Abstract and Applied Analysis

Volume 2010, Article ID 108920, 12 pages

http://dx.doi.org/10.1155/2010/108920

## Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means

^{1}Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China^{2}Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 29 March 2010; Revised 23 July 2010; Accepted 29 July 2010

Academic Editor: Lance Littlejohn

Copyright © 2010 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

We answer the question: for , what are the greatest value and the least value such that the double inequality holds for all with . Here, , , and denote the power of order , Seiffert, and geometric means of two positive numbers and , respectively.

#### 1. Introduction

For , the power mean of order and the Seiffert mean of two positive numbers and are defined by The main properties of the power mean are given in [1]. It is well known that is strictly increasing with respect to for fixed with . Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for the power mean can be found in the literature [2–16].

The Seiffert mean was introduced by Seiffert in [17], it can be rewritten in the following symmetric form (see [18, ]):

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

In [9], Alzer and Janous presented the sharp power mean bounds for the sum as follows: for all with .

In [17], Seiffert proved that for all with .

The following power mean bounds for the Seiffert mean was given by Jagers [19]: for all with .

In [20, 21], the authors presented the bounds for the Seiffert mean in terms of and as follows: for all with .

The following sharp lower power mean bounds for , , and can be found in [4, 6]: for all with .

The purpose of this paper is to answer the question: for , what are the greatest value and the least value such that the double inequality holds for all with .

#### 2. Lemmas

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

Lemma 2.1. *Let , and . Then there exists such that for , for and .*

*Proof. *Simple computations lead to

Inequality (2.5) implies that is strictly increasing in . Then (2.3) and (2.4) lead to that there exists such that for and for . Hence, is strictly decreasing in and strictly increasing in .

Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the monotonicity of .

Lemma 2.2. *If , then the following statements are true:*(1)*;*(2)*;*(3)*.*

*Proof. *Simple computations lead to + = ; + + = ; + + = .

Lemma 2.3. *If , then
**
holds for all with .*

*Proof. *Without loss of generality, we assume that . Let and . Then
Let
Then simple computations lead to
where / + ,
where + + ,
Let and . Then

From (2.21)–(2.23) we clearly see that for , hence is strictly increasing in . Then (2.20) implies that for , hence is strictly increasing in .

It follows from (2.18) and the monotonicity of that for , hence is strictly increasing in . Then (2.17) implies that for , therefore is strictly increasing in .

Equation (2.16) and the monotonicity of lead to that for , so is strictly increasing in .

From (2.9)–(2.14) and the monotonicity of we can deduce that
for .

Therefore, Lemma 2.3 follows from (2.7) and (2.8) together with (2.24).

*Remark 2.4. *In [22, Theorem], the authors proved that
for all with .

Obviously, (2.6) is the generalization of (2.25)–(2.27).

*Remark 2.5. *If , then and (2.17) leads to

Inequality (2.28) and the continuity of imply that there exists such that
for .

From (2.29) and (2.9)–(2.16) we can deduce that
for .

Equations (2.7) and (2.8) together with (2.30) lead to
for all with .

Therefore, is the largest value in such that inequality (2.6) holds for .

#### 3. Main Result

Theorem 3.1. *If , then
**
holds for all with , and and are the best possible lower and upper power mean bounds for the product .*

*Proof. *For all with , from (1.3), (1.5) and Lemma 2.1 we clearly see that
for all , and
for .

Next, we prove that (3.3) is also true for and all with .

Without loss of generality, we assume that . Let and . Then (1.1) leads to
Let
Then simple computations lead to
From Lemma 2.1 we know that there exists such that
for , and
for .

We divide two cases to prove that
for .

*Case 1. *. Then (3.11) follows from (3.7) and (3.8) together with (3.10).

*Case 2. *. Then (3.7) can be written as
where
Let , then (3.13) leads to
where
Let , then (3.15) leads to

From and , we clearly see that , , and . These inequalities and Lemma 2.2 lead to
From (3.16)–(3.18) we can deduce that
for .

Equation (3.14) together with (3.19) imply that
for .

Therefore, (3.11) follows from (3.9) and (3.12) together with (3.20).

It follows from (3.4)–(3.6) and (3.11) that
for and all with .

At last, we prove that and are the best possible lower- and upper-power mean bounds for the product , respectively.

For any and , from (1.1) one has
where (1/2) .

Let , making use of the Taylor expansion we get

Equation (3.22) implies that for any there exists such that for .

Equations (3.23) and (3.24) imply that for any there exists such that for .

#### Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. This paper is partly supported by N S Foundation of China under Grant 60850005, N S Foundation of Zhejiang Province under Grants Y7080106 and Y607128, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

#### References

- P. S. Bullen, D. S. Mitrinović, and P. M. Vasić,
*Means and Their Inequalities*, vol. 31 of*Mathematics and Its Applications*, D. Reidel Publishing, Dordrecht, The Netherlands, 1988. View at Zentralblatt MATH · View at MathSciNet - 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 · View at Scopus - M.-Y. Shi, Y.-M Chu, and Y.-P. Jiang, “Optimal inequalities among various means of two arguments,”
*Abstract and Applied Analysis*, Article ID 694394, 10 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-M Chu and W.-F. Xia, “Two sharp inequalities for power mean, geometric mean, and harmonic mean,”
*Journal of Inequalities and Applications*, Article ID 741923, 6 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B.-Y. Long and Y.-M Chu, “Optimal power mean bounds for the weighted geometric mean of classical means,”
*Journal of Inequalities and Applications*, Article ID 905679, 6 pages, 2010. 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - F. Burk, “The geometric, logarithmic, and arithmetic mean inequality,”
*The American Mathematical Monthly*, vol. 94, no. 6, pp. 527–528, 1987. View at Publisher · View at Google Scholar · View at MathSciNet - H. Alzer and W. Janous, “Solution of problem ${8}^{\ast}$,”
*Crux Mathematicorum with Mathematical Mayhem*, vol. 13, pp. 173–178, 1987. 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 · View at MathSciNet - H. Alzer, “Ungleichungen für ${(e/a)}^{a}{(b/e)}^{b}$,”
*Elemente der Mathematik*, vol. 40, pp. 120–123, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. O. Imoru, “The power mean and the logarithmic mean,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 5, no. 2, pp. 337–343, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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, 1981. View at Google Scholar - 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, 1981. 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 - 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 · View at MathSciNet - H.-J. Seiffert, “Problem 887,”
*Nieuw Archief voor Wiskunde*, vol. 11, no. 2, pp. 176–176, 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 - A. A. Jagers, “Solution of problem 887,”
*Nieuw Archief voor Wiskunde*, vol. 12, pp. 230–231, 1994. View at Google Scholar · View at MathSciNet - H.-J. Seiffert, “Ungleichungen für einen bestimmten Mittelwert,”
*Nieuw Archief voor Wiskunde. Vierde Serie*, vol. 13, no. 2, pp. 195–198, 1995. 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 - M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, “Three best inequalities for means in two variables,”
*International Mathematical Forum*, vol. 5, no. 22, pp. 1059–1066, 2010. View at Google Scholar