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Journal of Function Spaces
Volume 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

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 [210].

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 [1820], 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 inequalitieshold 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 [2528] 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 thatfor 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 towhere whereWe 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 13 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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