Abstract

We extend the classical notions of translativity and homogeneity of means to ๐น-homogeneity, that is, invariance with respect to an operation ๐นโˆถ๐ผร—๐ผโ†’๐ผ. We find the shape of ๐น for the arithmetic weighted mean and then the general form of ๐น for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by ๐น-homogeneity with respect to a single operation ๐น, one needs to take two of such operations in order to characterize a mean.

1. Introduction

Definition 1.1. Let ๐ผโŠ‚โ„ be an interval. A function ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ such that min{๐‘ฅ,๐‘ฆ}โ‰ค๐‘€(๐‘ฅ,๐‘ฆ)โ‰คmax{๐‘ฅ,๐‘ฆ}๐‘ฅ,๐‘ฆโˆˆ๐ผ,(1.1) is called a mean (on ๐ผ2).

Note that every mean is reflexive, that is๐‘€(๐‘ฅ,๐‘ฅ)=๐‘ฅโˆ€๐‘ฅโˆˆ๐ผ.(1.2) A mean ๐‘€ is called symmetric, if๐‘€(๐‘ฅ,๐‘ฆ)=๐‘€(๐‘ฆ,๐‘ฅ)โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ผ.(1.3)

In [1], Aczรฉl and Dhombres distinguished two special types of means, defined in ๐ผร—๐ผ, where ๐ผโŠ‚โ„+.

(i) Translative: if the function is translation invariant, that is for all ๐‘ฅ,๐‘ฆโˆˆ๐ผ,๐‘กโˆˆโ„+๐‘ฅ+๐‘ก,๐‘ฆ+๐‘กโˆˆ๐ผโŸน๐‘€(๐‘ฅ+๐‘ก,๐‘ฆ+๐‘ก)=๐‘€(๐‘ฅ,๐‘ฆ)+๐‘ก.(1.4)

(ii) Homogeneous: if the function is multiplication invariant, that is for all ๐‘ฅ,๐‘ฆโˆˆ๐ผ,๐‘ขโˆˆโ„+๐‘ฅ๐‘ข,๐‘ฆ๐‘ขโˆˆ๐ผโŸน๐‘€(๐‘ฅ๐‘ข,๐‘ฆ๐‘ข)=๐‘€(๐‘ฅ,๐‘ฆ)๐‘ข.(1.5) For more information about means, see, for instance, [1โ€“6].

In the present paper, we present three approaches to the question of translativity and homogeneity of means. First, we โ€œdiscoverโ€ 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 ๐ผโŠ‚โ„ be an interval and let ๐นโˆถ๐ผร—๐ผโ†’๐ผ be a function. A mean ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ such that ๐‘€(๐น(๐‘ก,๐‘ฅ),๐น(๐‘ก,๐‘ฆ))=๐น(๐‘ก,๐‘€(๐‘ฅ,๐‘ฆ)),(1.6) for every ๐‘ฅ,๐‘ฆ,๐‘กโˆˆ๐ผ is called F-homogeneous.

In the last section, we determine all the operations ๐น with respect to which quasi-linear means are ๐น-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 ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ on an interval ๐ผ satisfying ๐ผ+๐ผโŠ‚๐ผ, that is the equation๐‘€(๐‘ +๐‘ฅ,๐‘ +๐‘ฆ)=๐‘ +๐‘€(๐‘ฅ,๐‘ฆ),(2.1) and the homogeneous mean equation ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ on an interval ๐ผ satisfying ๐ผโ‹…๐ผโŠ‚๐ผ, that is the equation๐‘€(๐‘ ๐‘ฅ,๐‘ ๐‘ฆ)=๐‘ ๐‘€(๐‘ฅ,๐‘ฆ),(2.2) may be treated as conditional forms of the following equations:๐‘€(๐‘ +๐‘ฅ,๐‘ก+๐‘ฆ)=๐œ™(๐‘ ,๐‘ก)+๐‘€(๐‘ฅ,๐‘ฆ),(2.3)๐‘€(๐‘ ๐‘ฅ,๐‘ก๐‘ฆ)=๐œ™(๐‘ ,๐‘ก)๐‘€(๐‘ฅ,๐‘ฆ),(2.4) respectively.

Here, the word โ€œconditionโ€ means that (2.3) and (2.4) have to be satisfied for pairs (๐‘ ,๐‘ก) of the set ๐‘Š={(๐‘ ,๐‘ก)โˆˆ๐ผร—๐ผโˆถ๐‘ =๐‘ก}.

To show that (2.1) actually is the conditional equation (2.3), we put ๐‘ก=๐‘ ,๐‘ฆ=๐‘ฅ into (2.3) and use reflexivity of ๐‘€. Then,๐‘ +๐‘ฅ=๐‘€(๐‘ +๐‘ฅ,๐‘ +๐‘ฅ)=๐œ™(๐‘ ,๐‘ )+๐‘€(๐‘ฅ,๐‘ฅ)=๐œ™(๐‘ ,๐‘ )+๐‘ฅ,(2.5) whence ๐œ™(๐‘ ,๐‘ )=๐‘ ,๐‘ โˆˆ๐ผ.

Putting ๐‘ =(๐‘ ,๐‘ก) and ๐‘ฅ=(๐‘ฅ,๐‘ฆ), we can rewrite (2.3) in the form๐‘€๎€ท๐‘ +๐‘ฅ๎€ธ๎€ท๐‘ =๐œ™๎€ธ๎€ท๐‘ฅ+๐‘€๎€ธ,(2.6) which is satisfied for ๐‘ โˆˆ๐ผร—๐ผ and ๐‘ฅโˆˆ๐ผร—๐ผ. This is Pexider conditional equation (in this case the condition refers to the fact that (2.6) holds for pairs (๐‘ ,๐‘ฅ)โˆˆ(๐ผร—๐ผ)ร—(๐ผร—๐ผ)). To solve (2.6) let us note that fixing arbitrarily ๐‘ฅ,๐‘ฆ,๐‘ 1,๐‘ 2,๐‘ก1,๐‘ก2โˆˆ๐ผ, and letting ๐‘ฅ=(๐‘ฅ,๐‘ฆ),๐‘ =(๐‘ 1,๐‘ 2),๐‘ก=(๐‘ก1,๐‘ก2), we get, from (2.6)๐œ™๎€ท๐‘ +๐‘ก๎€ธ๎€ท๐‘ =๐‘€+๐‘ก+๐‘ฅ๎€ธ๎€ท๐‘ฅโˆ’๐‘€๎€ธ๎€ท๐‘ =๐œ™๎€ธ๎€ท๐‘ก+๐‘€+๐‘ฅ๎€ธ๎€ท๐‘ฅโˆ’๐‘€๎€ธ๎€ท๐‘ =๐œ™๎€ธ๎€ท๐‘ก+๐œ™๎€ธ๎€ท๐‘ฅ+๐‘€๎€ธ๎€ท๐‘ฅโˆ’๐‘€๎€ธ๎€ท๐‘ =๐œ™๎€ธ๎€ท๐‘ก+๐œ™๎€ธ.(2.7)Thus ๐œ™ satisfies the Cauchy equation for ๐‘ ,๐‘กโˆˆ๐ผร—๐ผ. Moreover, from (2.6) and properties of mean ๐‘€, we obtain (๐‘ฅโˆˆ๐ผ is arbitrarily fixed)๎€ท๐‘ min1,๐‘ 2๎€ธ๎€ท๐‘ โ‰ค๐‘€1+๐‘ฅ,๐‘ 2๎€ธ๎€ท๐‘ +๐‘ฅโˆ’๐‘ฅ=๐‘€+๎€ธ๎€ท๐‘ (๐‘ฅ,๐‘ฅ)โˆ’๐‘€(๐‘ฅ,๐‘ฅ)=๐œ™๎€ธ๎€ท๐‘ โ‰คmax1,๐‘ 2๎€ธ,(2.8) so ๐œ™ is majorized (and minorized) on ๐ผร—๐ผ by a continuous function. The well known results (cf. for instance [1, 2, 7]) imply that ๐œ™ is a linear mapping, that is there exist constants ๐›ผ,๐œ”โˆˆโ„ such that๐œ™๎€ท๐‘ 1,๐‘ 2๎€ธ=๐›ผ๐‘ 1+๐œ”๐‘ 2,๎€ท๐‘ 1,๐‘ 2๎€ธโˆˆ๐ผร—๐ผ.(2.9) From (2.8), we infer that ๐‘ =๐œ™(๐‘ ,๐‘ )=(๐›ผ+๐œ”)๐‘ ,๐‘ โˆˆ๐ผ, so ๐›ผ=1โˆ’๐œ”. Moreover, from (2.8), we have for ๐‘ 1<๐‘ 2,๐‘ 1,๐‘ 2โˆˆ๐ผ๐‘ 1โ‰ค(1โˆ’๐œ”)๐‘ 1+๐œ”๐‘ 2โ‰ค๐‘ 2,(2.10) that is, ๐œ”(๐‘ 2โˆ’๐‘ 1)โ‰ฅ0 and (1โˆ’๐œ”)(๐‘ 2โˆ’๐‘ 1)โ‰ฅ0, so ๐œ”โˆˆ[0,1]. In other words, we have proved that there exists an ๐œ”โˆˆ[0,1] such that๐œ™๎€ท๐‘ 1,๐‘ 2๎€ธ=(1โˆ’๐œ”)๐‘ 1+๐œ”๐‘ 2,๎€ท๐‘ 1,๐‘ 2๎€ธโˆˆ๐ผร—๐ผ.(2.11) Now, put ๐‘ =๐‘ฆ and ๐‘ก=๐‘ฅ into (2.3). Using the reflexivity of ๐‘€ and (2.11), we obtain๐‘ฅ+๐‘ฆ=๐‘€(๐‘ฆ+๐‘ฅ,๐‘ฅ+๐‘ฆ)=๐œ™(๐‘ฆ,๐‘ฅ)+๐‘€(๐‘ฅ,๐‘ฆ)=(1โˆ’๐œ”)๐‘ฆ+๐œ”๐‘ฅ+๐‘€(๐‘ฅ,๐‘ฆ),(2.12) whence๐‘€(๐‘ฅ,๐‘ฆ)=(1โˆ’๐œ”)๐‘ฅ+๐œ”๐‘ฆ,(๐‘ฅ,๐‘ฆ)โˆˆ๐ผร—๐ผ.(2.13) Thus we obtain the following.

Theorem 2.1. Let ๐ผโŠ‚โ„ be a non-degenerate interval such that ๐ผ+๐ผโŠ‚๐ผ. Then, a mean ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ and a function ๐œ™โˆถ๐ผร—๐ผโ†’โ„ satisfy (2.3) if and only if there exists a constant ๐œ”โˆˆ[0,1] such that

๐‘€(๐‘ฅ,๐‘ฆ)=๐œ™(๐‘ฅ,๐‘ฆ)=(1โˆ’๐œ”)๐‘ฅ+๐œ”๐‘ฆ,(๐‘ฅ,๐‘ฆ)โˆˆ๐ผร—๐ผ.(2.14) Now, we define in โ„2 an operation โ€ข by๎€ท๐‘ 1,๐‘ 2๎€ธโ€ข๎€ท๐‘ (๐‘ฅ,๐‘ฆ)=1๐‘ฅ,๐‘ 2๐‘ฆ๎€ธ.(2.15) Let ๐ผโŠ‚(0,โˆž) be an interval such that ๐ผโ‹…๐ผโŠ‚๐ผ. Suppose that ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ is a mean and ๐œ™โˆถ๐ผร—๐ผโ†’โ„ is an arbitrary function. We consider the equation (๐‘ =(๐‘ 1,๐‘ 2),๐‘ฅ=(๐‘ฅ,๐‘ฆ))๐‘€๎€ท๐‘ โ€ข๐‘ฅ๎€ธ๎€ท๐‘ =๐œ™๎€ธ๐‘€๎€ท๐‘ฅ๎€ธ.(2.16) From the assumption ๐ผโŠ‚(0,โˆž), hence both ๐‘€ and ๐œ™ take on positive values. Putting ๐‘ข1=ln๐‘ 1,๐‘ข2=ln๐‘ 2,๐‘ฃ=ln๐‘ฅ,๐‘ค=ln๐‘ฆ and defining ๎‚‹๎‚๐‘€โˆถln๐ผร—ln๐ผโ†’ln๐ผ,๐œ™โˆถln๐ผร—ln๐ผโ†’โ„ by๎‚‹๎‚๐œ™๎€ท๐‘ข๐‘€(๐‘ฃ,๐‘ค)=ln๐‘€(exp(๐‘ฃ),exp(๐‘ค)),1,๐‘ข2๎€ธ๎€ท๎€ท๐‘ข=ln๐œ™exp1๎€ธ๎€ท๐‘ข,exp2,๎€ธ๎€ธ(2.17) we see that๎‚‹๐‘€๎€ท๐‘ข1+๐‘ฃ,๐‘ข2๎€ธ=๎‚๐œ™๎€ท๐‘ข+๐‘ค1,๐‘ข2๎€ธ+๎‚‹๐‘€(๐‘ฃ,๐‘ค),(2.18) for all ๐‘ข1,๐‘ข2,๐‘ฃ,๐‘คโˆˆln๐ผ. It is easy to show that ๎‚‹๐‘€ is a mean, because so was ๐‘€. By Theorem 2.1, we have, for some constant ๐œ”โˆˆ[0,1],๎‚‹๎‚๐‘€(๐‘ฃ,๐‘ค)=๐œ™(๐‘ฃ,๐‘ค)=(1โˆ’๐œ”)๐‘ฃ+๐œ”๐‘ค,(๐‘ฃ,๐‘ค)โˆˆln๐ผร—ln๐ผ,(2.19) hence๐‘€(๐‘ฅ,๐‘ฆ)=๐œ™(๐‘ฅ,๐‘ฆ)=๐‘ฅ1โˆ’๐œ”๐‘ฆ๐œ”,(๐‘ฅ,๐‘ฆ)โˆˆ๐ผร—๐ผ.(2.20) We obtain the following.

Theorem 2.2. Let ๐ผโŠ‚(0,โˆž) be a non-degenerate interval such that ๐ผโ‹…๐ผโŠ‚๐ผ. Then a mean ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ and a function ๐œ™โˆถ๐ผร—๐ผโ†’โ„ satisfy (2.16) if and only if there exists a constant ๐œ”โˆˆ[0,1] such that ๐‘€ and ๐œ™ are given by (2.20).

Remark 2.3. Let us note that two other Pexider equations, ๐‘€๎€ท๐‘ +๐‘ฅ๎€ธ๎€ท๐‘ =๐œ™๎€ธ๐‘€๎€ท๐‘ฅ๎€ธ,๐‘€๎€ท๐‘ โ€ข๐‘ฅ๎€ธ๎€ท๐‘ =๐œ™๎€ธ๎€ท๐‘ฅ+๐‘€๎€ธ,(2.21) have no solutions (๐‘€,๐œ™) such that ๐‘€ is reflexive. In fact, putting ๐‘ =(๐‘ ,๐‘ ) and ๐‘ฅ=(๐‘ฅ,๐‘ฅ), we get ๐‘ +๐‘ฅ=๐œ™(๐‘ ,๐‘ )๐‘ฅ,๐‘ ๐‘ฅ=๐œ™(๐‘ ,๐‘ )+๐‘ฅ,(2.22) respectively. The equalities cannot be satisfied for all ๐‘ ,๐‘ฅโˆˆ๐ผ.

3. Further Generalizations

Now, we come back to (2.1). We notice that it is also a special case of the following equation:๐‘€(๐‘ +๐‘ฅ,๐œ“(๐‘ )+๐‘ฆ)=๐‘€(๐‘ ,๐œ“(๐‘ ))+๐‘€(๐‘ฅ,๐‘ฆ),(3.1) that is, the Cauchy equation for ๐‘€, but with one of variables belonging to the graph of some fixed function ๐œ“โˆถ๐ผโ†’๐ผ. We will consider also the equation๐‘€(๐‘ ๐‘ฅ,๐œ“(๐‘ )๐‘ฆ)=๐‘€(๐‘ ,๐œ“(๐‘ ))๐‘€(๐‘ฅ,๐‘ฆ).(3.2) Note that if we take ๐œ“=id๐ผ, we get (2.1) or (2.2).

We will prove results concerning the above equations. Let us start with the โ€œadditiveโ€ case.

Theorem 3.1. Let ๐ผ be a non-degenerate interval such that ๐ผ+๐ผโŠ‚๐ผ. Let ๐œ“โˆถ๐ผโ†’๐ผ be a function such that the mapping ฮจโˆถ๐ผร—๐ผโ†’๐ผร—๐ผ given by ฮจ๎€ท๐‘ 1,๐‘ 2๎€ธ=๎€ท๐‘ 1๎€ท๐‘ +๐œ“2๎€ธ,๐‘ 2๎€ท๐‘ +๐œ“1๎€ธ๎€ธ(3.3) is a surjection. Then, a symmetric and reflexive mapping ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ satisfies (3.1) if and only if ๐‘€ is the arithmetic mean.

Proof. Fix a ๐‘ โˆˆ๐ผ and put ๐‘ฅ=๐œ“(๐‘ ),๐‘ฆ=๐‘  into (3.1). Using the reflexivity and the symmetry of ๐‘€, we obtain ๐‘ +๐œ“(๐‘ )=๐‘€(๐‘ +๐œ“(๐‘ ),๐œ“(๐‘ )+๐‘ )=๐‘€(๐‘ ,๐œ“(๐‘ ))+๐‘€(๐œ“(๐‘ ),๐‘ )=2๐‘€(๐‘ ,๐œ“(๐‘ )),(3.4) whence ๐‘€(๐‘ ,๐œ“(๐‘ ))=๐‘ +๐œ“(๐‘ )2,๐‘ โˆˆ๐ผ.(3.5) Now, fix a (๐‘ฅ,๐‘ฆ)โˆˆ๐ผร—๐ผ. Let (๐‘ 1,๐‘ 2)โˆˆ๐ผร—๐ผ be chosen so that ฮจ(๐‘ 1,๐‘ 2)=(๐‘ฅ,๐‘ฆ). From (3.1) and (3.5), we get ๐‘€๎€ท๐‘ (๐‘ฅ,๐‘ฆ)=๐‘€1๎€ท๐‘ +๐œ“2๎€ธ๎€ท๐‘ ,๐œ“1๎€ธ+๐‘ 2๎€ธ๎€ท๐‘ =๐‘€1๎€ท๐‘ ,๐œ“1๎€ท๐œ“๎€ท๐‘ ๎€ธ๎€ธ+๐‘€2๎€ธ,๐‘ 2๎€ธ=๐‘ 1๎€ท๐‘ +๐œ“1๎€ธ๎€ท๐‘ +๐œ“2๎€ธ+๐‘ 22=๐‘ฅ+๐‘ฆ2.(3.6) This completes the proof.

Analogously as in the proof of Theorem 2.2, we obtain the following.

Theorem 3.2. Let ๐ผโŠ‚(0,โˆž) be an interval such that ๐ผโ‹…๐ผโŠ‚๐ผ. Let ๐œ“โˆถ๐ผโ†’๐ผ be a function such that ฮจโˆถ๐ผร—๐ผโ†’๐ผร—๐ผ defined by ฮจ๎€ท๐‘ 1,๐‘ 2๎€ธ=๎€ท๐‘ 1๐œ“๎€ท๐‘ 2๎€ธ,๐‘ 2๐œ“๎€ท๐‘ 1๎€ธ๎€ธ(3.7) is a bijection. Then, a symmetric and reflexive mapping ๐‘€โˆถ๐ผร—๐ผโ†’๐ผ is a solution of (3.2) if and only if ๐‘€ is the geometric mean.

Example 3.3. The assumption of a surjectivity of ฮจ is essential. This is shown by example of the function ๐œ“=id for which we have ฮจ(๐‘ 1,๐‘ 2)=(๐‘ 1+๐‘ 2,๐‘ 1+๐‘ 2)(ฮจ(๐‘ 1,๐‘ 2)=(๐‘ 1๐‘ 2,๐‘ 1๐‘ 2)), 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 ๐‘โˆˆ(0,โˆž)โงต{1} be fixed. The arithmetic mean is the only symmetric mean satisfying the equation ๐‘€(๐‘ +๐‘ฅ,๐‘๐‘ +๐‘ฆ)=๐‘€(๐‘ ,๐‘๐‘ )+๐‘€(๐‘ฅ,๐‘ฆ),(3.8) for all ๐‘ ,๐‘ฅ,๐‘ฆโˆˆ(0,โˆž).

Proof. The function (0,โˆž)2โˆ‹๎€ท๐‘ 1,๐‘ 2๎€ธโŸถ๎€ท๐‘ 1+๐‘๐‘ 2,๐‘ 2+๐‘๐‘ 1๎€ธ(3.9) is a bijection of (0,โˆž)2 onto itself; it is enough to use Theorem 3.1.

Similarly, using Theorem 3.2, we obtain the following.

Corollary 3.5. Let a ๐‘โˆˆโ„โงต{โˆ’1,1} be fixed. The geometric mean is the only symmetric mean satisfying the equation ๐‘€(๐‘ ๐‘ฅ,๐‘ ๐‘๐‘ฆ)=๐‘€(๐‘ ,๐‘ ๐‘)๐‘€(๐‘ฅ,๐‘ฆ),(3.10) for all ๐‘ ,๐‘ฅ,๐‘ฆโˆˆ(0,โˆž).

4. ๐น-Homogeneity of Quasi-Linear Means

We will now consider the following problem. What are operations ๐น with respect to which quasi-linear means are ๐น-homogeneous?

Let us begin with the weighted arithmetic mean defined on an interval ๐ผ. Let ๐‘โˆˆ[0,1] be fixed and let ๐นโˆถ๐ผร—๐ผโ†’๐ผ be an operation satisfying the equation(1โˆ’๐‘)๐น(๐‘ ,๐‘ฅ)+๐‘๐น(๐‘ ,๐‘ฆ)=๐น(๐‘ ,(1โˆ’๐‘)๐‘ฅ+๐‘๐‘ฆ),(4.1) for all ๐‘ ,๐‘ฅ,๐‘ฆโˆˆ๐ผ.

First, we will suppose that ๐นโˆถ๐ผร—๐ผโ†’๐ผ is a function continuous in second variable.

Let us define ๐‘“๐‘ โˆถ=๐น(๐‘ ,โ‹…) for every ๐‘ โˆˆ๐ผ. Then, for every ๐‘ , the function ๐‘“๐‘  solves the equation ๐‘“๐‘ ((1โˆ’๐‘)๐‘ฅ+๐‘๐‘ฆ)=(1โˆ’๐‘)๐‘“๐‘ (๐‘ฅ)+๐‘๐‘“๐‘ (๐‘ฆ).(4.2) In view of the equality (cf. [8]) ๐‘ข+๐‘ฃ2๎‚ƒ=(1โˆ’๐‘)(1โˆ’๐‘)๐‘ข+๐‘ฃ2๎‚„๎‚ƒ+๐‘๐‘ข+๐‘(1โˆ’๐‘)๐‘ฃ+๐‘๐‘ข+๐‘ฃ2๎‚„,(4.3) which holds for all ๐‘ข,๐‘ฃโˆˆโ„ and ๐‘โˆˆ[0,1], it follows that ๐‘“๐‘ ๎‚€๐‘ฅ+๐‘ฆ2๎‚=๐‘“๐‘ (๐‘ฅ)+๐‘“๐‘ (๐‘ฆ)2.(4.4) The assumption of continuity in the second variable implies (cf. e.g., [2] or [7]) the existence of functions ๐ด,๐ตโˆถ๐ผโ†’โ„ such that๐‘“๐‘ (๐‘ฅ)=๐น(๐‘ ,๐‘ฅ)=๐ด(๐‘ )๐‘ฅ+๐ต(๐‘ ),(4.5) for all ๐‘ ,๐‘ฅโˆˆ๐ผ.

Now, let us additionally assume that ๐น is associative, that is, for all ๐‘ ,๐‘ก,๐‘ฅโˆˆ๐ผ we have๐น(๐น(๐‘ ,๐‘ก),๐‘ฅ)=๐น(๐‘ ,๐น(๐‘ก,๐‘ฅ)).(4.6) In view of (4.5), we obtain the equivalent equality ๐ด(๐ด(๐‘ )๐‘ก+๐ต(๐‘ ))๐‘ฅ+๐ต(๐ด(๐‘ )๐‘ก+๐ต(๐‘ ))=๐ด(๐‘ )๐ด(๐‘ก)๐‘ฅ+๐ด(๐‘ )๐ต(๐‘ก)+๐ต(๐‘ ),(4.7) for all ๐‘ ,๐‘ก,๐‘ฅโˆˆ๐ผ. Thus we see that the operation given by (4.5) is associative if and only if the following system of equations ๐ด(๐ด(๐‘ )๐‘ก+๐ต(๐‘ ))=๐ด(๐‘ )๐ด(๐‘ก),(4.8)๐ต(๐ด(๐‘ )๐‘ก+๐ต(๐‘ ))=๐ด(๐‘ )๐ต(๐‘ก)+๐ต(๐‘ ),(4.9) holds for all ๐‘ ,๐‘กโˆˆ๐ผ. Assume now that ๐น is symmetric (hence it is also continuous with respect to the first variable). This means that, by (4.9),๐ด(๐‘ )๐ต(๐‘ก)+๐ต(๐‘ )=๐ด(๐‘ก)๐ต(๐‘ )+๐ต(๐‘ก)(4.10) or equivalently(๐ด(๐‘ )โˆ’1)๐ต(๐‘ก)=(๐ด(๐‘ก)โˆ’1)๐ต(๐‘ )(4.11) for all ๐‘ ,๐‘กโˆˆ๐ผ.

We consider the following two cases.

(a) ๐ด=1. Then ๐น(๐‘ ,๐‘ก)=๐‘ก+๐ต(๐‘ ), and the symmetry of ๐น yields the equality๐‘ก+๐ต(๐‘ )=๐‘ +๐ต(๐‘ก),(4.12)

that is๐ต(๐‘ )โˆ’๐‘ =๐ต(๐‘ก)โˆ’๐‘ก,(4.13) for all ๐‘ ,๐‘กโˆˆ๐ผ. So, there exists a constant ๐›ผ0โˆˆโ„ such that๐ต(๐‘ )=๐‘ +๐›ผ0,๐‘ โˆˆ๐ผ.(4.14) Thus,๐น(๐‘ ,๐‘ก)=๐‘ก+๐‘ +๐›ผ0=๐›พ๐›ผโˆ’10๎€ท๐›พ๐›ผ0(๐‘ )+๐›พ๐›ผ0๎€ธ(๐‘ก),๐‘ก,๐‘ โˆˆ๐ผ,(4.15) where ๐›พ๐›ผ0โˆถ๐ผโ†’๐ผ is defined by๐›พ๐›ผ0(๐‘ )=๐‘ +๐›ผ0.(4.16) Note that the condition ๐›พ๐›ผ0(๐ผ)โŠ‚๐ผ is a restriction imposed both on ๐ผ and ๐›ผ0.

(b) ๐ดโ‰ 1. Let ๐ด(๐‘ 0)โ‰ 1 for some ๐‘ 0โˆˆ๐ผ. Putting ๐‘ =๐‘ 0 into (4.11), we calculate๐ต(๐‘ก)=๐‘‘(๐ด(๐‘ก)โˆ’1),(4.17) where ๐‘‘โˆถ=๐ต(๐‘ 0)/(๐ด(๐‘ 0)โˆ’1). Using the symmetry of ๐น, we get๐ด(๐‘ )๐‘ก+๐‘‘(๐ด(๐‘ )โˆ’1)=๐น(๐‘ ,๐‘ก)=๐น(๐‘ก,๐‘ )=๐ด(๐‘ก)๐‘ +๐‘‘(๐ด(๐‘ก)โˆ’1).(4.18) We have๐ด(๐‘ )(๐‘ก+๐‘‘)=๐ด(๐‘ก)(๐‘ +๐‘‘),๐‘ ,๐‘กโˆˆ๐ผ,(4.19) whence๐ด(๐‘ )=๐‘ +๐‘‘๐ด(๐‘ก)๐‘ก+๐‘‘,๐‘ ,๐‘กโˆˆ๐ผโงต{โˆ’๐‘‘}.(4.20) In other words,๐ด(๐‘ )=๐›ผ2๐‘ +๐›ผ2๐‘‘=๐›ผ2๐‘ +๐›ผ1,(4.21) for all ๐‘ โ‰ โˆ’๐‘‘, where ๐›ผ2 is a constant and ๐›ผ1=๐›ผ2๐‘‘. From (4.19), it results that ๐‘ =โˆ’๐‘‘ implies ๐ด(๐‘ )=0, so (4.21) holds for all ๐‘ โˆˆ๐ผ. Hence,๐ต๎€ท๐›ผ(๐‘ก)=๐‘‘2๐‘ก+๐›ผ1๎€ธโˆ’1=๐›ผ2๎€ท๐›ผ๐‘‘๐‘ก+๐‘‘1๎€ธโˆ’1=๐›ผ1๎€ท๐›ผ๐‘ก+๐‘‘1๎€ธโˆ’1,๐‘กโˆˆ๐ผ,(4.22) and, finally, denoting ๐‘‘(๐›ผ1โˆ’1) by ๐›ผ0, we get๐น(๐‘ ,๐‘ก)=๐›ผ2๐‘ ๐‘ก+๐›ผ1(๐‘ +๐‘ก)+๐›ผ0,๐‘ ,๐‘กโˆˆ๐ผ.(4.23) Now, it is easy to check that associativity of ๐น is equivalent to๐›ผ0๐›ผ2=๐›ผ21โˆ’๐›ผ1.(4.24) We have the following possibilities.(i)๐›ผ2=0 and ๐›ผ1=0. In this case, we have ๐น(๐‘ ,๐‘ก)=๐›ผ0,(4.25) and it is enough to assume that ๐›ผ0โˆˆ๐ผ.(ii)๐›ผ2=0 and ๐›ผ1=1. Then, similarly as in case (a), ๐น(๐‘ ,๐‘ก)=๐‘ +๐‘ก+๐›ผ0=๐›พ๐›ผโˆ’10๎€ท๐›พ๐›ผ0(๐‘ )+๐›พ๐›ผ0๎€ธ,(๐‘ก)(4.26) where ๐›พ๐›ผ0โˆถ๐ผโ†’๐ผ is defined by (4.16).(iii)๐›ผ2โ‰ 0. Now, we have1๐น(๐‘ ,๐‘ก)=๐›ผ2๐›ผ๎€ท๎€ท2๐‘ +๐›ผ1๐›ผ๎€ธ๎€ท2๐‘ก+๐›ผ1๎€ธโˆ’๐›ผ1๎€ธ=๐›ฟ๐›ผโˆ’11,๐›ผ2๎€ท๐›ฟ๐›ผ1,๐›ผ2(๐‘ )๐›ฟ๐›ผ1,๐›ผ2๎€ธ,(๐‘ก)(4.27) where ๐›ฟ๐›ผ1,๐›ผ2โˆถ๐ผโ†’๐ผ is given by๐›ฟ๐›ผ1,๐›ผ2(๐‘ )=๐›ผ1+๐›ผ2๐‘ .(4.28) Note that ๐›ฟ๐›ผ1,๐›ผ2(๐ผ)โŠ‚๐ผ is a condition imposed on the interval ๐ผ and constants ๐›ผ1,๐›ผ2 as well.

Thus, we have proved.

Proposition 4.1. Let ๐ผ be a non-degenerate interval, let ๐นโˆถ๐ผร—๐ผโ†’๐ผ be a function which is associative, symmetric, and continuous in each variable. Let ๐‘โˆˆ[0,1] be fixed. Then, the weighted arithmetic mean is ๐น-homogeneous (that is (4.1) holds) if and only if there exist constants ๐›ผ0,๐›ผ1, and ๐›ผ2 such that ๐น is given by (4.23), the condition (4.24) is satisfied and (i)(๐›ผ2=๐›ผ1=0โ‡’๐›ผ0โˆˆ๐ผ), (ii)((๐›ผ2=0โˆง๐›ผ1=1)โ‡’(๐›พ๐›ผ0(๐ผ)+๐›พ๐›ผ0(๐ผ)โŠ‚๐›พ๐›ผ0(๐ผ)โŠ‚๐ผ)) where ๐›พ๐›ผ0 is given by (4.16), (iii)(๐›ผ2โ‰ 0โ‡’(๐›ฟ๐›ผ1,๐›ผ2(๐ผ)โ‹…๐›ฟ๐›ผ1,๐›ผ2(๐ผ)โŠ‚๐›ฟ๐›ผ1,๐›ผ2(๐ผ)โŠ‚๐ผ)) where ๐›ฟ๐›ผ1,๐›ผ2 is given by (4.28).

Remark 4.2. We see that among the operations listed in Proposition 4.1 are the following, (i)๐น(๐‘ ,๐‘ก)=๐‘ +๐‘ก(๐›ผ0=๐›ผ2=0,๐›ผ1=1), ๐ผโˆˆ{โ„,(โˆ’โˆž,๐‘Ž),(โˆ’โˆž,๐‘Ž],(๐‘,โˆž),[๐‘,โˆž)โˆถ๐‘Žโ‰ค0,๐‘โ‰ฅ0}, (ii)๐น(๐‘ ,๐‘ก)=๐‘ โ‹…๐‘ก(๐›ผ2=1,๐›ผ1=๐›ผ0=0),(๐ผ,โ‹…) is a subsemigroup of (โ„,โ‹…), (iii)๐น(๐‘ ,๐‘ก)=2๐‘ ๐‘กโˆ’(๐‘ +๐‘ก)+1=1/2((2๐‘ โˆ’1)(2๐‘กโˆ’1)+1)(๐›ผ2=2,๐›ผ1=โˆ’1,๐›ผ0=1), in this case ๐ผโˆˆ{โ„,(๐‘Ž,โˆž),[๐‘Ž,โˆž),๐‘Žโ‰ฅ1}.

Now, let us generalize Proposition 4.1 to the case of arbitrary quasi-linear means.

We admit the following definition.

Definition 4.3. Let ๐ผ and ๐ฝ be non-degenerate intervals, let ๐นโˆถ๐ผร—๐ผโ†’๐ผ be an arbitrary function, let ๐‘“โˆถ๐ผโ†’๐ฝ be a bijection. Then, we define the function ๐น๐‘“โˆถ๐ฝร—๐ฝโ†’๐ฝ by ๐น๐‘“๎€ท๐น๎€ท๐‘“(๐‘ข,๐‘ฃ)=๐‘“โˆ’1(๐‘ข),๐‘“โˆ’1.(๐‘ฃ)๎€ธ๎€ธ(4.29)

Remark 4.4. We can easily see that ๐น is associative and symmetric if and only if ๐น๐‘“ has the same properties. Moreover, if ๐‘“ is continuous, then ๐น is continuous in each variable if and only if ๐น๐‘“ is continuous in each variable.

We obviously have the following.

Lemma 4.5. Let ๐ผ and ๐ฝ be non-degenerate intervals. Let ๐‘“โˆถ๐ผโ†’๐ฝ be a continuous bijection, let ๐นโˆถ๐ผร—๐ผโ†’๐ผ be a function, and let ๐‘โˆˆ[0,1] be fixed. Then the quasi-linear mean ๐‘€๐‘“โˆถ๐ผร—๐ผโ†’๐ผ given by ๐‘€๐‘“(๐‘ฅ,๐‘ฆ)=๐‘“โˆ’1((1โˆ’๐‘)๐‘“(๐‘ฅ)+๐‘๐‘“(๐‘ฆ)),(4.30) is ๐น-homogeneous if and only if the weighted arithmetic mean on the interval ๐ฝ is ๐น๐‘“-homogeneous.

From Proposition 4.1 and Lemma 4.5, we obtain the following.

Theorem 4.6. Let ๐ผ and ๐ฝ be non-degenerate intervals and let ๐‘“โˆถ๐ผโ†’๐ฝ be a continuous bijection, let ๐นโˆถ๐ผร—๐ผโ†’๐ผ be a function which is associative, symmetric, and continuous in each variable. Let ๐‘โˆˆ[0,1] be fixed. Then, the quasi-linear mean ๐‘€๐‘“โˆถ๐ผร—๐ผโ†’๐ผ is F- homogeneous if and only if there exist constants ๐›ผ0,๐›ผ1, and ๐›ผ2 such that ๐น is given by ๐น(๐‘ ,๐‘ก)=๐‘“โˆ’1๎€ท๐›ผ2๐‘“(๐‘ )๐‘“(๐‘ก)+๐›ผ1(๐‘“(๐‘ )+๐‘“(๐‘ก))+๐›ผ0๎€ธ,๐‘ ,๐‘กโˆˆ๐ผ,(4.31) the condition (4.24) is satisfied, and (i)(๐›ผ2=๐›ผ1=0โ‡’๐›ผ0โˆˆ๐ผ), (ii)((๐›ผ2=0โˆง๐›ผ1=1)โ‡’(๐›พ๐›ผ0(๐ฝ)+๐›พ๐›ผ0(๐ฝ)โŠ‚๐›พ๐›ผ0(๐ฝ)โŠ‚๐ฝ)) where ๐›พ๐›ผ0โˆถ๐ฝโ†’๐ฝ is given by ๐›พ๐›ผ0(๐‘ข)=๐‘ข+๐›ผ0,(4.32)(iii)(๐›ผ2โ‰ 0โ‡’(๐›ฟ๐›ผ1,๐›ผ2(๐ฝ)โ‹…๐›ฟ๐›ผ1,๐›ผ2(๐ฝ)โŠ‚๐›ฟ๐›ผ1,๐›ผ2(๐ฝ)โŠ‚๐ฝ)) where ๐›ฟ๐›ผ1,๐›ผ2โˆถ๐ฝโ†’๐ฝ is given by ๐›ฟ๐›ผ1,๐›ผ2(๐‘ข)=๐›ผ1+๐›ผ2๐‘ข.(4.33)

Remark 4.7. Among operations ๐น, for which ๐‘€๐‘“ is homogeneous, are the following (๐‘ ,๐‘ก)โŸถ๐‘“โˆ’1((๐‘“(๐‘ )๐‘“(๐‘ก)),(4.34)๐‘ ,๐‘ก)โŸถ๐‘“โˆ’1(๐‘“(๐‘ )+๐‘“(๐‘ก)).(4.35) It is possible to see that ๐‘€๐‘“ 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, ๐น-homogeneity alone does not characterize the mean ๐‘€๐‘“. In order to see it, define ๐นโˆถโ„+ร—โ„+โ†’โ„+ by ๐น(๐‘ ,๐‘ก)=๐‘ ๐‘ก+2(๐‘ +๐‘ก)+2=(๐‘ +2)(๐‘ก+2)โˆ’2=๐›ฟโˆ’12,1๎€ท๐›ฟ2,1(๐‘ )๐›ฟ2,1๎€ธ.(๐‘ก)(4.36)

We show that there exist many ๐น-homogeneous quasi-linear means, even when we restrict our attention to symmetric ones. In fact, suppose that ๐‘€ is an F-homogeneous quasi-arithmetic mean, that is ๐‘€(๐น(๐‘ ,๐‘ก),๐น(๐‘ ,๐‘ข))=๐น(๐‘ ,๐‘€(๐‘ก,๐‘ข)),๐‘ ,๐‘ก,๐‘ขโˆˆโ„+,(4.37) or ๐‘€๎€ท๐›ฟโˆ’12,1๎€ท๐›ฟ2,1(๐‘ )๐›ฟ2,1๎€ธ(๐‘ก),๐›ฟโˆ’12,1๎€ท๐›ฟ2,1(๐‘ )๐›ฟ2,1(๐‘ข)๎€ธ๎€ธ=๐›ฟโˆ’12,1๎€ท๐›ฟ2,1(๐‘ )๐›ฟ2,1๎€ธ.(๐‘€(๐‘ก,๐‘ข))(4.38) Substituting ๐‘ฅ=๐›ฟ2,1(๐‘ ),๐‘ฆ=๐›ฟ2,1(๐‘ก), and ๐‘ง=๐›ฟ2,1(๐‘ข), and defining ๎‚‹๐‘€โˆถ[2,โˆž)ร—[2,โˆž)โ†’[2,โˆž) by ๎‚‹๐‘€(๐‘ฅ,๐‘ฆ)=๐›ฟ2,1๎€ท๐‘€๎€ท๐›ฟโˆ’12,1(๐‘ฅ),๐›ฟโˆ’12,1(,๐‘ฆ)๎€ธ๎€ธ(4.39) we see that ๎‚‹๐‘€ is homogeneous on [2,โˆž). It follows (cf. for instance [1]) that either ๎‚‹โˆš๐‘€(๐‘ฅ,๐‘ฆ)=๐‘ฅโ‹…๐‘ฆ(4.40) or there is a ๐‘˜โ‰ 0 such that ๎‚‹๎‚ต๐‘ฅ๐‘€(๐‘ฅ,๐‘ฆ)=๐‘˜+๐‘ฆ๐‘˜2๎‚ถ1/๐‘˜,(4.41) whence either โˆš๐‘€(๐‘ ,๐‘ก)=(๐‘ +2)(๐‘ก+2)โˆ’2(4.42) or ๎‚ต๐‘€(๐‘ ,๐‘ก)=(๐‘ +2)๐‘˜+(๐‘ก+2)๐‘˜2๎‚ถ1/๐‘˜โˆ’2.(4.43)