 Research
 Open access
 Published:
Three families of twoparameter means constructed by trigonometric functions
Journal of Inequalities and Applications volume 2013, Article number: 541 (2013)
Abstract
In this paper, we establish three families of trigonometric functions with two parameters and prove their monotonicity and bivariate logconvexity. Based on them, three twoparameter families of means involving trigonometric functions, which include SchwabBorchardt 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 twoparameter hyperbolic means, which similarly contain many new means, are also presented without proofs.
MSC: 26E60, 26D05, 33B10, 26A48.
1 Introduction
Let {\mathbb{R}}_{+} denote the set of positive real numbers and a,b\in {\mathbb{R}}_{+}. A twovariable continuous function M:{\mathbb{R}}_{+}^{2}\to {\mathbb{R}}_{+} is called a mean on {\mathbb{R}}_{+} if
holds. For convenience, however, we assume that a\ne b in what follows unless otherwise stated.
There exist many elementary means. They can be divided into three classes according to main categories of basic elementary functions by their composition. The first class is mainly constructed by power functions, like the Stolarsky means [1] defined by
and Gini means [2] defined by
It is well known that the Stolarsky and Gini means are very important, they contain many famous means, for instance, {S}_{1,0}(a,b)=L(a,b)  the logarithmic mean, {S}_{1,1}(a,b)=I(a,b)  the identric (exponential) mean, {S}_{2,1}(a,b)={G}_{1,0}(a,b)=A(a,b), {S}_{2p,p}(a,b)={G}_{p,0}(a,b)={A}^{1/p}({a}^{p},{b}^{p})={A}_{p}  the porder power mean, {S}_{p,0}(a,b)={L}^{1/p}({a}^{p},{b}^{p})={L}_{p}  the porder logarithmic mean, {S}_{p,p}(a,b)={I}^{1/p}({a}^{p},{b}^{p})={I}_{p}  the porder identric (exponential) mean; {G}_{2,0}(a,b)=Q(a,b)  the quadratic mean, {G}_{1,1}(a,b)=Z(a,b)  the powerexponential mean, {G}_{p,p}(a,b)={Z}^{1/p}({a}^{p},{b}^{p})={Z}_{p}  the porder powerexponential mean, etc. The second class is mainly made up of exponential and logarithmic functions, such as the second part of SchwabBorchardt mean (see [3], [[4], Section 3, equation (2.3)], [5]) defined by
the logarithmic mean L(a,b), the exponential mean defined by
given in [6] (also see [7, 8]) by Sándor and Toader, and the NeumanSándor mean defined in [5] by
It should be noted that NS is actually a SchwabBorchardt mean since NS(a,b)=SB(Q,A) mentioned by Neuman and Sándor in [5].
The third class is mainly composed of trigonometric functions and their inverses, for example, the first part of SchwabBorchardt mean defined by (1.3), the first and second Seiffert means [9, 10] defined by
respectively, and the new mean presented recently by Sándor in [11, 12] defined as
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 SchwabBorchardt 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 twoparameter means constructed by trigonometric functions, which include the SchwabBorchardt 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 35. In Section 6, we establish some new inequalities for twoparameter trigonometric means. In the last section, two families of twoparameter 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,cx) defined by (2.1) increases for positive x if c\ge 0 and decreases if c\le 0 provided c+x,cx\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,cx)>0 for x\in (0,m) provided c+x,cx\in (m,m) if {f}^{\prime} is convex on (0,m).
To this end, we first show two facts. Firstly, application of Lemma 2.2 leads to
The second one states that if 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.
Now we can prove the desired result. When x\in (0,m), application of the two previous facts and notice that {f}^{\prime} is odd on (m,m) lead to
Hence, {F}^{\prime}(c+x,cx)={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 wellknown 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.227229]
Let 0<x<\pi. Then we have
where {B}_{n} is the Bernoulli number.
3 Twoparameter sine means
3.1 Twoparameter sine functions
We begin with the form of hyperbolic functions of Stolarsky means defined by (1.1) to introduce the twoparameter sine functions. Let t=ln\sqrt{b/a}. Then the Stolarsky means can be expressed in hyperbolic functions as
where
We call Sh(p,q,t) twoparameter 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 twoparameter sine function for short.
Now let us observe its properties.
Proposition 3.1 Let the twoparameter sine function \tilde{S} be defined by (3.2). Then

(i)
\tilde{S} is decreasing in p, q on [2,2], and is logconcave in (p,q) for (p,q)\in {[0,2]}^{2} and logconvex for (p,q)\in {[2,0]}^{2};

(ii)
\tilde{S} is decreasing and logconcave in t on (0,\pi /2) for p+q>0, and is increasing and logconvex for p+q<0.
Proof We have
where

(i)
We prove that \tilde{S} is decreasing in p, q on [2,2], and is logconcave in (p,q) for (p,q)\in {[0,2]}^{2} and logconvex 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].
Differentiation and employing (2.2), (2.6) yield that for t\in (0,\pi /2),
which prove part one.

(ii)
Now we show that \tilde{S} is decreasing and logconcave in t on (0,\pi /2) for p+q>0, and increasing and logconvex 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}{pq}{\int}_{q}^{p}{f}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}dx=\frac{p+q}{p+q}\frac{1}{pq}{\int}_{q}^{p}{f}^{\prime}(x)\phantom{\rule{0.2em}{0ex}}dx.(3.7)
Differentiation and application of (2.2) and (2.3) give
It follows from (3.7) that
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 cx=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(2,22c)\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,2cp,t) is decreasing on [2,c) and increasing on (c,2c+2] for c\in (2,0], and is increasing on [2c2,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,2p,t) is increasing in p on [0,1) and decreasing on [1,2], we have
Due to \tilde{S}(1,q,t) is decreasing in q on [2,2], we get
Simplifying leads to (3.8).

(ii)
Similarly, since p\mapsto \tilde{S}(p,1p,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 twoparameter 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
Using power series extension gives
Hence 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).
Case 1: p,q\ge 0 and p+q\le 3. This case can be divided into two subcases. In the first subcase of (p,q)\in [0,2]\times [0,1] or [0,1]\times [0,2], by the monotonicity of \tilde{S}(p,q,t) in p, q on [2,2], we get
In the second subcase of (p,q)\in {[1,2]}^{2} and p+q\le 3, it is derived that
From Proposition 3.2 it is seen that \tilde{S}(p,3p,t) is increasing on [1,3/2] and decreasing on [3/2,2], which reveals that \tilde{S}(p,3p,t)>\tilde{S}(2,1,t)=cost, that is, the desired result.
Case 2: p\ge 0, q\le 0 or p\le 0, q\ge 0 and p+q\le 3. Because of the symmetry of 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(3q,2)=2. Using the monotonicity of \tilde{S}(p,q,t) in p, q on [2,2] again leads us to
On the other hand, from p+q\ge 0, that is, p\ge q, it is acquired that
which proves Case 2 and the sufficiency is complete. □
Now we can give the definition of the twoparameter 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 twoparameter sine mean of a and b.
As a family of means, the twoparameter 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 SchwabBorchardt mean defined by (1.3).
To generate more means involving a twoparameter sine function, we need to note a simple fact: If {M}_{1}, {M}_{2}, 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 twoparameter 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 twoparameter sine means
From Propositions 3.1, 3.2 and Theorem 3.1, we easily obtain the properties of twoparameter sine means.
Property 3.1 The twoparameter sine means {\mathcal{S}}_{p,q}(a,b) are symmetric with respect to parameters p and q.
Property 3.2 The twoparameter sine means {\mathcal{S}}_{p,q}(a,b) are decreasing in p and q.
Property 3.3 The twoparameter sine means {\mathcal{S}}_{p,q}(a,b) are logconcave in (p,q) for p,q>0.
Property 3.4 The twoparameter sine means {\mathcal{S}}_{p,q}(a,b) are homogeneous and symmetric with respect to a and b.
Now we prove the monotonicity of twoparameter sine means in a and b.
Property 3.5 Suppose that 0<a<b. Then, for fixed b>0, the twoparameter 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).
(ii) Now we prove the monotonicity of {\mathcal{S}}_{p,q}(a,b) in b. We have ln{\mathcal{S}}_{p,q}(a,b):=lnalncost+ln\tilde{S}(p,q,t). Differentiation leads to
where
here
is an even function on [2,2]. Hence, to prove \partial (ln{\mathcal{S}}_{p,q}(a,b))/\partial b>0, it suffices to prove that for p,q\in [2,2] with 0\le p+q\le 3, the inequality H(p,q)+tant>0 is valid for t\in (0,\pi /2). Differentiation again gives
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,3p) 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.
Case 1: p,q\ge 0 and p+q\le 3. This case can be divided into two subcases. In the first subcase of (p,q)\in [0,2]\times [0,1] or [0,1]\times [0,2], by the monotonicity of H(p,q) in p, q on [2,2], we have
In the second subcase of (p,q)\in {[1,2]}^{2} and p+q\le 3, it is derived from the monotonicities of H(p,q) and H(p,3p) that
Case 2: p\ge 0, q\le 0 or p\le 0, q\ge 0 and p+q\le 3. Because of the symmetry of 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(3q,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
which implies that
Similarly, we have
4 Twoparameter cosine means
4.1 Twoparameter cosine functions
In the same way, the Gini means defined by (1.2) can be expressed in hyperbolic functions by letting t=ln\sqrt{b/a}:
where
We call Ch(p,q,t) twoparameter hyperbolic cosine functions. Analogously, we can define the twoparameter cosine functions as follows.
Definition 4.1 The function \tilde{C} is called a twoparameter 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 twoparameter cosine function \tilde{C} be defined by (4.2). Then

(i)
\tilde{C} is decreasing in p, q on [1,1], and is logconcave in (p,q) for (p,q)\in {[0,1]}^{2} and logconvex for (p,q)\in {[1,0]}^{2};

(ii)
\tilde{C} is decreasing and logconcave in t on (0,\pi /2) for p+q>0, and is increasing and logconvex for p+q<0.
Proposition 4.2 For fixed c\in (1,1), let min(1,12c)\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,2cp,t) is decreasing on [1,c) and increasing on (c,2c+1] for c\in (1,0], and is increasing on [2c1,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,1p,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 twoparameter 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
Using power series extension gives
Hence 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).
Case 1: p,q\ge 0 and p+q\le 1. By Proposition 4.1 it is obtained that
From Proposition 4.2 it is seen that \tilde{C}(p,1p,t) is increasing on [0,1/2] and decreasing on [1/2,1], which yields \tilde{C}(p,1p,t)>\tilde{C}(1,0,t)=cost, which proves Case 1.
Case 2: p\ge 0, q\le 0 or p\le 0, q\ge 0 and p+q\le 1. We assume that p\ge q. Then p\ge 0, q\le 0. Due to p\in [0,1] and p+q\le 1, we have p\le min(1q,1)=1. Using the monotonicity of \tilde{C}(p,q,t) in p, q on [1,1] gives
At the same time, since p+q\ge 0, that is, p\ge q, we have
which proves Case 2 and the sufficiency is complete. □
Thus the twoparameter 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 twoparameter cosine mean of a and b.
The twoparameter cosine means similarly include many new means, for example, when 0<a<b,
is a mean, where SB(a,b) is the SchwabBorchardt mean defined by (1.3).
Additionally, 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), and they lie between G and A, G and Q, A and Q, respectively.
4.3 Properties of twoparameter cosine means
From Propositions 4.1, 4.2 and Theorem 4.1, we can deduce the properties of twoparameter 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 logconcave 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 twoparameter 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 Twoparameter tangent means
5.1 Twoparameter tangent functions
Now we define the twoparameter 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 twoparameter tangent function if \tilde{\mathcal{T}} is defined on {[1,1]}^{2}\times (0,\pi /2) by
Proposition 5.1 Let the twoparameter 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 logconvex in (p,q) for p,q>0 and logconvex for p,q<0;

(ii)
\tilde{\mathcal{T}} is increasing and logconvex in t for p+q>0, and is decreasing and logconcave for p+q<0.
Proof We have
where
(i) To prove part one, by Lemmas 2.1 and 2.2 it suffices to check that g is convex on [1,1] and {g}^{\prime} is convex on [0,1]. In fact, differentiation and application of (2.8) yield
Differentiation again gives
Thus part one is proved.
(ii) For proving part two, we have to check that \partial {g}^{\prime}/\partial t\ge 0 and {\partial}^{2}{g}^{\prime}/\partial {t}^{2}\ge 0 for x\in [0,1]. Differentiating {g}^{\prime}(x) given in (5.3) for 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(1,12c)\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,2cp,t) is increasing on [1,c) and decreasing on (c,1+2c] for c\in (1,0], and is decreasing on [2c1,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,1p,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 twoparameter tangent means and examples
Before giving the definition of twoparameter 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.
Case 1: p,q\ge 0 and p+q\le 1. From the monotonicity of {\mathcal{T}}_{p,q}(a,b) in p, q on [1,1], it is deduced that
By Proposition 5.2 we can see that \tilde{\mathcal{T}}(p,1p,t) is decreasing on [0,1/2] and increasing on [1/2,1], which yields
that is, the desired result.
Case 2: p\ge 0, q\le 0 or p\le 0, q\ge 0 and p+q\le 1. We assume that p\ge q. Analogously, there must be p\le min(1q,1)=1. Using the monotonicity of \tilde{\mathcal{T}}(p,q,t) in p, q on [1,1] gives
Noticing that p+q\ge 0, that is, p\ge q, we have
which proves Case 2 and the proof is finished. □
We are now in a position to define the twoparameter 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 twoparameter tangent mean of a and b.
Here are some examples of twoparameter tangent means.
Example 5.1 For 0<a<b, both the following
are means of a and b, where SB(a,b) is the SchwabBorchardt 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 twoparameter tangent means
From Propositions 5.1 and 5.2 and Theorem 5.1, we see that the properties of twoparameter 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 logconvex 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 twoparameter trigonometric means in a and b.
Property 5.5 Let 0<a<b. Then, for fixed a>0, the twoparameter tangent mean {\mathcal{T}}_{p,q}(a,b) is increasing in b on (a,\mathrm{\infty}). For fixed b>0, the twoparameter 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
Application of Proposition 5.1 yields \partial (ln{\mathcal{T}}_{p,q}(a,b))/\partial b>0, which proves part one.

(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}
where
here
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).
Utilizing (2.8) and differentiating again give
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,1p) 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.
Case 1: p,q\ge 0 and p+q\le 1. By the monotonicity of J(p,q) in p, q on [1,1] and of J(p,1p) in p on [0,1], we have
Case 2: p\ge 0, q\le 0 or p\le 0, q\ge 0 and p+q\le 1. We assume that p\ge q. Then p\ge 0, q\le 0 and p\le min(1q,1)=1. Therefore, we get
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
which indicates that
Additionally, {\mathcal{T}}_{p,q}(a,b) has a unique property which shows the relation among twoparameter 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 twoparameter trigonometric means
As shown in the previous sections, by using Propositions 3.15.2 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 twoparameter trigonometric means from these ones derived by using Propositions 3.15.2, 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 SchwabBorchardt 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 twoparameter trigonometric means by using their monotonicity and logconvexity. Our steps are as follows.
Step 1: Obtaining an inequality (I_{1}) for trigonometric functions sint, cost and tant by using the monotonicity and logconvexity of twoparameter 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).
Step 2: For 0<a<b, letting t=arccos(a/b) and next multiplying each side of (3.8) by b and simplifying yield
Step 3: With (a,b)\to (G,A) yields
With (a,b)\to (G,Q) yields
With (a,b)\to (A,Q) yields
Example 6.2
Step 1: For t\in (0,\pi /2), from (3.9) it is derived that
Step 2: For 0<a<b, letting t=arccos(a/b) and next multiplying each side of (6.1) by b and simplifying yield
Step 3: With (a,b)\to (G,A) yields
With (a,b)\to (G,Q) yields
With (a,b)\to (A,Q) yields
Example 6.3
Step 1: For t\in (0,\pi /2), we have (4.3).
Step 2: For 0<a<b, letting t=arccos(a/b) and next multiplying each side of (4.3) by b and simplifying yield
Step 3: With (a,b)\to (G,A) yields
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((ab)/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).
Step 2: For 0<a<b, letting t=arccos(a/b) and next multiplying each side of (5.4) by a and simplifying give
Step 3: With (a,b)\to (G,A) yields
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),
If for 0<a<b, put t=arccos(a/b), then we have
In a similar way, the following inequalities
can be changed into
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
While the second one in (6.1) is weaker than the first one in (6.8) since
7 Families of twoparameter hyperbolic means
After three families of twoparameter trigonometric means have been successfully constructed, we are encouraged to establish further twoparameter 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.
To prove the above theorems, it suffices to use comparison theorems given in [15, 16] by Páles because both S{h}_{p,q}(b,a) and C{h}_{p,q}(b,a) are means if and only if
respectively. Here we omit further details.
The monotonicities and logconvexities 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].
The above theorems indicate that for 0<a<b, all the following
are means of a and b, where SB(b,a) is the SchwabBorchardt mean defined by (1.3).
It is easy to verify that
where L(b,a) and I(b,a) are logarithmic and identric means, respectively, while {Z}_{p}(b,a)={Z}^{1/p}({a}^{p},{b}^{p}) is the porder powerexponential mean. Also, all the following
are means lying in G and Q. Likewise, all the following
are also means between A and Q, where NS(a,b) is the NeumanSándor mean defined by (1.4).
It should be noted that the new mean V(a,b) is similar to NS(a,b).
Similar to (5.1), for p,q\in \mathbb{R}, we can define the twoparameter hyperbolic tangent function as follows:
By some verifications, however, Th(p,q,t) does not have good properties like monotonicity in parameters p and q, and therefore, we fail to define a family of twoparameter 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,
is clearly a mean of 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.
References
Stolarsky KB: Generalizations of the logarithmic mean. Math. Mag. 1975, 48: 87–92. 10.2307/2689825
Gini C: Di una formula comprensiva delle medie. Metron 1938, 13: 3–22.
Brenner JL, Carlson BC: Homogeneous mean values: weights and asymptotes. J. Math. Anal. Appl. 1987, 123: 265–280. 10.1016/0022247X(87)903088
Carlson BC: Algorithms involving arithmetic and geometric means. Am. Math. Mon. 1971, 78: 496–505. 10.2307/2317754
Neuman E, Sándor J: On the SchwabBorchardt mean. Math. Pannon. 2003, 14(2):253–266.
Toader G: Some mean values related to the arithmeticgeometric mean. J. Math. Anal. Appl. 1998, 218(2):358–368. 10.1006/jmaa.1997.5766
Sándor, J, Toader, G: On some exponential means. Seminar on Mathematical Analysis (ClujNapoca, 1989–1990), Preprint, 90, “BabesBolyai” Univ., Cluj, 35–40 (1990)
Sándor J, Toader G: On some exponential means. Part II. Int. J. Math. Math. Sci. 2006., 2006: Article ID 51937
Seiffert HJ: Werte zwischen dem geometrischen und dem arithmetischen Mittel zweier Zahlen. Elem. Math. 1987, 42: 105–107.
Seiffert HJ: Aufgabe 16. Die Wurzel 1995, 29: 221–222.
Sándor, J: Trigonometric and hyperbolic inequalities. Available online at arXiv:1105.0859 (2011). eprintatarXiv.org
Sándor J: Two sharp inequalities for trigonometric and hyperbolic functions. Math. Inequal. Appl. 2012, 15(2):409–413.
Leach EB, Sholander MC: Extended mean values. Am. Math. Mon. 1978, 85: 84–90. 10.2307/2321783
Leach EB, Sholander MC: Extended mean values II. J. Math. Anal. Appl. 1983, 92: 207–223. 10.1016/0022247X(83)902809
Páles Z: Inequalities for sums of powers. J. Math. Anal. Appl. 1988, 131: 265–270. 10.1016/0022247X(88)902041
Páles Z: Inequalities for differences of powers. J. Math. Anal. Appl. 1988, 131: 271–281. 10.1016/0022247X(88)902053
Qi F: Logarithmic convexities of the extended mean values. Proc. Am. Math. Soc. 2002, 130(6):1787–1796. 10.1090/S000299390106275X
Yang ZH: On the homogeneous functions with two parameters and its monotonicity. J. Inequal. Pure Appl. Math. 2005., 6(4): Article ID 101. Available online at http://www.emis.de/journals/JIPAM/images/155_05_JIPAM/155_05.pdf
Yang ZH: On the logconvexity of twoparameter homogeneous functions. Math. Inequal. Appl. 2007, 10(3):499–516.
Yang ZH: On the monotonicity and logconvexity of a fourparameter homogeneous mean. J. Inequal. Appl. 2008., 2008: Article ID 149286. Available online at http://www.journalofinequalitiesandapplications.com/content/2008/1/149286
Yang ZH: The logconvexity of another class of oneparameter means and its applications. Bull. Korean Math. Soc. 2012, 49(1):33–47. Available online at http://www.mathnet.or.kr/mathnet/thesis_file/BKMS49–133–47.pdf 10.4134/BKMS.2012.49.1.033
Yang ZH: New sharp bounds for logarithmic mean and identric mean. J. Inequal. Appl. 2013. 10.1186/1029242X2013116
Zhu L: Generalized Lazarevićs inequality and its applications  Part II. J. Inequal. Appl. 2009., 2009: Article ID 379142
Zhu L: New inequalities for hyperbolic functions and their applications. J. Inequal. Appl. 2012. 10.1186/1029242X2012303
Bullen PS: Handbook of Means and Their Inequalities. Kluwer Academic, Dordrecht; 2003.
Merkle M: Conditions for convexity of a derivative and some applications to the Gamma function. Aequ. Math. 1998, 55: 273–280. 10.1007/s000100050036
Mitrinović DS: Analytic Inequalities. Springer, Berlin; 1970.
Mitrinović DS: Elementary Inequalities. Noordhoff, Groningen; 1964.
Group of compilation: Handbook of Mathematics. Peoples’ Education Press, Beijing; 1979. (Chinese)
Yang ZH: New sharp bounds for identric mean in terms of logarithmic mean and arithmetic mean. J. Math. Inequal. 2012, 6(4):533–543. 10.7153/jmi0651
Neuman E: Inequalities for the SchwabBorchardt mean and their applications. J. Math. Inequal. 2011, 5(4):601–609.
Neuman E, Sándor J: On certain means of two arguments and their extensions. Int. J. Math. Math. Sci. 2003, 16: 981–993.
Neuman E, Sándor J: On the SchwabBorchardt mean. Math. Pannon. 2006, 17(1):49–59.
Witkowski A: Interpolations of SchwabBorchardt mean. Math. Inequal. Appl. 2012, 16(1):193–206.
Jagers AA: Solution of problem 887. Nieuw Arch. Wiskd. 1994, 12(2):30–31.
Sándor J: On certain inequalities for means III. Arch. Math. 2001, 76: 34–40. 10.1007/s000130050539
Hästö PA: A monotonicity property of ratios of symmetric homogeneous means. J. Inequal. Pure Appl. Math. 2002., 3(5): Article ID 71
Sándor J, Neuman E: On certain means of two arguments and their extensions. Int. J. Math. Math. Sci. 2003, 2003(16):981–993. 10.1155/S0161171203208103
Chu YM, Qiu YF, Wang MK: Sharp power mean bounds for the combination of Seiffert and geometric means. Abstr. Appl. Anal. 2010., 2010: Article ID 108920
He D, Shen ZJ: Advances in research on Seiffert mean. Commun. Inequal. Res. 2010., 17(4): Article ID 26. Available online at http://old.irgoc.org/Article/UploadFiles/201010/20101026104515652.pdf
Wang SS, Chu YM: The best bounds of the combination of arithmetic and harmonic means for the Seiffert’s mean. Int. J. Math. Anal. 2010, 4(21–24):1079–1084.
Wang MK, Qiu YF, Chu YM: Sharp bounds for Seiffert means in terms of Lehmer means. J. Math. Inequal. 2010, 4(4):581–586.
Chu YM, Qiu YF, Wang MK, Wang GD: The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert’s mean. J. Inequal. Appl. 2010., 2010: Article ID 436457
Liu H, Meng XJ: The optimal convex combination bounds for Seiffert’s mean. J. Inequal. Appl. 2011., 2011: Article ID 686834
Chu YM, Wang MK, Gong WM: Two sharp double inequalities for Seiffert mean. J. Inequal. Appl. 2011., 2011: Article ID 44 10.1186/1029242X201144
Chu YM, Hou SW: Sharp bounds for Seiffert mean in terms of contraharmonic mean. Abstr. Appl. Anal. 2012., 2012: Article ID 425175
Chu YM, Long BY, Gong WM, Song YQ: Sharp bounds for Seiffert and NeumanSándor means in terms of generalized logarithmic means. J. Inequal. Appl. 2013., 2013: Article ID 10. Available online at http://www.journalofinequalitiesandapplications.com/content/2013/1/10
Jiang WD, Qi F: Some sharp inequalities involving Seiffert and other means and their concise proofs. Math. Inequal. Appl. 2012, 15(4):1007–1017.
Hästö PA: Optimal inequalities between Seiffert’s mean and power mean. Math. Inequal. Appl. 2004, 7(1):47–53.
Costin I, Toader G: A nice separation of some Seiffert type means by power means. Int. J. Math. Math. Sci. 2012., 2012: Article ID 430692
Yang, ZH: Sharp bounds for the second Seiffert mean in terms of power means. Available online at arXiv:1206.5494v1 (2012). eprintatarXiv.org
Yang, ZH: The monotonicity results and sharp inequalities for some powertype means of two arguments. Available online at arXiv:1210.6478 (2012). eprintatarXiv.org
Yang, ZH: Sharp bounds for Seiffert mean in terms of weighted power means of arithmetic mean and geometric mean. Math. Inequal. Appl. (2013, in print)
Chu YM, Wang MK, Qiu YF: An optimal double inequality between powertype Heron and Seiffert means. J. Inequal. Appl. 2010., 2010: Article ID 146945
Costin I, Toader G: Optimal evaluations of some Seifferttype means by power means. Appl. Math. Comput. 2013, 219: 4745–4754. 10.1016/j.amc.2012.10.091
Neuman E, Sándor J: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the CusaHuygens, Wilker and Huygens inequalities. Math. Inequal. Appl. 2010, 13(4):715–723.
Lv YP, Wang GD, Chu YM: A note on Jordan type inequalities for hyperbolic functions. Appl. Math. Lett. 2012, 25: 505–508.
Yang, ZH: Refinements of MitrinovicCusa inequality. Available online at arXiv:1206.4911 (2012). eprintatarXiv.org
Yang ZH: Refinements of a twosided inequality for trigonometric functions. J. Math. Inequal. 2013, 7(4):601–615. 10.7153/jmi0757
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Yang, ZH. Three families of twoparameter means constructed by trigonometric functions. J Inequal Appl 2013, 541 (2013). https://doi.org/10.1186/1029242X2013541
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2013541