# Generalized Homogeneity of Means

Authors: Małgorzata Pałys; Maciej Sablik
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We extend the classical notions of translativity and homogeneity of means toF-homogeneity, that is,  invariance with respect to an operation
F:I×I→I. We find the shape of F for the arithmetic weighted mean and then the general form of
F for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by F-homogeneity with respect to a single operation
F, one needs to take two of such operations in order to characterize a mean.


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## Body

## 1. Introduction

Definition 1.1.
LetI⊂ℝ be an interval. A function M:I×I→I such that
(1.1)min⁡{x,y}≤M(x,y)≤max⁡{x,y}x,y∈I,
is called a mean (on I2).Note that every mean isreflexive, that is(1.2)M(x,x)=x∀x∈I.
A mean M is called symmetric, if(1.3)M(x,y)=M(y,x)∀x,y∈I.In [1], Aczél and Dhombres distinguished two special types of means, defined in I×I, where I⊂ℝ+.(i)  Translative: if the function is translation invariant, that is for allx,y∈I,t∈ℝ+(1.4)x+t,y+t∈I⟹M(x+t,y+t)=M(x,y)+t.(ii) Homogeneous: if the function is multiplication invariant, that is for allx,y∈I,u∈ℝ+(1.5)xu,yu∈I⟹M(xu,yu)=M(x,y)u.
For more information about means, see, for instance, [1–6].In the present paper, we present three approaches to the question of translativity and homogeneity of means. First, we “discover” 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.
LetI⊂ℝ be an interval and let F:I×I→I be a function. A mean M:I×I→I such that
(1.6)M(F(t,x),F(t,y))=F(t,M(x,y)),
for every x,y,t∈I is called F-homogeneous.In the last section, we determine all the operationsF with respect to which quasi-linear means are F-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 equationM:I×I→I on an interval I satisfying I+I⊂I, that is the equation(2.1)M(s+x,s+y)=s+M(x,y),
and the homogeneous mean equation M:I×I→I on an interval I satisfying I·I⊂I, that is the equation(2.2)M(sx,sy)=sM(x,y),
may be treated as conditional forms of the following equations:(2.3)M(s+x,t+y)=ϕ(s,t)+M(x,y),(2.4)M(sx,ty)=ϕ(s,t)M(x,y),
respectively.Here, the word “condition” means that (2.3) and (2.4) have to be satisfied for pairs (s,t) of the set W={(s,t)∈I×I:s=t}.To show that (2.1) actually is the conditional equation (2.3), we put t=s,y=x into (2.3) and use reflexivity of M. Then,(2.5)s+x=M(s+x,s+x)=ϕ(s,s)+M(x,x)=ϕ(s,s)+x,
whence ϕ(s,s)=s,s∈I.Puttings̲=(s,t) and x̲=(x,y), we can rewrite (2.3) in the form(2.6)M(s̲+x̲)=ϕ(s̲)+M(x̲),
which is satisfied for s̲∈I×I and x̲∈I×I. This is Pexider conditional equation (in this case the condition refers to the fact that (2.6) holds for pairs (s̲,x̲)∈(I×I)×(I×I)). To solve (2.6) let us note that fixing arbitrarily x,y,s1,s2,t1,t2∈I, and letting x̲=(x,y),s̲=(s1,s2),t̲=(t1,t2), we get, from (2.6)(2.7)ϕ(s̲+t̲)=M(s̲+t̲+x̲)-M(x̲)=ϕ(s̲)+M(t̲+x̲)-M(x̲)=ϕ(s̲)+ϕ(t̲)+M(x̲)-M(x̲)=ϕ(s̲)+ϕ(t̲).Thus ϕ satisfies the Cauchy equation for s̲,t̲∈I×I. Moreover, from (2.6) and properties of mean M, we obtain (x∈I is arbitrarily fixed)(2.8)min⁡(s1,s2)≤M(s1+x,s2+x)-x=M(s̲+(x,x))-M(x,x)=ϕ(s̲)≤max⁡(s1,s2),
so ϕ is majorized (and minorized) on I×I by a continuous function. The well known results (cf. for instance [1, 2, 7]) imply that ϕ is a linear mapping, that is there exist constants α,ω∈ℝ such that(2.9)ϕ(s1,s2)=αs1+ωs2,(s1,s2)∈I×I.
From (2.8), we infer that s=ϕ(s,s)=(α+ω)s,s∈I, so α=1-ω. Moreover, from (2.8), we have for s1<s2,s1,s2∈I(2.10)s1≤(1-ω)s1+ωs2≤s2,
that is, ω(s2-s1)≥0 and (1-ω)(s2-s1)≥0, so ω∈[0,1]. In other words, we have proved that there exists an ω∈[0,1] such that(2.11)ϕ(s1,s2)=(1-ω)s1+ωs2,(s1,s2)∈I×I.
Now, put s=y and t=x into (2.3). Using the reflexivity of M and (2.11), we obtain(2.12)x+y=M(y+x,x+y)=ϕ(y,x)+M(x,y)=(1-ω)y+ωx+M(x,y),
whence(2.13)M(x,y)=(1-ω)x+ωy,(x,y)∈I×I.
Thus we obtain the following.Theorem 2.1.
LetI⊂ℝ be a non-degenerate interval such that I+I⊂I. Then, a  mean M:I×I→I and a function ϕ:I×I→ℝ satisfy (2.3) if and only if there exists a constant ω∈[0,1] such that(2.14)M(x,y)=ϕ(x,y)=(1-ω)x+ωy,(x,y)∈I×I.
Now, we define in ℝ2 an operation • by(2.15)(s1,s2)•(x,y)=(s1x,s2y).
Let I⊂(0,∞) be an interval such that I·I⊂I. Suppose that M:I×I→I is a mean and ϕ:I×I→ℝ is an arbitrary function. We consider the equation (s̲=(s1,s2),x̲=(x,y))(2.16)M(s̲•x̲)=ϕ(s̲)M(x̲).
From the assumption I⊂(0,∞), hence both M and ϕ take on positive values. Putting u1=lns1,u2=lns2,v=lnx,w=lny and defining M̃:lnI×lnI→lnI,ϕ̃:lnI×lnI→ℝ by(2.17)M̃(v,w)=ln⁡M(exp⁡(v),exp⁡(w)),ϕ̃(u1,u2)=ln⁡ϕ(exp⁡(u1),exp⁡(u2)),
we see that(2.18)M̃(u1+v,u2+w)=ϕ̃(u1,u2)+M̃(v,w),
for all u1,u2,v,w∈lnI. It is easy to show that M̃ is a mean, because so was M. By Theorem 2.1, we have, for some constant ω∈[0,1],(2.19)M̃(v,w)=ϕ̃(v,w)=(1-ω)v+ωw,(v,w)∈ln⁡I×ln⁡I,
hence(2.20)M(x,y)=ϕ(x,y)=x1-ωyω,(x,y)∈I×I.
We obtain the following.Theorem 2.2.
LetI⊂(0,∞) be a non-degenerate interval such that I·I⊂I.  Then a mean M:I×I→I and a function ϕ:I×I→ℝ satisfy (2.16) if and only if there exists a constant ω∈[0,1] such that M and ϕ are given by (2.20).Remark 2.3.
Let us note that two other Pexider equations,(2.21)M(s̲+x̲)=ϕ(s̲)M(x̲),M(s̲•x̲)=ϕ(s̲)+M(x̲),
have no solutions (M,ϕ) such that M is reflexive. In fact, putting s̲=(s,s) and x̲=(x,x), we get
(2.22)s+x=ϕ(s,s)x,sx=ϕ(s,s)+x,
respectively. The equalities cannot be satisfied for all s,x∈I.


## 3. Further Generalizations

Now, we come back to (2.1). We notice that it is also a special case of the following equation:(3.1)M(s+x,ψ(s)+y)=M(s,ψ(s))+M(x,y),
that is, the Cauchy equation for M, but with one of variables belonging to the graph of some fixed function ψ:I→I. We will consider also the equation(3.2)M(sx,ψ(s)y)=M(s,ψ(s))M(x,y).
Note that if we take ψ=idI, we get (2.1) or (2.2).We will prove results concerning the above equations. Let us start with the “additive” case.Theorem 3.1.
LetI be a non-degenerate interval such that I+I⊂I. Let ψ:I→I be a function such that the mapping Ψ:I×I→I×I given by
(3.3)Ψ(s1,s2)=(s1+ψ(s2),s2+ψ(s1))
is a surjection. Then, a symmetric and reflexive mapping M:I×I→I satisfies (3.1) if and only if M is the arithmetic mean.Proof.
Fix as∈I and put x=ψ(s),y=s into (3.1). Using the reflexivity and the symmetry of M, we obtain
(3.4)s+ψ(s)=M(s+ψ(s),ψ(s)+s)=M(s,ψ(s))+M(ψ(s),s)=2M(s,ψ(s)),
whence
(3.5)M(s,ψ(s))=s+ψ(s)2,s∈I.
Now, fix a (x,y)∈I×I. Let (s1,s2)∈I×I be chosen so that Ψ(s1,s2)=(x,y). From (3.1) and (3.5), we get
(3.6)M(x,y)=M(s1+ψ(s2),ψ(s1)+s2)=M(s1,ψ(s1))+M(ψ(s2),s2)=s1+ψ(s1)+ψ(s2)+s22=x+y2.
This completes the proof.Analogously as in the proof of Theorem2.2, we obtain the following.Theorem 3.2.
LetI⊂(0,∞) be an interval such that I·I⊂I. Let ψ:I→I be a function such that  Ψ:I×I→I×I defined by
(3.7)Ψ(s1,s2)=(s1ψ(s2),s2ψ(s1))
is a bijection. Then, a symmetric and reflexive mapping M:I×I→I is a solution of (3.2) if and only if M is the geometric mean.Example 3.3.
The assumption of a surjectivity ofΨ is essential. This is shown by example of the function ψ=id for which we have Ψ(s1,s2)=(s1+s2,s1+s2)(Ψ(s1,s2)=(s1s2,s1s2)), 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.
Letc∈(0,∞)∖{1} be fixed. The arithmetic mean is the only symmetric mean satisfying the equation
(3.8)M(s+x,cs+y)=M(s,cs)+M(x,y),
for all s,x,y∈(0,∞).Proof.
The function(3.9)(0,∞)2∋(s1,s2)⟶(s1+cs2,s2+cs1)
is a bijection of (0,∞)2 onto itself; it is enough to use Theorem 3.1.Similarly, using Theorem3.2, we obtain the following.Corollary 3.5.
Let ac∈ℝ∖{-1,1} be fixed. The geometric mean is the only symmetric mean satisfying the equation
(3.10)M(sx,scy)=M(s,sc)M(x,y),
for all s,x,y∈(0,∞).


## 4.F-Homogeneity of Quasi-Linear Means

We will now consider the following problem. What are operationsF with respect to which quasi-linear means are F-homogeneous?Let us begin with the weighted arithmetic mean defined on an intervalI. Let p∈[0,1] be fixed and let F:I×I→I be an operation satisfying the equation(4.1)(1-p)F(s,x)+pF(s,y)=F(s,(1-p)x+py),
for all s,x,y∈I.First, we will suppose thatF:I×I→I is a function continuous in second variable.Let us definefs:=F(s,·) for every s∈I. Then, for every s, the function fs solves the equation (4.2)fs((1-p)x+py)=(1-p)fs(x)+pfs(y).
In view of the equality (cf. [8]) (4.3)u+v2=(1-p)[(1-p)u+v2+pu]+p[(1-p)v+pu+v2],
which holds for all u,v∈ℝ and p∈[0,1], it follows that (4.4)fs(x+y2)=fs(x)+fs(y)2.
The assumption of continuity in the second variable implies (cf. e.g., [2] or [7]) the existence of functions A,B:I→ℝ such that(4.5)fs(x)=F(s,x)=A(s)x+B(s),
for all s,x∈I.Now, let us additionally assume thatF is associative, that is, for all s,t,x∈I we have(4.6)F(F(s,t),x)=F(s,F(t,x)).
In view of (4.5), we obtain the equivalent equality (4.7)A(A(s)t+B(s))x+B(A(s)t+B(s))=A(s)A(t)x+A(s)B(t)+B(s),
for all s,t,x∈I. Thus we see that the operation given by (4.5) is associative if and only if the following system of equations (4.8)A(A(s)t+B(s))=A(s)A(t),(4.9)B(A(s)t+B(s))=A(s)B(t)+B(s),
holds for all s,t∈I. Assume now that F is symmetric (hence it is also continuous with respect to the first variable). This means that, by (4.9),(4.10)A(s)B(t)+B(s)=A(t)B(s)+B(t)
or equivalently(4.11)(A(s)-1)B(t)=(A(t)-1)B(s)
for all s,t∈I.We consider the following two cases.(a)A=1. Then F(s,t)=t+B(s), and the symmetry of F yields the equality(4.12)t+B(s)=s+B(t),that is(4.13)B(s)-s=B(t)-t,
for all s,t∈I. So, there exists a constant α0∈ℝ such that(4.14)B(s)=s+α0,s∈I.
Thus,(4.15)F(s,t)=t+s+α0=γα0-1(γα0(s)+γα0(t)),t,s∈I,
where γα0:I→I is defined by(4.16)γα0(s)=s+α0.
Note that the condition γα0(I)⊂I is a restriction imposed both on I and α0.(b)A≠1. Let A(s0)≠1 for some s0∈I. Putting s=s0 into (4.11), we calculate(4.17)B(t)=d(A(t)-1),
where d:=B(s0)/(A(s0)-1). Using the symmetry of F, we get(4.18)A(s)t+d(A(s)-1)=F(s,t)=F(t,s)=A(t)s+d(A(t)-1).
We have(4.19)A(s)(t+d)=A(t)(s+d),s,t∈I,
whence(4.20)A(s)s+d=A(t)t+d,s,t∈I∖{-d}.
In other words,(4.21)A(s)=α2s+α2d=α2s+α1,
for all s≠-d, where α2 is a constant and α1=α2d. From (4.19), it results that s=-d implies A(s)=0, so (4.21) holds for all s∈I. Hence,(4.22)B(t)=d(α2t+α1-1)=α2dt+d(α1-1)=α1t+d(α1-1),t∈I,
and, finally, denoting d(α1-1) by α0, we get(4.23)F(s,t)=α2st+α1(s+t)+α0,s,t∈I.
Now, it is easy to check that associativity of F is equivalent to(4.24)α0α2=α12-α1.
We have the following possibilities.(i)
α2=0 and α1=0. In this case, we have(4.25)F(s,t)=α0,
and it is enough to assume that α0∈I.(ii)
α2=0 and α1=1. Then, similarly as in case (a),(4.26)F(s,t)=s+t+α0=γα0-1(γα0(s)+γα0(t)),
where γα0:I→I is defined by (4.16).(iii)
α2≠0. Now, we have(4.27)F(s,t)=1α2((α2s+α1)(α2t+α1)-α1)=δα1,α2-1(δα1,α2(s)δα1,α2(t)),
where δα1,α2:I→I is given by(4.28)δα1,α2(s)=α1+α2s.
Note that δα1,α2(I)⊂I is a condition imposed on the interval I and constants α1,α2 as well.Thus, we have proved.Proposition 4.1.
LetI be a non-degenerate interval, let F:I×I→I be a function which is associative, symmetric, and continuous in each variable. Let p∈[0,1] be fixed. Then, the weighted arithmetic mean is F-homogeneous (that is (4.1) holds) if and only if there exist constants α0,α1, and α2 such that F is given by (4.23), the condition (4.24) is satisfied and  (i)
(α2=α1=0⇒α0∈I),(ii)
((α2=0∧α1=1)⇒(γα0(I)+γα0(I)⊂γα0(I)⊂I)) where γα0 is given by (4.16),(iii)
(α2≠0⇒(δα1,α2(I)·δα1,α2(I)⊂δα1,α2(I)⊂I)) where δα1,α2 is given by (4.28).Remark 4.2.
We see that among the operations listed in Proposition4.1 are the following,  (i)
F(s,t)=s+t(α0=α2=0,α1=1), I∈{ℝ,(-∞,a),(-∞,a],(b,∞),[b,∞):a≤0,b≥0},(ii)
F(s,t)=s·t(α2=1,α1=α0=0),(I,·) is a subsemigroup of (ℝ,·),(iii)
F(s,t)=2st-(s+t)+1=1/2((2s-1)(2t-1)+1)(α2=2,α1=-1,α0=1), in this case I∈{ℝ,(a,∞),[a,∞),a≥1}.Now, let us generalize Proposition4.1 to the case of arbitrary quasi-linear means.We admit the following definition.Definition 4.3.
LetI and J be non-degenerate intervals, let F:I×I→I be an arbitrary function, let f:I→J be a bijection. Then, we define the function Ff:J×J→J by
(4.29)Ff(u,v)=f(F(f-1(u),f-1(v))).Remark 4.4.
We can easily see thatF is associative and symmetric if and only if Ff has the same properties. Moreover, if f is continuous, then F is continuous in each variable if and only if Ff is continuous in each variable.We obviously have the following.Lemma 4.5.
LetI and J be non-degenerate intervals. Let f:I→J be a continuous bijection, let F:I×I→I be a function, and let p∈[0,1] be fixed. Then the quasi-linear mean Mf:I×I→I given by
(4.30)Mf(x,y)=f-1((1-p)f(x)+pf(y)),
is F-homogeneous if and only if the weighted arithmetic mean on the interval J is Ff-homogeneous.From Proposition4.1 and Lemma 4.5, we obtain the following.Theorem 4.6.
LetI and J be non-degenerate intervals and let f:I→J be a continuous bijection, let F:I×I→I be a function which is associative, symmetric, and continuous in each variable. Let p∈[0,1] be fixed. Then, the quasi-linear mean Mf:I×I→I is F- homogeneous if and only if there exist constants α0,α1, and α2 such that F is given by
(4.31)F(s,t)=f-1(α2f(s)f(t)+α1(f(s)+f(t))+α0),s,t∈I,
the condition (4.24) is satisfied, and  (i)
(α2=α1=0⇒α0∈I),(ii)
((α2=0∧α1=1)⇒(γα0(J)+γα0(J)⊂γα0(J)⊂J)) where γα0:J→J is given by(4.32)γα0(u)=u+α0,(iii)
(α2≠0⇒(δα1,α2(J)·δα1,α2(J)⊂δα1,α2(J)⊂J))  where δα1,α2:J→J is given by(4.33)δα1,α2(u)=α1+α2u.Remark 4.7.
Among operationsF, for which Mf is homogeneous, are the following
(4.34)(s,t)⟶f-1(f(s)f(t)),(4.35)(s,t)⟶f-1(f(s)+f(t)).
It is possible to see that Mf 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 Theorem4.6 does not hold. More exactly, F-homogeneity alone does not characterize the mean Mf. In order to see it, define F:ℝ+×ℝ+→ℝ+ by
(4.36)F(s,t)=st+2(s+t)+2=(s+2)(t+2)-2=δ2,1-1(δ2,1(s)δ2,1(t)).We show that there exist manyF-homogeneous quasi-linear means, even when we restrict our attention to symmetric ones. In fact, suppose that M is an F-homogeneous quasi-arithmetic mean, that is (4.37)M(F(s,t),F(s,u))=F(s,M(t,u)),s,t,u∈R+,
or (4.38)M(δ2,1-1(δ2,1(s)δ2,1(t)),δ2,1-1(δ2,1(s)δ2,1(u)))=δ2,1-1(δ2,1(s)δ2,1(M(t,u))).
Substituting x=δ2,1(s),y=δ2,1(t), and z=δ2,1(u), and defining M̃:[2,∞)×[2,∞)→[2,∞) by (4.39)M̃(x,y)=δ2,1(M(δ2,1-1(x),δ2,1-1(y))),
we see that M̃ is homogeneous on [2,∞). It follows (cf. for instance [1]) that either (4.40)M̃(x,y)=x⋅y
or there is a k≠0 such that (4.41)M̃(x,y)=(xk+yk2)1/k,
whence either (4.42)M(s,t)=(s+2)(t+2)-2
or (4.43)M(s,t)=((s+2)k+(t+2)k2)1/k-2.



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