- Research
- Open access
- Published:
Optimal bounds for Neuman means in terms of geometric, arithmetic and quadratic means
Journal of Inequalities and Applications volume 2014, Article number: 175 (2014)
Abstract
In this paper, we present sharp bounds for the two Neuman means and derived from the Schwab-Borchardt mean in terms of convex combinations of either the weighted arithmetic and geometric means or the weighted arithmetic and quadratic means, and the mean generated either by the geometric or by the quadratic mean.
MSC:26E60.
1 Introduction
Let with , then the Schwab-Borchardt mean is defined by
where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.
It is well known that is strictly increasing in both a and b, nonsymmetric and homogeneous of degree 1 with respect to a and b. Many symmetric bivariate means are special cases of the Schwab-Borchardt mean, for example,
where , and denote the classical geometric mean, arithmetic mean and quadratic mean of a and b, respectively. The Schwab-Borchardt mean was investigated in [1, 2].
Let , be the harmonic and contraharmonic means of two positive numbers a and b, respectively. Then it is well known that
for with .
Recently, the Schwab-Borchardt mean and its special cases have been the subject of intensive research. Neuman and Sándor [3, 4] proved that the inequalities
hold for all with . In [5], the author proved that the double inequalities
and
hold for all with if and only if  , , and . Chu and Long [6] found that the double inequality
holds for all with if and only if and  , where () and is the p th power mean of a and b. Zhao et al. [7] presented the least values , , and the greatest values , , such that the double inequalities
and
hold for all with .
Very recently, the bivariate means , , and derived from the Schwab-Borchardt mean have been defined by Neuman [8, 9] as follows:
We call the means , , and given in (1.3) the Neuman means. Moreover, let , then the following explicit formulas for , , and have been found by Neuman [8]:
where p, q, r and s are defined implicitly as , , and , respectively. Clearly, , , and .
In [8], Neuman proved that the inequalities
hold for with .
He et al. [10] found the greatest values , and the least values , such that the double inequalities
and
hold for all with .
It follows from (1.2) and (1.6) together with (1.7) that
for all with .
For fixed with , let , ,
Then it is not difficult to verify that and are continuous and strictly increasing on and , respectively. Note that
Motivated by (1.8)-(1.12), in the article we present the best possible parameters , and such that the double inequalities
hold for all with .
Our main results are the following Theorems 1.1-1.4. All numerical computations are carried out using Mathematica software.
Theorem 1.1 The double inequality
holds for all with if and only if and .
Theorem 1.2 The two-sided inequality
holds true for all with if and only if and  .
Theorem 1.3 Let , then the double inequality
holds for all with if and only if and  .
Theorem 1.4 Let , then the two-sided inequality
holds true for all with if and only if and  .
2 Two lemmas
In order to prove our main results, we need two lemmas, which we present in this section.
Lemma 2.1 Let and
Then the following statements are true:
-
(1)
If , then for all and for all ;
-
(2)
If , then there exists such that for and for ;
-
(3)
If , then there exists such that for and for .
Proof For part (1), if , then (2.1) becomes
Therefore, part (1) follows easily from (2.2).
For part (2), if , then simple computations lead to
It follows from (2.3) and (2.4) together with (2.7) that is strictly increasing on . Therefore, part (2) follows from (2.5) and (2.6) together with the monotonicity of .
For part (3), if  , then numerical computations lead to
It follows from (2.7) and (2.8) that
for .
Therefore, part (3) follows easily from (2.9)-(2.11). □
Lemma 2.2 Let and
Then the following statements are true:
-
(1)
If , then for all ;
-
(2)
If , then for all ;
-
(3)
If , then there exists such that for and for ;
-
(4)
If , then there exists such that for and for .
Proof For parts (1) and (2), if or , then (2.12) becomes
Therefore, parts (1) and (2) follow from (2.13).
For part (3), if , then numerical computations show that
From (2.14), (2.15) and (2.18) we clearly see that is strictly increasing on . Therefore, part (3) follows from (2.16) and (2.17) together with the monotonicity of .
For part (4), if , then numerical computations lead to
It follows from (2.18), (2.19), (2.20) and (2.23) that
for .
Therefore, part (4) follows from (2.21) and (2.22) together with (2.24). □
3 Proofs of Theorems 1.1-1.4
Proof of Theorem 1.1 Without loss of generality, we assume that . Let , , and . Then and (1.4) leads to
where
and
where is defined as in Lemma 2.1.
We divide the proof into two cases.
Case 1: . Then from Lemma 2.1(1) and (3.5) we clearly see that is strictly decreasing on . Therefore,
for all with follows from (3.2) and (3.4) together with the monotonicity of .
Case 2: . Then from (3.3), (3.5) and Lemma 2.1(2) we know that
and there exists such that is strictly decreasing on and strictly increasing on . Therefore,
for all with follows from (3.2) and (3.4) together with (3.7) and the piecewise monotonicity of .
Note that
and
Therefore, Theorem 1.1 follows from (3.6) and (3.8) together with the following statements.
-
If , then equations (3.1) and (3.9) imply that there exists small enough such that for all with .
-
If , then equations (3.1) and (3.10) imply that there exists large enough such that for all with .
 □
Proof of Theorem 1.2 Without loss of generality, we assume that . Let , , and . Then , , and (1.5) leads to
where
where is defined as in Lemma 2.1.
We divide the proof into two cases.
Case 1:  . Then from (3.14) and (3.15) together with Lemma 2.1(3) we clearly see that there exists such that is strictly decreasing on and strictly increasing on , and
Therefore,
for all with follows easily from (3.12) and (3.13) together with (3.16) and the piecewise monotonicity of .
Case 2: . Then Lemma 2.1(1) and (3.15) lead to the conclusion that is strictly increasing on . Therefore,
for all with follows from (3.12) and (3.13) together with the monotonicity of .
Note that
and
Therefore, Theorem 1.2 follows from (3.11) and (3.17)-(3.20). □
Proof of Theorem 1.3 Without loss of generality, we assume that . Let , , and . Then and (1.4) leads to
where
and
where
where is defined as in Lemma 2.2.
We divide the proof into four cases.
Case 1: . Then Lemma 2.2(1) and (3.24) together with (3.25) lead to the conclusion that is strictly increasing on . Therefore,
for all with follows easily from (3.21) and (3.22) together with the monotonicity of .
Case 2: . Let and , then and power series expansions lead to
Equations (3.21) and (3.26) imply that there exists small enough such that for all with .
Case 3: . Then from Lemma 2.2(3) and (3.23)-(3.25) we clearly see that there exists such that is strictly increasing on and strictly decreasing on , and
Therefore,
for all with follows easily from (3.21) and (3.22) together with (3.27) and the piecewise monotonicity of .
Case 4: . Then
Equation (3.21) and inequality (3.28) imply that there exists large enough such that for all with . □
Proof of Theorem 1.4 Without loss of generality, we assume that . Let , , and . Then , , and (1.5) leads to
where
where is defined as in Lemma 2.2.
We divide the proof into four cases.
Case 1: . Then Lemma 2.2(2) and (3.32) lead to the conclusion that is strictly increasing on . Therefore,
for all with follows easily from (3.29) and (3.30) together with the monotonicity of .
Case 2: . Let and , then and power series expansions lead to
Equations (3.29) and (3.33) imply that there exists small enough such that for all with .
Case 3: . Then (3.31) and (3.32) together with Lemma 2.2(4) lead to the conclusion that there exists such that is strictly increasing on and strictly decreasing on , and
Therefore,
for all with follows easily from (3.29) and (3.30) together with (3.34) and the piecewise monotonicity of .
Case 4: . Then
Equation (3.29) and inequality (3.35) imply that there exists large enough such that for all with . □
References
Carlson BC: Algorithms involving arithmetic and geometric means. Am. Math. Mon. 1971, 78: 496–505. 10.2307/2317754
Borwein JM, Borwein PB: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York; 1987.
Neuman E, Sándor J: On the Schwab-Borchardt mean. Math. Pannon. 2003,14(2):253–266.
Neuman E, Sándor J: On the Schwab-Borchardt mean II. Math. Pannon. 2006,17(1):49–59.
Neuman E: A note on a certain bivariate mean. J. Math. Inequal. 2012,6(4):637–643.
Chu Y-M, Long B-Y: Bounds of the Neuman-Sándor mean using power and identric mean. Abstr. Appl. Anal. 2013., 2013: Article ID 832591
Zhao T-H, Chu Y-M, Liu B-Y: Optimal bounds for Neuman-Sándor mean in terms of the convex combinations of harmonic, geometric, quadratic, and contraharmonic means. Abstr. Appl. Anal. 2013., 2013: Article ID 302635
Neuman E: On some means derived from the Schwab-Borchardt mean. J. Math. Inequal. 2014,8(1):171–183.
Neuman E: On some means derived from the Schwab-Borchardt mean II. J. Math. Inequal. 2014,8(2):361–370.
He Z-Y, Chu Y-M, Wang M-K: Optimal bounds for Neuman means in terms of harmonic and contraharmonic means. J. Appl. Math. 2013., 2013: Article ID 807623
Acknowledgements
The authors would like to express their deep gratitude to the referees for giving many valuable suggestions. The research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307, the Natural Science Foundation of the Open University of China under Grant Q1601E-Y and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-13Z04.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
W-MQ provided the main idea and carried out the proof of Lemmas 2.1 and 2.2. Y-MC carried out the proof of Theorems 1.1-1.4. All authors read and approved the final manuscript.
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
Qian, WM., Chu, YM. Optimal bounds for Neuman means in terms of geometric, arithmetic and quadratic means. J Inequal Appl 2014, 175 (2014). https://doi.org/10.1186/1029-242X-2014-175
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-175