Abstract and Applied AnalysisVolume 2011 (2011), Article ID 605259, 8 pagesdoi:10.1155/2011/605259

Research Article

Sharp Generalized Seiffert Mean Bounds for Toader Mean

Yu-Ming Chu,¹ Miao-Kun Wang,¹ Song-Liang Qiu,² and Ye-Fang Qiu¹

¹Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China²Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 4 June 2011; Revised 10 August 2011; Accepted 11 August 2011

Academic Editor: Detlev Buchholz

Copyright © 2011 Yu-Ming Chu 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 , the generalized Seiffert mean of two positive numbers and is defined by . In this paper, we find the greatest value and least value such that the double inequality holds for all with , and give new bounds for the complete elliptic integrals of the second kind. Here, denotes the Toader mean of two positive numbers and .

1. Introduction

For , the generalized Seiffert mean of two positive numbers and is defined by

It is well known that is continuous and strictly increasing with respect to for fixed with . In particular, if , then the generalized Seiffert mean reduces to the Seiffert mean

Recently, the Seiffert mean and its generalization have been the subject of intensive research, many remarkable inequalities for these means can be found in the literature [1–5].

In [6], Toader introduced the Toader mean of two positive numbers and as follows: where , is the complete elliptic integral of the second kind.

Vuorinen [7] conjectured that for all with , where is the power mean of order of two positive numbers and . This conjecture was proved by Barnard et al. [8].

In [9], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: for all with .

The main purpose of this paper is to find the greatest value and least value such that the double inequality holds for all with and give new bounds for the complete elliptic integrals of the second kind.

2. Lemmas

In order to establish our main result, we need several formulas and lemmas, which we present in this section.

The following formulas were presented in [10, Appendix E, pages 474-475]: Let , then

Lemma 2.1 (see [10, Theorem 1.25]). For , let be continuous on and be differentiable on , let for all . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2. (1)   is strictly increasing from onto ; (2)    is strictly increasing from onto ; (3)    is strictly increasing from onto ; (4)    is strictly increasing from onto ; (5)    is strictly increasing from onto ; (6)    is strictly increasing from onto .

Proof. Parts (1)–(4) can be found in [10, Theorem , Theorem  , and Exercise and ].For part (5), clearly (. Let and , then , and It follows from (2.3) and part (1) together with Lemma 2.1 that is strictly increasing in and .For part (6), clearly . Let and , then , , and From (2.4), parts (2) and (5) together with Lemma 2.1, we know that is strictly increasing in , and .

Lemma 2.3. (1) is strictly increasing from onto . (2) for .

Proof. For part (1), clearly and . Simple computation leads to where .Making use of Lemma 2.2 (1), (2), and (6), we get Therefore, part (1) follows from (2.5) and (2.6) together with the limiting values of at and .For part (2), simple computations yield that where . Note that where .From Lemma 2.2(1), (3), and (4) together with the monotonicity of we know that is strictly increasing in . Moreover, Equations (2.11)–(2.13) and the monotonicity of lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in .It follows from (2.8)–(2.10) and the piecewise monotonicity of that there exists such that is strictly decreasing in and strictly increasing in .Therefore, part follows from (2.7) and the piecewise monotonicity of .

3. Main Result

Theorem 3.1. Inequality holds for all with , and and are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean , respectively.

Proof. Firstly, we prove that for all with .Without loss of generality, we assume that . Let , . Then (1.1) and (1.3) lead to where and are defined as in Lemma 2.3.Therefore, inequality (3.1) follows from (3.2) and (3.3) together with Lemma 2.3.Next, we prove that and are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean , respectively.For any and , from (1.1) and (1.3) one has where .Letting and making use of Taylor expansion, we get Inequality (3.4) and equations (3.5) and (3.6) imply that for any there exist and , such that for and for .

From Theorem 3.1, we get new bounds for the complete elliptic integrals of the second kind as follows.

Corollary 3.2. The inequality holds for all .

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grant 11071069 and Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924. The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.

References

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

G. Toader, “Seiffert type means,” Nieuw Archief voor Wiskunde, vol. 17, no. 3, pp. 379–382, 1999.

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, p. 23, 2002.

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.

J. Sándor, “On certain inequalities for means. III,” Archiv der Mathematik, vol. 76, no. 1, pp. 34–40, 2001.

G. Toader, “Some mean values related to the arithmetic-geometric mean,” Journal of Mathematical Analysis and Applications, vol. 218, no. 2, pp. 358–368, 1998.

M. Vuorinen, “Hypergeometric functions in geometric function theory,” in Special Functions and Differential Equations (Madras, 1997), pp. 119–126, Allied, New Delhi, India, 1998.

R. W. Barnard, K. Pearce, and K. C. Richards, “A monotonicity property involving and comparisons of the classical approximations of elliptical arc length,” SIAM Journal on Mathematical Analysis, vol. 32, no. 2, pp. 403–419, 2000.

H. Alzer and S.-L. Qiu, “Monotonicity theorems and inequalities for the complete elliptic integrals,” Journal of Computational and Applied Mathematics, vol. 172, no. 2, pp. 289–312, 2004.

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.