Journal of Function Spaces
Volume 2015 (2015), Article ID 517647, 5 pages
http://dx.doi.org/10.1155/2015/517647

Research Article

Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

Yu-Ming Chu,¹ Li-Min Wu,² and Ying-Qing Song¹

¹School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China
²Department of Mathematics, Huzhou University, Huzhou 313000, China

Received 3 November 2014; Accepted 27 April 2015

Academic Editor: David R. Larson

Copyright © 2015 Yu-Ming Chu 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.

Abstract.

We present the best possible parameters and such that the double inequality holds for all and with , where    and and are the power and one-parameter harmonic means of and , respectively.

1. Introduction

For and , the th power mean of and is defined by

It is well known that is strictly increasing with respect to for fixed with , symmetric and homogeneous of degree 1. Many classical means are special cases of the power mean: for example, is the harmonic mean, is the geometric mean, is the arithmetic mean, and is the quadratic mean. The main properties of the power mean are given in [1]. Recently, the power mean has attracted the attention of many researchers. In particular, many remarkable inequalities for the power mean can be found in the literature [2–10].

Let , , , , and be the logarithmic, first Seiffert, identric, second Seiffert, and contraharmonic means of two distinct positive real numbers and , respectively. Then it is well known that the inequalities

hold for all with .

Lin [11] proved that the double inequality

holds for all with if and only if and .

In [12], Pittenger presented the best possible parameters and such that the double inequality

holds for all with , where , , and is the generalized logarithmic mean of and .

Jagers [13] and Seiffert [14] proved that the double inequalities

hold for all with .

In [15, 16], the authors proved that the double inequalities

hold for all with .

Costin and Toader [17] proved that the double inequality

holds for all with .

In [18–20], the authors proved that the double inequalities

hold for all with if and only if , , , and .

Čizmesija [21] proved that and are the best possible parameters such that the double inequality holds for all and with .

In [22, 23], the authors proved that the inequalities

hold for all if and only if , , and , where and are, respectively, the complete elliptic integrals of the first and second kinds.

Let and be the bivariate symmetric mean. Then, the one-parameter mean was defined by Neuman [24] as follows:

Let and . Then, the authors in [25–28] proved that the inequalities

hold for all with if and only if , , , , , , , , , , and , where is the unique solution of the equation .

The main purpose of this paper is to present the best possible parameters and such that the double inequality holds for all and with .

2. Lemmas

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

Lemma 1. The inequality

holds for all .

Proof. It is not difficult to verify that for all . Therefore, we only need to prove that for , where . Simple computations lead to

where

for all .
Inequality (15) implies that for all . Then, from (13) we clearly see that for all .

Lemma 2. The inequality

holds for all .

Proof. Let and . Then, it is not difficult to verify that

for all .
It follows from Lemma 1 that

for all .
Inequalities (17) and (18) lead to

for all .

Lemma 3. The inequality

holds for all .

Proof. Let and . Then, it follows from (17) and (18) that

for all .

3. Main Results

Theorem 4. The double inequality

holds for all and with if and only if and .

Proof. Without loss of generality, we assume that and . Let , where

Then, simple computations lead to

where

where

We divide the proof into two cases.
Case 1 ( ). We divide the discussion into two subcases.
Subcase 1.1 ( and ). Then, we clearly see that , and (28), (29), (32), and (33) lead to

for .
It follows easily from (24), (25), (27), (30), and (34)–(36) that

for all .
Subcase 1.2 ( and ). Then, , and (37) follows from

Case 2 ( ). Then, we clearly see that , and Lemmas 1–3 and (23), (26), (28), (29), and (32) lead to

and (36) again holds.
It follows from (30), (36), and (45) that is strictly increasing on . Then, (43) and (44) lead to the conclusion that there exists such that is strictly decreasing on and strictly increasing on .
From (41) and (42) together with the piecewise monotonicity of we clearly see that there exists such that is strictly decreasing on and strictly increasing on . Then, (25), (27), and (40) lead to the conclusion that there exists such that is strictly decreasing on and strictly increasing on . Therefore,

for all follows from (24) and (39) together with the piecewise monotonicity of .
Next, we prove that and are the best possible parameters such that the double inequality

holds for all and with .
Let , , , , and . Then, we have

Let and make use of the Taylor expansion; then, (48) leads to

Inequality (49) and equation (50) imply that for any and there exist and such that for and for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grants 61374086 and 11371125, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Natural Science Foundation of Hunan Province under Grant 14JJ2127.

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