Journal of Applied Mathematics

Volume 2013 (2013), Article ID 807623, 4 pages

http://dx.doi.org/10.1155/2013/807623

Research Article

Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means

Zai-Yin He,¹ Yu-Ming Chu,² and Miao-Kun Wang²

¹Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

²School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China

Received 7 August 2013; Accepted 11 November 2013

Academic Editor: Chong Lin

Copyright © 2013 Zai-Yin He 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.

For with , the Schwab-Borchardt mean is defined as . In this paper, we find the greatest values of and and the least values of and in such that and . Similarly, we also find the greatest values of and and the least values of and in such that and . Here, , , and are the harmonic, arithmetic, and contraharmonic means, respectively, and , , , and are four Neuman means derived from the Schwab-Borchardt mean.

1. Introduction

For with , the Schwab-Borchardt mean is defined as

It is well known that the mean is strictly increasing in both and , nonsymmetric, and homogeneous of degree 1 in its variables. Several 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, respectively.

The Schwab-Borchardt mean was firstly investigated in [1–4]. In [3], the authors pointed out that the logarithmic mean, two Seiffert means, and the Neuman-Sándor mean are particular cases of the Schwab-Borchardt mean. Later, and its special cases have been the subject of intensive research. In particular, many inequalities for them can be found in the literature [3–13].

Let , be the harmonic and contraharmonic means of two positive numbers and , respectively. Then, it is well known that

for with .

Recently, the second author of this paper reviewed two elegant papers [14, 15] by Neuman and found that the bivariate means , , , and , derived from the Schwab-Borchardt mean are very interesting. They are defined as follows:

We call the means , , , and , defined in (4) the Neuman means. Moreover, if we let , then explicit formulas for , , , and are in the following:

where , , , and are defined implicitly as , , and , respectively. Clearly, , , , and .

Neuman [14, 15] presented several optimal bounds for , , , and . The bounding quantities are arithmetic convex, geometric convex, and harmonic convex combinations of their generating means. Besides, he also proved that

for with .

For fixed with , and . Let

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 are the greatest values of and and the least values of and in such that and ? And what are the greatest values of and and the least values of and in such that and ? The main purpose of this paper is to answer these questions. Our main results are in Theorems 1 and 2.

Theorem 1. Let . Then, the double inequality

holds for all with if and only if and . Also the double inequality

holds for all with if and only if and .

Theorem 2. Let . Then, the double inequality

holds for all with if and only if and . Also the double inequality

holds for all with if and only if and .

2. Two Lemmas

In order to prove the desired theorems, we first give two lemmas.

Lemma 1 (see [16, Theorem 1.25]). For , let be continuous on , and be 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. (1) The function is strictly increasing from onto .

(2) The function is strictly increasing from onto .

(3) The function is strictly decreasing from onto .

(4) The function is strictly decreasing from onto .

Proof. From part (1), let and . Then, , , and

It is well known that is strictly decreasing on . Then, Lemma 1 and (14) lead to the conclusion that is strictly increasing on . Moreover, by l’Hôptial’s rule we have and .

From part (2), similarly let and . Then , and

It is well known that is strictly increasing on . Then, by Lemma 1 and (15) we know that is strictly increasing on . Clearly, , while by l’Hôptial’s rule we have .

Parts (3) and (4) have been proven in [14, Theorem 3].

3. Proofs of Theorems 1 and 2

Proof of Theorem 1. Without loss of generality, we assume that . Let and ; then,

provided that . Thus, inequality (9) follows from (16) and Lemma 2(1). Similarly,

provided that . Thus, inequality (10) follows from (17) and Lemma 2(2).

Proof of Theorem 2. Without loss of generality, we assume that . Let and , then

provided that . Thus, inequality (11) follows from (18) and Lemma 2(3). Similarly,

provided that . Thus, inequality (12) follows from (19) and Lemma 2(4).

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Natural Science Foundation of Huzhou Teachers College under Grant KX21063.

References

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.

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

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.

B. C. Carlson and J. L. Gustafson, “Total positivity of mean values and hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 14, no. 2, pp. 389–395, 1983.

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.

Y.-M. Chu, M.-K. Wang, and Z.-K. Wang, “Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means,” Mathematical Inequalities & Applications, vol. 15, no. 2, pp. 415–422, 2012.

E. Neuman, “Inequalities for the Schwab-Borchardt mean and their applications,” Journal of Mathematical Inequalities, vol. 5, no. 4, pp. 601–609, 2011.

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

E. Neuman, “Sharp inequalities involving Neuman-Sándor and logarithmic means,” Journal of Inequalities and Applications, vol. 7, no. 3, pp. 413–419, 2013.

A. Witkowski, “Interpolations of Schwab-Borchardt means,” Mathematical Inequalities & Applications, vol. 16, no. 1, pp. 193–206, 2013.

M.-K. Wang, Y.-F. Qiu, and Y.-M. Chu, “Sharp bounds for Seiffert means in terms of Lehmer means,” Journal of Mathematical Inequalities, vol. 4, no. 4, pp. 581–586, 2010.

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. In press

E. Neuman, “On some means derived from the Schwab-Borchardt mean II,” Journal of Mathematical Inequalities. In press

G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, NY, USA, 1997.