Abstract
In this paper, we present the sharp bounds of the ratios \(U(a,b)/L_{4}(a,b)\), \(P_{2}(a,b)/U(a,b)\), \(NS(a,b)/P_{2}(a,b)\) and \(B(a,b)/NS(a,b)\) for all \(a, b>0\) with \(a\neq b\), where \(L_{4}(a,b)=[(b^{4}-a^{4})/(4(\log b-\log a))]^{1/4}\), \(U(a,b)=(b-a)/[\sqrt{2}\arctan((b-a)/\sqrt{2ab})]\), \(P_{2}(a,b)=[(b^{2}-a^{2})/(2\arcsin ((b^{2}-a^{2})/(b^{2}+a^{2})))]^{1/2}\), \(NS(a,b)=(b-a)/[2\sinh ^{-1}((b-a)/(b+a))]\), \(B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}\), \(A(a,b)=(a+b)/2\), \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\), and \(T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]\).
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1 Introduction
For \(r\in\mathbb{R}\), the rth power mean \(M(a,b; r)\) of two distinct positive real numbers a and b is defined by
It is well known that \(M(a,b; r)\) is continuous and strictly increasing with respect to \(r\in\mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\). Many classical means are the special cases of the power mean, for example, \(M(a,b; -1)=2ab/(a+b)=H(a,b)\) is the harmonic mean, \(M(a,b; 0)=\sqrt{ab}=G(a,b)\) is the geometric mean, \(M(a,b; 1)=(a+b)/2=A(a,b)\) is the arithmetic mean, and \(M(a,b; 2)=\sqrt {(a^{2}+b^{2})/2}=Q(a,b)\) is the quadratic mean. The main properties for the power mean are given in [1].
Let
be, respectively, the logarithmic mean, first Seiffert mean [2], Yang mean [3], Neuman-Sándor mean [4, 5], second Seiffert mean [6], Sándor-Yang mean [3, 7] of two distinct positive real numbers a and b.
Recently, the sharp bounds for certain bivariate means in terms of the power mean have attracted the attention of many mathematicians.
Radó [8] and Lin [9], Jagers [10] and Hästö [11, 12] proved that the double inequalities
hold for all \(a, b>0\) and \(a\neq b\) with the best possible parameters 0, \(1/3\), \(\log2/\log\pi\), and \(2/3\).
In [13–17], the authors proved that the double inequalities
hold for all \(a, b>0\) and \(a\neq b\) if and only if \(\alpha\leq\log 2/\log[2\log(1+\sqrt{2})]\), \(\beta\geq4/3\), \(\lambda\leq2\log 2/(2\log\pi-\log2)\) and \(\mu\geq4/3\).
Very recently, Yang and Chu [18] presented that \(p=4\log2/(4+2\log 2-\pi)\) and \(q=4/3\) are the best possible parameters such that the double inequality
holds for all \(a, b>0\) and \(a\neq b\).
Let
and
be, respectively, the fourth-order logarithmic and second-order first Seiffert means of a and b.
Then from (1.4)-(1.10) we clearly see that \(M(a, b; 4/3)\) is the common sharp upper power mean bound for \(L_{4}(a,b)\), \(U(a,b)\), \(P_{2}(a,b)\), \(NS(a,b)\), and \(B(a,b)\). Therefore, it is natural to ask what are the size relationships among these means? The main purpose of this paper is to answer this question.
2 Lemmas
In order to prove our main results we need several lemmas, which we present in this section.
Lemma 2.1
(See Lemma 7 of [19])
Let \(\{a_{k}\}_{k=0}^{\infty}\) be a nonnegative real sequence with \(a_{m}>0\) and \(\sum_{k=m+1}^{\infty }a_{k}>0\), and
be a convergent power series on the interval \((0, \infty)\). Then there exists \(t_{m+1}\in(0, \infty)\) such that \(P(t_{m+1})=0\), \(P(t)<0\) for \(t\in(0, t_{m+1})\) and \(P(t)>0\) for \(t\in(t_{m+1}, \infty)\).
Lemma 2.2
Let \(n\in\mathbb{N}\). Then
for all \(n\geq6\).
Proof
Let
Then we clearly see that
for \(n\geq6\).
It follows from (2.2) and (2.4) that
for \(n\geq6\).
Therefore, Lemma 2.2 follows easily from (2.1), (2.3), and (2.5). □
Lemma 2.3
Let \(t>0\) and
Then there exists a unique \(t_{0}\in(0, \infty)\) such that \(g_{1}(t)<0\) for \(t\in(0, t_{0})\), \(g_{1}(t_{0})=0\), and \(g_{1}(t)>0\) for \(t\in(t_{0}, \infty)\).
Proof
It follows from (2.6) that
where
Making use of power series formulas, (2.9) gives
where \(v_{n}\) is defined by (2.1).
Note that
From Lemma 2.1, (2.8), (2.10), and (2.11) we know that there exists \(t_{1}\in(0, \infty)\) such that \(g_{1}(t)\) is strictly decreasing on \((0, t_{1}]\) and strictly increasing on \([t_{1}, \infty)\).
Therefore, Lemma 2.3 follows easily from (2.7) and the piecewise monotonicity of \(g_{1}(t)\). □
Lemma 2.4
The inequality
holds for all \(x\in(0, \pi/2)\).
Proof
Simple computations lead to
Let
Then
It follows from (2.13), (2.14), (2.16), and (2.17) that
Inequalities (2.18)-(2.20) imply that the sequence \(\{\omega_{n}\}\) is strictly decreasing for \(n\geq3\), \(\lim_{n\rightarrow\infty}\omega _{n}=0\) and \(\sum_{n=2}^{\infty}(-1)^{n-1}\omega_{n}\) is a Leibniz series. Therefore, Lemma 2.4 follows from (2.12), (2.13), and (2.15). □
Lemma 2.5
The inequality
hold for all \(t\in(0, \infty)\).
Proof
Let \(x=\arcsin(\tanh(2t))\in(0, \pi/2)\) and
Then
Therefore, Lemma 2.5 follows easily from (2.21)-(2.23) and Lemma 2.4. □
3 Main results
Theorem 3.1
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda_{1}\leq c_{0}\) and \(\mu_{1}=\infty\), where
and \(t_{0}\in(0, \infty)\) is defined by Lemma 2.3. Moreover, numerical computations show that \(t_{0}=1.1336\ldots\) and \(c_{0}=0.9991\ldots\) .
Proof
Since \(U(a,b)\) and \(L_{4}(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(b>a>0\). Let \(t=\log\sqrt{b/a}>0\), then (1.2) and (1.9) lead to
Let
Then
where \(g_{1}(t)\) is defined by (2.6).
It follows from Lemma 2.3 and (3.5) that there exists a unique \(t_{0}\in(0, \infty)\) such that \(g_{1}(t_{0})=0\), \(g(t)\) is strictly decreasing on \((0, t_{0}]\) and strictly increasing on \([t_{0}, \infty)\).
Therefore, Theorem 3.1 follows from (3.2)-(3.4) and the piecewise monotonicity of \(g(t)\). □
Theorem 3.2
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda_{2}\leq 1\) and \(\mu_{2}\geq\sqrt{\pi/2}=1.2533\ldots\) .
Proof
Since \(U(a,b)\) and \(P_{2}(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(b>a>0\). Let \(t=\log\sqrt{b/a}>0\), then (1.10) and (3.1) lead to
Let
Then simple computations lead to
It follows from Lemma 2.5 and (3.10) that \(h(t)\) is strictly increasing on \((0, \infty)\). Therefore, Theorem 3.2 follows easily from (3.7)-(3.9) and the monotonicity of \(h(t)\). □
Remark 3.1
Let \(b>a>0\) and \(t=\log\sqrt{b/a}>0\). Then
It follows from Lemma 2.5 that
Equations (3.1), (3.6), and (3.11) together with inequality (3.12) lead to the conclusion that the inequality
holds for all \(a, b>0\) with \(a\neq b\).
Theorem 3.3
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda_{3}\leq 1\) and \(\mu_{3}\geq\sqrt{\pi}/[2\log(1+\log2)]=1.0055\ldots\) .
Proof
Since \(NS(a,b)\) and \(P_{2}(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(b>a>0\). Let \(t=\log\sqrt{b/a}>0\), then (1.2) and (3.6) lead to
Let
Then simple computations lead to
where
Let \(x=\arcsin(\tanh(2t))\in(0, \pi/2)\). Then
From (3.17)-(3.22) we clearly see that \(h_{2}(t)\) is strictly increasing on \((0, \infty)\). Therefore, Theorem 3.3 follows from (3.14)-(3.16) and the monotonicity of \(h_{2}(t)\). □
Remark 3.2
From the proof of Theorem 3.2 we know that
which is equivalent to
Equations (3.6), (3.11), and (3.13) together with inequality (3.23) lead to the conclusion that the inequality
holds for all \(a, b>0\) with \(a\neq b\).
Theorem 3.4
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda_{4}\leq 1\) and \(\mu_{4}\geq\sqrt{2}e^{\pi/4-1}\log(1+\sqrt {2})=1.0057\ldots\) .
Proof
Since \(NS(a,b)\) and \(B(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(b>a>0\). Let \(t=\log\sqrt{b/a}>0\), then (1.3) and (3.13) lead to
Let
Then simple computations lead to
where
Let \(x=\tanh(t)\in(0, 1)\). Then
for \(x\in(0, 1)\).
It follows from (3.27)-(3.34) that \(f(t)\) is strictly increasing on \((0, \infty)\). Therefore, Theorem 3.4 follows easily from (3.24)-(3.26) and the monotonicity of \(f(t)\). □
Remark 3.3
From the proof of Theorem 3.4 we know that the inequalities
hold for all \(x\in(0, \infty)\). Inequalities (3.35)-(3.37) lead to the conclusion that the inequalities
hold for all \(a, b>0\) with \(a\neq b\).
Remark 3.4
Let \(I(a,b)=(b^{b}/a^{a})^{1/(b-a)}/e\) be the identric mean of two distinct positive real numbers a and b, and \(I_{2}(a,b)=I^{1/2} (a^{2}, b^{2} )\) be the second-order identric mean. Then from Theorems 3.1-3.4 and the inequalities \(M(a,b;2/3)< I(a,b)< M(a, b; \log2)\) [20, 21] and \(P_{2}(a,b)>L_{4}(a,b)\) [22] we get two inequalities chains as follows:
and
for all \(a, b>0\) with \(a\neq b\).
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Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
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Yang, ZH., Chu, YM. Inequalities for certain means in two arguments. J Inequal Appl 2015, 299 (2015). https://doi.org/10.1186/s13660-015-0828-8
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DOI: https://doi.org/10.1186/s13660-015-0828-8