Abstract
We present the best possible parameters such that the double inequalities , , , hold for all with , where , , , are the Neuman means, and , , , are the one-parameter means.
MSC:26E60.
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1 Introduction
Let with . Then the Schwab-Borchardt mean [1–3], and the Neuman means , , , and [4, 5] of a and b are given by
respectively. Here, and are, respectively, the inverse cosine and inverse hyperbolic cosine functions, and , , and are, respectively, the classical harmonic, arithmetic, and contraharmonic means of a and b.
Let , and , , , and be the parameters such that and . Then the following explicit formulas were found by Neuman [4]:
Let and N be a bivariate symmetric mean. Then the one-parameter bivariate mean was defined by Neuman [6] as follows:
Recently, the Neuman means , , , and , and the one-parameter bivariate mean have been the subject of intensive research. He et al. [7] found the greatest values , and , and the least values , and such that the double inequalities
hold for all with .
In [4, 5], Neuman proved that the inequalities
hold for all with , where , , , and are, respectively, the logarithmic, first Seiffert, quadratic, and second Seiffert means of a and b.
Qian and Chu [8] proved that the double inequalities
hold for all with if and only if , , , and , where is the geometric mean of a and b.
In [9], the authors proved that the double inequalities
hold for all with if and only if , , , , , , , and .
Let (). Then Neuman [6, 10] proved that the inequalities
hold for all with if and only if , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , where is the Neuman-Sándor mean of a and b.
The main purpose of this paper is to present the best possible parameters , , , , , , , on the interval such that the double inequalities
hold for all with .
2 Main results
Theorem 2.1 Let . Then the double inequality
holds for all with if and only if and .
Proof Without loss of generality, we assume that . Let , , and . Then , and (1.1) and (1.3) lead to
where
where
We divide the discussion into two cases.
Case 1 . Then (2.6) becomes
From (2.5) and (2.8) we clearly see that is strictly decreasing on , then (2.4) leads to the conclusion that
for all .
Therefore,
for all with follows from (2.2) and (2.9).
Case 2 . Then numerical computations lead to
It follows from (2.7) and (2.11) together with (2.12) that is strictly decreasing on . Then inequalities (2.13) and (2.14) together with (2.5) lead to the conclusion that there exists such that is strictly decreasing on and strictly increasing on .
Note that inequality (2.4) becomes
From (2.2), (2.15), and the piecewise monotonicity of we clearly see that the inequality
holds for all with .
Note that
Therefore, Theorem 2.1 follows from (2.10) and (2.16)-(2.18) together with the fact that inequality (2.1) is equivalent to the inequality (2.19) as follows:
□
Theorem 2.2 Let . Then the double inequality
holds for all with if and only if and .
Proof Without loss of generality, we assume that . Let , , , and . Then , , , and (1.2) and (1.3) lead to
where
where is defined by (2.6).
We divide the discussion into two cases.
Case 1 . Then it follows from (2.6) that
for all .
Therefore,
for all with follows easily from (2.21)-(2.24).
Case 2 . Then numerical computations lead to
It follows from (2.7) and (2.26)-(2.28) that
for .
Equation (2.23) and inequalities (2.29)-(2.31) lead to the conclusion that there exists such that is strictly decreasing on and strictly increasing on .
Note that (2.22) becomes
Therefore,
for all with follows from (2.21) and (2.32) together with the piecewise monotonicity of .
Note that
Therefore, Theorem 2.2 follows from (2.25) and (2.33)-(2.35) together with the fact that inequality (2.20) is equivalent to the inequality (2.36) as follows:
□
Theorem 2.3 Let . Then the double inequality
holds for all with if and only if and .
Proof Without loss of generality, we assume that . Let , , and . Then , and (1.1) and (1.3) lead to
where
where
We divide the discussion into two cases.
Case 1 . Then (2.41) leads to
for .
Therefore,
for all with follows easily from (2.38)-(2.40) and (2.42).
Case 2 . Then it follows from (1.3) and (1.4) that
for all with .
Note that
Therefore, Theorem 2.3 follows from (2.43)-(2.46) and the fact that inequality (2.37) is equivalent to
□
Theorem 2.4 Let . Then the double inequality
holds for all with if and only if and .
Proof Without loss of generality, we assume that . Let , , , and . Then , , , and (1.2) and (1.3) lead to
where
where is defined by (2.41).
We divide the discussion into two cases.
Case 1 . Then (2.41) leads to
for .
Therefore,
for all with follows easily from (2.48)-(2.51).
Case 2 . Then numerical computations lead to
From (2.41) and (2.50) together with (2.53)-(2.55) we clearly see that there exists such that is strictly increasing on and strictly decreasing on .
Note that (2.49) becomes
It follows from (2.56) and the piecewise monotonicity of that
for all .
Therefore,
for all with follows from (2.48) and (2.58).
Note that
Therefore, Theorem 2.4 follows from (2.52) and (2.58)-(2.60) together with the fact that inequality (2.47) is equivalent to
□
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Acknowledgements
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.
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Authors’ contributions
Z-HS provided the main idea and carried out the proof of Theorem 2.1. W-MQ carried out the proof of Theorem 2.2. Y-MC carried out the proof of Theorems 2.3 and 2.4. All authors read and approved the final manuscript.
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Shao, ZH., Qian, WM. & Chu, YM. Sharp bounds for Neuman means in terms of one-parameter family of bivariate means. J Inequal Appl 2014, 468 (2014). https://doi.org/10.1186/1029-242X-2014-468
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DOI: https://doi.org/10.1186/1029-242X-2014-468