Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 694394 | 10 pages | https://doi.org/10.1155/2009/694394

Optimal Inequalities among Various Means of Two Arguments

Academic Editor: Dumitru Baleanu
Received10 Sep 2009
Accepted19 Oct 2009
Published10 Nov 2009

Abstract

We establish two optimal inequalities among power mean , arithmetic mean , logarithmic mean , and geometric mean .

1. Introduction

For , the power mean of order of two positive numbers and is defined by

In the recent past, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for the mean can be found in literature [111]. It is well-known that is continuous and strictly increasing with respect to for fixed and .

If we denote by

, and the identric mean, logarithmic mean, arithmetic mean, geometric mean and the harmonic mean, of two positive real numbers and , respectively, then

with equality if and only if in each inequality.

In [12], Alzer and Janous established the following sharp double inequality (see also [13, page 350]):

for all real numbers .

In [14], Mao proved

for all real numbers , and the constant in the left side inequality cannot be improved.

In [1517], the authors presented the bounds for and in terms of and as follows:

for all with .

Alzer [18] proved

In [5], Alzer and Qiu established

for , and with .

The main purpose of this paper is to present the optimal bounds for and for all in terms of the power mean . Moreover, two optimal inequalities among , , and are proved.

2. Lemmas

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

Lemma 2.1. If , then for .

Proof. Let , , , , , , , , and , then simple computation yields
From (2.10) we clearly see that is strictly increasing in . Therefore, Lemma 2.1 follows from (2.2)–(2.9) and (2.11) together with the monotonicity of .

Lemma 2.2. If , then for .

Proof. Let , , , , , and , then simple computation leads to
From (2.18) we clearly see that is strictly increasing in . Therefore, Lemma 2.2 follows from (2.13)–(2.17) and (2.19) together with the monotonicity of .

3. Main Results

Theorem 3.1. If , then with equality if and only if , and the parameter cannot be improved.

Proof. If , then we clearly see that
If , then without loss of generality we assume that and let ; hence elementary calculation yields

Let
then where
From Lemma 2.1 and (3.6) we know that
for .
Therefore, we get
for that follows from (3.3)–(3.5) and (3.8).
Next, we prove that the parameter cannot be improved.
For any , let , then (1.1) leads to
where
Making use of the Taylor expansion we get

Equations (3.10) and (3.12) imply that for any and , there exists , such that
for .

Remark 3.2. For any we have for all , with equality if and only if , and the parameter in the lower bound cannot be improved.
In fact, if , then we clearly see that . If , then follows from .
Next, we prove that the parameter in the lower bound cannot be improved.
For any , we have

Equation (3.15) implies that for any , there exists , such that
for .

Theorem 3.3. If , then for all , with equality if and only if , and the parameter cannot be improved.

Proof. If , then we clearly see that
If , then without loss of generality, we assume that , let ; hence simple computation leads to
Let then where
From Lemma 2.2 and (3.22) we know that
for .
Therefore, we get
for that follows from (3.19)–(3.21) and (3.24).
Next, we prove that the constants cannot be improved.
For any , let , then (1.1) leads to
where
Making use of the Taylor expansion we get

Equations (3.26) and (3.28) imply that for any and , there exists , such that
for .

Remark 3.4. For any , we have Therefore, (3.30) implies that inequality holds for all and , with equality if and only if and the parameter in the lower bound cannot be improved.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 60850005) and the Natural Science Foundation of Zhejiang Province (Grant no. D7080080 and no. Y607128).

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Copyright © 2009 Ming-yu Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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