Some Comparison Inequalities for Generalized Muirhead and Identric Means

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 .

1. Introduction

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

(1.1)

(1.2)

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,

(1.3)

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

(1.4)

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

(1.5)

and be the arithmetic, logarithmic, geometric, and harmonic means of two positive numbers and . Then it is well known that

(1.6)

for all with .

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

(1.7)

for all with .

Pittenger [16] proved that

(1.8)

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

In [8, 9], Sándor established that

(1.9)

for all with .

Alzer and Qiu [5] proved the inequalities

(1.10)

for all with if and only if and .

In [3], Chu and Xia proved that

(1.11)

for all and , and

(1.12)

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.

Let and be two real numbers such that and . Let one define the function as follows:

(2.1)

then the following statements hold.

(1)If and , then for

• (2) If , , and , then for

(3) If , then for .

Proof.

Simple computations lead to

(2.2)

(2.3)

(2.4)

(2.5)

where

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

where

(2.11)

(2.12)

(2.13)

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

Case.

, and . From (2.13), (2.12), (2.9), and (2.4), we clearly see that

(2.14)

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

Case 2.

, and we conclude that

(2.15)

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

Equation (2.15) and imply that

(2.16)

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

(2.17)

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

(2.18)

Therefore, for easily follows from (2.2), (2.5), (2.7), and (2.10) together with (2.18).

• (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.

For convenience, we introduce the following classified regions in :

(3.1)

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

(3.2)

Let

(3.3)

Then simple computations yield

(3.4)

where

(3.5)

Note that

(3.6)

(3.7)

where is defined as in Lemma 2.1.

We divide the proof into three cases.

Case.

. We divide our discussion into two subcases.

Subcase 1.

. From Lemma 2.1(2) we get

(3.8)

for .

Equations (3.3)–(3.8) imply that

(3.9)

for .

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

Subcase 2.

. Then from (1.1), (1.4), and (1.6) together with the monotonicity of the power mean with respect to for fixed , we get

(3.10)

Case.

. We divide our discussion into four subcases.

Subcase 3.

. Then Lemma 2.1(1) leads to

(3.11)

for .

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

Subcase 4.

. Then from (1.1), (1.4), and (1.8) together with the monotonicity of the power mean with respect to for fixed we clearly see that

(3.12)

Subcase 5.

. Then from Lemma 2.1(3) we know that (3.11) holds again; hence, .

Subcase 6.

. Then (1.6) leads to

(3.13)

Case.

. We divide our discussion into two subcases.

Subcase 7.

. Then (2.4) leads to

(3.14)

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

(3.15)

for .

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

(3.16)

for .

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

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

(3.17)

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

Subcase 8.

. Then (2.2) and (2.4) together with the continuity of imply that there exists such that

(3.18)

for .

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

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

(3.19)

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.