Journal of Function Spaces
Volume 2016 (2016), Article ID 5131907, 7 pages
http://dx.doi.org/10.1155/2016/5131907

Research Article

Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

Ying-Qing Song,¹ Wei-Mao Qian,² and Yu-Ming Chu¹

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

Received 13 September 2015; Accepted 11 January 2016

Academic Editor: Henryk Hudzik

Copyright © 2016 Ying-Qing Song 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 inequalities and hold for all with and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, , , , and are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of and , and .

1. Introduction

Let with . Then, the Schwab-Borchardt mean [1–7] of and is given by

where is the inverse hyperbolic cosine functions.

It is well known that the Schwab-Borchardt mean is strictly increasing in both and , nonsymmetric, and homogeneous of degree 1 with respect to and . Many symmetric bivariate means are special cases of the Schwab-Borchardt mean. For example, is the first Seiffert mean, is the second Seiffert mean, is the Neuman-Sándor mean, and is the logarithmic mean, where is the inverse hyperbolic sine function, is the inverse hyperbolic tangent function, , , and are, respectively, the geometric, arithmetic, and quadratic means of and , and the inequalities

hold for all with (see [1]), where is the harmonic mean of and .

Recently, the Schwab-Borchardt mean and its generated means have been the subject intensive research. In particular, many remarkable inequalities for these means can be found in the literature [1–5, 8–16].

Liu and Meng [17] proved that the double inequality

holds for all with if and only if and , where

is the centroidal mean of and .

Very recently, Neuman [18] introduced the Neuman mean

presented the explicit formula for as

and proved that the double inequality

holds for all with , where .

From (2), (3), and (7), we clearly see that the double inequality

holds for all with .

Let . Then, it is not difficult to verify that the function is continuous and strictly increasing on for fixed with . Note that

for all with .

Motivated by inequalities (8) and (9), it is natural to ask what are best possible parameters and such that the double inequalities

hold for all with ? The main purpose of this paper is to answer these questions.

2. Lemmas

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

Lemma 1 (see [19–21]). For , let be continuous on and be 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 2 (see [22–24]). Let and be two real power series converging on with for all . Then, the following statements are true:

(1)If the nonconstant sequence is increasing (decreasing) for all , then the function is strictly increasing (decreasing) on .(2)If there exists such that the nonconstant sequence is increasing (decreasing) for and decreasing (increasing) for , then there exists such that the function is strictly increasing (decreasing) on and strictly decreasing (increasing) on .

Lemma 3. The function

is strictly increasing from onto .

Proof. Making use of the power series formulas and , we have

Let

Then, we clearly see that

for all and

Let

Then, simple computations lead to

for all .
It follows from (16)–(18) that the nonconstant sequence is increasing for and decreasing for . Then, from (13)–(15) and Lemma 2(2) we know that there exists such that is strictly increasing on and strictly decreasing on .
Differentiating gives

Note that

It follows from (19) and (20) together with the piecewise monotonicity of that and is strictly increasing on .
Therefore, Lemma 3 follows from (21) and the monotonicity of on the interval .

Lemma 4. The function

is strictly increasing from onto .

Proof. Making use of the power series formulas and together with the identity , one has

Let

Then,

for all . Note that

Therefore, Lemma 4 follows easily from (23)–(26) and Lemma 2(1).

3. Main Results

Theorem 5. The double inequality

holds for all with if and only if and .

Proof. Since all the means , , and are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let . Then, (4) leads to

From (6) and (28), we clearly see that inequality (27) is equivalent to

Let . Then, and

Therefore, Theorem 5 follows easily from (29) and (30) together with Lemma 4.

Theorem 6. The double inequality

holds for all with if and only if and .

Proof. Without loss of generality, we assume that . Let . Then, from (6) and (28) we clearly see that inequality (31) is equivalent to

Let . Then, , and

Let

Then, simple computations lead to

It follows from (34)–(37) and Lemmas 1 and 3 that the function

is strictly increasing on the interval . Note that

where and are defined by (14).
Therefore, Theorem 6 follows easily from (32)–(34) and (39) together with the monotonicity of on the interval .

Theorem 7. Let . Then, the double inequality

holds for all with if and only if and .

Proof. Without loss of generality, we assume that . Let and . Then, (4) and (6) lead to

Let . Then, and

where is defined by Lemma 4.
Therefore, Theorem 7 follows easily from (41) and (42) together with Lemma 4.

Let and . Then, we clearly see that

and (4) and (6) lead to

From (44) and Theorems 5–7, we get Corollary 8 immediately.

Corollary 8. Let and . Then, the double inequalities

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

Let . Then, we clearly see that

Corollary 9. Let and . Then, the double inequalities

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

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11371125 and 61374086, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, the Natural Science Foundation of Hunan Province under Grant 12C0577, and the Natural Science Foundation of the Zhejiang Broadcast and TV University under Grant XKT-15G17.

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