• Research Article
• Open Access

# An Optimal Double Inequality for Means

Journal of Inequalities and Applications20102010:578310

https://doi.org/10.1155/2010/578310

• Accepted: 27 September 2010
• Published:

## Abstract

For , the generalized logarithmic mean , arithmetic mean and geometric mean of two positive numbers and are defined by , ; , , , ; , , ; , , ; and , respectively. In this paper, we give an answer to the open problem: for , what are the greatest value and the least value , such that the double inequality holds for all ?

## Keywords

• Real Number
• Simple Computation
• Taylor Expansion
• Recent Past
• Positive Real Number

## 1. Introduction

For , the generalized logarithmic mean of two positive numbers and is defined by
(1.1)

It is wellknown that is continuous and increasing with respect to for fixed and . In the recent past, the generalized logarithmic mean has been the subject of intensive research. Many remarkable inequalities and monotonicity results can be found in the literature [19]. It might be surprising that the generalized logarithmic mean, has applications in physics, economics, and even in meteorology [1013].

If we denote by , , , and the arithmetic mean, identric mean, logarithmic mean, geometric mean and harmonic mean of two positive numbers and , respectively, then
(1.2)
For , the th power mean of two positive numbers and is defined by
(1.3)
In [14], Alzer and Janous established the following sharp double inequality (see also [15], Page 350):
(1.4)

for all .

For , Janous [16] found the greatest value and the least value such that
(1.5)

for all .

In [1719] the authors present bounds for and in terms of and .

Theorem A.

For all positive real numbers and with , one has
(1.6)

The proof of the following Theorem B can be found in [20].

Theorem B.

For all positive real numbers and with , one has
(1.7)

The following Theorems C–E were established by Alzer and Qiu in [21].

Theorem C.

The inequalities
(1.8)

hold for all positive real numbers and with if and only if and .

Theorem D.

Let and be real numbers with . If , then
(1.9)
And if , then
(1.10)

Theorem E.

For all real numbers and with , one has
(1.11)

with the best possible parameter .

However, the following problem is still open: for , what are the greatest value and the least value , such that the double inequality
(1.12)

holds for all ? The purpose of this paper is to give the solution to this open problem.

## 2. Lemmas

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

Lemma 2.1.

If , then
(2.1)

Proof.

Let , then simple computation yields
(2.2)
(2.3)
where
(2.4)
where
(2.5)
(2.6)
(2.7)
If , then from (2.7) we clearly see that
(2.8)

Therefore, Lemma 2.1 follows from (2.3)–(2.6) and (2.8).

Lemma 2.2.

If , then
(2.9)

Proof.

Let , then simple computation leads to
(2.10)
where
(2.11)
where
(2.12)
where
(2.13)
(2.14)
If , then from (2.14) we clearly see that
(2.15)

From (2.10)–(2.13) and (2.15) we know that for .

## 3. Main Results

Theorem 3.1.

If , then for all , with equality if and only if   , and the constant in , cannot be improved.

Proof.

If , then we clearly see that .

If , without loss of generality, we assume that . Let and
(3.1)

Firstly, we prove . The proof is divided into three cases.

Case 1.

. We note that (1.1) leads to the following identity:
(3.2)

From (3.2) and Lemma 2.1 we clearly see that for and .

Case 2.

. Equation (1.1) leads to the following identity:
(3.3)

From (3.3) and Lemma 2.2 we clearly see that for and .

Case 3.

. From (1.1) we have the following identity:
(3.4)
Equation (3.4) and elementary computation yields
(3.5)
(3.6)
where
(3.7)
(3.8)
If , then (3.8) implies
(3.9)

for . Therefore, follows from (3.5)–(3.7) and (3.9).

If , then (3.8) leads to
(3.10)

for . Therefore, follows from (3.5)–(3.7) and (3.10).

Next, we prove that the constant in the inequality cannot be improved. The proof is divided into five cases.

Case 1.

. For any , let , then (1.1) leads to
(3.11)

where .

Making use of Taylor expansion we get
(3.12)

Case 2.

. For any , let , then
(3.13)

where .

Using Taylor expansion we have
(3.14)

Case 3.

. For any , let , then
(3.15)

where .

Making use of Taylor expansion and elaborated calculation we have
(3.16)

Case 4.

. For any , let , then
(3.17)

where .

Using Taylor expansion and elaborated calculation we have
(3.18)

Case 5.

. For any , let , then
(3.19)

where .

Using Taylor expansion and elaborated calculation we get
(3.20)

Cases 1–5 show that for any , there exists , for any there exists such that for .

Theorem 3.2.

If , then for all , with equality if and only if , and the constant in cannot be improved.

Proof.

If , then we clearly see that .

If , without loss of generality, we assume that . Let and
(3.21)
Firstly, we prove for . Simple computation leads to
(3.22)
where
(3.23)

for and .

From (3.23) we clearly see that
(3.24)

for .

Since , we have for . Therefore, follows from (3.22) and (3.24).

Next, we prove that the constant cannot be improved.

For any , we have
(3.25)

Equation (3.25) imply that for any there exists , such that for .

## Declarations

### Acknowledgment

This work was supported by the Natural Science Foundation of Zhejiang Broadcast and TV University (Grant no. XKT-09G21).

## Authors’ Affiliations

(1)
Huzhou Broadcast and TV University, Huzhou, 313000, China

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