- Research Article
- Open access
- Published:
The Optimal Convex Combination Bounds for Seiffert's Mean
Journal of Inequalities and Applications volume 2011, Article number: 686834 (2011)
Abstract
We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values ,
and the least values
,
such that the double inequalities
and
hold for all
with
. Here,
,
,
, and
denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers
and
, respectively.
1. Introduction
For with
, the Seiffert't mean
was introduced by Seiffert [1] as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ1_HTML.gif)
Recently, the inequalities for means have been the subject of intensive research. In particular, many remarkable inequalities for can be found in the literature [2–6]. Seiffert's mean
can be rewritten as (see [5, equation (2.4)])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ2_HTML.gif)
Let , and
be the contraharmonic, arithmetic, geometric and harmonic means of two positive real numbers
and
with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ3_HTML.gif)
In [7], Seiffert proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ4_HTML.gif)
for all with
.
In [8], the authors found the greatest value and the least value
such that the double inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ5_HTML.gif)
holds for all with
.
For more results, see [9–23].
The purpose of the present paper is to find the greatest values and the least values
such that the double inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ6_HTML.gif)
hold for all with
.
2. Main Results
Firstly, we present the optimal convex combination bounds of contraharmonic and geometric means for Seiffert's mean as follows.
Theorem 2.1.
The double inequality holds for all
with
if and only if
and
.
Proof.
Firstly, we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ7_HTML.gif)
for all with
.
Without loss of generality, we assume that . Let
and
. Then (1.1) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ8_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ9_HTML.gif)
Simple computations lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ10_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ11_HTML.gif)
We divide the proof into two cases.
Case 1 ().
In this case,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ12_HTML.gif)
Therefore, the second inequality in (2.1) follows from (2.2)–(2.6). Notice that in this case, the second equality in (2.4) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ13_HTML.gif)
Case 2 ().
From (2.5), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ14_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ17_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ21_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ24_HTML.gif)
From (2.17) and (2.18), we clearly see that for
; hence
is strictly increasing in
, which together with (2.16) implies that there exists
such that
for
and
for
; and hence
is strictly decreasing in
and strictly increasing for
. From (2.14) and the monotonicity of
, there exists
such that
for
and
for
; hence
is strictly decreasing in
and strictly increasing for
. As this goes on, there exists
such that
is strictly decreasing in
and strictly increasing in
. Note that if
, then the second equality in (2.4) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ25_HTML.gif)
Thus for all
. Therefore, the first inequality in (2.1) follows from (2.2) and (2.3).
Secondly, we prove that is the best possible lower convex combination bound of the contraharmonic and geometric means for Seiffert's mean.
If , then (2.5) (with
in place of
) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ26_HTML.gif)
From this result and the continuity of we clearly see that there exists
such that
for
. Then the last equality in (2.4) implies that
for
. Thus
is decreasing for
. Due to (2.4),
for
, which is equivalent to, by (2.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ27_HTML.gif)
for .
Finally, we prove that is the best possible upper convex combination bound of the contraharmonic and geometric means for Seiffert's mean.
If , then from (1.1) one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ28_HTML.gif)
Inequality (2.22) implies that for any there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ29_HTML.gif)
for .
Secondly, we present the optimal convex combination bounds of the contraharmonic and harmonic means for Seiffert's mean as follows.
Theorem 2.2.
The double inequality holds for all
with
if and only if
and
.
Proof.
Firstly, we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ30_HTML.gif)
for all with
.
Without loss of generality, we assume that . Let
and
. Then (1.1) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ31_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ32_HTML.gif)
Simple computations lead to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ33_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ34_HTML.gif)
We divide the proof into two cases.
Case 1 ().
In this case,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ35_HTML.gif)
Therefore, the first inequality in (2.24) follows from (2.25)–(2.29). Notice that in this case, the second equality in (2.27) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ36_HTML.gif)
Case 2 ().
From (2.28) we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ38_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ41_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ43_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ44_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ45_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ46_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ47_HTML.gif)
From (2.40) and (2.41) we clearly see that for
; hence
is strictly decreasing in
, which together with (2.39) implies that there exists
such that
for
and
for
, and hence
is strictly increasing in
and strictly decreasing for
. From (2.37) and the monotonicity of
, there exists
such that
for
and
for
; hence
is strictly increasing in
and strictly decreasing for
. As this goes on, there exists
such that
is strictly increasing in
and strictly decreasing in
. Notice that if
, then the second equality in (2.27) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ48_HTML.gif)
Thus for all
. Therefore, the second inequality in (2.24) follows from (2.25) and (2.26).
Secondly, we prove that is the best possible upper convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.
If , then (2.28) (with
in place of
) leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ49_HTML.gif)
From this result and the continuity of we clearly see that there exists
such that
for
. Then the last equality in (2.27) implies that
for
. Thus
is increasing for
. Due to (2.27),
for
, which is equivalent to, by (2.25),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ50_HTML.gif)
for .
Finally, we prove that is the best possible lower convex combination bound of the contraharmonic and harmonic means for Seiffert's mean.
If , then from (1.1) one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ51_HTML.gif)
Inequality (2.45) implies that for any there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F686834/MediaObjects/13660_2010_Article_2355_Equ52_HTML.gif)
for .
References
Seiffert H-J: Problem 887. Nieuw Archief voor Wiskunde 1993,11(2):176.
Seiffert H-J: Aufgabe 16. Die Wurzel 1995, 29: 221–222.
Hästö PA: Optimal inequalities between Seiffert's mean and power means. Mathematical Inequalities & Applications 2004,7(1):47–53.
Neuman E, Sándor J: On certain means of two arguments and their extensions. International Journal of Mathematics and Mathematical Sciences 2003, (16):981–993.
Neuman E, Sándor J: On the Schwab-Borchardt mean. Mathematica Pannonica 2003,14(2):253–266.
Hästö PA: A monotonicity property of ratios of symmetric homogeneous means. Journal of Inequalities in Pure and Applied Mathematics 2002,3(5, article 71):1–54.
Seiffert H-J: Ungleichungen für einen bestimmten mittelwert. Nieuw Archief voor Wiskunde 1995,13(2):195–198.
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, -7.
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.
Wang M-K, Qiu Y-F, Chu Y-M: Sharp bounds for Seiffert means in terms of Lehmer means. Journal of Mathematical Inequalities 2010,4(4):581–586.
Wang S, Chu Y: The best bounds of the combination of arithmetic and harmonic means for the Seiffert's mean. International Journal of Mathematical Analysis 2010,4(22):1079–1084.
Zong C, Chu Y: An inequality among identric, geometric and Seiffert's means. International Mathematical Forum 2010,5(26):1297–1302.
Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10.
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.
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.
Chu Y-M, Long B-Y: Best possible inequalities between generalized logarithmic mean and classical means. Abstract and Applied Analysis 2010, 2010:-13.
Shi M-Y, Chu Y-M, Jiang Y-P: Optimal inequalities among various means of two arguments. Abstract and Applied Analysis 2009, 2009:-10.
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.
Chu Y-M, Xia W-F: Inequalities for generalized logarithmic means. Journal of Inequalities and Applications 2009, 2009:-7.
Wen J, Wang W-L: The optimization for the inequalities of power means. Journal of Inequalities and Applications 2006, 2006:-25.
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/S1025583498000253
Neuman E, Sándor J: On the Schwab-Borchardt mean. Mathematica Pannonica 2006,17(1):49–59.
Jagers AA: Solution of problem 887. Nieuw Archief voor Wiskunde 1994, 12: 230–231.
Acknowledgments
The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. This research is partly supported by N S Foundation of Hebei Province (Grant A2011201011), and the Youth Foundation of Hebei University (Grant 2010Q24).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Liu, H., Meng, XJ. The Optimal Convex Combination Bounds for Seiffert's Mean. J Inequal Appl 2011, 686834 (2011). https://doi.org/10.1155/2011/686834
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/686834