Journal of Applied Mathematics

Volume 2012, Article ID 480689, 8 pages

http://dx.doi.org/10.1155/2012/480689

## Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

^{1}School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China^{2}School of Mathematics Science, Anhui University, Hefei 230039, China

Received 29 January 2012; Revised 19 February 2012; Accepted 12 March 2012

Academic Editor: Yuri Sotskov

Copyright © 2012 Wei-Mao Qian and Bo-Yong Long. 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.

#### Abstract

We present sharp upper and lower generalized logarithmic mean bounds for the geometric weighted mean of the geometric and harmonic means.

#### 1. Introduction

For the generalized logarithmic mean of two positive numbers and is defined by

It is well-known that is continuous and strictly increasing with respect to for fixed and with . In the recent past, the generalized logarithmic mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in the literature [1–23]. The generalized logarithmic mean has applications in convex function, economics, physics, and even in meteorology [24–27]. In [26] the authors study a variant of Jensen’s functional equation involving , which appear in a heat conduction problem. Let , , , and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers and with , respectively. Then it is well known that

In [28–30], the authors present bounds for and in terms of and .

Proposition 1.1. *For all positive real numbers and with , one has
*

The proof of the following Proposition 1.2 can be found in [31].

Proposition 1.2. *For all positive real numbers and with , we have
*

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

The main properties of these means are given in [32]. Several authors discussed the relationship of certain means to . The following sharp bounds for , , , and in terms of power means are proved in [31, 33–37].

Proposition 1.3. *For all positive real numbers and with one has
*

The following three results were established by Alzer and Qiu in [38].

Proposition 1.4. *The inequalities
**
hold for all positive real numbers and with if and only if
*

Proposition 1.5. *Let and be real numbers with . If , then
**
And, if , then
*

Proposition 1.6. *For all positive real numbers and with , one has
**
with the best possible parameter *

In [39] the authors presented inequalities between the generalized logarithmic mean and the product for all with and with .

It is the aim of this paper to give a solution to the problem: for , what are the greatest value and the least value , such that the inequality holds for all ?

#### 2. Main Result

Theorem 2.1. *For and all , one has the following:*(1)* for ,*(2)* for , and for , with equality if and only if , and the parameters and in each inequality cannot be improved.*

*Proof. * (1) If and , then (1.1) implies that .

If and , then (1.1) leads to

(2) If , then from (1.1) we clearly see that for any .

If , without loss of generality, we assume . Let and
Then (1.1) and simple computations yield
where ,

If , then (2.7) implies
for .

From (2.3)–(2.6) and (2.8) we know that for .

If , then (2.7) leads to
for . Therefore for follows from (2.3)–(2.6) and (2.9).

Let
for ; then (1.1) and elementary calculations lead to
where ,

If , then (2.15) implies
for .

From (2.11)–(2.14) and (2.16) we know that for .

If , then (2.15) leads to
for . Therefore, for follows from (2.11)–(2.14) and (2.17).

Next, we prove that the parameters and in either case cannot be improved. The proof is divided into two cases.*Case 1 (). *For any and , from (1.1) one has
where .

Let ; making use of the Taylor expansion, we get

Equations (2.18) and (2.19) imply that for any and there exists , such that for .

On the other hand, for any we have

From (2.20) we know that for any and there exists , such that for .*Case 2 (). *For any and , from (1.1) one has
where .

Let ; making use of the Taylor expansion, we have

Equations (2.21) and (2.22) imply that for any and there exists , such that for .

On the other hand, for any , we have

From (2.23) we know that for any and there exists , such that for .

#### Acknowledgment

This work was supported by the Natural Science Foundation of Zhejiang Broad-cast and TV University under Grant XKT-09G21.

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