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

Abstract

In this paper, we prove that the double inequalities

$$\begin{array}{c}\alpha N_{QA}(a,b)+(1-\alpha )G(a,b)<TD[A(a,b),G(a,b)]<\beta N_{QA}(a,b)+(1-\beta )G(a,b),\hfill \\ \lambda N_{AQ}(a,b)+(1-\lambda )G(a,b)<TD[A(a,b),G(a,b)]<\mu N_{AQ}(a,b)+(1-\mu )G(a,b)\hfill \end{array}$$

hold for all $a,b>0$ with $a\ne b$ if and only if $\alpha \le 3/8$, $\beta \ge 4/[\pi (log(1+\sqrt{2})+\sqrt{2})]=0.5546\cdots$ , $\lambda \le 3/10$ and $\mu \ge 8/[\pi (\pi +2)]=0.4952\cdots$ , 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 $x,y,z\ge 0$ with $xy+xz+yz\ne 0$ 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

$$\begin{array}{c}{R}_F(x,y,z)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{[(t+x)(t+y)(t+z)]}^{-1/2}dt,\hfill \\ R_G(x,y,z)=\frac{1}{4}{\int }_0^{\mathrm{\infty }}{[(t+x)(t+y)(t+z)]}^{-1/2}(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z})tdt,\hfill \\ \mathcal{K}(r)={\int }_0^{\pi /2}{[1-r^2{sin}^2(t)]}^{-1/2}dt,\mathcal{E}(r)={\int }_0^{\pi /2}{[1-r^2{sin}^2(t)]}^{1/2}dt,\hfill \end{array}$$

respectively.

The well-known identities

$$\mathcal{K}(r)=R_F(0,1-r^2,1),\mathcal{E}(r)=2R_G(0,1-r^2,1)$$

were established by Carlson in [1].

Let $a,b>0$ with $a\ne b$. Then the Toader mean $\mathit{TD}(a,b)$ [2] and the Schwab-Borchardt mean $SB(a,b)$ [3–5] are respectively defined by

$$\begin{array}{rcl}TD(a,b)& =& \frac{2}{\pi }{\int }_0^{\pi /2}\sqrt{a^2{cos}^2(t)+b^2{sin}^2(t)}dt\\ & =& \{\begin{array}{cc}2a\mathcal{E}(\sqrt{1-{(b/a)}^2})/\pi ,\hfill & a>b,\hfill \\ 2b\mathcal{E}(\sqrt{1-{(a/b)}^2})/\pi ,\hfill & a<b,\hfill \end{array}\end{array}$$ 1.1

and

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

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:

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

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])

$$\begin{array}{c}\begin{array}{rl}TD(a,b)& =\frac{4}{\pi }{R}_G(a^2,b^2,0)\\ & =\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{\left[(t+a^2)(t+b^2)\right]}^{-1/2}(\frac{a^2}{t+a^2}+\frac{b^2}{t+b^2})tdt,\end{array}\hfill \\ \begin{array}{rl}SB(a,b)& =1/R_F(a^2,b^2,b^2)\\ & =2/{\int }_0^{\mathrm{\infty }}{\left[(t+a^2)(t+b^2)(t+b^2)\right]}^{-1/2}dt,\end{array}\hfill \\ \begin{array}{rl}N(a,b)& =R_G(a^2,b^2,b^2)\\ & =\frac{1}{4}{\int }_0^{\mathrm{\infty }}{\left[(t+a^2)(t+b^2)(t+b^2)\right]}^{-1/2}(\frac{a^2}{t+a^2}+\frac{b^2}{t+b^2}+\frac{b^2}{t+b^2})tdt.\end{array}\hfill \end{array}$$

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

$$M_p(a,b)={[(a^p+b^p)/2]}^{1/p}(p\ne 0),M_0(a,b)=\sqrt{ab}.$$ 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 $a\ne b$, 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 $a\ne b$, 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:

$$N_{QA}(a,b)=\frac{1}{2}A(a,b)[\sqrt{1+v^2}+\frac{{sinh}^{-1}(v)}{v}],$$ 1.3

$$N_{AQ}(a,b)=\frac{1}{2}A(a,b)[1+(1+v^2)\frac{{tan}^{-1}(v)}{v}]$$ 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 $a\ne b$, 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 [8–41].

In [42], Vuorinen conjectured that

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

for all $a,b>0$ with $a\ne b$. 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:

$$TD(a,b)<M_{log2/log(\pi /2)}(a,b)$$

for all $a,b>0$ with $a\ne b$.

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 $a\ne b$ if and only if $\alpha \le 3/4$ and $\beta \ge 4(4-\pi )/[\pi (\pi -2)]=0.9573\cdots$ .

Note that

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

for all $a,b>0$ with $a\ne b$.

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

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

for all $a,b>0$ with $a\ne b$.

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

$$\begin{array}{c}\alpha N_{QA}(a,b)+(1-\alpha )G(a,b)<TD[A(a,b),G(a,b)]<\beta N_{QA}(a,b)+(1-\beta )G(a,b),\hfill \\ \lambda N_{AQ}(a,b)+(1-\lambda )G(a,b)<TD[A(a,b),G(a,b)]<\mu N_{AQ}(a,b)+(1-\mu )G(a,b)\hfill \end{array}$$

hold for all $a,b>0$ with $a\ne b$. 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

$$\mathcal{K}\left(0^{+}\right)=\mathcal{E}\left(0^{+}\right)=\pi /2,\mathcal{K}\left(1^{-}\right)=+\mathrm{\infty },\mathcal{E}\left(1^{-}\right)=1,$$

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

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

Lemma 2.1

see [21], Theorem 1.25

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

$$\frac{f(x)-f(a)}{g(x)-g(a)}\mathit{\text{and}}\frac{f(x)-f(b)}{g(x)-g(b)}.$$

If $f^{\prime }(x)/g^{\prime }(x)$ 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. The function $r\mapsto [\mathcal{E}(r)-(1-r^2)\mathcal{K}(r)]/r^2$ is strictly increasing from $(0,1)$ onto $(\pi /4,1)$;

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

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

Lemma 2.3

The function $r\mapsto {\phi }_1(r)=\{\frac{2}{\pi }\sqrt{1-r^2}[2\mathcal{E}(r)-\mathcal{K}(r)]+2r^2-1\}/r^2$ is strictly increasing from $(0,1)$ onto $(3/4,1)$.

Proof

Simple computations lead to

$${\phi }_1\left(0^{+}\right)=\frac{3}{4},{\phi }_1\left(1^{-}\right)=1,$$ 2.1

$${\phi }_1^{\prime }(r)=\frac{2}{\pi r^3}{\gamma }_1(r),$$ 2.2

where

$${\gamma }_1(r)=\frac{\mathcal{K}(r)-3\mathcal{E}(r)}{\sqrt{1-r^2}}+\pi ,$$ 2.3

$${\gamma }_1\left(0^{+}\right)=0,$$ 2.4

$${\gamma }_1^{\prime }(r)=\frac{r^3}{{(1-r^2)}^{3/2}}\frac{(2-r^2)\mathcal{K}(r)-2\mathcal{E}(r)}{r^4}.$$ 2.5

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

$${\gamma }_1^{\prime }(r)>\frac{\pi r^3}{16{(1-r^2)}^{3/2}}>0.$$ 2.6

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

Lemma 2.4

The function $r\mapsto {\phi }_2(r)=(2r^2+\sqrt{1-r^4}-1)/r^2$ is strictly decreasing from $(0,1)$ onto $(1,2)$.

Proof

It is easy to verify that

$${\phi }_2\left(0^{+}\right)=2,{\phi }_2\left(1^{-}\right)=1,$$ 2.7

$${\phi }_2^{\prime }(r)=\frac{2(\sqrt{1-r^4}-1)}{r^3\sqrt{1-r^4}}<0$$ 2.8

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

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

Lemma 2.5

The function $r\mapsto {\phi }_3(r)=[2r^2\mathcal{K}(r)-5\mathcal{E}(r)]/\sqrt{1-r^2}$ is strictly increasing from $(0,1)$ onto $(-5\pi /2,+\mathrm{\infty })$.

Proof

It is not difficult to verify that

$${\phi }_3\left(0^{+}\right)=-\frac{5}{2}\pi ,{\phi }_3\left(1^{-}\right)=+\mathrm{\infty },$$ 2.9

$${\phi }_3^{\prime }(r)=\frac{r}{{(1-r^2)}^{3/2}}[(5-3r^2)\frac{\mathcal{K}(r)-\mathcal{E}(r)}{r^2}-\mathcal{E}(r)].$$ 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

$${\phi }_3^{\prime }(r)>\frac{r}{{(1-r^2)}^{3/2}}[(5-3r^2)\times \frac{\pi }{4}-\frac{\pi }{2}]=\frac{3\pi }{4}\frac{r}{\sqrt{1-r^2}}>0$$ 2.11

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

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

Lemma 2.6

The function $r\mapsto {\phi }_4(r)=\{\frac{2}{\pi }\sqrt{1-r^2}[2\mathcal{E}(r)-(1+r^2)\mathcal{K}(r)]+3r^2-1\}/r^2$ is strictly increasing from $(0,1)$ onto $(3/4,2)$.

Proof

Let ${\varphi }_1(r)=\frac{2}{\pi }\sqrt{1-r^2}[2\mathcal{E}(r)-(1+r^2)\mathcal{K}(r)]+3r^2-1$, ${\varphi }_2(r)=r^2$. Then simple computations give

$${\varphi }_1\left(0^{+}\right)={\varphi }_2(0)=0,{\phi }_4(r)={\varphi }_1(r)/{\varphi }_2(r),$$ 2.12

$${\phi }_4\left(1^{-}\right)=2,$$ 2.13

$$\frac{{\varphi }_1^{\prime }(r)}{{\varphi }_2^{\prime }(r)}=3+\frac{1}{\pi \sqrt{1-r^2}}\left[\frac{\mathcal{E}(r)-(1-r^2)\mathcal{K}(r)}{r^2}\right]+\frac{1}{\pi }{\phi }_3(r).$$ 2.14

It follows from Lemma 2.2(1), Lemma 2.5 and the function $r\mapsto \sqrt{1-r^2}$ strictly decreasing that ${\varphi }_1^{\prime }(r)/{\varphi }_2^{\prime }(r)$ is strictly increasing on $(0,1)$ and

$${\phi }_4\left(0^{+}\right)=\underset{r\to 0^{+}}{lim}\frac{{\varphi }_1^{\prime }(r)}{{\varphi }_2^{\prime }(r)}=\frac{3}{4}.$$ 2.15

Therefore, Lemma 2.6 follows from Lemma 2.1, (2.12), (2.13) and (2.15) together with the monotonicity of ${\varphi }_1^{\prime }(r)/{\varphi }_2^{\prime }(r)$. □

Lemma 2.7

The function ${\phi }_5(r)=[3r^2+\sqrt{1-r^2}-1]/r^2$ is strictly decreasing from $(0,1)$ onto $(2,5/2)$.

Proof

We clearly see that

$${\phi }_5\left(0^{+}\right)=\frac{5}{2},{\phi }_5\left(1^{-}\right)=2,$$ 2.16

$${\phi }_5^{\prime }(r)=-\frac{{(1-\sqrt{1-r^2})}^2}{r^3\sqrt{1-r^2}}<0$$ 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

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

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

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

$$TD[A(a,b),G(a,b)]=\frac{2}{\pi }A(a,b)\mathcal{E}(r),$$ 3.2

$$G(a,b)=A(a,b)\sqrt{1-r^2},N_{QA}(a,b)=\frac{1}{2}A(a,b)[\sqrt{1+r^2}+\frac{{sinh}^{-1}(r)}{r}].$$ 3.3

It follows from (3.2)-(3.3) that

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

Let $f_1(r)=\frac{4}{\pi }r\epsilon (r)-2r\sqrt{1-r^2}$, $f_2(r)={sinh}^{-1}(r)+(r\sqrt{1+r^2}-2r\sqrt{1-r^2})$ and

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

Then simple computations lead to

$$f_1\left(0^{+}\right)=f_2(0)=0,$$ 3.6

$$\frac{f_1^{\prime }(r)}{f_2^{\prime }(r)}=\frac{\frac{2}{\pi }\sqrt{1-r^2}[2\epsilon (r)-\kappa (r)]+2r^2-1}{2r^2+\sqrt{1-r^4}-1}=\frac{{\phi }_1(r)}{{\phi }_2(r)},$$ 3.7

where ${\phi }_1(r)$ and ${\phi }_2(r)$ are defined as in Lemmas 2.3 and 2.4.

It follows from Lemmas 2.3-2.4 and (3.7) that $f_1^{\prime }(r)/f_2^{\prime }(r)$ 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,

$$\underset{r\to 0^{+}}{lim}\frac{\frac{4}{\pi }r\epsilon (r)-2r\sqrt{1-r^2}}{{sinh}^{-1}(r)+(r\sqrt{1+r^2}-2r\sqrt{1-r^2})}=\frac{3}{8},$$ 3.8

$$\underset{r\to 1^{-}}{lim}\frac{\frac{4}{\pi }r\epsilon (r)-2r\sqrt{1-r^2}}{{sinh}^{-1}(r)+(r\sqrt{1+r^2}-2r\sqrt{1-r^2})}=\frac{4}{\pi [\pi (log(1+\sqrt{2})+\sqrt{2})]}.$$ 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

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

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

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

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

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

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

$$g_1\left(0^{+}\right)=g_2(0)=0,$$ 3.14

$$\frac{g_1^{\prime }(r)}{g_2^{\prime }(r)}=\frac{\frac{2}{\pi }\sqrt{1-r^2}[2\epsilon (r)-(1+r^2)\kappa (r)]+3r^2-1}{3r^2+\sqrt{1-r^2}-1}=\frac{{\phi }_4(r)}{{\phi }_5(r)},$$ 3.15

where ${\phi }_4(r)$ and ${\phi }_5(r)$ are defined as in Lemmas 2.6 and 2.7.

It follows from Lemmas 2.6-2.7 and (3.15) that $g_1^{\prime }(r)/g_2^{\prime }(r)$ 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,

$$\underset{r\to 0^{+}}{lim}\frac{[\frac{4}{\pi }r\epsilon (r)-2r\sqrt{1-r^2}]/(1+r^2)}{{tan}^{-1}(r)+(r-2r\sqrt{1-r^2})/(1+r^2)}=\frac{3}{10},$$ 3.16

$$\underset{r\to 1^{-}}{lim}\frac{[\frac{4}{\pi }r\epsilon (r)-2r\sqrt{1-r^2}]/(1+r^2)}{{tan}^{-1}(r)+(r-2r\sqrt{1-r^2})/(1+r^2)}=\frac{8}{\pi (\pi +2)}.$$ 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

$$\begin{array}{c}\frac{1}{4}\pi \alpha [\sqrt{1+r^2}+\frac{{sinh}^{-1}(r)}{r}]+\frac{1}{2}\pi (1-\alpha )\sqrt{1-r^2}\hfill \\ <\mathcal{E}(r)<\frac{1}{4}\pi \beta [\sqrt{1+r^2}+\frac{{sinh}^{-1}(r)}{r}]+\frac{1}{2}\pi (1-\beta )\sqrt{1-r^2},\hfill \\ \frac{1}{4}\pi \lambda [1+(1+r^2)\frac{{tan}^{-1}(r)}{r}]+\frac{1}{2}\pi (1-\lambda )\sqrt{1-r^2}\hfill \\ <\mathcal{E}(r)<\frac{1}{4}\pi \mu [1+(1+r^2)\frac{{tan}^{-1}(r)}{r}]+\frac{1}{2}\pi (1-\mu )\sqrt{1-r^2}\hfill \end{array}$$

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.