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`Abstract and Applied AnalysisVolume 2013, Article ID 790783, 6 pageshttp://dx.doi.org/10.1155/2013/790783`
Research Article

On Certain Inequalities for Neuman-Sándor Mean

1School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
2School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

Received 2 March 2013; Accepted 14 April 2013

Copyright © 2013 Wei-Mao Qian and Yu-Ming Chu. 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 several new sharp bounds for Neuman-Sándor mean in terms of arithmetic, centroidal, quadratic, harmonic root square, and contraharmonic means.

1. Introduction

A binary map (where is the set of positive numbers) is said to be a bivariate mean if the following statements are satisfied for all : (reflexivity property); (symmetry property); (homogeneous of order one); is continuous and strictly increasing with respect to and .

Let , and be the bivariate means such that for all with . The problems to find the best possible parameters and such that the inequalities and hold for all with have attracted the interest of many mathematicians.

For with , the Neuman-Sándor mean [1] is defined by where is the inverse hyperbolic sine function.

Recently, the bounds for the Neuman-Sándor mean in terms of other bivariate means have been the subject of intensive research.

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

Neuman and Sándor [1, 2] proved that the inequalities hold for all with .

Let with , , and . Then, the Ky Fan inequalities can be found in [1].

Li et al. [3] proved that for all with , where , and , is the th generalized logarithmic mean of and , and is the unique solution of the equation .

In [4], the author proved that the double inequalities hold for all with if and only if , , , and .

In [5, 6], the authors proved that , , , and are the best possible constants such that the inequalities , , and hold for all with . In here, and are the th generalized logarithmic and th power means of and , respectively.

Zhao et al. [7, 8] found the least values , , , , and the greatest values , , , , such that the double inequalities hold for all with .

In [9], the authors proved that if , then the double inequality holds for all with if and only if and .

The aim of this paper is to present the sharp bounds for Neuman-Sándor mean in terms of the combinations of either arithmetic and centroidal means, or quadratic and harmonic root square means, contraharmonic and harmonic root square means. Our main results are shown in Theorems 14.

Theorem 1. The double inequality holds for all with if and only if and .

Theorem 2. The double inequality holds for all with if and only if and .

Theorem 3. The double inequality holds for all with if and only if and .

Theorem 4. The double inequality holds for all with if and only if and .

2. Lemmas

In order to establish our main results we need several lemmas, which we present in this section.

Lemma 5 (see [10, Theorem 1.25]). For , let be continuous on and differentiable on , and let on . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 6 (see [11, Lemma 1.1]). Suppose that the power series and have the radius of convergence and for all . Let . Then,(1)if the sequence is (strictly) increasing (decreasing), then is also (strictly) increasing (decreasing) on ;(2)if the sequence is (strictly) increasing (decreasing) for and (strictly) decreasing (increasing) for , then there exists such that is (strictly) increasing (decreasing) on and (strictly) decreasing (increasing) on .

Lemma 7. The function is strictly decreasing on , where and are the hyperbolic sine and cosine functions, respectively.

Proof. Let Then, making use of power series formulas, we have
It follows from (12)–(14) that where
It is not difficult to verify that is positive and strictly increasing in . Then, from (18), we get that
for . Note that
Equations (17) and (20) together with inequality (19) lead to the conclusion that the sequence is strictly decreasing for and strictly increasing for . Then, from Lemma 6(2) and (15), we clearly see that there exists such that is strictly decreasing on and strictly increasing on .
Let . Then, simple computations lead to
Differentiating (12) yields
From (13) together with (21) and (22), we get
From the piecewise monotonicity of and inequality (23) we clearly see that , and the proof of Lemma 7 is completed.

Lemma 8. The function is strictly increasing from onto .

Proof. Let Then, making use of power series formulas, we have
It follows from (24)–(26) that where
Note that
It is not difficult to verify that the function is strictly decreasing in . Then from (29), we know that the sequence is strictly increasing for . Hence, from Lemma 6(1), (24), and (27) the monotonicity of follows. Moreover, and

3. Proofs of Theorems 1–4

Proof of Theorem 1. Since , , and are symmetric and homogeneous of degree , without loss of generality, we assume that . Let and . Then, , , and
Let Then, simple computations lead to where and . Note that is strictly decreasing for . Hence, from Lemma 6(1) and (33), we know that is strictly decreasing in . Moreover,
Therefore, Theorem 1 follows from (31), (32), and (34) together with the monotonicity of .

Proof of Theorem 2. Without loss of generality, we assume that . Let and . Then, , , and
Let , , and Then, , , and where is defined as in Lemma 7.
It follows from Lemmas 5 and 7, (36), and (37) that is strictly decreasing in . Moreover,
Therefore, Theorem 2 follows easily from (35), (36), and (38) together with the monotonicity of .

Proof of Theorem 3. Since , , and are symmetric and homogeneous of degree , without loss of generality, we assume that . Let and . Then, , , and
Let
Then, simple computations lead to where , and . Note that is strictly increasing for . Hence, from Lemma 6(1) and (41), we know that is strictly increasing in . Moreover,
Therefore, Theorem 3 follows from (39), (40), and (42) together with the monotonicity of .

Proof of Theorem 4. Since , , and are symmetric and homogeneous of degree , without loss of generality, we assume that . Let and . Then, , , and where is defined as in Lemma 8.
Therefore, Theorem 4 follows from (43) and Lemma 8.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Huzhou City under Grant 2012YZ06, and the Natural Science Foundation of the Open University of China under Grant Q1601E-Y.

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