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.
Keywords:
trigonometric function; hyperbolic function; mean; inequality1 Introduction
Let denote the set of positive real numbers and . A twovariable continuous function is called a mean on if
holds. For convenience, however, we assume that 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,  the logarithmic mean,  the identric (exponential) mean, ,  the porder power mean,  the porder logarithmic mean,  the porder identric (exponential) mean;  the quadratic mean,  the powerexponential mean,  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 , 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 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 , , 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 , .
From the published literature, the first and second classes have been focused on and investigated, and there are a lot of references (see [1,1324]). 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]
Letfbe a differentiable function defined on an intervalI. Then the divided differences functionFdefined onby
is increasing (decreasing) in both variables if and only iffis convex (concave).
Lemma 2.2 [[26], Theorem 1]
Letfbe a differentiable function defined on an intervalI, and letFbe defined onby (2.1). Then the following statements are equivalent:
(iii) Fis bivariate convex (concave) on.
Lemma 2.3Ifis a differentiable even function such thatis convex in, then the functiondefined by (2.1) increases for positivexifand decreases ifprovided.
Proof Differentiation yields
Since f is an even and differentiable function, it is easy to verify that , . From this we only need to prove that for , for provided if is convex on .
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 () and is convex on , then, for with , the inequality
holds. Indeed, using the fact and the property of a convex function, the second one easily follows.
Now we can prove the desired result. When , application of the two previous facts and notice that is odd on lead to
This completes the proof. □
Lemma 2.4The 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 , while it is obviously true for . Inequality (2.7) can be found in [[28], Problem 5.11, 5.12]. This lemma is proved. □
Lemma 2.5 [[29], pp.227229]
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 . Then the Stolarsky means can be expressed in hyperbolic functions as
where
We call twoparameter hyperbolic sine functions. Accordingly, for suitable p, q, t, we can give the definition of sine versions of as follows.
Definition 3.1 The function is called a sine function with parameters if is defined on by
is said to be a twoparameter sine function for short.
Now let us observe its properties.
Proposition 3.1Let the twoparameter sine functionbe defined by (3.2). Then
(i) is decreasing inp, qon, and is logconcave inforand logconvex for;
(ii) is decreasing and logconcave intonfor, and is increasing and logconvex for.
Proof We have
where
(i) We prove that is decreasing in p, q on , and is logconcave in for and logconvex for . By Lemmas 2.1 and 2.2, it suffices to check that f is concave in and that is concave for and convex for .
Differentiation and employing (2.2), (2.6) yield that for ,
which prove part one.
(ii) Now we show that is decreasing and logconcave in t on for , and increasing and logconvex for . It is easy to verify that is an odd function on , and so can be written in the form of integral as
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 for given by (3.6). Let and . Then by Lemma 2.3 we immediately obtain the following.
Proposition 3.2For fixed, letand, and letbe defined by (3.2). Then the functionis decreasing onand increasing onfor, and is increasing onand decreasing onfor.
By Propositions 3.1 and 3.2 we can obtain some new inequalities for trigonometric functions.
Proof (i) By Proposition 3.2, is increasing in p on and decreasing on , we have
Due to is decreasing in q on , we get
Simplifying leads to (3.8).
(ii) Similarly, since is increasing on and decreasing on , we get
while follows by the monotonicity of in q on . 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 3.1Let, and letbe defined by (3.2). Then, for all, defined by
is a mean ofaandbif and only if.
Proof Without lost of generality, we assume that . Let . Then the statement in question is equivalent to that the inequalities
hold for if and only if , where is defined by (3.2).
Necessity. We prove that the condition is necessary. If (3.11) holds, then we have
Using power series extension gives
Hence we have
Sufficiency. We show that the condition is sufficient. Clearly, . Now we distinguish two cases to prove (3.11).
Case 1: and . This case can be divided into two subcases. In the first subcase of or , by the monotonicity of in p, q on , we get
In the second subcase of and , it is derived that
From Proposition 3.2 it is seen that is increasing on and decreasing on , which reveals that , that is, the desired result.
Case 2: , or , and . Because of the symmetry of p and q, we assume that . Then , . Due to and , we have . Using the monotonicity of in p, q on again leads us to
On the other hand, from , that is, , 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 and such that , and let be defined by (3.2). Then 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 , all the following
are means of a and b, where 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 are means of distinct positive numbers x and y with , then 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 .
Example 3.2 Let . Then both the following
are means of a and b, where is the first Seiffert mean defined by (1.5) and is Sándor’s mean defined by (1.7). Also, they lie between G and A.
Example 3.3 Let . Then both the following
are means of a and b, and between G and Q.
It is interesting that the new mean is somewhat similar to the second Seiffert mean .
Example 3.4 Let . Then both the following
are means of a and b, where 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 are symmetric with respect to parameters p and q.
Property 3.2 The twoparameter sine means are decreasing in p and q.
Property 3.3 The twoparameter sine means are logconcave in for .
Property 3.4 The twoparameter sine means 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 . Then, for fixed , the twoparameter sine means are increasing in a on . For fixed , they are increasing in b on .
Proof (i) Let . Then . Differentiation yields
which, by part two of Proposition 3.1, reveals that , that is, is increasing in a on .
(ii) Now we prove the monotonicity of in b. We have . Differentiation leads to
where
here
is an even function on . Hence, to prove , it suffices to prove that for with , the inequality is valid for . Differentiation again gives
It follows by Lemmas 2.1 and 2.3 that is decreasing in p and q on and is increasing on and decreasing on .
Next we distinguish two cases to prove for with .
Case 1: and . This case can be divided into two subcases. In the first subcase of or , by the monotonicity of in p, q on , we have
In the second subcase of and , it is derived from the monotonicities of and that
Case 2: , or , and . Because of the symmetry of p and q, we assume that . Then , . This together with with gives . Therefore, we have
which proves the monotonicity of in b on and the proof is complete. □
Remark 3.1 Suppose that . Then, by the monotonicity of 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 :
where
We call twoparameter hyperbolic cosine functions. Analogously, we can define the twoparameter cosine functions as follows.
Definition 4.1 The function is called a twoparameter cosine function if is defined on by
Similar to the proofs of Propositions 3.1 and 3.2, we give the following assertions without proofs.
Proposition 4.1Let the twoparameter cosine functionbe defined by (4.2). Then
(i) is decreasing inp, qon, and is logconcave inforand logconvex for;
(ii) is decreasing and logconcave intonfor, and is increasing and logconvex for.
Proposition 4.2For fixed, letand, and letbe defined by (4.2). Then the functionis decreasing onand increasing onfor, and is increasing onand decreasing onfor.
Propositions 4.1 and 4.2 also contain some new inequalities for trigonometric functions, as shown in the following corollary.
Proof By Propositions 4.1 and 4.2, we see that is decreasing in q on and is decreasing in p on . 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 4.1Let, and letbe defined by (4.2). Then, for all, defined by
is a mean ofaandbif and only if.
Proof We assume that and let . Then the desired assertion is equivalent to the inequalities
hold for if and only if , where is defined by (4.2).
Necessity. If (4.5) holds, then we have
Using power series extension gives
Hence we have
Sufficiency. We show that the condition is sufficient. Clearly, . Now we distinguish two cases to prove (4.5).
Case 1: and . By Proposition 4.1 it is obtained that
From Proposition 4.2 it is seen that is increasing on and decreasing on , which yields , which proves Case 1.
Case 2: , or , and . We assume that . Then , . Due to and , we have . Using the monotonicity of in p, q on gives
At the same time, since , that is, , 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 and such that , and let be defined by (4.2). Then 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 ,
is a mean, where is the SchwabBorchardt mean defined by (1.3).
Additionally, let . 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 are symmetric with respect to parameters p and q.
Property 4.2 are decreasing in p and q.
Property 4.3 are logconcave in for .
Property 4.4 are homogeneous and symmetric with respect to a and b.
Property 4.5 Suppose that . Then, for fixed , the twoparameter cosine means are increasing in a on . For fixed , they are increasing in b on .
The proof of Property 4.5 is similar to that of Property 3.5, which is left to readers.
Remark 4.1 Assume that . Then employing the monotonicity of 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 is called a twoparameter tangent function if is defined on by
Proposition 5.1Let the twoparameter tangent functionbe defined by (5.1). Then
(i) is increasing inp, qon, and is logconvex inforand logconvex for;
(ii) is increasing and logconvex intfor, and is decreasing and logconcave for.
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 and is convex on . 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 and for . Differentiating 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 5.2For fixed, letand, and letbe defined by (5.1). Then the functionis increasing onand decreasing onfor, and is decreasing onand increasing onfor.
As an application of Propositions 5.1 and 5.2, we give the following corollary.
Proof Propositions 5.1 and 5.2 indicate that is increasing in q on and is decreasing in p on and increasing on . 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 5.1Let, and letbe defined by (5.1). Then, for all, defined by
Proof We assume that and let . Then is a mean of a and b if and only if the inequalities
hold for , where is defined by (5.1). Similarly, it can be divided into two cases.
Case 1: and . From the monotonicity of in p, q on , it is deduced that
By Proposition 5.2 we can see that is decreasing on and increasing on , which yields
that is, the desired result.
Case 2: , or , and . We assume that . Analogously, there must be . Using the monotonicity of in p, q on gives
Noticing that , that is, , 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 and such that , and let be defined by (5.1). Then 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 , both the following
are means of a and b, where is the SchwabBorchardt mean defined by (1.3).
Example 5.2 Let . 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 is symmetric with respect to parameters p and q.
Property 5.2 is increasing in p and q.
Property 5.3 is logconvex in for .
Property 5.4 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 . Then, for fixed , the twoparameter tangent mean is increasing in b on . For fixed , the twoparameter tangent mean is increasing in a on .
Proof (i) Let . Then . Differentiation yields
Application of Proposition 5.1 yields , which proves part one.
(ii) Now we prove the monotonicity of in a. Since can be written as , we have
where
here
is even on . Thus, to prove , it suffices to prove that for with , the inequality holds for .
Utilizing (2.8) and differentiating again give
By Lemmas 2.1 and 2.3 we see that is increasing in p and q on , and is decreasing on and increasing on .
Now we distinguish two cases to prove for with .
Case 1: and . By the monotonicity of in p, q on and of in p on , we have
Case 2: , or , and . We assume that . Then , and . Therefore, we get
which proves the monotonicity of in a on . □
Remark 5.1 Utilizing the monotonicity property, we have
which indicates that
Additionally, has a unique property which shows the relation among twoparameter sine, cosine and tangent means.
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 [2224,30]). In fact, Neuman also offered some successful examples (see [31]).
The inequalities involving SchwabBorchardt mean SB are mainly due to Neuman and Sándor (see [3133]), 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 [11,33,3555]. 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 , letting 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 and be two means of a and b with for all . Making a change of variables and leads to another inequality (I_{3}) for means involving trigonometric functions.
Now we illustrate these steps.
Example 6.1
Step 2: For , letting and next multiplying each side of (3.8) by b and simplifying yield
Example 6.2
Step 1: For , from (3.9) it is derived that
Step 2: For , letting and next multiplying each side of (6.1) by b and simplifying yield
Example 6.3
Step 2: For , letting and next multiplying each side of (4.3) by b and simplifying yield
With can yield corresponding inequalities.
Remark 6.1 From inequalities (6.3) it is derived that
hold for , where 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, , while the second one is obtained by the first one in (6.3).
Example 6.4
Step 2: For , letting and next multiplying each side of (5.4) by a and simplifying give
With , 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 ,
In a similar way, the following inequalities
can be changed into
by letting for , where the left inequality in (6.8) is due to Neuman and Sándor [[56], (2.5)] (also see [5759]) 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 7.1Let, and letbe defined by (3.1). Then, for all, defined by
is a mean ofaandbif and only if
Theorem 7.2Let, and letbe defined by (4.1). Then, for all, defined by
is a mean ofaandbif and only if.
To prove the above theorems, it suffices to use comparison theorems given in [15,16] by Páles because both and are means if and only if
respectively. Here we omit further details.
The monotonicities and logconvexities of and of in parameters p and q are clearly the same as those of and of , 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 [13,14,1719,21].
The above theorems indicate that for , all the following
are means of a and b, where is the SchwabBorchardt mean defined by (1.3).
It is easy to verify that
where and are logarithmic and identric means, respectively, while 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 is the NeumanSándor mean defined by (1.4).
It should be noted that the new mean is similar to .
Similar to (5.1), for , we can define the twoparameter hyperbolic tangent function as follows:
By some verifications, however, 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 , it is showed that 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 , and let be defined by (7.3). Finding p, q such that is a mean of a and b.
Competing interests
The author declares that they have no competing interests.
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.
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