Open Access

An Optimal Double Inequality between Power-Type Heron and Seiffert Means

Journal of Inequalities and Applications20102010:146945

https://doi.org/10.1155/2010/146945

Received: 29 August 2010

Accepted: 31 October 2010

Published: 16 November 2010

Abstract

For , the power-type Heron mean and the Seiffert mean of two positive real numbers and are defined by , ; , and , ; , , respectively. In this paper, we find the greatest value and the least value such that the double inequality holds for all with .

1. Introduction

For , the power-type Heron mean and the Seiffert mean of two positive real numbers and are defined by
(1.1)
(1.2)

respectively.

Recently, the means of two variables have been the subject of intensive research [115]. In particular, many remarkable inequalities for and can be found in the literature [1620].

It is well known that is continuous and strictly increasing with respect to for fixed with . Let ,   , ,   , and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers and with , respectively. Then
(1.3)
For , the power mean of order of two positive numbers and is defined by
(1.4)

The main properties for power mean are given in [21].

In [16], Jia and Cao presented the inequalities
(1.5)

for all with , , and .

Sándor [22] proved that
(1.6)

for all with .

In [19], Seiffert established that
(1.7)

for all with .

The purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean . Our main result is the following Theorem 1.1.

Theorem 1.1.

For all with , one has
(1.8)

and and are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean , respectively.

2. Lemmas

In order to prove our main result, Theorem 1.1, we need two lemmas which we present in this section.

Lemma 2.1.

If and , then
(2.1)

Proof.

For , we clearly see that
(2.2)
Let
(2.3)
Then
(2.4)

and is strictly decreasing in because of for .

Therefore, Lemma 2.1 follows from (2.2)–(2.4) together with the monotonicity of .

Lemma 2.2.

If , , and , then there exists such that for and for .

Proof.

Let , , , , , , and . Then elaborated computations lead to
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(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.)
(2.22)
(2.23)
From the expression of and Lemma 2.1, we get
(2.24)

From (2.24), we know that is strictly decreasing in . Then (2.22) implies that is strictly decreasing in .

From (2.20) and (2.21) together with the monotonicity of , we clearly see that there exists such that is strictly increasing in and strictly decreasing in .

Inequality (2.17) and (2.18) together with the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in .

The piecewise monotonicity of together with (2.14) and (2.15) leads to the fact that there exists such that is strictly increasing in and strictly decreasing in .

From (2.11) and (2.12) together with the piecewise monotonicity of , we conclude that there exists such that is strictly increasing in and strictly decreasing in .

Equations (2.8) and (2.9) together with the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in .

Therefore, Lemma 2.2 follows from (2.5) and (2.6) together with the piecewise monotonicity of .

3. Proof of Theorem 1.1

Proof of Theorem 1.1.

Without loss of generality, we assume that . We first prove that . Let , then from (1.1) and (1.2) we have
(3.1)
Let
(3.2)
Then simple computations lead to
(3.3)
where . Note that
(3.4)
where
(3.5)

for .

Therefore, follows from (3.1)–(3.5).

Next, we prove that . Let and , then (1.1) and (1.2) lead to
(3.6)
Let
(3.7)
Then simple computations lead to
(3.8)
(3.9)
where . Note that
(3.10)
(3.11)
(3.12)
where
(3.13)

From (3.12) and (3.13) together with Lemma 2.2, we clearly see that there exists such that is strictly increasing in and strictly decreasing in .

Equations (3.9)–(3.11) and the piecewise monotonicity of imply that there exists such that is strictly increasing in and strictly decreasing in . Then from (3.8) we get
(3.14)

for .

Therefore, follows from (3.6) and (3.7) together with (3.14).

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

For any and , from (1.1) and (1.2), one has
(3.15)
(3.16)

where .

Let , making use of Taylor extension, we get
(3.17)

Equations (3.15) and (3.17) together with inequality (3.16) imply that for any , there exist and such that for and for .

Declarations

Acknowledgments

This work was supported by the Natural Science Foundation of China under Grant no. 11071069, the Natural Science Foundation of Zhejiang Province under Grant no. Y7080106, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no. T200924.

Authors’ Affiliations

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

References

  1. 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
  2. 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
  3. 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
  4. 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
  5. Chu Y-M, Qiu Y-F, Wang M-K, Wang G-D: The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert's mean. Journal of Inequalities and Applications 2010, 2010:-7.Google Scholar
  6. Long B, Xia W, Chu Y: An optimal inequality for power mean, geometric mean and harmonic mean. International Journal of Modern Mathematics 2010, 5(2):149–155.MathSciNetMATHGoogle Scholar
  7. Chu Y-M, Xia W-f: Two optimal double inequalities between power mean and logarithmic mean. Computers & Mathematics with Applications. 2010, 60(1):83–89. 10.1016/j.camwa.2010.04.032MathSciNetView ArticleMATHGoogle Scholar
  8. Zhang X-M, Xi B-Y, Chu Y-M: A new method to prove and find analytic inequalities. Abstract and Applied Analysis 2010, 2010:-19.Google Scholar
  9. Zhang X-M, Chu Y-M: A new method to study analytic inequalities. Journal of Inequalities and Applications 2010, 2010:-13.Google Scholar
  10. Chu Y-M, Xia W-f: Inequalities for generalized logarithmic means. Journal of Inequalities and Applications 2009, 2009:-7.Google Scholar
  11. 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
  12. Chu Y-M, Xia W-f: Two sharp inequalities for power mean, geometric mean, and harmonic mean. Journal of Inequalities and Applications 2009, -6.Google Scholar
  13. Chu Y, Xia W: Solution of an open problem for Schur convexity or concavity of the Gini mean values. Science in China A 2009, 52(10):2099–2106. 10.1007/s11425-009-0116-5MathSciNetView ArticleMATHGoogle Scholar
  14. Chu Y, Zhang X: Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave. Journal of Mathematics of Kyoto University 2008, 48(1):229–238.MathSciNetMATHGoogle Scholar
  15. Chu Y, Zhang X, Wang G: The Schur geometrical convexity of the extended mean values. Journal of Convex Analysis 2008, 15(4):707–718.MathSciNetMATHGoogle Scholar
  16. Jia G, Cao J: A new upper bound of the logarithmic mean. Journal of Inequalities in Pure and Applied Mathematics 2003., 4(4, article 80):Google Scholar
  17. Zhang Z-H, Lokesha V, Wu Y-d: The new bounds of the logarithmic mean. Advanced Studies in Contemporary Mathematics 2005, 11(2):185–191.MathSciNetMATHGoogle Scholar
  18. Zhang Z-H, Wu Y-d: The generalized Heron mean and its dual form. Applied Mathematics E-Notes 2005, 5: 16–23.MathSciNetMATHGoogle Scholar
  19. Seiffert J: Aufgabe 16. Die Wurzel 1995, 29: 221–222.Google Scholar
  20. Hästö PA: A monotonicity property of ratios of symmetric homogeneous means. Journal of Inequalities in Pure and Applied Mathematics 2002., 3(5, article 71):Google Scholar
  21. Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities. D. Reidel, , he Netherlands; 1988:xx+459.View ArticleMATHGoogle Scholar
  22. Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471–473. 10.1007/BF01200091MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Yu-Ming Chu et al. 2010

This article is published under license to BioMed Central Ltd. 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.