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.
Similar content being viewed by others
1. Introduction
For with , the Seiffert't mean was introduced by Seiffert [1] as follows:
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)])
Let , and be the contraharmonic, arithmetic, geometric and harmonic means of two positive real numbers and with . Then
In [7], Seiffert proved that
for all with .
In [8], the authors found the greatest value and the least value such that the double inequality
holds for all with .
The purpose of the present paper is to find the greatest values and the least values such that the double inequalities
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
for all with .
Without loss of generality, we assume that . Let and . Then (1.1) leads to
where
Simple computations lead to
where
We divide the proof into two cases.
Case 1 ().
In this case,
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
Case 2 ().
From (2.5), we have that
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
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
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),
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
Inequality (2.22) implies that for any there exists such that
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
for all with .
Without loss of generality, we assume that . Let and . Then (1.1) leads to
where
Simple computations lead to
where
We divide the proof into two cases.
Case 1 ().
In this case,
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
Case 2 ().
From (2.28) we have that
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
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
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),
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
Inequality (2.45) implies that for any there exists such that
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