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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 273653, 5 pages

http://dx.doi.org/10.1155/2013/273653

## Optimal Lower Generalized Logarithmic Mean Bound for the Seiffert Mean

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

Received 23 December 2012; Accepted 26 February 2013

Academic Editor: Francisco J. Marcellán

Copyright © 2013 Ying-Qing Song 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 present the greatest value such that the inequality holds for all with , where and denote the Seiffert and th generalized logarithmic means of and , respectively.

#### 1. Introduction

For , the th generalized logarithmic mean is defined by and the Seiffert mean [1] is defined by

It is well known that the generalized logarithmic mean is continuous and strictly increasing with respect to for fixed with . The special cases of the generalized logarithmic mean are, for example, is the geometric mean, is the logarithmic mean, is the identric mean, and is the arithmetic mean. The Seiffert mean can be rewritten as (see [2, equation (2.4)])

Recently, the bivariate means have been the subject of intensive research. In particular, many remarkable inequalities and properties for the generalized logarithmic and the Seiffert means can be found in the literature [3–13].

In [1, 11], Seiffert proved that the inequalities hold for all with .

Sándor [14] presented the bounds for the Seiffert mean in terms of the arithmetic mean and geometric mean as follows: for all with .

Hästö [15] proved that the double inequality holds for all with if and only if and , where is the th power mean of and .

In [16], the authors found the greatest value and least value such that the double inequality holds for all with , where is the contraharmonic mean of and .

Motivated by the first inequality in (4), Gao [17] gave the best possible constants and such that the double inequality holds for all with .

In [18], the author solved the following open problem proposed by Long and Chu [19]: what is the smallest (largest ) such that the inequality holds for and all with ?

Chu et al. [20] proved that the double inequality holds for all with if and only if and , where is the second Seiffert mean of and , and is the unique solution of the equation .

In [21], the authors answered the question: what are the greatest value and the least value such that the double inequality holds for any with and all with ?

Motivated by the first inequality in (4), it is natural to ask what are the best possible generalized logarithmic mean bounds for the Seiffert mean ? It is the aim of this paper to answer this question.

#### 2. Preliminaries

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

Lemma 1. *Let the function be defined with
**
Then, is a continuous and strictly increasing function.*

*Proof. *From (6), we clearly see that
If , then simple computation yields
If we define
then
Equation (11) implies that
Equations (10) and (12) lead to
for .

From (9) and (10) together with (13), we clearly see that
for .

Therefore, the continuity of follows from (6) and (7), and the strict monotonicity of follows from (8), (14) and the continuity of .

*Remark 2. *From Lemma 1, we clearly see that for any fixed , there exists a unique such . In particular, for , making use of Mathematica software, we get

Therefore, the unique solution of the equation belongs to the interval .

Lemma 3. *Let and let be defined with . Then, there exists such that for and for .*

*Proof. *Simple computations lead to

Let . Further computations lead to

Inequalities (34) and (35) imply that
for .

From (31) and (32) together with (36), we clearly see that there exists such that for and for . Hence, is strictly decreasing on and strictly increasing on .

It follows from (28) and (29) together with the monotonicity of that there exists such that is strictly decreasing on and strictly increasing on .

Equations (25)-(26) and the monotonicity of lead to the conclusion that there exists such that for and for . Therefore, is strictly decreasing on and strictly increasing on .

From (22) and (23) together with the monotonicity of , we clearly see that there exists such that is strictly decreasing on and strictly increasing on .

Equations (19)-(20) and the monotonicity of imply that there exists such that is strictly decreasing on and strictly increasing on .

Therefore, Lemma 3 follows from (16) and (17) together with the monotonicity of .

#### 3. Main Results

Theorem 4. *Let be as in Lemma 1 and let be the unique solution of the equation . Then, for all , , the inequality
**
holds, and is the best possible lower generalized logarithmic mean bound for the Seiffert mean .*

*Proof. *From Remark 2, it follows that

We first prove that the inequality (37) holds. Without loss of generality, we assume that . If , then, from (1) and (2), we have
If
then simple computations lead to
where
where

From (38) and (44)-(45) together with Lemma 3, we clearly see that there exists such that is strictly decreasing on and strictly increasing on . Then, (38) and (42)-(43) together with the monotonicity of imply that there exists such that is strictly decreasing on and strictly increasing on .

Therefore, follows from (39)–(41) and the monotonicity of .

Next, we prove that is the best possible lower generalized logarithmic mean bound for the Seiffert mean .

For any , from (1) and (2), we get
Lemma 1 and (46) lead to

Inequality (47) implies that, for , there exists such that for .

Theorem 5. * is the best possible upper generalized logarithmic mean bound for the Seiffert mean .*

*Proof. *For any and , from (1) and (2), we have
where .

When , then making use of the Taylor expansion, we get

Equations (48) and (49) imply that for any , there exists such that
for .

Therefore, Theorem 5 follows from inequalities (4) and (50).

#### Acknowledgments

This paper was supported by the Natural Science Foundation of China under Grant 61173123, the Natural Science Foundation of Zhejiang Province under Grants Z1110551 and LY12F02012, and the Natural Science Foundation of Huzhou City under Grant 2012YZ06.

#### References

- H. J. Seiffert, “Problem 887,”
*Nieuw Archief voor Wiskunde (4)*, vol. 11, no. 2, p. 176, 1993. View at Google Scholar - 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 - I. Costin and G. Toader, “Optimal evaluations of some Seiffert-type means by power means,”
*Applied Mathematics and Computation*, vol. 219, no. 9, pp. 4745–4754, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Google Scholar - W. M. Qian and B. Y. Long, “Sharp bounds by the generalized logarithmic mean for the geometric weighted mean of the geometric and harmonic means,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 480689, 8 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. N. Shi and S. H. Wu, “Refinement of an inequality for the generalized logarithmic mean,”
*Chinese Quarterly Journal of Mathematics*, vol. 23, no. 4, pp. 594–599, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. P. Chen, “The monotonicity of the ratio between generalized logarithmic means,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 1, pp. 86–89, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Qi, S. X. Chen, and C. P. Chen, “Monotonicity of ratio between the generalized logarithmic means,”
*Mathematical Inequalities & Applications*, vol. 10, no. 3, pp. 559–564, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Toader, “Seiffert type means,”
*Nieuw Archief voor Wiskunde (4)*, vol. 17, no. 3, pp. 379–382, 1999. View at Google Scholar · View at MathSciNet - C. E. M. Pearce, J. Pečarić, and V. Šimić, “On weighted generalized logarithmic means,”
*Houston Journal of Mathematics*, vol. 24, no. 3, pp. 459–465, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. J. Seiffert, “Ungleichungen für einen bestimmten Mittelwert,”
*Nieuw Archief voor Wiskunde (4)*, vol. 13, no. 2, pp. 195–198, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. E. M. Pearce and J. Pečarić, “Some theorems of Jensen type for generalized logarithmic means,”
*Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 40, no. 9-10, pp. 789–795, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. B. Stolarsky, “The power and generalized logarithmic means,”
*The American Mathematical Monthly*, vol. 87, no. 7, pp. 545–548, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Sándor, “On certain inequalities for means III,”
*Archiv der Mathematik*, vol. 76, no. 1, pp. 34–40, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. A. Hästö, “Optimal inequalities between Seiffert's mean and power means,”
*Mathematical Inequalities & Applications*, vol. 7, no. 1, pp. 47–53, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Liu and X. J. Meng, “The optimal convex combination bounds for Seiffert's mean,”
*Journal of Inequalities and Applications*, vol. 2011, Article ID 686834, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Q. Gao, “Inequalities for the Seiffert's means in terms of the identric mean,”
*Journal of Mathematical Sciences: Advances and Applications*, vol. 10, no. 1-2, pp. 23–31, 2011. View at Google Scholar · View at MathSciNet - L. Matejíčka, “Proof of one optimal inequality for generalized logarithmic, arithmetic, and geometric means,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 902432, 5 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Y. Long and Y. M. Chu, “Optimal inequalities for generalized logarithmic, arithmetic, and geometric means,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 806825, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. M. Chu, M. K. Wang, and G. D. Wang, “The optimal generalized logarithmic mean bounds for Seiffert's mean,”
*Acta Mathematica Scientia B*, vol. 32, no. 4, pp. 1619–1626, 2012. View at Google Scholar - Y. L. Jiang, Y. M. Chu, and B. Y. Long, “An optimal double inequality between logarithmic and generalized logarithmic means,”
*Journal of Applied Analysis*. In press.