#### Abstract

We present the sharp bounds for the Neuman means , , and in terms of the arithmetic, harmonic, and contraharmonic means. Our results are the refinements or improvements of the results given by Neuman.

#### 1. Introduction

For with , the Schwab-Borchardt mean of and is given by where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.

It is well-known that the mean is strictly increasing in both and , nonsymmetric and homogeneous of degree with respect to and . Many symmetric bivariate means are special cases of the Schwab-Borchardt mean; for example, where , , and denote the classical geometric mean, arithmetic mean, and quadratic mean of and , respectively. The Schwab-Borchardt mean was investigated in [1, 2].

Let and be the harmonic and contraharmonic means of two positive numbers and , respectively. Then, it is well-known that for with .

Recently, the Schwab-Borchardt mean and its special cases have been the subject of intensive research. Neuman and Sándor [3, 4] proved that the inequalities hold for all , with . In [5], the author proved that the double inequalities hold for all , with if and only if , , , and . Chu and Long [6] found that the double inequality holds for all with if and only if and , where and is the th power mean of and . Zhao et al. [7] presented the least values , , and and the greatest values , , and such that the double inequalities hold for all , with .

Very recently, the bivariate means , , , and derived from the Schwab-Borchardt mean are defined by Neuman [8, 9] as follows:

We call the means , , , and given in (8) the Neuman means. Moreover, let ; then the following explicit formulas for , , , and are found by Neuman [8]: where , , , and are defined implicitly as , , , and , respectively. Clearly, , , , and .

In [8, 9], Neuman proved that the inequalities hold for , with .

He et al. [10] found the greatest values , , , , and the least values , , , such that the double inequalities hold for all with .

Motivated by inequalities (12), it is natural to ask what the greatest values , , , and and the least values , , , and are such that the double inequalities hold for all , with .

The purpose of this paper is to answer these questions. All numerical computations are carried out using MATHEMATICA software. Our main results are the following Theorems 1–4.

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

Theorem 2. *The two-sided inequality
**
holds true 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 two-sided inequality
**
holds true for all with if and only if and .*

#### 2. Two Lemmas

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

Lemma 5. *Let and
**
Then, the following statements are true.*(1)*If , then for all and for all .*(2)*If , then there exists such that for and for .*(3)*If , then there exists such that for and for .*

*Proof. *For part (1), if , then (19) becomes
Therefore, part (1) follows easily from (20).

For part (2), if , then simple computations lead to

It follows from (21)–(23) and (29) that is strictly increasing on . Then, (27) and (28) lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in .

Therefore, part (2) follows from (24) and (25) together with the piecewise monotonicity of .

For part (3), if , then numerical computations lead to

It follows from (26) and (30)–(32) that
for .

Therefore, part (3) follows easily from (33)–(35).

Lemma 6. *Let and
**
Then, the following statements are true.*(1)*If , then for all and for all .*(2)*If , then there exists such that for and for .*

*Proof. *For part (1), if , then (36) becomes
Therefore, part (1) follows from (37).

For part (2), if , then numerical computations lead to

From (38) and (39) together with (42), we clearly see that
for .

Therefore, part (2) follows from (40) and (41) together with (43).

#### 3. Proofs of Theorems 1–4

*Proof of Theorem 1. *Without loss of generality, we assume that . Let , , , and . Then, , , ,
where
where is defined as in Lemma 5.

We divide the proof into two cases.*Case 1* *(**).* Then, from Lemma 5(1) and (49), we clearly see that is strictly decreasing in . Therefore,
for all with follows from (45) and (48) together with the monotonicity of . *Case 2 (**).* Then, from (47) and (49) and Lemma 5(2), we know that
and there exists such that is strictly decreasing in and strictly increasing in . Therefore,
for all with follows from (45) and (48) together with (51) and the piecewise monotonicity of .

Note that

Therefore, Theorem 1 follows from (50) and (52)–(54) together with the following statements.(i)If , then (44) and (53) imply that there exists small enough such that for all with .(ii)If , then (44) and (54) imply that there exists large enough such that for all with .

*Proof of Theorem 2. *Without loss of generality, we assume that . Let , , , and . Then, , , ,
where
where is defined as in Lemma 5.

We divide the proof into two cases.*Case 1 (**).* Then, from (59) and (60) together with Lemma 5(3), we clearly see that there exists such that is strictly increasing in and strictly decreasing in , and

Therefore,
for all with follows easily from (56) and (58) together with (61) and the piecewise monotonicity of .*Case 2 (**).* Then, Lemma 5(1) and (60) lead to the conclusion that is strictly decreasing in . Therefore,
for all with follows from (56) and (58) together with the monotonicity of .

Note that

Therefore, Theorem 2 follows from (55) and (62)–(65).

*Proof of Theorem 3. *Without loss of generality, we assume that . Let , , , and . Then, and (9) leads to

It follows from (66) that
where
where is defined as in Lemma 6.

If , then Lemma 6(1) and (71) lead to the conclusion that is strictly increasing in . Therefore,
for all , with follows from (68) and (70) together with the monotonicity of .

Note that

Therefore, Theorem 3 follows from (12) and (67) together with (72)–(74).

*Proof of Theorem 4. *Without loss of generality, we assume that . Let , , , and . Then, , , and and (10) leads to

It follows from (75) that
where
where is defined as in Lemma 6.

We divide the proof into two cases.*Case 1 (**).* Then, (81) and Lemma 6(1) lead to the conclusion that is strictly increasing in . Therefore,
for all with follows easily from (77) and (79) together with the monotonicity of .*Case 2 (**).* Then, (80) and (81) together with Lemma 6(2) lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in , and
Therefore,
for all with follows easily from (77) and (79) together with (83) and the piecewise monotonicity of .

Note that

Therefore, Theorem 4 follows from (76) and (82) together with (84)–(86).

#### 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 61374086 and 11171307, the Natural Science Foundation of the Open University of China under Grant Q1601E-Y, and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-13Z04.