Positive answer for a conjecture about stabilizable means
© Raïssouli; licensee Springer. 2013
Received: 24 May 2013
Accepted: 5 September 2013
Published: 7 November 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.
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.
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.
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.
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.
stable if ;
stabilizable if there exist two nontrivial stable means and satisfying the relation . We then say that m is -stabilizable.
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.
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.
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.
which concludes the proof. □
The above proposition implies again that every cross mean is stable. The next theorem is more interesting.
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.
we deduce that every p-increasing (resp. p-decreasing) sequence is p-convergent.
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.
hold for all . Consequently, the mean-sequences and both p-converge.
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.
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 .
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.
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.
which refines . The procedure can be continued for obtaining more iterative refinements for this latter double inequality.
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).
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