Abstract
In this paper, we present the best possible parameters \(\alpha(r)\) and \(\beta(r)\) such that the double inequality
holds for all \(r\leq 1\) and \(a, b>0\) with \(a\neq b\), and we provide new bounds for the complete elliptic integral \(\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta\) \((r\in (0, \sqrt{2}/2))\) of the second kind, where \(TD(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta\), \(A(a,b)=(a+b)/2\) and \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\) are the Toader, arithmetic, and quadratic means of a and b, respectively.
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1 Introduction
For \(p\in [0, 1]\), \(q\in \mathbb{R}\) and \(a,b>0\) with \(a\neq b\), the pth generalized Seiffert mean \(S_{p}(a, b)\), qth Gini mean \(G_{q}(a, b)\), qth power mean \(M_{q}(a, b)\), qth Lehmer mean \(L_{q}(a,b)\), harmonic mean \(H(a,b)\), geometric mean \(G(a,b)\), arithmetic mean \(A(a,b)\), quadratic mean \(Q(a,b)\), Toader mean \(TD(a,b)\) [1], centroidal mean \(\overline{C}(a,b)\), contraharmonic mean \(C(a,b)\) are, respectively, defined by
It is well known that \(S_{p}(a, b)\), \(G_{q}(a, b)\), \(M_{q}(a, b)\), and \(L_{q}(a,b)\) are continuous and strictly increasing with respect to \(p\in [0, 1]\) and \(q\in \mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\), and the inequalities
hold for all \(a, b>0\) with \(a\neq b\).
The Toader mean \(TD(a,b)\) has been well known in the mathematical literature for many years, it satisfies
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.
Let \(r\in (0, 1)\), \(\mathcal{K}(r)=\int _{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{-1/2}\,d\theta\) and \(\mathcal{E}(r)=\int _{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta\) be, respectively, the complete elliptic integrals of the first and second kind. Then \(\mathcal{K}(0^{+})=\mathcal{E}(0^{+})=\pi/2\), \(\mathcal{K}(r)\), and \(\mathcal{E}(r)\) satisfy the derivatives formulas (see [5], Appendix E, p.474-475)
the values \(\mathcal{K}(\sqrt{2}/2)\) and \(\mathcal{E}(\sqrt{2}/2)\) can be expressed as (see [6], Theorem 1.7)
where \(\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\,dt(\operatorname{Re}{x}>0)\) is the Euler gamma function, and the Toader mean \(TD(a,b)\) can be rewritten as
Recently, the Toader mean \(TD(a,b)\) has been the subject of intensive research. Vuorinen [7] conjectured that the inequality
holds for all \(a, b>0\) with \(a\neq b\). This conjecture was proved by Qiu and Shen [8], and Barnard, Pearce and Richards [9], respectively.
Alzer and Qiu [10] presented a best possible upper power mean bound for the Toader mean as follows:
for all \(a, b>0\) with \(a\neq b\).
Neuman [2], and Kazi and Neuman [3] proved that the inequalities
hold for all \(a, b>0\) with \(a\neq b\), where \(AGM(a,b)\) is the arithmetic-geometric mean of a and b.
In [11–13], the authors presented the best possible parameters \(\lambda_{1}, \mu_{1}\in [0, 1]\) and \(\lambda_{2}, \mu_{2}, \lambda_{3}, \mu_{3}\in \mathbb{R}\) such that the double inequalities \(S_{\lambda_{1}}(a,b)< TD(a,b)< S_{\mu_{1}}(a,b)\), \(G_{\lambda_{2}}(a,b)< TD(a,b)< G_{\mu_{2}}(a,b)\) and \(L_{\lambda_{3}}(a,b)< TD(a,b)< L_{\mu_{3}}(a,b)\) hold for all \(a, b>0\) with \(a\neq b\).
Let \(\lambda, \mu, \alpha, \beta\in (1/2, 1)\). Then Chu, Wang and Ma [14], and Hua and Qi [15] proved that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda\leq 3/4\), \(\mu\geq 1/2+\sqrt{\pi(4-\pi)}/(2\pi)\), \(\alpha\leq 1/2+\sqrt{3}/4\) and \(\beta\geq 1/2+\sqrt{12/\pi-3}/2\).
In [16–20], the authors proved that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\alpha_{1}\leq 1/2\), \(\beta_{1}\geq (4-\pi)/[(\sqrt{2}-1)\pi]\), \(\alpha_{2}\leq 1/2\), \(\beta_{2}\geq 4-2\log\pi/\log 2\), \(\alpha_{3}\leq 1/4\), \(\beta_{3}\geq 4/\pi-1\), \(\alpha_{4}\leq \pi/2-1\), \(\beta_{4}\geq 3/4\), \(\alpha_{5}\leq 5/8\), \(\beta_{5}\geq 2/\pi\), \(\alpha_{6}\leq 1/8\), \(\beta_{6}\geq 2/\pi-1/2\), \(\alpha_{7}\leq 3/4\), \(\beta_{7}\geq 12/\pi-3\), \(\alpha_{8}\leq \pi-3\), \(\beta_{8}\geq 1/4\), \(\alpha_{9}\leq 5/6\), \(\beta_{9}\geq 2\sqrt{2}/\pi\), \(\alpha_{10}\leq 0\), and \(\beta_{10}\geq 1/6\).
The main purpose of this paper is to present the best possible parameters \(\alpha(r)\) and \(\beta(r)\) such that the double inequality
holds for all \(r\leq 1\) and \(a, b>0\) with \(a\neq b\).
2 Lemmas
In order to prove our main result we need two lemmas, which we present in this section.
Lemma 2.1
Let \(p\in (0, 1)\), \(t\in (0, \sqrt{2}/2)\), \(\lambda=(2+\sqrt{2})[1-2\mathcal{E}(\sqrt{2}/2)/\pi]=0.478\ldots\) and
Then \(f(t)<0\) for all \(t\in (0, \sqrt{2}/2)\) if and only if \(p\geq 1/2\) and \(f(t)>0\) for all \(t\in (0, \sqrt{2}/2)\) if and only if \(p\leq \lambda\).
Proof
It follows from (2.1) that
where
for all \(t\in (0, \sqrt{2}/2)\).
It follows from (2.13) that \(f_{1}^{\prime\prime}(t)\) is strictly decreasing on \((0, \sqrt{2}/2)\).
We divide the proof into three cases.
Case 1 \(p\geq 1/2\). Then (2.11) leads to
From (2.14) and the monotonicity of \(f_{1}^{\prime\prime}(t)\) we clearly see that \(f_{1}^{\prime}(t)\) is strictly decreasing on \((0, \sqrt{2}/2)\). Therefore, \(f(t)<0\) for all \(t\in (0, \sqrt{2}/2)\) follows easily from (2.2), (2.4), (2.5), (2.8), and the monotonicity of \(f_{1}^{\prime}(t)\).
Case 2 \(0< p\leq \lambda\). Then from (2.11) and (2.12) together with \(4\mathcal{E}(\sqrt{2}/2)-3\mathcal{K}(\sqrt{2}/2)=-0.159\ldots\) we clearly see that
It follows from (2.15) and the monotonicity of \(f_{1}^{\prime\prime}(t)\) that there exists \(t_{0}\in (0, \sqrt{2}/2)\) such that \(f_{1}^{\prime}(t)\) is strictly increasing on \((0, t_{0}]\) and strictly decreasing on \([t_{0}, \sqrt{2}/2)\).
Let \(\lambda^{\ast}=\frac{\sqrt{2}}{\pi} [2\mathcal{E}(\frac{\sqrt{2}}{2})-\mathcal{K}(\frac{\sqrt{2}}{2}) ]=0.381\ldots\) and \(\lambda^{\ast\ast}=\frac{2\sqrt{2}}{\pi} [\mathcal{K}(\frac{\sqrt{2}}{2})-\mathcal{E}(\frac{\sqrt{2}}{2}) ]=0.453\ldots\) . We divide the proof into three subcases.
Subcase 2.1 \(0< p\leq\lambda^{\ast}\). Then (2.9) leads to
It follows from (2.8) and (2.16) together with the piecewise monotonicity of \(f_{1}^{\prime}(t)\) that
for all \(t\in (0, \sqrt{2}/2)\).
Therefore, \(f(t)>0\) for all \(t\in (0, \sqrt{2}/2)\) follows easily from (2.2), (2.4), (2.5), and (2.17).
Subcase 2.2 \(\lambda^{\ast}< p\leq\lambda^{\ast\ast}\). Then (2.6) and (2.9) lead to
It follows from (2.8) and (2.19) together with the piecewise monotonicity of \(f_{1}^{\prime}(t)\) that there exists \(t_{1}\in (0, \sqrt{2}/2)\) such that \(f_{1}(t)\) is strictly increasing on \((0, t_{1}]\) and strictly decreasing on \([t_{1}, \sqrt{2}/2)\).
Equation (2.5) and inequality (2.18) together with the piecewise monotonicity of \(f_{1}(t)\) lead to the conclusion that
for all \(t\in (0, \sqrt{2}/2)\).
Therefore, \(f(t)>0\) for all \(t\in (0, \sqrt{2}/2)\) follows easily from (2.2) and (2.4) together with (2.20).
Subcase 2.3 \(\lambda^{\ast\ast}< p\leq \lambda\). Then (2.3), (2.6), and (2.9) lead to
It follows from (2.8) and (2.23) together with the piecewise monotonicity of \(f_{1}^{\prime}(t)\) that there exists \(t_{2}\in (0, \sqrt{2}/2)\) such that \(f_{1}(t)\) is strictly increasing on \((0, t_{2}]\) and strictly decreasing on \([t_{2}, \sqrt{2}/2)\).
From (2.4), (2.5), and (2.22) together with the piecewise monotonicity of \(f_{1}(t)\) we clearly see that there exists \(t_{3}\in (0, \sqrt{2}/2)\) such that \(f(t)\) is strictly increasing on \((0, t_{3}]\) and strictly decreasing on \([t_{3}, \sqrt{2}/2)\).
Therefore, \(f(t)>0\) for all \(t\in (0, \sqrt{2}/2)\) follows easily from (2.2) and (2.21) together with the piecewise monotonicity of \(f(t)\).
Case 3 \(\lambda< p<1/2\). Then (2.3), (2.6), (2.9), (2.11), and (2.12) lead to
It follows from (2.27) and (2.28) together with the monotonicity of \(f_{1}^{\prime\prime}(t)\) that there exists \(t_{4}\in (0, \sqrt{2}/2)\) such that \(f_{1}^{\prime}(t)\) is strictly increasing on \((0, t_{4}]\) and strictly decreasing on \([t_{4}, \sqrt{2}/2)\).
Equation (2.8) and inequality (2.26) together with the piecewise monotonicity of \(f_{1}^{\prime}(t)\) lead to the conclusion that there exists \(t_{5}\in (0, \sqrt{2}/2)\) such that \(f_{1}(t)\) is strictly increasing on \((0, t_{5}]\) and strictly decreasing on \([t_{5}, \sqrt{2}/2)\).
From (2.4), (2.5), (2.25), and the piecewise monotonicity of \(f_{1}(t)\) we clearly see that there exists \(t_{6}\in (0, \sqrt{2}/2)\) such that \(f(t)\) is strictly increasing on \((0, t_{6}]\) and strictly decreasing on \([t_{6}, \sqrt{2}/2)\).
Therefore, there exists \(t_{7}\in (0, \sqrt{2}/2)\) such that \(f(t)>0\) for \(t\in (0, t_{7})\) and \(f(t)<0\) for \(t\in (t_{7}, \sqrt{2}/2)\) follows from (2.2) and (2.24) together with the piecewise monotonicity of \(f(t)\). □
Lemma 2.2
Let \(r\in \mathbb{R}\), \(a, b>0\) with \(1< b/a<\sqrt{2}\), \(c_{0}=2\mathcal{E}(\sqrt{2}/2)/\pi=0.859\ldots\) , \(c_{1}=\sqrt{2}/2\), \(\lambda(r)\) and \(U(r; a,b)\) be defined by
and
respectively. Then the function \(r\mapsto U(r; a,b)\) is strictly decreasing on \((-\infty, \infty)\).
Proof
Let \(x=b/a\in (1, \sqrt{2})\), \(r\neq 0\), and
Then from (2.29)-(2.31) one has
where inequalities (2.35) and (2.36) hold due to \(c_{0}>c_{1}\) and the function \(t\mapsto \log t/(t^{r}-1)\) is strictly decreasing on \((0, \infty)\).
Note that \(\lambda(r)\in (0, 1)\) and the function \(x\rightarrow V(r, x)\) is strictly increasing on \((1, \sqrt{2})\). Then (2.34)-(2.36) lead to the conclusion that there exists \(x_{0}\in (1, \sqrt{2})\) such that the function \(x\mapsto \partial \log U(r; a, b)/\partial r\) is strictly decreasing on \((1, x_{0})\) and strictly increasing on \((x_{0}, \sqrt{2})\).
It follows from (2.32) and (2.33) together with the piecewise monotonicity of the function \(x\mapsto \partial \log U(r; a, b)/\partial r\) on the interval \((1, \sqrt{2})\) that
for all \(a, b>0\) with \(1< b/a<\sqrt{2}\).
3 Main result
Theorem 3.1
Let \(c_{0}=2\mathcal{E}(\sqrt{2}/2)/\pi=0.859\ldots\) , \(c_{1}=\sqrt{2}/2\) and \(\lambda(r)\) be defined by (2.29). Then the double inequality
holds for all \(r\leq 1\) and \(a, b>0\) with \(a\neq b\) if and only if \(\alpha(r)\geq 1/2\) and \(\beta(r)\leq \lambda(r)\), where \(r=0\) is the limit value of \(r\rightarrow 0\).
Proof
We first prove that Theorem 3.1 holds for \(r=1\).
Since \(A(a,b)< TD[A(a,b), Q(a,b)]< Q(a,b)\) for all \(a, b>0\) with \(a\neq b\), and \(A(a,b)\), \(TD(a,b)\) and \(Q(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(\alpha(1)\), \(\beta(1)\in (0, 1)\) and \(a>b\). Let \(t=(a-b)/\sqrt{2(a^{2}+b^{2})}\in (0, \sqrt{2}/2)\) and \(p\in (0, 1)\). Then (1.1) and (1.2) lead to
where \(f(t)\) is defined as in Lemma 2.1.
Therefore, Theorem 3.1 for \(r=1\) follows easily from Lemma 2.1 and (3.1).
Next, let \(r<1\) and \(a, b>0\) with \(a\neq b\), then it follows from Theorem 3.1 for \(r=1\) that
Note that
Therefore, Theorem 3.1 for \(r<1\) follows from (3.2)-(3.6) and Lemma 2.2 together with the monotonicity of the function \(r\mapsto [(a^{r}+b^{r})/2]^{1/r}\). □
Let \(r=1\). Then Theorems 3.1 leads to Corollary 3.2 immediately.
Corollary 3.2
Let \(\lambda=(2+\sqrt{2})[1-2\mathcal{E}(\sqrt{2}/2)/\pi]\). Then the double inequality
holds for all \(t\in (0, \sqrt{2}/2)\).
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Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 61673169, 11371125, 11401191, and 61374086.
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Zhao, TH., Chu, YM. & Zhang, W. Optimal inequalities for bounding Toader mean by arithmetic and quadratic means. J Inequal Appl 2017, 26 (2017). https://doi.org/10.1186/s13660-017-1300-8
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DOI: https://doi.org/10.1186/s13660-017-1300-8