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# Positive answer for a conjecture about stabilizable means

- Mustapha Raïssouli
^{1, 2}Email author

**2013**:467

https://doi.org/10.1186/1029-242X-2013-467

© Raïssouli; licensee Springer. 2013

**Received:**24 May 2013**Accepted:**5 September 2013**Published:**7 November 2013

## Abstract

In an earlier paper (Raïssouli in Appl. Math. E-Notes 11:159-174, 2011), the author conjectured that for given stable means ${m}_{1}$ and ${m}_{2}$ such that ${m}_{1}\le {m}_{2}$, there exists a unique $({m}_{1},{m}_{2})$-stabilizable mean satisfying that ${m}_{1}\le m\le {m}_{2}$. In the present paper, a positive answer of this conjecture is given. Some examples, illustrating the theoretical study, are discussed.

**MSC:**26E60.

## Keywords

- means
- stable means
- stabilizable means

## 1 Introduction

*m*between positive real numbers satisfying the following statements:

- (i)
$m(a,a)=a$ for all $a>0$;

- (ii)
$m(a,b)=m(b,a)$ for all $a,b>0$;

- (iii)
$m(ta,tb)=tm(a,b)$ for all $a,b,t>0$;

- (iv)
$m(a,b)$ is an increasing function in

*a*(and in*b*); - (v)
$m(a,b)$ is a continuous function of

*a*and*b*.

and are known as the arithmetic, geometric, harmonic, logarithmic and identric means, respectively.

We say that *m* is a strict mean if $m(a,b)$ is strictly increasing in *a* and in *b*. Also, every strict mean *m* satisfies that $m(a,b)=a\u27f9a=b$. It is not hard to check that the trivial means min and max are not strict, while *A*, *G*, *H*, *L*, *I*, *S*, *C* are strict means.

For the sake of simplicity for the reader, we end this section by recalling some basic notions and results stated by the author in an earlier paper [1] and needed in the sequel.

**Definition 1.1**Let ${m}_{1}$, ${m}_{2}$ and ${m}_{3}$ be three given means. For all $a,b>0$, define

called the resultant mean-map of ${m}_{1}$, ${m}_{2}$ and ${m}_{3}$.

A study investigating the elementary properties of the resultant mean-map was stated in [1]. Here, we just recall the following result needed later.

**Proposition 1.1**

*Let*${m}_{1}$, ${m}_{2}$

*and*${m}_{3}$

*be three means*.

*Then the map*$(a,b)\mapsto \mathcal{R}({m}_{1},{m}_{2},{m}_{3})(a,b)$

*defines a mean*.

*Further the mean*-

*map*$({m}_{1},{m}_{2},{m}_{3})\mapsto \mathcal{R}({m}_{1},{m}_{2},{m}_{3})$

*is point*-

*wisely increasing with respect to each of its mean variables*,

*that is*,

In [1, 4], the author gives a lot of examples about computations of $\mathcal{R}({m}_{1},{m}_{2},{m}_{3})$ when ${m}_{1}$, ${m}_{2}$ and ${m}_{3}$ are means belonging to the set of the above standard means.

As proved in [1, 3, 4], and will be again shown throughout this paper, the resultant mean-map stems its importance in the fact that it is a good tool for introducing the stability and stabilizability concepts as recalled below.

**Definition 1.2**A mean

*m*is said to be:

- (a)
stable if $\mathcal{R}(m,m,m)=m$;

- (b)
stabilizable if there exist two nontrivial stable means ${m}_{1}$ and ${m}_{2}$ satisfying the relation $\mathcal{R}({m}_{1},m,{m}_{2})=m$. We then say that

*m*is $({m}_{1},{m}_{2})$-stabilizable.

In [1, 3], the author stated a developed study about the stability and stabilizability of the above standard means. In particular, he proved the following.

**Theorem 1.2**

*With the above*,

*the following assertions hold true*:

- (1)
*The arithmetic*,*geometric and harmonic means**A*,*G**and**H**are stable*. - (2)
*The logarithmic mean**L**is*$(H,A)$-*stabilizable and*$(A,G)$-*stabilizable*. - (3)
*The identric mean**I**is*$(G,A)$-*stabilizable*.

## 2 Two needed results

The next definition [2], recalling another concept for means, will be needed in the sequel.

**Definition 2.1**Let ${m}_{1}$ and ${m}_{2}$ be two means. The tensor product of ${m}_{1}$ and ${m}_{2}$ is the map, denoted ${m}_{1}\otimes {m}_{2}$, defined by

A mean *m* will be called cross mean if the map ${m}^{\otimes 2}:=m\otimes m$ is symmetric with its four variables.

It is proved in [1] that every cross mean is stable. The reverse of this latter assertion is still an open problem.

for all $a,b>0$.

Clearly, $\mathcal{M}({m}_{1},{m}_{2},{m}_{2})={m}_{2}$ for all means ${m}_{1}$ and ${m}_{2}$. If ${m}_{1}$ is a strict mean, then $\mathcal{M}({m}_{1},{m}_{2},{m}_{3})={m}_{2}$ (resp. $\mathcal{M}({m}_{1},{m}_{2},{m}_{3})={m}_{3}$) if and only if ${m}_{2}={m}_{3}$. Further, we have $\mathcal{M}({m}_{1},{m}_{2},{m}_{3})=\mathcal{M}({m}_{1},{m}_{3},{m}_{2})$ for all means ${m}_{2}$ and ${m}_{3}$.

The next result will be of interest later.

**Proposition 2.1**

*Let*${m}_{1}$

*and*${m}_{2}$

*be two given means*.

*Assume that*${m}_{1}$

*is a cross mean*,

*then we have*

*Proof*By Definition 1.1 and Definition 1.2, one has, for all $a,b>0$,

which concludes the proof. □

The above proposition implies again that every cross mean is stable. The next theorem is more interesting.

**Theorem 2.2**

*Let*${m}_{1}$, ${m}_{2}$, ${m}_{3}$

*and*${m}_{4}$

*be four given means*.

*Assume that*${m}_{1}$

*is a cross mean*,

*then the following holds*:

*Proof*By Definition 1.1, we have, for all $a,b>0$,

which completes the proof. □

In [1], for defining an $({m}_{1},{m}_{2})$-stabilizable mean, the author imposed that the means ${m}_{1}$ and ${m}_{2}$ should be nontrivial and stable. The fact that ${m}_{1}$ and ${m}_{2}$ are nontrivial is clear since the relation $\mathcal{R}(min,m,max)=m$ is valid for every mean *m*. However, the fact that ${m}_{1}$ and ${m}_{2}$ are stable was imposed only in the aim to characterize a stabilizable mean *m* (as *L* and *I*) in terms of ${m}_{1}$ and ${m}_{2}$ having simple expressions (as *A*, *G* and *H*). As example, we know that *L* is $(H,A)$-stabilizable, where *H* and *A* are (stable) means having expressions more simple as that of *L*. Analogous way for the fact that *L* is $(A,G)$-stabilizable and *I* is $(G,A)$-stabilizable can be stated.

## 3 Existence and uniqueness of a stabilizable mean

In [1], the author stated the following conjecture.

**Conjecture** *Let* ${m}_{1}$ *and* ${m}_{2}$ *be two nontrivial stable means such that* ${m}_{1}\le {m}_{2}$. *Then there exists one and only one mean* *m*, *which is* $({m}_{1},{m}_{2})$-*stabilizable*, *satisfying that* ${m}_{1}\le m\le {m}_{2}$.

The aim of this section is to prove that the above conjecture is true when we add convenient hypotheses for the means ${m}_{1}$ and ${m}_{2}$. Of course, following Definition 1.2, ${m}_{1}$ and ${m}_{2}$ will be assumed to be stable means. We can ask why it is interesting to solve the above conjecture. In fact, as we have seen before, the means *L* and *I*, having complicated expressions, are stabilizable with respect to *A*, *G*, *H* whose expressions are more simple. It follows that if for given (simple) means ${m}_{1}$ and ${m}_{2}$ we show that there exists a unique $({m}_{1},{m}_{2})$-stabilizable mean, we can then characterize new means in terms of known (simple) means. This can be also useful when we speak for means involving several variables or those with operator arguments, of course if the above conjecture can be extended for these classes of generalized means.

*p*-convergent) if, for all $a,b>0$, the real sequence ${({m}_{n}(a,b))}_{n}$ converges. Setting ${m}_{\mathrm{\infty}}(a,b)={lim}_{n}{m}_{n}(a,b)$, it is easy to see that ${m}_{\mathrm{\infty}}$ is a mean. Similarly, we define the point-wise monotonicity of ${({m}_{n})}_{n}$. By virtue of the double inequality

we deduce that every *p*-increasing (resp. *p*-decreasing) sequence ${({m}_{n})}_{n}$ is *p*-convergent.

By mathematical induction, it is not hard to check that ${\mathrm{\Theta}}_{n}$ and ${\mathrm{\Upsilon}}_{n}$ are means for every $n\ge 0$. In the following, we study the *p*-convergence of the mean-sequences ${({\mathrm{\Theta}}_{n})}_{n}$ and ${({\mathrm{\Upsilon}}_{n})}_{n}$. We may state the next result.

**Proposition 3.1**

*Let*${m}_{1}$

*and*${m}_{2}$

*be two stable means with*${m}_{1}\le {m}_{2}$.

*Then the following mean*-

*inequalities*

*hold for all* $n\ge 1$. *Consequently*, *the mean*-*sequences* ${({\mathrm{\Theta}}_{n})}_{n}$ *and* ${({\mathrm{\Upsilon}}_{n})}_{n}$ *both* *p*-*converge*.

*Proof*Since ${\mathrm{\Theta}}_{0}:={m}_{1}\le {m}_{2}:={\mathrm{\Upsilon}}_{0}$, we deduce by simple mathematical induction, with the help of (1.1), that ${\mathrm{\Theta}}_{n}\le {\mathrm{\Upsilon}}_{n}$ for each $n\ge 0$. Now, using the fact that ${m}_{1}$ is stable, we can write, again with help of (1.1),

This, with mathematical induction, shows that ${\mathrm{\Theta}}_{n-1}\le {\mathrm{\Theta}}_{n}$ for each $n\ge 1$. Analogously, we prove that ${\mathrm{\Upsilon}}_{n}\le {\mathrm{\Upsilon}}_{n-1}$ for every $n\ge 1$. Summarizing, we deduce that ${({\mathrm{\Theta}}_{n})}_{n}$ is a *p*-increasing sequence *p*-upper bounded by ${m}_{2}$, while ${({\mathrm{\Upsilon}}_{n})}_{n}$ is a *p*-decreasing sequence *p*-lower bounded by ${m}_{1}$. Then the desired result follows, and so this completes the proof. □

We explicitly notice that the above mean-sequences ${({\mathrm{\Theta}}_{n})}_{n}$ and ${({\mathrm{\Upsilon}}_{n})}_{n}$ *p*-converge for all comparable means ${m}_{1}$ and ${m}_{2}$, *i.e.*, ${m}_{1}\le {m}_{2}$ (or ${m}_{2}\le {m}_{1}$, see Remark 3.1 below). Now, a natural question arises from the above: under what conditions on ${m}_{1}$ and ${m}_{2}$ do the *p*-limits of ${({\mathrm{\Theta}}_{n})}_{n}$ and ${({\mathrm{\Upsilon}}_{n})}_{n}$ coincide? In what follows, we are interested in finding a positive answer to this question.

**Proposition 3.2**

*Let*${m}_{1}$

*and*${m}_{2}$

*be two means*.

*Assume that*${m}_{1}$

*is a cross mean*,

*then the mean*-

*sequences*${({\mathrm{\Theta}}_{n})}_{n}$

*and*${({\mathrm{\Upsilon}}_{n})}_{n}$

*satisfy the following relationship*:

*Proof*We use mathematical induction. For $n=0$, by (3.1) and Proposition 2.1, we have

*n*, by (3.1) we obtain

so proving the desired result. □

Now, we are in a position to state the next result.

**Theorem 3.3** *Let* ${m}_{1}$ *and* ${m}_{2}$ *be two stable means with* ${m}_{1}\le {m}_{2}$. *Assume further that* ${m}_{1}$ *is strict and a cross mean*. *Then*, *the mean*-*sequences* ${({\mathrm{\Theta}}_{n})}_{n}$ *and* ${({\mathrm{\Upsilon}}_{n})}_{n}$ *both* *p*-*converge to the same limit* *m* *which is* $({m}_{1},{m}_{2})$-*stabilizable and satisfying* ${m}_{1}\le m\le {m}_{2}$.

*Proof* According to Proposition 3.1, the sequences ${({\mathrm{\Theta}}_{n})}_{n}$ and ${({\mathrm{\Upsilon}}_{n})}_{n}$ both *p*-converge. Call their limits Θ and ϒ, respectively. By the continuity of ${m}_{1}$, relationship (3.3) gives, when $n\to \mathrm{\infty}$, $\mathrm{\Theta}={m}_{1}(\mathrm{\Theta},\mathrm{\Upsilon})$. This, with the fact that ${m}_{1}$ is strict, yields $\mathrm{\Theta}=\mathrm{\Upsilon}:=m$. Letting $n\to \mathrm{\infty}$ in the first (or second) relation of (3.1), we obtain, with the help of continuity of ${m}_{1}$, $m=\mathcal{R}({m}_{1},m,{m}_{2})$, which means that *m* is $({m}_{1},{m}_{2})$-stabilizable. Inequalities (3.2) imply that ${m}_{1}\le m\le {m}_{2}$, which completes the proof. □

**Corollary 3.4** *Let* ${m}_{1}$ *and* ${m}_{2}$ *be as in the above theorem*. *Let* *m* *be a* $({m}_{1},{m}_{2})$-*stabilizable mean such that* ${m}_{1}\le m\le {m}_{2}$. *Then* *m* *is the common* *p*-*limit of the above sequences* $({\mathrm{\Theta}}_{n})$ *and* $({\mathrm{\Upsilon}}_{n})$.

*Proof*We show, by mathematical induction, that ${\mathrm{\Theta}}_{n}\le m\le {\mathrm{\Upsilon}}_{n}$ for all $n\ge 0$. For $n=0$, it is true by virtue of ${\mathrm{\Theta}}_{0}={m}_{1}\le m\le {m}_{2}=\mathrm{\Upsilon}$. Assume that ${\mathrm{\Theta}}_{n}\le m\le {\mathrm{\Upsilon}}_{n}$, the recursive relations (3.1), with the help of (1.1), give

This, with the fact that *m* is $({m}_{1},{m}_{2})$-stabilizable, *i.e.*, $m=\mathcal{R}({m}_{1},m,{m}_{2})$, yields ${\mathrm{\Theta}}_{n+1}\le m\le {\mathrm{\Upsilon}}_{n+1}$. It follows that ${\mathrm{\Theta}}_{n}\le m\le {\mathrm{\Upsilon}}_{n}$ for all $n\ge 0$. Since the sequences $({\mathrm{\Theta}}_{n})$ and $({\mathrm{\Upsilon}}_{n})$ both *p*-converge to the same limit, we deduce the desired result. □

The above corollary tells us that every $({m}_{1},{m}_{2})$-stabilizable mean is the common *p*-limit of the above sequences $({\mathrm{\Theta}}_{n})$ and $({\mathrm{\Upsilon}}_{n})$. This, with the uniqueness of the *p*-limit, implies immediately the next result, which gives an affirmative answer of the above conjecture.

**Corollary 3.5** *Let* ${m}_{1}$ *and* ${m}_{2}$ *be as in the above theorem*. *Then there exists one and only one* $({m}_{1},{m}_{2})$-*stabilizable mean* *m* *such that* ${m}_{1}\le m\le {m}_{2}$.

**Remark 3.1** (i) If the means ${m}_{1}$ and ${m}_{2}$ are such that ${m}_{1}\ge {m}_{2}$, analogous results as those above can be stated in a similar way. We leave to the reader the task to formulate these results in a detailed manner. In particular, with convenient means ${m}_{1}$ and ${m}_{2}$, there exists one and only one $({m}_{1},{m}_{2})$-stabilizable mean satisfying that ${m}_{2}\le m\le {m}_{1}$.

(ii) For ${m}_{1}$ and ${m}_{2}$ as in the above theorem, the last corollary tells us that the map $m\mapsto \mathcal{R}({m}_{1},m,{m}_{2})$ has one and only one mean-fixed point.

**Example 3.1** As already pointed, the mean *L* is $(H,A)$-stabilizable. Following the above study, *L* is the unique $(H,A)$-stabilizable mean satisfying $H\le L\le A$, and so *L* can be characterized as the *p*-limit of an iterative algorithm involving the simple means *H* and *A*. The same can be said for the other stabilizable means mentioned in Theorem 2.2. We leave it for the reader to give more details about this latter point in a similar manner as previously explained.

- (a)
There exists one and only one $({m}_{1},{m}_{2})$-stabilizable mean for suitable means ${m}_{1}$ and ${m}_{2}$ as previously showed.

- (b)
A given mean

*m*can be $({m}_{1},{m}_{2})$-stabilizable and $({m}_{1}^{\prime},{m}_{2}^{\prime})$-stabilizable for two distinct couples $({m}_{1},{m}_{2})$ and $({m}_{1}^{\prime},{m}_{2}^{\prime})$. Indeed, as already pointed before, the logarithmic mean*L*is simultaneously $(A,G)$-stabilizable and $(H,A)$-stabilizable.

*m*, we set

*n*-iterate of ${\mathrm{\Lambda}}_{{m}_{1}}^{{m}_{2}}$,

*i.e.*, ${({\mathrm{\Lambda}}_{{m}_{1}}^{{m}_{2}})}^{(0)}(m)=m$ and, for $n\ge 1$,

*m*can be written as

**Example 3.2**

*L*is $(H,A)$-stabilizable. Then we have

which refines $H\le L\le A$. The procedure can be continued for obtaining more iterative refinements for this latter double inequality.

## Declarations

### Acknowledgements

Many thanks to the anonymous referee for bringing us some recent references. This work was supported by the Research Center of Taibah University (No. 4625/2013).

## Authors’ Affiliations

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