We extend the classical notions of translativity and homogeneity of means to -homogeneity, that is, invariance with respect to an operation
We find the shape of for the arithmetic weighted mean and then the general form of
for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by -homogeneity with respect to a single operation
, one needs to take two of such operations in order to characterize a mean.
1. Introduction
Definition 1.1. Let be an interval. A function such that
is called a mean (on ).
Note that every mean is reflexive, that is
A mean is called symmetric, if
In [1], Aczรฉl and Dhombres distinguished two special types of means, defined in , where .
(i) Translative: if the function is translation invariant, that is for all
(ii) Homogeneous: if the function is multiplication invariant, that is for all
For more information about means, see, for instance, [1โ6].
In the present paper, we present three approaches to the question of translativity and homogeneity of means. First, we โdiscoverโ some functional equations which generalize properties (1.4) and (1.5). Let us adopt the following concept of generalized homogeneity which in many instances covers both translativity and homogeneity.
Definition 1.2. Let be an interval and let be a function. A mean such that
for every is called F-homogeneous.
In the last section, we determine all the operations with respect to which quasi-linear means are -homogeneous. Also, we characterize means as solutions to systems of functional equations, thus generalizing a result from [1] (cf. Proposition 9, page 249).
2. Pexider Equations for Means
We notice that the translative mean equation on an interval satisfying , that is the equation
and the homogeneous mean equation on an interval satisfying , that is the equation
may be treated as conditional forms of the following equations:
respectively.
Here, the word โconditionโ means that (2.3) and (2.4) have to be satisfied for pairs of the set .
To show that (2.1) actually is the conditional equation (2.3), we put into (2.3) and use reflexivity of . Then,
whence .
Putting and , we can rewrite (2.3) in the form
which is satisfied for and . This is Pexider conditional equation (in this case the condition refers to the fact that (2.6) holds for pairs ). To solve (2.6) let us note that fixing arbitrarily , and letting , we get, from (2.6)Thus satisfies the Cauchy equation for . Moreover, from (2.6) and properties of mean , we obtain ( is arbitrarily fixed)
so is majorized (and minorized) on by a continuous function. The well known results (cf. for instance [1, 2, 7]) imply that is a linear mapping, that is there exist constants such that
From (2.8), we infer that , so . Moreover, from (2.8), we have for
that is, and , so . In other words, we have proved that there exists an such that
Now, put and into (2.3). Using the reflexivity of and (2.11), we obtain
whence
Thus we obtain the following.
Theorem 2.1. Let be a non-degenerate interval such that . Then, a mean and a function satisfy (2.3) if and only if there exists a constant such that
Now, we define in an operation by
Let be an interval such that . Suppose that is a mean and is an arbitrary function. We consider the equation
From the assumption , hence both and take on positive values. Putting and defining by
we see that
for all . It is easy to show that is a mean, because so was . By Theorem 2.1, we have, for some constant ,
hence
We obtain the following.
Theorem 2.2. Let be a non-degenerate interval such that . Then a mean and a function satisfy (2.16) if and only if there exists a constant such that and are given by (2.20).
Remark 2.3. Let us note that two other Pexider equations,
have no solutions such that is reflexive. In fact, putting and , we get
respectively. The equalities cannot be satisfied for all .
3. Further Generalizations
Now, we come back to (2.1). We notice that it is also a special case of the following equation:
that is, the Cauchy equation for , but with one of variables belonging to the graph of some fixed function . We will consider also the equation
Note that if we take , we get (2.1) or (2.2).
We will prove results concerning the above equations. Let us start with the โadditiveโ case.
Theorem 3.1. Let be a non-degenerate interval such that . Let be a function such that the mapping given by
is a surjection. Then, a symmetric and reflexive mapping satisfies (3.1) if and only if is the arithmetic mean.
Proof. Fix a and put into (3.1). Using the reflexivity and the symmetry of , we obtain
whence
Now, fix a . Let be chosen so that . From (3.1) and (3.5), we get
This completes the proof.
Analogously as in the proof of Theorem 2.2, we obtain the following.
Theorem 3.2. Let be an interval such that . Let be a function such that defined by
is a bijection. Then, a symmetric and reflexive mapping is a solution of (3.2) if and only if is the geometric mean.
Example 3.3. The assumption of a surjectivity of is essential. This is shown by example of the function for which we have , equations (3.1) and (3.2) take forms (2.1) and (2.2), respectively, which have other solutions, even in the class of quasi-arithmetic means (cf. [1]).
We have the following corollaries of theorems.
Corollary 3.4. Let be fixed. The arithmetic mean is the only symmetric mean satisfying the equation
for all .
Proof. The function
is a bijection of onto itself; it is enough to use Theorem 3.1.
Similarly, using Theorem 3.2, we obtain the following.
Corollary 3.5. Let a be fixed. The geometric mean is the only symmetric mean satisfying the equation
for all .
4. -Homogeneity of Quasi-Linear Means
We will now consider the following problem. What are operations with respect to which quasi-linear means are -homogeneous?
Let us begin with the weighted arithmetic mean defined on an interval . Let be fixed and let be an operation satisfying the equation
for all .
First, we will suppose that is a function continuous in second variable.
Let us define for every . Then, for every , the function solves the equation
In view of the equality (cf. [8])
which holds for all and , it follows that
The assumption of continuity in the second variable implies (cf. e.g., [2] or [7]) the existence of functions such that
for all .
Now, let us additionally assume that is associative, that is, for all we have
In view of (4.5), we obtain the equivalent equality
for all . Thus we see that the operation given by (4.5) is associative if and only if the following system of equations
holds for all . Assume now that is symmetric (hence it is also continuous with respect to the first variable). This means that, by (4.9),
or equivalently
for all .
We consider the following two cases.
(a) . Then , and the symmetry of yields the equality
that is
for all . So, there exists a constant such that
Thus,
where is defined by
Note that the condition is a restriction imposed both on and .
(b) . Let for some . Putting into (4.11), we calculate
where . Using the symmetry of , we get
We have
whence
In other words,
for all , where is a constant and . From (4.19), it results that implies , so (4.21) holds for all . Hence,
and, finally, denoting by , we get
Now, it is easy to check that associativity of is equivalent to
We have the following possibilities.(i) and . In this case, we have
and it is enough to assume that .(ii) and . Then, similarly as in case (a),
where is defined by (4.16).(iii). Now, we have
where is given by
Note that is a condition imposed on the interval and constants as well.
Thus, we have proved.
Proposition 4.1. Let be a non-degenerate interval, let be a function which is associative, symmetric, and continuous in each variable. Let be fixed. Then, the weighted arithmetic mean is -homogeneous (that is (4.1) holds) if and only if there exist constants , and such that is given by (4.23), the condition (4.24) is satisfied and (i),
(ii) where is given by (4.16), (iii) where is given by (4.28).
Remark 4.2. We see that among the operations listed in Proposition 4.1 are the following, (i), , (ii) is a subsemigroup of , (iii), in this case .
Now, let us generalize Proposition 4.1 to the case of arbitrary quasi-linear means.
We admit the following definition.
Definition 4.3. Let and be non-degenerate intervals, let be an arbitrary function, let be a bijection. Then, we define the function by
Remark 4.4. We can easily see that is associative and symmetric if and only if has the same properties. Moreover, if is continuous, then is continuous in each variable if and only if is continuous in each variable.
We obviously have the following.
Lemma 4.5. Let and be non-degenerate intervals. Let be a continuous bijection, let be a function, and let be fixed. Then the quasi-linear mean given by
is -homogeneous if and only if the weighted arithmetic mean on the interval is -homogeneous.
From Proposition 4.1 and Lemma 4.5, we obtain the following.
Theorem 4.6. Let and be non-degenerate intervals and let be a continuous bijection, let be a function which is associative, symmetric, and continuous in each variable. Let be fixed. Then, the quasi-linear mean is F- homogeneous if and only if there exist constants , and such that is given by
the condition (4.24) is satisfied, and (i),
(ii) where is given by (iii) where is given by
Remark 4.7. Among operations , for which is homogeneous, are the following
It is possible to see that is the unique quasi-linear mean which is homogeneous with respect to (4.34) and (4.35) (cf. [1], Theorem 15.8).
Remark 4.8. We see that the inverse to Theorem 4.6 does not hold. More exactly, -homogeneity alone does not characterize the mean . In order to see it, define by
We show that there exist many -homogeneous quasi-linear means, even when we restrict our attention to symmetric ones. In fact, suppose that is an F-homogeneous quasi-arithmetic mean, that is
or
Substituting , and , and defining by
we see that is homogeneous on . It follows (cf. for instance [1]) that either
or there is a such that
whence either
or
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