On a method of construction of new means with applications

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R E S E A R C H

Open Access

On a method of construction of new means

with applications

Mustapha Raïssouli

1,2*

and József Sándor

3

*Correspondence:

raissouli_10@hotmail.com

1Faculty of Science, Department of

Mathematics, Taibah University, P.O. Box 30097, Al Madinah Al Munawwarah, Zip Code 41477, Kingdom of Saudi Arabia

2Faculty of Sciences, Department of

Mathematics, Moulay Ismail University, Meknes, Morocco Full list of author information is available at the end of the article

Abstract

In the present paper, we would like to introduce a simple transformation for bivariate means from which we derive a lot of new means. Relationships between the standard means are also obtained. A simple link between the Stolarsky mean and the Gini mean is given. As applications, this transformation allows us to extend some means from two to three or more arguments.

MSC: 26E60

Keywords: means; decomposable means; mean-inequalities; Stolarsky mean; Gini mean

1 Introduction and basic notions

In the recent past, the theory of means has been the subject intensive research. It has proved to be a useful tool for theoretical viewpoint as well as for practical purposes. For the definition of a mean, various statements, more or less different, can be found in the literature; see [] and the references therein. Throughout this paper, we adopt the following definition.

Definition . A functionm: (,∞)×(,∞) –→(,∞) is called a mean if

a,b>  min(a,b)≤m(a,b)≤max(a,b).

From this, it is clear that every mean is with positive values and reflexive, that is,m(a,a) =

afor eacha> . The maps (a,b)–→min(a,b) and (a,b)–→max(a,b) are (trivial) means which will be denoted byminandmax, respectively. The standard examples of means are given as follows:

A:=A(a,b) =a+b

 ; G:=G(a,b) =

ab; H:=H(a,b) = ab

a+b;

L:=L(a,b) = ba

lnb–lna, L(a,a) =a;

I:=I(a,b) = 

e

bb

aa /(ba)

, I(a,a) =a;

S:=S(a,b) =aa/(a+b)bb/(a+b); C:=C(a,b) = a+b

a+b ;

P:=P(a,b) = ba

arctan√b/aπ =

ba

arcsinbb+aa, P(a,a) =a

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and are known as the arithmetic, geometric, harmonic, logarithmic, identric, weighted geometric, contraharmonic and (first) Seiffert means, respectively.

The set of all means can be equipped with a partial ordering, called a point-wise order, defined bym≤mif and only ifm(a,b)≤m(a,b) for everya,b> . We writem<mif and only ifm(a,b) <m(a,b) for alla,b>  witha=b. With this, the above means satisfy the known chain of inequalities

min<H<G<L<P<I<A<S<C<max.

A meanmis symmetric ifm(a,b) =m(b,a) and homogeneous ifm(ta,tb) =tm(a,b) for alla,b,t> . The above means are all symmetric and homogeneous. However, the mean (a,b)–→(a+b)/ is (homogeneous) not symmetric, while (a,b)–→ln(ea+eb) is (sym-metric) not homogeneous. The mean (a,b)–→ln(ea+eb) is neither symmetric nor homo-geneous.

A meanmis called monotone if (a,b)–→m(a,b) is increasing inaand inb, that is, if

a≤a(resp.b≤b), thenm(a,b)≤m(a,b) (resp.m(a,b)≤m(a,b)). There are many means which are not monotone. For example, it is easy to see that the meansA,G,H,L

are monotone butCis not. However, extending the above definitions of reflexivity and monotonicity from a mean to a general binary map, the following result is of interest [, ].

Proposition . Let m be a monotone and reflexive map.Then m is a mean.

For a given meanm, we setm*(a,b) = (m(a–,b–))–, and it is easy to see thatm*is also a mean, called the dual mean ofm. Ifmis homogeneous, then so ism* withm*(a,b) =

ab/m(b,a). Ifmis symmetric and homogeneous, then so ism*, and in this case, we have

m*(a,b) =ab/m(a,b). Every meanmsatisfiesm**:= (m*)*=m, and ifmandmare two means such thatm≤m, thenm*≥m*. One can check thatmin*=maxandmax*=min. Further, the arithmetic and harmonic means are mutually dual (i.e.,A*=H,H*=A) and the geometric mean is self-dual (i.e.,G*=G). The dual of the logarithmic and identric means has been studied by the second author in [].

The following inequalities are immediate from the above:

min<C*<S*<H<I*<P*<L*<G<L<P<I<A<S<C<max.

Remark . Letk: ], +∞[ –→], +∞[ be a monotone continuous function and denote by k–its inverse function. An extension of the dual mean can be given by m

k(a,b) =

k–(m(k(a),k(b))). It is easy to verify that mk is a mean. Fork(x) = /x, we obtain the classical dual. If we choose k(x) =xrwithr> , then we get the following generalized dual m*r(a,b) = (m(ar,br))–/r. Ifm is symmetric and homogeneous, thenm*r(a,b) =

ab/(m(ar,br))/r.

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f(x)≤max(x, ). Clearly,f() = , and if, moreover,mis symmetric, thenf(x) =xf(/x) for everyx> . It is obvious that a meanmis monotone if and only if its associated function is increasing. For example, the contraharmonic meanCis not monotone because its as-sociated functionf(x) = (x+ )/(x+ ) satisfies (x+ )f(x) =x+ x– , and it is easy to see thatf is not increasing for allx> , but only forx>√ – .

Now, let us observe the next question : under which sufficient condition a given function

f is the associated function of a certain mean? The following result gives an answer to this situation; see [], Remark .

Proposition . Let f : ], +∞[ –→],∞[ be a function such that min(x, ) ≤f(x)≤

max(x, )for every x> .Then m(a,b) =bf(a/b)defines a(homogeneous)mean.If, more-over,f is increasing with f(x) =xf(/x)for each x> ,then m is monotone symmetric.

The next result is also of interest to our present paper; see [, ].

Proposition . Let m be a symmetric and homogeneous mean having a strictly increas-ing associated function f.Then f*,the associated function to the dual mean m*,is strictly

increasing,too.

By virtue of the relationm**=m(and sof**=f), the result of the above proposition is in fact an equivalence. It follows that the associated function of the dualC*will be not always increasing since that ofCis not.

In the literature, there are some families of means which include the above familiar means. Precisely, letpandqbe two real numbers. The Stolarsky meanEp,qof order (p,q) is defined for alla,b>  such thata=bas well by

Ep,q(a,b) := ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

(pqbbqpaaqp)/(qp) ifpq(pq)= ,

exp(–p+aplnaapbbpplnb) ifp=q= , (plnbbplnapa)/p ifp= ,q= ,

ab ifp=q= 

withEp,q(a,a) =a, while the Gini meanGp,qof order (p,q) is defined by

Gp,q(a,b) := ⎧ ⎪ ⎨ ⎪ ⎩

(aapq++bbqp)/(qp) ifp=q,

exp(aplnaap++bbpplnb) ifp=q= ,

ab ifp=q= .

Clearly, the meansEp,qandGp,qare symmetric and homogeneous. Further,Ep,qandGp,q are symmetric inpandq. It is worth mentioning that

Ep,p(a,b) =G,p(a,b) =

(ap+bp)/p ifp= ,

ab ifp= 

is called the power mean of orderp. It is well known thatEp,qandGp,qare strictly increasing with bothpandq. For particular choices ofpandq, we find again

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2 A mean-transformation The following formula

L(a,b) =

n=

Aa/n,b/n:=

n=

a/n+b/n

 ()

is well known in the literature []; see also [, ] for another approach. The aim of this section is to observe () in a general point of view. The second side of () is an infinite product involving the arithmetic meanAand the geometric sequence (/n). Let us try to replaceAby an arbitrary meanmand (/n) by a general sequence. Two points arise from this situation: the corresponding infinite product should be convergent and we need its value to define a mean. By virtue of the double inequality

min(a,b)≤m(a,b)≤max(a,b),

which is valid for every meanm, the infinite product∞n=m(atn,btn) will be convergent provided that the sequence (tn) has a constant sign and the series

n=tnis convergent. We must choose (tn) positive and satisfying∞n=tn= , and so we have

min(a,b)≤

n=

matn,btnmax(a,b). ()

Summarizing the above, we may state the following.

Definition . Lett= (tn)n≥ be a positive sequence such that

n=tn=  andmbe a mean. For alla,b> , define

mπt(a,b) =

n=

matn,btn ()

which we call thet-transformation ofm.

We explicitly notice that the convergence of the infinite product in () is shown by the double inequality ().

The elementary properties of the mean-transformationm–→mπt are summarized in the next result.

Proposition . Let m,m,mbe given means and t= (tn)be a positive sequence such that

n=tn= .Then the following assertions are fulfilled:

(i) mπtis a mean.

(ii) Ifmis homogeneous(resp.symmetric,monotone),then so ismπtand the associated functionfπttomπtis given by

x>  fπt(x) =

n=

fxtn,

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(iii) (mπt)*s= (m*s)πt for eachs> ,where*

sdenotes the generalized dual mean(see Remark.).

(iv) m<m=⇒t<m

πt  .

(v) ∀α∈[, ] (m–α

)πt= (m

πt  )–α(m

πt )α.

Proof

(i) It is already proved by ().

(ii) The symmetry ofmπtfrom that ofmis obvious, while the monotonicity ofmπt fol-lows from the fact thatmis monotone andmπtis defined as (infinite) product of positive terms. Now, assume thatmis homogeneous. By definition, we have for alla,b,α> 

mπt(αa,αb) =

n=

m(αa)tn, (αb)tn=

n=

αtn

n=

matn,btn.

The homogeneity ofmπt follows sincen=αtn=α

n=tn=α.

The homogeneity ofmandmπtimplies, with (), that the associated function ofmπtis given by

x>  fπt(x) =mπt(x, ) =

n=

mxtn, =

n=

fxtn.

(iii) By definition, we have successively

mπt*s

(a,b) =mπtas,bs–/s=

n=

mastn,bstn –/s

=

n=

matns,btns–/s=

n=

m*satn,btn=m*sπt.

(iv) and (v) are not difficult. Details are omitted for the reader as a simple exercise.

We now present the following examples. In all these examples, the sequencet= (tn) is as in the above.

Example . It is easy to verify thatGπt=G. Otherwise, we have

Aπt(a,b) =

n=

atn+btn

 ≤

n=

a+b

tn

=a+b

 =A(a,b),

with strict inequality fora=bsince  <tn<  and the mapx–→is strictly concave for  <α< . We then haveAπt<A, and by Proposition .(iii) and (iv), we deduceH<Hπt. In summary, we have

H<Hπt<Gπt=G<Aπt<A. ()

Example . According to the definition, with (), we have

Lπt(a,b) =

n=

Latn,btn=

n=

i=

atn/i+btn/i

 ≤ ∞ i= ∞ n=

a/i+b/i

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for the same reason as in the above example. Then

Lπt(a,b)

i=

a/i +b/i

n=tn =

i=

a/i +b/i

 =L(a,b).

Further, the above inequalities are strict fora=b. It follows thatLπt<Land so

L*<L*πt<Gπt=G<Lπt<L. ()

Example . It is well known that

I(a,b) =exp

log( –s)a+sbds.

With this we have

Iπt(a,b) =

n=

exp

log( –s)atn+sbtnds

=exp

n= 

log( –s)atn+sbtnds

≤exp

n= 

log( –s)a+sbtnds

.

We deduce thatIπt(a,b)I(a,b) with strict inequality fora=b. It follows thatIπt<Iand so

I*<I*πt<Gπt=G<Iπt<I. ()

Reduction of the three chains of inequalities (), () and () in one chain does not appear to be obvious. However, for the particular casetn= /n, the above three chains can be reduced into one chain; see Corollary . in Section  below.

Example . Letrbe a real number such that  <r< . Settingtn= ( –r)rn–, we have

n=tn= , and so the hypotheses of the above definition are satisfied. In this case, we write

mπt(a,b) :=mπr(a,b) =

n=

ma(–r)rn–,b(–r)rn–.

The situation of this example will be developed below.

In what precedes, starting from a given meanm, we have defined a new class of means

mπtprovided that the positive sequencet= (t

n) satisfies∞n=tn= . Our procedure can be recursively continued: forkpositive sequencest= (tn,),t= (tn,), . . . ,tk= (tn,k) such that∞n=tn,i=  fori= , , . . . ,k, we can define the following:

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where we setTk:= (t,t, . . . ,tk), or explicitly,

mπTk(a,b) =

n=

maTn,k,bTn,k,

withTn,k:= k

i=tn,i. In particular, iftn,i= ( –ri)rni–with  <ri<  fori= , , . . . ,k, we obtain

mπRk(a,b) =

n=

maRn,k,bRn,k,

where we put

Rk:= (r,r, . . . ,rk) and Rn,k:= k

i= ( –ri

k

i=

ri n–

.

In what follows, we will explore this latter situation in more detail. For the sake of sim-plicity, we restrict ourselves to the caser= /. The general case can be stated in a similar manner and we leave it to the reader. Precisely, we put the following.

Definition . Letmbe a given mean. We set=m,mπ=mπ and, for all integer

k≥,

mπk+=mπkπ=mππk.

Clearly,k is a mean for allk. The next example may be stated.

Example . We haveGπk=Gfor eachk. Formula () written in a brief formL=Aπ, with a mathematical induction, yields

Lπk–(a,b) =Aπk(a,b) =

n,n,...,nk=

Aa/Nk,a/Nk,

where we putNk:=ki=nifor everyk≥.

If we apply the above definition to the standard means, we obtain the following iterative inequalities:

G<Lπk<Pπk<Iπk<Aπk=Lπk–<Pπk–<Iπk–<Aπk–=Lπk–<· · ·. ()

A sequence (mk)kof means will be called point-wise convergent (in shortp-convergent) if, for alla,b> , the real sequence (mk(a,b))kconverges. Settingm∞(a,b) =limkmk(a,b), it is easy to see thatm∞ is a mean. Similarly, we define the point-wise monotonicity of

(mk)k. By virtue of the double inequality

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we deduce that everyp-increasing (resp.p-decreasing) sequence (mk)kisp-convergent. This together with inequalities () implies that the mean-sequences (Lπk)

k, (Pπk)k, (Iπk)k and (Aπk)

kare decreasinglyp-convergent. Then, what are their limits? The answer to this latter question will be presented in the next section after stating some needed results.

3 Study of a special case

As already pointed before, this section will be devoted to studying the family of meansmπt for the casetn= ( –r)rn–with  <r< . We recall the following formula:

mπt(a,b) :=mπr(a,b) =

n=

ma(–r)rn–,b(–r)rn–, ()

which we call ther-decomposition ofm. Ifr= /, we simply write, that is,

(a,b) :=

n=

ma/n,b/n.

In what follows, we will see that the meanmπrsatisfies good properties, the first of which is announced as well.

Proposition . With the above,the following assertions hold true:

(i) For alla,b> and <r< ,we have

mπra/(–r),b/(–r)=mπrar/(–r),br/(–r)m(a,b). ()

In particular(forr= /),we obtainmπ(a,b) =mπ(a,b)m(a,b).

(ii) Assume thatmis homogeneous and letf andfπrbe the associated functions ofm andmπr,respectively.Then,for everyx> ,one has

fπrx/(–r)=fπrxr/(–r)f(x). ()

In particular(forr= /),we obtainfπ(x) =fπ(x)f(x).

Proof

(i) By () we have successively

mπra/(–r),b/(–r)=

n=

marn–,brn–

=m(a,b)

n=

marn–,brn–=m(a,b)

n=

marn,brn.

The desired result follows after a simple manipulation.

(ii) Follows from the fact thatm(a,b) =bf(a/b) when combined with (i). The proof is

completed.

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Corollary . Let m be a symmetric and homogeneous mean such that (mπk) k is

p-convergent.Then its limit is mπ=G,the geometric mean.In particular, (Lπk)

k, (Pπk)k, (k)

kand(Aπk)kare decreasingly p-convergent to the same limit G.

Proof If (mπk)

kisp-convergent, then the sequence (fπk)kis alsop-convergent, wherefπk is the associated function of mπk. That is, there exists a function (corresponding to a symmetric homogeneous mean) such thatg(x) =limkfπk(x) for eachx> . According to Proposition .(ii) and the recursive definition ofmπk, we can write

x>  fπkx=fπk(x)fπk–(x).

Then we deduce, by lettingk→ ∞,

x>  gx=g(x)g(x) =g(x).

It follows thatg(x) =√x, which is the associated function ofG, in this way proving the first part of the proposition. For the second part, as already pointed before, the sequences (Lπk)

k, (Pπk)k, (Iπk)kand (Aπk)karep-decreasing. It follows that theyp-converge and by the first part they haveGas a common limit. The proof of the proposition is complete.

Corollary . Let mand mbe two homogeneous means such that mπr=m

πr

for a certain

r∈], [.Then m=m.

Proof Letf andg be the associated functions ofmandm, respectively. Assume that

mπr  =m

πr

 for somer∈], [, thenfπr(x) =gπr(x) for allx>  and so

fπrxr/(–r)=gπrxr/(–r).

It follows that

fπrxr/(–r)f(x) =gπrxr/(–r)f(x),

and by Proposition .(ii), we obtain

fπrx/(–r)=gπrxr/(–r)f(x).

Sincefπr(x) =gπr(x) for allx> , then

gπrx/(–r)=gπrxr/(–r)f(x),

or by Proposition .(ii) again,

gπrxr/(–r)g(x) =gπrxr/(–r)f(x).

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Now, let us observe the next question: Does () (resp ()) characterizemπr for a given mean (resp. homogeneous mean)m? For the sake of simplicity, we assume thatmis ho-mogeneous and we will prove the following theorem.

Theorem . Let m be a homogeneous mean with its associated function f and let <r< .

Assume that there exists a continuous function frsuch that fr() = and

x>  fr

=fr

–f(x), withα:= /( –r) > . ()

Then fris the associated function of mπrdefined by().

Proof Assume that () holds. It is equivalent to

x>  fr(x) =fr

xrfx–r. ()

But we can apply this to write

x>  fr

xr=fr

xrfxrr,

which when substituted in () yields

x>  fr(x) =fr

xrfxr(–r)fx–r.

By a simple mathematical induction, we can establish that for all integerN≥ we have

x>  fr(x) =fr

xrN

N

n=

fx(–r)rn–.

Since  <r< , thenrN tends to  whenNgoes to +. LettingN+in the previous equality, with the fact thatfris continuous andfr() = , we obtain

fr(x) =

n=

fx(–r)rn–,

which, following Proposition .(ii), is the associated function ofmπr, in this way proving

the desired result.

We now present some examples illustrating the above. In all these examples,randαare such that  <r< ,α:= /( –r) > .

Example . Letf(x) =√xbe the associated function ofG. We havef()/f(xα–) =x, that is,Gπr=Gfor all  <r< , which has been already pointed before.

Example . Letf(x) = (x+ )/ be the associated function ofA. Clearly, we have

f()

f(–)=

+ 

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which is the associated function of–,α. Otherwise,Gαπr–,α=A. In particular, withr= /

(and soα= ), we obtain=A. By Proposition ., we deduceC*π=H.

Example . Letf(x) = (x– )/lnx,x> , withf() =  be the associated function ofL. A simple computation leads to

f()

f(–)=

α– 

α

– 

–– ,

which is the associated function of–,α, that is,E πr

α–,α=L. In particular (ifr= /,α= ),

we find=Land soHπ=L*.

Example . Letf(x) =e–xx/(x–), x> , with f() =  be the associated function of I. Similarly, we obtain

f()

f(–)=exp

αxα

– 

(α– )–

–– 

lnx

.

To find out if this latter function corresponds to a certain homogeneous mean for allα>  is left to the reader. For the particular caser= /,α= , the answer to this question is obviously positive sincef(x)/f(x) =xx/(x+)is the associated function ofS. We then have

=Iand soS*π=I*.

Example . Letf(x) = x–

arcsinxx–+,x> , withf() =  be the associated function ofP. Ob-viously, we have

f(x)

f(x) = (x+ )

arcsinx–

x+

arcsinx–

x+

.

The fact that this latter function is the associated function of a certain homogeneous mean does not appear to be obvious. See the section below for a general point of view.

4 Decomposable means

For the sake of convenience, for concrete examples, we may introduce the following no-tion.

Definition . Letmbe a mean such that there exists a meanm and somer∈], [ satisfyingm=mπr

 . Then we say thatmis (m,r)-decomposable. In the caser= /, we simply saymism-decomposable.

Proposition . With the above,the following properties are met:

(i) Ifmis(m,r)-decomposable,then for alls> ,the generalized dual meanm*s is (m*s

,r)-decomposable.

(ii) Ifmis(m,r)-decomposable andmis(m,r)-decomposable,then for allα∈[, ],

m–αmαis(m–α

,r)-decomposable.

Proof

(i) Comes from Proposition .(ii).

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Now, we will illustrate the above notions and results by some examples.

Example . We have already seen thatGπr =Gfor eachr], [. We say thatGis self-decomposable. Also, we can see thatminπr=minandmaxπr=maxfor everyr], [.

Example . The relationship (), written in a brief form =L, says that L is A

-decomposable. By Proposition .(i), we deduce thatL*isH-decomposable. We have also seen =I, that is,IisS-decomposable and soI*isS*-decomposable. We leave it to the reader to see thatAisC-decomposable and soHisC*-decomposable. Other examples, in a more general point of view, will be stated below.

Example . In this example, we are interested in the link between two double-power means, namely the Stolarsky and Gini means. By virtue of this interest, we state its content as an explicit result from which we will derive some interesting consequences.

Theorem . For all real numbers p,q,the Stolarsky mean Ep,qis Gp,q-decomposable,that

is,the relationship

Ep,q= (Gp,q)π ()

holds for all real numbers p and q.

Proof Let

fp,q(x) =

xq– 

xp–  /(qp)

, x> , withfp,q() = 

be the associated function ofEp,q, with convenient forms forp=q=  andp= ,q= . Using Proposition ., we obtain

p,q(x)

p,q(x) =

xq– 

xp– .

xp– 

xq–  /(qp)

,

which after simplification remains

p,q(x)

p,q(x) =

xq+ 

xp+  /(qp)

.

This latter function is that associated toGp,q, that is,Ep,q=Gπp,q. The proof is completed.

Corollary . The following chain of inequalities holds true:

=G<Lπ<Pπ<Iπ<Aπ=L<P<Sπ=I<Cπ=A<S<C. ()

Proof According to the above theorem, we immediately deduce the following:

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SinceG<L<P<I<A, Proposition .(iii) yields =G<Lπ <Pπ <Iπ<Aπ =L. The

desired inequalities follow by combining the two above chains of inequalities.

Another interesting consequence of the above theorem is that the increase monotonicity ofGp,qwith bothp,qimplies that ofEp,q. In fact, by Proposition .(iii) and Theorem ., we successively obtain

(p<p,q<q) =⇒ Gp,q<Gp,q =⇒ G

π

p,q<G

π

p,q

=⇒ Ep,q<Ep,q,

in this way proving the desired aim.

Theorem . contains more new applications: some extensions forGp,qcan imply anal-ogous ones forEp,q. See Section  below for more details concerning this latter situation.

Example . Similarly to the above, we leave it to the reader to establish that

p =Ep,, whereMprefers to the Lehmer mean defined by

Mp(a,b) =

ap+ap/bp/+bp

/p ,

withM(a,b) =G(a,b). In particular, takingp= , we obtain (another decomposition of

L):

L=

 A+

 G

π

.

We leave to the reader the routine task of formulating further examples in the aim to obtain some links between other special means.

Now, a question arises naturally from the above: Is it true that every mean is (m,r )-decomposable? In other words, letmbe a given mean, do a meanm and a real number  <r<  such thatmπr

 =mexist? The answer to this latter question appears to be inter-esting. In fact, for reason of simplicity, assume thatmis homogeneous and we search for a homogeneous meanmsuch that =m. According to Proposition ., it is equivalent to have f() =f(xα–)g(x) for all x> , wheref denotes the associated function of the given meanmandg will be that of the unknown meanm. That is to say, the function

x–→f()/f(xα–) is the associated function to a certain meanm

for someα> . Com-bining this with Proposition ., we can state the next result, which gives an answer to the above question.

Proposition . Let m be a homogeneous mean with its associated function f.For some

 <r< ,we put

gr(x) =f

/fxα–, withα= /( –r) > .

Then the following assertions are equivalent:

(i) There exists a homogeneous meanmsuch thatmis(m,r)-decomposable,that is,

m=

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(ii) The functionx–→gr(x)is the associated function of a certain homogeneous mean.

(iii) The following inequalities:

fxα–≤fxαxfxα–

hold for everyx≥,with reversed inequalities for eachx≤. If in the above the function x–→gr(x)is increasing and f is such that

x>  fxαfx–α=xfxαfxα–,

then the homogeneous mean mis symmetric and monotone.

The following corollary is immediate from the above proposition when combined with Proposition ..

Corollary . Let m be a homogeneous mean with its associated function f and let us put g(x) =f(x)/f(x).Then the following properties are equivalent:

(i) mism-decomposable.

(ii) gis the associated function of a certain mean. (iii) The double inequality

f(x)≤fx≤xf(x)

holds for allx≥,with reversed inequalities for eachx≤. If,moreover,g is increasing and

x>  fxf(/x) =xf(x)f/x, ()

then mis a symmetric and monotone mean.

Corollary . Let m be a(symmetric)homogeneous monotone mean and let <r< be a given real number. Then there exists a homogeneous mean msuch that m is(m,r)

-decomposable.In particular,every(symmetric)homogeneous monotone mean m is m

-decomposable for some homogeneous mean m.

Proof Letf be the associated function ofmand setα= /( –r) > . Sincemis monotone, then

m(x, )≤mxα, ≤mxα,x

for allx≥, with reversed inequalities forx≤. It follows that

f(x)≤fxαxfxα–

for everyx≥, with reversed inequalities ifx≤. By virtue of the above proposition, we can conclude the first part of the announced result. Takingr= /,α= , we obtain the

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In the above corollary, we explicitly notice that the meanmshould be monotone in order to ensure thatmism-decomposable for some meanm, butmis not necessary mono-tone. As an example, we have already seen thatAisC-decomposable withAmonotone butCis not monotone.

We can formulate the above in another way. LetM,Mh,m, Ms,h,m denote the set of means, homogeneous monotone means and symmetric homogeneous monotone means, respectively. For fixed  <r< , letr:M–→Mbe defined byr(m) =mπr for every meanm. Clearly,r(M) =Dris the set ofr-decomposable means and the following mean-chain of strict inclusions holds:

Ms,h,mMh,mDrM.

Example . The standard meansA,H,G,L,I are all (symmetric) homogeneous and

monotone, then we find again that these means are decomposable. However, as already pointed before, the contraharmonic mean C is not monotone and so we cannot apply the above corollary. Let us try to apply directly Proposition .. We can easily verify that the associated functionf(x) = (x+ )/(x+ ) ofC does not satisfy (iii) and soC is not decomposable.

We leave it to the reader to check if the meansSandPare decomposable or not. The reader can easily verify that all the above standard meansA,H,G,L,I,P,C,Ssatisfy relationship (). However, for the corresponding functionsg, it is not always monotone as in the case forAandH. In the general case, the following result gives a sufficient condition for ensuring the increase monotonicity ofg.

Proposition . Let m be a symmetric homogeneous monotone mean and f be its

asso-ciated function.Suppose that the function F(x) =f(x)/√x is strictly increasing.Then the function g(x) =f(x)/f(x)will be strictly increasing.

Proof One hasg(x) =F(x)k(x), wherek(x) =x/f(x). Sincef(/x) =f(x)/xis decreasing, we get thatkis increasing. SinceF(x) is strictly increasing, we get thatgis strictly increasing (as a product of an increasing and a strictly increasing positive functions).

Example . The meansGandLsatisfy the conditions of the above proposition, whereas

the meansAandHdo not. This rejoins the fact thatAisC-decomposable andHisC* -decomposable withCandC*not monotone means. We leave it to the reader to verify if the meansIandPsatisfy or not the conditions of the above proposition.

5 Application: means with several arguments

In this section, we investigate an application of our above theoretical study. This appli-cation turns out the extension of some means from two variables to three or more argu-ments.

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geometric and harmonic means with several variables are, respectively, given by

A(a,a, . . . ,ak) =

k

i=ai

k ,

G(a,a, . . . ,ak) = k

k

i=

ai, H(a,a, . . . ,ak) =

k

i=a–i

k

– .

Extensions of the weighted geometric meanS(a,b) and contraharmonic meanC(a,b) are immediately given by

S(a,a, . . . ,ak) = k

i=

aai/Sk

i withSk=

k

i=

ai,

C(a,a, . . . ,ak) =

k

i=ai

k

i=ai .

The following result is well known in the literature.

Proposition . The following inequalities:

H(a,a, . . . ,ak) <G(a,a, . . . ,ak) <A(a,a, . . . ,ak)

<S(a,a, . . . ,ak) <C(a,a, . . . ,ak) ()

hold true for all distinct real numbers a,a, . . . ,ak.

However, the extension of the logarithm, identric and Seiffert means from two to three or more variables does not appear to be obvious from the above expressions of these means. In this sense, we refer the reader to [, –] for some extensions about the logarithmic and identric means. Here, we will derive other extensions of these latter means from our above study. In fact, the above transformation for means with two variables can be imme-diately stated in a similar manner for means involving several variables. For instance, we can define

(a

,a, . . . ,ak) =

n=

ma/ n,a/n, . . . ,a/k n. ()

It is also simple to see that(a

,a, . . . ,ak) =G(a,a, . . . ,ak) for alla,a, . . . ,ak. The relationshipL=, which has been stated as a result for two variables, allows us

to consider it as a definition for the logarithmic mean involving three or more arguments. That is, we can suggest that

L(a,a, . . . ,ak) :=

n=

Aa/ n,a/n, . . . ,a/k n

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] seems to be an interesting problem. We omit the details about this latter point which is beyond our aim here.

Similarly, by virtue of the relationI=, we can put a definition for identric mean with

several variables

I(a,a, . . . ,ak) =

n=

Sa/ n,a/n, . . . ,a/k n.

The comparison of this definition ofI(a,a, . . . ,ak) with that given in [, ] appears also to be an interesting problem.

To give more justification for our above extensions, the following result, which extends the inequalitiesG<L<I<Afrom two variables to several arguments, may be stated.

Proposition . With the above,the following inequalities:

G(a,a, . . . ,ak) <L(a,a, . . . ,ak) <I(a,a, . . . ,ak) <A(a,a, . . . ,ak)

hold for all distinct real numbers a,a, . . . ,ak.

Proof For the sake of simplicity, we write () in a brief formG<A<S<C. According to (), we easily show that <Aπ <Sπ <Cπ. SinceGπ =G,Aπ =L,Sπ =I, we then

obtainG<L<I<. Using () again, a simple computation yieldsCπ=A. The proof is

complete.

More generally, the relationshipEp,q=Gπp,qappears to be a good tool for extending the Stolarsky meanEp,q(a,b) from two to three or more arguments. In fact, the Gini mean

Gp,q(a,b) seems simple to extend for several arguments as well:

Gp,q(a,a, . . . ,ak) =

k

i=a

q i

k

i=a

p i

/(qp) ,

with the convenient cases

Gp,p(a,a, . . . ,ak) =exp

k

i=a

p ilnai

k

i=a

p i

and G,(a,a, . . . ,ak) =G(a,a, . . . ,ak).

With this, we can suggest that the Stolarsky mean with several variables can be defined by

Ep,q(a,a, . . . ,ak) =

n=

Gp,q

a/ n,a/ n, . . . ,a/k n.

Now, the question concerning the comparison of this definition ofEp,q(a,a, . . . ,ak) with some other ones given in [, ] can be considered as an interesting purpose for future research.

Competing interests

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Authors’ contributions

Both authors jointly worked, read and approved the final version of the paper.

Author details

1Faculty of Science, Department of Mathematics, Taibah University, P.O. Box 30097, Al Madinah Al Munawwarah, Zip Code

41477, Kingdom of Saudi Arabia.2Faculty of Sciences, Department of Mathematics, Moulay Ismail University, Meknes,

Morocco.3Department of Mathematics, Babes-Bolyai University, Str. Kogalniceanu nr.1, Cluj-Napoca, 400084, Romania.

Received: 25 August 2012 Accepted: 17 January 2013 Published: 5 March 2013

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doi:10.1186/1029-242X-2013-89

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