Abstract
In this paper, we prove that and are the best possible constants such that the double inequality
holds for all with , where ,
and are the quadratic, Neuman and second Seiffert means of a and b, respectively.
MSC:26E60.
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1 Introduction
For with , the Neuman mean [1, 2] derived from the Schwab-Borchardt mean [3, 4], the quadratic mean and the second Seiffert mean [5] are given by
and
respectively, where is the inverse hyperbolic sine function. Recently, the Neuman, quadratic and second Seiffert means have been the subject of intensive research. In particular, many remarkable inequalities for these means can be found in the literature [1–4, 6–15].
Let and be the arithmetic and contraharmonic means of a and b, respectively. Then Neuman [1] proved that the inequalities
hold for any with .
In [1, 2], Neuman found that , , , , and are the best possible constants such that the double inequalities
and
hold for any with .
He et al. [16] proved that and are the best possible constants in such that the double inequality
holds for any with .
In [17, 18], the authors proved that the double inequalities
and
hold for any with if and only if , , and .
The main purpose of this paper is to present the best possible constants α and β such that the double inequality
holds for any with . All numerical computations are carried out using MATHEMATICA software.
2 Lemmas
In order to prove our main results, we need several lemmas, which we present in this section.
Lemma 2.1 The double inequality
holds for .
Proof Let
Then we only need to show that and for .
Taking the differentiation of yields
where
where
It is well known that the inequality
holds for all .
Equation (2.9) and inequality (2.10) lead to the conclusion that
for .
Therefore, for follows easily from (2.4)-(2.8) and (2.11).
Differentiating leads to
where
It is well known that the inequality
holds for all .
Equation (2.14) and inequality (2.15) lead to the conclusion that
for .
Therefore, for follows from (2.12) and (2.13) together with (2.16). □
Lemma 2.2 The double inequality
holds for .
Proof A simple computation leads to
for . This implies
for .
From Lemma 2.1 and (2.18) we clearly see that
for . □
Lemma 2.3 The inequality
holds for .
Proof Let
Then we only need to show that for .
Differentiating (2.20) leads to
where
We claim that
for . Indeed, let
then for follows from the fact that
It follows from (2.23) and (2.24) that
for .
Therefore, for follows from (2.21) and (2.22) together with (2.25). □
Lemma 2.4 The inequality
holds for .
Proof Let
Then simple computations lead to
From (2.32) and together with , we clearly see that is strictly decreasing on and strictly increasing on . This in conjunction with (2.31) implies that there exists such that for and for . Then equation (2.29) leads to the conclusion that is strictly increasing on and strictly decreasing on .
Therefore, for follows from (2.28) and the piecewise monotonicity of . Moreover, the second inequality in (2.26) follows from
□
Lemma 2.5 The inequality
holds for .
Proof Let
Then it suffices to show for .
Differentiating yields
where
It is well known that
for .
For , it follows from (2.36) and (2.37) that
where
Differentiating yields
for .
Therefore, for follows from (2.40) and (2.41). This in conjunction with (2.35) and (2.38) implies that is strictly decreasing on . Therefore, we get for .
It follows from Lemma 2.4 that
for , where
Differentiating yields
It follows from (2.48) and (2.49) that there exists such that is strictly decreasing on and strictly increasing on . This in conjunction with (2.46) implies that there exists such that is strictly decreasing on and strictly increasing on . From (2.44) and the piecewise monotonicity of , we know that for ; this in conjunction with (2.42) implies for . □
Lemma 2.6 The function
is strictly decreasing on . Moreover, for .
Proof Differentiating yields
where
From Lemma 2.5 and (2.51) we clearly see that
for , where
Differentiating leads to
It follows from (2.58)-(2.61) that there exists such that is strictly decreasing on and strictly increasing on . This in conjunction with (2.55)-(2.57) implies that there exists such that is strictly decreasing on and strictly increasing on . Then from (2.54) we clearly see that for .
Therefore, it follows from (2.50) and (2.52) that is strictly decreasing on . Moreover, for . □
Lemma 2.7 The function
is strictly decreasing on . Moreover, for .
Proof Simple computations lead to
where
We claim that
for . Indeed, let
Then we clearly see that
Therefore, the double inequality (2.65) follows easily from (2.68)-(2.72).
Equation (2.64) and inequality (2.65) imply that
where
Let , then , and becomes
Equation (2.75) leads to
From (2.79) we clearly see that for and for . This in conjunction with (2.77) implies that is strictly decreasing on and strictly increasing on . Thus for follows from (2.78) and the piecewise monotonicity of .
Therefore, follows from (2.76). This in conjunction with (2.63) and (2.73) implies that is strictly decreasing on . Moreover, it follows from (2.62) that for . □
Lemma 2.8 The function
for .
Proof We first prove
for . Let
Then follows from and the fact that
where the second term follows from (2.65).
From Lemma 2.5 and (2.10) we clearly see that
for .
It follows from (2.80) and (2.81) that
for .
Simple computation yields
where
From (2.85)-(2.87) we know that there exists such that for and for . This in conjunction with (2.84) implies that is strictly decreasing on and strictly increasing on .
Therefore, follows from (2.83) and the piecewise monotonicity of . □
3 Main result
Theorem 3.1 The double inequality
holds for all with if and only if and .
Proof Since the Neuman mean , the quadratic mean and the second Seiffert mean are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let , then from (1.1)-(1.3) one has
Equations (3.2) and (3.3) lead to
It is easy to find that
We investigate the difference between the convex combination of , and as follows:
Let
Then simple computations lead to
where , and are defined as in Lemmas 2.6, 2.7 and 2.8, respectively.
From Lemmas 2.1-2.3 and (3.10) we clearly see that
for .
It follows from Lemmas 2.6-2.8 and (3.11) that
for . Then from and we know that there exists such that for and for . This in conjunction with (3.13) leads to the conclusion that is strictly increasing on and strictly decreasing on .
Therefore, for follows from (3.9) and the monotonicity of . In other words, we obtain
for with .
Obviously, if , then (1.4) gives
for with .
Therefore, Theorem 3.1 follows from (3.14) and (3.15) together with the following statements:
-
If , then (3.4) and (3.5) imply that there exists such that for all with .
-
If , then (3.4) and (3.6) imply that there exists such that for all with .
□
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Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 11301127 and 61374086, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
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Authors’ contributions
Y-MC provided the main idea and carried out the proof of Theorem 3.1. HW carried out the proof of Lemmas 2.1-2.4. T-HZ carried out the proof of Lemmas 2.5-2.8 and drafted the manuscript. All authors read and approved the final manuscript.
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Chu, YM., Wang, H. & Zhao, TH. Sharp bounds for the Neuman mean in terms of the quadratic and second Seiffert means. J Inequal Appl 2014, 299 (2014). https://doi.org/10.1186/1029-242X-2014-299
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DOI: https://doi.org/10.1186/1029-242X-2014-299