Abstract and Applied Analysis

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 .