Univ-bejaia.dz
132 (2013), p. 139 -149.
ALGEBRAIC AND TOPOLOGICAL STRUCTURES ON
THE SET OF MEAN FUNCTIONS AND
GENERALIZATION OF THE AGM MEAN
Abstract. In this paper, we present new structures and results on theset
MD of mean functions on a given symmetric domain
D in R2. First,we construct on
MD a structure of abelian group in which the neutralelement is the
arithmetic mean; then we study some symmetries in thatgroup. Next, we construct on
MD a structure of metric space underwhich
MD is the closed ball with center the
arithmetic mean and radius1
/2. We show in particular that the
geometric and
harmonic means lieon the boundary of
MD. Finally, we give two theorems generalizing theconstruction of the AGM mean. Roughly speaking, those theorems showthat for any two given means
M1 and
M2, which satisfy some regular-ity conditions, there exists a unique mean
M satisfying the functionalequation
M (
M1
, M2) =
M .
Let
D be a nonempty symmetric domain in R2. A
mean function (or
simply a
mean) on
D is a function
M :
D → R satisfying the followingthree axioms:
(i)
M is symmetric, that is,
M (
x, y) =
M (
y, x) for all (
x, y)
∈ D.
(ii) For all (
x, y)
∈ D, we have min(
x, y)
≤ M (
x, y)
≤ max(
x, y).
(iii) For all (
x, y)
∈ D, we have
M (
x, y) =
x =
⇒ x =
y.
Note that because of (ii), the implication in (iii) is actually an equivalence.
Among the most known examples of mean functions, we cite:
• The arithmetic mean A defined on R2 by: A(
x, y) =
x+
y.
The geometric mean G defined on (0
, +
∞)2 by: G(
x, y) =
• The harmonic mean H defined on (0
, +
∞)2 by: H(
x, y) = 2
xy .
• The Gauss arithmetic-geometric mean AGM defined on (0
, +
∞)2 by thefollowing process:
Given positive real numbers
x, y, AGM(
x, y) is the common limit of the two
2010
Mathematics Subject Classification. Primary 20K99, 54E35; Secondary 39B22.
Key words and phrases. Means, abelian groups, metric spaces, symmetries.
sequences (
xn)
n∈N and (
yn)
n∈N defined by
x0 =
x , y0 =
y
xn+1 =
xn+
yn
For a survey on mean functions, we refer to Chapter 8 of the book [Bor] in
which AGM takes the principal place. However, there are some differences
between that reference and the present paper. Indeed, in [Bor], only axiom
(ii) is taken to define a mean function; (iii) is added to obtain the so called
strict mean while (i) is not considered. In this paper, we shall see that the
three axioms (i), (ii) and (iii) are both necessary and sufficient to define a
good mean or a
good set of mean functions on a given domain. In particular,
axiom (iii), absent in [Bor], is necessary for the foundation of our algebraic
and topological structures (see Sections 2 and 3).
Given a nonempty symmetric domain
D in R2, we denote by
MD the
set of mean functions on
D. The purpose of this paper is on the one handto establish some algebraic and topological structures on
MD and to studysome of their properties and on the other hand to generalize in a natural
way the arithmetic-geometric mean AGM.
In the first section, we define on
MD a structure of abelian group in
which the neutral element is the arithmetic mean. The study of this group
reveals that the arithmetic, geometric and harmonic means lie in a particular
class of mean functions that we call
normal mean functions. We then study
symmetries on
MD and we discover that the symmetry with respect toeach of the three means A, G and H oddly coincides with another type
of symmetry (with respect to the same means) which we call
functional
symmetry. The problem of describing the set of all means realizing that
In the second section, we define on
MD a structure of metric space which
turns out to be a closed ball with center A and radius 1
/2. We then use
the group structure to calculate the distance between two means on
D;this permits us in particular to establish a simple characterization of the
In the third section, we introduce the concept of
functional middle of
two mean functions on
D which generalizes in a natural way the arithmetic-geometric mean, so that the latter is the functional middle of the arithmetic
and geometric means. We establish two sufficient conditions for the existence
and uniqueness of the functional middle of two means. The first one uses the
metric space structure of
MD by imposing on the two means in question
the condition that the distance between them is less than 1. The second
requires the two means in question to be continuous on
D. In the proof ofthe latter one, axiom (iii) plays a vital role.
2. An abelian group structure on
MD
Given a nonempty symmetric domain
D in R2, we denote by
AD the set
of asymmetric maps on
D, that is, maps
f :
D → R, satisfying
f (
x, y) =
−f (
y, x)
(
∀(
x, y)
∈ D)
.
It is clear that (
AD, +) (where + is the usual addition of maps from
D intoR) is an abelian group with neutral element the null map.
Now, consider
φ :
MD → R
D defined by: (
log
−M(
x,y)
−x
D , ∀(
x, y)
∈ D :
φ(
M )(
x, y) := 0
The axioms (i)-(iii) ensure that
−M(
x,y)
−x (for
x ̸=
y) is well-defined and
Theorem 2.1. We have φ(
MD) =
AD. In addition, the map φ :
M →
φ(
M )
is a bijection from MD to AD and its inverse is given by
∀f ∈ AD , ∀(
x, y)
∈ D :
φ−1(
f)(
x, y) =
Proof. Axiom (i) ensures that for all
M ∈ MD, we have
φ(
M)
∈ AD.
Next, if
f is an asymmetric map on
D, we easily verify that
M :
D → Rdefined by
M (
x, y) :=
x+
yef(
x,y) (
∀(
x, y)
∈ D) is a mean on
D and
φ(
M ) =
f .
Since obviously
φ is injective, the proof is finished.
We now transport, by
φ, the abelian group structure (
AD, +) onto
MD,
that is, we define on
MD the following composition law
∗:
∀M1
, M2
∈ MD :
M1
∗ M2 =
φ−1 (
φ(
M1) +
φ(
M2))
.
So (
MD, ∗) is an abelian group and
φ is a group isomorphism from (
MD, ∗)to (
AD, +). Furthermore, since the null map on
D is the neutral element of(
AD, +) and
φ−1(0) = A, the arithmetic mean A is the neutral element of(
MD, ∗).
By calculating explicitly
M1
∗ M2 (for
M1
, M2
∈ MD), we obtain:
Proposition 2.2. The composition law ∗ on MD is defined by:
{
x(
M1(
x,y)
−y)(
M2(
x,y)
−y)+
y(
M1(
x,y)
−x)(
M2(
x,y)
−x)
if x ̸=
y
(
M1(
x,y)
−x)(
M2(
x,y)
−x)+(
M1(
x,y)
−y)(
M2(
x,y)
−y)
for M1
, M2
∈ MD and (
x, y)
∈ D.
Now, it is easy to verify that the images of the geometric and harmonic
means under the isomorphism
φ are given by
(
∀(
x, y)
∈ (0
, +
∞)2)
,
φ(H)(
x, y) = log
x − log
y
(
∀(
x, y)
∈ (0
, +
∞)2)
.
From (2.2) and (2.3), we see that
φ(G) and
φ(H) (and trivially also
φ(A))
have a particular form: each can be written as
h(
x)
− h(
y), where
h is a realfunction of one variable.
To generalize, we define a
normal mean as a mean function
M :
I2
→ R
(
I ⊂ R) such that
φ(
M ) has the form
h(
x)
− h(
y) for some map
h :
I → R.
Equivalently, a
normal mean function is a function
M :
I2
→ R (
I ⊂ R)which can be written as
xP (
x) +
yP (
y)
P (
x) +
P (
y)
where
P :
I → R is a positive function on
I.
Study of some symmetries on the group (
MD, ∗)
. We are now inter-
ested in the symmetric image of a given mean
M1 with respect to another
mean
M0 via the group structure (
MD, ∗). Denote by
SM the symmetry
with respect to
M0 in the group (
MD, ∗), defined by
∀M1
, M2
∈ MD :
SM (
M
1) =
M2
⇐⇒ M1
∗ M2 =
M0
∗ M0
.
Using the group isomorphism
φ, we obtain by a simple calculation the ex-
Proposition 2.3. For any M0
, M1
∈ MD,
1
− x)(
M0
− y)2
− y(
M0
− x)2(
M1
− y)
(
M1
− x)(
M0
− y)2
− (
M0
− x)2(
M1
− y)
where, for simplicity, we have written M0
for M0(
x, y)
, M1
for M1(
x, y)
andSM (
M
As an application, we get the following immediate corollary:
Corollary 2.4. For any M ∈ MD, we have:
(1)
SA(
M ) =
x +
y − M = 2A
− M .
(2)
SG(
M ) =
xy = G2
(when D ⊂ (0
, +
∞)2
).
(when D ⊂ (0
, +
∞)2
).
(4)
SH =
SG
◦ SA
◦ SG
.
Now, we are going to define another symmetry on
MD (for
D of a certain
form), independent of the group structure (
MD, ∗). This new symmetry is
defined by solving a functional equation but it curiously coincides, in many
cases, with the symmetry defined above.
Definition 2.5. Let
I be a nonempty interval of R,
D =
I2 and
M0,
M1
and
M2 be three mean functions on
D such that
M1 and
M2 take their
values in
I. We say that
M2 is the
functional symmetric mean of
M1 with
respect to
M0 if the following functional equation is satisfied:
M0(
M1(
x, y)
, M2(
x, y)) =
M0(
x, y)
(
∀(
x, y)
∈ D)
.
Equivalently, we also say that
M0 is the
functional middle of
M1 and
M2.
According to axiom (iii), it is immediate that if the functional symmetric
mean exists then it is unique. This justifies the following notation:
Notation 2.6. Given two mean functions
M0 and
M1 on
D =
I2 with values
in
I (where
I is an interval of R), we denote by
σM (
M
symmetric mean (if it exists) of
M1 with respect to
M0.
A simple calculation establishes the following:
Proposition 2.7. Let M be a mean function on a suitable symmetric do-
σA(
M ) =
x +
y − M,
(
for D ⊂ (0
, +
∞)2)
,
(
for D ⊂ (0
, +
∞)2)
.
The remarkable phenomenon of the coincidence of the two symmetries
defined on
MD in the particular cases of the means A, G and H leads tothe following question:
Open question. For which mean functions
M on
D = (0
, +
∞)2 the two
symmetries with respect to
M (in the sense of the group law introduced on
MD and in the functional sense) coincide?
Example. Using the definition of AGM (see Section 1), it is easy to show
that A and G are symmetric in the functional sense with respect to AGM.
Throughout this section, we fix a nonempty symmetric domain
D in R2.
We suppose that
D contains at least one point (
x0
, y0) of R2 such that
x0
̸=
y0 (otherwise
MD reduces to a unique element). For all couples (
M1
, M2)
1(
x, y)
− M2(
x, y)
Proposition 3.1. The map d
of M2
D into [0
, +
∞]
is a distance on MD.
In addition, the metric space (
MD, d)
is the closed ball with center A
(the
arithmetic mean) and radius 1
.
Proof. First let us show that d(
M1
, M2) is finite for all
M1
, M2. For all(
x, y)
∈ D,
x ̸=
y, the numbers
M1(
x, y) and
M2(
x, y) lie in the interval[min(
x, y)
, max(
x, y)], so
|M1(
x, y)
− M2(
x, y)
| ≤ max(
x, y)
− min(
x, y) =
|x − y|.
1(
x, y)
− M2(
x, y)
that is, d(
M1
, M2)
≤ 1. Further, since the three axioms of a distance aretrivially satisfied, d is a distance on
MD.
Now, given
M ∈ MD, let us show that d(
M, A)
≤ 1. For all (
x, y)
∈ D,
x ̸=
y, the number
M (
x, y) lies in the closed interval with endpoints
x and
y, so
|M(
x, y)
− A(
x, y)
| ≤ max (
x − A(
x, y)
, y − A(
x, y))
x − x +
y , y − x +
y
M (
x, y)
− A(
x, y)
that is, d(
M, A)
≤ 1, as required.
Remark 3.2. Given
M1
, M2
∈ MD, since the map (
x, y)
→ M1(
x,y)
−M2(
x,y)
is obviously asymmetric (on the set
{(
x, y)
∈ D :
x ̸=
y}), we also have
1(
x, y)
− M2(
x, y)
We now establish a practical formula for the distance between two mean
Proposition 3.3. Let M1
and M2
be two mean functions on D. Set f1 =
φ(
M1)
and f2 =
φ(
M2)
. Then
(
x,y)
∈D (
ef1 + 1)(
ef2 + 1)
Proof. Using (2.1), for all (
x, y)
∈ D we have
M1(
x, y) =
φ−1(
f1)(
x, y) =
x+
yef1(
x,y) and
M
2(
x, y) =
φ−1(
f2)(
x, y) =
x+
yef2(
x,y)
As an application, we get the following immediate corollary:
Corollary 3.4. Let M be a mean function on D and f :=
φ(
M )
. Then,
setting s := sup
D f ∈ [0
, +
∞]
, we have
(We naturally suppose that es−1 = 1
when s = +
∞).
In particular, the mean M lies on the boundary of MD (that is, on the circlewith center A
and radius 1
) if and only if sup
Examples: The two means G and H lie on the boundary of
MD.
4. Construction of a functional middle of two means
Let
I ⊂ R (
I ̸=
∅) and let
D =
I2. The aim of this section is to
prove, under some
regularly conditions, the existence and uniqueness of the
functional middle of two given means
M1 and
M2 on
D; that is, the existenceand uniqueness of a new mean
M on
D satisfying the functional equation
M (
M1
, M2) =
M.
In this context, we obtain two results which only differ in the condition
imposed on
M1 and
M2. The first one requires d(
M1
, M2)
̸= 1 (where d isthe distance defined in Section 3) while the second requires
M1 and
M2 tobe continuous on
D (by taking
I an interval of R). Notice further that ourway of establishing the existence of the functional middle is constructive
and generalizes the idea of the AGM mean. Our first result is the following:
Theorem 4.1. Let M1
and M2
be two mean functions on D =
I2
, with
values in I and such that d(
M1
, M2)
< 1
. Then there exists a unique mean
function M on D satisfying the functional equation
M (
M1
, M2) =
M.
Moreover, for all (
x, y)
∈ D, M (
x, y)
is the common limit of the two realsequences (
xn)
and (
y
x0 =
x , y0 =
y,
xn+1 =
M1(
xn, yn)
Proof. Let
k := d(
M1
, M2). By hypothesis, we have
k < 1. Let (
xn) and
(
yn) be as in the statement and let (
u
un := min(
xn, yn) and
vn := max(
xn, yn) (
∀n ∈ N)
.
un+1 = min(
xn+1
, yn+1) = min(
M1(
xn, yn)
, M2(
xn, yn))
≥ min(
xn, yn) =
un
(because
M1(
xn, yn)
≥ min(
xn, yn) and
M2(
xn, yn)
≥ min(
xn, yn)).
Similarly, for all
n ∈ N,
vn+1 = max(
xn+1
, yn+1) = max(
M1(
xn, yn)
, M2(
xn, yn))
≤ max(
xn, yn) =
vn.
|vn+1
− un+1
| =
|max(
xn+1
, yn+1)
− min(
xn+1
, yn+1)
|
=
|M1(
xn, yn)
− M2(
xn, yn)
|≤ k|xn − yn|
|vn − un| ≤ kn|v0
− u0
|
It follows (since
k ∈ [0
, 1)) that (
vn − un) tends to 0 as
n tends to infinity.
Thus the bounded monotonic sequences (
un) and (
v
un ≤ xn ≤ vn and
un ≤ yn ≤ vn
also converge to the same limit. Denote the
common limit of the four sequences by
M (
x, y).
Now we show that the map
M :
D → R just defined is a mean function
on
D and satisfies
M (
M1
, M2) =
M . First we check the three axioms of amean function.
(i) Given (
x, y)
∈ D, on changing (
x, y) to (
y, x) in the definition of the
sequences (
xn) and (
y
, they remain unchanged except their first terms
(since
M1 and
M2 are symmetric). So,
M (
x, y) =
M (
y, x)
(
∀(
x, y)
∈ D)
.
(ii) Given (
x, y)
∈ D, since the corresponding sequences (
un) and (
v
are respectively non-decreasing and non-increasing and since
M (
x, y) is their
common limit, we have
u0
≤ M (
x, y)
≤ v0, that is,
min(
x, y)
≤ M (
x, y)
≤ max(
x, y)
.
(iii) Fix (
x, y)
∈ D. Suppose that
M (
x, y) =
x and, towards a contradiction,
x ̸=
y. Since
M1 and
M2 are means, we have (by axiom (iii))
M1(
x, y)
̸=
x and
M2(
x, y)
̸=
x.
Then
M (
x, y) =
x = min(
x, y) =
u0. So (
un) is non-decreasing and con-
verges to
u0. It follows that (
un) is necessarily constant and in particular
min(
M1(
x, y)
, M2(
x, y)) =
x,
Then
M (
x, y) =
x = max(
x, y) =
v0. So (
vn) is non-increasing and con-
verges to
v0. It follows that (
vn) is constant and in particular
v
max(
M1(
x, y)
, M2(
x, y)) =
x,
which again contradicts (4.1), proving (iii).
To prove
M (
M1
, M2) =
M , note that changing in the definition of (
xn)
n
and (
yn) the couple (
x, y) of
D to (
M
1(
x, y)
, M2(
x, y)) just amounts to shift-
ing these sequences (namely we obtain (
xn+1) instead of (
x
instead of (
yn) ). Consequently, the common limit (which is
M (
x, y)) re-
M (
M1(
x, y)
, M2(
x, y)) =
M (
x, y)
.
It remains to show that
M is the unique mean satisfying the func-
tional equation
M (
M1
, M2) =
M . Let
M ′ be any mean function satis-fying
M ′(
M1
, M2) =
M ′ and fix (
x, y)
∈ D. We associate to (
x, y) thesequence (
xn, yn)
n∈N in the statement of the theorem. Using the relation
M ′(
M1
, M2) =
M ′, we have
M ′(
x, y) =
M ′(
x1
, y1) =
M ′(
x2
, y2) =
· · · =
M ′(
xn, yn) =
· · ·
But since
M ′ is a mean, it follows that for all
n ∈ N,
min(
xn, yn)
≤ M ′(
x, y)
≤ max(
xn, yn)
,
M ′(
x, y) =
M (
x, y)
,
From Theorem 4.1, we derive the following corollary:
Corollary 4.2. Let M
be a mean function on D =
I2
, with values in I.
Then there exists a unique mean on D satisfying the functional equation
=
M (
x, y) (
∀(
x, y)
∈ D)
.
In addition, for all (
x, y)
∈ D, M (
x, y)
is the common limit of the two realsequences (
xn)
and (
y
Proof. Since the metric space (
MD, d) is the closed ball with center A andradius 1
/2 (see Proposition 3.1), we have d(M
, A)
≤ 1
/2
< 1. The corollarythen immediately follows from Theorem 4.1.
In the following theorem, we establish another sufficient condition for
the existence and uniqueness of the functional middle of two means.
Theorem 4.3. Suppose that I is an interval of R
and let M1
and M2
be
two mean functions on D =
I2
with values in I. Suppose that M1
and M2
are continuous on D. Then there exists a unique mean function M on D
satisfying the functional equation
M (
M1
, M2) =
M.
In addition, for all (
x, y)
∈ D, M (
x, y)
is the common limit of the two realsequences (
xn)
and (
y
Proof. Fix (
x, y)
∈ D and define (
un) and (
v
are convergent. Let
u =
u(
x, y) and
v =
v(
x, y) denote their respective limits (so
u and
v lie in [
u0
, v0] =[min(
x, y)
, max(
x, y)]
⊂ I).
Now, since
M1 and
M2 are symmetric on
D, we have, for all
n ∈ N,
xn+1 =
M1(
un, vn) and
yn+1 =
M2(
un, vn)
.
By continuity, the sequences (
xn) and (
y
respective limits are
M1(
u, v) and
M2(
u, v). Letting
n → ∞ in
xn+1 =
M1(
xn, yn), we obtain
M1(
u, v) =
M1 (
M1(
u, v)
, M2(
u, v))
,
M1(
u, v) =
M2(
u, v)
.
converge to the same limit. Denoting by
M (
x, y) this
common limit, we show as in the proof of Theorem 4.1 that
M is a mean
function on
D and that it is the unique mean on
D which satisfies thefunctional equation
M (
M1
, M2) =
M .
[Bor] J. M. Borwein and P. B. Borwein,
Pi and the AGM, (A study in
Analytic Number Theory and Computational Complexity), Wiley., New
Department of Mathematics, University of B´
E-mail address:
[email protected]
Source: http://univ-bejaia.dz/staff/photo/pubs/1104-532-means.pdf
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