A new order preserving average function on a quotient space of strictly monotone functions and its applications

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R E S E A R C H A R T I C L E

Open Access

A new order-preserving average function on

a quotient space of strictly monotone

functions and its applications

Yasuo Nakasuji

1*

_{and Sin-Ei Takahasi}

2

*_{Correspondence:}

[email protected]

1_{The Open University of Japan,}

Chiba, 261-8586, Japan Full list of author information is available at the end of the article

Abstract

We introduce an order in a quotient space of strictly monotone continuous functions on a real interval and show that a new average function on this quotient space is order-preserving. We also apply this new order-preserving function to derive a ﬁnite form of Jensen type inequality with negative weights.

MSC: Primary 39B62; secondary 26B25; 26A51

Keywords: Jensen’s inequality; strictly monotone function; order-preserving average function

1 Introduction and main results

This is meant as a continuation of our paper [] related to Jensen’s inequality []. The reader should refer to the recent paper of József [] on Jensen’s inequality. Further the paper is related to the notion of quasi-arithmetic means, so the reader should refer to the recent paper of Janusz [].

LetIbe a ﬁnite closed interval [m,M] onRandC(I) the space of all continuous real-valued functions deﬁned onI. Moreover, letC+

sm(I) (resp.C–sm(I)) be the set of all functions

inC(I) which are strictly monotone increasing (resp. decreasing) onI. Put

Csm(I) =Csm+ (I)∪Csm– (I).

ThenCsm(I) is equal to the space of all strictly monotone continuous functions onI. For

anyϕ,ψ ∈Csm(I), we writeϕ∼=ψ if there exist two numbersa,b∈Rsuch thatϕ(x) =

aψ(x) +b for allx∈I. Then it is clear that∼= is an equivalence relation inCsm(I). Let

˜

Csm(I) be the quotient space ofCsm(I) by∼= and we denote byϕ˜the coset ofϕ∈Csm(I).

We introduce an orderinC˜sm(I) in the next section (see Theorem ).

Let (,μ) be a probability space andfa function inL₍_,_μ_{) such that}_f₍_ω₎_∈_I_{for almost}

allω∈. Then we see thatϕ◦f ∈L₍_,_μ_{) for all} _ϕ_∈_C

sm(I) becauseϕ is a bounded

continuous function andμis a ﬁnite measure. Put

Mϕ(f) =ϕ–

ϕ◦f dμ

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for eachϕ ∈Csm(I). Then [, Theorem ] which gives a new interpretation of Jensen’s

inequality is restated as ϕ˜ ˜ψ ⇒Mϕ(f)≤Mψ(f). In this paper, we give a new

order-preserving average functionN[I,f]on the quotient spaceC˜sm(I), according to this idea. We

also apply this functionN[I,f]to derive a ﬁnite form of Jensen type inequality with negative

weights.

Letϕbe an arbitrary function ofCsm(I). Sinceϕ(I) is an interval ofRandμis a

proba-bility measure on, it follows that

ϕ(m) +ϕ(M) –

ϕ◦f dμ∈ϕ(I),

and hence we have

ϕ–

ϕ(m) +ϕ(M) –

ϕ◦f dμ

∈I.

Note that a simple computation implies that ifϕ,ψ∈Csm(I) satisfyϕ˜=ψ˜, then

ϕ–

ϕ(m) +ϕ(M) –

ϕ◦f dμ

=ψ–

ψ(m) +ψ(M) –

ψ◦f dμ

holds. Then denote byN[I,f](ϕ˜) the above value.

In this case, our main result can be stated as follows.

Theorem  N[I,f]is an order-preserving real-valued function on the quotient spaceC˜sm(I)

with order,that is,ϕ˜ ˜ψ⇒N[I,f](ϕ˜)≤N[I,f](ψ˜).

The above theorem easily implies the following result, which is a ﬁnite form of Jensen type inequality with negative weights.

Corollary  Letϕ,ψ∈Csm(I)withϕ˜ ˜ψand t, . . . ,tn∈Rwith

n

i=ti= ,  <t,tn< ,

and t, . . . ,tn–< .Then

ϕ–

_n

i=

tiϕ(xi)

≤ψ–

_n

i=

tiψ(xi)

holds for all x, . . . ,xn∈I with x≤x, . . . ,xn–≤xn. Finally, we give concrete examples of Corollary .

2 An order in the quotient spaceC˜sm(I)

Let us start with the following two lemmas.

Lemma  Letϕ∈Csm(I).Then:

(i) ϕis increasing and convex onIif and only ifϕ–is increasing and concave onϕ(I). (ii) ϕis increasing and concave onIif and only ifϕ–is increasing and convex onϕ(I). (iii) ϕis decreasing and convex onIif and only ifϕ–_{is decreasing and convex on}_ϕ₍_I_).

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Proof Straightforward.

Lemma 

(i) Ifϕis a convex function onIandψ is an increasing convex function onϕ(I),then ψ◦ϕis convex onI.

(ii) Ifϕis a convex function onIandψ is a decreasing concave function onϕ(I),then ψ◦ϕis concave onI.

(iii) Ifϕis a concave function onIandψ is an increasing concave function onϕ(I),then ψ◦ϕis concave onI.

(iv) Ifϕis a concave function onIandψ is a decreasing convex function onϕ(I),then ψ◦ϕis convex onI.

Proof Straightforward.

For anyϕ,ψ∈Csm(I), we writeϕψif any of the following four conditions holds:

(i) ϕ,ψ∈C+

sm(I)andϕ◦ψ–is concave onψ(I).

(ii) ϕ∈C_sm– (I),ψ∈C_sm+ (I)andϕ◦ψ–is convex onψ(I). (iii) ϕ,ψ∈C–

sm(I)andϕ◦ψ–is convex onψ(I).

(iv) ϕ∈C+

sm(I),ψ∈C–sm(I)andϕ◦ψ–is concave onψ(I).

Remark Lemma  guarantees that the aboveϕψ is a restatement of the concepts ap-pearing in [, Lemma ].

Lemma  Letϕ,ϕ,ψ,ψ ∈Csm(I).Ifϕ∼=ϕ,ψ∼=ψ,andϕψ,thenϕ ψ .

Proof Assume thatϕ∼=ϕ,ψ∼=ψ, andϕψ. Then we must showϕ ψ. Sinceϕ∼=ϕ, ψ∼=ψ, we can writeϕ andψ as follows:

ϕ =aϕ+b and ψ =cψ+d

for somea,b,c,d∈R. Then we havea=  andc= . Put

ζ(x) =ax+b and η(x) =cx+d

for eachx∈R. In the case ofϕ,ψ∈C+_sm(I) anda,c> , we ﬁnd thatϕ◦ψ–is concave on ψ(I) becauseϕψ. Thenζ◦ϕ◦ψ–_{is increasing and concave on}_ψ₍_I_{) from Lemma -(iii)}

and henceϕ ◦ψ–₌_ζ_◦_ϕ_◦_ψ–_◦_η–_{is also concave on}_ψ₍_I_{) from Lemma -(iii). However,}

sinceϕ,ψ ∈C+

sm(I), we obtainϕ ψ as required. Moreover, we can easily see thatϕ

ψ holds in the other  cases:

ϕ∈C+

sm(I),ψ∈C–sm(I),a> ,c> 

, . . . ,

ϕ∈C_sm– (I),ψ∈C–_sm(I),a< ,c< .

For anyϕ˜,ψ˜ ∈ ˜Csm(I), we writeϕ˜ ˜ψ by the same notation ifϕψ holds. This is well

deﬁned by Lemma . In this case, we have the following.

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Proof We show the theorem by dividing into three steps. (I) It is evident thatsatisﬁes the reﬂexivity.

(II) Assume thatϕ˜ ˜ψandψ˜ ˜ϕ. Thenϕψ andψϕhold. In the case ofϕ,ψ∈

C+

sm(I), we ﬁnd thatϕ◦ψ–is concave onψ(I) andψ◦ϕ–is concave onϕ(I). Sinceψ◦ϕ–

is increasing and concave onϕ(I), it follows from Lemma -(ii) thatϕ◦ψ–_{= (}_ψ_◦_ϕ–₎–

is convex onψ(I). Thereforeϕ◦ψ–is aﬃne onψ(I) and henceϕ∼=ψ, that is,ϕ˜=ψ˜. By the same method, we can easily see thatϕ˜=ψ˜ holds in the other three cases:

ϕ∈C_sm+ (I),ψ∈C_sm– (I), ϕ∈C–_sm(I),ψ∈C_sm+ (I) and ϕ,ψ∈C–_sm(I).

Thereforesatisﬁes the symmetry law.

(III) Assume thatϕ˜ ˜ψandψ˜ ˜λ. Thenϕψandψλhold. In the case ofϕ,ψ,λ∈

C+

sm(I), we ﬁnd thatϕ◦ψ–is increasing and concave onψ(I) andψ◦λ–is concave on

λ(I). Then it follows from Lemma -(iii) thatϕ◦λ–= (ϕ◦ψ–)◦(ψ◦λ–) is concave on λ(I), and henceϕλ, that is,ϕ˜ ˜λholds. By the same method, we can easily see that

˜

ϕ ˜λholds in the other seven cases:

ϕ∈C_sm+ (I),ψ∈C+_sm(I),λ∈C_sm– (I), . . . , ϕ∈C_sm– (I),ψ∈C–_sm(I),λ∈C_sm– (I).

Thereforesatisﬁes the transitive law.

3 Proofs of Theorem 1 and Corollary 1

Letϕbe an arbitrary function ofCsm(I). Then an easy observation implies that

(–ϕ)–(y) =ϕ–(–y) ()

for ally∈–ϕ(I) and that

N[I,f](–ϕ) =N[I,f](ϕ˜). ()

Lemma  Let ϕ∈Csm(I).If eitherϕ is increasing and concave on I or decreasing and

convex on I,then

N[I,f](ϕ˜)≤

ϕ–◦ϕ(m) +ϕ(M) –ϕ◦fdμ≤m+M–

f dμ

holds.If eitherϕ is increasing and convex on I or decreasing and concave on I,then the above inequalities are reversed.

Proof (I) Suppose thatϕis increasing and concave onI. Thenϕ–is increasing and convex onϕ(I) by Lemma -(ii), and hence the ﬁrst inequality in Lemma  follows from Jensen’s inequality. Put

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for eachx∈I. Then it follows from Lemma -(i) thatϕ is a convex function onI such thatϕ (m) =ϕ (M) =m+M. Therefore we have

ϕ–ϕ(m) +ϕ(M) –ϕf(ω)≤m+M–f(ω) ()

for almost allω∈. By integrating () with respect toω, we obtain the second inequality in Lemma . We next suppose thatϕis decreasing and convex onI. Then –ϕis increasing and concave onI. Therefore the desired inequality follows from (), (), and the above argument.

(II) Suppose thatϕis increasing and convex onI. Thenϕ–is increasing and concave on ϕ(I) by Lemma -(i), and hence the ﬁrst inequality in Lemma  is reversed from Jensen’s inequality. Also sinceϕ is concave onIby Lemma -(iii), it follows that the second in-equality in Lemma  is reversed from a consideration in (I). Similarly for the decreasing

and concave case.

Proof of Theorem Letϕ˜,ψ˜ ∈ ˜Csm(I) withϕ˜ ˜ψ, whereϕ,ψ∈Csm(I).

(I-i) In the case ofϕ,ψ∈C+

sm(I), we ﬁnd thatϕ◦ψ–is increasing and concave onψ(I) =

[ψ(m),ψ(M)] becauseϕψ. Therefore we have from Lemma 

ψN[I,f](ϕ˜)

=ψ◦ϕ–ϕ(m) +ϕ(M) –

ϕ◦f dμ

=ϕ◦ψ––ϕ◦ψ–ψ(m)+ϕ◦ψ–ψ(M)– ϕ◦ψ–◦(ψ◦f)dμ

=N[ψ(I),ψ◦f]

ϕ◦ψ–

≤ψ(m) +ψ(M) –

ψ◦f dμ

=ψN[I,f](ψ˜)

,

so we obtainN[I,f](ϕ˜)≤N[I,f](ψ˜) sinceψis strictly increasing onI.

(I-ii) In the case ofϕ∈C_sm– (I) andψ∈C_sm+ (I), we ﬁnd thatϕ◦ψ–is decreasing and convex onψ(I) because ϕψ. Then –ϕ,ψ ∈C+

sm(I) and (–ϕ)◦ψ– is increasing and

concave onψ(I). Therefore we have from (I-i) and ()

N[I,f](ϕ˜) =N[I,f](–ϕ)≤N[I,f](ψ˜).

(I-iii) In the case ofϕ,ψ∈C–

sm(I), we ﬁnd thatϕ◦ψ–is increasing and convex onψ(I)

becauseϕψ. Thenϕ∈C–

sm(I), –ψ∈C+sm(I), andϕ◦(–ψ)–is decreasing and convex

on –ψ(I) by (). Therefore we have from (I-ii) and ()

N[I,f](ϕ˜)≤N[I,f](–ψ) =N[I,f](ψ˜).

(I-iv) In the case ofϕ∈C+

sm(I) andψ∈Csm– (I), we ﬁnd thatϕ◦ψ–is decreasing and

con-cave onψ(I) becauseϕψ. Then –ϕ,ψ∈C–

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onψ(I). Therefore we have from (I-iii) and ()

N[I,f](ϕ˜) =N[I,f](–ϕ)≤N[I,f](ψ˜).

This completes the proof.

Remark Letϕ,ψ∈Csm(I). We see from Theorem  and Lemma  thatψϕand then

N[I,f](ϕ˜)≥N[I,f](ψ˜) if any of the following four conditions holds:

(v) ϕ,ψ∈C_sm+ (I)andϕ◦ψ–is convex onψ(I). (vi) ϕ∈C–

sm(I),ψ∈Csm+ (I), andϕ◦ψ–is concave onψ(I).

(vii) ϕ,ψ∈C–

sm(I)andϕ◦ψ–is concave onψ(I).

(viii) ϕ∈C+

sm(I),ψ∈Csm– (I), andϕ◦ψ–is convex onψ(I).

Throughout the remainder of the paper, we assume that=Iandf(x) =xfor allx∈I.

Proof of Corollary Letϕ,ψ∈Csm(I) withϕψandt, . . . ,tn∈Rwith

n

i=ti= ,  <t,

tn<  andt, . . . ,tn–< . Letx, . . . ,xn∈Ibe such thatx≤x, . . . ,xn–≤xn. Puts=  –

t,s= –t, . . . ,sn–= –tn–,sn=  –tn. Then we have

n

i=si=  ands, . . . ,sn> . So μ≡sδx+· · ·+snδxn

is a probability measure onI, whereδxdenotes the Dirac measure atx∈I. Taking [x,xn] instead ofIin Theorem , we obtain

ϕ–

ϕ(x) +ϕ(xn) – n

i=

siϕ(xi)

≤ψ–

ψ(x) +ψ(xn) – n

i=

siψ(xi)

,

which implies the desired inequality

ϕ–

_n

i=

tiϕ(xi)

≤ψ–

_n

i=

tiψ(xi)

.

This completes the proof.

Remark Letϕ, ψ be in Csm(I) such that any of (v), (vi), (vii), and (viii) holds. Then

ψ ϕholds from Lemma . Therefore ift, . . . ,tn∈Rwith

n

i=ti= ,  <t,tn< , and

t, . . . ,tn–< , then

ϕ–

_n

i=

tiϕ(xi)

≥ψ–

_n

i=

tiψ(xi)

holds from Corollary .

Example  Putϕ(x) =logxandψ(x) =xfor each positive numberx> . Then Corollary  easily implies that

n

i=

xti i ≤

n

i=

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holds for allt, . . . ,tn∈Rwith

n

i=ti= ,  <t,tn< , and t, . . . ,tn–< , and all

pos-itive numbersx, . . . ,xnwithx≤x, . . . ,xn–≤xn. This is a geometric-arithmetic mean inequality with negative weights.

Example  Putϕ(x) = 

xandψ(x) =logxfor each positive numberx> . Then Corollary  easily implies that

_n

i=

ti

xi

–

≤ n

i=

xti i

holds for allt, . . . ,tn∈Rwith

n

i=ti= ,  <t,tn< , andt, . . . ,tn–< , and all positive

numbersx, . . . ,xnwithx≤x, . . . ,xn–≤xn. This is a harmonic-geometric mean inequal-ity with negative weights.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{The Open University of Japan, Chiba, 261-8586, Japan.}2_{Toho University, Funabashi, 273-0866, Japan.}

Acknowledgements

The authors are grateful to the referees for careful reading of the paper and for helpful suggestions and comments. The second author is partially supported by Grant-in-Aid for Scientiﬁc Research, Japan Society for the Promotion of Science.

Received: 23 June 2014 Accepted: 21 October 2014 Published:06 Nov 2014 References

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10.1186/1029-242X-2014-450