Sharp Geometric Mean Bounds for Neuman Means

Abstract

We find the best possible constants and such that the double inequalities + + + + + + + + hold for all with , where , , and are, respectively, the geometric, arithmetic, and quadratic means and , , , and are the Neuman means.

1. Introduction

For with , the Schwab-Borchardt mean [1, 2] of and is given by where and are the inverse cosine and inverse hyperbolic cosine functions, respectively. Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for Schwab-Borchardt mean and its generated means can be found in the literature [1–6].

Very recently, Neuman [7] found a new bivariate mean derived from the Schwab-Borchardt mean as follows:

Let , , , and be the Neuman means, where , , and are the classical geometric, arithmetic, and quadratic means of and , respectively. Then the inequalities for all with , were established by Neuman [7].

Let and . Then the following explicit formulas for , , , and are presented in [7] where , , , and are the inverse hyperbolic tangent, inverse sine, inverse hyperbolic sine, and inverse tangent functions, respectively.

In [7], Neuman also proved that the double inequalities hold for all with if and only if , , , , , , , and .

Let with , , , Then, it is not difficult to verify that and are continuous and strictly increasing on and , respectively. Note that

Therefore, it is natural to ask what the best possible constants and are such that the double inequalities hold for all with . The main purpose of this paper is to answer this question.

2. Lemmas

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

Lemma 1 (see [8, Theorem 1.25]). Let , be continuous on and differentiable on , and 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 [9, 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 increasing from onto .

Proof. Making use of the power series expansion, we have Let Then for all . Note that
Therefore, Lemma 3 follows easily from Lemma 2 and (14)–(17).

Lemma 4. The function is strictly increasing from onto .

Proof. It is not difficult to verify that Let and . Then, From Lemma 1 and (19)-(20), we know that we just need to prove that the function is strictly increasing on .
Let , , , and . Then Note that is strictly increasing on . Therefore, the monotonicity of follows easily from (22) and (23) together with the monotonicity of .

Lemma 5. The function is strictly increasing from onto .

Proof. Simple computations lead to Let and . Then Let Then for all .
It, from Lemma 2 and (28)–(31), that is strictly increasing on . Therefore, Lemma 5 follows easily from (26) and (27) together with Lemma 1 and the monotonicity of .

Lemma 6. The function is strictly increasing from on .

Proof. Differentiating gives Let Then, simple computations lead to for .
Therefore, Lemma 6 follows easily from (33)–(39) together with the fact that and .

3. Main Results

Theorem 7. Let . Then the double inequality holds for all with if and only if and .

Proof. Since both the geometric mean and arithmetic mean are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let and . Then, from (4), one has Let . Then , and where is defined as in Lemma 3.
Therefore, Theorem 7 follows easily from (41) and (42) together with Lemma 3.

Theorem 8. Let . Then the double inequality holds for all with if and only if and .

Proof. We follow the same idea in the proof of Theorem 7. Without loss of generality, we assume that . Let and . Then, from (5), we get Let . Then, and simple computation leads to where is defined as in Lemma 4.
Therefore, Theorem 8 follows easily from (44) and (45) together with Lemma 4.

Theorem 9. Let . Then the double inequality holds for all with if and only if and .

Proof. Since both the quadratic mean and arithmetic mean are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let and . Then, (6) gives Let . Then, and elementary computations lead to where is defined as in Lemma 5.
Therefore, Theorem 9 follows easily from (47) and (48) together with Lemma 5.

Theorem 10. Let . Then the double inequality holds for all with if and only if and .

Proof. We follow the same idea in the proof of Theorem 9. Let , and . Then, from (7), we have Let . Then and where is defined as in Lemma 6.
Therefore, Theorem 10 follows easily from (50) and (51) together with Lemma 6.

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 61370173 and 61374086, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Applied Research Major Project of Public Welfare Technology of Huzhou City under Grant 2013GZ02.