#### Abstract

We find the least value and the greatest value such that for and all with , where , , and are the harmonic, logarithmic, and -th power means of two positive numbers and , respectively.

#### 1. Introduction

For , the -th power mean and logarithmic mean of two positive numbers and are defined by respectively.

It is well known that is continuous and strictly increasing with respect to for fixed , with . In the recent past, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for and can be found in the literature [1–17]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [18–20]. In [18], the authors study a variant of Jensen's functional equation involving , which appears in a heat conduction problem. A representation of as an infinite product and an iterative algorithm for computing the logarithmic mean as the common limit of two sequences of special geometric and arithmetic means are given in [8]. In [21, 22], it is shown that can be expressed in terms of Gauss's hypergeometric function . And, in [21], the authors prove that the reciprocal of the logarithmic mean is strictly totally positive, that is, every determinant with elements , where and , is positive for all .

Let (), , and be the arithmetic, identric, geometric, and harmonic means of two positive numbers and , respectively, then it is well known that for all , with .

In [23], Alzer and Janous established the following best possible inequality: for all with .

In [8, 11, 24], the authors presented bounds for in terms of and for all with .

The following companion of (1.3) provides inequalities for the geometric and arithmetic means of and . A proof can be found in [25] for all with .

The following sharp bounds for , , , and in terms of the power means are proved in [4, 5, 7, 9, 16, 25, 26]: for all with .

Alzer and Qiu [27] found the sharp bound of in terms of the power mean as follows: for all with , with the best possible parameter .

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

#### 2. Lemmas

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

Lemma 2.1. *Let , , and . Then for .*

*Proof. *Simple computations lead to
where
for .

Inequality (2.6) implies that is strictly decreasing in , then from (2.4) and (2.5) we know that exists such that for and for . Hence, equation (2.3) leads to the conclusion that is strictly increasing in and strictly decreasing in .

Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the piecewise monotonicity of .

Lemma 2.2. *Let , , and , then for .*

*Proof. *Let . Then simple computations lead to
,
,
where
for .

From (2.13) and (2.14), we clearly see that for , then (2.12) leads to the conclusion that is strictly in .

Therefore, Lemma 2.2 follows from (2.7)–(2.11) and the monotonicity of .

Lemma 2.3. *Let , , and , then for .*

*Proof. *Let and , then simple computations lead to
,
,
,
,
where and are defined as in Lemmas 2.1 and 2.2, respectively.

From (2.19) and (2.10) together with Lemmas 2.1 and 2.2, we clearly see that is strictly increasing in .

Therefore, Lemma 2.3 follows from (2.15)–(2.18) and the monotonicity of .

#### 3. Main Result

Theorem 3.1. *Inequality
**
holds for and all with , and is the best possible lower power mean bound for the sum .*

*Proof. *We divide the proof of inequality (3.1) into two cases.*Case 1 (). *Without loss of generality, we assume that and put , then from (1.1) and (1.2), we have
Let
then simple computations lead to
where ,
where ,
for .

Therefore, inequality (3.1) follows easily from (3.2)–(3.6).*Case 2 (). *Without loss of generality, we assume that . Let and , then from (1.1) and (1.2), one has
Let
Then simple computations lead to
where is defined as in Lemma 2.3.

From Lemma 2.3 and (3.10), we clearly see that is strictly increasing in .

Therefore, inequality (3.1) follows from (3.7)–(3.9) and the monotonicity of .

Next, we prove that is the best possible lower power mean bound for the sum if .

For any , , and , one has where . Letting and making use of Taylor expansion, we have

Equations (3.11) and (3.12) imply that for any and there exists , such that for .

*Remark 3.2. *If , then from (1.1) and (1.2), we have

Equation (3.13) implies that for any , there exists , such that for . Therefore, is the least value of in such that inequality (3.1) holds for all with .

#### Acknowledgments

This research is supported by the N. S. Foundation of China under Grant 11071069, N. S. Foundation of Zhejiang province under Grants nos. Y7080106 and Y6100170, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no. T200924.