Abstract and Applied Analysis

Volume 2012 (2012), Article ID 302635, 9 pages

doi:10.1155/2012/302635

Research Article

Optimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means

Tie-Hong Zhao,¹ Yu-Ming Chu,² and Bao-Yu Liu³

¹Department of Mathematics, Hangzhou Normal University, Hangzhou 313036, China

²School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

³School of Science, Hangzhou Dianzi University, Hangzhou 310018, China

Received 15 October 2012; Accepted 5 December 2012

Academic Editor: Julian López-Gómez

Copyright © 2012 Tie-Hong Zhao 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 lower and upper bounds for the Neuman-Sándor mean in terms of the convex combinations of either the harmonic and quadratic means or the geometric and quadratic means or the harmonic and contraharmonic means.

1. Introduction

For with , the Neuman-Sándor mean [1] is defined by

where is the inverse hyperbolic sine function.

Recently, the theory of bivariate means have been the subject of intensive research [2–17]. In particular, many remarkable inequalities for the Neuman-Sándor mean can be found in the literature [1, 18–20].

Let , , , , , , and be the harmonic, geometric, logarithmic, first Seiffert, arithmetic, second Seiffert, quadratic, and contraharmonic means of and , respectively. Then it is well known that the inequalities

hold for all with .

In [1, 18], Neuman and Sándor proved that the double inequalities

hold for all with .

Let , with , and . Then the following Ky Fan inequalities

were presented in [1].

The double inequality for all with was established by Li et al. in [19], where (), and is the th generalized logarithmic mean of and , and is the unique solution of the equation .

Neuman [20] proved that the double inequalities

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

The main purpose of this paper is to find the least values , , , and the greatest values , , , such that the double inequalities

hold true for all with .

Our main results are presented in Theorems 1.1–1.3.

Theorem 1.1. The double inequality

holds for all with if and only if and .

Theorem 1.2. The double inequality

holds for all with if and only if and .

Theorem 1.3. The double inequality

holds for all with if and only if and .

2. Lemmas

In order to prove our main results we need two Lemmas, which we present in this section.

Lemma 2.1 (see [21, Lemma 1.1]). Suppose that the power series and have the radius of convergence and for all . Let , then the following statements are true.

(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 2.2. Let , and

Then and for all .

Proof. From (2.1), one has

where

We divide the proof into two cases.

Case 1. (). Then (2.5) leads to

where

We clearly see that the function is strictly decreasing in . Then from (2.8), we get

for .

Therefore, for all follows easily from (2.2), (2.4), (2.6), (2.7), and (2.9).

Case 2. (). Then (2.3) and (2.5) yield

where

We divide the discussion of this case into two subcases, and all computations are carried out using MATHEMATICA software.

Subcase A. . Then from (2.12) and the fact that

we know that

for .

Subcase B. . Then from (2.12), one has

where

We conclude that

for all . Indeed, if , then (2.17) follows from (2.16) and the inequality

If , then (2.17) follows from (2.16) and the inequalities

From (2.15) together with (2.17) we clearly see that there exists such that for and for .

Subcases A and B lead to the conclusion that for and for . Thus from (2.11), we know that is strictly increasing in and strictly decreasing in .

It follows from (2.4) and (2.10) together with the piecewise monotonicity of that there exists such that is strictly increasing in and strictly decreasing in .

Therefore, for follows from (2.2) and (2.10) together with the piecewise monotonicity of .

3. Proof of Theorems 1.1–1.3

Proof of Theorem 1.1. Since , and are symmetric and homogeneous of degree 1. Hence, without loss of generality, we assume that . Let and . Then , , , , and

Making use of power series and , we can express (3.1) as follows:

Let and Then . Moreover, by a simple calculation, we see that

for .

Equations (3.1) and (3.2) together with inequality (3.3) and Lemma 2.1 lead to the conclusion that is strictly decreasing in . This in turn implies that

Therefore, Theorem 1.1 follows from (3.1) and (3.4) together with the monotonicity of .

Proof of Theorem 1.2. Since , and are symmetric and homogeneous of degree 1. Hence, without loss of generality, we assume that . Let , and . Then making use of gives

Moreover, we obtain

We take the difference between the additive convex combination of , , and as follows:

where is defined as in Lemma 2.2.

Therefore, for all with follows from (3.8) and Lemma 2.2. This conjunction with the following statement gives the asserted result.

(i)If , then (3.5) and (3.6) imply that there exists such that for all with .(ii)If , then (3.5) and (3.7) imply that there exists such that for all with .

Proof of Theorem 1.3. We will follow, to some extent, lines in the proof of Theorem 1.1. First we rearrange terms of (1.9) to obtain

Use of followed by a substitution gives

where

Since the function is an even function, it suffices to investigate its behavior on the interval .

Using power series of and , then (3.11) can be rewritten as

Let and . Then

It follows from (3.13) that the sequence is strictly increasing for .

Equations (3.12) and (3.13) together with Lemma 2.1 and the monotonicity of lead to the conclusion that is strictly increasing in . Moreover,

Making use of (3.14) and (3.10) together with the monotonicity of gives the asserted result.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11271105, and Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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