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Optimal convex combination bounds of geometric and Neuman means for Toader-type mean

Abstract

In this paper, we prove that the double inequalities

$$\begin{aligned}& {\alpha }N_{QA}(a,b)+({1-\alpha })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\beta }N_{QA}(a,b)+({1-\beta })G(a,b), \\& {\lambda }N_{AQ}(a,b)+({1-\lambda })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\mu }N_{AQ}(a,b)+({1-\mu })G(a,b) \end{aligned}$$

hold for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha \leq 3/8\), \(\beta \geq 4/ [\pi ( \log (1+\sqrt{2})+\sqrt{2}) ]=0.5546 \cdots \) , \(\lambda \leq 3/10\) and \(\mu \geq 8/ [\pi (\pi +2) ]=0.4952 \cdots \) , where \(TD(a,b)\), \(G(a,b)\), \(A(a,b)\) and \(N_{QA}(a,b)\), \(N_{AQ}(a,b)\) are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.

Introduction

For \(x,y,z \geq 0\) with \(xy+xz+yz\neq 0\) and \(r\in (0,1)\), the symmetric integrals \(R_{F}(x,y,z)\) and \(R_{G}(x,y,z)\) [1] of the first and second kinds, and the complete elliptic integrals \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) of the first and second kinds are defined by

$$\begin{aligned}& R_{F}(x,y,z) =\frac{1}{2} \int_{0}^{\infty } \bigl[(t+x) (t+y) (t+z) \bigr]^{-1/2}\,dt, \\& R_{G}(x,y,z) =\frac{1}{4} \int_{0}^{\infty } \bigl[(t+x) (t+y) (t+z) \bigr]^{-1/2} \biggl(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z} \biggr)t\,dt, \\& \mathcal{K}(r)= \int_{0}^{\pi /2} \bigl[1-r^{2} \sin^{2}(t) \bigr]^{-1/2}\,dt, \qquad \mathcal{E}(r)= \int_{0}^{\pi /2} \bigl[1-r^{2} \sin^{2}(t) \bigr]^{1/2}\,dt, \end{aligned}$$

respectively.

The well-known identities

$$\mathcal{K}(r)=R_{F}\bigl(0,1-r^{2},1\bigr),\qquad \mathcal{E}(r)=2 R_{G}\bigl(0,1-r^{2},1\bigr) $$

were established by Carlson in [1].

Let \(a,b>0\) with \(a\neq b\). Then the Toader mean \(\mathit{TD}(a,b)\) [2] and the Schwab-Borchardt mean \(SB(a,b)\) [35] are respectively defined by

$$\begin{aligned} TD(a,b) =& \frac{2}{\pi } \int_{0}^{\pi /2}\sqrt{a^{2} \cos^{2}(t)+b ^{2}\sin^{2}(t)}\,dt \\ =& \textstyle\begin{cases} 2a\mathcal{E} (\sqrt{1-(b/a)^{2}} )/\pi, & a>b, \cr 2b\mathcal{E} (\sqrt{1-(a/b)^{2}} )/\pi, & a< b, \end{cases}\displaystyle \end{aligned}$$
(1.1)

and

$$SB(a,b)= \textstyle\begin{cases} \frac{\sqrt{b^{2}-a^{2}}}{\cos^{-1}(a/b)}, & a< b, \\ \frac{\sqrt{a^{2}-b^{2}}}{\cosh^{-1}(a/b)}, & a>b, \end{cases} $$

where \(\cos^{-1}(x)\) and \(\cosh^{-1}(x)=\log (x+\sqrt{x^{2}-1})\) are the inverse cosine and inverse hyperbolic cosine functions, respectively.

Very recently, Neuman [6] introduced the Neuman mean \(N(a,b)\) of the second kind as follows:

$$N(a,b)=\frac{1}{2} \biggl[a+\frac{b^{2}}{SB(a,b)} \biggr]. $$

It is well known that the Toader mean \(TD(a,b)\), the Schwab-Borchardt mean \(SB(a,b)\) and the Neuman mean of the second kind \(N(a,b)\) satisfy the identities (see [6, 7])

$$\begin{aligned}& \begin{aligned} TD(a,b) &=\frac{4}{\pi }R_{G}\bigl(a^{2},b^{2},0 \bigr) \\ &=\frac{1}{\pi } \int_{0} ^{\infty } \bigl[\bigl(t+a^{2}\bigr) \bigl(t+b^{2}\bigr) \bigr]^{-1/2} \biggl(\frac{a^{2}}{t+a ^{2}}+ \frac{b^{2}}{t+b^{2}} \biggr)t\,dt, \end{aligned} \\& \begin{aligned} SB(a,b) &=1/R_{F}\bigl(a^{2},b^{2},b^{2} \bigr) \\ &=2/ \int_{0}^{\infty } \bigl[\bigl(t+a^{2}\bigr) \bigl(t+b ^{2}\bigr) \bigl(t+b^{2}\bigr) \bigr]^{-1/2}\,dt, \end{aligned} \\& \begin{aligned} N(a,b) &=R_{G}\bigl(a^{2},b^{2},b^{2} \bigr) \\ &=\frac{1}{4} \int_{0}^{\infty } \bigl[\bigl(t+a^{2}\bigr) \bigl(t+b^{2}\bigr) \bigl(t+b^{2}\bigr) \bigr]^{-1/2} \biggl(\frac{a^{2}}{t+a^{2}}+\frac{b^{2}}{t+b^{2}}+ \frac{b^{2}}{t+b^{2}} \biggr)t\,dt. \end{aligned} \end{aligned}$$

Let \(p\in \mathbb{R}\) and \(a,b>0\). Then the pth power mean \(M_{p}(a,b)\) is defined by

$$ M_{p}(a,b) = \bigl[\bigl(a^{p}+b^{p} \bigr)/2 \bigr]^{1/p}(p\neq 0),\qquad M_{0}(a,b)= \sqrt{ab}. $$
(1.2)

We clearly see that \(M_{p}(a,b)\) is symmetric and homogeneous of degree one with respect to a and b, strictly increasing with respect to \(p\in \mathbb{R}\) for fixed \(a,b>0\) with \(a\neq b\), and the inequalities

$$G(a,b)=M_{0}(a,b)< A(a,b)=M_{1}(a,b)< Q(a,b)=M_{2}(a,b) $$

hold for \(a,b>0\) with \(a\neq b\), 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.

In [6], Neuman presented the explicit formula for \(N_{QA}(a,b) \equiv N[Q(a,b),A(a,b)]\) and \(N_{AQ}(a,b)\equiv N[A(a,b),Q(a,b)]\) as follows:

$$\begin{aligned}& N_{QA}(a,b)=\frac{1}{2}A(a,b) \biggl[ \sqrt{1+v^{2}}+ \frac{\sinh^{-1}(v)}{v} \biggr], \end{aligned}$$
(1.3)
$$\begin{aligned}& N_{AQ}(a,b)=\frac{1}{2}A(a,b) \biggl[1+ \bigl(1+v^{2}\bigr)\frac{\tan^{-1}(v)}{v} \biggr] \end{aligned}$$
(1.4)

and proved that the inequalities

$$ A(a,b)< N_{QA}(a,b)< N_{AQ}(a,b)< Q(a,b) $$
(1.5)

hold for \(a,b>0\) with \(a\neq b\), where \(v=(a-b)/(a+b)\).

Recently, the Toader mean has been the subject of intensive research. In particular, many remarkable inequalities for Toader mean and other related means can be found in the literature [841].

In [42], Vuorinen conjectured that

$$TD(a,b)>M_{3/2}(a,b) $$

for all \(a,b>0\) with \(a\neq b\). This conjecture was proved by Qiu and Shen [43], and Barnard et al. [44], respectively, and Alzer and Qiu [45] presented the best possible upper power mean bound for the Toader mean as follows:

$$TD(a,b)< M_{\log 2/\log (\pi /2)}(a,b) $$

for all \(a,b>0\) with \(a\neq b\).

Li, Qian and Chu [46] proved that the inequality

$$\alpha N_{AQ}(a,b)+(1-\alpha)A(a,b)< TD(a,b)< \beta N_{AQ}(a,b)+(1- \beta)A(a,b) $$

holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha \leq 3/4\) and \(\beta \geq 4(4-\pi)/ [\pi (\pi -2) ]=0.9573\cdots \) .

Note that

$$ G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< A(a,b) $$
(1.6)

for all \(a,b>0\) with \(a\neq b\).

From inequalities (1.5) and (1.6) we clearly see that

$$G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< N_{QA}(a,b)< N_{AQ}(a,b) $$

for all \(a,b>0\) with \(a\neq b\).

The main purpose of this paper is to find the greatest values α, λ and the least values β, μ such that the double inequalities

$$\begin{aligned}& \alpha N_{QA}(a,b)+(1-\alpha)G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< \beta N_{QA}(a,b)+(1-\beta)G(a,b), \\& \lambda N_{AQ}(a,b)+(1-\lambda)G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< \mu N_{AQ}(a,b)+(1-\mu)G(a,b) \end{aligned}$$

hold for all \(a,b>0\) with \(a\neq b\). As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions.

Lemmas

In order to prove our main results, we need several lemmas, which we present in this section.

For \(r\in (0,1)\), we clearly see that

$$\mathcal{K}\bigl(0^{+}\bigr)=\mathcal{E}\bigl(0^{+}\bigr)= \pi /2, \qquad \mathcal{K}\bigl(1^{-}\bigr)=+ \infty,\qquad \mathcal{E} \bigl(1^{-}\bigr)=1, $$

and \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) satisfy the formulas (see[21], Appendix E, pp.474-475)

$$\begin{aligned}& \frac{d\mathcal{K}(r)}{dr}=\frac{\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)}{r(1-r ^{2})}, \qquad \frac{d\mathcal{E}(r)}{dr}= \frac{\mathcal{E}(r)-\mathcal{K}(r)}{r}, \\& \frac{d [\mathcal{E}(r)-\mathcal{K}(r) ]}{dr}=-\frac{r \mathcal{E}(r)}{1-r^{2}}. \end{aligned}$$

Lemma 2.1

see [21], Theorem 1.25

For \(-\infty < a< b<+\infty \), let \(f,g:[a,b]\rightarrow \mathbb{R}\) be continuous on \([a,b]\) and differentiable on \((a,b)\), and \(g'(x)\neq 0\) on \((a,b)\). If \(f'(x)/g'(x)\) is increasing (decreasing) on \((a,b)\), then so are

$$\frac{f(x)-f(a)}{g(x)-g(a)}\quad \textit{and}\quad \frac{f(x)-f(b)}{g(x)-g(b)}. $$

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), Exercise 3.43(11) and Exercise 3.43(29)

  1. (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. (2)

    The function \(r\mapsto [\mathcal{K}(r)-\mathcal{E}(r) ]/r ^{2}\) is strictly increasing from \((0,1)\) onto \((\pi /4,+\infty)\);

  3. (3)

    The function \(r\mapsto [(2-r^{2})\mathcal{K}(r)-2\mathcal{E}(r) ]/r^{4}\) is strictly increasing from \((0,1)\) onto \((\pi /16,+ \infty)\).

Lemma 2.3

The function \(r\mapsto \varphi_{1}(r)= \{\frac{2}{\pi }\sqrt{1-r ^{2}} [2\mathcal{E}(r)-\mathcal{K}(r) ]+2r^{2}-1 \}/r^{2}\) is strictly increasing from \((0,1)\) onto \((3/4,1)\).

Proof

Simple computations lead to

$$\begin{aligned}& \varphi_{1}\bigl(0^{+}\bigr)= \frac{3}{4},\qquad \varphi_{1}\bigl(1^{-}\bigr)=1, \end{aligned}$$
(2.1)
$$\begin{aligned}& \varphi_{1}'(r)=\frac{2}{\pi r^{3}} \gamma_{1}(r), \end{aligned}$$
(2.2)

where

$$\begin{aligned}& \gamma_{1}(r)=\frac{\mathcal{K}(r)-3\mathcal{E}(r)}{\sqrt{1-r^{2}}}+ \pi, \end{aligned}$$
(2.3)
$$\begin{aligned}& \gamma_{1}\bigl(0^{+}\bigr)=0, \end{aligned}$$
(2.4)
$$\begin{aligned}& \gamma_{1}'(r)=\frac{r^{3}}{(1-r^{2})^{3/2}} \frac{(2-r^{2}) \mathcal{K}(r)-2\mathcal{E}(r)}{r^{4}}. \end{aligned}$$
(2.5)

From (2.5) and Lemma 2.2(3) we get

$$ \gamma_{1}'(r)>\frac{\pi r^{3}}{16(1-r^{2})^{3/2}}>0. $$
(2.6)

Therefore, Lemma 2.3 follows easily from (2.1), (2.2), (2.4) and (2.6). □

Lemma 2.4

The function \(r\mapsto \varphi_{2}(r)=(2r^{2}+\sqrt{1-r^{4}}-1)/r ^{2}\) is strictly decreasing from \((0,1)\) onto \((1,2)\).

Proof

It is easy to verify that

$$\begin{aligned}& \varphi_{2}\bigl(0^{+}\bigr)=2,\qquad \varphi_{2}\bigl(1^{-}\bigr)=1, \end{aligned}$$
(2.7)
$$\begin{aligned}& \varphi_{2}'(r)=\frac{2(\sqrt{1-r^{4}}-1)}{r^{3} \sqrt{1-r^{4}}}< 0 \end{aligned}$$
(2.8)

for \(r\in (0,1)\).

Therefore, Lemma 2.4 follows easily from (2.7) and (2.8). □

Lemma 2.5

The function \(r\mapsto \varphi_{3}(r)= [2r^{2}\mathcal{K}(r)-5 \mathcal{E}(r) ]/\sqrt{1-r^{2}}\) is strictly increasing from \((0,1)\) onto \((-5\pi /2,+\infty)\).

Proof

It is not difficult to verify that

$$\begin{aligned}& \varphi_{3}\bigl(0^{+}\bigr)=- \frac{5}{2}\pi,\qquad \varphi_{3}\bigl(1^{-}\bigr)=+\infty, \end{aligned}$$
(2.9)
$$\begin{aligned}& \varphi_{3}'(r)=\frac{r}{(1-r^{2})^{3/2}} \biggl[\bigl(5-3r^{2}\bigr)\frac{ \mathcal{K}(r)-\mathcal{E}(r)}{r^{2}}-\mathcal{E}(r) \biggr]. \end{aligned}$$
(2.10)

From (2.10) and Lemma 2.2(2) together with the monotonicity of \(\mathcal{E}(r)\) on \((0,1)\) we clearly see that

$$ \varphi_{3}'(r)>\frac{r}{(1-r^{2})^{3/2}} \biggl[\bigl(5-3r^{2}\bigr)\times \frac{ \pi }{4}-\frac{\pi }{2} \biggr]=\frac{3\pi }{4} \frac{r}{\sqrt{1-r^{2}}}>0 $$
(2.11)

for \(r\in (0,1)\).

Therefore, Lemma 2.5 follows from (2.9) and (2.11). □

Lemma 2.6

The function \(r\mapsto \varphi_{4}(r)= \{\frac{2}{\pi }\sqrt{1-r ^{2}} [2\mathcal{E}(r)-(1+r^{2})\mathcal{K}(r) ]+3r^{2}-1 \}/r^{2}\) is strictly increasing from \((0,1)\) onto \((3/4,2)\).

Proof

Let \(\phi_{1}(r)=\frac{2}{\pi }\sqrt{1-r^{2}} [2\mathcal{E}(r)-(1+r ^{2})\mathcal{K}(r) ]+3r^{2}-1\), \(\phi_{2}(r)=r^{2}\). Then simple computations give

$$\begin{aligned}& \phi_{1}\bigl(0^{+}\bigr)= \phi_{2}(0)=0,\qquad \varphi_{4}(r)=\phi_{1}(r)/\phi _{2}(r), \end{aligned}$$
(2.12)
$$\begin{aligned}& \varphi_{4}\bigl(1^{-}\bigr)=2, \end{aligned}$$
(2.13)
$$\begin{aligned}& \frac{\phi_{1}'(r)}{\phi_{2}'(r)}=3+\frac{1}{\pi \sqrt{1-r^{2}}} \biggl[\frac{\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)}{r^{2}} \biggr]+\frac{1}{ \pi }\varphi_{3}(r). \end{aligned}$$
(2.14)

It follows from Lemma 2.2(1), Lemma 2.5 and the function \(r\mapsto \sqrt{1-r^{2}}\) strictly decreasing that \(\phi_{1}'(r)/ \phi_{2}'(r)\) is strictly increasing on \((0,1)\) and

$$ \varphi_{4}\bigl(0^{+}\bigr)=\lim _{r\rightarrow 0^{+}}\frac{\phi_{1}'(r)}{ \phi_{2}'(r)}=\frac{3}{4}. $$
(2.15)

Therefore, Lemma 2.6 follows from Lemma 2.1, (2.12), (2.13) and (2.15) together with the monotonicity of \(\phi_{1}'(r)/\phi_{2}'(r)\). □

Lemma 2.7

The function \(\varphi_{5}(r)= [3r^{2} +\sqrt{1-r^{2}}-1 ]/r ^{2}\) is strictly decreasing from \((0,1)\) onto \((2,5/2)\).

Proof

We clearly see that

$$\begin{aligned}& \varphi_{5}\bigl(0^{+}\bigr)= \frac{5}{2},\qquad \varphi_{5}\bigl(1^{-}\bigr)=2, \end{aligned}$$
(2.16)
$$\begin{aligned}& \varphi_{5}'(r)=-\frac{(1-\sqrt{1-r^{2}})^{2}}{r^{3} \sqrt{1-r ^{2}}}< 0 \end{aligned}$$
(2.17)

for \(r\in (0,1)\).

Therefore, Lemma 2.7 follows easily from (2.16) and (2.17). □

Main results

Theorem 3.1

The double inequality

$$ \alpha N_{QA}(a,b)+(1-\alpha)G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< \beta N_{QA}(a,b)+(1-\beta)G(a,b) $$
(3.1)

holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha \leq 3/8\) and \(\beta \geq 4/ [\pi (\log (1+\sqrt{2})+\sqrt{2} ) ]=0.5546\cdots \) .

Proof

Since \(G(a,b)\), \(TD(a,b)\) and \(N_{QA}(a,b)\) are symmetric and homogenous of degree 1, without loss of generality, we assume that \(a>b>0\) and let \(r=(a-b)/(a+b)\in (0,1)\). Then (1.1)-(1.3) lead to

$$\begin{aligned}& TD \bigl[A(a,b),G(a,b) \bigr]=\frac{2}{\pi }A(a,b) \mathcal{E}(r), \end{aligned}$$
(3.2)
$$\begin{aligned}& G(a,b)=A(a,b)\sqrt{1-r^{2}},\qquad N_{QA}(a,b)= \frac{1}{2}A(a,b) \biggl[\sqrt{1+r ^{2}}+\frac{\sinh^{-1}(r)}{r} \biggr]. \end{aligned}$$
(3.3)

It follows from (3.2)-(3.3) that

$$\begin{aligned}& \frac{T [A(a,b),G(a,b) ]-G(a,b)}{N_{QA}(a,b)-G(a,b)} \\& \quad =\frac{\frac{2}{ \pi }\varepsilon (r)-\sqrt{1-r^{2}}}{\frac{1}{2} [\sqrt{1+r ^{2}}+\frac{\sinh^{-1}(r)}{r} ]-\sqrt{1-r^{2}}} \\& \quad =\frac{\frac{4}{\pi }r\varepsilon (r)-2r \sqrt{1-r^{2}}}{\sinh^{-1}(r)+ (r\sqrt{1+r^{2}}-2r \sqrt{1-r^{2}} )}. \end{aligned}$$
(3.4)

Let \(f_{1}(r)=\frac{4}{\pi }r\varepsilon (r)-2r \sqrt{1-r^{2}}\), \(f_{2}(r)=\sinh^{-1}(r)+ (r\sqrt{1+r^{2}}-2r \sqrt{1-r^{2}} )\) and

$$ f(r)=\frac{\frac{4}{\pi }r\varepsilon (r)-2r \sqrt{1-r^{2}}}{\sinh ^{-1}(r)+ (r\sqrt{1+r^{2}}-2r \sqrt{1-r^{2}} )}. $$
(3.5)

Then simple computations lead to

$$\begin{aligned}& f_{1}\bigl(0^{+}\bigr)=f_{2}(0)=0, \end{aligned}$$
(3.6)
$$\begin{aligned}& \frac{f_{1}'(r)}{f_{2}'(r)}=\frac{\frac{2}{\pi }\sqrt{1-r^{2}} [2\varepsilon (r)-\kappa (r) ]+2r^{2}-1 }{2r^{2}+\sqrt{1-r ^{4}}-1}=\frac{\varphi_{1}(r)}{\varphi_{2}(r)}, \end{aligned}$$
(3.7)

where \(\varphi_{1}(r)\) and \(\varphi_{2}(r)\) are defined as in Lemmas 2.3 and 2.4.

It follows from Lemmas 2.3-2.4 and (3.7) that \(f_{1}'(r)/f_{2}'(r)\) is strictly increasing on \((0,1)\). Then (3.5), (3.6) and Lemma 2.1 lead to the conclusion that \(f(r)\) is strictly increasing.

Moreover,

$$\begin{aligned}& \lim_{r\rightarrow 0^{+}}\frac{\frac{4}{\pi }r\varepsilon (r)-2r \sqrt{1-r^{2}}}{\sinh^{-1}(r)+ (r\sqrt{1+r^{2}}-2r \sqrt{1-r ^{2}} )}= \frac{3}{8}, \end{aligned}$$
(3.8)
$$\begin{aligned}& \lim_{r\rightarrow 1^{-}}\frac{\frac{4}{\pi }r\varepsilon (r)-2r \sqrt{1-r^{2}}}{\sinh^{-1}(r)+ (r\sqrt{1+r^{2}}-2r \sqrt{1-r ^{2}} )}= \frac{4}{\pi [\pi (\log (1+\sqrt{2})+\sqrt{2} ) ]}. \end{aligned}$$
(3.9)

Therefore, Theorem 3.1 follows easily from (3.4), (3.8) and (3.9) together with the monotonicity of \(f(r)\). □

Theorem 3.2

The double inequality

$$\begin{aligned} \lambda N_{AQ}(a,b)+(1-\lambda)G(a,b) < &TD \bigl[A(a,b),G(a,b) \bigr] \\ < & \mu N_{AQ}(a,b)+(1-\mu)G(a,b) \end{aligned}$$
(3.10)

holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\lambda \leq 3/10\) and \(\mu \geq 8/ [\pi (\pi +2) ]=0.4952\cdots \) .

Proof

Without loss of generality, we assume that \(a>b>0\) and let \(r=(a-b)/(a+b) \in (0,1)\). Then from (1.4) we get

$$ N_{AQ}(a,b)=\frac{1}{2}A(a,b) \biggl[1+ \bigl(1+r^{2}\bigr) \frac{\tan^{-1}(r)}{r} \biggr]. $$
(3.11)

It follows from (3.2), (3.11) and \(G(a,b)=A(a,b)\sqrt{1-r ^{2}}\) that

$$\begin{aligned}& \frac{TD [A(a,b),G(a,b) ]-G(a,b)}{N_{AQ}(a,b)-G(a,b)} \frac{\frac{2}{ \pi }\mathcal{E}(r)-\sqrt{1-r^{2}}}{\frac{1}{2} [1+(1+r^{2})\frac{ \tan^{-1}(r)}{r} ]-\sqrt{1-r^{2}}} \\& \quad =\frac{ [\frac{4}{\pi }r\mathcal{E}(r)-2r \sqrt{1-r^{2}} ]/(1+r ^{2})}{\tan^{-1}(r)+ (r-2r \sqrt{1-r^{2}} )/(1+r^{2})}. \end{aligned}$$
(3.12)

Let \(g_{1}(r)= [\frac{4}{\pi }r\mathcal{E}(r)-2r \sqrt{1-r^{2}} ]/(1+r^{2})\), \(g_{2}(r)=\tan^{-1}(r)+ (r-2r \sqrt{1-r^{2}} ) /(1+r^{2})\) and

$$ g(r)=\frac{ [\frac{4}{\pi }r\mathcal{E}(r)-2r\sqrt{1-r^{2}} ]/(1+r^{2})}{\tan^{-1}(r)+ (r-2r \sqrt{1-r^{2}} )/(1+r ^{2})}. $$
(3.13)

Then simple computations lead to

$$\begin{aligned}& g_{1}\bigl(0^{+}\bigr)=g_{2}(0)=0, \end{aligned}$$
(3.14)
$$\begin{aligned}& \frac{g_{1}'(r)}{g_{2}'(r)}=\frac{\frac{2}{\pi }\sqrt{1-r^{2}} [2\varepsilon (r)-(1+r^{2})\kappa (r) ]+3r^{2}-1 }{3r^{2}+\sqrt{1-r ^{2}}-1}=\frac{\varphi_{4}(r)}{\varphi_{5}(r)}, \end{aligned}$$
(3.15)

where \(\varphi_{4}(r)\) and \(\varphi_{5}(r)\) are defined as in Lemmas 2.6 and 2.7.

It follows from Lemmas 2.6-2.7 and (3.15) that \(g_{1}'(r)/g_{2}'(r)\) is strictly increasing on \((0,1)\). Then (3.13), (3.14) and Lemma 2.1 lead to the conclusion that \(g(r)\) is strictly increasing.

Moreover,

$$\begin{aligned}& \lim_{r\rightarrow 0^{+}}\frac{ [\frac{4}{\pi }r\varepsilon (r)-2r \sqrt{1-r^{2}} ]/(1+r^{2})}{\tan^{-1}(r)+ (r-2r \sqrt{1-r ^{2}} )/(1+r^{2})}= \frac{3}{10}, \end{aligned}$$
(3.16)
$$\begin{aligned}& \lim_{r\rightarrow 1^{-}}\frac{ [\frac{4}{\pi }r\varepsilon (r)-2r \sqrt{1-r^{2}} ]/(1+r^{2})}{\tan^{-1}(r)+ (r-2r \sqrt{1-r ^{2}} )/(1+r^{2})}= \frac{8}{\pi (\pi +2)}. \end{aligned}$$
(3.17)

Therefore, Theorem 3.2 follows from (3.12), (3.16) and (3.17) together with the monotonicity of \(g(r)\). □

From Theorems 3.1-3.2 we get the following Corollary 3.3 immediately.

Corollary 3.3

Let \(\alpha =3/8\), \(\beta =4/ [\pi (\log (1+\sqrt{2})+ \sqrt{2} ) ]=0.5546\cdots \) , \(\lambda =3/10\) and \(\mu =8/ [ \pi (\pi +2) ]=0.4952\cdots \) . Then the double inequalities

$$\begin{aligned}& \frac{1}{4}\pi \alpha \biggl[\sqrt{1+r^{2}}+\frac{\sinh^{-1}(r)}{r} \biggr]+\frac{1}{2}\pi (1- \alpha)\sqrt{1-r^{2}} \\& \quad < \mathcal{E}(r)< \frac{1}{4}\pi \beta \biggl[\sqrt{1+r^{2}}+\frac{\sinh^{-1}(r)}{r} \biggr]+\frac{1}{2}\pi (1- \beta)\sqrt{1-r^{2}}, \\& \frac{1}{4}\pi \lambda \biggl[1+\bigl(1+r^{2}\bigr) \frac{\tan^{-1}(r)}{r} \biggr]+ \frac{1}{2}\pi (1- \lambda) \sqrt{1-r^{2}} \\& \quad < \mathcal{E}(r)< \frac{1}{4}\pi \mu \biggl[1+\bigl(1+r^{2}\bigr) \frac{\tan^{-1}(r)}{r} \biggr]+ \frac{1}{2}\pi (1- \mu)\sqrt{1-r^{2}} \end{aligned}$$

hold for all \(r\in (0,1)\).

Results and discussion

In this paper, we provide the sharp bounds for the Toader-type mean in terms of the convex combination of geometric and Neuman means. As applications, we find new bounds for the complete elliptic integral of the second kind.

Conclusion

In the article, we present the optimal convex combination bounds of the geometric and Neuman means for the Toader-type mean, and give several new upper and lower bounds for the complete elliptic integral of the second kind. The given results are the improvements of some previously known results.

References

  1. Carlson, BC: Special Functions of Applied Mathematics. Academic Press, New York (1977)

    MATH  Google Scholar 

  2. Toader, GH: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358-368 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  3. Neuman, E, Sándor, J: On the Schwab-Borchardt mean. Math. Pannon. 14(2), 253-266 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Neuman, E, Sándor, J: On the Schwab-Borchardt mean II. Math. Pannon. 17(1), 49-59 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Neuman, E: Inequalities for the Schwab-Borchardt mean and their applications. J. Math. Inequal. 5(4), 601-609 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  6. Neuman, E: On a new bivariate mean. Aequ. Math. 88(3), 277-289 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  7. Neuman, E: Bounds for symmetric elliptic integrals. J. Approx. Theory 122(2), 249-259 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  8. Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals. J. Approx. Theory 146(2), 212-226 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  9. Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals II. In: Special Functions and Orthogonal Polynomials. Contemp. Math., vol. 471, pp. 127-138. Am. Math. Soc., Providence (2008)

    Chapter  Google Scholar 

  10. 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).

    MathSciNet  MATH  Google Scholar 

  11. Chu, Y-M, Wang, M-K: Inequalities between arithmetic-geometric, Gini, and Toader means. Abstr. Appl. Anal. 2012, Article ID 830585 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Chu, Y-M, Wang, M-K: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  13. Chu, Y-M, Wang, M-K, Qiu, S-L: Optimal combination bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41-51 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  14. Hua, Y, Qi, F: The best bounds for Toader mean in terms of the centroidal and arithmetic means. Filomat 28(4), 775-780 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  15. Song, Y-Q, Jiang, W-D, Chu, Y-M, Yan, D-D: Optimal bounds for Toader mean in terms of arithmetic and contraharmonic means. J. Math. Inequal. 7(4), 751-757 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  16. Li, W-H, Zheng, M-M: Some inequalities for bounding Toader mean. J. Funct. Spaces Appl. 2013, Article ID 394194 (2013).

    MathSciNet  MATH  Google Scholar 

  17. Sun, H, Chu, Y-M: Bounds for Toader mean by quadratic and harmonic means. Acta Math. Sci. 35A(1), 36-42 (2015) (in Chinese)

    MATH  Google Scholar 

  18. Hua, Y, Qi, F: A double inequality for bounding Toader mean by the centroidal mean. Proc. Indian Acad. Sci. Math. Sci. 124(4), 527-531 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  19. 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)

    MathSciNet  Article  MATH  Google Scholar 

  20. Zhao, T-H, Chu, Y-M, Zhang, W: Optimal inequalities for bounding Toader mean by arithmetic and quadratic means. J. Inequal. Appl. 2017, Article ID 26 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  21. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)

    MATH  Google Scholar 

  22. Wang, M-K, Li, Y-M, Chu, Y-M: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. doi:10.1007/s11139-017-9888-3

  23. Wang, M-K, Chu, Y-M: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607-622 (2017)

    MathSciNet  Article  Google Scholar 

  24. 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 meas. J. Nonlinear Sci. Appl. 9(5), 3424-3432 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Song, Y-Q, Zhao, T-H, Chu, Y-M, Zhang, X-H: Optimal evaluation of a Toader-type mean by power mean. J. Inequal. Appl. 2015, Article ID 408 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  26. Qian, W-M, Song, Y-Q, Zhang, X-H, Chu, Y-M: Sharp bounds for Toader mean in terms of arithmetic and second contraharmonic means. J. Funct. Spaces 2015, Article ID 452832 (2015)

    MathSciNet  MATH  Google Scholar 

  27. 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)

    MathSciNet  Article  MATH  Google Scholar 

  28. 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)

    MathSciNet  MATH  Google Scholar 

  29. 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)

    MathSciNet  Article  MATH  Google Scholar 

  30. 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)

    MathSciNet  Article  MATH  Google Scholar 

  31. 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)

    MathSciNet  Article  MATH  Google Scholar 

  32. 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)

    MathSciNet  Article  MATH  Google Scholar 

  33. 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)

    MathSciNet  Article  MATH  Google Scholar 

  34. 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)

    MathSciNet  Article  MATH  Google Scholar 

  35. Chu, Y-M, Qiu, S-L, Wang, M-K: Sharp inequalities involving the power mean and complete elliptic integral of the first kind. Rocky Mt. J. Math. 43(5), 1489-1496 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  36. Wang, G-D, Zhang, Z-H, Chu, Y-M: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661-1667 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  37. Yang, Z-H, Chu, Y-M, Zhang, W: Accurate approximations for the complete elliptic integral of the second kind. J. Math. Anal. Appl. 438(2), 875-888 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  38. Yang, Z-H, Chu, Y-M, Zhang, X-H: Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind. J. Nonlinear Sci. Appl. 10(3), 929-936 (2017)

    MathSciNet  Article  Google Scholar 

  39. Yang, Z-H, Chu, Y-M, Zhang, W: Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean. J. Inequal. Appl. 2016, Article ID 176 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  40. Yang, Z-H, Chu, Y-M, Wang, M-K: Monotonicity criterion for the quotient of power series with application. J. Math. Anal. Appl. 428(1), 587-604 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  41. Wang, M-K, Wang, Z-K, Chu, Y-M: An optimal double inequality between geometric and identric means. Appl. Math. Lett. 25(3), 471-475 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  42. Vuorinen, M: Hypergeometric functions in geometric function theory. In: Special Functions and Differential Equations, Madras, 1997, pp. 119-126. Allied Publ., New Delhi (1998)

    Google Scholar 

  43. Qiu, S-L, Shen, J-M: On two problems concerning means. J. Hangzhou Inst. Electron. Eng. 17(3), 1-7 (1997) (in Chinese)

    Google Scholar 

  44. Barnard, RW, Pearce, K, Richards, KC: An inequality involving the generalized hypergeometric function and the arc length of an ellipse. SIAM J. Math. Anal. 31(3), 693-699 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  45. Alzer, H, Qiu, S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289-312 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  46. Li, J-F, Qian, W-M, Chu, Y-M: Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means. J. Inequal. Appl. 2015, Article ID 277 (2015)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This research was supported by the Natural Science Foundation of Zhejiang Province under Grant LY13A010004 and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-15G17.

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Yang, YY., Qian, WM. Optimal convex combination bounds of geometric and Neuman means for Toader-type mean. J Inequal Appl 2017, 201 (2017). https://doi.org/10.1186/s13660-017-1473-1

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  • DOI: https://doi.org/10.1186/s13660-017-1473-1

MSC

  • 26E60
  • 33E05

Keywords

  • Toader mean
  • geometric mean
  • Neuman mean