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# Optimal power mean bounds for the second Yang mean

Journal of Inequalities and Applications20162016:31

https://doi.org/10.1186/s13660-016-0970-y

• Received: 21 July 2015
• Accepted: 14 January 2016
• Published:

## Abstract

In this paper, we present the best possible parameters p and q such that the double inequality
$$M_{p}(a,b)< V(a,b)< M_{q}(a,b)$$
holds for all $$a, b>0$$ with $$a\neq b$$, where $$M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$$ ($$r\neq0$$) and $$M_{0}(a,b)= \sqrt {ab}$$ is the rth power mean and $$V(a,b)=(a-b)/[\sqrt{2}\sinh^{-1}((a-b)/\sqrt{2ab})]$$ is the second Yang mean.

## Keywords

• power mean
• second Yang mean
• arithmetic mean
• geometric mean
• Lehmer mean
• first Seiffert mean
• logarithmic mean

• 26E60

## 1 Introduction

For $$r\in\mathbb{R}$$, the rth power mean $$M_{r}(a,b)$$ of two distinct positive real numbers a and b is defined by
$$M_{r}(a,b)= \textstyle\begin{cases} (\frac{a^{r}+b^{r}}{2} )^{1/r},& r\neq0, \\ \sqrt{ab}, & r=0. \end{cases}$$
(1.1)

It is well known that $$M_{r}(a,b)$$ is continuous and strictly increasing with respect to $$r\in\mathbb{R}$$ for fixed $$a, b>0$$ with $$a\neq b$$. Many classical means are special cases of the power mean, for example, $$M_{-1}(a,b)=2ab/(a+b)=H(a,b)$$ is the harmonic mean, $$M_{0}(a,b)=\sqrt{ab}=G(a,b)$$ is the geometric mean, $$M_{1}(a,b)=(a+b)/2=A(a,b)$$ is the arithmetic mean, and $$M_{2}(a,b)=\sqrt{(a^{2}+b^{2})/2}=Q(a,b)$$ is the quadratic mean. The main properties for the power mean are given in .

Let
\begin{aligned}& L(a,b)=\frac{a-b}{\log a-\log b},\quad\quad I(a,b)=\frac{1}{e} \biggl(\frac{a^{a}}{b^{b}} \biggr)^{1/(a-b)},\quad\quad P(a,b)=\frac{a-b}{2\arcsin(\frac{a-b}{a+b} )}, \\& U(a,b)=\frac{a-b}{\sqrt{2}\arctan(\frac{a-b}{\sqrt{2ab}} )},\quad\quad T^{\ast }(a,b)=\frac{2}{\pi} \int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2} \theta+b^{2}\sin^{2}\theta}d\theta, \\& NS(a,b)=\frac{a-b}{2\sinh^{-1} (\frac{a-b}{a+b} )},\quad\quad X(a,b)=A(a,b)e^{G(a,b)/P(a,b)-1}, \\& T(a,b)=\frac{a-b}{2\arctan(\frac{a-b}{a+b} )},\quad\quad B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}, \end{aligned}
and
$$V(a,b)=\frac{a-b}{\sqrt{2}\sinh^{-1} (\frac{a-b}{\sqrt{2ab}} )}$$
(1.2)
be, respectively, the logarithmic mean, identric mean, first Seiffert mean , first Yang mean , Toader mean , Neuman-Sándor mean [5, 6], Sándor mean , second Seiffert mean , Sándor-Yang mean , and second Yang mean  of two distinct positive real numbers a and b, where $$\sinh^{-1}(x)=\log(x+\sqrt {x^{2}+1})$$ is the inverse hyperbolic sine function.
Recently, the bounds for certain bivariate means in terms of the power mean have attracted the attention of many mathematicians. Radó  (see also ) proved that the double inequalities
\begin{aligned} &M_{p}(a,b)< L(a,b)< M_{q}(a,b), \\ &M_{\lambda}(a,b)< I(a,b)< M_{\mu}(a,b) \end{aligned}
(1.3)
hold for all $$a, b>0$$ with $$a\neq b$$ if and only if $$p\leq0$$, $$q\geq 1/3$$, $$\lambda\leq2/3$$, and $$\mu\geq\log2$$.
In , the authors proved that the double inequality
$$M_{p}(a,b)< T^{\ast}(a,b)< M_{q}(a,b)$$
holds for all $$a, b>0$$ with $$a\neq b$$ if and only if $$p\leq3/2$$ and $$q\geq\log2/(\log\pi-\log2)$$.
Jagers , Hästö [18, 19], Yang , and Costin and Toader  proved that $$p_{1}=\log2/\log\pi$$, $$q_{1}=2/3$$, $$p_{2}=\log 2/(\log\pi-\log2)$$, and $$q_{2}=5/3$$ are the best possible parameters such that the double inequalities
\begin{aligned} &M_{p_{1}}(a,b)< P(a,b)< M_{q_{1}}(a,b), \\ &M_{p_{2}}(a,b)< T(a,b)< M_{q_{2}}(a,b) \end{aligned}
(1.4)
hold for all $$a, b>0$$ with $$a\neq b$$.
In , the authors proved that the double inequalities
\begin{aligned}& M_{\lambda_{1}}(a,b)< NS(a,b)< M_{\mu_{1}}(a,b), \\& M_{\lambda_{2}}(a,b)< U(a,b)< M_{\mu_{2}}(a,b), \\& M_{\lambda_{3}}(a,b)< X(a,b)< M_{\mu_{3}}(a,b), \end{aligned}
hold for all $$a, b>0$$ with $$a\neq b$$ if and only if $$\lambda_{1}\leq \log2/\log[2\log(1+\sqrt{2})]$$, $$\mu_{1}\geq4/3$$, $$\lambda_{2}\leq 2\log2/(2\log\pi-\log2)$$, $$\mu_{2}\geq4/3$$, $$\lambda_{3}\leq1/3$$, and $$\mu_{3}\geq\log2/(1+\log2)$$.
Very recently, Yang and Chu  showed that $$p=4\log2/(4+2\log2-\pi )$$ and $$q=4/3$$ are the best possible parameters such that the double inequality
$$M_{p}(a,b)< B(a,b)< M_{q}(a,b)$$
holds for all $$a, b>0$$ with $$a\neq b$$.
The main purpose of this paper is to present the best possible parameters p and q such that the double inequality
$$M_{p}(a,b)< V(a,b)< M_{q}(a,b)$$
holds for all $$a, b>0$$ with $$a\neq b$$.

## 2 Lemmas

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

### Lemma 2.1

Let $$t>0$$, $$p\in\mathbb{R}$$, and
\begin{aligned} f(t, p) =& 2\sinh\bigl[2(p-1)t\bigr] +\sinh\bigl[2(p+1)t\bigr]+\sinh\bigl[2(p-2)t \bigr] \\ &{} +p\sinh(4t)-\sinh(2t). \end{aligned}
(2.1)
Then the following statements are true:
1. (i)

$$f(t,p)>0$$ for all $$t>0$$ if and only if $$p\geq2/3$$;

2. (ii)

$$f(t,p)<0$$ for all $$t>0$$ if and only if $$p\leq0$$.

### Proof

It follows from (2.1) that
\begin{aligned} \frac{\partial f(t,p)}{\partial t} =&\sinh(4t)+4t\cosh\bigl[2(p-1)t\bigr] +2t\cosh\bigl[2(p+1)t\bigr]+2t\cosh\bigl[2(p-2)t\bigr] \\ >&0 \end{aligned}
(2.2)
for all $$t>0$$ and $$p\in\mathbb{R}$$.
(i) If $$f(t,p)>0$$ for all $$t>0$$, then (2.1) leads to
$$\lim_{t\rightarrow0^{+}}\frac{f(t, p)}{t}=12 \biggl(p-\frac{2}{3} \biggr)\geq0,$$
which gives $$p\geq2/3$$.
If $$p\geq2/3$$, then (2.1) and (2.2) lead to the conclusion that
\begin{aligned} f(t,p) \geq& f \biggl(t, \frac{2}{3} \biggr)=\frac{2}{3}\sinh(4t)- \sinh(2t)-2\sinh\biggl(\frac{2}{3}t \biggr) -\sinh\biggl( \frac{8}{3}t \biggr)+\sinh\biggl(\frac{10}{3}t \biggr) \\ =&\frac{8}{3}\sinh^{3} \biggl(\frac{2}{3}t \biggr) \cosh\biggl(\frac{2}{3}t \biggr) \biggl[8\cosh^{2} \biggl( \frac{2}{3}t \biggr) +6\cosh\biggl(\frac{2}{3}t \biggr)-3 \biggr]>0 \end{aligned}
for all $$t>0$$.
(ii) If $$f(t, p)<0$$ for all $$t>0$$, then from part (i) we know that $$p<2/3$$. We assert that $$p\leq0$$, otherwise $$0< p<2/3$$ and (2.1) leads to
\begin{aligned}& \lim_{t\rightarrow+\infty}\frac{f(t, p)}{e^{4t}} \\& \quad=\lim_{t\rightarrow+\infty}\frac{-2\sinh[2(1-p)t]+\sinh[2(1+p)t]-\sinh [2(2-p)t]+p\sinh(4t)-\sinh(2t)}{e^{4t}} \\& \quad=\frac{p}{2}>0, \end{aligned}
which contradicts with $$f(t, p)<0$$ for all $$t>0$$.
If $$p\leq0$$, then from (2.1) and (2.2) we have
$$f(t,p)\leq f(t, 0)=-2\sinh(2t)-\sinh(4t)< 0$$
for all $$t>0$$. □

### Lemma 2.2

The double inequality
$$\bigl[\cosh(pt)\bigr]^{1/p}< \frac{\sqrt{2}\sinh(t)}{\sinh^{-1}[\sqrt {2}\sinh(t)]}< \bigl[\cosh(qt) \bigr]^{1/q}$$
(2.3)
holds for all $$t>0$$ if and only if $$p\leq0$$ and $$q\geq2/3$$. Here
$$\bigl[\cosh(pt)\bigr]^{1/p}\big|_{p=0}:=\lim_{p\rightarrow0}\bigl[\cosh(pt)\bigr]^{1/p}.$$

### Proof

Let $$t>0$$, $$p\in\mathbb{R}$$ and $$F(t, p)$$ be defined by
$$F(t, p)=\log\biggl[\frac{\sqrt{2}\sinh(t)}{\sinh^{-1} (\sqrt{2}\sinh (t) )} \biggr]-\frac{1}{p}\log\bigl[ \cosh(pt)\bigr].$$
(2.4)
Then making use of the power series formulas
\begin{aligned}& \sinh(t)=t+\frac{t^{3}}{3!}+\frac{t^{5}}{5!}+\frac{t^{7}}{7!}+\cdots =\sum _{n=0}^{\infty}\frac{t^{2n+1}}{(2n+1)!}, \\& \cosh(t)=1+\frac{t^{2}}{2!}+\frac{t^{4}}{4!}+\frac{t^{6}}{6!}+\cdots =\sum _{n=0}^{\infty}\frac{t^{2n}}{(2n)!}, \\& \sinh^{-1}(t)=t-\frac{1}{2}\times\frac{t^{3}}{3}+ \frac{1\times3}{2\times4}\times\frac{t^{5}}{5}-\frac{1\times3\times 5}{2\times4\times6}\times \frac{t^{7}}{7}+\cdots \\& \hphantom{\sinh^{-1}(t)}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n)!t^{2n+1}}{2^{2n}(n!)^{2}(2n+1)} \end{aligned}
we get
$$\log\biggl[\frac{\sqrt{2}\sinh(t)}{\sinh^{-1} (\sqrt{2}\sinh(t) )} \biggr]=\frac{t^{2}}{3}+o \bigl(t^{2} \bigr),\quad\quad \frac{1}{p}\log\bigl[\cosh(pt)\bigr]=- \frac{1}{2}pt^{2}+o\bigl(t^{2}\bigr)$$
(2.5)
for $$t\rightarrow0^{+}$$.
It follows from (2.4) and (2.5) that
\begin{aligned}& F\bigl(0^{+}, p\bigr)=0, \end{aligned}
(2.6)
\begin{aligned}& \frac{\partial F(t, p)}{\partial t}=\frac{\cosh[(p-1)t]}{\sinh(t)\cosh (pt)\sinh^{-1}[\sqrt{2}\sinh(t)]}f_{1}(t, p), \end{aligned}
(2.7)
where
\begin{aligned}& f_{1}(t, p)=\sinh^{-1}\bigl[\sqrt{2}\sinh(t)\bigr]- \frac{\sqrt{2}\sinh(t)\cosh(pt)\cosh(t)}{\sqrt{\cosh(2t)}\cosh[(p-1)t]}, \end{aligned}
(2.8)
\begin{aligned}& f_{1}(0, p)=0, \end{aligned}
(2.9)
\begin{aligned}& \frac{\partial f_{1}(t, p)}{\partial t}=-\frac{\sqrt{2}\sinh(t)}{4[\cosh (2t)]^{3/2}\cosh^{2}[(p-1)t]}f(t,p), \end{aligned}
(2.10)
where $$f(t,p)$$ is defined by Lemma 2.1.
$$\lim_{t\rightarrow0}\frac{F(t,p)}{t^{2}}=-\frac{1}{2} \biggl(p- \frac{2}{3} \biggr)$$
(2.11)
and
$$\lim_{t\rightarrow+\infty}F(t,p)=-\infty$$
(2.12)
if $$p>0$$.
We first prove that the inequality
$$\frac{\sqrt{2}\sinh(t)}{\sinh^{-1}[\sqrt{2}\sinh(t)]}< \bigl[\cosh (pt)\bigr]^{1/p}$$
(2.13)
holds for all $$t>0$$ if and only if $$p\geq2/3$$.

If $$p\geq2/3$$, then inequality (2.13) holds for all $$t>0$$ follows easily from Lemma 2.1(i), (2.4), (2.6), (2.7), (2.9), and (2.10).

If inequality (2.13) holds for all $$t>0$$, then (2.4) and (2.11) lead to $$p\geq2/3$$.

Next, we prove that the inequality
$$\frac{\sqrt{2}\sinh(t)}{\sinh^{-1}[\sqrt{2}\sinh(t)]}>\bigl[\cosh (pt)\bigr]^{1/p}$$
(2.14)
holds for all $$t>0$$ if and only if $$p\leq0$$.

If $$p\leq0$$, then that inequality (2.14) holds for all $$t>0$$ follows easily from Lemma 2.1(ii), (2.4), (2.6), (2.7), (2.9), and (2.10).

If inequality (2.14) holds for all $$t>0$$, then (2.4) leads to $$F(t,p)>0$$. We assert that $$p\leq0$$, otherwise $$p>0$$ and (2.12) implies that there exists large enough $$T_{0}>0$$ such that $$F(t, p)<0$$ for $$t\in(T_{0}, \infty)$$. □

### Lemma 2.3

Let $$t>0$$, $$p\in\mathbb{R}$$, and $$f_{1}(t,p)$$ be defined by (2.8). Then the following statements are true:
1. (i)

$$f_{1}(t,p)<0$$ for all $$t>0$$ if and only if $$p\geq2/3$$;

2. (ii)

$$f(t,p)>0$$ for all $$t>0$$ if and only if $$p\leq0$$.

### Proof

(i) If $$p\geq2/3$$, then $$f_{1}(t,p)<0$$ for all $$t>0$$ follows easily from (2.9) and (2.10) together with Lemma 2.1(i).

If $$f_{1}(t,p)<0$$ for all $$t>0$$, then (2.8) leads to
$$\lim_{t\rightarrow0}\frac{f_{1}(t,p)}{t^{3}}=\frac{-\sqrt{2} (p-\frac {2}{3} )t^{3}+o(t^{3})}{t^{3}}=-\sqrt{2} \biggl(p-\frac{2}{3} \biggr)\leq0,$$
which gives $$p\geq2/3$$.

(ii) If $$p\leq0$$, then $$f_{1}(t,p)>0$$ for all $$t>0$$ follows easily from (2.9) and (2.10) together with Lemma 2.1(ii).

Note that
\begin{aligned}& \frac{f_{1}(t,p)}{e^{(|p|-|p-1|)t}\sinh(t)} \\& \quad=\frac{\sinh^{-1}[\sqrt{2}\sinh(t)]}{e^{(|p|-|p-1|)t}\sinh(t)}-\frac {\sqrt{2}\cosh(t)\cosh(pt)}{e^{(|p|-|p-1|)t}\cosh[(p-1)t]\sqrt{\cosh(2t)}} \\& \quad=\frac{\log[\sqrt{2}\sinh(t)+\sqrt{\cosh(2t)} ]}{e^{(|p|-|p-1|)t}\sinh (t)}-\frac{\sqrt{2} (1+e^{-2|p|t} )\cosh(t)}{ (1+e^{-2|p-1|t} )\sqrt{\cosh(2t)}}. \end{aligned}
(2.15)
If $$f_{1}(t,p)>0$$ for all $$t>0$$, then
$$\lim_{t\rightarrow+\infty}\frac{f_{1}(t,p)}{e^{(|p|-|p-1|)t}\sinh (t)}\geq0$$
and we assert that $$p\leq0$$. Otherwise, equation (2.15) leads to
$$\lim_{t\rightarrow+\infty}\frac{f_{1}(t,p)}{e^{(|p|-|p-1|)t}\sinh (t)}=-\frac{\sqrt{2}}{2}< 0$$
if $$p=1$$ and
$$\lim_{t\rightarrow+\infty}\frac{f_{1}(t,p)}{e^{(|p|-|p-1|)t}\sinh (t)}=-\sqrt{2}< 0$$
if $$p\in(0, 1)\cup(1, \infty)$$. □

## 3 Main results

### Theorem 3.1

The double inequality
$$M_{p}(a,b)< V(a,b)< M_{q}(a,b)$$
holds for all $$a, b>0$$ with $$a\neq b$$ if and only if $$p\leq0$$ and $$q\geq2/3$$.

### Proof

Since both $$M_{r}(a,b)$$ and $$V(a,b)$$ are symmetric and homogeneous of degree 1, without loss of generality, we assume that $$a>b>0$$. Let $$t=\frac{1}{2}\log(a/b)>0$$ and $$r\in\mathbb{R}$$, then (1.1) and (1.2) lead to
$$V(a,b)=\sqrt{ab}V \biggl(\sqrt{\frac{a}{b}}, \sqrt{\frac{b}{a}} \biggr)=\sqrt{ab}V \bigl(e^{t}, e^{-t} \bigr) = \frac{\sqrt{2ab}\sinh(t)}{\sinh^{-1}[\sqrt{2}\sinh(t)]}$$
(3.1)
and
$$M_{r}(a,b)=\sqrt{ab}M_{r} \biggl(\sqrt{\frac{a}{b}}, \sqrt{\frac{b}{a}} \biggr)=\sqrt{ab}M_{r} \bigl(e^{t}, e^{-t} \bigr) =\sqrt{ab}\bigl[\cosh(rt)\bigr]^{1/r}.$$
(3.2)

Therefore, Theorem 3.1 follows easily from (3.1) and (3.2) together with Lemma 2.2. □

### Theorem 3.2

The double inequality
$$\frac{a^{p-1}+b^{p-1}}{a^{p}+b^{p}}\frac{ab\sqrt{2 (a^{2}+b^{2} )}}{a+b}< V(a,b) < \frac{a^{q-1}+b^{q-1}}{a^{q}+b^{q}} \frac{ab\sqrt{2 (a^{2}+b^{2} )}}{a+b}$$
holds for all $$a, b>0$$ with $$a\neq b$$ if and only if $$p\geq2/3$$ and $$q\leq0$$.

### Proof

Without loss of generality, we assume that $$a>b>0$$. Let $$t=\frac{1}{2}\log(a/b)>0$$ and $$r\in\mathbb{R}$$, then
$$\frac{a^{r-1}+b^{r-1}}{a^{r}+b^{r}}\frac{ab\sqrt{2 (a^{2}+b^{2} )}}{a+b}=\frac{\sqrt{ab}\cosh[(r-1)t]\sqrt{\cosh(2t)}}{\cosh(t)\cosh(rt)}.$$
(3.3)

Therefore, Theorem 3.2 follows easily from (3.1) and (3.3) together with Lemma 2.3. □

Let $$p\in\mathbb{R}$$ and $$a, b>0$$. Then the pth Lehmer mean  $$L_{p}(a,b)=\frac{a^{p+1}+b^{p+1}}{a^{p}+b^{p}}$$ is strictly increasing with respect to $$p\in\mathbb{R}$$ for fixed $$a, b>0$$ with $$a\neq b$$. From Theorem 3.2 we get Corollary 3.3 immediately.

### Corollary 3.3

The double inequality
$$\frac{Q(a,b)G^{2}(a,b)}{A(a,b)L_{p-1}(a,b)}< V(a,b)< \frac {Q(a,b)G^{2}(a,b)}{A(a,b)L_{q-1}(a,b)}$$
holds for all $$a, b>0$$ with $$a\neq b$$ if and only if $$p\geq2/3$$ and $$q\leq0$$.

Let $$p=2/3, 1, 2, +\infty$$ and $$q=0, -1/2, -1, -2, -\infty$$. Then Corollary 3.3 leads to

### Corollary 3.4

The inequalities
\begin{aligned} \min\{a, b\}\frac{Q(a,b)}{A(a,b)} < &\frac{G^{2}(a,b)}{Q(a,b)}< \frac {G^{2}(a,b)Q(a,b)}{A^{2}(a,b)} \\ < &\frac {Q(a,b)G^{4/3}(a,b)M^{1/3}_{1/3}(a,b)}{A(a,b)M^{2/3}_{2/3}(a,b)}< V(a,b) < Q(a,b) \\ < &\frac{Q(a,b)[2A(a,b)-G(a,b)]}{A(a,b)} \\ < &\frac{Q^{3}(a,b)}{A^{2}(a,b)} < \frac {2Q^{2}(a,b)-G^{2}(a,b)}{Q(a,b)}< \max\{a, b\}\frac{Q(a,b)}{A(a,b)} \end{aligned}
hold for all $$a, b>0$$ with $$a\neq b$$.
From (1.3), (1.4), and Theorem 3.1 we clearly see that $$M_{2/3}(a,b)$$ is the sharp upper power mean bound for the 2-order generalized logarithmic mean $$L^{1/2}(a^{2}, b^{2})$$, the first Seiffert mean $$P(a,b)$$, and the second Yang mean $$V(a,b)$$. In , Theorem 3, Yang and Chu proved that the inequality
$$P(a,b)>L^{1/r} \bigl(a^{r}, b^{r} \bigr)$$
(3.4)
holds for all $$a, b>0$$ with $$a\neq b$$ if and only if $$r\leq2$$.

As a result of comparing $$V(a,b)$$ with $$L^{1/2} (a^{2}, b^{2} )$$, we have the following.

### Theorem 3.5

The inequality
$$V(a,b)< L^{1/2} \bigl(a^{2}, b^{2} \bigr)$$
holds for all $$a, b>0$$ with $$a\neq b$$.

### Proof

We assume that $$a>b$$. Let $$t=\frac{1}{2}\log(a/b)>0$$, then
$$L^{1/2} \bigl(a^{2}, b^{2} \bigr)= \biggl( \frac{a^{2}-b^{2}}{2(\log a-\log b)} \biggr)^{1/2}=\sqrt{ab}\sqrt{\frac {\sinh(2t)}{2t}}.$$
(3.5)
It follows from (3.1) and (3.5) that
\begin{aligned}& L^{1/2} \bigl(a^{2}, b^{2} \bigr)-V(a,b) \\& \quad=\frac{\sqrt{ab}\sqrt{\sinh(2t)}}{\sqrt{2t}\sinh^{-1} (\sqrt{2}\sinh(t) )} \bigl[\sinh^{-1} \bigl(\sqrt{2}\sinh(t) \bigr)- \sqrt{2t}\tanh(t) \bigr]. \end{aligned}
(3.6)
Let
$$g(t)=\sinh^{-1} \bigl(\sqrt{2}\sinh(t) \bigr)-\sqrt{2t}\tanh(t).$$
(3.7)
Then simple computation leads to
\begin{aligned}& g(0)=0, \end{aligned}
(3.8)
\begin{aligned}& g^{\prime}(t)=\sqrt{2} \biggl(\frac{\cosh(t)}{\sqrt{\cosh(2t)}}-\frac {t+\sinh(t)\cosh(t)}{2\cosh^{2}(t)\sqrt{t\tanh(t)}} \biggr), \end{aligned}
(3.9)
\begin{aligned}& \biggl(\frac{\cosh(t)}{\sqrt{\cosh(2t)}} \biggr)^{2}- \biggl( \frac{t+\sinh(t)\cosh(t)}{2\cosh^{2}(t)\sqrt{t\tanh(t)}} \biggr)^{2} \\& \quad=\frac{\cosh^{2}(t)}{\cosh(2t)}-\frac{(t+\sinh(t)\cosh(t))^{2}}{4t\sinh (t)\cosh^{3}(t)} \\& \quad=\frac{(2t\cosh(2t)-\sinh(2t))(\sinh(2t)\cosh(2t)-2t)}{16t\sinh(t)\cosh (2t)\cosh^{3}(t)} \\& \quad =\frac{\sinh(4t)-4t}{16t\sinh(t)\cosh(2t)\cosh^{3}(t)} \biggl(\cosh(2t)- \frac{\sinh(2t)}{2t} \biggr)>0 \end{aligned}
(3.10)
for $$t>0$$.

Therefore, Theorem 3.5 follows easily from (3.6)-(3.10). □

### Remark 3.6

From (1.4), (3.4), Theorems 3.1, and 3.5 we get the inequalities
$$M_{0}(a,b)< V(a,b)< L^{1/2} \bigl(a^{2}, b^{2} \bigr)< P(a,b)< M_{2/3}(a,b)$$
for all $$a, b>0$$ with $$a\neq b$$.

## Declarations

### Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. The research was supported by the Major Project Foundation of the Department of Education of Hunan Province under Grant 12A026. 