Journal of Applied Mathematics

Volume 2013 (2013), Article ID 438971, 3 pages

http://dx.doi.org/10.1155/2013/438971

Research Article

Optimal Two Parameter Bounds for the Seiffert Mean

Hui Sun, Ying-Qing Song, and Yu-Ming Chu

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

Received 30 April 2013; Revised 18 July 2013; Accepted 18 July 2013

Academic Editor: Zhihua Zhang

Copyright © 2013 Hui Sun 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 obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and Applied Analysis (2012).

1. Introduction

For with , the Seiffert mean , root mean square , and contraharmonic mean are defined by

respectively. It is well known that the inequalities

hold for all with .

Recently, , , and have been the subject of intensive research. In particular, many remarkable inequalities and properties for these means can be found in the literature [1–8].

For , very recently Chu et al. [9, 10] proved that the inequalities

hold for all with if and only if , , , and .

Let , , and

where is the classical arithmetic mean of and . Then from (2), (3), and (7) we clearly see that

and is strictly increasing with respect to for fixed with .

It is natural to ask what are the greatest value and the least value in such that the double inequality

holds for all with and . The aim of this paper is to answer this question; our main result is the following Theorem 1.

Theorem 1. If and , then the double inequality

holds for all with if and only if and .

Remark 2. If we take and in Theorem 1, then inequality (10) reduces to inequalities (5) and (6), respectively.

2. Proof of Theorem 1

In order to prove Theorem 1 we need two lemmas, which we present in this section.

Lemma 3 (see [11, 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 4. Let , and

Then

(1) for if and only if ;(2) for if and only if .

Proof. By (12) and simple computations one has

where

Let and . Then is strictly increasing in :

It is not difficult to verify that the function is strictly increasing from onto ; hence (18) implies that is strictly decreasing in . Therefore, is strictly decreasing in follows from Lemma 3 and (17) together with the monotonicity of and . Moreover, making use of l'Hôpital's rule we get

We divide the proof into three cases.

Case  1 (). Then from (15) and (19) together with the monotonicity of we conclude that is strictly increasing in . Therefore for 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 in . Therefore for follows from (13) and the monotonicity of .

Case  3 (). Then from (15), (19), and (20) together with the monotonicity of we know that there exists such that is strictly decreasing in and strictly increasing in . Therefore, in if and only if follows from (13) and the piecewise monotonicity of , which (14) gives immediately .

Proof of Theorem 1. Since both and are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let . Then from (1) and (7) we get

Therefore, Theorem 1 follows from Lemma 4 and (21).

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant nos. 11071069 and 11171307) and the Natural Science Foundation of Zhejiang Province (Grant no. LY13A010004).

References

H. J. Seiffert, “Aufgabe β16,” Die Wurzel, vol. 29, pp. 221–222, 1995.

P. A. Hästö, “A monotonicity property of ratios of symmetric homogeneous means,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 5, article 71, 23 pages, 2002.

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.

H. Liu and X.-M. Ju, “The optimal convex combination bounds for Seiffert's mean,” Journal of Inequalities and Applications, vol. 2011, Article ID 686834, 9 pages, 2011.

W.-D. Jiang, “Some sharp inequalities involving reciprocals of the Seiffert and other means,” Journal of Mathematical Inequalities, vol. 6, no. 4, pp. 593–599, 2012.

W.-D. Jiang and F. Qi, “Some sharp inequalities involving Seiffert and other means and their concise proofs,” Mathematical Inequalities & Applications, vol. 15, no. 4, pp. 1007–1017, 2012.

J. Sandor, “Trigonometric and hyperbolic inequalities,” submitted, http://arxiv.org/abs/1105.0859

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.

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.

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