Abstract and Applied Analysis

Volume 2013 (2013), Article ID 903982, 5 pages

http://dx.doi.org/10.1155/2013/903982

## Bounds for the Combinations of Neuman-Sándor, Arithmetic, and Second Seiffert Means in terms of Contraharmonic Mean

^{1}Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China^{2}School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China^{3}School of Information and Engineering, Huzhou Teachers College, Huzhou 313000, China^{4}School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China

Received 4 January 2013; Accepted 27 February 2013

Academic Editor: Salvatore A. Marano

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

We give the greatest values , and the least values , in (1/2, 1) such that the double inequalities and hold for any and all with , where , , and are the arithmetic, Neuman-Sándor, contraharmonic, and second Seiffert means of and , respectively.

#### 1. Introduction

For with , the Neuman-Sándor mean [1], second Seiffert mean [2] are defined by respectively. Herein, is the inverse hyperbolic sine function.

Let , , , , , , , and be the harmonic, geometric, logarithmic, first Seiffert, identric, arithmetic, quadratic, and contraharmonic means of two distinct positive real numbers and , respectively. Then it is well known that the inequalities hold for all with .

Among means of two variables, the Neuman-Sándor, contraharmonic, and second Seiffert means have attracted the attention of several researchers. In particular, many remarkable inequalities and applications for these means can be found in the literature [3–15].

Neuman and Sándor [1, 16] proved that the inequalities hold for all with .

Let with , and . Then the Ky Fan inequalities can be found in [1].

Li et al. [17] proved that the double inequality holds for all with , where , and is the th generalized logarithmic mean of and , and is the unique solution of the equation .

In [18], Neuman proved that the inequalities hold for all with if and only if , , and .

Zhao et al. [19] found the least values and the greatest values such that the double inequalities hold for all with .

In [20, 21], the authors proved that the double inequalities hold for all with if and only of , , , , , and .

For , Chu et al. [22, 23] proved that the inequalities hold for all with if and only if , , and .

The aim of this paper is to find the greatest values , and the least values , such that the double inequalities hold for any and all with .

#### 2. Lemmas

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

Lemma 1 (see [24, Theorem 1.25]). *For , let be continuous on and 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. * Let and
**
Then for all if and only if and for all if and only if .*

* Proof. *From (12), one has
where

Let and , then

It is not difficult to verify that the function is strictly increasing on . Then (17) and (18) together with Lemma 1 lead to the conclusion that is strictly decreasing on . Moreover, making use of L'Hôpital's rule, we get

We divide the proof into four cases.*Case **1. *. Then from (15) and (19) together with the monotonicity of , we clearly see that is strictly increasing on . Therefore, for all follows from (13) and the monotonicity of .*Case **2. *. Then from (15) and (20) together with the monotonicity of , we clearly see that is strictly decreasing on . Therefore, for all follows from (13) and the monotonicity of .*Case **3. *. Then (14) leads to

From (15), (19), and (20) together with the monotonicity of , we clearly see that there exists unique such that is strictly decreasing on and strictly increasing on . Therefore, for all follows from (13) and (21) together with the piecewise monotonicity of .*Case **4**. *. Then (14) leads to

It follows from (15), (19), and (20) together with the monotonicity of , there exists unique such that is strictly decreasing on and strictly increasing on . Equation (13) and inequality (22) together with the piecewise monotonicity of lead to the conclusion that there exists such that for and for .

Lemma 3. * Let and
**
Then for all if and only if and for all if and only if .*

* Proof. *From (23) we get
where

Let and , then

It is not difficult to verify that the function is strictly increasing on . Then (28) together with Lemma 1 leads to the conclusion that is strictly decreasing on . Moreover, making use of L'Hôpital's rule, we have

We divide the proof into four cases.*Case **1. *. Then from (26) and (29) together with the monotonicity of , we clearly see that is strictly increasing on . Therefore, for all follows from (24) and the monotonicity of .*Case **2*. . Then from (26) and (30) together with the monotonicity of , we clearly see that is strictly decreasing on . Therefore, for all follows from (24) and the monotonicity of .*Case **3. *. Then (25) leads to

From (26), (29), and (30) together with the monotonicity of , we clearly see that there exists such that is strictly decreasing on and strictly increasing on . Therefore, for all follows from (24) and (31) together with the piecewise monotonicity of .*Case **4. *. Then (25) leads to

It follows from (26), (29), and (30) together with the monotonicity of , there exists such that is strictly decreasing on and strictly increasing on . Equation (24) and inequality (32) together with the piecewise monotonicity of lead to the conclusion that there exists such that for and for .

#### 3. Main Results

Theorem 4. *If and , then the double inequality
**
holds for all with if and only if and .*

* Proof. *Since , , and are symmetric and homogeneous of degree one, without loss of generality, we assume that . Let and , then and

Therefore, Theorem 4 follows easily from Lemma 2 and (34).

Theorem 5. *If and , then the double inequality
**
holds for all with if and only if and .*

* Proof. *Since , , and are symmetric and homogeneous of degree one, without loss of generality, we assume that . Let and , then and

Therefore, Theorem 5 follows easily from Lemma 3 and (36).

*Remark 6. * If , then Theorem 4 reduces to the first double inequality in (8).

Corollary 7. * If , then the double inequality
**
holds for all with if and only if and .*

* Proof. *Corollary 7 follows easily from Theorem 5 with .

#### Acknowledgments

This research was supported by the Natural Science Foundation of China (Grants no.11171307, 61173123), the Natural Science Foundation of Zhejiang Province (Grants no. Z1110551, LY12F02012), and the Natural science Foundation of Huzhou City (Grant no. 2012YZ06).

#### References

- E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,”
*Mathematica Pannonica*, vol. 14, no. 2, pp. 253–266, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-J. Seiffert, “Aufgabe $\beta $ 16,”
*Die Wurzel*, vol. 29, pp. 221–222, 1995. View at Google Scholar - Y.-M. Chu and B.-Y. Long, “Bounds of the Neuman-Sándor mean using power and identric means,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 832591, 6 pages, 2012. View at Publisher · View at Google Scholar - Y.-M. Chu, B.-Y. Long, W.-M. Gong, and Y.-Q. Song, “Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means,”
*Journal of Inequalities and Applications*, vol. 2013, article 10, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - F. R. Villatoro, “Local error analysis of Evans-Sanugi, nonlinear one-step methods based on $\theta $-means,”
*International Journal of Computer Mathematics*, vol. 87, no. 5, pp. 1009–1022, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - F. R. Villatoro, “Stability by order stars for non-linear theta-methods based on means,”
*International Journal of Computer Mathematics*, vol. 87, no. 1–3, pp. 226–242, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Pahikkala, “On contraharmonic mean and Phythagorean triples,”
*Elemente der Mathematik*, vol. 65, no. 2, pp. 62–67, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - O. Y. Ababneh and R. Rozita, “New third order Runge Kutta based on contraharmonic mean for stiff problems,”
*Applied Mathematical Sciences*, vol. 3, no. 5–8, pp. 365–376, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Toader and G. Toader, “Complementaries of Greek means with respect to Gini means,”
*International Journal of Applied Mathematics & Statistics*, vol. 11, no. N07, pp. 187–192, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Lim, “The inverse mean problem of geometric and contraharmonic means,”
*Linear Algebra and Its Applications*, vol. 408, pp. 221–229, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Toader and G. Toader, “Generalized complementaries of Greek means,”
*Pure Mathematics and Applications*, vol. 15, no. 2-3, pp. 335–342, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M. Chu, M.-K. Wang, and Y.-F. Qiu, “Optimal Lehmer mean bounds for the geometric and arithmetic combinations of arithmetic and Seiffert means,”
*Azerbaijan Journal of Mathematics*, vol. 2, no. 1, pp. 3–9, 2012. View at Google Scholar · View at MathSciNet - P. A. Hästö, “A monotonicity property of ratios of symmetric homogeneous means,”
*JIPAM*, vol. 3, no. 5, article 71, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M. Chu, M.-K. Wang, and Y.-F. Qiu, “An optimal double inequality between power-type Heron and Seiffert means,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 146945, 11 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Neuman and J. Sándor, “On the Schwab-Borchardt mean. II,”
*Mathematica Pannonica*, vol. 17, no. 1, pp. 49–59, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M. Li, B.-Y. Long, and Y.-M. Chu, “Sharp bounds for the Neuman-Sándor mean in terms of generalized logarithmic mean,”
*Journal of Mathematical Inequalities*, vol. 6, no. 4, pp. 567–577, 2012. View at Google Scholar - E. Neuman, “A note on a certain bivariate mean,”
*Journal of Mathematical Inequalities*, vol. 6, no. 4, pp. 637–643, 2012. View at Google Scholar - 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. View at Publisher · View at Google Scholar - Y.-M. Chu, C. Zong, and G.-D. Wang, “Optimal convex combination bounds of Seiffert and geometric means for the arithmetic mean,”
*Journal of Mathematical Inequalities*, vol. 5, no. 3, pp. 429–434, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M. Chu, M.-K. Wang, and W.-M. Gong, “Two sharp double inequalities for Seiffert mean,”
*Journal of Inequalities and Applications*, vol. 2011, article 44, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-M. Chu and S.-W. Hou, “Sharp bounds for Seiffert mean in terms of contraharmonic mean,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 425175, 6 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M. Chu, S.-W. Hou, and Z.-H. Shen, “Sharp bounds for Seiffert mean in terms of root mean square,”
*Journal of Inequalities and Applications*, vol. 2012, article 11, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorinen,
*Conformal Invariants, Inequalities, and Quasiconformal Maps*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1997. View at MathSciNet