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Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters

Journal of Inequalities and Applications20172017:274

https://doi.org/10.1186/s13660-017-1550-5

• Accepted: 22 October 2017
• Published:

Abstract

In the article, we present the best possible parameters $$\lambda=\lambda (p)$$ and $$\mu=\mu(p)$$ on the interval $$[0, 1/2]$$ such that the double inequality
\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}
holds for any $$p\in[1, \infty)$$ and all $$a, b>0$$ with $$a\neq b$$, where $$A(a, b)=(a+b)/2$$, $$G(a,b)=\sqrt{ab}$$ and $$E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$$ are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.

Keywords

• quasi-arithmetic mean
• complete elliptic integral
• Gaussian hypergeometric function
• arithmetic mean
• geometric mean

• 26E60
• 33E05

1 Introduction

Let $$r\in(0,1)$$. Then the Legendre complete elliptic integrals $$\mathcal {K}(r)$$ and $$\mathcal{E}(r)$$ [1, 2] of the first and second kinds are defined as
$$\mathcal{K}(r)= \int_{0}^{\pi/2}\frac{dt}{\sqrt{1-r^{2}\sin^{2}(t)}}, \qquad\mathcal{E}(r)= \int_{0}^{\pi/2}\sqrt{1-r^{2} \sin^{2}(t)}\,dt,$$
respectively. It is well known that the function $$r\rightarrow\mathcal {K}(r)$$ is strictly increasing from $$(0, 1)$$ onto $$(\pi/2, \infty)$$ and the function $$r\rightarrow\mathcal{E}(r)$$ is strictly decreasing from $$(0, 1)$$ onto $$(1, \pi/2)$$, and they satisfy the formulas (see [3, Appendix E, pp. 474,475])
$$\begin{gathered} \frac{d{\mathcal{K}(r)}}{dr}=\frac{{\mathcal{E}(r)}-{r'}^{2}{\mathcal {K}}(r)}{r{r'}^{2}},\qquad \frac{d{\mathcal{E}(r)}}{dr}=\frac{{\mathcal{E}(r)}-{\mathcal{K}(r)}}{r}, \\ \mathcal{K} \biggl(\frac{2\sqrt{r}}{1+r} \biggr)=(1+r)\mathcal{K}(r),\qquad \mathcal{E} \biggl(\frac{2\sqrt{r}}{1+r} \biggr)=\frac{2\mathcal {E}(r)-{r'}^{2}\mathcal{K}}{1+r}, \end{gathered}$$
where $$r'=\sqrt{1-r^{2}}$$.
The complete elliptic integrals $$\mathcal{K}(r)$$ and $$\mathcal{E}(r)$$ are the particular cases of the Gaussian hypergeometric function 
$$F(a,b;c;x)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac {x^{n}}{n!}\quad (-1< x< 1),$$
where $$(a)_{0}=1$$ for $$a\neq0$$, $$(a)_{n}=a(a+1)(a+2)\cdots (a+n-1)=\Gamma(a+n)/\Gamma(a)$$ is the shifted factorial function and $$\Gamma(x)=\int_{0}^{\infty }t^{x-1}e^{-t}\,dt$$ ($$x>0$$) is the gamma function . Indeed,
$$\begin{gathered} \mathcal{K}(r)=\frac{\pi}{2}F \biggl( \frac{1}{2},\frac{1}{2};1;r^{2} \biggr) = \frac{\pi}{2}\sum_{n=0}^{\infty} \frac{ (\frac{1}{2} )_{n}^{2}}{(n!)^{2}}r^{2n}, \\ \mathcal{E}(r)=\frac{\pi}{2}F \biggl(-\frac{1}{2}, \frac{1}{2};1;r^{2} \biggr) =\frac{\pi}{2}\sum _{n=0}^{\infty}\frac{ (-\frac{1}{2} )_{n} (\frac{1}{2} )_{n}}{(n!)^{2}}r^{2n}. \end{gathered}$$

Recently, the bounds for the complete elliptic integrals have attracted the attention of many researchers. In particular, many remarkable inequalities and properties for $$\mathcal{K}(r)$$, $$\mathcal{E}(r)$$ and $$F(a,b;c;x)$$ can be found in the literature .

In 1998, a class of quasi-arithmetic mean was introduced by Toader  which is defined by
$$M_{p,n}(a,b)=p^{-1} \biggl(\frac{1}{\pi} \int_{0}^{\pi}p\bigl(r_{n}(\theta )\,d \theta\bigr) \biggr)=p^{-1} \biggl(\frac{2}{\pi} \int_{0}^{\pi/2}p\bigl(r_{n}(\theta )\,d \theta\bigr) \biggr),$$
where $$r_{n}(\theta)=(a^{n}\cos^{2}\theta+b^{n}\sin^{2}\theta)^{1/n}$$ for $$n\neq0$$, $$r_{0}(\theta)=a^{\cos^{2}\theta}b^{\sin^{2}\theta}$$, and p is a strictly monotonic function. It is well known that many important means are the special cases of the quasi-arithmetic mean. For example,
\begin{aligned} M_{1/x,2}(a,b)= \frac{\pi}{2 \int_{0}^{\pi /2}{\frac{d\theta}{\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}}}} = \textstyle\begin{cases} {\pi a} / [2{\mathcal{K}} (\sqrt{1-(b/a)^{2}} ) ],&a\geq b,\\ {\pi b} / [2{\mathcal{K}} (\sqrt{1-(a/b)^{2}} ) ],&a< b, \end{cases}\displaystyle \end{aligned}
is the arithmetic-geometric mean of Gauss ,
$$M_{x,2}(a,b))=\frac{2}{\pi} \int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2} \theta +b^{2}\sin^{2}\theta}\,d\theta = \textstyle\begin{cases}2a{\mathcal{E}} (\sqrt{1-(b/a)^{2}} )/\pi,&a\geq b,\\ 2b{\mathcal{E}} (\sqrt{1-(a/b)^{2}} )/\pi,&a< b, \end{cases}$$
is the Toader mean , and
$$M_{x,0}(a,b))=\frac{2}{\pi} \int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin ^{2}\theta}\,d \theta$$
Let $$p=\sqrt{x}$$ and $$n=1$$. Then $$M_{p,n}(a,b)$$ reduces to a special quasi-arithmetic mean
$$E(a,b)=M_{\sqrt{x},1}(a,b))= \textstyle\begin{cases}4a [{\mathcal{E}} (\sqrt{1-b/a} ) ]^{2}/\pi ^{2},&a\geq b,\\ 4b [{\mathcal{E}} (\sqrt{1-a/b} ) ]^{2}/\pi^{2},&a< b. \end{cases}$$
(1.1)
Let
$$\begin{gathered} A(a,b)=\frac{a+b}{2}, \qquad G(a,b)=\sqrt{ab}, \\ M_{p}(a,b)= \biggl(\frac{a^{p}+b^{p}}{2} \biggr)^{1/p} (p\neq0), \qquad M_{0}(a,b)=\sqrt{ab}, \end{gathered}$$
be the arithmetic, geometric and pth power means of a and b, respectively. Then it is well known that the inequality
$$G(a,b)=M_{0}(a,b)< A(a,b)=M_{1}(a,b)$$
(1.2)
holds for all $$a, b>0$$ with $$a\neq b$$, and the double inequality
$$\frac{\pi}{2}M_{3/2}\bigl(1, r^{\prime}\bigr)< \mathcal{E}(r)< \frac{\pi }{2}M_{2}\bigl(1, r^{\prime}\bigr)$$
(1.3)
holds for all $$r\in(0, 1)$$ (see [75, 19.9.4]).
From (1.1)-(1.3) we clearly see that
$$G(a,b)< E(a,b)< A(a,b)$$
for all $$a, b>0$$ with $$a\neq b$$.
Let $$p\in[1, \infty)$$ and
$$f(x; p; a, b)=G^{p}\bigl[xa+(1-x)b, xb+(1-x)a\bigr]A^{1-p}(a,b).$$
Then it is not difficult to verify that the function $$x\rightarrow f(x; p; a, b)$$ is strictly increasing on $$[0, 1/2]$$ for fixed $$p\in[1, \infty)$$ and $$a, b>0$$ with $$a\neq b$$. Note that
\begin{aligned}[b] f(0; p; a, b)&=G^{p}(a,b)A^{1-p}(a,b) \leq G(a,b) \\ &< E(a,b)< A(a,b)=f(1/2; p; a, b) \end{aligned}
(1.4)
for all $$p\in[1, \infty)$$ and $$a, b>0$$ with $$a\neq b$$.
Motivated by inequalities (1.4) and the monotonicity of the function $$x\rightarrow f(x; p; a, b)$$ on the interval $$[0, 1/2]$$, in the article, we shall find the best possible parameters $$\lambda=\lambda(p), \mu=\mu(p)$$ on the interval $$[0, 1/2]$$ such that the double inequality
\begin{aligned} &G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a\bigr]A^{1-p}(a,b) \\ &\quad< E(a,b)< G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}
holds for any $$p\in[1, \infty)$$ and all $$a, b>0$$ with $$a\neq b$$.

2 Lemmas

Lemma 2.1

(see [3, Theorem 1.25])

Let $$-\infty< a< b<+\infty$$, $$f, g: [a, b]\rightarrow\mathbb{R}$$ be continuous on $$[a, b]$$ and differentiable on $$(a,b)$$, and $$g^{\prime}(x)\neq0$$ on $$(a, b)$$. If $$f^{\prime}(x)/g^{\prime}(x)$$ is increasing (decreasing) on $$(a,b)$$, then so are the functions
$$\frac{f(x)-f(a)}{g(x)-g(a)}, \qquad\frac{f(x)-f(b)}{g(x)-g(b)}.$$
If $$f^{\prime}(x)/g^{\prime}(x)$$ is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2

The inequality
$$\frac{1}{4p}+ \biggl(\frac{2\sqrt{2}}{\pi} \biggr)^{4/p}< 1$$
holds for all $$p\in[1, \infty)$$.

Proof

Let
$$f(p)=\frac{1}{4p}+ \biggl(\frac{2\sqrt{2}}{\pi} \biggr)^{4/p}.$$
(2.1)
\begin{aligned}& \lim_{p\rightarrow\infty}f(p)=1, \end{aligned}
(2.2)
\begin{aligned}& \begin{aligned}[b] f^{\prime}(p)&=\frac{4}{p^{2}}\log \biggl(\frac{\sqrt{2}\pi}{4} \biggr) \biggl[ \biggl(\frac{2\sqrt{2}}{\pi} \biggr)^{4/p}-\frac{1}{16\log (\frac{\sqrt{2}\pi}{4} )} \biggr] \\ &\geq\frac{4}{p^{2}}\log \biggl(\frac{\sqrt{2}\pi}{4} \biggr) \biggl[ \biggl( \frac{2\sqrt{2}}{\pi} \biggr)^{4}-\frac{1}{16\log (\frac {\sqrt{2}\pi}{4} )} \biggr] \\ &=\frac{1024\log (\frac{\sqrt{2}\pi}{4} )-\pi^{4}}{4\pi^{4}p^{2}}>0 \end{aligned} \end{aligned}
(2.3)
for $$p\in[1, \infty)$$.

Therefore, Lemma 2.2 follows easily from (2.1)-(2.3). □

Lemma 2.3

The following statements are true:
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 $$r\mapsto[\mathcal{E}(r)+(1-r^{2})\mathcal {K}(r)]/(1-r^{2})$$ is strictly increasing from $$(0, 1)$$ onto $$(\pi, \infty)$$.

4. (4)

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

5. (5)

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

Proof

Parts (1) and (2) can be found in the literature [3, Theorem 3.21(1) and Exercise 3.43(11)].

For part (3), let $$f_{1}(r)=[\mathcal{E}(r)+(1-r^{2})\mathcal {K}(r)]/(1-r^{2})$$. Then simple computations lead to
\begin{aligned}& f_{1}\bigl(0^{+}\bigr)=\pi, \qquad f_{1} \bigl(1^{-}\bigr)=\infty, \end{aligned}
(2.4)
\begin{aligned}& f^{\prime}_{1}(r)=\frac{r}{(1-r^{2})^{2}} \biggl[\frac{2}{r^{2}} \bigl(\mathcal {E}(r)-\bigl(1-r^{2}\bigr)\mathcal{K}(r)\bigr)+ \bigl(1-r^{2}\bigr)\mathcal{K}(r) \biggr]. \end{aligned}
(2.5)
It follows from part (1) and (2.5) that
$$f^{\prime}_{1}(r)>0$$
(2.6)
for all $$r\in(0, 1)$$. Therefore, part (3) follows from (2.4) and (2.6).
For part (4), let $$f_{2}(r)=[2\mathcal{E}(r)-(1-r^{2})\mathcal {K}(r)]/(1+r^{2})$$, then one has
\begin{aligned}& f_{2}\bigl(0^{+}\bigr)=\frac{\pi}{2}, \qquad f_{1}\bigl(1^{-}\bigr)=1, \end{aligned}
(2.7)
\begin{aligned}& f^{\prime}_{2}(r)=\frac{r}{(1+r^{2})^{2}} \biggl[ \bigl(1-r^{2}\bigr)\frac{\mathcal {E}(r)-(1-r^{2})\mathcal{K}(r)}{r^{2}}-2\mathcal{E}(r) \biggr]. \end{aligned}
(2.8)
From part (1) and (2.8) we clearly see that
$$f^{\prime}_{2}(r)< -\frac{r}{(1+r^{2})}< 0$$
(2.9)
for all $$r\in(0, 1)$$. Therefore, part (4) follows from (2.7) and (2.9).
For part (5), let $$f_{3}(r)=r^{2}[2\mathcal{E}(r)-(1-r^{2})\mathcal {K}(r)]/ [(1+r^{2})^{2}(\mathcal{K}(r)-\mathcal{E}(r)) ]$$, then $$f_{3}(r)$$ can be rewritten as
$$f_{3}(r)=\frac{2\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)}{1+r^{2}} \times\frac{1}{\frac{\mathcal{K}(r)-\mathcal{E}(r)}{r^{2}}}\times \frac {1}{1+r^{2}}.$$
(2.10)
Therefore, part (5) follows easily from parts (2) and (4) together with (2.10). □

Lemma 2.4

The function
$$g(r)=\frac{r^{2}\mathcal{K}(r)}{(1+r^{2})[\mathcal{K}(r)-\mathcal{E}(r)]}$$
is strictly decreasing from $$(0, 1)$$ onto $$(1/2, 2)$$.

Proof

Let $$g_{1}(r)=r^{2}\mathcal{K}(r)$$ and $$g_{2}(r)=(1+r^{2})[\mathcal {K}(r)-\mathcal{E}(r)]$$. Then we clearly see that
\begin{aligned}& g_{1}\bigl(0^{+}\bigr)=g_{2} \bigl(0^{+}\bigr)=0, \qquad g(r)=\frac{g_{1}(r)}{g_{2}(r)}, \end{aligned}
(2.11)
\begin{aligned}& g\bigl(1^{-}\bigr)=\frac{1}{2}, \end{aligned}
(2.12)
\begin{aligned}& \frac{g^{\prime}_{1}(r)}{g^{\prime}_{2}(r)}=\frac{1}{2-\frac{3\mathcal {E}(r)}{\frac{\mathcal{E}(r)+(1-r^{2})\mathcal{K}(r)}{1-r^{2}}}}. \end{aligned}
(2.13)
From Lemma 2.3(3), (2.11) and (2.13) we know that
$$g\bigl(0^{+}\bigr)=\lim_{r\rightarrow0^{+}}\frac{g^{\prime}_{1}(r)}{g^{\prime}_{2}(r)}=2$$
(2.14)
and the function $$g^{\prime}_{1}(r)/g^{\prime}_{2}(r)$$ is strictly decreasing on $$(0, 1)$$.

Therefore, Lemma 2.4 follows easily from Lemma 2.1, (2.11), (2.12) and (2.14) together with the monotonicity of the function $$g^{\prime}_{1}(r)/g^{\prime}_{2}(r)$$. □

Lemma 2.5

Let $$u\in[0, 1]$$, $$r\in(0, 1)$$, $$p\in[1, \infty)$$ and
$$h(u, p; r)=\frac{1}{2}p\log \biggl[1-\frac{4ur^{2}}{(1+r^{2})^{2}} \biggr] -\log \biggl[\frac{4 (2\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r) )^{2}}{\pi^{2}(1+r^{2})} \biggr].$$
(2.15)
Then one has
1. (1)

$$h(u, p; r)>0$$ for all $$r\in(0, 1)$$ if and only if $$u\leq1/4p$$;

2. (2)

$$h(u, p; r)<0$$ for all $$r\in(0, 1)$$ if and only if $$u\geq 1-(2\sqrt{2}/\pi)^{4/p}$$.

Proof

It follows from (2.15) that
\begin{aligned}& h\bigl(u, p; 0^{+}\bigr)=0, \end{aligned}
(2.16)
\begin{aligned}& h\bigl(u, p; 1^{-}\bigr)=\frac{p}{2}\log(1-u)+\log \biggl( \frac{\pi^{2}}{8} \biggr), \end{aligned}
(2.17)
\begin{aligned}& \begin{aligned}[b] \frac{\partial h(u, p; r)}{\partial r}&=\frac{2(1-r^{2})[\mathcal {K}(r)-\mathcal{E}(r)]}{ r(1+r^{2})[2\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)]} - \frac{4pur(1-r^{2})}{(1+r^{2}) [(1+r^{2})^{2}-4ur^{2} ]} \\ &=\frac{2(1-r^{2}) [2(\mathcal{K}(r)-\mathcal {E}(r))+p(2\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r)) ]}{(1+r^{2}) [(1+r^{2})^{2}-4ur^{2} ][2\mathcal {E}(r)-(1-r^{2})\mathcal{K}(r)]}\bigl[h_{1}(p; r)-2u\bigr], \end{aligned} \end{aligned}
(2.18)
where
\begin{aligned}[b] h_{1}(p; r)&=\frac{(1+r^{2})^{2}[\mathcal{K}(r)-\mathcal{E}(r)]}{r^{2} [2(\mathcal{K}(r)-\mathcal{E}(r))+p(2\mathcal {E}(r)-(1-r^{2})\mathcal{K}(r)) ]} \\ &=\frac{1}{g(r)+(p-1)f_{3}(r)}, \end{aligned}
(2.19)
where $$f_{3}(r)$$ and $$g(r)$$ are defined by (2.10) and Lemma 2.4, respectively.
From Lemma 2.3(5) and Lemma 2.4 together with (2.19) we clearly see that the function $$r\rightarrow h_{1}(p; r)$$ is strictly increasing on $$(0, 1)$$ and
\begin{aligned}& h_{1}\bigl(p; 0^{+}\bigr)=\frac{1}{2p}, \end{aligned}
(2.20)
\begin{aligned}& h_{1}\bigl(p; 1^{-}\bigr)=2. \end{aligned}
(2.21)
From Lemma 2.2 we know that $$1-(2\sqrt{2}/\pi)^{4/p}>1/(4p)$$. Therefore, we only need to divide the proof into three cases as follows.

Case 1 $$u\leq1/(4p)$$. Then Lemma 2.3(4), (2.18), (2.20) and the monotonicity of the function $$r\rightarrow h_{1}(p; r)$$ on the interval $$(0, 1)$$ lead to the conclusion that the function $$r\rightarrow h(u, p; r)$$ is strictly increasing on $$(0, 1)$$. Therefore, $$h(u, p; r)>0$$ for all $$r\in(0, 1)$$ follows from (2.16) and the monotonicity of the function $$r\rightarrow h(u, p; r)$$.

Case 2 $$u\geq1-(2\sqrt{2}/\pi)^{4/p}$$. Then from Lemma 2.2, Lemma 2.3(5), (2.17), (2.18), (2.20), (2.21) and the monotonicity of the function $$r\rightarrow h_{1}(p; r)$$ on the interval $$(0, 1)$$ we clearly see that there exists $$r_{0}\in(0, 1)$$ such that the function $$r\rightarrow h(u, p; r)$$ is strictly decreasing on $$(0, r_{0})$$ and strictly increasing on $$(r_{0}, 1)$$, and
$$h\bigl(u, p; 1^{-}\bigr)\leq0.$$
(2.22)
Therefore, $$h(u, p; r)<0$$ for all $$r\in(0, 1)$$ follows from (2.16) and (2.22) together with the piecewise monotonicity of the function $$r\rightarrow h(u, p; r)$$ on the interval $$(0, 1)$$.
Case 3 $$1/(4p)< u<1-(2\sqrt{2}/\pi)^{4/p}$$. Then (2.17) leads to
$$h\bigl(u, p; 1^{-}\bigr)>0.$$
(2.23)
It follows from Lemma 2.3(5), (2.18), (2.20), (2.21) and the monotonicity of the function $$r\rightarrow h_{1}(p; r)$$ on the interval $$(0, 1)$$ that there exists $$r^{\ast}\in(0, 1)$$ such that the function $$r\rightarrow h(u, p; r)$$ is strictly decreasing on $$(0, r^{\ast})$$ and strictly increasing on $$(r^{\ast}, 1)$$. Therefore, there exists $$\lambda \in(0, 1)$$ such that $$h(u, p; r)<0$$ for $$r\in(0, \lambda)$$ and $$h(u, p; r)>0$$ for $$r\in(\lambda, 1)$$. □

3 Main result

Theorem 3.1

Let $$\lambda, \mu\in[0, 1/2]$$. Then the double inequality
$$\begin{gathered} G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a\bigr]A^{1-p}(a,b) \\ \quad< E(a,b)< G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{gathered}$$
holds for any $$p\in[1, \infty)$$ and all $$a, b>0$$ with $$a\neq b$$ if and only if $$\lambda\leq1/2-\sqrt{1-(2\sqrt{2}/\pi)^{4/p}}/2$$ and $$\mu\geq1/2-\sqrt{p}/(4p)$$.

Proof

Let $$t\in[0, 1/2]$$, since $$G^{p}[ta+(1-t)b, tb+(1-t)a]A^{1-p}(a,b)$$ and $$E(a,b)$$ are symmetric and homogeneous of degree one, without loss of generality, we assume that $$a>b>0$$. Let $$r\in(0, 1)$$ and $$b/a=(1-r)^{2}/(1+r)^{2}$$. Then (1.1) leads to
$$\begin{gathered} E(a,b)=\frac{4(1+r)^{2}}{\pi^{2}(1+r^{2})}A(a,b){\mathcal{E}}^{2} \biggl( \frac{2\sqrt{r}}{1+r} \biggr) =\frac{4}{\pi^{2}}A(a,b)\frac{ [2\mathcal{E}(r)-(1-r^{2})\mathcal {K}(r) ]^{2}}{1+r^{2}}, \\ \log \bigl[G^{p}\bigl(ta+(1-t)b, tb+(1-t)a \bigr)A^{1-p}(a,b) \bigr]-\log E(a,b) \\ \quad=\log \biggl[\frac{G^{p}(ta+(1-t)b, tb+(1-t)a)A^{1-p}(a,b)}{A(a,b)} \biggr]-\log \biggl[\frac {E(a,b)}{A(a,b)} \biggr] \\ \quad=\frac{1}{2}p\log \biggl[1-\frac {4(1-2t)^{2}r^{2}}{(1+r^{2})^{2}} \biggr] -\log \biggl[ \frac{4 (2\mathcal{E}(r)-(1-r^{2})\mathcal{K}(r) )^{2}}{\pi^{2}(1+r^{2})} \biggr]. \end{gathered}$$
(3.1)
Therefore, Theorem 3.1 follows easily from Lemma 2.5 and (3.1). □

Let $$p=1, 2$$, then Theorem 3.1 leads to Corollary 3.2 immediately.

Corollary 3.2

Let $$\lambda_{1}, \mu_{1}, \lambda_{2}, \mu_{2}\in[0, 1/2]$$. Then the double inequalities
$$\begin{gathered} H\bigl[\lambda_{1}a+(1- \lambda_{1})b, \lambda_{1}b+(1-\lambda _{1})a \bigr]< E(a,b)< H\bigl[\mu_{1}a+(1-\mu_{1})b, \mu_{1}b+(1-\mu_{1})a\bigr], \\ G\bigl[\lambda_{2}a+(1-\lambda_{2})b, \lambda_{2}b+(1- \lambda _{2})a\bigr]< E(a,b)< G\bigl[\mu_{2}a+(1- \mu_{2})b, \mu_{2}b+(1-\mu_{2})a\bigr] \end{gathered}$$
hold for all $$a, b>0$$ with $$a\neq b$$ if and only if $$\lambda_{1}\leq 1/2-\sqrt{1-8/\pi^{2}}/2=0.2823\ldots$$ , $$\mu_{1}\geq1/2-\sqrt{2}/8=0.3232\ldots$$ , $$\lambda_{2}\leq1/2-\sqrt {1-64/\pi^{4}}/2=0.2071\ldots$$ and $$\mu_{2}\geq1/4$$.

Let $$p\in[1, \infty)$$, $$r\in(0, 1)$$, $$a=r$$, $$b=1-r^{2}={r^{\prime }}^{2}$$, $$\lambda=1/2-\sqrt{1-(2\sqrt{2}/\pi)^{4/p}}/2$$ and $$\mu=1/2-\sqrt{p}/(4p)$$. Then (1.1) and Theorem 3.1 lead to Corollary 3.3 immediately.

Corollary 3.3

The double inequality
$$\begin{gathered} \frac{\sqrt{2}\pi}{4} \bigl(1+{r^{\prime}}^{2} \bigr)^{(1-p)/2} \biggl[4{r^{\prime}}^{2} + \biggl( \frac{8}{\pi^{2}} \biggr)^{2/p}r^{4} \biggr]^{p/4} \\ \quad< \mathcal{E}(r) < \frac{\sqrt{2}\pi}{4} \bigl(1+{r^{\prime }}^{2} \bigr)^{(1-p)/2} \biggl[\bigl(1+{r^{\prime}}^{2} \bigr)^{2}-\frac {r^{4}}{4p} \biggr]^{p/4} \end{gathered}$$
holds for all $$r\in(0, 1)$$ and $$p\in[1, \infty)$$.

4 Results and discussion

In this paper, we provide the sharp bounds for the special quasi-arithmetic mean $$E(a,b)$$ in terms of the arithmetic mean $$A(a,b)$$ and geometric mean $$G(a,b)$$ with two parameters. As consequences, we present the best possible one-parameter harmonic and geometric means bounds for $$E(a,b)$$ and find new bounds for the complete elliptic integral of the second kind.

5 Conclusion

In the article, we derive a new bivariate mean $$E(a,b)$$ from the quasi-arithmetic mean and provide its sharp upper and lower bounds in terms of the concave combination of arithmetic and geometric means.

Declarations

Acknowledgements

The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191) and the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101).

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. 