,{\delta}_{3}). □

**Remark 3.3** It follows from (1.1) and (1.4) that

for all \nu >0.

Equation (3.12) implies that \nu >0 such that {L}_{\nu}(a,b)>M(a,b) for all a,b>0 does not exist.

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## Acknowledgements

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003 and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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The authors declare that they have no competing interests.

### Authors’ contributions

YMC provided the main idea and carried out the proof of Remarks 3.1-3.3 in this article. BYL carried out the proof of Lemma 2.1 and Theorem 3.1 in this article. WMG carried out the proof of Lemma 2.2 and Theorem 3.2 in this article. YQS carried out the proof of Lemma 2.3 and Theorem 3.3 in this article. All authors read and approved the final manuscript.

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Chu, YM., Long, BY., Gong, WM. *et al.* Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means.
*J Inequal Appl* **2013**, 10 (2013). https://doi.org/10.1186/1029-242X-2013-10

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DOI: https://doi.org/10.1186/1029-242X-2013-10

### Keywords

- generalized logarithmic mean
- Neuman-Sándor mean
- first Seiffert mean
- second Seiffert mean