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

- Yu-Ming Chu
^{1}Email author, - Bo-Yong Long
^{2}, - Wei-Ming Gong
^{1}and - Ying-Qing Song
^{1}

**2013**:10

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

© Chu et al.; licensee Springer 2013

**Received: **15 June 2012

**Accepted: **18 December 2012

**Published: **7 January 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.

## Keywords

## 1 Introduction

*a*and

*b*are defined by

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].

*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.

*a*and

*b*is defined as the common limit of sequences $\{{a}_{n}\}$ and $\{{b}_{n}\}$, which are given by

*a*and

*b*with $a\ne b$, respectively. Then it is well known that the inequalities

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

*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 $a\ne b$.

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

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

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

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*.

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 ${\beta}_{3}\ge \pi /4$.

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

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 $[1,+\mathrm{\infty})$.

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

for $x>1$.

for $x>1$.

for $x>1$.

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

for all $x>1$.

for all $x>1$.

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 $P(a,b)$.

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 $T(a,b)$.

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 $M(a,b)$.

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.

## Declarations

### 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.

## Authors’ Affiliations

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