Characterization of homogeneous symmetric monotone bivariate means
- Mustapha Raïssouli^{1, 2}Email authorView ORCID ID profile and
- Anis Rezgui^{1, 3}
https://doi.org/10.1186/s13660-016-1150-9
© Raïssouli and Rezgui 2016
Received: 20 May 2016
Accepted: 16 August 2016
Published: 13 September 2016
Abstract
In this paper, we introduce a class of bivariate means generated by an integral of a continuous increasing function on \((0,+\infty)\). This class of means widens the spectrum of possible means and leads to many easy and interesting mean-inequalities. We show that this class of means characterizes the large class of homogeneous symmetric monotone means.
Keywords
MSC
1 Introduction
In recent few years, many authors have introduced plenty of classes of bivariate means in terms of functions or integrals of functions; see, for instance, [1–5]. Most of these classes of means require very specific assumptions and/or conditions on the function, which in fact restricts the range of the underlined class of means. In this paper, we also introduce a class of bivariate means defined as an integral of functions, where the underlying function is just continuous and increasing on \((0,\infty)\), which is not restrictive and so gives much more possibilities to define new means and new mean-inequalities.
The main result of the paper is a characterization result; in fact, we show that our class of means characterizes a large class of means, the class of homogeneous symmetric monotone means.
The paper is organized as follows: Section 2 is devoted to the introduction of the new class of means with some examples. In Section 3, we focus ourselves on comparison results, that is, on how we could obtain mean-inequalities of interest using this new formulation. In the last section, we prove a characterization result; in fact, we show that every homogeneous symmetric strict monotone mean can be seen as an element of our class of means.
2 A new class of bivariate means
Proposition 2.1
For any continuous and strictly monotonic function f on \((0,+\infty )\), the binary map \(m_{f}\) defined by (2.1) is a homogeneous bivariate mean.
Proof
It is straightforward and therefore omitted here. □
Remark 2.1
The mean \(m_{f}\) is not always symmetric; see Example 2.3 and Example 2.2. Otherwise, it is not hard to check that \(m_{f+c}=m_{f}\) for each constant c and that \(m_{\alpha,f}=m_{f}\) for every \(\alpha\neq0\). In particular, \(m_{-f}=m_{f}\). Due to this, without loss of generality, we only consider functions f that are continuous strictly increasing on \((0,\infty)\) and satisfying \(f(1)=0\).
The following result gives other equivalent forms of \(m_{f}\).
Proposition 2.2
- (i)For all \(a,b>0\), we have$$ m_{f}(a,b)=b \int_{0}^{1}f^{-1} \bigl(tf(a/b) \bigr) \,dt. $$(2.2)
- (ii)If, moreover, f is differentiable, then we have (for all \(a,b>0\), \(a\neq b\))$$\begin{aligned}& m_{f}(a,b)=\frac{b}{f(a/b)} \int_{1}^{a/b}uf'(u)\,du, \end{aligned}$$(2.3)$$\begin{aligned}& m_{f}(a,b)=a-\frac{b}{f(a/b)} \int_{1}^{a/b}f(u)\,du. \end{aligned}$$(2.4)
Proof
(i) If in (2.1) we make the change of variables \(u=tf(a/b)\) with \(0\leq t\leq1\), then we obtain (2.2) by simple topics of integration.
(ii) Setting \(u=f(s)\) in (2.1), we obtain (2.3) after an elementary manipulation. By integration by parts, (2.4) follows from (2.3). □
The previous forms of \(m_{f}\) lead to the following regularity result.
Corollary 2.3
Let \(f\in C^{0}_{\uparrow}(0,\infty)\) be such that \(f(1)=0\). Then the mean \(m_{f}\) is continuous strict monotone.
Proof
Since f is continuous and \(m_{f}\) is a mean, the continuity of \(m_{f}\) follows from (2.1) with standard topics of real analysis. The fact that \(m_{f}\) is strict monotone follows from (2.2). The details are simple and therefore omitted here. □
We now present some examples that illustrate the previous mean \(m_{f}\). The first example shows that the mean \(m_{f}\) includes the standard means A, H, G, and L.
Example 2.1
- (i)
With \(f(x)=x-1\), we easily verify that \(m_{f}=A\). If \(f(x)=\ln x\), then \(m_{f}=L\). Letting \(f(x)=1-1/\sqrt{x}\), simple computation leads to \(m_{f}=G\), and if \(f(x)=1-1/x^{2}\), then \(m_{f}=H\).
- (ii)Let \(f(x)=-1/x+1\). By (2.4) simple computation leads to (for \(a,b>0\), \(a\neq b\))which is the dual logarithmic mean, that is, \(L^{*}(a,b)= (L(a^{-1},b^{-1}) )^{-1}\).$$m_{f}(a,b)=\frac{ab}{b-a} (\ln b-\ln a )=\frac{ab}{L(a,b)}:=L^{*}(a,b), $$
We now state the following example, which, in its turn, includes a lot of the most interesting standard means.
Example 2.2
Remark 2.2
As already pointed in the Introduction, the Schwab-Borchardt mean SB is of great interest since it includes a lot of symmetric means. Our approach includes, in turn, the mean SB. The following example explains this latter situation and that of the previous remark. It also shows that \(m_{f}\) is not always symmetric.
Example 2.3
After discussing some examples, we are now in a position to state the following result, which shows that the mean map \(f\longmapsto m_{f}\) is one-to-one modulus multiplication by positive real numbers.
Theorem 2.4
Let \(f,g\in C^{1}_{\uparrow}(0,\infty)\) be such that \(f(1)=g(1)=0\) and \(g'(1)\neq0\). Then, \(m_{f}=m_{g}\) if and only if \(f=\alpha\cdot g\) for some real number \(\alpha>0\).
Proof
3 Mean-inequalities
In the ongoing section we shall state some mean-inequalities that are either new or easy to obtain using the class of means introduced in the previous section. Despite its general interest in mathematical analysis, it remains true that the major interest of introducing new classes of means is the obtention of mean-inequalities.
Now, we may state the following propositions.
Proposition 3.1
- (i)If f is convex, then for all \(a,b>0\), we havewith reversed inequalities if f is concave.$$ \frac{a+b}{2}\leq m_{f}(a,b)\leq b f^{-1} \biggl(\frac{1}{2}f(a/b) \biggr), $$(3.2)
- (ii)If f is convex and differentiable, then, for all \(a,b>0\), \(a\neq b\), we havewith reversed inequalities if f is concave.$$ \frac{a+b}{2}\leq m_{f}(a,b)\leq a- \frac{a-b}{f(a/b)}f \biggl(\frac {a+b}{2b} \biggr), $$(3.3)
Proof
Similarly, (2.3) with (3.1) immediately yields the following result.
Proposition 3.2
Remark 3.1
If in the previous results the considered functions are strictly convex (resp. strictly concave), then the associated inequalities are strict (resp. reversed).
Remark 3.2
Inequalities (3.4) become equalities for \(f(u)=u-1\) and \(g(u)=\ln u\), which correspond to \(m_{f}=A\) and \(m_{f}(a,b)=L(a,b)\), respectively. This is because the real function \(u\longmapsto uf'(u)\) is linear affine if and only if \(f(u)=c_{1}u+c_{2}\ln u+c_{3}\) for some constants \(c_{1}\), \(c_{2}\), and \(c_{3}\) to be chosen to ensure \(f\in C^{0}_{\uparrow}(0,\infty)\) with \(f(1)=0\), that is, \(c_{1},c_{2}\geq0\), \(c_{1}+c_{2}\neq0\), and \(c_{1}+c_{3}=0\).
We now present some examples that illustrate the previous results.
Example 3.1
Let \(f(x)=\ln x\), which is strictly increasing and strictly concave on \((0,\infty)\). We have seen that \(m_{f}=L\). By Proposition 3.1 and Remark 3.1, (3.2) and (3.3) are here reversed. After simple computation, (3.2) gives the known double inequality \(G< L< A\). See also [3] for a direct way.
Example 3.2
It appears to be interesting to determine convenient conditions on f for which the left- and right-hand sides of (3.4) are binary means.
The next result shows that direct comparison of two functions \(f,g\in C^{0}_{\uparrow}(0,\infty)\) leads to inequalities involving \(m_{f}\) and \(m_{g}\).
Proposition 3.3
- (i)If \(f(x)< g(x)\) for all \(x>0\), \(x\neq1\), then, for all \(a,b>0\), \(a\neq b\), we have$$\bigl(f(a/b) \bigr)^{2} \bigl(a-m_{f}(a,b) \bigr)< \bigl(g(a/b) \bigr)^{2} \bigl(a-m_{g}(a,b) \bigr). $$
- (ii)If \(f'(x)\leq g'(x)\) for all \(x>0\), then, for all \(a,b>0\), we have$$\bigl\vert f(a/b)\bigr\vert m_{f}(a,b)\leq\bigl\vert g(a/b) \bigr\vert m_{g}(a,b). $$
Proof
The following example illustrates the previous proposition.
Example 3.3
Proposition 3.4
- (i)
If the composed function \(g\circ f^{-1}\) is convex (resp. concave), then \(m_{f}\leq m_{g}\) (resp. \(m_{f}\geq m_{g}\)).
- (ii)
If f is concave and g is convex, then \(m_{f}\leq A\leq m_{g}\).
Proof
(ii) Let id be the identity map of \((0,\infty)\). If f is concave (resp. g is convex), then \(\mathrm{id}\circ f^{-1}\) is convex (resp. \(g\circ \mathrm{id}^{-1}\) is convex). To conclude, we apply (i) knowing that \(m_{\mathrm{id}-1}=A\). □
Remark 3.3
If, in the previous proposition, we replace the word ‘convex’ (resp. ‘concave’) by ‘strictly convex’ (resp. ‘strictly concave’), then all the related inequalities are strict.
The following example illustrates the previous result.
Example 3.4
The next corollary can be considered as an example of the previous proposition.
Corollary 3.5
Proof
We now state the following example explaining how to use the previous corollary. Although it is a very simple example, we will use it as a good tool for obtaining a symmetric homogeneous bivariate mean that appears to us to be new.
Example 3.5
Summarizing the previous discussion, we have the following result.
Proposition 3.6
4 Characterization of homogeneous symmetric monotone means
We preserve the same notation. For \(f\in C^{0}_{\uparrow}(0,\infty)\), we have seen that \(m_{f}\) is a continuous and strict monotone mean. Inversely, given a mean m, under what condition there exists \(f\in C^{0}_{\uparrow}(0,\infty)\) such that \(m=m_{f}\)? We should note here that the answer of the latter question has to be seen as a characterization of bivariate means that could be fitted in the class of means introduced in (2.1).
Before giving an answer to that question, we state the following technical lemmas.
Lemma 4.1
Proof
Lemma 4.2
- (i)
If ϕ is differentiable, then \(\phi'(1)=1/2\).
- (ii)If ϕ is continuously differentiable then$$ \lim_{x\rightarrow1}\frac{(x-1)\phi'(x)}{x-\phi(x)}=1. $$(4.1)
- (iii)If (4.1) holds, then we have$$ \lim_{x\rightarrow1}\exp \biggl( \int\frac{\phi'(x)}{x-\phi (x)}\,dx \biggr)=0. $$(4.2)
Proof
Lemma 4.3
Proof
Now, we are in a position to state our characterization theorem, which answers the question stated at the beginning of this section.
Theorem 4.4
Let m be a symmetric homogeneous strict monotone mean, and ϕ its associate function. Assume that ϕ is continuously differentiable. Then there exists \(f_{m}:=f\in C^{1}_{\uparrow}(0,\infty )\) with \(f(1)=0\) such that \(m=m_{f}\). Further, such f, called here the intrinsic function of m, is the solution of the ODE (4.3) in Lemma 4.3.
Proof
Corollary 4.5
The mapping Ψ defined by (4.5) is well defined and realizes an injection between the set of symmetric homogeneous strict monotone means, \(\mathcal{M}_{\mathrm{shsm}}\), and the set \(\mathcal{F}/_{\equiv}\).
Proof
Due to Theorem 4.4, for every \(m\in\mathcal{M}_{\mathrm{shsm}}\), there exists \(f\in\mathcal{F}\) such that \(m=m_{f}\). To check that Ψ is a map, consider two means such that \(m_{1}=m_{2}\) and let \(f_{1}\), \(f_{2}\) be the two associated intrinsic functions, \(m_{1}=m_{f_{1}}\) and \(m_{2}=m_{f_{2}}\). We need to check that \(c(f_{1})=c(f_{2})\). Since \(m_{f_{1}}=m_{f_{2}}\), Theorem 2.4 implies that there exists \(\alpha>0\) such that \(f_{1}=\alpha\cdot f_{2}\), and so \(c(f_{1})=c(f_{2})\).
To check the injectivity of Ψ, let \(m_{1}\) and \(m_{2}\) be two means such that \(\Psi(m_{1})=\Psi(m_{2})\). This means that \(c(f_{1})=c(f_{2})\), where \(f_{1}\), \(f_{2}\) are respectively the two associated intrinsic functions of \(m_{1}\) and \(m_{2}\). By the definition of the equivalence relation ≡, \(c(f_{1})=c(f_{2})\) implies that there exists \(\alpha>0\) such that \(f_{1}=\alpha\cdot f_{2}\), which again by Theorem 2.4 implies that \(m_{1}=m_{2}\). Note here that Ψ is not onto because for a given \(f\in\mathcal{F}\), the mean \(m_{f}\) is not necessarily symmetric. The proof is finished. □
Now, we will state some examples that illustrate our theoretical results and show the generality of our approach.
Example 4.1
Example 4.2
Example 4.3
Example 4.4
Example 4.5
Example 4.6
The next example shows that the assumption ‘m is a monotone mean’ in Theorem 4.4 is necessary.
Example 4.7
Finally, we state the following example, which includes a lot of particular situations previously discussed.
Example 4.8
The particular cases \(p=1\), \(p=0\), \(p=-2\), \(p=-1\), and \(p=-1/2\) were previously discussed; see Example 2.1.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their comments and suggestions, which have been included in the final version of this manuscript.
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Authors’ Affiliations
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