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Generalized stabilizability for bivariate means
Journal of Inequalities and Applications volume 2013, Article number: 233 (2013)
The stabilizability concept for bivariate means has been introduced and studied in (Raïssouli in Appl. Math. E-Notes, 11:159-174, 2011). It has been proved to be a useful tool for theoretical viewpoint as well as for practical purposes (Raïssouli in Appl. Math. E-Notes, 11:159-174, 2011). In the present paper, we give an extension of the stabilizability concept, the so-called generalized stabilizability. Our paper will be completed by some applications and examples illustrating the above extension and showing the interest of this work.
In this section, we state some basic notions about bivariate means, see . A function is called a (bivariate) mean if
It is clear that every mean is with positive values and reflexive, that is, for all . The maps and are (trivial) means which will be denoted by min and max, respectively. The standard examples of means are as follows :
they are known in the literature as the arithmetic, geometric, harmonic, logarithmic, identric, weighted geometric and contra-harmonic means, respectively. It is well known that
for all with equalities if and only if .
A mean m is symmetric if and homogeneous if for all . The above means are all symmetric and homogeneous. A mean m is called monotone if is increasing in a and in b, that is, and imply and . It is not hard to check that the means A, G, H, L, I are monotone but S and C are not.
Let m be a homogeneous mean. Writing , we then associate to m a unique positive function f defined by for all . The function f will be called an associated function to the mean m, or we simply say that f corresponds to the mean m. It follows that f corresponds to a homogeneous mean if and only if for every . Clearly, and if, moreover, m is symmetric, then for each . It is obvious that a mean m is monotone if and only if its associated function is increasing. For example, and as pointed in , the contraharmonic mean C is not monotone because its associated function satisfies , and it is easy to see that f is not increasing for all , but only for .
In the literature, there are some families of means, called power means, which include the above familiar means. Let p and q be two real numbers. The Stolarsky mean of order is defined by 
It is understood that this family of means includes some interesting cases as well:
• The power binomial mean
• The power logarithmic mean
It is easy to see that is homogeneous and symmetric for all fixed real numbers p and q. Further, it is clear that is symmetric in p and q, and it is well known that is strictly increasing in both p and q. In particular the power means and are strictly increasing in p.
The reminder of this paper is organized as follows. Section 2 displays a class of means, the so-called parameterized means that will be needed later. Section 3 is focused on introducing a new definition that includes a lot of standard means. Section 4 is devoted to extending the resultant mean-map, introduced by the second author in an earlier paper for means, to positive maps in the aim to state conveniently the generalized stabilizability in Section 5. This latter concept is then applied to standard (power) means. In Section 6, application to mean inequalities is investigated and interesting examples are discussed.
2 Parameterized bivariate means
This section is focused on stating a special class of means depending on a real parameter, the so-called parameterized means, defined as follows.
Definition 2.1 Let be a family of maps with for fixed . We say that is a parameterized (or weighted) mean if the following assertions are satisfied:
(i) is a mean for all fixed ,
(ii) For all , and or, and ,
(iii) for all and .
From the above definition, we deduce that is a symmetric mean which we call the associated (symmetric) mean of the parameterized mean . The following
are known in the literature as the parameterized arithmetic, geometric and harmonic means, respectively. For all , , and are homogeneous monotone, but not symmetric unless . Clearly, , and which are the associated symmetric means of , and respectively. The parameterized weighted geometric and contra-harmonic means are given by
It is easy to see that and .
The parameterized logarithmic mean was introduced in  as follows:
while the parameterized identric mean can be inspired from  as well
The above parameterized means satisfy the following chain of inequalities (see ):
with strict inequalities if and only if and .
The parameterized power binomial mean can be immediately given by
for , with . The parameterized Stolarsky mean can be also inspired from  by setting
where refers to the parameterized Gini mean defined through
In particular, the parameterized power logarithmic mean is given by
where stands for the parameterized mean defined through
Let be a (closed or open) interval of ℝ. Since there exists a homeomorphism between and , we then can define a parameterized mean indexed by in the following sense:
(j) is a mean for all fixed ,
(jj) for all , and or, and ,
(jjj) for all and .
The associated symmetric mean of is . Let us observe the next example explaining this latter situation.
Example 2.1 Let p be a fixed real number. For and , we set
It is easy to verify that satisfies (j),(jj) and (jjj), then it is a parameterized mean. The associated symmetric mean is the power (binomial) mean .
By the simple transformation , it is always possible to reduce into . Henceforth, when we consider a parameterized mean , it will be indexed by .
3 Class of -means
Let be a given parameterized mean. Assume that is continuous with respect to the variable , then it is easy to sketch that the maps
define two bivariate symmetric means. The following examples illustrate this situation.
Example 3.1 It is well known that
refers to the dual of the logarithmic mean L.
Example 3.2 The following
defines a symmetric mean. It is proved in  that
See also  for a general approach including the above expansion.
Example 3.3 For and , we set
It is easy to verify that defines a parameterized mean. Then the expression
defines a symmetric mean. Taking , we see that is the Toader mean introduced in .
In what follows we are interested in extending the situation of the two above examples. We may then state the next definition.
Definition 3.1 Let be a parameterized mean, continuous with respect to . A mean M, for which there exist two continuous strictly monotonic functions and such that
will be called a -mean.
We explicitly notice that, by virtue of Definition 2.1(iii) and with the change of variable in (3.6), we deduce that every -mean is symmetric. For this, we need to assume that the involved means, in the announced theoretical results, are symmetric.
If in the above definition we have , then (3.6) becomes
In particular, with we obtain
as a special class of means known in the literature as quasi-arithmetic means. The power binomial mean is quasi-arithmetic with for and for . In summary, the means A, G, H are included in the above definition. Following (3.2) and (3.3) the means L and I are also included in the above. More generally, the power mean , for all real numbers p and q, can be also obtained as a particular case of the above definition when convenient functions Φ and ϕ are chosen. Let us observe the next example explaining this latter situation.
Example 3.4 Let p and q be two fixed real numbers. Assume that and .
(i) Let us choose and , . In this case the symmetric mean obtained through (3.6) is given by
A simple computation shows that coincides with . This, with Definition 3.1, means that the Stolarsky mean is a -mean with Φ, ϕ and m previously defined.
(ii) We can obtain the same as in (i) by choosing and , and so is a -mean.
For or , we left the reader to choose the convenient functions ϕ and Φ for and in a similar manner as a previous one.
Remark 3.1 Formula (3.6) in the particular case was considered by Toader and Sándor  in the aim to obtain some mean-inequalities when convenient hypotheses on ϕ and Φ are assumed. Here, we consider (3.6) in its general form in the aim to introduce some mean-concepts and investigate some related applications.
4 Resultant functional-map
In , the author defined the resultant mean-map concept for bivariate means as a good tool for introducing the stability and stabilizability notions. This concept can be extended for functionals instead of means as well.
Definition 4.1 Let be three given functions. For , define
which we call the resultant functional-map of f, g and h.
We explicitly notice that ℛ is a map with three functionals variables f, g, h (which justifies the chosen terminology), while is a functional with two positive real variables a, b. If f, g, h are bivariate symmetric means, the above definition coincides with that introduced in . In our next study, we will be restricted by the case where the functionals f and h are bivariate symmetric means, while g is such that , where m is a symmetric mean and is a continuous strictly monotonic function. For the sake of simplicity, we write instead of . For all three given means , , , we also use the following notation:
With this, the next result may be stated.
Proposition 4.1 Let , , be three symmetric means with and monotone, and let be a continuous strictly monotonic function. Then the following assertions are satisfied:
(i) is a mean.
(ii) If , , are three means such that , , , then we have
Proof We limit our attention in this proof to Φ strictly increasing, since the case of Φ strictly decreasing can be stated in a similar manner.
(i) Since is a mean, then by definition, for all , we have
This, with the fact that Φ is increasing, yields
By the monotonicity of , we get
Using the fact that is increasing too, we obtain
which means that is a mean.
(ii) Since , we deduce, with the monotonicity of ,
It follows that
The increased monotonicity of Φ implies that
Now, from with the monotonicity of , we infer that
Finally, using the fact that is increasing with the definition of , we obtain the desired result, which completes the proof. □
Now, let us observe the following example illustrating the above.
Example 4.1 Let . An elementary computation gives . Note that here the function Φ is the inverse of the associated function of the homogeneous mean A, i.e., . Similarly, we verify that with , and an analogous remark as the previous one holds.
Example 4.2 (1) Let , . Then relationships (3.4) and (3.5) can be summarized by .
(2) Let , . Expression (3.3) can be written as .
These relationships, and a lot of others, will be seen and interpreted in a general point of view. See section below.
Now, let us state the next example which will be needed later. We omit the computation details for the reader.
Example 4.3 Take , . It is not hard to establish that
5 Generalized stabilizability
As already pointed before, this section will be focused on presenting an extension of the stabilizability concept already introduced in . We then may state the following definition.
Definition 5.1 Let M be a given symmetric mean. Assume that there exist two nontrivial symmetric means , and a continuous strictly monotonic function such that
Then we say that M is -stabilizable.
If , that is, , we then say M is Φ-stable.
In other words, M is -stabilizable if and only if M is a mean-fixed point of the map .
If the identity function, then the above definition coincides with that of stabilizability, and in this case, we simply say M is -stabilizable.
Now, let us observe the next example illustrating the above definition.
Example 5.1 Following Example 4.1 we can say that A is Φ-stable with and, G is Φ-stable with . We left the reader to show that H is Φ-stable with a convenient function Φ to be defined.
Example 5.2 According to Example 4.2, the logarithmic mean L is -stabilizable with , while the identric mean I is -stabilizable with .
We now are in a position to state the next result which will be with interest of giving other examples of means satisfying the situation of the above definition.
Theorem 5.1 Let be a parameterized mean such that
Let Φ, ϕ be as in Definition 3.1 and let us set
Then every -mean M is -stabilizable, that is, .
Proof Definition 3.1 with (5.2) yields
Making the change of variable in (5.3), we get
This, with condition (5.1), yields
Similarly, the change of variable leads to the following:
According to (iii) of Definition 2.1, we can write
Using (5.1), we obtain
Combining (5.5) with (5.6), we get
The proof of the theorem is completed. □
Remark 5.1 If the parameterized mean is homogeneous with its associated function , i.e., , then condition (5.1) can be simplified and is equivalent to
It is easy to see that the parameterized mean of Example 3.4 satisfies the above condition.
Notation In what follows, if for some real number , then we write instead of and instead of . For , we write and . With Example 4.2, we then have L is -stabilizable, i.e., , and I is -stabilizable, i.e., .
Corollary 5.2 Let p, q be two real numbers. Then we have
(1) If , then is -stabilizable.
(2) is -stabilizable.
In particular, for the power logarithmic mean is -stabilizable.
Proof It is sufficient to combine Theorem 5.1 with Example 3.4. Details are omitted here for the reader. □
We end this section by stating the following remark which highlights the interest of the above results.
Remark 5.2 To characterize a mean M as an -mean is of great importance since the involved arithmetic mean A is the simplest analytic mean further with a linear affine character. In particular, in the computation context, the relationship
holds for all functions , all means m and every vector . See section below for a more explicit explanation.
6 Application for mean-inequalities
In this section we present some applications of the above theoretical study for obtaining mean-inequalities. Following Theorem 5.1 and Remark 5.2, we may state the following.
Theorem 6.1 Let M be a -mean where Φ, ϕ and m are as in Theorem 5.1. Let be n symmetric means such that
for some probability vector . Then the following holds:
Proof According to Theorem 5.1, the mean M is -stabilizable. This, when combined with relationship (5.7) and Proposition 4.1(ii), yields the desired result after simple manipulation. □
Theorem 6.1, when combined with Corollary 5.2, immediately gives the next result.
Corollary 6.2 Let be a fixed real number. Let be n symmetric means such that
for some probability vector . Then we have
Taking in the above corollary, with the fact that , we immediately obtain the next result.
Corollary 6.3 Let be n symmetric means such that
for some probability vector . Then one has
Now, we present an example illustrating the above results.
Example 6.1 The inequality
is well known in the literature, see . If we apply Corollary 6.3 with , , and , , we obtain, after simple computation,
Using the convexity of the real-function on , the reader can easily verify that the inequality (6.8) refines (6.7), in this way proving the desired aim.
More importance to this example is given by Corollary 6.3. We can apply it again with , , and , , . We then obtain
We left to the reader the routine task of computing the right-hand side of (6.9), to ensure that the obtained upper bound of refines that of (6.8) and to show how we can repeat the application of Corollary 6.3 for with convenient means and related coefficients.
Example 6.2 The inequality
proved in  by Neuman and Sándor, is a refinement of (6.7). If we apply Corollary 6.3 to (6.10), with , , and , , we obtain
This, with the help of Example 4.3 and a simple reduction, yields the following inequality:
It is easy to verify that this latter inequality refines (6.10), in this way proving the interest of our approach.
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The authors thank the anonymous referees for their comments and suggestions which have been included in the final version of this paper. This work was supported by the Research Center of Taibah University 433/1698.
The authors declare that they have no competing interests.
Both authors jointly worked, read and approved the final version of the paper.
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Gasmi, A., Raïssouli, M. Generalized stabilizability for bivariate means. J Inequal Appl 2013, 233 (2013). https://doi.org/10.1186/1029-242X-2013-233
- Monotonic Function
- Generalize Stabilizability
- Probability Vector
- Parameterized Power
- Associate Function