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Sub-stabilizability and super-stabilizability for bivariate means
Journal of Inequalities and Applications volume 2014, Article number: 28 (2014)
The stability and stabilizability concepts for means in two variables have been introduced in (Raïssouli in Appl. Math. E-Notes 11:159-174, 2011). It has been proved that the arithmetic, geometric, and harmonic means are stable, while the logarithmic and identric means are stabilizable. In the present paper, we introduce new concepts, the so-called sub-stabilizability and super-stabilizability, and we apply them to some standard means.
In this section, we recall some basic notions about means in two variables that will be needed later. Throughout the following, we understand by a (bivariate) mean a binary map m between positive real numbers satisfying the following statement:
Every mean satisfies for each . The maps and are (trivial) means, which will be denoted by min and max, respectively. The standard examples of means are given in the following (see  for instance and the related references cited therein):
and are known as the arithmetic, geometric, harmonic, logarithmic, and identric means, respectively.
There are more means of interest known in the literature. For instance, the following:
A mean m is symmetric if for all , and monotone if is increasing in a and in b, that is, if (resp. ) then (resp. ). For more details as regards monotone means, see .
For two means and we write if and only if for every and, if and only if for all with . Two means and are comparable if or , and we say that m is between two comparable means and if . If the above inequalities are strict then we say that m is strictly between and . The above means are all comparable with the well-known chain of inequalities
For a given mean m, we set , and it is easy to see that is also a mean, called the dual mean of m. Every mean m satisfies , and if and are two means such that then . Further, the arithmetic and harmonic means are mutually dual (i.e. , ) and the geometric mean is self-dual (i.e. ).
Let p be a real number. The next means are of interest.
The power (binomial) mean:
The power logarithmic mean:
We end this section by recalling the next result which will be needed in the sequel.
Theorem 1.1 The following mean-inequalities hold:
Further these inequalities are the best possible i.e. , , are the best power logarithmic means lower bounds of P, M, T, while , , are the best power (binomial) means upper bounds of P, M, T, respectively. Otherwise, there is no such that P, M or T is strictly less that .
2 Needed tools
For the sake of simplicity for the reader, we recall here more basic notions and results that will be needed in the sequel, see  for more details. We begin by the next definition.
Definition 2.1 Let , , and be three given symmetric means. For all , define
called the resultant mean-map of , , and .
Example 2.1 It is not hard to verify that
A study investigating the elementary properties of the resultant mean-map has been stated in . In particular, if , , and are three symmetric monotone means then the map defines a mean, where we have the relationship
We also recall the next result, see .
Theorem 2.1 Let , , , , , and be strict symmetric monotone means such that
Then we have
If moreover there exists such that , then one has
Definition 2.2 A symmetric 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.
A developed study about the stability and stabilizability of the standard means was presented in . In particular the next result has been proved there.
Theorem 2.2 With the above, the following assertions are met:
The power binomial mean is stable for all real number p. In particular, the arithmetic, geometric, and harmonic means A, G, and H are stable.
The power logarithmic mean is -stabilizable for all real number p.
The logarithmic mean L is -stabilizable and -stabilizable while the identric mean I is -stabilizable.
Remark 2.1 The symmetry character of the above involved mean is, by definition, taken as essential hypothesis. In fact, if we attempt to extend the above concepts to non-symmetric means by keeping the same definitions (Definition 2.1 and Definition 2.2), the simple means , with , , do not satisfy . In another way, the definition of ℛ, together with that related to the stability and stabilizability concepts, is not exactly the same as above, but must be investigated for non-symmetric means. We leave the details as regards the latter point to a later time.
The next definition is also needed here .
Definition 2.3 Let and be two symmetric means. The tensor product of and is the map, denoted , defined by
A symmetric mean m will be called cross mean if the map is symmetric in its four variables.
It is proved in  that every cross mean is stable. The reverse of the latter assertion is still an open problem. Otherwise, it is conjectured  that the first Seiffert mean P is not stabilizable and such a problem is also still open. We also conjecture here that the second Seiffert mean and the Neuman-Sándor mean are not stabilizable either.
The next result needed here has also been proved in .
Theorem 2.3 Let and be two nontrivial stable symmetric monotone means such that (resp. ). Assume that is moreover a cross mean. Then there exists one and only one -stabilizable mean m such that (resp. ).
Recently, Raïssouli and Sándor  introduced a mean-transformation defined in the following way: for a given mean m (symmetric or not) they set
This allowed them to construct a lot of new means and to obtain good relationships between some standard means. In particular, they obtained , , , and for every real number p, where S and C refer, respectively, to the weighted geometric mean and contra-harmonic mean defined by
3 Two special subsets of means
Let be the set of all symmetric means. For fixed , we set
It is clear that and , that is, these sets are nonempty. Moreover, by equation (2.1) the relationship
is obvious. By virtue of this equivalence, it will be sufficient to study the properties of one the sets and and to deduce that of the other by duality.
Example 3.1 With the help of Theorem 2.1, it is simple to see that and . So and . We can also verify that and . Other more interesting examples will be seen later.
The next result is of interest.
Proposition 3.1 Let , be two nontrivial monotone (symmetric) stable means where is a cross mean. Then the intersection between and is reduced to the unique mean m which is the -stabilizable mean.
Proof Following Theorem 2.3, let m be the unique -stabilizable mean. Then and so and . Inversely, let ; then and so m is the unique -stabilizable mean. □
Now, we are in a position to state the next result ensuring the existence of a maximal super-stabilizable (resp. minimal sub-stabilizable) mean.
Theorem 3.2 Let , be two symmetric monotone means. Then the set has at least a maximal element.
Before giving the proof of the last theorem we state the next corollary, which is immediate from the above.
Corollary 3.3 Let , be as in the above theorem. Then the set has at least a minimal element.
Proof For proving the theorem, we will show that the set is (nonempty) inductively ordered. Let us equip with the point-wise order induced by that of the set of all means. Let be a nonempty total ordered set and we get . Then, is a mean. Clearly, is an upper bound of E and we wish to establish that . Indeed, for all , we have
Since and are monotone, we deduce by Theorem 2.1, for all and so , that is, . It follows that every nonempty totally ordered subset of has an upper bound in , that is, is inductive. We can then apply the classical Zorn lemma to conclude and the proof of the theorem is complete. □
Remark 3.1 A question arises from the above: Let and be two given symmetric means. Is it true that
Proposition 3.4 For all given symmetric mean m, we have:
The sets and are (linearly) convex.
The sets and are geometrically convex.
Proof (1) follows from the linear-affine character of A with the definition of ℛ, while (2) comes from the geometric character of G. The details are simple and omitted here. □
4 Sub-stabilizability and super-stabilizability
The next definition may be stated.
Definition 4.1 Let , be two nontrivial stable comparable means. A mean m is called:
-sub-stabilizable if and m is between and ,
-super-stabilizable if and m is between and .
Following Theorem 2.3, the above definition extends that of stabilizability in the sense that a mean m is -stabilizable if and only if (a) and (b) hold. It follows that the above concepts bring something new for non-stable and non-stabilizable means. For this, we say that m is strictly -sub-stabilizable if and m is strictly -super-stabilizable if , with in both cases m being strictly between and .
With the notation of the above section we have
Example 4.1 We can easily see that G is -super-stabilizable (but not strictly) while A is -sub-stabilizable. However, T and M are not -sub-stabilizable, since they are not between G and A. More interesting examples, presented as main results, will be stated in the section below.
Theorem 4.1 Let m be a continuous symmetric mean. Then the following assertions are met:
If there exists a symmetric mean such that m is -sub-stabilizable then .
If there exists a symmetric mean such that m is -super-stabilizable then .
Proof (1) Assume that m is -sub-stabilizable, that is,
or, according to the definition of ℛ,
This, with a simple mathematical induction, implies that the inequality
holds true for each integer . Letting in the latter inequality and using the fact that m is continuous we infer that
which with equation (2.2) means that .
It is similar to that the above. The details are omitted here. □
The above theorem has various consequences, which we will state in what follows.
Corollary 4.2 Let m be a continuous symmetric mean. Then the next statements hold true:
If m is -sub-stabilizable for some then . In particular, if m is -sub-stabilizable then .
If m is -super-stabilizable for some then . In particular, if m is -super-stabilizable then .
Proof It is immediate by combining the above theorem with the fact that for each real number p, and , . □
Remark 4.1 (i) The above corollary tells us that L is a minimal element of and it is a maximal element of : this rejoins the fact that L is -stabilizable.
The above corollary implies that I is not -super-stabilizable, but it is perhaps -sub-stabilizable. See more details as regards the latter point in the section below.
Corollary 4.3 Let be a strictly -sub-stabilizable mean. Then , where q is the greatest number such that and r is the smallest number such that .
Proof If is strictly -sub-stabilizable then, by definition, and, by the above corollary, . Combining these latter mean-inequalities we deduce the desired result. □
Corollary 4.4 (i) If there exists p such that P is strictly -sub-stabilizable then .
If M is strictly -sub-stabilizable for some p then .
If T is strictly -sub-stabilizable then .
There is no such that P, M or T is -super-stabilizable.
Proof Combining the above corollary with Theorem 1.1, we immediately deduce the assertions (i), (ii), and (iii).
Assertion (iv) follows from Corollary 4.2(ii) with Theorem 1.1 again. Details are omitted here. □
5 Application to some standard means
This section will be devoted to an application of the above concepts to some known means. We begin with the next result.
Theorem 5.1 The logarithmic mean L is strictly -super-stabilizable.
Proof First, the reader will do well to distinguish between the two next statements: ‘L is strictly -super-stabilizable’ to prove here and ‘L is -stabilizable’ already shown in . By definition and by a simple reduction, we have to prove
for all with . We will present two different proofs for equation (5.1). By the symmetric character of the involved means, we can assume, without loss the generality, that .
The first method is much more natural: Since , we have
Then by the inequality
valid for all real numbers x, y with , one has
This gives equation (5.1), so it completes the proof of the first method.
The second method is based on the fact that we can always set and with . A simple computation leads to
Substituting these in equation (5.1) we are in a position to show that
for all , which clearly holds and inequality (5.1) is again proved.
In summary, we have shown that L is strictly -super-stabilizable. □
Remark 5.1 We can also see that L is strictly -sub-stabilizable. In fact, since L is -stabilizable and , we obtain (with the help of Theorem 2.1)
which, with , means that L is strictly -sub-stabilizable.
Theorem 5.2 The identric mean I is strictly -sub-stabilizable.
Proof We will present here two different methods for proving our claim: The first is direct and based on some mean-inequalities already stated in the literature, while the second one is similar to above.
First method: We have to show(5.2)
for all with . If we recall that  the function is concave upon both variables, we immediately deduce that
Otherwise, it is well known that (see  for example) and for all , . We then obtain
Second method: To show equation (5.2) is equivalent to proving that(5.4)
As previously, we can easily verify that
Substituting these in the above and using the identity
valid for each , the desired inequality is reduced to showing that
for all . A simple computation leads to
It follows that Φ is strictly decreasing for and so . The second method is complete. □
Remark 5.2 Another method for proving equation (5.4) can be stated as follows: It is well known (and easy to verify) that for all , where is the so-called weighted geometric mean. With this, equation (5.4) is equivalent to i.e. , which is a well-known mean-inequality.
As a consequence of the above, the next result gives a double inequality refining and involving the four standard means G, L, I, and A.
Corollary 5.3 We have
Proof The above theorem means that , which, with Theorem 2.1 and the fact that L is -stabilizable, yields
This, with Example 2.1 and a simple manipulation, gives the desired result. □
Of course, the above theorems when combined with the properties of sub-super-stabilizability imply that is, simultaneously, strictly -sub-stabilizable and strictly -super-stabilizable, while is strictly -super-stabilizable.
As already pointed out before, whether the first Seiffert mean P is stabilizable still is an open problem. However, the next result may be stated.
Theorem 5.4 The first Seiffert mean P is strictly -sub-stabilizable.
Proof Explicitly, we have to prove that
holds for all with . We also present here two different methods.
First method: this method is analogous to the above. Simple computation leads to
for each . After simple substitution and reduction we are in a position to show that
for every . We can easily obtain (after computation and reduction)
for all . The desired inequality follows in the same way as previously.
Second method: this method is based on an integral form of . It is easy to see that, for all (with without loss the generality), we have(5.7)
This, with a simple manipulation, yields
If we use the variable of change , in the left integral of equation (5.9) our aim is then reduced to showing that
It is very easy to verify that
from which equation (5.10) follows. The proof is complete. □
which is exactly equation (5.6).
6 Some open problems
In the above section, we have proved that P is strictly -sub-stabilizable. The fact that P is strictly -super-stabilizable is not proved yet. This is equivalent to showing that
holds for all with . As above, and setting , , we are in a position to show that
for all . We then present the following.
Problem 1: Prove or disprove that the first Seiffert mean P is strictly -super-stabilizable.
Problem 2: Find the best real numbers and for which P is strictly -sub-stabilizable.
Problem 3: Are the means T and M strictly -sub-stabilizable for some real numbers , ?
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The present work was supported by the Deanship of Scientific Research of Taibah University.
The authors declare that they have no competing interests.
Both authors jointly worked, read and approved the final manuscript.
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Raïssouli, M., Sándor, J. Sub-stabilizability and super-stabilizability for bivariate means. J Inequal Appl 2014, 28 (2014). https://doi.org/10.1186/1029-242X-2014-28
- stable means
- stabilizable means
- sub-stabilizable means
- super-stabilizable means