Abstract and Applied Analysis

Volume 2014 (2014), Article ID 949815, 6 pages

http://dx.doi.org/10.1155/2014/949815

Research Article

Sharp Geometric Mean Bounds for Neuman Means

Yan Zhang,¹ Yu-Ming Chu,¹ and Yun-Liang Jiang²

¹School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China

²School of Information Engineering, Huzhou Teachers College, Huzhou 313000, China

Received 31 March 2014; Accepted 27 April 2014; Published 6 May 2014

Academic Editor: Alberto Fiorenza

Copyright © 2014 Yan Zhang 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 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.

References

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