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# Three families of two-parameter means constructed by trigonometric functions

- Zhen-Hang Yang
^{1}Email author

**2013**:541

https://doi.org/10.1186/1029-242X-2013-541

© Yang; licensee Springer. 2013

**Received:**16 April 2013**Accepted:**10 October 2013**Published:**19 November 2013

## Abstract

In this paper, we establish three families of trigonometric functions with two parameters and prove their monotonicity and bivariate log-convexity. Based on them, three two-parameter families of means involving trigonometric functions, which include Schwab-Borchardt mean, the first and second Seiffert means, Sándor’s mean and many other new means, are defined. Their properties are given and some new inequalities for these means are proved. Lastly, two families of two-parameter hyperbolic means, which similarly contain many new means, are also presented without proofs.

**MSC:** 26E60, 26D05, 33B10, 26A48.

## Keywords

- trigonometric function
- hyperbolic function
- mean
- inequality

## 1 Introduction

holds. For convenience, however, we assume that $a\ne b$ in what follows unless otherwise stated.

*p*-order power mean, ${S}_{p,0}(a,b)={L}^{1/p}({a}^{p},{b}^{p})={L}_{p}$ - the

*p*-order logarithmic mean, ${S}_{p,p}(a,b)={I}^{1/p}({a}^{p},{b}^{p})={I}_{p}$ - the

*p*-order identric (exponential) mean; ${G}_{2,0}(a,b)=Q(a,b)$ - the quadratic mean, ${G}_{1,1}(a,b)=Z(a,b)$ - the power-exponential mean, ${G}_{p,p}(a,b)={Z}^{1/p}({a}^{p},{b}^{p})={Z}_{p}$ - the

*p*-order power-exponential mean,

*etc.*The second class is mainly made up of exponential and logarithmic functions, such as the second part of Schwab-Borchardt mean (see [3], [[4], Section 3, equation (2.3)], [5]) defined by

It should be noted that *NS* is actually a Schwab-Borchardt mean since $NS(a,b)=SB(Q,A)$ mentioned by Neuman and Sándor in [5].

where $A=(a+b)/2$, $G=\sqrt{ab}$, *P* is defined by (1.5). As Neuman and Sándor pointed out in [5], the first and second Seiffert means are generated by the Schwab-Borchardt mean, because $SB(G,A)=P(a,b)$, $SB(A,Q)=T(a,b)$.

From the published literature, the first and second classes have been focused on and investigated, and there are a lot of references (see [1, 13–24]). While the third class is relatively little known.

The aim of this paper is to define three families of two-parameter means constructed by trigonometric functions, which include the Schwab-Borchardt mean *SB*, the first and second Seiffert means *P*, *T*, and Sándor’s mean *X*.

The paper is organized as follows. In Section 2, some useful lemmas are given. Three families of trigonometric functions and means with two parameters and their properties are presented in Sections 3-5. In Section 6, we establish some new inequalities for two-parameter trigonometric means. In the last section, two families of two-parameter hyperbolic means are also presented in the same way without proofs.

## 2 Lemmas

For later use, we give the following lemmas.

**Lemma 2.1** [[25], p.26]

*Let*

*f*

*be a differentiable function defined on an interval*

*I*.

*Then the divided differences function*

*F*

*defined on*${I}^{2}$

*by*

*is increasing* (*decreasing*) *in both variables if and only if* *f* *is convex* (*concave*).

**Lemma 2.2** [[26], Theorem 1]

*Let*

*f*

*be a differentiable function defined on an interval*

*I*,

*and let*

*F*

*be defined on*${I}^{2}$

*by*(2.1).

*Then the following statements are equivalent*:

- (i)
${f}^{\prime}$

*is convex*(*concave*)*on**I*, - (ii)
$F(x,y)\le (\ge )\frac{{f}^{\prime}(x)+{f}^{\prime}(y)}{2}$

*for all*$x,y\in I$, - (iii)
*F**is bivariate convex*(*concave*)*on*${I}^{2}$.

**Lemma 2.3** *If* $f:(-m,m)\to \mathbb{R}$ *is a differentiable even function such that* ${f}^{\prime}$ *is convex in* $(0,m)$, *then the function* $x\mapsto F(c+x,c-x)$ *defined by* (2.1) *increases for positive* *x* *if* $c\ge 0$ *and decreases if* $c\le 0$ *provided* $c+x,c-x\in (-m,m)$.

*Proof*Differentiation yields

Since *f* is an even and differentiable function, it is easy to verify that ${g}_{c}(-x)={g}_{c}(x)$, ${g}_{-c}(x)=-{g}_{c}(x)$. From this we only need to prove that for $c\ge 0$, ${F}^{\prime}(c+x,c-x)>0$ for $x\in (0,m)$ provided $c+x,c-x\in (-m,m)$ if ${f}^{\prime}$ is convex on $(0,m)$.

*h*is a continuous and odd function on $[-d,d]$ ($d>0$) and is convex on $[0,d]$, then, for $u,v\in (0,d]$ with $u>v$, the inequality

holds. Indeed, using the fact $h(0)=0$ and the property of a convex function, the second one easily follows.

Hence, ${F}^{\prime}(c+x,c-x)={x}^{-1}{g}_{c}(x)>0$ for $c,x\ge 0$.

This completes the proof. □

**Lemma 2.4**

*The following inequalities are true*:

*Proof* Inequalities (2.2)-(2.5) easily follow by the elementary differential method, and we omit all details here. Inequality (2.6) can be derived from a well-known inequality given in [[27], p.238]) by Adamović and Mitrinović for $0<|x|<\pi /2$, while it is obviously true for $\pi /2\le |x|<\pi $. Inequality (2.7) can be found in [[28], Problem 5.11, 5.12]. This lemma is proved. □

**Lemma 2.5** [[29], pp.227-229]

*Let*$0<|x|<\pi $.

*Then we have*

*where* ${B}_{n}$ *is the Bernoulli number*.

## 3 Two-parameter sine means

### 3.1 Two-parameter sine functions

We call $Sh(p,q,t)$ two-parameter hyperbolic sine functions. Accordingly, for suitable *p*, *q*, *t*, we can give the definition of sine versions of $Sh(p,q,t)$ as follows.

**Definition 3.1**The function $\tilde{S}$ is called a sine function with parameters if $\tilde{S}$ is defined on ${[-2,2]}^{2}\times (0,\pi /2)$ by

$\tilde{S}$ is said to be a two-parameter sine function for short.

Now let us observe its properties.

**Proposition 3.1**

*Let the two*-

*parameter sine function*$\tilde{S}$

*be defined by*(3.2).

*Then*

- (i)
$\tilde{S}$

*is decreasing in**p*,*q**on*$[-2,2]$,*and is log*-*concave in*$(p,q)$*for*$(p,q)\in {[0,2]}^{2}$*and log*-*convex for*$(p,q)\in {[-2,0]}^{2}$; - (ii)
$\tilde{S}$

*is decreasing and log*-*concave in**t**on*$(0,\pi /2)$*for*$p+q>0$,*and is increasing and log*-*convex for*$p+q<0$.

*Proof*We have

- (i)
We prove that $\tilde{S}$ is decreasing in

*p*,*q*on $[-2,2]$, and is log-concave in $(p,q)$ for $(p,q)\in {[0,2]}^{2}$ and log-convex for $(p,q)\in {[-2,0]}^{2}$. By Lemmas 2.1 and 2.2, it suffices to check that*f*is concave in $p,q\in [-2,2]$ and that ${f}^{\prime}$ is concave for $p,q\in [0,2]$ and convex for $p,q\in [-2,0]$.

- (ii)Now we show that $\tilde{S}$ is decreasing and log-concave in
*t*on $(0,\pi /2)$ for $p+q>0$, and increasing and log-convex for $p+q<0$. It is easy to verify that ${f}^{\prime}$ is an odd function on $[-2,2]$, and so $ln\tilde{S}(p,q,t)$ can be written in the form of integral as$ln\tilde{S}(p,q,t)=\frac{1}{p-q}{\int}_{q}^{p}{f}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}dx=\frac{p+q}{|p|+|q|}\frac{1}{|p|-|q|}{\int}_{|q|}^{|p|}{f}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}dx.$(3.7)

which proves part two and, consequently, the proof is completed. □

From the proof of Proposition 3.1, we see that *f* defined by (3.3) is an even function and ${f}^{\u2034}(x)<0$ for $x\in [0,2]$ given by (3.6). Let $m=2$ and $c-x=p\in (-2,2)$. Then by Lemma 2.3 we immediately obtain the following.

**Proposition 3.2** *For fixed* $c\in (-2,2)$, *let* $-min(2,2-2c)\le p\le min(2,2c+2)$ *and* $t\in (0,\pi /2)$, *and let* $\tilde{S}(p,q,t)$ *be defined by* (3.2). *Then the function* $p\mapsto \tilde{S}(p,2c-p,t)$ *is decreasing on* $[-2,c)$ *and increasing on* $(c,2c+2]$ *for* $c\in (-2,0]$, *and is increasing on* $[2c-2,c)$ *and decreasing on* $(c,2]$ *for* $c\in (0,2)$.

By Propositions 3.1 and 3.2 we can obtain some new inequalities for trigonometric functions.

**Corollary 3.1**

*For*$t\in (0,\pi /2)$,

*we have*

*Proof*(i) By Proposition 3.2, $\tilde{S}(p,2-p,t)$ is increasing in

*p*on $[0,1)$ and decreasing on $[1,2]$, we have

*q*on $[-2,2]$, we get

- (ii)Similarly, since $p\mapsto \tilde{S}(p,1-p,t)$ is increasing on $[-1,1/2)$ and decreasing on $(1/2,2]$, we get$\begin{array}{rcl}\tilde{S}(2,-1,t)& <& \tilde{S}(\frac{3}{2},-\frac{1}{2},t)<\tilde{S}(\frac{4}{3},-\frac{1}{3},t)<\tilde{S}(1,0,t)\\ <& \tilde{S}(\frac{4}{5},\frac{1}{5},t)<\tilde{S}(\frac{3}{4},\frac{1}{4},t)<\tilde{S}(\frac{2}{3},\frac{1}{3},t)<\tilde{S}(\frac{1}{2},\frac{1}{2},t),\end{array}$

while $\tilde{S}(\frac{1}{2},\frac{1}{2},t)<\tilde{S}(\frac{1}{2},\frac{1}{4},t)$ follows by the monotonicity of $\tilde{S}(1/2,q,t)$ in *q* on $[-2,2]$. Simplifying yields inequalities (3.9). □

### 3.2 Definition of two-parameter sine means and examples

Being equipped with Propositions 3.1, 3.2, we can easily establish a family of means generated by (3.2). To this end, we have to prove the following statement.

**Theorem 3.1**

*Let*$p,q\in [-2,2]$,

*and let*$\tilde{S}(p,q,t)$

*be defined by*(3.2).

*Then*,

*for all*$a,b>0$, ${\mathcal{S}}_{p,q}(a,b)$

*defined by*

*is a mean of* *a* *and* *b* *if and only if* $0\le p+q\le 3$.

*Proof*Without lost of generality, we assume that $0<a\le b$. Let $t=arccos(a/b)$. Then the statement in question is equivalent to that the inequalities

hold for $t\in (0,\pi /2)$ if and only if $0\le p+q\le 3$, where $\tilde{S}(p,q,t)$ is defined by (3.2).

*Necessity*. We prove that the condition $0\le p+q\le 3$ is necessary. If (3.11) holds, then we have

which implies that $0\le p+q\le 3$.

*Sufficiency*. We show that the condition $0\le p+q\le 3$ is sufficient. Clearly, $max(p,q)\ge 0$. Now we distinguish two cases to prove (3.11).

*p*,

*q*on $[-2,2]$, we get

From Proposition 3.2 it is seen that $\tilde{S}(p,3-p,t)$ is increasing on $[1,3/2]$ and decreasing on $[3/2,2]$, which reveals that $\tilde{S}(p,3-p,t)>\tilde{S}(2,1,t)=cost$, that is, the desired result.

*p*and

*q*, we assume that $p\ge q$. Then $p\ge 0$, $q\le 0$. Due to $p\in [0,2]$ and $p+q\le 3$, we have $p\le min(3-q,2)=2$. Using the monotonicity of $\tilde{S}(p,q,t)$ in

*p*,

*q*on $[-2,2]$ again leads us to

which proves Case 2 and the sufficiency is complete. □

Now we can give the definition of the two-parameter sine means as follows.

**Definition 3.2** Let $a,b>0$ and $p,q\in [-2,2]$ such that $0\le p+q\le 3$, and let $\tilde{S}(p,q,t)$ be defined by (3.2). Then ${\mathcal{S}}_{p,q}(a,b)$ defined by (3.10) is called a two-parameter sine mean of *a* and *b*.

As a family of means, the two-parameter sine means contain many known and new means.

**Example 3.1**Clearly, for $0<a<b$, all the following

are means of *a* and *b*, where $SB(a,b)$ is the Schwab-Borchardt mean defined by (1.3).

*M*are means of distinct positive numbers

*x*and

*y*with ${M}_{1}<{M}_{2}$, then $M({M}_{1},{M}_{2})$ is also a mean and satisfies inequalities

Applying the fact to Definition 3.2, we can obtain more means involving a two-parameter sine function, in which, as mentioned in Section 1, *G*, *A* and *Q* denote the geometric, arithmetic and quadratic means, respectively, and we have $G<A<Q$.

**Example 3.2**Let $(a,b)\to (G,A)$. Then both the following

are means of *a* and *b*, where $P=P(a,b)$ is the first Seiffert mean defined by (1.5) and $X(a,b)$ is Sándor’s mean defined by (1.7). Also, they lie between *G* and *A*.

**Example 3.3**Let $(a,b)\to (G,Q)$. Then both the following

are means of *a* and *b*, and between *G* and *Q*.

It is interesting that the new mean $U(a,b)$ is somewhat similar to the second Seiffert mean $T(a,b)$.

**Example 3.4**Let $(a,b)\to (A,Q)$. Then both the following

are means of *a* and *b*, where $T=T(a,b)$ is the second Seiffert mean defined by (1.6). Moreover, they are between *A* and *Q*.

### 3.3 Properties of two-parameter sine means

From Propositions 3.1, 3.2 and Theorem 3.1, we easily obtain the properties of two-parameter sine means.

**Property 3.1** The two-parameter sine means ${\mathcal{S}}_{p,q}(a,b)$ are symmetric with respect to parameters *p* and *q*.

**Property 3.2** The two-parameter sine means ${\mathcal{S}}_{p,q}(a,b)$ are decreasing in *p* and *q*.

**Property 3.3** The two-parameter sine means ${\mathcal{S}}_{p,q}(a,b)$ are log-concave in $(p,q)$ for $p,q>0$.

**Property 3.4** The two-parameter sine means ${\mathcal{S}}_{p,q}(a,b)$ are homogeneous and symmetric with respect to *a* and *b*.

Now we prove the monotonicity of two-parameter sine means in *a* and *b*.

**Property 3.5** Suppose that $0<a<b$. Then, for fixed $b>0$, the two-parameter sine means ${\mathcal{S}}_{p,q}(a,b)$ are increasing in *a* on $(0,b)$. For fixed $a>0$, they are increasing in *b* on $(a,\mathrm{\infty})$.

*Proof*(i) Let $t=arccos(a/b)$. Then $ln{\mathcal{S}}_{p,q}(a,b):=lnb+ln\tilde{S}(p,q,t)$. Differentiation yields

which, by part two of Proposition 3.1, reveals that $\partial (ln{\mathcal{S}}_{p,q}(a,b))/\partial a>0$, that is, ${\mathcal{S}}_{p,q}(a,b)$ is increasing in *a* on $(0,b)$.

*b*. We have $ln{\mathcal{S}}_{p,q}(a,b):=lna-lncost+ln\tilde{S}(p,q,t)$. Differentiation leads to

It follows by Lemmas 2.1 and 2.3 that $H(p,q)$ is decreasing in *p* and *q* on $[-2,2]$ and $H(p,3-p)$ is increasing on $[1,3/2)$ and decreasing on $(3/2,2]$.

Next we distinguish two cases to prove $H(p,q)>0$ for $p,q\in [-2,2]$ with $0\le p+q\le 3$.

*p*,

*q*on $[-2,2]$, we have

*p*and

*q*, we assume that $p\ge q$. Then $p\ge 0$, $q\le 0$. This together with $p,q\in [-2,2]$ with $p+q\le 3$ gives $p\le min(3-q,2)=2$. Therefore, we have

which proves the monotonicity of ${\mathcal{S}}_{p,q}(a,b)$ in *b* on $(a,\mathrm{\infty})$ and the proof is complete. □

**Remark 3.1**Suppose that $0<a<b$. Then, by the monotonicity of ${\mathcal{S}}_{p,q}(a,b)$ in

*a*and

*b*, we see that

## 4 Two-parameter cosine means

### 4.1 Two-parameter cosine functions

We call $Ch(p,q,t)$ two-parameter hyperbolic cosine functions. Analogously, we can define the two-parameter cosine functions as follows.

**Definition 4.1**The function $\tilde{C}$ is called a two-parameter cosine function if $\tilde{C}$ is defined on ${[-1,1]}^{2}\times (0,\pi /2)$ by

Similar to the proofs of Propositions 3.1 and 3.2, we give the following assertions without proofs.

**Proposition 4.1**

*Let the two*-

*parameter cosine function*$\tilde{C}$

*be defined by*(4.2).

*Then*

- (i)
$\tilde{C}$

*is decreasing in**p*,*q**on*$[-1,1]$,*and is log*-*concave in*$(p,q)$*for*$(p,q)\in {[0,1]}^{2}$*and log*-*convex for*$(p,q)\in {[-1,0]}^{2}$; - (ii)
$\tilde{C}$

*is decreasing and log*-*concave in**t**on*$(0,\pi /2)$*for*$p+q>0$,*and is increasing and log*-*convex for*$p+q<0$.

**Proposition 4.2** *For fixed* $c\in (-1,1)$, *let* $-min(1,1-2c)\le p\le min(1,1+2c)$ *and* $t\in (0,\pi /2)$, *and let* $\tilde{C}(p,q,t)$ *be defined by* (4.2). *Then the function* $p\mapsto \mathcal{C}(p,2c-p,t)$ *is decreasing on* $[-1,c)$ *and increasing on* $(c,2c+1]$ *for* $c\in (-1,0]$, *and is increasing on* $[2c-1,c)$ *and decreasing on* $(c,1]$ *for* $c\in (0,1)$.

Propositions 4.1 and 4.2 also contain some new inequalities for trigonometric functions, as shown in the following corollary.

**Corollary 4.1**

*For*$t\in (0,\pi /2)$,

*we have*

*Proof*By Propositions 4.1 and 4.2, we see that $\tilde{C}(1/2,q,t)$ is decreasing in

*q*on $[-1,1]$ and $\tilde{C}(p,1-p,t)$ is decreasing in

*p*on $[0,1/2)$. It is obtained that

which by some simplifications yields the desired inequalities. □

### 4.2 Definition of two-parameter cosine means and examples

Similarly, by Propositions 4.1, 4.2, we can easily present a family of means generated by (4.2). Of course, we need to prove the following theorem.

**Theorem 4.1**

*Let*$p,q\in [-1,1]$,

*and let*$\tilde{C}(p,q,t)$

*be defined by*(4.2).

*Then*,

*for all*$a,b>0$, ${\mathcal{C}}_{p,q}(a,b)$

*defined by*

*is a mean of* *a* *and* *b* *if and only if* $0\le p+q\le 1$.

*Proof*We assume that $0<a\le b$ and let $t=arccos(a/b)$. Then the desired assertion is equivalent to the inequalities

hold for $t\in (0,\pi /2)$ if and only if $0\le p+q\le 1$, where $\tilde{C}(p,q,t)$ is defined by (4.2).

*Necessity*. If (4.5) holds, then we have

which yields $0\le p+q\le 1$.

*Sufficiency*. We show that the condition $0\le p+q\le 1$ is sufficient. Clearly, $max(p,q)\ge 0$. Now we distinguish two cases to prove (4.5).

From Proposition 4.2 it is seen that $\tilde{C}(p,1-p,t)$ is increasing on $[0,1/2]$ and decreasing on $[1/2,1]$, which yields $\tilde{C}(p,1-p,t)>\tilde{C}(1,0,t)=cost$, which proves Case 1.

*p*,

*q*on $[-1,1]$ gives

which proves Case 2 and the sufficiency is complete. □

Thus the two-parameter cosine means can be defined as follows.

**Definition 4.2** Let $a,b>0$ and $p,q\in [-1,1]$ such that $0\le p+q\le 1$, and let $\tilde{C}(p,q,t)$ be defined by (4.2). Then ${\mathcal{C}}_{p,q}(a,b)$ defined by (4.4) is called a two-parameter cosine mean of *a* and *b*.

is a mean, where $SB(a,b)$ is the Schwab-Borchardt mean defined by (1.3).

are means of *a* and *b*, where *P*, *T* are the first and second Seiffert mean defined by (1.5) and (1.6), *U* is defined by (3.16), and they lie between *G* and *A*, *G* and *Q*, *A* and *Q*, respectively.

### 4.3 Properties of two-parameter cosine means

From Propositions 4.1, 4.2 and Theorem 4.1, we can deduce the properties of two-parameter cosine means as follows.

**Property 4.1** ${\mathcal{C}}_{p,q}(a,b)$ are symmetric with respect to parameters *p* and *q*.

**Property 4.2** ${\mathcal{C}}_{p,q}(a,b)$ are decreasing in *p* and *q*.

**Property 4.3** ${\mathcal{C}}_{p,q}(a,b)$ are log-concave in $(p,q)$ for $p,q>0$.

**Property 4.4** ${\mathcal{C}}_{p,q}(a,b)$ are homogeneous and symmetric with respect to *a* and *b*.

**Property 4.5** Suppose that $0<a<b$. Then, for fixed $b>0$, the two-parameter cosine means ${\mathcal{C}}_{p,q}(a,b)$ are increasing in *a* on $(0,b)$. For fixed $a>0$, they are increasing in *b* on $(a,\mathrm{\infty})$.

The proof of Property 4.5 is similar to that of Property 3.5, which is left to readers.

**Remark 4.1**Assume that $0<a<b$. Then employing the monotonicity of ${\mathcal{C}}_{p,q}(a,b)$ in

*a*and

*b*, we have

## 5 Two-parameter tangent means

### 5.1 Two-parameter tangent functions

Now we define the two-parameter tangent function and prove its properties, proofs of which are also the same as those of Propositions 3.1 and 3.2.

**Definition 5.1**The function $\tilde{\mathcal{T}}$ is called a two-parameter tangent function if $\tilde{\mathcal{T}}$ is defined on ${[-1,1]}^{2}\times (0,\pi /2)$ by

**Proposition 5.1**

*Let the two*-

*parameter tangent function*$\tilde{\mathcal{T}}$

*be defined by*(5.1).

*Then*

- (i)
$\tilde{\mathcal{T}}$

*is increasing in**p*,*q**on*$[-1,1]$,*and is log*-*convex in*$(p,q)$*for*$p,q>0$*and log*-*convex for*$p,q<0$; - (ii)
$\tilde{\mathcal{T}}$

*is increasing and log*-*convex in**t**for*$p+q>0$,*and is decreasing and log*-*concave for*$p+q<0$.

*Proof*We have

*g*is convex on $[-1,1]$ and ${g}^{\prime}$ is convex on $[0,1]$. In fact, differentiation and application of (2.8) yield

Thus part one is proved.

*t*, we have

In the same method as the proof of part two in Proposition 3.1, part two in this proposition easily follows.

This completes the proof. □

The following proposition is a consequence of Lemma 2.3, the proof of which is also the same as that of Proposition 3.2 and is left to readers.

**Proposition 5.2** *For fixed* $c\in (-1,1)$, *let* $-min(1,1-2c)\le p\le min(1,1+2c)$ *and* $t\in (0,\pi /2)$, *and let* $\tilde{\mathcal{T}}(p,q,t)$ *be defined by* (5.1). *Then the function* $p\mapsto \tilde{\mathcal{T}}(p,2c-p,t)$ *is increasing on* $[-1,c)$ *and decreasing on* $(c,1+2c]$ *for* $c\in (-1,0]$, *and is decreasing on* $[2c-1,c)$ *and increasing on* $(c,1]$ *for* $c\in (0,1)$.

As an application of Propositions 5.1 and 5.2, we give the following corollary.

**Corollary 5.1**

*For*$t\in (0,\pi /2)$,

*we have*

*Proof*Propositions 5.1 and 5.2 indicate that $\tilde{\mathcal{T}}(1/2,q,t)$ is increasing in

*q*on $[-1,1]$ and $\tilde{\mathcal{T}}(p,1-p,t)$ is decreasing in

*p*on $[0,1/2)$ and increasing on $(1/2,1]$. It follows that

which, by some simplifications, yields the required inequalities. □

### 5.2 Definition of two-parameter tangent means and examples

Before giving the definition of two-parameter tangent means, we firstly prove the following statement.

**Theorem 5.1**

*Let*$p,q\in [-1,1]$,

*and let*$\tilde{\mathcal{T}}(p,q,t)$

*be defined by*(5.1).

*Then*,

*for all*$a,b>0$, ${\mathcal{T}}_{p,q}(a,b)$

*defined by*

*is a mean of* *a* *and* *b* *if* $0\le p+q\le 1$.

*Proof*We assume that $0<a\le b$ and let $t=arccos(a/b)$. Then ${\mathcal{T}}_{p,q}(a,b)$ is a mean of

*a*and

*b*if and only if the inequalities

hold for $t\in (0,\pi /2)$, where $\tilde{\mathcal{T}}(p,q,t)$ is defined by (5.1). Similarly, it can be divided into two cases.

*p*,

*q*on $[-1,1]$, it is deduced that

that is, the desired result.

*p*,

*q*on $[-1,1]$ gives

which proves Case 2 and the proof is finished. □

We are now in a position to define the two-parameter tangent means by (5.1).

**Definition 5.2** Let $a,b>0$ and $p,q\in [-1,1]$ such that $0\le p+q\le 1$, and let $\tilde{\mathcal{T}}(p,q,t)$ be defined by (5.1). Then ${\mathcal{T}}_{p,q}(a,b)$ defined by (5.5) is called a two-parameter tangent mean of *a* and *b*.

Here are some examples of two-parameter tangent means.

**Example 5.1**For $0<a<b$, both the following

are means of *a* and *b*, where $SB(a,b)$ is the Schwab-Borchardt mean defined by (1.3).

**Example 5.2**Let $(a,b)\to (G,A),(G,Q),(A,Q)$. Then all the following

are means of *a* and *b*, where *P*, *T* are the first and second Seiffert mean defined by (1.5) and (1.6), *U* is defined by (3.16). Also, they lie between *G* and *A*, *G* and *Q*, *A* and *Q*, respectively.

### 5.3 Properties of two-parameter tangent means

From Propositions 5.1 and 5.2 and Theorem 5.1, we see that the properties of two-parameter tangent means are similar to those of sine ones.

**Property 5.1** ${\mathcal{T}}_{p,q}(a,b)$ is symmetric with respect to parameters *p* and *q*.

**Property 5.2** ${\mathcal{T}}_{p,q}(a,b)$ is increasing in *p* and *q*.

**Property 5.3** ${\mathcal{T}}_{p,q}(a,b)$ is log-convex in $(p,q)$ for $p,q>0$.

**Property 5.4** ${\mathcal{T}}_{p,q}(a,b)$ is homogeneous and symmetric with respect to *a* and *b*.

Now we prove the monotonicity of two-parameter trigonometric means in *a* and *b*.

**Property 5.5** Let $0<a<b$. Then, for fixed $a>0$, the two-parameter tangent mean ${\mathcal{T}}_{p,q}(a,b)$ is increasing in *b* on $(a,\mathrm{\infty})$. For fixed $b>0$, the two-parameter tangent mean ${\mathcal{T}}_{p,q}(a,b)$ is increasing in *a* on $(0,b)$.

*Proof*(i) Let $t=arccos(a/b)$. Then $ln{\mathcal{T}}_{p,q}(a,b):=lna+ln\tilde{\mathcal{T}}(p,q,t)$. Differentiation yields

- (ii)Now we prove the monotonicity of ${\mathcal{T}}_{p,q}(a,b)$ in
*a*. Since $ln{\mathcal{T}}_{p,q}(a,b)$ can be written as $ln{\mathcal{T}}_{p,q}(a,b)=lnb+ln\tilde{\mathcal{T}}(p,q,t)+lncost$, we have$\begin{array}{rl}\frac{\partial}{\partial a}ln{\mathcal{T}}_{p,q}(a,b)& =(\frac{\partial}{\partial t}ln\tilde{\mathcal{T}}(p,q,t)-\frac{sint}{cost})\times \frac{\partial t}{\partial a}\\ :=-\frac{1}{\sqrt{{b}^{2}-{a}^{2}}}(J(p,q)-tant),\end{array}$

is even on $[-1,1]$. Thus, to prove $\partial (ln{\mathcal{T}}_{p,q}(a,b))/\partial a>0$, it suffices to prove that for $p,q\in [-1,1]$ with $0\le p+q\le 1$, the inequality $J(p,q)-tant<0$ holds for $t\in (0,\pi /2)$.

By Lemmas 2.1 and 2.3 we see that $J(p,q)$ is increasing in *p* and *q* on $[-1,1]$, and $J(p,1-p)$ is decreasing on $[0,1/2)$ and increasing on $(1/2,1]$.

Now we distinguish two cases to prove $J(p,q)-tant<0$ for $p,q\in [-1,1]$ with $0\le p+q\le 1$.

*p*,

*q*on $[-1,1]$ and of $J(p,1-p)$ in

*p*on $[0,1]$, we have

which proves the monotonicity of ${\mathcal{T}}_{p,q}(a,b)$ in *a* on $(0,b)$. □

**Remark 5.1**Utilizing the monotonicity property, we have

Additionally, ${\mathcal{T}}_{p,q}(a,b)$ has a unique property which shows the relation among two-parameter sine, cosine and tangent means.

**Property 5.6** For $0<a<b$, if $0\le p+q\le 1$, then ${\mathcal{T}}_{p,q}(a,b)=a{\mathcal{S}}_{p,q}(a,b)/{\mathcal{C}}_{p,q}(a,b)$. In particular, ${\mathcal{T}}_{1,0}(a,b)={\mathcal{S}}_{1,0}(a,b)=SB(a,b)$.

## 6 Inequalities for two-parameter trigonometric means

As shown in the previous sections-, by using Propositions 3.1-5.2 we can establish a series of new inequalities for trigonometric functions and reprove some known ones. However, we are more interested in how to establish new inequalities for two-parameter trigonometric means from these ones derived by using Propositions 3.1-5.2, as obtaining an inequality for bivariate mans from the corresponding one for hyperbolic functions (see [22–24, 30]). In fact, Neuman also offered some successful examples (see [31]).

The inequalities involving Schwab-Borchardt mean *SB* are mainly due to Neuman and Sándor (see [31–33]), and Witkowski [34] also has some contributions to them. More often, however, inequalities for means constructed by trigonometric functions seem to be related to the first and second Seiffert means, see [11, 33, 35–55]. In this section, we establish some new inequalities for two-parameter trigonometric means by using their monotonicity and log-convexity. Our steps are as follows.

*Step 1*: Obtaining an inequality (I_{1}) for trigonometric functions sin*t*, cos*t* and tan*t* by using the monotonicity and log-convexity of two-parameter trigonometric functions and simplifying them.

*Step 2:* For $0<a<b$, letting $t=arccos(a/b)$ in inequalities (I_{1}) obtained in Step 1 and next multiplying both sides by *b* or *a* and simplifying yield an inequality (I_{2}) for means involving trigonometric functions.

*Step 3*: Let $m=m(a,b)$ and $M=M(a,b)$ be two means of *a* and *b* with $m(a,b)<M(a,b)$ for all $a,b>0$. Making a change of variables $a\to m(a,b)$ and $b\to M(a,b)$ leads to another inequality (I_{3}) for means involving trigonometric functions.

Now we illustrate these steps.

**Example 6.1**

Step 1: For $t\in (0,\pi /2)$, we have (3.8).

*b*and simplifying yield

**Example 6.2**

*b*and simplifying yield

**Example 6.3**

Step 1: For $t\in (0,\pi /2)$, we have (4.3).

*b*and simplifying yield

With $(a,b)\to (G,Q),(A,Q)$ can yield corresponding inequalities.

**Remark 6.1**From inequalities (6.3) it is derived that

hold for $0<a<b$, where $L(x,y)$ is the logarithmic mean of positive numbers *x* and *y*. The first inequality of (6.5) follows from the relation between the second and fourth terms, that is, $bexp((a-b)/SB(a,b))>a$, while the second one is obtained by the first one in (6.3).

**Example 6.4**

Step 1: For $t\in (0,\pi /2)$, we have (5.4).

*a*and simplifying give

With $(a,b)\to (G,Q),(A,Q)$, we can derive corresponding inequalities.

**Remark 6.2**Applying our method in establishing inequalities for means to certain known ones involving trigonometric functions, we can obtain corresponding inequalities which are possibly related to means. For example, the Wilker inequality states that for $t\in (0,\pi /2)$,

by letting $t=arccos(a/b)$ for $0<a<b$, where the left inequality in (6.8) is due to Neuman and Sándor [[56], (2.5)] (also see [57–59]) and the right one is known as Cusa’s inequality.

**Remark 6.3**The third inequality in (6.1) is clearly superior to Cusa’s inequality (the right one of (6.8)) because

## 7 Families of two-parameter hyperbolic means

After three families of two-parameter trigonometric means have been successfully constructed, we are encouraged to establish further two-parameter means of a hyperbolic version. They are included in the following theorems.

**Theorem 7.1**

*Let*$p,q\in \mathbb{R}$,

*and let*$Sh(p,q,t)$

*be defined by*(3.1).

*Then*,

*for all*$a,b>0$, $S{h}_{p,q}(b,a)$

*defined by*

*is a mean of*

*a*

*and*

*b*

*if and only if*

**Theorem 7.2**

*Let*$p,q\in \mathbb{R}$,

*and let*$Ch(p,q,t)$

*be defined by*(4.1).

*Then*,

*for all*$a,b>0$, $C{h}_{p,q}(b,a)$

*defined by*

*is a mean of* *a* *and* *b* *if and only if* $0\le p+q\le 1$.

respectively. Here we omit further details.

The monotonicities and log-convexities of $S{h}_{p,q}(b,a)$ and of $C{h}_{p,q}(b,a)$ in parameters *p* and *q* are clearly the same as those of $Sh(p,q,t)$ and of $Ch(p,q,t)$, which are in turn equivalent to those of Stolarsky means defined by (1.1) and of Gini means defined by (1.2), respectively. These properties can be found in [13, 14, 17–19, 21].

are means of *a* and *b*, where $SB(b,a)$ is the Schwab-Borchardt mean defined by (1.3).

*p*-order power-exponential mean. Also, all the following

*G*and

*Q*. Likewise, all the following

are also means between *A* and *Q*, where $NS(a,b)$ is the Neuman-Sándor mean defined by (1.4).

It should be noted that the new mean $V(a,b)$ is similar to $NS(a,b)$.

*p*and

*q*, and therefore, we fail to define a family of two-parameter hyperbolic tangent means. However, for certain

*p*,

*q*and $0<a<b$, it is showed that $bTh(p,q,arccosh(b/a))$ is a mean of

*a*and

*b*, for example,

*a*and

*b*. It is also proved that

is also a mean of *a* and *b*. For this reason, we pose an open problem as the end of this paper.

**Problem 7.1** Let $0<a<b$, and let $Th(p,q,t)$ be defined by (7.3). Finding *p*, *q* such that $b\times Th(p,q,arccosh(b/a))$ is a mean of *a* and *b*.

## Declarations

### Acknowledgements

The author would like to thank Ms. Jiang Yiping for her help. The author also wishes to thank the reviewer(s) who gave some important and valuable advice.

## Authors’ Affiliations

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