# New inequalities for hyperbolic functions and their applications

## Abstract

In this paper, we obtain some new inequalities in the exponential form for the whole of the triples about the four functions $\left\{1,\left(sinht\right)/t,exp\left(tcotht-1\right),cosht\right\}$. Then we generalize some well-known inequalities for the arithmetic, geometric, logarithmic, and identric means to obtain analogous inequalities for their p th powers, where $p>0$.

MSC: 26E60, 26D07.

## 1 Introduction

Let sinht, cosht, and cotht be the hyperbolic sine, hyperbolic cosine, and hyperbolic cotangent, respectively. It is well known that (see )

$1<\frac{sinht}{t}<{e}^{tcotht-1}
(1.1)

holds for all $t\ne 0$.

In the recent paper , we have established the following Cusa-type inequalities of exponential type for the triple $\left\{1,\left(sinht\right)/t,cosht\right\}$ described as follows.

Theorem 1.1 (Cusa-type inequalities [, Part (i) of Theorem 1.1])

Let $p\ge 4/5$, and $t\ne 0$. Then the double inequality

$\left(1-\lambda \right)+\lambda {\left(cosht\right)}^{p}<{\left(\frac{sinht}{t}\right)}^{p}<\left(1-\eta \right)+\eta {\left(cosht\right)}^{p}$
(1.2)

holds if and only if $\eta \ge 1/3$ and $\lambda \le 0$.

On the other hand, the author of this paper  obtains the following inequalities of exponential type for the triple $\left\{1,exp\left(tcotht-1\right),cosht\right\}$.

Theorem 1.2 ([, Theorem 2])

Let $p>0$, and $t\ne 0$. Then

(1) if $0, the double inequality

$\alpha {\left(cosht\right)}^{p}+\left(1-\alpha \right)<{e}^{p\left(tcotht-1\right)}<\beta {\left(cosht\right)}^{p}+\left(1-\beta \right)$
(1.3)

holds if and only if $\alpha \le 2/3$ and $\beta \ge {\left(2/e\right)}^{p}$;

(2) if $p\ge 2$, the double inequality

$\alpha {\left(cosht\right)}^{p}+\left(1-\alpha \right)<{e}^{p\left(tcotht-1\right)}<\beta {\left(cosht\right)}^{p}+\left(1-\beta \right)$
(1.4)

holds if and only if $\alpha \le {\left(2/e\right)}^{p}$ and $\beta \ge 2/3$.

Next, we do the work for each of the triples $\left\{\left(sinht\right)/t,exp\left(tcotht-1\right),cosht\right\}$ and $\left\{1,\left(sinht\right)/t,exp\left(tcotht-1\right)\right\}$, and obtain the following two new results.

Theorem 1.3 Let $0, and $t\ne 0$. Then

$\alpha {\left(cosht\right)}^{p}+\left(1-\alpha \right){\left(\frac{sinht}{t}\right)}^{p}<{e}^{p\left(tcotht-1\right)}<\beta {\left(cosht\right)}^{p}+\left(1-\beta \right){\left(\frac{sinht}{t}\right)}^{p}$
(1.5)

holds if and only if $\alpha \le 1/2$ and $\beta \ge {\left(2/e\right)}^{p}$.

Theorem 1.4 Let $p\ge 286/693$, and $t\ne 0$. Then

$\alpha +\left(1-\alpha \right){e}^{p\left(tcotht-1\right)}<{\left(\frac{sinht}{t}\right)}^{p}<\beta +\left(1-\beta \right){e}^{p\left(tcotht-1\right)}$
(1.6)

holds if and only if $\beta \le 1/2$ and $\alpha \ge 1$.

In this paper, we shall give the elementary proofs of Theorem 1.3 and Theorem 1.4. In the last section, we apply Theorems 1.1-1.4 to obtain some new results for four classical means.

## 2 Lemmas

Lemma 2.1 ()

Let $f,g:\left[a,b\right]\to \mathbb{R}$ be two continuous functions which are differentiable on $\left(a,b\right)$. Further, let ${g}^{\prime }\ne 0$ on $\left(a,b\right)$. If ${f}^{\prime }/{g}^{\prime }$ is increasing (or decreasing) on $\left(a,b\right)$, then the functions $\left(f\left(x\right)-f\left({b}^{-}\right)\right)/\left(g\left(x\right)-g\left({b}^{-}\right)\right)$ and $\left(f\left(x\right)-f\left({a}^{+}\right)\right)/\left(g\left(x\right)-g\left({a}^{+}\right)\right)$ are also increasing (or decreasing) on $\left(a,b\right)$.

Lemma 2.2 Let $t\in \left(0,+\mathrm{\infty }\right)$. Then the inequality

$D\left(t\right)\triangleq t{sinh}^{5}t+2t{sinh}^{3}t+{t}^{4}cosht-{sinh}^{4}tcosht-{t}^{3}{sinh}^{3}t-2{t}^{3}sinht>0$

holds.

Proof Using the power series expansions of the functions sinh5t, sinh3t, cosht, ${sinh}^{4}tcosht$, and sinht, we have

$\begin{array}{rcl}D\left(t\right)& =& \frac{1}{16}t\left(sinh5t-5sinh3t+10sinht\right)+\frac{1}{2}t\left(sinh3t-3sinht\right)+{t}^{4}cosht\\ -\frac{1}{16}\left(cosh5t-3cosh3t+2cosht\right)-\frac{1}{4}{t}^{3}\left(sinh3t-3sinht\right)-2{t}^{3}sinht\\ =& \frac{1}{16}\sum _{n=0}^{\mathrm{\infty }}\frac{{5}^{2n+1}-5\cdot {3}^{2n+1}+10}{\left(2n+2\right)!}{t}^{2n+1}+\frac{1}{2}\sum _{n=0}^{\mathrm{\infty }}\frac{{3}^{2n+1}-3}{\left(2n+1\right)!}{t}^{2n+2}+\sum _{n=0}^{\mathrm{\infty }}\frac{1}{\left(2n\right)!}{t}^{2n+4}\\ -\frac{1}{16}\sum _{n=0}^{\mathrm{\infty }}\frac{{5}^{2n}-3\cdot {3}^{2n}+2}{\left(2n\right)!}{t}^{2n}-\frac{1}{4}\sum _{n=0}^{\mathrm{\infty }}\frac{{3}^{2n+1}-3}{\left(2n+1\right)!}{t}^{2n+4}-2\sum _{n=0}^{\mathrm{\infty }}\frac{1}{\left(2n+1\right)!}{t}^{2n+4}\\ =& \frac{1}{16}\sum _{n=3}^{\mathrm{\infty }}\frac{{l}_{n}}{\left(2n+4\right)!}{t}^{2n+4},\end{array}$

where

$\begin{array}{rcl}{l}_{n}& =& \left(2n+4\right)\left({5}^{2n+3}-5\cdot {3}^{2n+3}+10\right)+8\left(2n+4\right)\left({3}^{2n+3}-3\right)\\ +16\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)\left(2n+1\right)-\left({5}^{2n+4}-3\cdot {3}^{2n+4}+2\right)\\ -4\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)\left({3}^{2n+1}-3\right)-32\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)\\ =& \left(250n-125\right){25}^{n}+\left(279-462n-432{n}^{2}-96{n}^{3}\right){9}^{n}\\ +256{n}^{4}+1,120{n}^{3}+1,520{n}^{2}+532n-154,\phantom{\rule{1em}{0ex}}n=3,4,\dots .\end{array}$

Using a basic differential method, we can easily prove

$\begin{array}{rcl}f\left(x\right)& \triangleq & \left(250x-125\right){25}^{x}+\left(279-462x-432{x}^{2}-96{x}^{3}\right){9}^{x}\\ +256{x}^{4}+1,120{x}^{3}+1,520{x}^{2}+532x-154>0\end{array}$

on $\left[3,\mathrm{\infty }\right)$. This leads to ${l}_{n}>0$ for $n=3,4,\dots$ , and $D\left(t\right)>0$. So, the proof of Lemma 2.2 is complete. □

## 3 Proof of Theorem 1.3

Let

$F\left(t\right)\equiv \frac{{\left(\frac{t}{sinht}{e}^{tcotht-1}\right)}^{p}-1}{{\left(tcotht\right)}^{p}-1}=\frac{{f}_{1}\left(t\right)-{f}_{1}\left({0}^{+}\right)}{{g}_{1}\left(t\right)-{g}_{1}\left({0}^{+}\right)},$

where ${f}_{1}\left(t\right)={\left(\frac{t}{sinht}{e}^{tcotht-1}\right)}^{p}$ and ${g}_{1}\left(t\right)={\left(tcotht\right)}^{p}$. Then

${k}_{1}\left(t\right)\triangleq \frac{{f}_{1}^{\prime }\left(t\right)}{{g}_{1}^{\prime }\left(t\right)}=\frac{{e}^{p\left(tcotht-1\right)}}{{\left(cosht\right)}^{p-1}}\cdot \frac{{sinh}^{2}t-{t}^{2}}{sinht\left(sinhtcosht-t\right)}.$

We compute

${k}_{1}^{\prime }\left(t\right)=\frac{{e}^{p\left(tcotht-1\right)}}{{\left(cosht\right)}^{p}}\cdot \frac{{u}_{1}\left(t\right)}{{\left(sinht\right)}^{3}{\left(sinhtcosht-t\right)}^{2}},$

where

$\begin{array}{rcl}{u}_{1}\left(t\right)& =& 2{t}^{2}{sinh}^{4}tcosht+{sinh}^{4}tcosht-4t{sinh}^{5}t\\ -3t{sinh}^{3}t+3{t}^{2}{sinh}^{2}tcosht-{t}^{3}sinht\\ -p\left(t{sinh}^{5}t+2t{sinh}^{3}t+{t}^{4}cosht-{sinh}^{4}tcosht-{t}^{3}{sinh}^{3}t-2{t}^{3}sinht\right)\\ =& 2{t}^{2}{sinh}^{4}tcosht+{sinh}^{4}tcosht-4t{sinh}^{5}t\\ -3t{sinh}^{3}t+3{t}^{2}{sinh}^{2}tcosht-{t}^{3}sinht-pD\left(t\right).\end{array}$

If $0, by Lemma 2.2 we have

$\begin{array}{rcl}5{u}_{1}\left(t\right)& \ge & 10{t}^{2}{sinh}^{4}tcosht+13{sinh}^{4}tcosht-28t{sinh}^{5}t\\ -46t{sinh}^{3}t+30{t}^{2}{sinh}^{2}tcosht+6{t}^{3}sinht-8{t}^{4}cosht+8{t}^{3}{sinh}^{3}t\\ =& \sum _{n=3}^{\mathrm{\infty }}\frac{{h}_{n}}{16\left(2n+4\right)!}{t}^{2n+4},\end{array}$

where

$\begin{array}{rcl}{h}_{n}& =& 10\left(2n+4\right)\left(2n+3\right)\left({5}^{2n+2}-3\cdot {3}^{2n+2}+2\right)+13\left({5}^{2n+4}-3\cdot {3}^{2n+4}+2\right)\\ -28\left(2n+4\right)\left({5}^{2n+3}-5\cdot {3}^{2n+3}+10\right)-184\left(2n+4\right)\left({3}^{2n+3}-3\right)\\ +120\left(2n+4\right)\left(2n+3\right)\left({3}^{2n+2}-1\right)+96\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)\left(2n+1\right)2n\\ -128\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)\left(2n+1\right)+96\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)\left({3}^{2n}-1\right)\\ =& \left(1,000{n}^{2}-3,500n-2,875\right){25}^{n}+\left(768{n}^{3}+6,696{n}^{2}+13,956n+4,113\right){9}^{n}\\ +\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)\left(2n+1\right)\left(192n-128\right)\\ -96\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)-100\left(2n+4\right)\left(2n+3\right)+272\left(2n+4\right)+26\\ >& 0\end{array}$

for $n=3,4,\dots$ .

We have ${u}_{1}\left(t\right)>0$ for $0. So, ${k}_{1}^{\prime }\left(t\right)>0$ for $t>0$, and ${f}_{1}^{\prime }\left(t\right)/{g}_{1}^{\prime }\left(t\right)={k}_{1}\left(t\right)$ is increasing on $\left(0,+\mathrm{\infty }\right)$. Hence, $F\left(t\right)$ is increasing on $\left(0,+\mathrm{\infty }\right)$ by Lemma 2.1. At the same time, ${lim}_{t\to {0}^{+}}F\left(t\right)=1/2$ and ${lim}_{t\to +\mathrm{\infty }}F\left(t\right)={\left(2/e\right)}^{p}$. So, the proof of Theorem 1.3 is complete.

## 4 Proof of Theorem 1.4

Let

$S\left(t\right)\equiv \frac{{\left(\frac{sinht}{t}{e}^{1-tcotht}\right)}^{p}-1}{{e}^{p\left(1-tcotht\right)}-1}=\frac{{f}_{2}\left(t\right)-{f}_{2}\left({0}^{+}\right)}{{g}_{2}\left(t\right)-{g}_{2}\left({0}^{+}\right)},$

where ${f}_{2}\left(t\right)={\left(\frac{sinht}{t}{e}^{1-tcotht}\right)}^{p}$ and ${g}_{2}\left(t\right)={e}^{p\left(1-tcotht\right)}$. Then

${k}_{2}\left(t\right)\triangleq \frac{{f}_{2}^{\prime }\left(t\right)}{{g}_{2}^{\prime }\left(t\right)}={\left(\frac{sinht}{t}\right)}^{p-1}\frac{{\left(sinht\right)}^{3}-{t}^{2}sinht}{{t}^{2}\left(sinhtcosht-t\right)},$

and

${k}_{2}^{\prime }\left(t\right)={\left(\frac{sinht}{t}\right)}^{p-2}\frac{{u}_{2}\left(t\right)}{{t}^{4}{\left(sinhtcosht-t\right)}^{2}},$

where where ${e}_{n}=1-\left({d}_{n}/{c}_{n}\right)$ and Let Then

${e}_{n}=1-\frac{{d}_{n}}{{c}_{n}}=\frac{j\left(n\right)}{i\left(n\right)}.$

Let $\mathrm{\Delta }\left(n\right)=286i\left(n\right)-693j\left(n\right)$. Then

$\begin{array}{rcl}\mathrm{\Delta }\left(n\right)& =& \left(741,313n-5,759,424\right){36}^{n}+{16}^{n}\left[2,275,328\left(2n+5\right)+4,009,984\\ +70,400\left(2n+5\right)\left(2n+4\right)\left(2n+3\right)-532,224\left(2n+5\right)\left(2n+4\right)\right]\\ +{4}^{n}\left[9,152\left(2n+5\right)\left(2n+4\right)\left(2n+3\right)\left(2n+2\right)-62,656\left(2n+5\right)\left(2n+4\right)\left(2n+3\right)\\ +133,056\left(2n+5\right)\left(2n+4\right)-610,016\left(2n+5\right)-156,640\right].\end{array}$

First, we check that $\mathrm{\Delta }\left(n\right)>0$ for $n=3,4,5,6,7$; second, we can easily obtain that $\mathrm{\Delta }\left(n\right)>0$ for $n\ge 8$. So, we have that $\mathrm{\Delta }\left(n\right)>0$ for $n=3,4,\dots$ .

So, we have ${u}_{2}\left(t\right)>0$ for $p\ge 286/693$. So, ${k}_{2}^{\prime }\left(t\right)>0$ for $t>0$, and ${f}_{2}^{\prime }\left(t\right)/{g}_{2}^{\prime }\left(t\right)={k}_{2}\left(t\right)$ is increasing on $\left(0,+\mathrm{\infty }\right)$. Hence, $S\left(t\right)$ is increasing on $\left(0,+\mathrm{\infty }\right)$ by Lemma 2.1 when $p\ge 286/693$. At the same time, ${lim}_{t\to {0}^{+}}S\left(t\right)=1/2$ and ${lim}_{t\to +\mathrm{\infty }}S\left(t\right)=1$. So, the proof of Theorem 1.4 is complete.

## 5 Applications of theorems

In this section, we assume that x and y are two different positive numbers. Let $A\left(x,y\right)$, $G\left(x,y\right)$, $L\left(x,y\right)$, and $I\left(x,y\right)$ be the arithmetic, geometric, logarithmic, and identric means, respectively. Without loss of generality, we set $0. By the transformation $t=\left(log\left(y/x\right)\right)/2$, we can compute and obtain where $t>0$.

Now, the four results in Section 1 are equivalent to the following ones for four classical means.

Theorem 5.1 Let $p\ge 4/5$, and x and y be positive real numbers with $x\ne y$. Then

$\alpha {A}^{p}\left(x,y\right)+\left(1-\alpha \right){G}^{p}\left(x,y\right)<{L}^{p}\left(x,y\right)<\beta {A}^{p}\left(x,y\right)+\left(1-\beta \right){G}^{p}\left(x,y\right)$
(5.1)

holds if and only if $\alpha \le 0$ and $\beta \ge 1/3$.

Theorem 5.1 can deduce the following one, which is from Zhu .

Corollary 5.2 ([, Theorem 1])

Let $p\ge 1$, and x and y be positive real numbers with $x\ne y$. Then

$\alpha {A}^{p}\left(x,y\right)+\left(1-\alpha \right){G}^{p}\left(x,y\right)<{L}^{p}\left(x,y\right)<\beta {A}^{p}\left(x,y\right)+\left(1-\beta \right){G}^{p}\left(x,y\right)$
(5.2)

holds if and only if $\alpha \le 0$ and $\beta \ge 1/3$.

When letting $p=1$ in Theorem 5.1, one can obtain the result (see , [, Theorem 1]).

Corollary 5.3 Let x and y be positive real numbers with $x\ne y$. Then

$\alpha A\left(x,y\right)+\left(1-\alpha \right)G\left(x,y\right)
(5.3)

holds if and only if $\alpha \le 0$ and $\beta \ge 1/3$.

When letting $\beta =1/3$ in the right-hand inequality of (5.3), one can obtain the well-known inequality by Carlson 

$L\left(x,y\right)<\frac{1}{3}A\left(x,y\right)+\frac{2}{3}G\left(x,y\right).$
(5.4)

Theorem 5.4 Let $p>0$. Then

(1) if $0, the double inequality

$\alpha {A}^{p}\left(x,y\right)+\left(1-\alpha \right){G}^{p}\left(x,y\right)<{I}^{p}\left(x,y\right)<\beta {A}^{p}\left(x,y\right)+\left(1-\beta \right){G}^{p}\left(x,y\right)$
(5.5)

holds if and only if $\alpha \le 2/3$ and $\beta \ge {\left(2/e\right)}^{p}$;

(2) if $p\ge 2$, the double inequality

$\alpha {A}^{p}\left(x,y\right)+\left(1-\alpha \right){G}^{p}\left(x,y\right)<{I}^{p}\left(x,y\right)<\beta {A}^{p}\left(x,y\right)+\left(1-\beta \right){G}^{p}\left(x,y\right)$
(5.6)

holds if and only if $\alpha \le {\left(2/e\right)}^{p}$ and $\beta \ge 2/3$.

The part (2) of Theorem 5.4 is a result of Trif .

When letting $p=2$ and $\beta =2/3$ in the right-hand inequality of (5.6), one can obtain the following result, which is from Sándor and Trif .

${I}^{2}\left(x,y\right)<\frac{2}{3}{A}^{2}\left(x,y\right)+\frac{1}{3}{G}^{2}\left(x,y\right).$
(5.7)

When letting $p=1$ in the double inequality (5.5), one can obtain the following result (see , [, Theorem 2]).

Corollary 5.5 Let x and y be positive real numbers with $x\ne y$. Then

$\alpha A\left(x,y\right)+\left(1-\alpha \right)G\left(x,y\right)
(5.8)

holds if and only if $\alpha \le 2/3$ and $\beta \ge 2/e$.

When letting $\alpha =2/3$ in the left-hand inequality in (5.8), one can obtain the following result, which is from Sándor .

$\frac{2}{3}A\left(x,y\right)+\frac{1}{3}G\left(x,y\right)
(5.9)

Theorem 5.6 Let $0, x and y be positive real numbers with $x\ne y$. Then

$\alpha {A}^{p}\left(x,y\right)+\left(1-\alpha \right){L}^{p}\left(x,y\right)<{I}^{p}\left(x,y\right)<\beta {A}^{p}\left(x,y\right)+\left(1-\beta \right){L}^{p}\left(x,y\right)$
(5.10)

holds if and only if $\alpha \le 1/2$ and $\beta \ge {\left(2/e\right)}^{p}$.

Theorem 5.6 can deduce the following result (see Zhu ).

Corollary 5.7 ([, Theorem 3])

Let x and y be positive real numbers with $x\ne y$. Then

$\alpha A\left(x,y\right)+\left(1-\alpha \right)L\left(x,y\right)
(5.11)

holds if and only if $\alpha \le 1/2$ and $\beta \ge 2/e$.

When letting $\alpha =1/2$ in the left-hand inequality of (5.11), one can obtain the following result, which is from Sándor [4, 19].

$I\left(x,y\right)>\frac{A\left(x,y\right)+L\left(x,y\right)}{2}.$
(5.12)

Finally, we give the bounds for ${L}^{p}\left(x,y\right)$ in terms of ${G}^{p}\left(x,y\right)$ and ${I}^{p}\left(x,y\right)$, and obtain the following new result.

Theorem 5.8 Let x and y be positive real numbers with $x\ne y$, and $p\ge 286/693$. Then

$\alpha {G}^{p}\left(x,y\right)+\left(1-\alpha \right){I}^{p}\left(x,y\right)<{L}^{p}\left(x,y\right)<\beta {G}^{p}\left(x,y\right)+\left(1-\beta \right){I}^{p}\left(x,y\right)$
(5.13)

holds if and only if $\beta \le 1/2$ and $\alpha \ge 1$.

Theorem 5.8 can deduce a result of Zhu :

Corollary 5.9 ([, Theorem 4])

Let x and y be positive real numbers with $x\ne y$. Then

$\alpha G\left(x,y\right)+\left(1-\alpha \right)I\left(x,y\right)
(5.14)

holds if and only if $\beta \le 1/2$ and $\alpha \ge 1$.

Obviously, the right-hand side of (5.14) is an extension of the following inequality:

$L\left(x,y\right)<\frac{1}{2}\left(G\left(x,y\right)+I\left(x,y\right)\right),$
(5.15)

which was given by Alzer .

## References

1. Mitrinović DS: Analytic Inequalities. Springer, Berlin; 1970.

2. Ostle B, Terwilliger HL: A comparison of two means. Proc. Mont. Acad. Sci. 1957, 17: 69–70.

3. Leach EB, Sholander MC: Extended mean values. J. Math. Anal. Appl. 1983, 92: 207–223. 10.1016/0022-247X(83)90280-9

4. Sándor J: On the identric and logarithmic means. Aequ. Math. 1990, 40: 261–270. 10.1007/BF02112299

5. Alzer H: Ungleichungen für Mittelwerte. Arch. Math. 1986, 47: 422–426. 10.1007/BF01189983

6. Stolarsky KB: The power mean and generalized logarithmic means. Am. Math. Mon. 1980, 87: 545–548. 10.2307/2321420

7. Zhu L: Inequalities for hyperbolic functions and their applications. J. Inequal. Appl. 2010., 2010: Article ID 130821

8. Zhu L: Some new inequalities for means in two variables. Math. Inequal. Appl. 2008, 11(3):443–448.

9. Vamanamurthy K, Vuorinen M: Inequalities for means. J. Math. Anal. Appl. 1994, 183: 155–166. 10.1006/jmaa.1994.1137

10. Anderson GD, Qiu S-L, Vamanamurthy MK, Vuorinen M: Generalized elliptic integral and modular equations. Pac. J. Math. 2000, 192: 1–37. 10.2140/pjm.2000.192.1

11. Pinelis I: L’Hospital type results for monotonicity, with applications. J. Inequal. Pure Appl. Math. 2002., 3: Article ID 5 (electronic)

12. Alzer H, Qiu S-L: Inequalities for means in two variables. Arch. Math. 2003, 80: 201–215. 10.1007/s00013-003-0456-2

13. Zhu L, Wu JH: The weighted arithmetic and geometric means of the arithmetic mean and the geometric mean. J. Math. Technol. 1998, 14: 150–154. (in Chinese)

14. Zhu L: From chains for mean value inequalities to Mitrinovic’s problem II. Int. J. Math. Educ. Sci. Technol. 2005, 36: 118–125. 10.1080/00207390412331314971

15. Zhu L: New inequalities for means in two variables. Math. Inequal. Appl. 2008, 11(2):229–235.

16. Carlson BC: The logarithmic mean. Am. Math. Mon. 1972, 79: 615–618. 10.2307/2317088

17. Trif T: Note on certain inequalities for means in two variables. J. Inequal. Pure Appl. Math. 2005., 6: Article ID 43 (electronic)

18. Sándor J, Trif T: Some new inequalities for means of two arguments. Int. J. Math. Math. Sci. 2001, 25: 525–532. 10.1155/S0161171201003064

19. Sándor J: A note on some inequalities for means. Arch. Math. 1991, 56: 471–473. 10.1007/BF01200091

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Correspondence to Ling Zhu.

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Zhu, L. New inequalities for hyperbolic functions and their applications. J Inequal Appl 2012, 303 (2012). https://doi.org/10.1186/1029-242X-2012-303

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• DOI: https://doi.org/10.1186/1029-242X-2012-303

### Keywords

• hyperbolic sine
• hyperbolic cosine
• hyperbolic cotangent
• geometric mean
• logarithmic mean
• identric mean
• arithmetic mean
• best constants 