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Several sharp inequalities about the first Seiffert mean
Journal of Inequalities and Applications volume 2018, Article number: 174 (2018)
Abstract
In this paper, we deal with the problem of finding the best possible bounds for the first Seiffert mean in terms of the geometric combination of logarithmic and the Neuman–Sándor means, and in terms of the geometric combination of logarithmic and the second Seiffert means.
1 Introduction
A mean is a function \(f: \mathbb{R}_{+}^{2}\rightarrow \mathbb{R}_{+}\) which satisfies
Each mean is reflexive, namely
That is also used as the definition of \(f(a,a)\).
A mean is symmetric if
it is homogeneous (of degree 1) if
We shall refer here to some symmetric and homogeneous means as follows.
For \(a,b > 0\) with \(a \ne b\), the Neuman–Sándor mean \(M(a,b)\) [16], the first Seiffert mean \(P(a,b)\) [18], the second Seiffert mean \(T(a,b)\) [19] and the logarithmic mean \(L(a,b)\) are defined by
and
respectively.
Let \(M_{p}(a,b) = {(({a^{p}} + {b^{p}})/2)^{1/p}}\) (\(p\neq 0\)) stand for the pth power means. The mean \(M_{1}=A\) is the arithmetic mean, and the mean \(M_{2}=Q\) is the root-square mean. The geometric mean is given by \(G(a,b) = \sqrt{ab}\), but verifying also the property \(\lim_{p\rightarrow 0}M_{p}(a,b)=M_{0}(a,b)=G(a,b)\).
As Carlson remarked in [2], the logarithmic mean can be rewritten as
thus the means M, P, T and L are very similar. In [16] it is also proven that these means can be defined using the non-symmetric Schwab–Borchardt mean \(\mathcal{SB}\) given by
see [1]. It has been established in [16] that
For two means \(\mathcal{M}\) and \(\mathcal{N}\) we write \(\mathcal{M < N}\) if \(\mathcal{M}(a,b) < \mathcal{N}(a, b)\) for \(\forall a, b>0\), \(a\neq b\). It is well known that the inequalities
Recently, the inequalities for means have been the subject of intensive research. Many remarkable inequalities can be found in the literature [5, 7, 9, 14, 15, 17, 20].
In [6], Costin and Toader presented
holding for all \(a,b > 0\) with \(a \ne b\).
The following sharp power mean bounds for the first Seiffert mean \(P(a,b)\) are given by Jagers in [13]:
for all \(a,b > 0\) with \(a \ne b\). Hästö [11] improved the results of [13] and the sharp result was found that
In [8, 12], the authors proved that the double inequalities
hold for all \(a,b > 0\) with \(a \ne b\) if and only if \({\alpha _{1}}\geq 1/3,{\beta _{1}}\leq (1/2)\pi (4/\pi -1/\sinh ^{-1}(1))\), \({\alpha _{2}}\geq 1/3\), \({\beta _{2}}\leq {\log (4\log (1+\sqrt{2})/\pi )}/\log 2\).
In [3, 4, 10], the authors proved that the double inequalities
hold for all \(a,b > 0\) with \(a \ne b\) if and only if \({\alpha _{3}} \le (4 - \pi )/[(\sqrt{2} - 1)\pi ]\), \({\beta _{3}} \ge 2/3\), \({\alpha _{4}} \le 2/3\), \({\beta _{4}} \ge 4 - 2\log \pi /\log 2\), \({\alpha _{5}} \le 1/2\), \({\beta _{5}} \le [2\log (\log (1 + \sqrt{2} ) + \log 2)]/(2\log \pi - \log 2)\) and \({\alpha _{6}} \ge 1/2\), \({\beta _{6}} \le [\pi (\sqrt{2} \log (1 + \sqrt{2} ) - 1)]/[(\sqrt{2} \pi - 2)\log (1 + \sqrt{2} )]\).
The main purpose of this paper is to find the least values α and β such that the inequalities
and
hold for all \(a,b > 0\) with \(a \ne b\). Moreover, we find that both upper bounds for \(P(a,b)\) are trivial cases. That is to say, the inequalities \(P(a,b) < {L^{{\lambda _{1}}}}(a,b){T^{1 - {\lambda _{1}}}}(a,b)\) and \(P(a,b) < {L^{{\lambda _{2}}}}(a,b){M^{1 - {\lambda _{2}}}}(a,b)\) hold for all \(a,b > 0\) with \(a \ne b\) if and only if \({\lambda _{1}}=0\) and \({\lambda _{2}}=0\), which we will address at the end of this paper.
2 Lemmas
To establish our main results, we need several lemmas, which we present in this section.
For \(x \in (0,1)\), the following power series expansions of the functions \(\sin ^{ - 1}(x)\), \(\sinh ^{ - 1}(x)\), \(\tan ^{-1}(x)\) and \({\tanh ^{ - 1}}(x)\) are presented:
Lemma 2.1
Let \({f_{1}}(x) = 4(1 + {x^{2}})\sqrt{1 - {x^{2}}}{\tanh ^{ - 1}}(x) \tan ^{-1}(x)\). Then
for \(x \in (0,1)\).
Proof
Let
then
where \(h(x)=1 - x{\tanh ^{ - 1}}(x) - \sqrt{1 - {x^{2}}} (1 - \frac{1}{2}{x^{2}})\). Noting that, for any \(x\in (0,1)\),
we can get
It follows from (2.7) and (2.10) that
for \(x\in (0,1)\). Considering \(g(0) = 0\), then we have \(g(x)<0\) for \(x\in (0,1)\). That is,
for \(x\in (0,1)\).
Considering (2.3), we have
for \(x\in (0,1)\).
Therefore, it follows from (2.12) and (2.13) that
for \(x \in (0,1)\). □
Lemma 2.2
Let \({f_{2}}(x)=3\tan ^{-1}(x)\sin ^{-1}(x)(1+{x^{2}})\), \({f_{3}}(x)=\tanh ^{-1}(x)\sin ^{-1}(x)(1-{x^{2}})\). Then
for \(x \in (0,1)\).
Proof
Noticing that
we obtain
for \(x \in (0,1)\). Therefore, (2.15) holds.
for \(x \in (0,1)\). Therefore, (2.16) holds. □
Lemma 2.3
If \(x \in (0,1)\), then one has
Proof
Let
It follows that
Notice that \(\sqrt{1 - x} > 1 - \frac{1}{2}x - \frac{1}{2}{x^{2}}\) for \(x \in (0,1)\), therefore \(\sqrt{1 - {x^{4}}} > 1 - \frac{1}{2}{x^{4}} - \frac{1}{2}{x^{8}}\) for \(x \in (0,1)\). Considering \({\tanh ^{ - 1}}(x) > x + \frac{1}{3}{x^{3}} + \frac{1}{5}{x^{5}}\) for \(x \in (0,1)\), we have
for any \(x \in (0,1)\).
Equation (2.23) and inequality (2.24) lead to \({g'_{1}}(x) < 0\) for any \(x \in (0,1)\). Noting that \({g_{1}}(0) = 0\), thus we have \({g_{1}}(x) < 0\) for any \(x \in (0,1)\). Inequality (2.20) is proved.
Let
Then one has
Because \(\sqrt{1 + {x^{2}}} < 1 + \frac{1}{2}{x^{2}}\) and \(\sinh ^{ - 1}(x) > x - \frac{1}{6}{x^{3}}\) for \(x\in (0,1)\), it follows that
for any \(x \in (0,1)\).
Equation (2.26) and inequality (2.27) lead to \({g'_{2}}(x) > 0\) for any \(x \in (0,1)\). Note that \(g_{2}(0) = 0\). So \({g_{2}}(x) > 0\) for any \(x \in (0,1)\). Inequality (2.21) is established. □
Lemma 2.4
Let \({f_{4}}(x)=3\sinh ^{-1}(x)\tanh ^{-1}(x)\sqrt{1- {x^{4}}}\) and \({f_{5}}(x)=2\sin ^{-1}(x)\sinh ^{-1}(x)\sqrt{1+ {x^{2}}}\). Then
for any \(x \in (0,1)\).
Proof
Because
for \(x \in (0,1)\), we can get
for any \(x \in (0,1)\). This is inequality (2.28).
Observe \(\sin ^{ - 1}(x) > x + \frac{1}{6}{x^{3}} + \frac{3}{{40}}{x^{5}}\) for \(x \in (0,1)\). It follows that
for any \(x \in (0,1)\). Inequality (2.29) holds. □
3 Main results
Note that \(L(a,b)\), \(P(a,b)\), \(T(a,b)\) and \(M(a,b)\) are symmetric and homogeneous of degree 1. In this section, without loss of generality, we can assume that \(a > b\), then \(x: = (a - b)/(a + b) \in (0,1)\). We have the following two theorems.
Theorem 3.1
The inequality
holds for all \(a,b > 0\) with \(a \ne b\) if and only if \(\alpha \ge \frac{3}{4}\).
Proof
Noting that
we have
Direct computations lead to
Next, we take the logarithm of (3.1) and consider the difference between the convex combination of \(\log L(a,b),\log P(a,b) \) and \(\log T(a,b)\) as follows:
It follows that
where \({f_{1}}(x)\), \({f_{2}}(x)\), and \({f_{3}}(x)\) are defined as in Lemmas 2.1 and 2.2, respectively. Thus, from Lemmas 2.1 and 2.2 one deduces that
Therefore, it follows from (3.6)–(3.8) that
for \(x \in (0,1)\).
According to (3.5) and (3.9), we conclude that \(P(a,b) > {L^{\frac{3}{4}}}(a,b){T^{\frac{1}{4}}}(a,b)\) for all \(a,b > 0\) with \(a \ne b\). Considering \(L(a,b)< P(a,b)< T(a,b)\) holds for all \(a,b > 0\) with \(a \ne b\), we can see that (3.1) holds for all \(a,b > 0\) with \(a \ne b\) and \(\alpha \ge \frac{3}{4}\).
If \(\alpha < \frac{3}{4}\), then Eqs. (3.3) and (3.4) imply that there exists \(0 < {\sigma _{1}} < 1\) such that \(P(a,b) < {L^{\alpha }}(a,b){T^{1 - \alpha }}(a,b)\) for all a, b with \((a - b)/(a + b) \in (0,{\sigma _{1}})\). The proof is completed. □
Theorem 3.2
The inequality
holds for all \(a,b > 0\) with \(a \ne b\) if and only if \(\beta \ge \frac{2}{3}\).
Proof
Noting that
we have
Direct computations lead to
Let
It follows that
where \({f_{3}}(x)\), \({f_{4}}(x)\), and \({f_{5}}(x)\) are defined as in Lemmas 2.2 and 2.4, respectively. Thus, from Lemmas 2.2 and 2.4 one deduces that
Therefore, it follows from (3.15)–(3.17) that
for any \(x \in (0,1)\).
According to (3.14) and (3.18), we conclude that \(P(a,b) > {L^{\frac{2}{3}}}(a,b){M^{\frac{1}{3}}}(a, b)\) for all \(a,b > 0\) with \(a \ne b\). Considering \(L(a,b)< P(a,b)< M(a,b)\) holds for all \(a,b > 0\) with \(a \ne b\), we can see that (3.10) holds for all \(a,b > 0\) with \(a \ne b\) and \(\beta \ge \frac{2}{3}\).
If \(\beta < \frac{2}{3}\), then Eqs. (3.12) and (3.13) imply that there exists \(0 < {\sigma _{2}} < 1\) such that \(P(a,b) < {L^{\beta }}(a,b){M^{1 - \beta }}(a,b)\) for all a, b, with \((a - b)/(a + b) \in (0,{\sigma _{2}})\). The proof is completed. □
Remark
Let us show there is no \({\lambda _{1}},{\lambda _{2}} \in (0,1)\) such that \(P(a,b) < {L^{{\lambda _{1}}}}(a,b){T^{1 - {\lambda _{1}}}}(a,b)\) and \(P(a,b) < {L^{{\lambda _{2}}}}(a,b){M^{1 - {\lambda _{2}}}}(a,b)\) hold for all \(a,b > 0\) with \(a \ne b\). Firstly, we assume that \({\lambda _{1}} > 0\), then Eq. (3.3) and \(\lim _{x \to {1^{-} }} \frac{{\log [\sin^{ - 1}(x)] - \log [\tan^{ - 1}(x)]}}{{\log [\tanh^{ - 1}(x)] - \log [\tan^{ - 1}(x)]}} = 0\) imply that there exists \(0<{\sigma _{3}}<1\) such that \(P(a,b) > {L^{{\lambda _{1}}}}(a,b){T^{1 - {\lambda _{1}}}}(a,b)\) for all a, b with \((a - b)/(a + b) \in (1 - {\sigma _{3}},1)\). This in conjunction with the well-known inequality \(P(a,b) < T(a,b)\), which is the case of \({\lambda _{1}} = 0\), indicates that \(P(a,b) < {L^{{\lambda _{1}}}}(a,b){T^{1 - {\lambda _{1}}}}(a,b)\) if and only if \({\lambda _{1}} =0\). With the same method, we can obtain \(P(a,b) < {L^{{\lambda _{2}}}}(a,b){M^{1 - {\lambda _{2}}}}(a,b)\) if and only if \({\lambda _{2} =0}\).
4 Conclusion
In the article, we give the best possible bounds for the first Seiffert mean in terms of the geometric combination of logarithmic and the Neuman–Sándor means, and in terms of the geometric combination of logarithmic and the second Seiffert means.
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Funding
This research was supported by NSFC (No. 11501001), Foundations of Anhui Educational Committee (KJ2017A029) and Anhui University (J01006023, Y01002428), China.
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Long, B., Xu, L. & Wang, Q. Several sharp inequalities about the first Seiffert mean. J Inequal Appl 2018, 174 (2018). https://doi.org/10.1186/s13660-018-1763-2
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DOI: https://doi.org/10.1186/s13660-018-1763-2