Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

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 inequalitieshold 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 inequalityholds for all with if and only if and , whereis the centroidal mean of and .

Very recently, Neuman [18] introduced the Neuman mean presented the explicit formula for as and proved that the double inequalityholds for all with , where .

From (2), (3), and (7), we clearly see that the double inequalityholds for all with .

Let . Then, it is not difficult to verify that the function is continuous and strictly increasing on for fixed with . Note thatfor 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 haveLetThen, we clearly see thatfor all andLetThen, simple computations lead tofor 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 thatIt 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 hasLetThen,for all . Note thatTherefore, Lemma 4 follows easily from (23)–(26) and Lemma 2(1).

3. Main Results

Theorem 5. The double inequalityholds 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 toFrom (6) and (28), we clearly see that inequality (27) is equivalent toLet . Then, andTherefore, Theorem 5 follows easily from (29) and (30) together with Lemma 4.

Theorem 6. The double inequalityholds 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 toLet . Then, , andLetThen, simple computations lead toIt follows from (34)–(37) and Lemmas 1 and 3 that the function is strictly increasing on the interval . Note thatwhere 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 toLet . Then, andwhere 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.