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Positive answer for a conjecture about stabilizable means
Journal of Inequalities and Applications volume 2013, Article number: 467 (2013)
In an earlier paper (Raïssouli in Appl. Math. E-Notes 11:159-174, 2011), the author conjectured that for given stable means and such that , there exists a unique -stabilizable mean satisfying that . In the present paper, a positive answer of this conjecture is given. Some examples, illustrating the theoretical study, are discussed.
In the recent past, the theory of means has been the subject of intensive research. Stability and stabilizability concepts for means have been recently introduced by the author in . It has been proved to be a useful tool for theoretical viewpoint as well as for practical purposes [2–4]. In this section, we recall some basic notions about means in two variables that will be needed later. We understand by (bivariate) mean a binary map m between positive real numbers satisfying the following statements:
for all ;
for all ;
for all ;
is an increasing function in a (and in b);
is a continuous function of a and b.
and are known as the arithmetic, geometric, harmonic, logarithmic and identric means, respectively.
For two means and , we write if and only if for every and if and only if for all with . The above means satisfy the known chain of inequalities
We say that m is a strict mean if is strictly increasing in a and in b. Also, every strict mean m satisfies that . 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  and needed in the sequel.
Definition 1.1 Let , and be three given means. For all , define
called the resultant mean-map of , and .
A study investigating the elementary properties of the resultant mean-map was stated in . Here, we just recall the following result needed later.
Proposition 1.1 Let , and be three means. Then the map defines a mean. Further the mean-map is point-wisely increasing with respect to each of its mean variables, that is,
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:
stable if ;
stabilizable if there exist two nontrivial stable means and satisfying the relation . We then say that m is -stabilizable.
Theorem 1.2 With the above, the following assertions hold true:
The arithmetic, geometric and harmonic means A, G and H are stable.
The logarithmic mean L is -stabilizable and -stabilizable.
The identric mean I is -stabilizable.
2 Two needed results
The next definition , recalling another concept for means, will be needed in the sequel.
Definition 2.1 Let and be two means. The tensor product of and is the map, denoted , defined by
A mean m will be called cross mean if the map is symmetric with its four variables.
It is proved in  that every cross mean is stable. The reverse of this latter assertion is still an open problem.
Now, let , and be three given means. For the sake of simplicity, we set
in the sense that
for all .
Clearly, for all means and . If is a strict mean, then (resp. ) if and only if . Further, we have for all means and .
The next result will be of interest later.
Proposition 2.1 Let and be two given means. Assume that is a cross mean, then we have
Proof By Definition 1.1 and Definition 1.2, one has, for all ,
Since is a cross mean, then
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 , , and be four given means. Assume that is a cross mean, then the following holds:
Proof By Definition 1.1, we have, for all ,
which with (2.1) gives
Since is a cross mean, then we have
Again, by Definition 1.1 and (2.1), respectively, we obtain
which completes the proof. □
In , for defining an -stabilizable mean, the author imposed that the means and should be nontrivial and stable. The fact that and are nontrivial is clear since the relation is valid for every mean m. However, the fact that and are stable was imposed only in the aim to characterize a stabilizable mean m (as L and I) in terms of and having simple expressions (as A, G and H). As example, we know that L is -stabilizable, where H and A are (stable) means having expressions more simple as that of L. Analogous way for the fact that L is -stabilizable and I is -stabilizable can be stated.
3 Existence and uniqueness of a stabilizable mean
In , the author stated the following conjecture.
Conjecture Let and be two nontrivial stable means such that . Then there exists one and only one mean m, which is -stabilizable, satisfying that .
The aim of this section is to prove that the above conjecture is true when we add convenient hypotheses for the means and . Of course, following Definition 1.2, and 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 and we show that there exists a unique -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.
Before giving an affirmative response that we are waiting for, we state some needed notions. A sequence of means will be called point-wise convergent (in short, p-convergent) if, for all , the real sequence converges. Setting , it is easy to see that is a mean. Similarly, we define the point-wise monotonicity of . By virtue of the double inequality
we deduce that every p-increasing (resp. p-decreasing) sequence is p-convergent.
Now, let and be two given means and define the following two mean-sequences:
By mathematical induction, it is not hard to check that and are means for every . In the following, we study the p-convergence of the mean-sequences and . We may state the next result.
Proposition 3.1 Let and be two stable means with . Then the following mean-inequalities
hold for all . Consequently, the mean-sequences and both p-converge.
Proof Since , we deduce by simple mathematical induction, with the help of (1.1), that for each . Now, using the fact that is stable, we can write, again with help of (1.1),
This, with mathematical induction, shows that for each . Analogously, we prove that for every . Summarizing, we deduce that is a p-increasing sequence p-upper bounded by , while is a p-decreasing sequence p-lower bounded by . Then the desired result follows, and so this completes the proof. □
We explicitly notice that the above mean-sequences and p-converge for all comparable means and , i.e., (or , see Remark 3.1 below). Now, a natural question arises from the above: under what conditions on and do the p-limits of and coincide? In what follows, we are interested in finding a positive answer to this question.
Proposition 3.2 Let and be two means. Assume that is a cross mean, then the mean-sequences and satisfy the following relationship:
Proof We use mathematical induction. For , by (3.1) and Proposition 2.1, we have
Assume that (3.3) is true for n, by (3.1) we obtain
This, with Theorem 2.2, yields
so proving the desired result. □
Now, we are in a position to state the next result.
Theorem 3.3 Let and be two stable means with . Assume further that is strict and a cross mean. Then, the mean-sequences and both p-converge to the same limit m which is -stabilizable and satisfying .
Proof According to Proposition 3.1, the sequences and both p-converge. Call their limits Θ and ϒ, respectively. By the continuity of , relationship (3.3) gives, when , . This, with the fact that is strict, yields . Letting in the first (or second) relation of (3.1), we obtain, with the help of continuity of , , which means that m is -stabilizable. Inequalities (3.2) imply that , which completes the proof. □
Corollary 3.4 Let and be as in the above theorem. Let m be a -stabilizable mean such that . Then m is the common p-limit of the above sequences and .
Proof We show, by mathematical induction, that for all . For , it is true by virtue of . Assume that , the recursive relations (3.1), with the help of (1.1), give
This, with the fact that m is -stabilizable, i.e., , yields . It follows that for all . Since the sequences and both p-converge to the same limit, we deduce the desired result. □
The above corollary tells us that every -stabilizable mean is the common p-limit of the above sequences and . 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 and be as in the above theorem. Then there exists one and only one -stabilizable mean m such that .
Remark 3.1 (i) If the means and are such that , 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 and , there exists one and only one -stabilizable mean satisfying that .
(ii) For and as in the above theorem, the last corollary tells us that the map has one and only one mean-fixed point.
Example 3.1 As already pointed, the mean L is -stabilizable. Following the above study, L is the unique -stabilizable mean satisfying , 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.
It is worth mentioning that the reader will do well in distinguishing between the following two situations:
There exists one and only one -stabilizable mean for suitable means and as previously showed.
A given mean m can be -stabilizable and -stabilizable for two distinct couples and . Indeed, as already pointed before, the logarithmic mean L is simultaneously -stabilizable and -stabilizable.
Finally, the following is of interest: Let and be as in the above theorem. For every mean m, we set
Then, for fixed means and , defines a map from the set of means into itself. If denotes the n-iterate of , i.e., and, for ,
then the above study tells us that every -stabilizable mean m can be written as
for the point-wise limit. Further, the following iterative inequalities hold true (if ):
Example 3.2 L is -stabilizable. Then we have
Simple computations lead to
We then obtain
which refines . The procedure can be continued for obtaining more iterative refinements for this latter double inequality.
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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).
The author declares that he has no competing interests.
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Raïssouli, M. Positive answer for a conjecture about stabilizable means. J Inequal Appl 2013, 467 (2013). https://doi.org/10.1186/1029-242X-2013-467
- stable means
- stabilizable means