- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Journal of Applied Mathematics

Volume 2012 (2012), Article ID 182905, 8 pages

http://dx.doi.org/10.1155/2012/182905

## Optimal Inequalities for Power Means

^{1}Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China^{2}School of Mathematics Science, Anhui University, Hefei 230039, China^{3}Department of Mathematics, Hunan City University, Yiyang 413000, China

Received 9 December 2011; Revised 30 January 2012; Accepted 2 February 2012

Academic Editor: Hector Pomares

Copyright © 2012 Yong-Min Li 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 best possible power mean bounds for the product for any , , and all with . Here, is the th power mean of two positive numbers and .

#### 1. Introduction

For , the th power 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 . Many classical means are special cases of the power mean, for example, , and are the harmonic, geometric and arithmetic means of and , respectively. Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities and properties for the power mean can be found in literature [1–22].

Let , and be the logarithmic, Seiffert and identric means of two positive numbers and with , respectively. Then it is well known that for all with .

In [23–29], the authors presented the sharp power mean bounds for , , and as follows: for all with .

Alzer and Qiu [12] proved that the inequality holds for all with if and only if .

The following sharp bounds for the sum , and the products and in terms of power means were proved in [5, 8]: for any and all with .

In [2, 7] the authors answered the questions: for any , what are the greatest values , , , and , and the least values , , , and , such that the inequalities hold for all with ?

It is the aim of this paper to present the best possible power mean bounds for the product for any , and all with .

#### 2. Main Result

Theorem 2.1. *Let , and with . Then*(1)* for ,*(2)* for and for , and the bounds and for the product in either case are best possible. *

*Proof. *From (1.1) we clearly see that is symmetric and homogenous of degree 1. Without loss of generality, we assume that , .(1)If , then (1.1) leads to (2)Firstly, we compare the value of to the value of for . From (1.1) we have

Let
then simple computations lead to
where
where

If , then (2.9) implies that is strictly decreasing in . Therefore, follows easily from (2.2)–(2.8) and the monotonicity of .

If , then (2.9) leads to the conclusion that is strictly increasing in . Therefore, follows easily from (2.2)–(2.8) and the monotonicity of .

Secondly, we compare the value of to the value of . It follows from (1.1) that

Let
then simple computations lead to

If , then (2.13) implies that is strictly increasing in . Therefore, follows easily from (2.10)–(2.12) and the monotonicity of .

If , then (2.13) leads to the conclusion that is strictly decreasing in . Therefore, follows easily from (2.10)–(2.12) and the monotonicity of .

Next, we prove that the bound for the product in either case is best possible.

If , then for any and we have

Letting and making use of Taylor’s expansion, one has

Equations (2.14) and (2.15) imply that for any and there exists , such that for .

If , then for any and we have

Letting and making use of Taylor’s expansion, one has

Equations (2.16) and (2.17) imply that for any and there exists , such that for .

Finally, we prove that the bound for the product in either case is best possible.

If , then for any we clearly see that

Equation (2.18) implies that for any and there exists , such that for .

If , then for any we have

Equation (2.19) implies that for any and there exists , such that for .

#### Acknowledgments

This paper was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

#### References

- Y.-M. Chu, S.-S. Wang, and C. Zong, “Optimal lower power mean bound for the convex combination of harmonic and logarithmic means,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 520648, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Chu and B. Long, “Sharp inequalities between means,”
*Mathematical Inequalities & Applications*, vol. 14, no. 3, pp. 647–655, 2011. View at Google Scholar · View at Zentralblatt MATH - M.-K. Wang, Y.-M. Chu, Y.-F. Qiu, and S.-L. Qiu, “An optimal power mean inequality for the complete elliptic integrals,”
*Applied Mathematics Letters*, vol. 24, no. 6, pp. 887–890, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B.-Y. Long and Y.-M. Chu, “Optimal power mean bounds for the weighted geometric mean of classical means,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 905679, 6 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W.-F. Xia, Y.-M. Chu, and G.-D. Wang, “The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 604804, 9 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y.-M. Chu and W.-F. Xia, “Two optimal double inequalities between power mean and logarithmic mean,”
*Computers & Mathematics with Applications*, vol. 60, no. 1, pp. 83–89, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y.-M. Chu, Y.-F. Qiu, and M.-K. Wang, “Sharp power mean bounds for the combination of Seiffert and geometric means,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 108920, 12 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, “Optimal inequalities among various means of two arguments,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 694394, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y.-M. Chu and W.-F. Xia, “Two sharp inequalities for power mean, geometric mean, and harmonic mean,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 741923, 6 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Wu, “Generalization and sharpness of the power means inequality and their applications,”
*Journal of Mathematical Analysis and Applications*, vol. 312, no. 2, pp. 637–652, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 Google Scholar · View at Zentralblatt MATH - H. Alzer and S.-L. Qiu, “Inequalities for means in two variables,”
*Archiv der Mathematik*, vol. 80, no. 2, pp. 201–215, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Alzer, “A power mean inequality for the gamma function,”
*Monatshefte für Mathematik*, vol. 131, no. 3, pp. 179–188, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. E. Pečarić, “Generalization of the power means and their inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 161, no. 2, pp. 395–404, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. S. Bullen, D. S. Mitrinović, and P. M. Vasić,
*Means and Their Inequalities*, vol. 31 of*Mathematics and Its Applications (East European Series)*, D. Reidel Publishing Co., Dordrecht, The Netherlands, 1988. - S.-H. Wu and H.-N. Shi, “A relation of weak majorization and its applications to certain inequalities for means,”
*Mathematica Slovaca*, vol. 61, no. 4, pp. 561–570, 2011. View at Publisher · View at Google Scholar - S. Wu and L. Debnath, “Inequalities for differences of power means in two variables,”
*Analysis Mathematica*, vol. 37, no. 2, pp. 151–159, 2011. View at Publisher · View at Google Scholar - S. Wu, “On the weighted generalization of the Hermite-Hadamard inequality and its applications,”
*The Rocky Mountain Journal of Mathematics*, vol. 39, no. 5, pp. 1741–1749, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-N. Shi, M. Bencze, S.-H. Wu, and D.-M. Li, “Schur convexity of generalized Heronian means involving two parameters,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 879273, 9 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Wu and L. Debnath, “Inequalities for convex sequences and their applications,”
*Computers & Mathematics with Applications*, vol. 54, no. 4, pp. 525–534, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-N. Shi, S.-H. Wu, and F. Qi, “An alternative note on the Schur-convexity of the extended mean values,”
*Mathematical Inequalities & Applications*, vol. 9, no. 2, pp. 219–224, 2006. View at Google Scholar · View at Zentralblatt MATH - S. Wu, “Some results on extending and sharpening the Weierstrass product inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 308, no. 2, pp. 689–702, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Burk, “The geomeric, logarithmic, and arithmetic mean inequality,”
*The American Mathematical Monthly*, vol. 94, no. 6, pp. 527–528, 1987. View at Publisher · View at Google Scholar - H. Alzer, “Ungleichungen für Mittelwerte,”
*Archiv der Mathematik*, vol. 47, no. 5, pp. 422–426, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Alzer, “Ungleichungen für
*(e/a)*^{a}*(b/e)*^{b},”*Elemente der Mathematik*, vol. 40, pp. 120–123, 1985. View at Google Scholar - 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 - A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,”
*Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika*, no. 678–715, pp. 15–18, 1980. View at Google Scholar · View at Zentralblatt MATH - A. O. Pittenger, “The symmetric, logarithmic and power means,”
*Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika*, no. 678–715, pp. 19–23, 1980. View at Google Scholar · View at Zentralblatt MATH - T. P. Lin, “The power mean and the logarithmic mean,”
*The American Mathematical Monthly*, vol. 81, pp. 879–883, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH