Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means
Wei-Mao Qian1and Zhong-Hua Shen2
Academic Editor: Md. Sazzad Chowdhury
Received10 Oct 2011
Accepted01 Dec 2011
Published22 Feb 2012
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 [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 .
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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