Abstract and Applied Analysis

Volume 2014 (2014), Article ID 354132, 8 pages

http://dx.doi.org/10.1155/2014/354132

Research Article

Refinements of Bounds for Neuman Means

Yu-Ming Chu¹ and Wei-Mao Qian²

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

²School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China

Received 26 December 2013; Accepted 13 February 2014; Published 18 March 2014

Academic Editor: Alberto Fiorenza

Copyright © 2014 Yu-Ming Chu and Wei-Mao Qian. 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 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.

References

B. C. Carlson, “Algorithms involving arithmetic and geometric means,” The American Mathematical Monthly, vol. 78, pp. 496–505, 1971.

J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, John Wiley & Sons, New York, NY, USA, 1987.

E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,” Mathematica Pannonica, vol. 14, no. 2, pp. 253–266, 2003.

E. Neuman and J. Sándor, “On the Schwab-Borchardt mean. II,” Mathematica Pannonica, vol. 17, no. 1, pp. 49–59, 2006.

E. Neuman, “A note on a certain bivariate mean,” Journal of Mathematical Inequalities, vol. 6, no. 4, pp. 637–643, 2012.

Y.-M. Chu and B.-Y. Long, “Bounds of the Neuman-Sándor mean using power and identric means,” Abstract and Applied Analysis, vol. 2013, Article ID 832591, 6 pages, 2013.

T.-H. Zhao, Y.-M. Chu, and B.-Y. Liu, “Optimal bounds for Neuman-Sándor mean in terms of the convex combinations of harmonic, geometric, quadratic, and contraharmonic means,” Abstract and Applied Analysis, vol. 2012, Article ID 302635, 9 pages, 2012.

E. Neuman, “On some means derived from the Schwab-Borchardt mean,” Journal of Mathematical Inequalities, Preprint, http://files.ele-math.com/preprints/jmi-1210-pre.pdf

E. Neuman, “On some means derived from the Schwab-Borchardt mean II,” Journal of Mathematical Inequalities, Preprint, http://files.ele-math.com/preprints/jmi-1289-pre.pdf

Z.-Y. He, Y.-M. Chu, and M.-K. Wang, “Optimal Bounds for Neuman Means in terms of harmonic and contraharmonic means,” Journal of Applied Mathematics, vol. 2013, Article ID 807623, 4 pages, 2013.