Abstract
In this paper, we present the best possible parameters \(\alpha, \beta \in\mathbb{R}\) and \(\lambda, \mu\in(1/2, 1)\) such that the double inequalities \(\alpha N_{AQ}(a,b)+(1-\alpha)A(a,b)< T^{\ast}(a,b)<\beta N_{AQ}(a,b)+(1-\beta)A(a,b)\), \(Q[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a]< T^{\ast}(a,b)< Q[\mu a+(1-\mu)b, \mu b+(1-\mu)a] \) hold for all \(a, b>0\) with \(a\neq b\), where \(T^{\ast}(a,b)\), \(A(a,b)\), \(Q(a,b)\) and \(N_{QA}(a,b)\) are the Toader, arithmetic, quadratic, and Neuman means of a and b, respectively.
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1 Introduction
For \(a,b>0\) the Toader mean \(T^{\ast}(a,b)\) [1] is given by
It is well known that the Toader mean satisfies
for all \(a, b>0\), where
stands for the symmetric complete elliptic integral of the second kind (see [2–4]), therefore it cannot be expressed in terms of the elementary transcendental functions.
Recently, the Toader mean \(T^{\ast}(a,b)\) has been the subject of intensive research. In particular, many remarkable inequalities for the Toader mean can be found in the literature [5–12].
Let \(p\in\mathbb{R}\) and \(a, b>0\). Then the pth power mean \(M_{p}(a,b)\) is defined by
It is well known that \(M_{p}(a,b)\) is continuous and strictly increasing with respect to \(p\in\mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\).
Vuorinen [13] conjectured that the inequality
holds for all \(a, b>0\) with \(a\neq b\). This conjecture was proved by Qiu and Shen [14], and Barnard et al. [15], respectively.
Alzer and Qiu [16] presented a best possible upper power mean bound for the Toader mean as follows:
for all \(a, b>0\) with \(a\neq b\).
Chu et al. [17] proved that the inequality
holds for all \(a, b>0\) with \(a\neq b\), where \(T(a,b)=(a-b)/[2\arctan ((a-b)/(a+b))]\) is the second Seiffert mean.
Another important mean of two positive real numbers a and b is the Schwab-Borchardt mean [18, 19]
where \(\cosh^{-1}(x)=\log(x+\sqrt{x^{2}-1})\) is the inverse hyperbolic cosine function.
It is well known that the Schwab-Borchardt mean \(SB(a,b)\) is strictly increasing in both a and b, nonsymmetric and homogeneous of degree 1. Many symmetric bivariate means are special cases of the Schwab-Borchardt mean. For example, \(P(a,b)=(a-b)/[2\arcsin((a-b)/(a+b))]=SB[G(a,b), A(a,b)]\) is the first Seiffert mean, \(T(a,b)=(a-b)/[2\arctan ((a-b)/(a+b))]=SB[A(a,b), Q(a,b)]\) is the second Seiffert mean, \(M(a,b)=(a-b)/[2\sinh^{-1}((a-b)/(a+b))]=SB[Q(a,b), A(a,b)]\) is the Neuman-Sándor mean, \(L(a,b)=(a-b)/[2\tanh ^{-1}((a-b)/(a+b))]=SB[A(a,b), G(a,b)]\) is the logarithmic mean, where \(G(a,b)=\sqrt{ab}\), \(A(a,b)=(a+b)/2\) and \(Q(a,b)=\sqrt {(a^{2}+b^{2})/2}\) are the geometric, arithmetic, and quadratic means of a and b, respectively.
Very recently, Neuman [20] introduced the Neuman mean,
and presented the explicit formula for \(N_{AQ}(a,b)\equiv N[A(a,b), Q(a,b)]\) as follows:
and proved that the double inequality
holds for all \(a, b>0\) with \(a\neq b\), where \(v=(a-b)/(a+b)\).
Inequalities (1.2), (1.3), and (1.5) lead to
for all \(a, b>0\) with \(a\neq b\).
Let \(a, b>0\) with \(a\neq b\) be fixed and \(f(x)=Q[xa+(1-x)b, xb+(1-x)a]\). Then it is not difficult to verify that \(f(x)\) is continuous and strictly increasing on \([1/2, 1]\). Note that
Motivated by inequalities (1.6) and (1.7), it is natural to ask: what are the best possible parameters \(\alpha, \beta\in\mathbb{R}\) and \(\lambda, \mu\in(1/2, 1)\) such that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\)? 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.
For \(r\in(0, 1)\) the complete elliptic integrals \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) of the first and second kinds are defined by
and
respectively. We clearly see that
and the Toader mean \(T^{\ast}(a,b)\) given by (1.1) can be expressed as
\(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) satisfy the formulas (see [21], Appendix E, p.474,475)
Lemma 2.1
(See [21], Theorem 1.25)
Let \(-\infty< a< b<\infty\), \(f, g: [a, b]\rightarrow(-\infty, \infty)\) be continuous on \([a,b]\) and differentiable on \((a, b)\), and \(g^{\prime}(x)\neq0\) on \((a,b)\). If \(f'(x)/g'(x)\) is increasing (decreasing) on \((a,b)\), then so are
If \(f'(x)/g'(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2
(See [21], Theorem 3.21)
(1) The function \(r\mapsto[\mathcal{E}(r)-(1-r^{2})\mathcal {K}(r)]/r^{2}\) is strictly increasing from \((0, 1)\) onto \((\pi/4, 1)\).
(2) The function \(r\mapsto(1-r^{2})^{\lambda}\mathcal{K}(r)\) is strictly decreasing from \((0, 1)\) onto \((0, \pi/2)\) if \(\lambda\geq1/4\).
Lemma 2.3
The function \(r\mapsto[2(1-r^{2})\mathcal{E}(r)-\pi]/r^{2}\) is strictly increasing from \((0, 1)\) onto \((-5\pi/4, -\pi)\).
Proof
Let \(f_{1}(r)=2(1-r^{2})\mathcal{E}(r)-\pi\), \(f_{2}(r)=r^{2}\) and \(f(r)=[2(1-r^{2})\mathcal{E}(r)-\pi]/r^{2}\). Then
and simple computations lead to
It follows from Lemma 2.2(1), (2.2), and (2.4) that \(f^{\prime }_{1}(r)/f^{\prime}_{2}(r)\) is strictly increasing on \((0, 1)\) and
Therefore, Lemma 2.3 follows from Lemma 2.1, (2.2), (2.3), (2.5), and the monotonicity of \(f^{\prime}_{1}(r)/f^{\prime}_{2}(r)\). □
Lemma 2.4
Let \(p\in(0, 1)\), \(r\in(0, 1)\) and
Then the following statements are true:
-
(1)
If \(p=3/4\), then \(f(r)>0\) for all \(r\in(0, 1)\);
-
(2)
If \(p=4(4-\pi)/[\pi(\pi-2)]=0.9573\cdots\), then there exists \(r_{0}\in(0, 1)\) such that \(f(r)<0\) for \(r\in(0, r_{0})\) and \(f(r)>0\) for \(r\in(r_{0}, 1)\).
Proof
For part (1), if \(p=3/4\), then (2.6) becomes
and Lemma 2.2(1) and Lemma 2.3 lead to
for all \(r\in(0, 1)\).
For part (2), if \(p=4(4-\pi)/[\pi(\pi-2)]\), then it follows from Lemma 2.2(1), Lemma 2.3, and (2.6) that
and \(f(r)\) is strictly increasing on \((0, 1)\).
Therefore, part (2) follows from (2.7) and (2.8) together with the monotonicity of \(f(r)\). □
3 Main results
Theorem 3.1
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\alpha\leq3/4\) and \(\beta\geq4(4-\pi)/[\pi(\pi-2)]=0.9573\cdots\).
Proof
Since \(A(a,b)\), \(T^{\ast}(a,b)\) and \(N_{AQ}(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(a>b\). Let \(r=(a-b)/(a+b)\in(0, 1)\) and \(p\in(0, 1)\). Then (2.1) leads to
It follows from (1.4), Lemma 2.2(2), and (3.2) that
where
where \(f(r)\) is defined as in Lemma 2.4.
We divide the proof into two cases.
Case 1 \(p=3/4\). Then Lemma 2.4(1) and (3.7) lead to the conclusion that \(F(r)\) is strictly increasing on \((0, 1)\). Therefore,
follows from (3.4) and (3.5) together with the monotonicity of \(F(r)\).
Case 2 \(p=4(4-\pi)/[\pi(\pi-2)]\). Then (3.6) becomes
and Lemma 2.4(2) and (3.7) imply that there exists \(r_{0}\in(0, 1)\) such that \(F(r)\) is strictly decreasing on \((0, r_{0}]\) and strictly increasing on \([r_{0}, 1)\). Therefore,
follows from (3.4), (3.5), (3.8), and the piecewise monotonicity of \(F(r)\).
Next, we prove that \(\alpha=3/4\) and \(\beta=4(4-\pi)/[\pi(\pi-2)]\) are the best possible parameters such that the double inequality (3.1) holds for all \(a, b>0\) with \(a\neq b\). It is not difficult to verify that
If \(\alpha>3/4\), then (3.3) and (3.9) imply that there exists \(0<\delta _{1}<1\) such that
for all \(a>b>0\) with \((a-b)/(a+b)\in(0, \delta_{1})\).
If \(\beta<4(4-\pi)/[\pi(\pi-2)]\), then (3.3) and (3.10) imply that there exists \(0<\delta_{2}<1\) such that
for all \(a>b>0\) with \((a-b)/(a+b)\in(1-\delta_{2}, 1)\). □
Theorem 3.2
Let \(\lambda, \mu\in(1/2, 1)\). Then the double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda\leq 1/2+\sqrt{2}/4=0.8535\cdots\) and \(\mu\geq1/2+\sqrt{16/\pi ^{2}-1}/2=0.8940\cdots\).
Proof
Without loss of generality, we assume that \(a>b>0\). Let \(r=(a-b)/(a+b)\in(0, 1)\) and \(p\in(0, 1)\). Then from (3.2) and
we get
where
Let
Then (3.13) and Lemma 2.2 lead to
We divide the proof into two cases.
Case 1 \(p=1/2+\sqrt{2}/4\). Then (3.18) becomes
From Lemma 2.2(1) and \(d[2\mathcal{E}(r)-(1-r^{2})\mathcal {K}(r)]/dr=[\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)]/r\) we know that the function \(r\mapsto2\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)\) is strictly increasing on \((0, 1)\). Then from Lemma 2.2(1) and (3.17) together with (3.20) we know that \(g_{1}(r)\) is strictly increasing on \((0, 1)\) and
for \(r\in(0, 1)\). Therefore,
follows from (3.12), (3.14), (3.16), and (3.21).
Case 2 \(p=1/2+\sqrt{16/\pi^{2}-1}/2\). Then (3.15), (3.18), and (3.19) lead to
It follows from (3.16), (3.23), and (3.24) together with the monotonicity of \(g_{1}(r)\) that there exists \(r^{\ast}\in(0, 1)\) such that \(g(r)\) is strictly decreasing on \((0, r^{\ast}]\) and strictly increasing on \([r^{\ast}, 1)\). Therefore,
follows from (3.12), (3.14), (3.22), and the piecewise monotonicity of \(g(r)\).
Next, we prove that \(\lambda=1/2+\sqrt{2}/4\) and \(\mu=1/2+\sqrt{16/\pi ^{2}-1}/2\) are the best possible parameters in \((1/2, 1)\) such that the double inequality (3.11) holds for all \(a, b>0\) with \(a\neq b\).
If \(1/2+\sqrt{2}/4< p<1\), then (3.18) leads to
Equations (3.12), (3.14), and (3.16) and inequality (3.25) imply that there exists \(\delta_{3}\in(0, 1)\) such that
for all \(a>b>0\) with \((a-b)/(a+b)\in(0, \delta_{3})\).
If \(1/2< p<1/2+\sqrt{16/\pi^{2}-1}/2\), then (3.15) leads to
Equation (3.12) and inequality (3.26) imply that there exists \(\delta _{4}\in(0, 1)\) such that
for all \(a>b>0\) with \((a-b)/(a+b)\in(1-\delta_{4}, 1)\). □
Let \(r\in(0, 1)\), \(r^{\ast}=r^{2}/(1+\sqrt{1-r^{2}})^{2}\), \(a=1\), \(b=\sqrt{1-r^{2}}\), \(\alpha=3/4\), \(\beta=4(4-\pi)/[\pi(\pi-2)]\), \(\lambda=1/2+\sqrt{2}/4\), and \(\mu=1/2+\sqrt{16/\pi^{2}-1}/2\). Then Theorems 3.1 and 3.2 lead to Corollary 3.3 as follows.
Corollary 3.3
The double inequalities
and
hold for all \(r\in(0, 1)\).
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Acknowledgements
The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. The research was supported by the Major Project Foundation of the Department of Education of Hunan Province under Grant 12A026.
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Li, JF., Qian, WM. & Chu, YM. Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means. J Inequal Appl 2015, 277 (2015). https://doi.org/10.1186/s13660-015-0800-7
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DOI: https://doi.org/10.1186/s13660-015-0800-7