Abstract
In this paper, we find the least value α and the greatest value β such that the double inequality
holds for all \(a,b>0\) with \(a\neq b\), where \(M(a,b)\), \(P(a,b)\), and \(T(a,b)\) are the Neuman-Sándor, the first and second Seiffert means of two positive numbers a and b, respectively.
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1 Introduction
For \(a,b>0\) with \(a\neq b\), the Neuman-Sándor mean \(M(a,b)\) [1], the first Seiffert mean \(P(a,b)\) [2], and the second Seiffert mean \(T(a,b)\) [3] are defined by
respectively. It can be observed that the first Seiffert mean \(P(a,b)\) can be rewritten as (see [1])
where \(\sinh^{-1}(x)=\log(x+\sqrt{x^{2}+1})\), \(\tan^{-1}(x)=\arctan (x)\), and \(\sin^{-1}(x)=\arcsin(x)\) are the inverse hyperbolic sine, inverse tangent, inverse sine functions, respectively.
Recently, the means M, P, and T and other means have been the subject of intensive research. Many remarkable inequalities for means can be found in the literature [4–10].
Let \(H(a,b)=2ab/(a+b)\), \(G(a,b)=\sqrt{ab}\), \(L(a, b)=(b-a)/(\log b-\log a)\), \(I(a,b)=1/e(b^{b}/ a^{a})^{1/(b-a)}\), \(A(a,b)=(a+b)/2\), \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\) and
denote the harmonic, geometric, logarithmic, identric, arithmetic, root-square, and the pth power means of two positive numbers a and b with \(a\neq b\), respectively. Then it is well known that the inequalities
hold for \(a,b>0\) with \(a\neq b\).
Neuman and Sándor [1] established
for all \(a,b>0\) with \(a\neq b\).
Gao [11] proved that the optimal double inequalities
hold for all \(a,b>0\) with \(a\neq b\).
The following bounds for the Seiffert means \(P(a,b)\) and \(T(a,b)\) in terms of the power mean were presented by Jagers in [12]:
for all \(a,b>0\) with \(a\neq b\). Hästö [13] improved the results of [12] and found the sharp lower power mean bound for the Seiffert mean \(P(a,b)\) as follows:
for all \(a,b>0\) with \(a\neq b\).
In [14], the authors proved that the sharp double inequality
holds for all \(a,b>0\) with \(a\neq b\).
Let \(\overline{L}_{p}(a,b)=(a^{p+1}+b^{p+1})/(a^{p}+b^{p})\) be the Lehmer mean of two positive numbers a and b with \(a\neq b\). In [15], the authors presented the following best possible Lehmer mean bounds for the Seiffert means \(P(a,b)\) and \(T(a,b)\):
for all \(a,b>0\) with \(a\neq b\).
Let u, v, and w be bivariate means such that \(u(a,b)< v(a,b)< w(a,b)\) for all \(a,b>0\) with \(a\neq b\). The problems of finding the best possible parameters α and β such that the inequalities \(\alpha u(a,b)+(1-\alpha)v(a,b)< w(a,b)<\beta u(a,b)+(1-\beta)v(a,b)\) and \(u(a,b)^{\alpha}v^{1-\alpha}(a,b)< w(a,b)< u(a,b)^{\beta}v^{1-\beta }(a,b)\) hold for all \(a,b>0\) with \(a\neq b\) have attracted the interest of many mathematicians.
In [16] and [17], the authors proved that the double inequalities
hold for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha_{1}\leq (4-\pi)/[(\sqrt{2}-1)\pi]\), \(\beta_{1}\geq2/3\), \(\alpha_{2}\leq2/3\), \(\beta_{2}\geq4-2\log\pi/\log2\).
In [1], Neuman and Sándor gave the inequality
In [8], Sándor proved the inequality
In [18] and [19], the authors proved that the double inequalities
hold for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha_{3}\leq 1/3\), \(\beta_{3}\geq2(\log(2+\sqrt{2})-\log3)/\log2\), \(\alpha _{4}\leq2/3\), \(\beta_{4}\geq1/[\sqrt{2}\log(1+\sqrt{2})]\).
In [20], the authors proved that the double inequality
holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha_{5}\leq \pi/2\), \(\beta_{5}\geq2/3\).
The main purpose of this paper is to find the least value α and the greatest value β such that the double inequality
holds for all \(a,b>0\) with \(a\neq b\).
Theorem 3.1 and Theorem 3.3 in [21] provide the inequality
following which one can get \(P^{\frac{1}{3}}(a,b)T^{\frac{2}{3}}\leq M(a,b)\). Then the lower bound of α in Theorem 3.1 of Section 3 is achieved.
2 Lemmas
To establish our main result, we need several lemmas, which we present in this section.
For \(x\in(0,1)\), the power series expansions of the functions \(\tan ^{-1}(x)\) and \(\sinh^{-1} (x)\) are presented as follows:
Lemma 2.1
If \(x\in(0,1)\), then one has
and
Proof
Square every terms of inequality (2.3) at the same time, then it is easy to prove it. Inequality (2.4) was proved in Lemma 2.3 of [22]. Inequalities (2.5) and (2.6) follow immediately from equations (2.1) and (2.2), respectively.
Let \(\Phi(x)=\tan^{-1}(x)-(x-\frac{x^{3}}{3}+\frac{x^{5}}{9})\). Then \(\Phi'(x)=\frac{x^{4}(4-5x^{2})}{9(1+x^{2})}\). Thus, \(\Phi(x)\) is strictly increasing on \((0,\frac{2\sqrt{5}}{5}]\) and strictly decreasing on \([\frac{2\sqrt{5}}{5},1)\). Considering \(\Phi(0)=0\) and \(\Phi(1)=0.0076\ldots>0\), we can get \(\Phi(x)>0\) for \(x \in(0,1)\). Therefore, inequality (2.7) holds. □
Lemma 2.2
If \(x\in(0,0.7)\), then one has
and
Proof
Let
Then
where
Differentiating \(\gamma_{1}^{*}(x)\) and \(\gamma_{3}^{*}(x)\), we have
Furthermore, direct or numerical computations lead to
and
From (2.19), we can easy to see that \(\gamma_{1}^{*}(x)\) is strictly increasing on \((0,\frac{1}{2}]\) and strictly decreasing on \([\frac {1}{2},1)\). This fact and (2.22) together with (2.14) imply that there exists \(x_{0} \in(\frac{1}{2},1)\), such that \(\gamma _{1}'(x)>0\) on \((0,x_{0})\) and \(\gamma_{1}'(x)<0\) on \((x_{0},1)\). The monotonicity of \(\gamma_{1}(x)\) and (2.21) lead to
for x \(\in(0,0.7)\). Therefore, inequality (2.8) holds.
Equation (2.15) shows that \(\gamma_{2}(x)>0\) on \((0,\frac{\sqrt {10}}{5})\) and \(\gamma_{2}(x)<0\) on \((\frac{\sqrt{10}}{5},1)\). This fact and (2.23) lead to
for x \(\in(0,0.7)\). That is to say inequality (2.9) holds.
By (2.20), we know that \(\gamma_{3}^{*}(x)\) is strictly increasing on \((0,\frac{\sqrt{6}}{6}]\) and strictly decreasing on \([\frac{\sqrt {6}}{6},1)\). This fact and (2.25) together with (2.16) imply that there must exist \(x_{1} \in(\frac{\sqrt{6}}{6},1)\), such that \(\gamma_{3}'(x)>0\) on \((0,x_{1})\) and \(\gamma_{3}'(x)<0\) on \((x_{1},1)\). It follows from the monotonicity of \(\gamma_{3}(x)\) and (2.24) that
for x \(\in(0,0.7)\). This means the inequality (2.10) holds. □
Lemma 2.3
If x \(\in(0.7,1)\), the double inequality
holds.
Proof
Let
Then
Equality (2.27) implies that \(\xi_{1}(x)\) is strictly increasing on \([\frac{\sqrt{30}}{10},1)\). Additional numerical computations lead to \(\frac{\sqrt{30}}{10}<0.7\) and \(\xi_{1}(0.7)=0.0007976\ldots>0\). Therefore, we can get \(\xi_{1}(x)>0\) for x \(\in(0.7,1)\). This implies the right hand side of the double inequality (2.26) holds.
Equality (2.28) implies \(\xi_{2}(x)\) is strictly decreasing on \((0,\frac{\sqrt{70}}{10}]\) and strictly increasing on \([\frac{\sqrt {70}}{10},1)\). Because of \(\xi_{2}(0)=0\) and \(\xi _{2}(1)=-0.0011\ldots<0\), it leads to \(\xi_{2}(x)<0\) for \(x \in(0,1)\). Specially, for \(x\in(0.7,1)\). This means the left hand side of the double inequality (2.26) holds. □
Lemma 2.4
Let
and
Then, for any \(x\in(0.7,1)\), we have
and
Proof
From Lemmas 2.6 and 2.7 of [22], for any \(x\in [0.7,1)\), we can get \(\mu_{1}'(x)\leq0.167\ldots<0.17 \) and \(\mu _{2}'(x)\leq-1.48798\ldots<-1.48\), respectively.
Differentiating \(\mu_{3}(x)\), we have
where
and
For any \(x \in[0.7,1)\),
and
follow from inequalities (2.7) and (2.26), respectively. Because \(-663+1\text{,}300x^{2}-637x^{4}<0\) and \(-4\text{,}743+4\text{,}836x^{2}-936x^{4}<0\) for x \(\in (0.7,1)\), we have
for \(x\in(0.7,1)\). Thus \(\mu_{3}'(x)<0\) for \(x\in(0.7,1)\) follows from (2.34), (2.35), and (2.36). Therefore, we obtain \(\mu_{3}(x)>\mu_{3}(1)=-0.0419\ldots>-0.05\) for \(x\in(0.7,1)\). □
Lemma 2.5
Let \(f(x)=\frac{1}{\sqrt{1+x^{2}}\sinh^{-1}(x)}-(1-\lambda_{0})\frac {1}{(1+x^{2})\tan^{-1}(x)}-\lambda_{0}\frac{1}{\sqrt {1-x^{2}}\sin^{-1}(x)}\), where \(\lambda_{0}=\frac{\log(\frac{4\log (1+\sqrt{2})}{\pi})}{\log2}=0.1663\ldots\) . Then the function \(f(x)\) is strictly decreasing on \((0.7,1)\).
Proof
It is obvious that
Differentiating \(f(x)\), we have
where \(\mu_{1}(x)\), \(\mu_{2}(x)\), and \(\mu_{3}(x)\) are defined as in Lemma 2.4. Therefore, Lemma 2.4 and equation (2.37) yield
for \(x\in(0.7,1)\). The proof is completed. □
3 Main results
Theorem 3.1
The double inequality
holds for all \(a,b > 0\) with \(a\neq b\) if and only if \(a\geq1/3\) and \(\beta\leq\frac{\log(\frac{4\log(1+\sqrt{2})}{\pi})}{\log 2}=0.1663\ldots\) .
Proof
Because \(P(a,b)\), \(M(a,b)\), and \(T(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(a>b\). Let \(p\in(0,1)\), \(\lambda_{0}=\frac{\log(\frac{4\log (1+\sqrt{2})}{\pi})}{\log2}\), and \(x=(a-b)/(a+b)\). Then \(x\in(0,1)\) and
It follows that
Differentiating \(D_{p}(x)\), we have
where
On one hand, when \(p=\frac{1}{3}\), Lemma 2.1 and equation (3.4) lead to
for \(x\in(0,1)\). According to (3.3) and (3.5), we can see that
for \(x\in(0,1)\).
On the other hand, when \(p=\lambda_{0}\), the inequalities (2.3) and (2.6) and Lemma 2.2 together with equation (3.4) lead to
for \(x\in(0,0.7)\), where
Because of \(42\lambda_{0}-33=-26.0142\ldots<0\) and \(105\lambda _{0}-90=-72.5354\ldots<0\), it follows that
and
for \(x\in(0,0.7)\). Thus, we can get
for \(x\in(0,0.7)\). Therefore, equation (3.3) and inequalities (3.7)-(3.9) imply
for \(x\in(0,0.7)\).
It follows from equation (3.3) and Lemma 2.5 that \(D_{\lambda _{0}}'(x)\) is strictly decreasing on \((0.7,1)\). Then from equation (3.10) and \(D_{\lambda_{0}}'(1^{-})=-\infty\), we know that there exists \(x_{*} \in(0.7,1)\) such that \(D_{\lambda_{0}}(x)\) is strictly increasing on \((0,x_{*}]\) and strictly decreasing on \([x_{*},1)\). This in conjunction with (3.2) means that
for x \(\in(0,1)\).
Therefore, for all \(a,b>0\) with \(a\neq b\),
follows from equations (3.1), (3.2), and (3.6) as well as
follows from equations (3.1), (3.2), and (3.11).
Finally, by easy computations, equations (1.1), (1.2), and (1.3) lead to
and
Thus, we have the following two claims.
Claim 1
If \(\alpha<\frac{1}{3}\), then from (3.14) and (3.15), there must exist \(\delta_{1}\in(0,1)\) such that \(M(a,b)< P^{\alpha}(a,b)T^{1-\alpha}(a,b)\) for all \(a,b >0\) with \((a-b)/(a+b)\in(0,\delta_{1})\).
Claim 2
If \(\beta>\lambda_{0}\), then from (3.14) and (3.16), there must exist \(\delta_{2} \in(0,1)\) such that \(M(a,b)>P^{\beta}(a,b)T^{1-\beta}(a,b)\) for all \(a,b >0\) with \((a-b)/(a+b)\in(1-\delta_{2},1)\).
Inequalities (3.12) and (3.13) in conjunction with the above two claims mean the proof is completed. □
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Acknowledgements
This research was supported by the Program of the National Science Foundation of China (Grant No. 11501002), Doctoral Scientific Research Foundation of Anhui University and Scientific Research Training for Undergraduate of Anhui University (KYXL2014002), China.
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Authors’ contributions
H-YH carried out the proof of Theorem 3.1. NW carried out the proof of Lemmas 2.1-2.3. B-YL provided the main idea and carried out the proof of Lemmas 2.4 and 2.5. All authors read and approved the final manuscript.
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Huang, HY., Wang, N. & Long, BY. Optimal bounds for Neuman-Sándor mean in terms of the geometric convex combination of two Seiffert means. J Inequal Appl 2016, 14 (2016). https://doi.org/10.1186/s13660-015-0955-2
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DOI: https://doi.org/10.1186/s13660-015-0955-2