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Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean

Journal of Inequalities and Applications20192019:168

https://doi.org/10.1186/s13660-019-2124-5

  • Received: 16 April 2019
  • Accepted: 4 June 2019
  • Published:

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
$$\begin{aligned}& C^{\nu }\bigl[\lambda _{1}x+(1-\lambda _{1})y, \lambda _{1}y+(1-\lambda _{1})x\bigr]A ^{1-\nu }(x, y) \\& \quad < \mathcal{R}_{QA}(x, y)< C^{\nu }\bigl[\mu _{1}x+(1-\mu _{1})y, \mu _{1}y+(1-\mu _{1})x\bigr]A^{1-\nu }(x, y), \\& C^{\nu }\bigl[\lambda _{2}x+(1-\lambda _{2})y, \lambda _{2}y+(1-\lambda _{2})x\bigr]A ^{1-\nu }(x, y) \\& \quad < \mathcal{R}_{AQ}(x, y)< C^{\nu }\bigl[\mu _{2}x+(1-\mu _{2})y, \mu _{2}y+(1-\mu _{2})x\bigr]A^{1-\nu }(x, y) \end{aligned}$$
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.

Keywords

  • Arithmetic mean
  • Quadratic mean
  • Contraharmonic mean
  • Schwab–Borchardt mean
  • Neuman mean
  • Two-parameter contraharmonic and arithmetic mean

MSC

  • 26E60

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
$$ \begin{aligned} &A(x, y)=\frac{x+y}{2},\qquad Q(x, y)=\sqrt{\frac{x^{2}+y^{2}}{2}}, \qquad C(x, y)= \frac{x^{2}+y^{2}}{x+y}, \\ &\operatorname{SB}(x, y)= \textstyle\begin{cases} \frac{\sqrt{y^{2}-x^{2}}}{\arccos {(x/y)}}, & x< y, \\ x, & x=y, \\ \frac{\sqrt{x^{2}-y^{2}}}{\cosh ^{-1}{(x/y)}}, & x>y, \end{cases}\displaystyle \end{aligned} $$
(1.1)
respectively, where \(\cosh ^{-1}(\sigma )=\log (\sigma +\sqrt{\sigma ^{2}-1})\) is the inverse hyperbolic cosine function.
The Gaussian arithmetic–geometric mean \(\operatorname{AGM}(x, y)\) [57] 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
$$ x_{0}=x,\qquad y_{0}=y, \qquad x_{n+1}=\frac{x_{n}+y_{n}}{2},\qquad y _{n+1}=\sqrt{x_{n}y_{n}}. $$
It is well known that the bivariate means have wide applications in mathematics, physics, engineering, and other natural sciences [855], many special functions can be expressed using bivariate means, for example, the complete elliptic integral
$$ \mathcal{K}(r)= \int _{0}^{\pi /2} \frac{dt}{\sqrt{1-r^{2}\sin ^{2}(t)}} \quad (0< r< 1) $$
of the first kind [5661] 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
$$ \mathcal{E}(r)= \int _{0}^{\pi /2}\sqrt{1-r^{2}\sin ^{2}(t)}\,dt $$
of the second kind [6470] can be given in terms of the Toader mean [7174]
$$ T(a,b)=\frac{2}{\pi } \int _{0}^{\pi /2}\sqrt{a^{2}\cos ^{2}(t)+b^{2} \sin ^{2}(t)}\,dt. $$
Indeed, we have
$$\begin{aligned}& \mathcal{K}(r)=\frac{\pi }{2}\frac{1}{\operatorname{AGM}(1, \sqrt{1-r^{2}})},\qquad \mu (r)=\frac{\pi }{2} \frac{\operatorname{AGM}(1, \sqrt{1-r^{2}})}{\operatorname{AGM}(1, r)}, \\& L(x, y)=2\pi T(x, y),\qquad \mathcal{E}(r)=\frac{\pi }{2}T \bigl(1, \sqrt{1-r ^{2}} \bigr). \end{aligned}$$
Recently, the inequalities for bivariate means have attracted the attention of many mathematicians. Neuman [75] introduced the Neuman means
$$\begin{aligned}& \mathcal{R}_{QA}(x, y)=\frac{1}{2} \biggl[Q(x, y)+ \frac{A^{2}(x, y)}{\operatorname{SB}(Q(x,y), A(x,y))} \biggr], \\& \mathcal{R}_{AQ}(x, y)=\frac{1}{2} \biggl[A(x, y)+ \frac{Q^{2}(x, y)}{\operatorname{SB}(A(x,y), Q(x,y))} \biggr] \end{aligned}$$
and provided the formulas
$$\begin{aligned}& \mathcal{R}_{QA}(x, y)=\frac{1}{2}A(x, y) \biggl[ \sqrt{1+u^{2}}+\frac{ \sinh ^{-1}(u)}{u} \biggr], \end{aligned}$$
(1.2)
$$\begin{aligned}& \mathcal{R}_{AQ}(x, y)=\frac{1}{2}A(x, y) \biggl[1+ \frac{(1+u^{2}) \arctan (u)}{u} \biggr] \end{aligned}$$
(1.3)
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
$$ A(x, y)< \mathcal{R}_{QA}(x, y)< \mathcal{R}_{AQ}(x, y)< Q(x, y) $$
(1.4)
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
$$\begin{aligned}& Q\bigl[\alpha _{1}x+(1-\alpha _{1})y, \alpha _{1}y+(1-\alpha _{1})x\bigr] \\& \quad < \mathcal{R}_{QA}(x, y)< Q\bigl[\beta _{1}x+(1-\beta _{1})y, \beta _{1}y+(1-\beta _{1})x\bigr], \end{aligned}$$
(1.5)
$$\begin{aligned}& Q\bigl[\alpha _{2}x+(1-\alpha _{2})y, \alpha _{2}y+(1-\alpha _{2})x\bigr] \\& \quad < \mathcal{R}_{AQ}(x, y)< Q\bigl[\beta _{2}x+(1-\beta _{2})y, \beta _{2}y+(1-\beta _{2})x\bigr] \end{aligned}$$
(1.6)
hold for \(x, y>0\) with \(x\neq y\).
In [77], Yang et al. proved that the double inequalities
$$\begin{aligned}& \alpha \biggl[\frac{C(x,y)}{3}+\frac{2A(x,y)}{3} \biggr]+(1-\alpha )C ^{1/3}(x, y)A^{2/3}(x, y) \\& \quad < \mathcal{R}_{AQ}(x, y)< \beta \biggl[\frac{C(x,y)}{3}+\frac{2A(x,y)}{3} \biggr]+(1-\beta )C ^{1/3}(x, y)A^{2/3}(x, y), \\& \lambda \biggl[\frac{C(x,y)}{6}+\frac{5A(x,y)}{6} \biggr]+(1-\lambda )C^{1/6}(x, y)A^{5/6}(x, y) \\& \quad < \mathcal{R}_{QA}(x, y)< \mu \biggl[\frac{C(x,y)}{6}+\frac{5A(x,y)}{6} \biggr]+(1-\mu )C^{1/6}(x, y)A^{5/6}(x, y) \end{aligned}$$
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:
$$ W_{\lambda , \nu }(x, y)=C^{\nu }\bigl[\lambda x+(1-\lambda )y, \lambda y+(1- \lambda )x\bigr]A^{1-\nu }(x, y), $$
(1.7)
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
$$\begin{aligned}& W_{\lambda , 1/2}(x, y)=Q\bigl[\lambda x+(1-\lambda )y, \lambda y+(1- \lambda )x \bigr], \end{aligned}$$
(1.8)
$$\begin{aligned}& W_{\lambda , 1}(x, y)=C\bigl[\lambda x+(1-\lambda )y, \lambda y+(1-\lambda )x \bigr], \end{aligned}$$
(1.9)
$$\begin{aligned}& W_{1/2, \nu }(x, y)=A(x, y), \\& W_{1, \nu }(x, y)=C^{\nu }(x, y)A^{1-\nu }(x, y)=A(x, y) \biggl[\frac{Q(x,y)}{A(x,y)} \biggr] ^{2\nu }\geq Q(x, y), \\& W_{1/2, \nu }(x, y)< \mathcal{R}_{QA}(x,y)< \mathcal{R}_{AQ}(x,y)< W_{1, \nu }(x, y). \end{aligned}$$
(1.10)
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
$$\begin{aligned}& W_{\lambda _{1}, \nu }(x,y)< \mathcal{R}_{QA}(x, y)< W_{\mu _{1}, \nu }(x,y), \\& W_{\lambda _{2}, \nu }(x,y)< \mathcal{R}_{AQ}(x, y)< W_{\mu _{2}, \nu }(x,y) \end{aligned}$$
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
$$ \frac{\varGamma (\tau )-\varGamma (\alpha )}{\varPsi (\tau )-\varPsi (\alpha )}, \qquad \frac{\varGamma (\tau )-\varGamma (\beta )}{\varPsi (\tau )-\varPsi (\beta )} $$
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
$$ \phi (t)=\frac{\sqrt{1+t^{2}}\sinh ^{-1}(t)}{t} $$
is strictly increasing from \((0, 1)\) onto \((1, \sqrt{2}\log (1+ \sqrt{2}) )\).

Proof

Differentiating \(\phi (t)\) gives
$$ \phi '(t)=\frac{\phi _{1}(t)}{t\sqrt{1+t^{2}}}, $$
(2.1)
where
$$ \phi _{1}(t)=t\sqrt{1+t^{2}}-\sinh ^{-1}(t). $$
(2.2)
It follows from (2.2) that
$$\begin{aligned}& \phi _{1}\bigl(0^{+}\bigr)=0, \end{aligned}$$
(2.3)
$$\begin{aligned}& \phi _{1}'(t)=\frac{2t^{2}}{\sqrt{1+t^{2}}}>0 \end{aligned}$$
(2.4)
for all \(t\in (0, 1)\).
Note that
$$ \phi \bigl(0^{+}\bigr)=1,\qquad \phi \bigl(1^{-}\bigr)=\sqrt{2} \log (1+\sqrt{2}). $$
(2.5)

Therefore, Lemma 2.2 follows from (2.1) and (2.3)–(2.5). □

Lemma 2.3

The function
$$ \varphi (t)=\frac{t^{3}}{(1+t^{2})\arctan (t)-t} $$
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
$$\begin{aligned}& \varphi _{1}\bigl(0^{+}\bigr)=\varphi _{2} \bigl(0^{+}\bigr),\qquad \varphi (t)=\frac{\varphi _{1}(t)}{\varphi _{2}(t)}, \end{aligned}$$
(2.6)
$$\begin{aligned}& \frac{\varphi '_{1}(t)}{\varphi '_{2}(t)}=\frac{3t}{2\arctan (t)}. \end{aligned}$$
(2.7)

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
$$ \varphi \bigl(0^{+}\bigr)=\frac{3}{2},\qquad \varphi \bigl(1^{-}\bigr)=\frac{2}{\pi -2}. $$
(2.8)

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
$$ f_{\theta , \nu }(t)=\nu \log \bigl(1+\theta t^{2}\bigr)-\log \bigl[t \sqrt{1+t ^{2}}+\sinh ^{-1}(t) \bigr]+\log t+\log 2. $$
(2.9)
Then we have the following two conclusions:
  1. (1)

    \(f_{\theta , \nu }(t)>0\) for all \(t\in (0, 1)\) if and only if \(\theta \geq 1/(6\nu )\);

     
  2. (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
$$\begin{aligned}& f_{\theta , \nu }\bigl(0^{+}\bigr)=0, \end{aligned}$$
(2.10)
$$\begin{aligned}& f_{\theta , \nu }\bigl(1^{-}\bigr)=\nu \log (1+\theta )-\log \bigl[ \sqrt{2}+ \log (1+\sqrt{2}) \bigr]+\log 2, \end{aligned}$$
(2.11)
$$\begin{aligned}& f^{\prime }_{\theta , \nu }(t)=\frac{t [(2\nu -1)(t\sqrt{1+t ^{2}}-\sinh ^{-1}(t)) +4\nu \sinh ^{-1}(t) ]}{(1+\theta t^{2}) [t\sqrt{1+t^{2}}+\sinh ^{-1}(t) ]} \bigl[\theta -f_{ \nu }(t) \bigr], \end{aligned}$$
(2.12)
where
$$ f_{\nu }(t)=\frac{t\sqrt{1+t^{2}}-\sinh ^{-1}(t)}{(2\nu -1)t^{2}[t\sqrt{1+t ^{2}}-\sinh ^{-1}(t)]+4\nu t^{2}\sinh ^{-1}(t)}. $$
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
$$\begin{aligned}& \psi _{1}\bigl(0^{+}\bigr)=\psi _{2} \bigl(0^{+}\bigr)=0,\qquad f_{\nu }(t)=\frac{\psi _{1}(t)}{ \psi _{2}(t)}, \end{aligned}$$
(2.13)
$$\begin{aligned}& \frac{\psi '_{1}(t)}{\psi '_{2}(t)}=\frac{1}{(2\nu +1)\phi (t)+2(2 \nu -1)t^{2}+4\nu -1}, \end{aligned}$$
(2.14)
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
$$\begin{aligned}& f_{\nu }\bigl(0^{+}\bigr)=\frac{1}{6\nu }, \end{aligned}$$
(2.15)
$$\begin{aligned}& f_{\nu }\bigl(1^{-}\bigr)=\frac{\sqrt{2}-\log (1+\sqrt{2})}{(2\nu -1) \sqrt{2}+(2\nu +1)\log (1+\sqrt{2})}=:\theta _{0}. \end{aligned}$$
(2.16)

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
$$ f_{\theta , \nu }\bigl(1^{-}\bigr)>0. $$
(2.17)

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
$$ f_{\theta , \nu }\bigl(1^{-}\bigr)\leq 0. $$
(2.18)

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
$$ g_{\vartheta , \nu }(t)=\nu \log \bigl(1+\vartheta t^{2}\bigr)-\log \bigl[t+\bigl(1+t ^{2}\bigr)\arctan (t) \bigr]+\log (t)+\log 2. $$
(2.19)
Then the following statements are true:
  1. (1)

    \(g_{\vartheta , \nu }(t)>0\) for all \(t\in (0, 1)\) if and only if \(\vartheta \geq 1/(3\nu )\);

     
  2. (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
$$\begin{aligned}& g_{\vartheta , \nu }\bigl(0^{+}\bigr)=0, \end{aligned}$$
(2.20)
$$\begin{aligned}& g_{\vartheta , \nu }\bigl(1^{-}\bigr)=\nu \log (1+\vartheta )-\log \biggl(\frac{ \pi +2}{4} \biggr), \end{aligned}$$
(2.21)
$$\begin{aligned}& g^{\prime }_{\vartheta , \nu }(t)=\frac{t [((2\nu -1)t^{2}+2 \nu +1)\arctan (t)+(2\nu -1)t ]}{(1+\vartheta t^{2}) [t+(1+t ^{2})\arctan (t) ]}\bigl[\vartheta -g_{\nu }(t)\bigr], \end{aligned}$$
(2.22)
where
$$ g_{\nu }(t)=\frac{t-(1-t^{2})\arctan (t)}{t^{2} [((2\nu -1)t^{2}+2 \nu +1)\arctan (t)+(2\nu -1)t ]}. $$
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
$$\begin{aligned}& \omega _{1}\bigl(0^{+}\bigr)=\omega _{2} \bigl(0^{+}\bigr)=0,\qquad g_{\nu }(t)=\frac{\omega _{1}(t)}{\omega _{2}(t)}, \end{aligned}$$
(2.23)
$$\begin{aligned}& \frac{\omega '_{1}(t)}{\omega '_{2}(t)}=\frac{1}{2[(2\nu -1)t^{2}+ \nu ]\varphi (t)+(2\nu -1)t^{4}} , \end{aligned}$$
(2.24)
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
$$\begin{aligned}& g_{\nu }\bigl(0^{+}\bigr)=\frac{1}{3v}, \end{aligned}$$
(2.25)
$$\begin{aligned}& g_{\nu }\bigl(1^{-}\bigr)=\frac{1}{(\pi +2)\nu -1}. \end{aligned}$$
(2.26)

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
$$ g_{\vartheta , \nu }\bigl(1^{-}\bigr)>0. $$
(2.27)

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
$$ g_{\vartheta , \nu }\bigl(1^{-}\bigr)\leq 0. $$
(2.28)

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
$$ W_{\lambda _{1}, \nu }(x, y)< \mathcal{R}_{QA}(x, y)< W_{\mu _{1}, \nu }(x, y) $$
(3.1)
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
$$\begin{aligned}& \frac{W_{\theta , \nu }(x, y))}{A(x, y)}= \bigl[1+(2\theta -1)^{2}t ^{2} \bigr]^{\nu }, \end{aligned}$$
(3.2)
$$\begin{aligned}& \frac{\mathcal{R}_{QA}(x, y)}{A(x, y)}=\frac{1}{2} \biggl[\sqrt{1+t ^{2}}+ \frac{\sinh ^{-1}(t)}{t} \biggr]. \end{aligned}$$
(3.3)
It follows from (3.2) and (3.3) that
$$\begin{aligned} \log \biggl[\frac{W_{\theta , \nu }(x, y)}{\mathcal{R}_{QA}(x, y)} \biggr] =& \log \biggl[\frac{W_{\theta , \nu }(x, y)}{A(x, y)} \biggr]- \log \biggl[\frac{\mathcal{R}_{QA}(x, y)}{A(x, y)} \biggr] \\ =&\nu \log \bigl[1+(2\theta -1)^{2}t^{2} \bigr]-\log \bigl[t \sqrt{1+t ^{2}}+\sinh ^{-1}(t) \bigr] \\ &{}+\log (t)+\log 2. \end{aligned}$$
(3.4)

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
$$ W_{\lambda _{2}, \nu }(x, y)< \mathcal{R}_{AQ}(x, y)< W_{\mu _{2}, \nu }(x, y) $$
(3.5)
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
$$\begin{aligned}& \frac{W_{\vartheta , \nu }(x, y))}{A(x, y)}= \bigl[1+(2\vartheta -1)^{2}t ^{2} \bigr]^{\nu }, \end{aligned}$$
(3.6)
$$\begin{aligned}& \frac{\mathcal{R}_{AQ}(x,y)}{A(x, y)}=\frac{1}{2} \biggl[1+\frac{(1+t ^{2})\arctan (t)}{t} \biggr]. \end{aligned}$$
(3.7)
From (3.6) and (3.7) we have
$$\begin{aligned} \log \biggl[\frac{W_{\vartheta , \nu }(x, y))}{\mathcal{R}_{AQ}(x,y)} \biggr] =& \log \biggl[\frac{W_{\vartheta , \nu }(x, y)}{A(x, y)} \biggr]- \log \biggl[\frac{\mathcal{R}_{AQ}(x,y)}{A(x,y)} \biggr] \\ =&\nu \log \bigl[1+(2\vartheta -1)^{2} t^{2} \bigr]-\log \bigl[t+\bigl(1+t ^{2}\bigr)\arctan (t) \bigr] \\ &{}+\log (t)+\log 2. \end{aligned}$$
(3.8)

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
$$\begin{aligned}& C\bigl[\lambda _{1}x+(1-\lambda _{1})y, \lambda _{1}y+(1-\lambda _{1})x\bigr]< \mathcal{R}_{QA}(x, y)< C\bigl[\mu _{1}x+(1-\mu _{1})y, \mu _{1}y+(1- \mu _{1})x\bigr], \\& C\bigl[\lambda _{2}x+(1-\lambda _{2})y, \lambda _{2}y+(1-\lambda _{2})x\bigr]< \mathcal{R}_{AQ}(x, y)< C\bigl[\mu _{2}x+(1-\mu _{2})y, \mu _{2}y+(1-\mu _{2})x\bigr] \end{aligned}$$
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
$$\begin{aligned}& 2 \biggl[\bigl(1-u^{2}\bigr)+ \biggl(\frac{\sqrt{2}+\log (1+\sqrt{2})}{2} \biggr) ^{1/\nu }u^{2} \biggr]^{\nu }-\sqrt{1+u^{2}} \\& \quad < \frac{\sinh ^{-1}(u)}{u}< 2 \biggl(1+\frac{u^{2}}{6\nu } \biggr)^{ \nu }- \sqrt{1+u^{2}}, \\& \frac{2 [ (1-u^{2} )+(\frac{2+\pi }{4})^{1/\nu }u^{2} ] ^{\nu }-1}{1+u^{2}}< \frac{\arctan (u)}{u}< \frac{2 (1+\frac{1}{3 \nu }u^{2} )^{\nu }-1}{1+u^{2}} \end{aligned}$$
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
$$ \mathcal{R}_{QA}(x, y)=\frac{1}{2} \biggl[Q(x, y)+ \frac{A^{2}(x, y)}{\operatorname{SB}(Q(x,y), A(x,y))} \biggr] $$
and
$$ \mathcal{R}_{AQ}(x, y)=\frac{1}{2} \biggl[A(x, y)+ \frac{Q^{2}(x, y)}{\operatorname{SB}(A(x,y), Q(x,y))} \biggr] $$
in terms of the two-parameter contraharmonic and arithmetic mean
$$ W_{\lambda , \nu }(x, y)=C^{\nu }\bigl[\lambda x+(1-\lambda )y, \lambda y+(1- \lambda )x\bigr]A^{1-\nu }(x, y), $$
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
$$ W_{\lambda _{1}, \nu }(x, y)< \mathcal{R}_{QA}(x, y)< W_{\mu _{1}, \nu }(x, y),\qquad W_{\lambda _{2}, \nu }(x, y)< \mathcal{R}_{AQ}(x, y)< W_{\mu _{2}, \nu }(x, y) $$
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.

Declarations

Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

Availability of data and materials

Not applicable.

Funding

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).

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Continuing Education, Huzhou Vocational & Technical College, Huzhou, China
(2)
College of Mathematics and Econometrics, Hunan University, Changsha, China
(3)
School of Mathematics and Statistics, Changsha University of Science & Technology, Changsha, China
(4)
Department of Mathematics, Huzhou University, Huzhou, China

References

  1. Chu, H.-H., Qian, W.-M., Chu, Y.-M., Song, Y.-Q.: Optimal bounds for a Toader-type mean in terms of one-parameter quadratic and contraharmonic means. J. Nonlinear Sci. Appl. 9(5), 3424–3432 (2016) MathSciNetMATHGoogle Scholar
  2. Chu, Y.-M., Hou, S.-W.: Sharp bounds for Seiffert mean in terms of contraharmonic mean. Abstr. Appl. Anal. 2012, Article ID 425175 (2012) MathSciNetMATHGoogle Scholar
  3. Chu, Y.-M., Wang, M.-K., Ma, X.-Y.: Sharp bounds for Toader mean in terms of contraharmonic mean with applications. J. Math. Inequal. 7(2), 161–166 (2013) MathSciNetMATHGoogle Scholar
  4. Neuman, E., Sándor, J.: On the Schwab–Borchardt mean. Math. Pannon. 14(2), 253–266 (2003) MathSciNetMATHGoogle Scholar
  5. Chu, Y.-M., Wang, M.-K.: Inequalities between arithmetic–geometric, Gini, and Toader means. Abstr. Appl. Anal. 2012, Article ID 830585 (2012) MathSciNetMATHGoogle Scholar
  6. Qian, W.-M., Chu, Y.-M.: Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters. J. Inequal. Appl. 2017, Article ID 274 (2017) MathSciNetMATHGoogle Scholar
  7. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic–geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(2), 1714–1726 (2018) MathSciNetMATHGoogle Scholar
  8. Lin, L., Liu, Z.-Y.: An alternating projected gradient algorithm for nonnegative matrix factorization. Appl. Math. Comput. 217(24), 9997–10002 (2011) MathSciNetMATHGoogle Scholar
  9. Liu, Z.-Y., Santos, J., Ralha, R.: On computing complex square roots of real matrices. Appl. Math. Lett. 25(10), 1565–1568 (2012) MathSciNetMATHGoogle Scholar
  10. Jiang, Y.-J., Ma, J.-T.: Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. J. Comput. Appl. Math. 244, 115–124 (2013) MathSciNetMATHGoogle Scholar
  11. Li, X.-F., Tang, G.-J., Tang, B.-Q.: Stress field around a strike–slip fault in orthotropic elastic layers via a hypersingular integral equation. Comput. Math. Appl. 66(11), 2317–2326 (2013) MathSciNetMATHGoogle Scholar
  12. Qin, G.-X., Huang, C.-X., Xie, Y.-Q., Wen, F.-H.: Asymptotic behavior for third-order quasi-linear differential equations. Adv. Differ. Equ. 2013, Article ID 305 (2013) MathSciNetMATHGoogle Scholar
  13. Tang, W.-S., Sun, Y.-J.: Construction of Runge–Kutta type methods for solving ordinary differential equations. Appl. Math. Comput. 234, 179–191 (2014) MathSciNetMATHGoogle Scholar
  14. Huang, C.-X., Yang, Z.-C., Yi, T.-S., Zou, X.-F.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014) MathSciNetMATHGoogle Scholar
  15. Huang, C.-X., Guo, S., Liu, L.-Z.: Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón–Zygmund kernel. J. Math. Inequal. 8(3), 453–464 (2014) MathSciNetMATHGoogle Scholar
  16. Xie, D.-Q., Li, J.: A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent. Nonlinear Anal., Real World Appl. 21, 185–196 (2015) MathSciNetMATHGoogle Scholar
  17. Dai, Z.-F., Chen, X.-H., Wen, F.-H.: A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378–386 (2015) MathSciNetMATHGoogle Scholar
  18. Wang, W.-S.: High order stable Runge–Kutta methods for nonlinear generalized pantograph equations on the geometric mesh. Appl. Math. Model. 39(1), 270–283 (2015) MathSciNetMATHGoogle Scholar
  19. Tang, Y.-X., Jing, K.: Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations. Math. Methods Appl. Sci. 39(11), 2821–2839 (2016) MathSciNetMATHGoogle Scholar
  20. Li, J.-L., Sun, G.-Y., Zhang, R.-M.: The numerical solution of scattering by infinite rough interfaces based on the integral equation method. Comput. Math. Appl. 71(7), 1491–1502 (2016) MathSciNetGoogle Scholar
  21. Dai, Z.-F.: Comments on a new class of nonlinear conjugate gradient coefficients with global convergence properties. Appl. Math. Comput. 276, 297–300 (2016) MathSciNetMATHGoogle Scholar
  22. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017) MathSciNetMATHGoogle Scholar
  23. Adil Khan, M., Chu, Y.-M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414–1430 (2017) MathSciNetMATHGoogle Scholar
  24. Li, J., Liu, F.-W., Feng, L.-B., Turner, I.W.: A novel finite volume method for the Riesz space distributed-order advection–diffusion equation. Appl. Math. Model. 46, 536–553 (2017) MathSciNetGoogle Scholar
  25. Liu, Z.-Y., Qin, X.-R., Wu, N.-C., Zhang, Y.-L.: The shifted classical circulant and skew circulant splitting iterative methods for Toeplitz matrices. Can. Math. Bull. 60(4), 807–815 (2017) MathSciNetMATHGoogle Scholar
  26. Duan, L., Huang, L.-H., Guo, Z.-Y., Fang, X.-W.: Periodic attractor for reaction–diffusion higher-order Hopfield neural networks with time-varying delays. Comput. Math. Appl. 73(2), 233–245 (2017) MathSciNetMATHGoogle Scholar
  27. Yang, C., Huang, L.-H.: New criteria on exponential synchronization and existence of periodic solutions of complex BAM networks with delays. J. Nonlinear Sci. Appl. 10(10), 5464–5482 (2017) MathSciNetMATHGoogle Scholar
  28. Duan, L., Huang, C.-X.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017) MathSciNetMATHGoogle Scholar
  29. Huang, C.-X., Liu, L.-Z.: Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest. Math. 40(3), 295–312 (2017) MathSciNetGoogle Scholar
  30. Xi, H.-Y., Huang, L.-H., Qiao, Y.-C., Li, H.-Y., Huang, C.-X.: Permanence and partial extinction in a delayed three-species food chain model with stage structure and time-varying coefficients. J. Nonlinear Sci. Appl. 10(12), 6177–6191 (2017) MathSciNetMATHGoogle Scholar
  31. Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, Article ID 118 (2018) MathSciNetGoogle Scholar
  32. Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Spaces 2018, Article ID 6928130 (2018) MathSciNetMATHGoogle Scholar
  33. Adil Khan, M., Khurshid, Y., Du, T.-S., Chu, Y.-M.: Generalization of Hermite–Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018) MathSciNetMATHGoogle Scholar
  34. Zhao, T.-H., Wang, M.-K., Zhang, W., Chu, Y.-M.: Quadratic transformation inequalities for Gaussian hypergeometric function. J. Inequal. Appl. 2018, Article ID 251 (2018) MathSciNetGoogle Scholar
  35. Tang, W.-S., Zhang, J.-J.: Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods. Appl. Math. Comput. 233, 204–219 (2018) Google Scholar
  36. Liu, Z.-Y., Wu, N.-C., Qin, X.-R., Zhang, Y.-L.: Trigonometric transform splitting methods for real symmetric Toeplitz systems. Comput. Math. Appl. 75(8), 2782–2794 (2018) MathSciNetGoogle Scholar
  37. Zhu, K.-X., Xie, Y.-Q., Zhou, F.: Pullback attractors for a damped semilinear wave equation with delays. Acta Math. Sin. 34(7), 1131–1150 (2018) MathSciNetMATHGoogle Scholar
  38. Zhang, Y.: On products of consecutive arithmetic progressions II. Acta Math. Hung. 156(1), 240–254 (2018) MathSciNetMATHGoogle Scholar
  39. Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146(11), 4667–4682 (2018) MathSciNetMATHGoogle Scholar
  40. Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018) MathSciNetGoogle Scholar
  41. Huang, C.-X., Qiao, Y.-C., Huang, L.-H., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, Article ID 186 (2018) MathSciNetMATHGoogle Scholar
  42. Duan, L., Fang, X.-W., Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(5), 1954–1965 (2018) MathSciNetMATHGoogle Scholar
  43. Tan, Y.-X., Huang, C.-X., Sun, B., Wang, T.: Dynamics of a class of delayed reaction–diffusion system with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018) MathSciNetMATHGoogle Scholar
  44. Qiu, S.-L., Ma, X.-Y., Chu, Y.-M.: Sharp Landen transformation inequalities for hypergeometric functions, with applications. J. Math. Anal. Appl. 474(2), 1306–1337 (2019) MathSciNetMATHGoogle Scholar
  45. Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601–617 (2019) MathSciNetGoogle Scholar
  46. Li, J., Ying, J.-Y., Xie, D.-X.: On the analysis and applications of an ion size-modified Poisson–Boltzmann equation. Nonlinear Anal., Real World Appl. 47, 188–203 (2019) MathSciNetMATHGoogle Scholar
  47. Li, Y., Li, J., Wen, P.H.: Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials. Appl. Math. Model. 68, 306–326 (2019) MathSciNetGoogle Scholar
  48. Jiang, Y.-J., Xu, X.-J.: A monotone finite volume methods for time fractional Fokker–Planck equations. Sci. China Math. 62(4), 783–794 (2019) MathSciNetGoogle Scholar
  49. Peng, J., Zhang, Y.: Heron triangles with figurate numbers sides. Acta Math. Hung. 157(2), 478–488 (2019) MathSciNetMATHGoogle Scholar
  50. Wang, J.-F., Chen, X.-Y., Huang, L.-H.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019) MathSciNetMATHGoogle Scholar
  51. Wang, J.-F., Huang, C.-X., Huang, L.-H.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019) MathSciNetGoogle Scholar
  52. Zhao, T.-H., Zhou, B.-C., Wang, M.-K., Chu, Y.-M.: On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, Article ID 42 (2019) MathSciNetGoogle Scholar
  53. Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Conformable integral inequalities of the Hermite–Hadamard type in terms of GG- and GA-convexities. J. Funct. Spaces 2019, Article ID 6926107 (2019) MathSciNetMATHGoogle Scholar
  54. Khurshid, Y., Adil Khan, M., Chu, Y.-M., Khan, Z.A.: Hermite–Hadamard–Fejér inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210 (2019) MATHGoogle Scholar
  55. Zaheer Ullah, S., Adil Khan, M., Khan, Z.A., Chu, Y.-M.: Integral majorization type inequalities for the function in the sense of strong convexity. J. Funct. Spaces 2019, Article ID 9487823 (2019) MathSciNetGoogle Scholar
  56. Wang, M.-K., Chu, Y.-M., Qiu, Y.-F., Qiu, S.-L.: An optimal power mean inequality for the complete elliptic integrals. Appl. Math. Lett. 24(6), 887–890 (2011) MathSciNetMATHGoogle Scholar
  57. Chu, Y.-M., Wang, M.-K., Qiu, Y.-F.: On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function. Abstr. Appl. Anal. 2011, Article ID 697547 (2011) MathSciNetMATHGoogle Scholar
  58. Wang, M.-K., Chu, Y.-M., Qiu, S.-L., Jiang, Y.-P.: Convexity of the complete elliptic integrals of the first kind with respect to Hölder means. J. Math. Anal. Appl. 388(2), 1141–1146 (2012) MathSciNetMATHGoogle Scholar
  59. Chu, Y.-M., Qiu, Y.-F., Wang, M.-K.: Hölder mean inequalities for the complete elliptic integrals. Integral Transforms Spec. Funct. 23(7), 521–527 (2012) MathSciNetMATHGoogle Scholar
  60. Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661–1667 (2014) MathSciNetMATHGoogle Scholar
  61. Yang, Z.-H., Qian, W.-M., Chu, Y.-M.: Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math. Inequal. Appl. 21(4), 1185–1199 (2018) MathSciNetMATHGoogle Scholar
  62. Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality for the Grötzsch ring function. Math. Inequal. Appl. 14(4), 833–837 (2011) MathSciNetMATHGoogle Scholar
  63. Qiu, S.-L., Qiu, Y.-F., Wang, M.-K., Chu, Y.-M.: Hölder mean inequalities for the generalized Grötzsch ring and Hersch–Pfluger distortion functions. Math. Inequal. Appl. 15(1), 237–245 (2012) MathSciNetMATHGoogle Scholar
  64. Wang, M.-K., Qiu, S.-L., Chu, Y.-M., Jiang, Y.-P.: Generalized Hersch–Pfluger distortion function and complete elliptic integrals. J. Math. Anal. Appl. 385(1), 221–229 (2012) MathSciNetMATHGoogle Scholar
  65. Chu, Y.-M., Wang, M.-K., Qiu, S.-L., Jiang, Y.-P.: Bounds for complete elliptic integrals of the second kind with applications. Comput. Math. Appl. 63(7), 1177–1184 (2012) MathSciNetMATHGoogle Scholar
  66. Chu, Y.-M., Wang, M.-K., Jiang, Y.-P., Qiu, S.-L.: Concavity of the complete elliptic integrals of the second kind with respect to Hölder means. J. Math. Anal. Appl. 395(2), 637–642 (2012) MathSciNetMATHGoogle Scholar
  67. Wang, M.-K., Chu, Y.-M.: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119–126 (2013) MathSciNetMATHGoogle Scholar
  68. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, Article ID 106 (2017) MathSciNetMATHGoogle Scholar
  69. Huang, T.-R., Tan, S.-Y., Ma, X.-Y., Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals. J. Inequal. Appl. 2018, Article ID 239 (2018) MathSciNetGoogle Scholar
  70. Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019) MathSciNetGoogle Scholar
  71. Chu, Y.-M., Wang, M.-K., Qiu, S.-L., Qiu, Y.-F.: Sharp generalized Seiffert mean bounds for Toader mean. Abstr. Appl. Anal. 2011, Article ID 605259 (2011) MathSciNetMATHGoogle Scholar
  72. Chu, Y.-M., Wang, M.-K., Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012) MathSciNetMATHGoogle Scholar
  73. Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3–4), 223–229 (2012) MathSciNetMATHGoogle Scholar
  74. Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Spaces 2019, Article ID 6082413 (2019) MathSciNetGoogle Scholar
  75. Neuman, E.: On a new bivariate mean. Aequ. Math. 88(3), 277–289 (2014) MathSciNetMATHGoogle Scholar
  76. Zhang, Y., Chu, Y.-M., Jiang, Y.-L.: Sharp geometric means bounds for Neuman means. Abstr. Appl. Anal. 2014, Article ID 949815 (2014) MathSciNetMATHGoogle Scholar
  77. Yang, Y.-Y., Qian, W.-M., Chu, Y.-M.: Refinements of bounds for Neuman means with applications. J. Nonlinear Sci. Appl. 9(4), 1529–1540 (2016) MathSciNetMATHGoogle Scholar
  78. Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997) MATHGoogle Scholar

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