#### Abstract

We give several sharp bounds for the Neuman means and ( and ) in terms of harmonic mean H (contraharmonic mean C) or the geometric convex combination of arithmetic mean A and harmonic mean H (contraharmonic mean C and arithmetic mean A) and present a new chain of inequalities for certain bivariate means.

#### 1. Introduction

For with , the Schwab-Borchardt mean [13] of and is defined as where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.

The Schwab-Borchardt mean can be expressed by the symmetric elliptic integral [4] of the first kind as follows [5] (see also [6, (3.21)]): where .

Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for the Schwab-Borchardt mean and its generated means can be found in the literature [13, 710].

Very recently, Neuman [11] found a new mean derived from the Schwab-Borchardt mean as follows:

Let , , , , , , , , and be, respectively, the geometric, harmonic, logarithmic, first Seiffert, arithmetic, Neuman-Sándor, second Seiffert, quadratic, and contraharmonic means, and let be the Neuman means. Then Neuman [11] proved that for all with , and the double inequalities hold for all with if and only if , , , , , , , and .

Zhang et al. [12] presented the best possible parameters and such that the double inequalities hold for all with .

In [13], the authors found the greatest values , , , , , , , and and the least values , , , , , , , and such that the double inequalities hold for all with .

Let , , , , , and be the parameters such that , . Then He et al. [14] proved that

Let , , , and . Then we clearly see that for all with , and the functions and are, respectively, strictly increasing on the intervals and for fixed with .

The main purpose of this paper is to find the best possible parameters , , , , , , , , , and such that the double inequalities hold, for all with , and present a new chain of inequalities for certain bivariate means.

#### 2. Lemmas

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

Lemma 1 (see [15, Theorem 1.25]). For , let be continuous on and differentiable on ; 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 [16, Lemma 1.1]). Suppose that the power series and have the radius of convergence and for all . If the sequence is (strictly) increasing (decreasing) for all , then the function is also (strictly) increasing (decreasing) on .

Lemma 3. The function is strictly decreasing from onto .

Proof. Making use of power series expansion we get
Let Then and is strictly decreasing for all .
Note that
Therefore, Lemma 3 follows easily from Lemma 2 and (18)–(21) together with the monotonicity of the sequence .

Lemma 4. The function is strictly increasing from onto .

Proof. Let and . Then simple computations lead to and is strictly increasing on .
Note that
Therefore, Lemma 4 follows from Lemma 1, (23), (24), and the monotonicity of .

Lemma 5. The function is strictly decreasing from onto .

Proof. Let and . Then simple computations lead to
Let Then it is not difficult to verify that for all , where and .
Note that
It follows from Lemma 2 and (27)–(29) that the function is strictly decreasing on . Therefore, Lemma 5 follows from (26) and (30) together with Lemma 1 and the monotonicity of .

Lemma 6. The function is strictly decreasing from onto .

Proof. Let , , , and . Then simple computations lead to
Since the function is strictly decreasing on , hence (34) leads to the conclusion that is strictly decreasing on .
Note that
Therefore, Lemma 6 follows easily from (32), (33), (35), and Lemma 1 together with the monotonicity of .

Lemma 7. The function is strictly increasing from onto .

Proof. Let and . Then simple computations lead to
Let
Then for all .
It follows from Lemma 2 and (38)–(40) that is strictly increasing on .
Note that
Therefore, Lemma 7 follows from Lemma 1, (37), and (41) together with the monotonicity of .

Lemma 8. The function is strictly increasing from onto .

Proof. Let , , , and . Then simple computations lead to
Since the function is strictly decreasing on , hence Lemma 1 and (43) lead to the conclusion that is strictly increasing on .
Note that
Therefore, Lemma 8 follows easily from (44) and (45) together with the monotonicity of .

#### 3. Main Results

Theorem 9. The double inequalities hold for all with if and only if , , , , , , , and .

Proof. Without loss of generality, we assume that . Let , , , , and be the parameters such that , . Then (9)–(12) lead to the conclusion that inequalities (46)–(49) are, respectively, equivalent to
Therefore, Theorem 9 follows easily from (50)–(53) and Lemmas 58.

Theorem 10. Let . Then the double inequalities, hold for all with if and only if , , , and .

Proof. Without loss of generality, we assume that . Let , , , and be the parameters such that . Then from (9) and (10) we have
Therefore, Theorem 10 follows easily from (55) and (56) together with Lemmas 3 and 4.

Theorem 11. Let . Then the double inequalities, hold for all with if and only if , , , and .

Proof. Without loss of generality, we assume that . Let , , , and be the parameters such that . Then from (11) and (12) one has where the functions and are defined as in Lemmas 3 and 4, respectively.
Note that
Therefore, Theorem 11 follows easily from Lemmas 3 and 4 together with (58)-(59).

Theorem 12. Let , , , and . Then the inequalities hold for all with .

Proof. It follows from [1, ] and [11, ] together with [9, , Theorems 2 and 5] that
Therefore, the second, fourth, ninth, and twelfth inequalities follow from (61) and the first, sixth, seventh, eighth, and eleventh inequalities follow from (62) immediately, while the fifth and thirteenth inequalities can be derived from and the fact that and are, respectively, the mean values of , , and , .
Next, we prove the third and tenth inequalities. In fact, the third inequality can be derived from the following inequalities (63) [14, Theorem 1.2] and (64) [9, Theorem 3] together with . Consider the following:
while the tenth inequality can be derived from the inequality in Theorem 9 and together with , where .

Remark 13. He et al. [14, Theorem 1.2], Xia and Chu [17, Theorem 3.1], and Chu et al. [18, Theorem 2.1] proved that the double inequalities hold for all with if and only if , , , , , , , and .
From the above results we clearly see that the mean values and and and are not comparable with each other.

#### 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, 11371125, and 11171307 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.