- Research Article
- Open Access

# Some Comparison Inequalities for Generalized Muirhead and Identric Means

- Miao-Kun Wang
^{1}, - Yu-Ming Chu
^{1}Email author and - Ye-Fang Qiu
^{1}

**2010**:295620

https://doi.org/10.1155/2010/295620

© Miao-Kun Wang et al. 2010

**Received:**21 December 2009**Accepted:**23 January 2010**Published:**28 January 2010

## Abstract

For , with , the generalized Muirhead mean with parameters and and the identric mean are defined by and , , , , respectively. In this paper, the following results are established: (1) for all with and ; (2) for all with and ; (3) if , then there exist such that and .

## Keywords

- Real Number
- Simple Computation
- Intensive Research
- Classified Region
- Sharp Inequality

## 1. Introduction

For , , with , the generalized Muirhead mean with parameters and and the identric mean are defined by

respectively.

The generalized Muirhead mean was introduced by Trif [1], the monotonicity of with respect to or was discussed, and a comparison theorem and a Minkowski-type inequality involving the generalized Muirhead mean were discussed.

It is easy to see that the generalized Muirhead mean is continuous on the domain and differentiable with respect to for fixed with . It is symmetric in and and in and . Many means are special cases of the generalized Muirhead mean, for example,

The well-known Muirhead inequality [2] implies that if are fixed, then is Schur convex on the domain and Schur concave on the domain . Chu and Xia [3] discussed the Schur convexity and Schur concavity of with respect to for fixed with .

Recently, the identric mean has been the subject of intensive research. In particular, many remarkable inequalities for the identric mean can be found in the literature [4–13].

The power mean of order of the positive real numbers and is defined by

The main properties of the power mean are given in [14]. In particular, is continuous and increasing with respect to for fixed . Let ,

for all with .

The following sharp inequality is due to Carlson [15]:

for all with .

Pittenger [16] proved that

for all with , and and are the optimal upper and lower power mean bounds for the identric mean .

In [8, 9], Sándor established that

for all with .

Alzer and Qiu [5] proved the inequalities

for all with if and only if and .

In [3], Chu and Xia proved that

for all and , and

for all and .

Our purpose in what follows is to compare the generalized Muirhead mean with the identric mean . Our main result is Theorem 1.1 which follows.

Theorem.

Suppose that , and . The following statements hold,

() If , then for all with

() If , then for all with

() If , then there exist such that and .

## 2. Lemma

In order to prove Theorem 1.1 we need Lemma 2.1 that follows.

Lemma.

then the following statements hold.

- (2)
If , , and , then for

(3) If , then for .

Proof.

( ) We divide the proof of Lemma 2.1( ) into two cases.

Case.

Therefore, for easily follows from (2.2), (2.5), (2.7), (2.10), and (2.14).

Case 2.

In fact, we clearly see that for , and for and .

Equation (2.15) and imply that

Therefore, for follows from (2.16) together with that can be rewritten as

( ) If , , and , then from (2.13), (2.12), (2.9), and (2.4) we get

- (3)
If , then we clearly see that inequalities (2.14) again hold, and for follows from (2.2), (2.5), (2.7), and (2.10) together with (2.14).

## 3. Proof of Theorem 1.1

Proof of Theorem 1.1.

Then we clearly see that , and .

Without loss of generality, we assume that . From the symmetry we clearly see that Theorem 1.1 is true if we prove that is positive, negative, and neither positive nor negative with respect to for , and .

Let , then (1.1) and (1.2) lead to

where is defined as in Lemma 2.1.

We divide the proof into three cases.

Case.

Subcase 1.

for .

Equations (3.3)–(3.8) imply that

for .

Therefore, follows from (3.2) and (3.9).

Subcase 2.

Case.

Subcase 3.

for .

Therefore, follows from (3.2)–(3.7) and (3.11).

Subcase 4.

Subcase 5.

Subcase 6.

Case.

Subcase 7.

Inequality (3.14) and the continuity of imply that there exists such that

for .

From (2.2) and (3.15) we clearly see that

for .

Therefore, for follows from (3.2)–(3.7) and (3.16).

On the other hand, from (3.3) we clearly see that

Equations (3.2) and (3.3) together with (3.17) imply that there exists sufficient large such that for .

Subcase 8.

for .

Therefore, for follows from (3.2)–(3.7) and (3.18).

On the other hand, from (3.3) we clearly see that

Equations (3.2) and (3.3) together with (3.19) imply that there exists sufficient large such that for .

Remark.

Let , then . Unfortunately, in this paper we cannot discuss the case of we leave it as an open problem to the readers.

## Declarations

### Acknowledgments

This research is partly supported by N. S. Foundation of China under grant no. 60850005 and the N. S. Foundation of Zhejiang Province under grants no. D7080080 and no. Y607128.

## Authors’ Affiliations

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