Abstract
In this paper, we prove that the double inequality holds for all with if and only if and , where and are the Yang and r th power means of a and b, respectively.
MSC:26E60.
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1 Introduction
Let and with . Then the p th power mean of a and b is given by
The main properties for the power mean are given in [1]. It is well known that is strictly increasing with respect to for fixed with . Many classical means are the special cases of the power mean, for example, is the harmonic mean, is the geometric mean, is the arithmetic mean, and is the quadratic mean.
Let , , , and be the logarithmic, first Seiffert, Neuman-Sándor, identric, and second Seiffert means of two distinct positive real numbers a and b, respectively. Then it is well known that the inequalities
hold for all with .
Recently, the bounds for certain bivariate means in terms of the power mean have been the subject of intensive research. Seiffert [2] proved that the inequalities
hold for all with .
Jagers [3] proved that the double inequality
holds for all with .
In [4, 5], Hästö established that
for all with .
Witkowski [6] proved that the double inequality
holds for all with .
In [7], Costin and Toader presented the result that
for all with .
Chu and Long [8] proved that the double inequality
holds for all with if and only if and .
The following sharp bounds for the logarithmic and identric means in terms of the power means can be found in the literature [9–16]:
for all with .
Recently, Yang [17] introduced the Yang mean of two distinct positive real numbers a and b as follows:
and he proved that the inequalities
hold for all with .
In [18], Yang et al. presented several sharp bounds for the Yang mean in terms of the geometric mean and quadratic mean .
The main purpose of this article is to find the greatest value p and the least value q such that the double inequality
holds for all with .
2 Lemmas
In order to prove our main results we need several lemmas, which we present in this section.
Lemma 2.1 Let be defined by
Then
-
(1)
is strictly decreasing with respect to x on if and only if ;
-
(2)
is strictly increasing with respect to x on if and only if .
Proof It follows from (2.1) that
where
-
(1)
If is strictly decreasing with respect to x on , then (2.2) leads to the conclusion that for all . In particular, from (2.3) we have
(2.4)
Therefore, follows from (2.4).
If , then it follows from (2.3) that
for all .
Equation (2.3) and inequality (2.5) lead to the conclusion that
for all .
Therefore, is strictly decreasing with respect to x on follows from (2.2) and (2.6).
-
(2)
If is strictly increasing with respect to x on , then (2.2) leads to the conclusion that for all . In particular, we have and we assert that . Indeed, from (2.3) we clearly see that for , , , for , and for .
If , then inequality (2.5) holds again. It follows from (2.3) and (2.5) that
for all .
Therefore, is strictly increasing with respect to x on follows from (2.2) and (2.7). □
Lemma 2.2 Let be defined by (2.1). Then
-
(1)
for all if and only if ;
-
(2)
for all if and only if .
Proof (1) If for all , then from (2.1) and the L’Hôpital rules we have
and .
If , then (2.1) and Lemma 2.1(1) lead to the conclusion that for all .
-
(2)
If for all , then . We claim that . Indeed, it follows from (2.1) that if .
If , then (2.1) and Lemma 2.1(2) lead to the conclusion that for all . □
Lemma 2.3 Let be defined by
Then for all if .
Proof It follows (2.8) that
where
From (2.11)-(2.13) and (2.16) together with (2.17) we get
for all .
Therefore, Lemma 2.3 follows easily from (2.9), (2.10), (2.14), (2.15), (2.18), and (2.19). □
Lemma 2.4 Let be defined by (2.8). Then for all if .
Proof It follows from (2.8) that
We divide the proof into two cases.
Case 1. . Then from
and (2.20) we clearly see that
for all .
Case 2. . Then (2.22) leads to
It follows from Lemma 2.3 and (2.23) that is strictly increasing with respect to x on .
Therefore, for all follows from (2.21) and the monotonicity of the with respect to x on the interval . □
Lemma 2.5 Let be defined by (2.1). Then there exists such that is strictly decreasing with respect to x on the interval and strictly increasing with respect to x on the interval if .
Proof Let and be defined by (2.3) and (2.8), respectively. Then from (2.8) we clearly see that
It follows from Lemma 2.4 and (2.25) that there exists such that is strictly increasing with respect to x on and strictly decreasing with respect to x on . This in conjunction with (2.24) leads to the conclusion that there exists such that for and for .
Note that
Therefore, Lemma 2.5 follows from (2.2) and (2.26) together with the piecewise positive and negative of on . □
Lemma 2.6 Let be defined by
Then the following statements are true:
-
(1)
is strictly increasing with respect to x on if and only if ;
-
(2)
is strictly decreasing with respect to x on if and only if ;
-
(3)
If , then there exists such that is strictly increasing with respect to x on and strictly decreasing with respect to x on .
Proof It follows from (2.27) and (2.28) that
where is defined by (2.1).
Therefore, parts (1) and (2) follow from Lemma 2.2 and (2.29).
Next, we prove part (3). If , then (2.1) leads to
From Lemma 2.5 and (2.30) we clearly see that there exists such that for and for .
Therefore, part (3) follows from (2.29) and the fact that for and for . □
3 Main results
Theorem 3.1 The double inequality
holds for all with if and only if and .
Proof Since both the Yang mean and the r th power mean are symmetric and homogeneous of degree 1, without loss of generality, we assume that and .
We first prove that the inequality holds for all if and only if .
If , then from (2.27) and Lemma 2.6(1) we get
for all .
Therefore, for all and follows from (3.1) and the monotonicity of the function .
If , then (2.27) and (2.28) lead to for all . In particular, we have
and .
Next, we prove that the inequality holds for all if and only if .
If holds for all , then (2.27) leads to for all . In particular, we have
We claim that . Indeed, follows from (3.2) if , and is obvious if .
If , then (2.27) leads to
It follows from (2.27) and (3.3) together with Lemma 2.6(3) that
for all .
Therefore, for all and follows from (3.4) and the monotonicity of the function . □
Theorem 3.2 Let with . Then the double inequality
holds with the best possible constants and .
Proof It follows from Lemma 2.6(1) and (2) together with (2.27) that
and
for all .
Therefore, for all follows from (3.5) and (3.6), and the optimality of the parameters and follows from the monotonicity of the functions and . □
Remark 3.1 For all with . Then from Lemma 2.6(1) and (2) together with (2.27) we clearly see that the Ky Fan type inequalities
hold if and only if and .
Let and be the p th Lehmer mean of two positive real numbers and a and b. Then the function defined by (2.1) can be rewritten as
From Lemma 2.2 and (3.7) we get Remark 3.2.
Remark 3.2 The double inequality
holds for all with if and only if and .
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Acknowledgements
This research was supported by the Natural Science Foundation of China under Grants 11171307 and 61374086, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
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Authors’ contributions
Z-HY provided the main idea and carried out the proof of Lemmas 2.1 and 2.2. L-MW carried out the proof of Lemmas 2.3-2.6. Y-MC carried out the proof of Theorems 3.1 and 3.2. All authors read and approved the final manuscript.
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Yang, ZH., Wu, LM. & Chu, YM. Optimal power mean bounds for Yang mean. J Inequal Appl 2014, 401 (2014). https://doi.org/10.1186/1029-242X-2014-401
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DOI: https://doi.org/10.1186/1029-242X-2014-401