Sub-super-stabilizability of certain bivariate means via mean-convexity
- Mustapha Raïssouli^{1, 2}Email authorView ORCID ID profile and
- József Sándor^{3}
https://doi.org/10.1186/s13660-016-1212-z
© Raïssouli and Sándor 2016
Received: 29 June 2016
Accepted: 14 October 2016
Published: 4 November 2016
Abstract
In this paper, we first show that the first Seiffert mean P is concave whereas the second Seiffert mean T and the Neuman-Sándor mean NS are convex. As applications, we establish the sub-stabilizability/super-stabilizability of certain bivariate means. Open problems are derived as well.
Keywords
bivariate mean convexity of mean sub-stabilizable mean super-stabilizable mean mean-inequalitiesMSC
26E601 Introduction
As usual, we identify a mean m with its value at \((x,y)\) by setting \(m=m(x,y)\) for simplicity. All the previous means are symmetric homogeneous monotone continuous.
If, moreover, \(m_{1}\) and \(m_{2}\) are comparable, then we say that m is \((m_{1},m_{2})\)-sub-stabilizable if \({\mathcal{R}}(m_{1},m,m _{2})\leq m\) and m is between \(m_{1}\) and \(m_{2}\); see [13]. If this latter mean inequality is strict, then we say that m is strictly \((m_{1},m_{2})\)-sub-stabilizable. The super-stabilizability of m is defined in an analogous manner (by reversing the previous mean inequalities). In short, we can say that m is \((m_{1},m_{2})\)-super-stabilizable if and only if \(m^{*}\) is \((m_{2}^{*},m_{1}^{*})\)-sub-stabilizable, where \(m^{*}\) refers to the dual mean of m defined by \(m^{*}(x,y)= (m(x^{-1},y^{-1} ))^{-1}\) for all \(x,y>0\). For a large study about sub-stabilizable and super-stabilizable means and the related results, see [13].
The following results will be also needed in the sequel.
Theorem 1.1
[11]
- (i)
The means A, H, G, and Q are stable.
- (ii)
The mean L is simultaneously \((A,G)\)-stabilizable and \((H,A)\)-stabilizable, whereas I is \((G,A)\)-stabilizable.
Theorem 1.2
Part (ii) of Theorem 1.2 was proved in [14] via a long way and later proved again by the authors in [15] via a simple and fast way. There is no result proved yet about stabilizability, sub-stabilizability, or super-stabilizability of the means T and NS.
The remainder of this paper is organized as follows. Section 2 contains basic notions about convexity of bivariate means and some needed lemmas. Section 3 is devoted to show the convexity/concavity of the three standard means P, NS, and T. Section 4 contains some applications of the previous results to the sub/super stabilizability of certain bivariate means. Some open problems of interest are derived as well.
2 Mean-convexity and needed tools
We say that m is partially convex (resp. concave) if the real functions \(x\longmapsto m(x,y)\) for fixed \(y>0\) and \(y\longmapsto m(x,y)\) for fixed \(x>0\) are convex (resp. concave) on \((0,\infty )\). If m is symmetric homogeneous, then m is partially convex (resp. concave) if and only if the map \(x\longmapsto m(x,1)\) is convex (resp. concave) on \((1,\infty )\). It is clear that every convex (resp. concave) mean is partially convex (resp. concave). The reverse of this latter property is not always true. However, for special class of regular means, it remains true, as confirmed by the following result.
Lemma 2.1
Let m be a homogeneous continuous mean. Then m is convex (resp. concave) if and only if the real function \(x\longmapsto m(x,1)\) is convex (resp. concave) on \((0,\infty )\).
Proof
It follows from [16], p.24. □
Remark 2.1
The next lemma will be needed in the sequel.
Lemma 2.2
- (i)
The maps \((x,y)\longmapsto A_{p}(x,y)\) for fixed \(p\in {\mathbb{R}}\) and \(p\longmapsto A_{p}(x,y)\) for fixed \(x,y>0\) are continuous and strictly increasing.
- (ii)
Let \(x,y>0\) be fixed. If \(p\geq 1\), then the map \(p\longmapsto A _{p}(x,y)\) is concave, that is (since \(p\longmapsto A_{p}(x,y)\) is continuous),
Now, we can state the following examples.
Example 2.1
Example 2.2
As proved in [23], the real functions \(x\longmapsto L(x,1)\) and \(x\longmapsto I(x,1)\) are (strictly) concave on \((0,\infty )\). By Lemma 2.1 we then deduce that L and I are also strictly concave.
Example 2.3
The following remark is worth to be stated.
Remark 2.2
The previous examples were just stated as direct illustrations of the related lemmas. However, their assertions are particular cases of Minkowski’s inequalities for difference means and Gini means, which were first proved in [24] and [25], respectively. It is also worth mentioning that the study of the log-convexity of these two families of means with respect to their parameters can be found, for instance, in [26–29].
Before proving that the means NS and T are strictly convex and that P is strictly concave, we need another lemma.
Lemma 2.3
Proof
It is a simple exercise of real analysis. Therefore, we leave the details to the reader. □
3 Convexity of P, NS, and T
We start this section by stating the following result.
Proposition 3.1
Let m be a symmetric homogeneous monotone mean. Assume that m is \(C^{2}\) and (strictly) convex. Then \(m^{*}\) is (strictly) concave.
Proof
Remark 3.1
The converse of the previous proposition is in general false, that is, the concavity of m does not imply the convexity of \(m^{*}\). In fact, G is concave, and \(G^{*}=G\) is also concave. Also, it is not hard to verify that \(L^{*}\) is concave, too, as is L.
Remark 3.2
Now, we discuss the convexity of the three standard means P, NS, and T.
Theorem 3.2
The first Seiffert mean P is strictly concave.
Proof
For the mean NS, we have the following:
Theorem 3.3
The Neuman-Sándor mean NS is strictly convex.
Proof
Finally, we state the following result.
Theorem 3.4
The second Seiffert mean T is strictly convex.
Proof
Remark 3.3
- (i)
According to Proposition 3.1, we deduce that \(NS^{*}\) and \(T^{*}\) are strictly concave.
- (ii)
Following their graphs, the means NS and T seem to be not log-convex. The mean P is log-concave since it is concave.
4 Application for sub-super-stabilizability
As already pointed before, this section displays some applications of the mean-convexity to the so-called sub/super-stabilizability of some standard means. Let \(m_{1}\), m, \(m_{2}\) be three means such that \(m_{1}< m< m_{2}\). In some situations, it may be of interest to show that \(m< m (m_{1},m_{2} )\) or \(m (m_{1},m_{2} )< m\), that is, \(m(x,y)<(>)\,m (m_{1}(x,y),m_{2}(x,y) )\) for all \(x,y>0\). Many inequalities of this type are well known in the literature, such as \(L< L(A,G)\), \(I(A,G)< I\), and \(T(A,Q)< T\); see, for instance, [13, 30]. In what follows, we will see that strict convexity/concavity of m, when combined with its sub/super-satbilizability, can be used for obtaining some of these (composed) mean-inequalities. This is described in the following result.
Theorem 4.1
- (i)
If m is strictly convex and \((A,A_{p})\)-sub-stabilizable for some \(p\in {\mathbb{R}}\), then we have \(m (A,A_{p} )< m\).
- (ii)
If m is strictly concave and \((A,A_{p})\)-super-stabilizable for some \(p\in {\mathbb{R}}\), then we have \(m< m (A,A_{p} )\).
Proof
(ii) It is analogous to (i) by similar arguments. □
It is worth mentioning that the sub-stabilizability and super-stabilizability in the previous theorem are not strict, and so we have the same conclusions when we replace both them by stabilizability. For example, L is strictly concave and \((A,G)\)-stabilizable. Then, Theorem 4.1(ii) immediately yields \(L< L(A,G)\).
Now, let us observe another example, which explains more how to use the mean-convexity for establishing the sub-stabilizability of a certain bivariate mean.
Theorem 4.2
Let m be a strictly concave mean. Assume that there exists \(r<1\) such that \(\frac{A+A_{r}}{2}\leq m< A\). Then m is strictly \((A,A_{r})\)-sub-stabilizable.
Proof
As a particular case of the previous theorem, we have the following result.
Corollary 4.3
The mean P is strictly \((A,G)\)-sub-stabilizable.
Proof
In [13] the authors proved this result by three different methods. We give here a fourth method based on the previous arguments. In fact, following [31, 32], we have \(\frac{2}{\pi }A+(1-\frac{2}{ \pi })G< P\), which implies that \(\frac{A+G}{2}< P\). The desired result follows from the previous theorem by taking \(m=P\) and \(r=0\). □
Another result of interest is presented in the following:
Theorem 4.4
Let m be a strictly convex mean. Assume that there exists \(r>1\) such that \(A< m\leq \frac{A+A_{r}}{2}\). Then m is strictly \((A,A_{r})\)-super-stabilizable.
Proof
The two following corollaries, whose proof will be deduced from that of the previous theorem, assert that the assumption \(m<\frac{A+A_{r}}{2}\), \(r>1\), is satisfied for the two particular cases \(m=NS\) and \(m=T\).
Corollary 4.5
The mean NS is strictly \((A,Q)\)-super-stabilizable.
Proof
Following [33], we have \(NS<\frac{2}{3}A+\frac{1}{3}Q\), which is stronger than \(NS<\frac{A+Q}{2}\). The desired result follows from the previous theorem by taking \(m=NS\) and \(r=2\) since \(A_{2}=Q\). □
Corollary 4.6
The mean T is strictly \((A,A_{3})\)-super-stabilizable.
Proof
Since T is convex, Theorem 4.4 immediately implies the desired result, provided that the inequality \((A{<})T<\frac{A+A_{3}}{2}\) holds. Such an inequality is established in the following: □
Proposition 4.7
Proof
Remark 4.1
From the three previous corollaries we immediately deduce that \(P^{*}\) is strictly \((H,G)\)-super-stabilizable, \(NS^{*}\) is strictly \((H,A_{-2})\)-sub-stabilizable, and \(T^{*}\) is strictly \((H,A_{-3})\)-sub-stabilizable, respectively.
Remark 4.2
In Theorem 4.4, the assumption \(m<\frac{A+A_{r}}{2}\) implies by Lemma 2.2 that \(m< A_{\frac{1+r}{2}}\), \(r>1\). Then if m has an upper bound as power mean, that is, \(m< A_{s}\), where s is the best possible, then we should have \(s\leq \frac{1+r}{2}\). For example, if \(m=NS\), then we know that \(NS< A_{4/3}\) with \(A_{4/3}\) the best power bound of NS (see [35]), and so we should have \(\frac{1+r}{2} \geq 4/3\), that is, \(r\geq 5/3\). Corollary 4.5 confirms that \(r=2\geq 5/3\) is a convenient case, but perhaps \(r=2\) is not the best possible. See more details in the next section.
Remark 4.3
Following Example 2.1, the mean \(\frac{A+A_{r}}{2}\) is strictly concave for \(r<1\) and strictly convex for \(r>1\). This, combined with Theorem 4.2 and Theorem 4.4, respectively, yields that \(\frac{A+A_{r}}{2}\) is strictly \((A,A_{r})\)-sub-stabilizable for \(r<1\) and strictly \((A,A_{r})\)-super-stabilizable for \(r>1\).
5 Some open problems
We end this paper by stating some open problems as the purpose for future research. These problems are derived from the previous theoretical results and their proofs.
Problem 1
Problem 2
Corollary 4.5 and Corollary 4.6 assert that Problem 2 has a positive answer when \(m=NS>A\) and \(m=T>A\), with \(r=2\) and \(r=3\), respectively. In parallel, Corollary 4.3 gives a positive answer for \(m=P< A\) with \(r=0\). Of course, we can then ask what is the best possible \(r>1\) such that \(NS<\frac{A+A_{r}}{2}\). A similar question can be stated for T and P. About this, we state the following conjectures.
Problem 3
Finally, we end this paper by mentioning the following. In the previous study, we have seen that the standard symmetric homogeneous monotone means that are less than A (such as min, H, G, L, \(L^{*}\), I, and P) are concave, whereas those that are greater than A (like max, C, S, T, and NS) are convex. We have seen that the nonmonotone mean \(C^{*}\) is neither convex nor concave with \(C>A\), and so \(C^{*}< A^{*}=H< A\). This, with Proposition 3.1, allows us to arise the following open problem.
Problem 4
- (i)
Prove or disprove that Proposition 3.1 holds for m not necessarily of class \(C^{2}\).
- (ii)
Prove or disprove that if \(m< A\), then m is strictly concave and if \(m>A\), then m is strictly convex.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their comments and for bringing us some references.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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