Journal of Applied MathematicsVolume 2012 (2012), Article ID 471096, 14 pagesdoi:10.1155/2012/471096

Research Article

Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means

Wei-Mao Qian¹ and Zhong-Hua Shen²

¹School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China²Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 10 October 2011; Accepted 1 December 2011

Academic Editor: Md. Sazzad Chowdhury

Copyright © 2012 Wei-Mao Qian and Zhong-Hua Shen. 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 prove that for and all with if and only if and if and only if , and the parameter is the best possible in either case. Here, , , and and are the harmonic, logarithmic, and pth power means of a and b, respectively.

1. Introduction

The classical logarithmic mean of two positive real numbers and with is defined by

In the recent past, the bivariate means have been the subject of intensive research. In particular, many remarkable inequalities for can be found in the literature [1–21]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [22–24]. In [22] the authors study a variant of Jensen's functional equation involving the logarithmic mean, which appears in a heat conduction problem. A representation of as an infinite product and an iterative algorithm for computing it as the common limit of two sequences of special geometric and arithmetic means are given in [4]. In [25, 26] it is shown that can be expressed in terms of Gauss hypergeometric function . And, in [26] the authors prove that the reciprocal of the logarithmic mean is strictly totally positive; that is, every determinant with elements , where and , is positive for all .

Let , , , , and , and be the geometric, harmonic, identric, arithmetic, th power, and th Lehmer means of two positive numbers and , respectively. Then it is well known that both and are continuous and strictly increasing with respect to for fixed with , and the inequalities hold for all with .

In [4], Carlson proves that the double inequality holds for all with .

In [5], Lin finds the best possible upper and lower power bounds for the logarithmic mean as follows: for all with .

In [9], Sándor establishes that for all with .

In [27], Alzer gives the optimal Lehmer mean bounds for , , and as follows: for all with .

The following sharp bounds for and in terms of power mean are presented in [28]: for all with .

In [29, 30], the authors obtain the sharp bounds for the products and and the sum in terms of power mean as follows: for any and all with .

In [31], Zhu presents some bounds for in terms of and and in terms of and .

In [32], Chu et al. prove that the double inequality holds for all with if and only if and .

It is the aim of this paper to give the optimal power mean bounds for the convex combination of harmonic and logarithmic means. Our main result is the following theorem.

Theorem 1.1. For and all with , one has (1) if and only if ;(2) if and only if . In particular, the parameter is the best possible in either case.

2. Lemmas

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

Lemma 2.1. Let , and . Then for .

Proof. Simple computations lead to where for .Inequality (2.6) implies that is strictly decreasing in . Then (2.4) and (2.5) lead to the conclusion that there exists such that for and for . It follows from (2.3) that is strictly increasing in and strictly decreasing in .Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the piecewise monotonicity of .

Lemma 2.2. Let , and . Then for .

Proof. Let Then simple computations lead to where for .From inequalities (2.13) we clearly see that for . Then (2.12) leads to the conclusion that is strictly increasing in .Therefore, Lemma 2.2 follows from (2.7)–(2.11) and the monotonicity of .

Lemma 2.3. Let , and . Then the following two statements are true: (1)if , then for ;(2)if , then for .

Proof. Let , , , and . Then simple computations lead to (1) If , then from (2.19) we note that where and are defined as in Lemmas 2.1 and 2.2, respectively.Lemmas 2.1 and 2.2 together with (2.20) imply that is strictly increasing in . Therefore, for follows from (2.14)–(2.18) and the monotonicity of .(2) If , then from (2.19) we have Inequalities (2.30) imply that is strictly decreasing in . Then (2.29) leads to the conclusion that is strictly decreasing in .It follows from (2.28) and the monotonicity of that is strictly decreasing in . Then inequalities (2.25)–(2.27) lead to the conclusion that for . Thus, is strictly decreasing in .From inequalities (2.22)–(2.24) and the monotonicity of we clearly see that for . Thus, is strictly decreasing in .It follows from (2.17) and (2.18) and inequality (2.21) together with the monotonicity of that for , which implies that is strictly decreasing in .Therefore, for follows from (2.14)–(2.16) and the monotonicity of .

Lemma 2.4. for .

Proof. Let Then simple computations lead to where , where , for .

Therefore, Lemma 2.4 follows from (2.31)–(2.36).

3. Proof of Theorem 1.1

Proof of Theorem 1.1. For all with , we first prove that for , for .Without loss of generality, we assume that and . We divide the proof into two cases.Case 1 (). Let . Then we clearly see that Therefore, inequality (3.1) follows from (3.3) and Lemma 2.4.Case 2 (). Then we have Let Then simple computations lead to where is defined as in Lemma 2.3.If , then inequality (3.1) follows from (3.4)–(3.6) and Lemma 2.3(1). If , then inequality (3.2) follows from (3.4)–(3.6) and Lemma 2.3(2).Next, we prove that the parameter in inequalities (3.1) and (3.2) is the best possible.For any , , and , one has where .Letting and making use of Taylor expansion, we get If and , then (3.7) and (3.8) imply that there exists such that for . If and , then (3.7) and (3.8) imply that there exists such that for .Finally, we prove that there exist with and such that and for any .If , then from the expression of in (2.21) we clearly see that , which leads to the conclusion that there exists such that for .From (2.14)–(2.18) and inequality (3.9) we know that for . Equations (3.4)–(3.6) and inequality (3.10) lead to the conclusion that for all .On the other hand, simple computations lead to Equation (3.11) implies that there exists such that for all .

Acknowledgment

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

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