Abstract
In the article, we prove that \(\lambda _{1}=1/2+\sqrt{ [ (\sqrt{2}+ \log (1+\sqrt{2}) )/2 ]^{1/\nu }-1}/2\), \(\mu _{1}=1/2+\sqrt{6 \nu }/(12\nu )\), \(\lambda _{2}=1/2+\sqrt{ [(\pi +2)/4 ] ^{1/\nu }-1}/2\) and \(\mu _{2}=1/2+\sqrt{3\nu }/(6\nu )\) are the best possible parameters on the interval \([1/2, 1]\) such that the double inequalities
hold for all \(x, y>0\) with \(x\neq y\) and \(\nu \in [1/2, \infty )\), where \(A(x, y)\) is the arithmetic mean, \(C(x, y)\) is the contraharmonic mean, and \(\mathcal{R}_{QA}(x, y)\) and \(\mathcal{R}_{AQ}(x, y)\) are two Neuman means.
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1 Introduction
Let \(x, y>0\). Then the arithmetic mean \(A(x, y)\), quadratic mean \(Q(x, y)\) [1], contraharmonic mean \(C(x, y)\) [2, 3], and Schwab–Borchardt mean \(\operatorname{SB}(x, y)\) [4] are given by
respectively, where \(\cosh ^{-1}(\sigma )=\log (\sigma +\sqrt{\sigma ^{2}-1})\) is the inverse hyperbolic cosine function.
The Gaussian arithmetic–geometric mean \(\operatorname{AGM}(x, y)\) [5,6,7] of two positive real numbers x and y is defined by the common limit of the sequences \(\{x_{n}\}_{n=0}^{\infty }\) and \(\{y_{n}\}_{n=0}^{\infty }\), which are given by
It is well known that the bivariate means have wide applications in mathematics, physics, engineering, and other natural sciences [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], many special functions can be expressed using bivariate means, for example, the complete elliptic integral
of the first kind [56,57,58,59,60,61] and the modulus \(\mu (r)\) of the plane Grötzsch ring [62, 63] can be expressed by the Gaussian arithmetic–geometric mean \(\operatorname{AGM}(x, y)\), the formula of the perimeter of an ellipse and the complete elliptic integral
of the second kind [64,65,66,67,68,69,70] can be given in terms of the Toader mean [71,72,73,74]
Indeed, we have
Recently, the inequalities for bivariate means have attracted the attention of many mathematicians. Neuman [75] introduced the Neuman means
and provided the formulas
if \(x>y>0\), where \(u=(x-y)/(x+y)\) and \(\sinh ^{-1}(\sigma )=\log ( \sigma +\sqrt{\sigma ^{2}+1})\) is the inverse hyperbolic sine function. Neuman [4] proved that the inequalities
hold for \(x, y>0\) with \(x\neq y\).
Zhang et al. [76] proved that \(\alpha _{1}=1/2+\sqrt{2\sqrt{2} \log (1+\sqrt{2})+\log ^{2}(1+\sqrt{2})-2}/4=0.7817\ldots \) , \(\beta _{1}=1/2+\sqrt{3}/6=0.7886\ldots \) , \(\alpha _{2}=1/2+\sqrt{\pi ^{2}+4\pi -12}/8=0.9038\ldots \) and \(\beta _{2}=1/2+\sqrt{6}/6=0.9082 \ldots \) are the best possible parameters on the interval \([1/2, 1]\) such that the double inequalities
hold for \(x, y>0\) with \(x\neq y\).
In [77], Yang et al. proved that the double inequalities
hold for for \(x, y>0\) with \(x\neq y\) if and only if \(\alpha \leq (3 \pi +6-12\sqrt[3]{2})/(16-12\sqrt[3]{2})=0.3470\ldots \) , \(\beta \geq 2/5\), \(\lambda \leq [3\sqrt{2}+3\log (1+\sqrt{2})-6 \sqrt[6]{2}]/(7-6\sqrt[6]{2})=0.5730\ldots \) and \(\mu \geq 16/25\).
The main purpose of the article is to generalize inequalities (1.5) and (1.6). To achieve this goal, we define the two-parameter contraharmonic and arithmetic mean \(W_{\lambda , \nu }(x, y)\) as follows:
where \(\lambda \in [1/2, 1]\) and \(\nu \in [1/2, \infty )\). We clearly see that the function \(\lambda \rightarrow W_{\lambda , \nu }(x, y)\) is strictly increasing on \([1/2, 1]\) for \(\nu \in [1/2, \infty )\) and \(x, y>0\) with \(x\neq y\).
It follows from (1.1), (1.4) and (1.7) that
Inequalities (1.5), (1.6), and (1.10) give us the motivation to discuss the question: What are the best possible parameters \(\lambda _{1}= \lambda _{1}(\nu )\), \(\mu _{1}=\mu _{1}(\nu )\), \(\lambda _{2}=\lambda _{2}( \nu )\) and \(\mu _{2}=\mu _{2}(\nu )\) on the interval \([1/2, 1]\) such that the double inequalities
hold for all \(x, y>0\) with \(x\neq y\) and \(\nu \in [1/2, \infty )\)?
2 Lemmas
In order to prove our main results, we need to introduce and establish five lemmas which we present in this section.
Lemma 2.1
([78, Theorem 1.25])
Let \(\alpha , \beta \in \mathbb{R}\) with \(\alpha <\beta \), \(\varGamma , \varPsi : [\alpha , \beta ]\rightarrow \mathbb{R}\) be continuous on \([\alpha , \beta ]\) and differentiable on \((\alpha , \beta )\) with \(\varPsi ^{\prime }(\tau )\neq 0\) on \((\alpha , \beta )\). Then the functions
are (strictly) increasing (decreasing) on \((\alpha , \beta )\) if \(\varGamma ^{\prime }(\tau )/\varPsi ^{\prime }(\tau )\) is (strictly) increasing (decreasing) on \((\alpha , \beta )\).
Lemma 2.2
The function
is strictly increasing from \((0, 1)\) onto \((1, \sqrt{2}\log (1+ \sqrt{2}) )\).
Proof
Differentiating \(\phi (t)\) gives
where
It follows from (2.2) that
for all \(t\in (0, 1)\).
Note that
Therefore, Lemma 2.2 follows from (2.1) and (2.3)–(2.5). □
Lemma 2.3
The function
is strictly increasing from \((0, 1)\) onto \((3/2, 2/(\pi -2) )\).
Proof
Let \(\varphi _{1}(t)=t^{3}\) and \(\varphi _{2}(t)=(1+t^{2})\arctan (t)-t\). Then we clearly see that
It is not difficult to verify that the function \(t\mapsto t/\arctan (t)\) is strictly increasing from \((0, 1)\) onto \((1, 4/\pi )\). Then equation (2.7) leads to the conclusion that \(\varphi '_{1}(t)/\varphi '_{2}(t)\) is strictly increasing on \((0, 1)\).
Note that
Therefore, Lemma 2.3 follows from Lemma 2.1, (2.6), (2.8), and the monotonicity of \(\varphi '_{1}(t)/\varphi '_{2}(t)\). □
Lemma 2.4
Let \(\theta \in [0, 1]\), \(\nu \in [1/2, \infty )\), \(t\in (0, 1)\) and
Then we have the following two conclusions:
-
(1)
\(f_{\theta , \nu }(t)>0\) for all \(t\in (0, 1)\) if and only if \(\theta \geq 1/(6\nu )\);
-
(2)
\(f_{\theta , \nu }(t)<0\) for all \(t\in (0, 1)\) if and only if \(\theta \leq [(\sqrt{2}+\log (1+\sqrt{2}))/2 ]^{1/ \nu }-1\).
Proof
It follows from (2.9) that
where
Let \(\psi _{1}(t)=t\sqrt{1+t^{2}}-\sinh ^{-1}(t)\) and \(\psi _{2}(t)=(2 \nu -1)t^{2}[t\sqrt{1+t^{2}}-\sinh ^{-1}(t)]+4\nu t^{2}\sinh ^{-1}(t)\). Then
where \(\phi (t)\) is defined in Lemma 2.2.
Equation (2.14) and Lemma 2.2 imply that \(\psi '_{1}(t)/\psi '_{2}(t)\) is strictly decreasing on \((0, 1)\). Therefore, the conclusion that \(f_{\nu }(t)\) is strictly decreasing on \((0, 1)\) follows from Lemma 2.1 and (2.13), together with the monotonicity of \(\psi '_{1}(t)/\psi '_{2}(t)\) on the interval \((0, 1)\). Moreover, making use of L’Hôpital’s rule, we have that
We divide the proof into three cases.
Case 1. \(\theta \geq 1/(6\nu )\). Then (2.12) and (2.15), together with the monotonicity of \(f_{\nu }(t)\) on the interval \((0, 1)\), lead to the conclusion that \(f_{\theta , \nu }(t)\) is strictly increasing on \((0, 1)\). Therefore, \(f_{\theta , \nu }(t)>0\) for all \(t\in (0, 1)\) follows from (2.10) and the monotonicity of \(f_{\theta , \nu }(t)\) on the interval \((0, 1)\).
Case 2. \(\theta \leq \theta _{0}\). Then from (2.12) and (2.16), together with the monotonicity of \(f_{\nu }(t)\) on the interval \((0, 1)\), we clearly see that \(f_{\theta , \nu }(t)\) is strictly decreasing on \((0, 1)\). Therefore, \(f_{\theta , \nu }(t)<0\) for all \(t\in (0, 1)\) follows from (2.10) and the monotonicity of \(f_{\theta , \nu }(t)\) on the interval \((0, 1)\).
Case 3. \(\theta _{0}<\theta <1/(6\nu )\). Then from (2.12), (2.15), (2.16), and the monotonicity of \(f_{\nu }(t)\) on the interval \((0, 1)\), we clearly see that there exists \(t_{0}\in (0, 1)\) such that \(f_{\theta , \nu }(t)\) is strictly decreasing on \((0, t_{0})\) and strictly increasing on \((t_{0}, 1)\).
We divide the proof into two subcases.
Subcase 3.1. \([(\sqrt{2}+\log (1+\sqrt{2}))/2 ] ^{1/\nu }-1<\theta <1/(6\nu )\). Then (2.11) leads to
Therefore, there exists \(t^{\ast }\in (t_{0}, 1)\) such that \(f_{\theta , \nu }(t)<0\) for \(t\in (0, t^{\ast })\) and \(f_{\theta , \nu }(t)>0\) for \(t\in (t^{\ast }, 1)\) follows from (2.10) and (2.17), together with the piecewise monotonicity of \(f_{\theta , \nu }(t)\) on the interval \((0, 1)\).
Subcase 3.2. \(\theta _{0}<\theta \leq [(\sqrt{2}+\log (1+ \sqrt{2}))/2 ]^{1/\nu }-1\). Then (2.11) leads to
Therefore, \(f_{\theta , \nu }(t)<0\) for all \(t\in (0, 1)\) follows from (2.10) and (2.18), together with the piecewise monotonicity of \(f_{\theta , \nu }(t)\) on the interval \((0, 1)\). □
Lemma 2.5
Let \(\vartheta \in [0, 1]\), \(\nu \in [1/2, \infty )\), \(t\in (0, 1)\) and
Then the following statements are true:
-
(1)
\(g_{\vartheta , \nu }(t)>0\) for all \(t\in (0, 1)\) if and only if \(\vartheta \geq 1/(3\nu )\);
-
(2)
\(g_{\vartheta , \nu }(t)<0\) for all \(t\in (0, 1)\) if and only if \(\vartheta \leq [(\pi +2)/4 ]^{1/\nu }-1\).
Proof
It follows from (2.19) that
where
Let \(\omega _{1}(t)=[t-(1-t^{2})\arctan (t)]/t^{2}\) and \(\omega _{2}(t)=[(2 \nu -1)t^{2}+2\nu +1]\arctan (t)+(2\nu -1)t\). Then elaborate computations lead to
where \(\varphi (t)\) is defined in Lemma 2.3.
From Lemma 2.3 and (2.24) we know that \(\omega '_{1}(t)/\omega '_{2}(t)\) is strictly decreasing on \((0, 1)\). Therefore, the conclusion that \(g_{\nu }(t)\) is strictly decreasing on \((0, 1)\) follows from Lemma 2.1 and (2.23), together with the monotonicity of \(\omega '_{1}(t)/\omega '_{2}(t)\) on the interval \((0, 1)\). Moreover, making use of L’Hôpital’s rule, we have that
We divide the proof into three cases.
Case 1. \(\vartheta \geq 1/(3\nu )\). Then (2.22) and (2.25), together with the monotonicity of \(g_{\nu }(t)\) on the interval \((0, 1)\), lead to the conclusion that \(g_{\vartheta , \nu }(t)\) is strictly increasing on \((0, 1)\). Therefore, \(g_{\vartheta , \nu }(t)>0\) for all \(t\in (0, 1)\) follows from (2.20) and the monotonicity of \(g_{\vartheta , \nu }(t)\) on the interval \((0, 1)\).
Case 2. \(\vartheta \leq 1/[(\pi +2)\nu -1]\). Then from (2.22) and (2.26), together with the monotonicity of \(g_{\nu }(t)\) on the interval \((0, 1)\), we clearly see that \(g_{\vartheta , \nu }(t)\) is strictly decreasing on \((0, 1)\). Therefore, \(g_{\vartheta , \nu }(t)<0\) for all \(t\in (0, 1)\) follows from (2.20) and the monotonicity of \(g_{\vartheta , \nu }(t)\) on the interval \((0, 1)\).
Case 3. \(1/[(\pi +2)\nu -1]<\vartheta <1/(6\nu )\). Then it follows from (2.22), (2.25), (2.26), and the monotonicity of \(g_{\nu }(t)\) on the interval \((0, 1)\) that there exists \(\rho _{0} \in (0, 1)\) such that \(g_{\vartheta , \nu }(t)\) is strictly decreasing on \((0, \rho _{0})\) and strictly increasing on \((\rho _{0}, 1)\).
We divide the proof into two subcases.
Subcase 3.1. \([(\pi +2)/4 ]^{1/\nu }-1<\vartheta <1/(6 \nu )\). Then (2.21) leads to
Therefore, there exists \(\rho ^{\ast }\in (\rho _{0}, 1)\) such that \(g_{\vartheta , \nu }(t)<0\) for \(t\in (0, \rho ^{\ast })\) and \(g_{\vartheta , \nu }(t)>0\) for \(t\in (\rho ^{\ast }, 1)\) follows from (2.20) and (2.27), together with the piecewise of \(g_{\vartheta , \nu }(t)\) on the interval \((0, 1)\).
Subcase 3.2. \(1/[(\pi +2)\nu -1]<\vartheta \leq [(\pi +2)/4 ] ^{1/\nu }-1\). Then (2.21) gives
Therefore, \(g_{\vartheta , \nu }(t)<0\) for all \(t\in (0, 1)\) follows from (2.20) and (2.28), together with the piecewise of \(g_{\vartheta , \nu }(t)\) on the interval \((0, 1)\). □
3 Main results
Theorem 3.1
Let \(\lambda _{1}, \mu _{1}\in [1/2, 1]\) and \(\nu \in [1/2, \infty )\). Then the double inequality
holds for all \(x, y>0\) with \(x\neq y\) if and only if \(\lambda _{1} \leq 1/2+\sqrt{ [ (\sqrt{2}+\log (1+\sqrt{2}) )/2 ] ^{1/\nu }-1}/2\) and \(\mu _{1}\geq 1/2+\sqrt{6\nu }/(12\nu )\).
Proof
Since both \(W_{\theta , \nu }(x, y)\) and \(\mathcal{R}_{QA}(x, y)\) are symmetric and homogenous of degree 1, without loss of generality, we assume that \(x>y>0\). Let \(t=(x-y)/(x+y)\in (0, 1)\) and \(\theta \in [1/2, 1]\). Then from (1.1), (1.2), and (1.7) we get
It follows from (3.2) and (3.3) that
Therefore, Theorem 3.1 follows easily from Lemma 2.4 and (3.4). □
Theorem 3.2
Let \(\lambda _{2}, \mu _{2}\in [1/2, 1]\) and \(\nu \in [1/2, \infty )\). Then the double inequality
holds for all \(x, y>0\) with \(x\neq y\) if and only if \(\lambda _{2} \leq 1/2+\sqrt{ [(\pi +2)/4 ]^{1/\nu }-1}/2\) and \(\mu _{2}\geq 1/2+\sqrt{3\nu }/(6\nu )\).
Proof
Since both \(W_{\vartheta , \nu }(x, y)\) and \(\mathcal{R}_{AQ}(x, y)\) are symmetric and homogenous of degree 1, without loss of generality, we assume that \(x>y>0\). Let \(t=(x-y)/(x+y)\in (0, 1)\) and \(\vartheta \in [1/2, 1]\). Then it follows from (1.1), (1.3), and (1.7) that
Therefore, Theorem 3.2 follows easily from Lemma 2.5 and (3.8). □
Remark 3.3
Let \(\nu =1/2\). Then from (1.8) we clearly see that Theorems 3.1 and 3.2 become (1.5) and (1.6), respectively.
Let \(\nu =1\). Then from (1.9) and Theorems 3.1 and 3.2 we get Corollary 3.4 immediately.
Corollary 3.4
Let \(\lambda _{1}, \mu _{1}, \lambda _{2}, \mu _{2}\in [1/2, 1]\). Then the double inequalities
hold for all \(x, y>0\) with \(x\neq y\) if and only if \(\lambda _{1} \leq 1/2+ \sqrt{ [ (\sqrt{2}+\log (1+\sqrt{2}) )/2 ]-1}/2=0.6922 \ldots \) , \(\mu _{1}\geq 1/2+\sqrt{6}/12=0.7041\ldots \) , \(\lambda _{2} \leq 1/2+\sqrt{ [(\pi +2)/4 ]-1}/2=0.7671\ldots \) and \(\mu _{2}\geq 1/2+\sqrt{3}/6=0.7886\ldots \) .
Let \(u\in (0, 1)\), \(x=1+u\), \(y=1-u\), \(\lambda _{1}=1/2+\sqrt{ [ (\sqrt{2}+ \log (1+\sqrt{2}) )/2 ]^{1/\nu }-1}/2\), \(\mu _{1}=1/2+\sqrt{6 \nu }/(12\nu )\), \(\lambda _{2}=1/2+\sqrt{ [(\pi +2)/4 ] ^{1/\nu }-1}/2\) and \(\mu _{2}=1/2+\sqrt{3\nu }/(6\nu )\). Then (1.2), (1.3), and Theorems 3.1 and 3.2 lead to Corollary 3.5.
Corollary 3.5
The double inequalities
hold for all \(u\in (0, 1)\) and \(\nu \in [1/2, \infty )\).
4 Results and discussion
In the article, we give the sharp bounds for the Neuman means
and
in terms of the two-parameter contraharmonic and arithmetic mean
and find new bounds for the functions \(\sinh (u)/u\) and \(\arctan (u)/u\) on the interval \((0, 1)\).
5 Conclusion
In the article, we prove that the double inequalities
hold for all \(x, y>0\) with \(x\neq y\) if and only if \(\lambda _{1} \leq 1/2+\sqrt{ [ (\sqrt{2}+\log (1+\sqrt{2}) )/2 ] ^{1/\nu }-1}/2\), \(\mu _{1}\geq 1/2+\sqrt{6\nu }/(12\nu )\), \(\lambda _{2}\leq 1/2+\sqrt{ [(\pi +2)/4 ]^{1/\nu }-1}/2\) and \(\mu _{2}\geq 1/2+\sqrt{3\nu }/(6\nu )\) if \(\lambda _{1}, \mu _{1}, \lambda _{2}, \mu _{2}\in [1/2, 1]\) and \(\nu \in [1/2, \infty )\). Our results are a natural generalization of some previously known results, and our approach may lead to many follow-up studies.
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The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.
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The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485) and the Natural Science Foundation of Huzhou City (Grant No. 2018YZ07).
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Qian, WM., He, ZY., Zhang, HW. et al. Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean. J Inequal Appl 2019, 168 (2019). https://doi.org/10.1186/s13660-019-2124-5
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DOI: https://doi.org/10.1186/s13660-019-2124-5