Bounds of the Neuman-Sándor Mean Using Power and Identric Means

Chu, Yu-Ming; Long, Bo-Yong

doi:https://doi.org/10.1155/2013/832591

Abstract

In this paper we find the best possible lower power mean bounds for the Neuman-Sándor mean and present the sharp bounds for the ratio of the Neuman-Sándor and identric means.

1. Introduction

For the th power mean , Neuman-Sándor Mean [1], and identric mean of two positive numbers and are defined by respectively, where is the inverse hyperbolic sine function.

The main properties for and are given in [2]. It is well known that is continuously and strictly increasing with respect to for fixed with . Recently, the power, Neuman-Sándor, and identric means have been a subject of intensive research. In particular, many remarkable inequalities for these means can be found in the literature [3–26].

Let , , , , , , , and be the harmonic, geometric, logarithmic, first Seiffert, arithmetic, second Seiffert, quadratic, and contraharmonic means of two positive numbers and with , respectively. Then, it is well known that the inequalities hold for all with .

The following sharp bounds for , , , and in terms of power means are presented in [27–32]: for all with .

Pittenger [31] found the greatest value and the least value such that the double inequality holds for all , where is the th generalized logarithmic means which is defined by

The following sharp power mean bounds for the first Seiffert mean are given in [10, 33]: for all with .

In [17], the authors answered the question: for , what are the greatest value and the least value such that the double inequality holds for all with ?

Neuman and Sándor [1] established that for all with .

Let with , and . Then, the Ky Fan inequalities were presented in [1].

In [24], Li et al. found the best possible bounds for the Neuman-Sándor mean in terms of the generalized logarithmic mean . Neuman [25] and Zhao et al. [26] proved that the inequalities hold for all with if and only if , , , , , , , and .

In [7], Sándor and Trif proved that the inequalities hold for all with .

Neuman and Sándor [15] and Gao [20] proved that , , , , , , , , , and are the best possible constants such that the double inequalities , , , , and hold for all with , where is the Heronian mean of and .

In [34], Sándor established that for all with .

It is not difficult to verify that the inequality holds for all with .

From inequalities (10), (14), and (15), one has for all with .

It is the aim of this paper to find the best possible lower power mean bound for the Neuman-Sándor mean and to present the sharp constants and such that the double inequality holds for all with .

2. Main Results

Theorem 1. is the greatest value such that the inequality holds for all with .

Proof. From (1) and (2), we clearly see that both and are symmetric and homogenous of degree one. Without loss of generality, we assume that and .
Let , then from (1) and (2) one has Let Then, simple computations lead to where where where for .
Equation (33) and inequality (34) imply that is strictly decreasing on . Then, the inequality (31) and (32) lead to the conclusion that there exists , such that is strictly increasing on and strictly decreasing on .
From (29) and (30) together with the piecewise monotonicity of , we clearly see that there exists , such that is strictly increasing on and strictly decreasing on .
It follows from (26)–(28) and the piecewise monotonicity of that there exists , such that , is strictly increasing on and strictly decreasing on .
From (23)–(25) and the piecewise monotonicity of we see that there exists , such that is strictly increasing on and strictly decreasing on .
Therefore, for follows easily from (19)–(22) and the piecewise monotonicity of .
Next, we prove that is the greatest value such that for all .
For any and , from (1) and (2), one has
Inequality (35) implies that for any , there exists , such that for .

Remark 2. is the least value such that inequality (16) holds for all with , namely, is the best possible upper power mean bound for the Neuman-Sándor mean .
In fact, for any and , one has
Letting and making use of Taylor expansion, we get
Equations (36) and (37) imply that for any there exists , such that for .

Theorem 3. For all with , one has with the best possible constants and .

Proof. From (2) and (3), we clearly see that both and are symmetric and homogenous of degree one. Without loss of generality, we assume that and . Let Then, simple computations lead to where where for .
From (46) and (47), we clearly see that is strictly increasing on . Then, (45) leads to the conclusion that is strictly increasing on .
Equations (43) and (44) together with the monotonicity of impliy that for . Then, (42) leads to the conclusion that is strictly increasing on .
It follows from equations (40) and (41) together with the monotonicity of that is strictly increasing on .
Therefore, Theorem 3 follows from (39) and the monotonicity of together with the facts that

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants nos. 11071069 and 11171307, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no. T200924.

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Copyright

Copyright © 2013 Yu-Ming Chu 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 and Applied Analysis