Open Access

The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

Journal of Inequalities and Applications20102010:436457

https://doi.org/10.1155/2010/436457

Received: 28 December 2009

Accepted: 22 April 2010

Published: 26 May 2010

Abstract

We find the greatest value and least value such that the double inequality holds for all with . Here , , and denote the arithmetic, harmonic, and Seiffert's means of two positive numbers and , respectively.

1. Introduction

For with the Seiffert's mean was introduced by Seiffert [1] as follows:
(1.1)
Recently, the inequalities for means have been the subject of intensive research [211]. In particular, many remarkable inequalities for the Seiffert's mean can be found in the literature [1217]. The Seiffert's mean can be rewritten as (see [14, ( )])
(1.2)
Let , , , , and be the arithmetic, geometric, harmonic, identric, and logarithmic means of two positive real numbers and with . Then
(1.3)
In [1], Seiffert proved that
(1.4)

for all with .

Later, Seiffert [18] established that
(1.5)

for all with .

In [19], Sándor proved that
(1.6)

for all with .

The following bounds for the Seiffert's mean in terms of the power mean were presented by Jagers in [17]:
(1.7)

for all with .

Hästö [13] found the sharp lower power bound for the Seiffert's mean as follows:
(1.8)

for all with .

The purpose of this paper is to find the greatest value and the least value such that the double inequality holds for all with .

2. Main Result

Theorem 2.1.

The double inequality holds for all with if and only if and .

Proof.

Firstly, we prove that
(2.1)
(2.2)

for all with .

Without loss of generality, we assume . Let and . Then (1.1) leads to
(2.3)
Let
(2.4)
Then simple computations lead to
(2.5)
(2.6)
(2.7)
where
(2.8)

We divide the proof into two cases.

Case 1.

If , then it follows from (2.8) that
(2.9)

for .

Therefore, inequality (2.1) follows from (2.3)–(2.5) and (2.7) together with (2.9).

Case 2.

If , then from (2.8) we have
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)

From (2.24) and (2.25) we clearly see that for , hence is strictly decreasing in . It follows from (2.22) and (2.23) together with the monotonicity of that there exists such that for and for , hence is strictly increasing in and strictly decreasing in .

From (2.19) and (2.20) together with the monotonicity of we know that there exists such that for ) and for , hence, is strictly increasing in and strictly decreasing in .

From (2.16) and (2.17) together with the monotonicity of we clearly see that there exists such that is strictly increasing in and strictly decreasing in . It follows from (2.13) and (2.14) together with the monotonicity of that there exists such that is strictly increasing in and strictly decreasing in . Then (2.7), (2.10) and (2.11) imply that there exists such that is strictly increasing in and strictly decreasing in .

Note that (2.6) becomes
(2.26)

for .

It follows from (2.5) and (2.26) together with the monotonicity of that
(2.27)

for .

Therefore, inequality (2.2) follows from (2.3) and (2.4) together with (2.27).

Secondly, we prove that is the best possible upper convex combination bound of arithmetic and harmonic means for the Seiffert's mean .

For any and , from (1.1) we have
(2.28)
where
(2.29)
It follows from (2.29) that
(2.30)
(2.31)
If , then (2.31) leads to
(2.32)
From (2.32) and the continuity of we clearly see that there exists such that
(2.33)
for . Then (2.30) and (2.33) imply that
(2.34)

for .

Therefore, for follows from (2.28) and (2.34).

Finally, we prove that is the best possible lower convex combination bound of arithmetic and harmonic means for the Seiffert's mean .

For , then from (1.1) one has
(2.35)

Inequality (2.35) implies that for any there exists such that for .

Declarations

Acknowledgments

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

Authors’ Affiliations

(1)
Department of Mathematics, Huzhou Teachers College
(2)
Department of Mathematics, Zhejiang Sci-Tech University

References

  1. Seiffert H-J: Problem 887. Nieuw Archief voor Wiskunde 1993, 11(2):176–176.MathSciNetGoogle Scholar
  2. Wang M-K, Chu Y-M, Qiu Y-F: Some comparison inequalities for generalized Muirhead and identric means. Journal of Inequalities and Applications 2010, 2010:-10.Google Scholar
  3. Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10.Google Scholar
  4. Long B-Y, Chu Y-M: Optimal power mean bounds for the weighted geometric mean of classical means. Journal of Inequalities and Applications 2010, 2010:-6.Google Scholar
  5. Xia W-F, Chu Y-M, Wang G-D: The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means. Abstract and Applied Analysis 2010, 2010:-9.Google Scholar
  6. Chu Y-M, Long B-Y: Best possible inequalities between generalized logarithmic mean and classical means. Abstract and Applied Analysis 2010, 2010:-13.Google Scholar
  7. Shi M-Y, Chu Y-M, Jiang Y-P: Optimal inequalities among various means of two arguments. Abstract and Applied Analysis 2009, 2009:-10.Google Scholar
  8. Chu Y-M, Xia W-F: Two sharp inequalities for power mean, geometric mean, and harmonic mean. Journal of Inequalities and Applications 2009, 2009:-6.Google Scholar
  9. Chu Y-M, Xia W-F: Inequalities for generalized logarithmic means. Journal of Inequalities and Applications 2009, 2009:-7.Google Scholar
  10. Wen J-J, Wang W-L: The optimization for the inequalities of power means. Journal of Inequalities and Applications 2006, 2006:-25.Google Scholar
  11. Hara T, Uchiyama M, Takahasi S-E: A refinement of various mean inequalities. Journal of Inequalities and Applications 1998, 2(4):387–395. 10.1155/S1025583498000253MathSciNetMATHGoogle Scholar
  12. Neuman E, Sándor J: On the Schwab-Borchardt mean. II. Mathematica Pannonica 2006, 17(1):49–59.MathSciNetMATHGoogle Scholar
  13. Hästö PA: Optimal inequalities between Seiffert's mean and power means. Mathematical Inequalities & Applications 2004, 7(1):47–53.MathSciNetView ArticleMATHGoogle Scholar
  14. Neuman E, Sándor J: On the Schwab-Borchardt mean. Mathematica Pannonica 2003, 14(2):253–266.MathSciNetMATHGoogle Scholar
  15. Neuman E, Sándor J: On certain means of two arguments and their extensions. International Journal of Mathematics and Mathematical Sciences 2003, (16):981–993.Google Scholar
  16. Hästö PA: A monotonicity property of ratios of symmetric homogeneous means. Journal of Inequalities in Pure and Applied Mathematics 2002, 3(5, article 71):1–23.MathSciNetMATHGoogle Scholar
  17. Jagers AA: Solution of problem 887. Nieuw Archief voor Wiskunde 1994, 12: 230–231.Google Scholar
  18. Seiffert H-J: Ungleichungen für einen bestimmten Mittelwert. Nieuw Archief voor Wiskunde 1995, 13(2):195–198.MathSciNetGoogle Scholar
  19. Sándor J: On certain inequalities for means. III. Archiv der Mathematik 2001, 76(1):34–40. 10.1007/s000130050539MathSciNetView ArticleMATHGoogle Scholar

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© Yu-Ming Chu et al. 2010

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