Quasilinearity of some functionals associated with monotonic convex functions

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Dragomir, Sever S (2012) Quasilinearity of some functionals associated with monotonic convex functions. Journal of Inequalities and Applications, 212 (276). pp. 1-13. ISSN 1029-242X

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R E S E A R C H Open Access

Quasilinearity of some functionals associated with monotonic convex functions

SS Dragomir^*

*Correspondence:

Sever.Dragomir@vu.edu.au;

sever.dragomir@wits.ac.za;

http://rgmia.org/dragomir/

School of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne City, MC 8001, Australia School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits, P.O. Box 14428, Johannesburg, 2050, South Africa

Abstract

Some quasilinearity properties of composite functionals generated by monotonic and convex/concave functions and their applications in improving some classical inequalities such as the Jensen, Hölder and Minkowski inequalities are given.

MSC: 26D15

Keywords: additive; superadditive and subadditive functionals; convex functions;

Jensen’s inequality; Hölder’s inequality; Minkowski’s inequality

1 Introduction

The problem of studying the quasilinearity properties of functionals associated with some celebrated inequalities such as the Jensen, Cauchy-Bunyakowsky-Schwarz, Hölder, Minkowski and other famous inequalities has been investigated by many authors during the last  years.

In the following, in order to provide a natural background that will enable us to construct composite functionals out of simple ones and to investigate their quasilinearity properties, we recall a number of concepts and simple results that are of importance for the task.

LetXbe a linear space. A subsetC⊆Xis called aconvex coneinXprovided the following conditions hold:

(i) x,y∈Cimplyx+y∈C;

(ii) x∈C,α≥implyαx∈C.

A functionalh:C→Ris calledsuperadditive(subadditive) onCif (iii) h(x+y)≥(≤)h(x) +h(y)for anyx,y∈C

andnonnegative(strictly positive) onCif, obviously, it satisfies (iv) h(x)≥(>) for eachx∈C.

The functionalhiss-positive homogeneousonCfor a givens>  if (v) h(αx) =α^sh(x)for anyα≥andx∈C.

Ifs= , we simply call it positive homogeneous.

In [], the following result has been obtained.

Theorem  Let x,y∈C and h:C→Rbe a nonnegative,superadditive and s-positive homogeneous functional on C.If M≥m≥are such that x–my and My–x∈C,then

M^sh(y)≥h(x)≥m^sh(y). (.)

©2012 Dragomir; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Now, considerv:C→Ranadditiveandstrictly positivefunctional onCwhich is also positive homogeneousonC,i.e.,

(vi) v(αx) =αv(x)for anyα> andx∈C.

In [] we obtained further results concerning the quasilinearity of some composite func- tionals.

Theorem  Let C be a convex cone in the linear space X and v:C→(,∞)be an additive functional on C.If h:C→[,∞)is a superadditive(subadditive)functional on C and p,q≥ ( <p,q< ),then the functional

p,q:C→[,∞), p,q(x) =h^q(x)v^{q(–p)}(x) (.)

is superadditive(subadditive)on C.

Theorem  Let C be a convex cone in the linear space X and v:C→(,∞)be an additive functional on C.If h:C→[,∞)is a superadditive functional on C and <p,q< ,then the functional

p,q:C→[,∞), p,q(x) =v^{q(–p)}(x)

h^q(x) (.)

is subadditive on C.

Another result similar to Theorem  has been obtained in [] as well, namely

Theorem  Let x,y∈C,h:C→Rbe a nonnegative,superadditive and s-positive homo- geneous functional on C and v be an additive,strictly positive and positive homogeneous functional on C.If p,q≥and M≥m≥are such that x–my,My–x∈C,then

M^{sq+q(–p)}p,q(y)≥p,q(x)≥m^{sq+q(–p)}p,q(y), (.)

wherep,qis defined by(.).

As shown in [] and [], the above results can be applied to obtain refinements of the Jensen, Hölder, Minkowski and Schwarz inequalities for weights satisfying certain condi- tions.

The main aim of the present paper is to study quasilinearity properties of other com- posite functionals generated by monotonic and convex/concave functions and to apply the obtained results to improving some classical inequalities as those mentioned above.

2 Some general results

We start with the following general result.

Theorem (Quasilinearity theorem) Let C be a convex cone in the linear space X and v:C→(,∞)be an additive functional on C.

(i) Ifh:C→[,∞)is a superadditive(subadditive)functional onCand

: [,∞)→Ris concave(convex)and monotonic nondecreasing on[,∞),then the composite functionalη:C→Rdefined by

η(x) :=v(x) h(x)

v(x)

(.) is superadditive(subadditive)onC.

(ii) Ifh:C→[,∞)is a superadditive(subadditive)functional onCand

: [,∞)→Ris convex(concave)and monotonic nonincreasing on[,∞),then the composite functionalηis subadditive(superadditive)onC.

Proof (i) Assume thathis superadditive and: [,∞)→Ris concave and monotonic nondecreasing on [,∞). Then

h(x+y)≥h(x) +h(y) for anyx,y∈C,

and sincev(x+y) =v(x) +v(y) for anyx,y∈C, by the monotonicity of, we have

h(x+y) v(x+y)

=

h(x+y) v(x) +v(y)

≥

h(x) +h(y) v(x) +v(y)

=

v(x)·^h(x)_v(x) +v(y)·^h(y)_v(y) v(x) +v(y)

(.) for anyx,y∈C.

Now, since: [,∞)→Ris concave,

v(x)·^h(x)_v(x)+v(y)·^h(y)_v(y) v(x) +v(y)

≥v(x)(^h(x)_v(x)) +v(y)(^h(y)_v(y)) v(x) +v(y)

=v(x)(^h(x)_v(x)) +v(y)(^h(y)_v(y))

v(x+y) (.)

for anyx,y∈C.

Utilizing (.) and (.), we get v(x+y)

h(x+y) v(x+y)

≥v(x) h(x)

v(x)

+v(y) h(y)

v(y)

(.) for anyx,y∈C, which shows that the functionalηis superadditive onC.

Now, ifh:C→Ris a subadditive functional onCand: [,∞)→[,∞) is convex and monotonic nondecreasing on [,∞), then the inequalities (.), (.) and (.) hold with the reverse sign for anyx,y∈C, which shows that the functionalηis subadditive onC.

(ii) Follows in a similar manner and the details are omitted.

Corollary (Boundedness property) Let C be a convex cone in the linear space X and v:C→(,∞)be an additive and positive homogeneous functional on C.Let x,y∈C and assume that there exist M≥m> such that x–my and My–x∈C.

(a) Ifh:C→[,∞)is a superadditive and positive homogeneous functional onCand : [,∞)→[,∞)is concave and monotonic nondecreasing on[,∞),then

Mη(y)≥η(x)≥mη(y). (.)

(aa) Ifh:C→[,∞)is a subadditive and positive homogeneous functional onCand : [,∞)→[,∞)is concave and monotonic nonincreasing on[,∞),then(.) is valid as well.

Proof We observe that ifvandhare positive homogeneous functionals, thenηis also a positive homogeneous functional, and by the quasilinearity theorem above, it follows that in both casesηis a superadditive functional onC. By applying Theorem  fors= , we

deduce the desired result.

Remark (Monotonicity property) LetCbe a convex cone in the linear spaceX. We say, forx,y∈X, thatx≥Cy(xis greater thanyrelative to the coneC) ifx–y∈C. Now, observe that ifx,y∈Candx≥Cy, then under the assumptions of Corollary , by (.), we have thatη(x)≥η(y), which is a monotonicity property for the functionalη.

There are various possibilities to build such functionals. For instance, for the finite fam- ilies of functionalsvi:C→(,∞) andhi:C→[,∞) withi∈I(I is afinite family of indices) and: [,∞)→Ris concave/convex and monotonic, then the composite func- tionalσ:C→Rdefined by

σ(x) :=

i∈I

vi(x) h_i(x)

v_i(x)

(.) has the same properties as the functionalη.

If, for a given coneC, we consider theCartesian product Cⁿ:=C× · · · ×C⊂Xⁿand define, for the vectorx:= (x, . . . ,xn)∈Cⁿ, the functional:Cⁿ→Rgiven by

(x) :=

n i=

v(xi) h(xi)

v(xi)

, (.)

wherevandhdefined onCare as above, then we observe thathas the same properties asη.

There are some natural examples of composite functionals that are embodied in the propositions below.

Proposition  Let C be a convex cone in the linear space X and v:C→(,∞)be an additive functional on C.

(i) Ifh:C→(,∞)is a superadditive functional onCandr> ,then the composite functionalηr:C→(,∞)defined by

ηr(x) :=[v(x)]^+r

[h(x)]^r (.)

is subadditive onC.In particular,η(x) = ^v_h(x)^(x) is subadditive.

(ii) Ifh:C→[,∞)is a superadditive functional onCandq∈(, ),then the composite functionalηq:C→[,∞)defined by

ηq(x) :=

v(x)–q h(x)q

(.) is superadditive onC.In particular,η/(x) =√

v(x)h(x)is superadditive.

(iii) Ifh:C→[,∞)is a subadditive functional onCandp≥,then the composite functionalηp:C→[,∞)defined by

ηp(x) := [h(x)]^p

[v(x)]^p– (.)

is subadditive onC.In particular,η(x) = ^h_v(x)^(x)is subadditive.

Proof Follows from Theorem  for the function: (,∞)→(,∞),(t) =t^swhich is convex and decreasing fors∈(–∞, ), concave and increasing fors∈(, ) and convex

and increasing fors∈[,∞). The details are omitted.

The following boundedness property also holds.

Corollary  Let C be a convex cone in the linear space X and v:C→(,∞)be an additive and positive homogeneous functional on C.Let x,y∈C and assume that there exist M≥ m> such that x–my and My–x∈C.If h:C→[,∞)is a superadditive and positive homogeneous functional on C and q∈(, ),then

M

v(y)–q h(y)q

≥

v(x)–q h(x)q

≥m v(y)–q

h(y)q

. (.)

In particular,

M^v(y)h(y)≥v(x)h(x)≥m^v(y)h(y). (.)

Proposition  Let C be a convex cone in the linear space X and v:C→(,∞)be an additive functional on C.

(i) Ifh:C→[,∞)is a subadditive functional onC,then the composite functional εα:C→(,∞)defined by

εα(x) :=v(x)exp αh(x)

v(x)

(.)

is subadditive onCprovidedα> .

(ii) Ifh:C→[,∞)is a superadditive functional onC,then the composite functionalεα

is also subadditive onCwhenα< .

The proof follows from Theorem . The details are omitted.

Remark  Similar composite functionals can be considered for the functions: [,∞)→ Rdefined as follows:

(t) =arctan(t), which is increasing and concave on [,∞);

(t) =sinh(t) :=



e^t–e^–t

, which is increasing and convex on [,∞);

(t) =cosh(t) := 



e^t+e^–t

, which is increasing and convex on [,∞);

(t) =tanh(t) :=e^t–e^–t

e^t+e^–t, which is increasing and concave on [,∞);

(t) =coth(t) :=e^t+e^–t

e^t–e^–t, which is decreasing and convex on (,∞).

For instance, if we consider the composite functional ηarctan(x) :=v(x)arctan

h(x) v(x)

,

whereh:C→[,∞) is a superadditive functional onC,v:C→(,∞) is an additive func- tional onCandCis a convex cone in the linear spaceX, then by the quasilinearity theorem, we conclude thatηarctan:C→[,∞) is superadditive onC. Moreover, ifv:C→(,∞) is an additive and positive homogeneous functional onC,h:C→[,∞) is a superadditive and positive homogeneous functional onCandx,y∈Csuch that there existM≥m>  with the property thatx–myandMy–x∈C, then

Mv(y)arctan h(y)

v(y)

≥v(x)arctan h(x)

v(x)

≥mv(y)arctan h(y)

v(y)

.

The same properties hold for the composite functional generated by the hyperbolic tan- gent function, namely

ηtanh(x) :=v(x) exp(^h(x)_v(x)) –exp(–^h(x)_v(x)) exp(^h(x)_v(x)) +exp(–^h(x)_v(x))

,

however, the details are omitted.

Taking into account the above result and its applications for various concrete examples of convex functions, it is therefore natural to investigate the corresponding results for the case oflog-convex functions, namely functions:I→(,∞),Iis an interval of real numbers for whichlnis convex.

We observe that such functions satisfy the elementary inequality

( –t)a+tb

≤

(a)–t (b)t

for anya,b∈Iandt∈[, ]. Also, due to the fact that the weighted geometric mean is less than the weighted arithmetic mean, it follows that any log-convex function is a convex function. However, obviously, there are functions that are convex but not log-convex.

Theorem (Quasimultiplicity theorem) Let C be a convex cone in the linear space X and v:C→(,∞)be an additive functional on C.

(i) Ifh:C→[,∞)is a superadditive(subadditive)functional onCand

: [,∞)→(,∞)is log-concave(log-convex)and monotonic nondecreasing on [,∞),then the composite functionalξ:C→(,∞)defined by

ξ(x) :=

h(x) v(x)

v(x)

(.)

is supermultiplicative(submultiplicative)onC,i.e.,we recall that

ξ(x+y)≥(≤)ξ(x)ξ(y) (.)

for anyx,y∈C.

(ii) Ifh:C→[,∞)is a superadditive(subadditive)functional onCand

: [,∞)→(,∞)is log-convex(log-concave)and monotonic nonincreasing on [,∞),then the composite functionalηis submultiplicative(supemultiplicative) onC.

Proof We observe that logξ(x) =v(x)log

h(x) v(x)

=ηlog()(x) for anyx∈C.

Applying now the quasilinearity theorem for the functionslog(), we deduce the desired result.

The details are omitted.

Corollary (Exponential boundedness) Let C be a convex cone in the linear space X and v:C→(,∞)be an additive and positive homogeneous functional on C.Let x,y∈C and assume that there exist M≥m> such that x–my and My–x∈C.

(a) Ifh:C→[,∞)is a superadditive and positive homogeneous functional onCand : [,∞)→[,∞)is log-concave and monotonic nondecreasing on[,∞),then

h(y)

v(y)

Mv(y)

≥ h(x)

v(x) v(x)

≥ h(y)

v(y)

mv(y)

. (.)

(aa) Ifh:C→[,∞)is a subadditive and positive homogeneous functional onCand : [,∞)→[,∞)is log-concave and monotonic nonincreasing on[,∞), then(.)is valid as well.

There are numerous examples of log-convex (log-concave) functions of interest that can provide some nice examples.

Following [], we consider the followingDirichlet series:

ψ(s) :=

∞ n=

an

n^s (.)

for which we assume that the coefficients an≥ forn≥ and the series is uniformly convergent fors> .

It is obvious that in this class we can find thezeta function ζ(s) :=

∞ n=

 n^s and thelambda function

λ(s) :=

∞ n=

 (n+ )^s =

 – ^–s ζ(s),

wheres> .

If(n) is thevon Mangoldt function, where

(n) :=

⎧⎨

⎩

logp, n=p^k(pprime,k≥),

, otherwise, then [, p.]:

–ζ(s) ζ(s) =

∞ n=

(n)

n^s , s> .

Ifd(n) is the number of divisors ofn, we have [, p.] the following relationships with the zeta function:

ζ^(s) = ∞

n=

d(n)

n^s , ζ^(s) ζ(s)=

∞ n=

d(n^)

n^s , ζ^(s) ζ(s)=

∞ n=

d^(n) n^s , and [, p.]

ζ^(s) ζ(s)=

∞ n=

^ω(n)

n^s , s> ,

whereω(n) is the number of distinct prime factors ofn.

We use the following result, see []

Lemma  The functionψdefined by(.)is nonincreasing and log-convex on(,∞).

Utilizing the quasimultiplicity theorem and this lemma, we can state the following result as well.

Proposition  Let C be a convex cone in the linear space X and v:C→(,∞)be an ad- ditive functional on C.If h:C→[,∞)is a subadditive functional on C andψ: (,∞)→ (,∞)is defined by(.),then the composite functionalξψ:C→(,∞)defined by

ξψ(x) := ψ

h(x) +v(x) v(x)

v(x)

(.) is submultiplicative on C.

Proof We observe that the function(t) :=ψ(t+ ) is well defined on (,∞) and is non- increasing and log-convex on this interval. Applying Theorem , we deduce the desired

result.

3 Applications

3.1 Applications for Jensen’s inequality

LetCbe a convex subset of the real linear spaceXand letf :C→Rbe a convex mapping.

Here we consider the following well-known form ofJensen’s discrete inequality:

f 

PI

i∈I

p_ix_i

≤  PI

i∈I

p_if(x_i), (.)

whereIdenotes a finite subset of the setNof natural numbers,x_i∈C,p_i≥ fori∈Iand P_I:=

i∈Ip_i> .

Let us fixI∈Pf(N) (the class of finite parts ofN) andxi∈C(i∈I). Now, consider the functionalJ:S+(I)→Rgiven by

JI(p) :=

i∈I

pif(xi) –PIf 

PI

i∈I

pixi

≥, (.)

whereS₊(I) :={p= (p_i)_i∈I|p_i≥,i∈IandP_I> }andf is convex onC.

We observe thatS+(I) is a convex cone and the functionalJIis nonnegative and positive homogeneous onS+(I).

Lemma ([]) The functional JI(·)is a superadditive functional on S+(I).

For a function: [,∞)→R, define the following functionalξ,I:S₊(I)→R: ξ,I(p) :=P_I

 PI

i∈I

p_if(x_i) –f 

PI

i∈I

p_ix_i

. (.)

By the use of Theorem , we can state the following proposition.

Proposition  If: [,∞)→Ris concave and monotonic nondecreasing on[,∞),then the composite functionalξ,I:S+(I)→Rdefined by(.)is superadditive on S+(I).

If: [,∞)→Ris convex and monotonic nonincreasing on[,∞),then the composite functionalξ,Iis subadditive on S+(I).

Proof Consider the functionalsv(p) :=P_I andh(p) :=J_I(p). We observe thatvis additive, his superadditive and

η(p) =v(p) h(p)

v(p)

=ξ,I(p).

Applying Theorem , we deduce the desired result.

Corollary  Ifp,q∈S+(I)and M≥m≥are such that Mp≥q≥mp,i.e.,Mpi≥qi≥ mpifor each i∈I,then

MPI

Q_I 

P_I

i∈I

pif(xi) –f 

P_I

i∈I

pixi

≥ 

Q_I

i∈I

qif(xi) –f 

Q_I

i∈I

qixi

≥mPI

Q_I 

P_I

i∈I

pif(xi) –f 

P_I

i∈I

pixi

≥ (.)

for any: [,∞)→[,∞)concave and monotonic nondecreasing function on[,∞).

The proof follows from Corollary  and the details are omitted.

On utilizing Proposition , statement (ii), we observe that the functionalξq,I:S+(I)→ [,∞), whereq∈(, ) and

ξq,I(p) :=PI

 PI

i∈I

pif(xi) –f 

PI

i∈I

pixi

q

=

P^q–_I

i∈I

p_if(x_i) –P^q_If 

PI

i∈I

p_ix_i q

, (.)

is superadditive and monotonic nondecreasing onS+(I).

Ifp,q∈S₊(I) andM≥m≥ are such thatMp≥q≥mp, then M^/q

P_I QI

/q

 PI

i∈I

pif(xi) –f 

PI

i∈I

pixi

≥  QI

i∈I

qif(xi) –f 

QI

i∈I

qixi

≥m^/q P_I

QI

/q

 PI

i∈I

p_if(x_i) –f 

PI

i∈I

p_ix_i

≥. (.)

Now, if we consider the following composite functionalξarctan,I:S+(I)→[,∞) given by ξarctan,I(p) =P_Iarctan

 PI

i∈I

p_if(x_i) –f 

PI

i∈I

p_ix_i

, (.)

then by utilizing Remark  we conclude thatξarctan,I issuperadditive and monotonic non- decreasingonS+(I).

Moreover, ifp,q∈S+(I) andM≥m≥ are such thatMp≥q≥mp, then M

P_I QI

arctan

 PI

i∈I

pif(xi) –f 

PI

i∈I

pixi

≥arctan 

QI

i∈I

qif(xi) –f 

QI

i∈I

qixi

≥m P_I

QI

arctan

 PI

i∈I

pif(xi) –f 

PI

i∈I

pixi

≥. (.)

It is also well known that iff :C→Ris astrictly convex mappingonCand, for a given sequence of vectorsxi∈C(i∈I), there exist at least two distinct indiceskandjinIso thatxk=xj, then

JI(p) :=

i∈I

pif(xi) –PIf 

P_I

i∈I

pixi

> 

for anyp∈S+(I) ={p= (pi)_i∈I|pi≥,i∈IandPI> }.

In this situation, for the functionf and the sequencex_i∈C(i∈I), we can define the functional

ηr,I(p) := P^+r_I [

i∈Ip_if(x_i) –P_If(_P^

I

i∈Ip_ix_i)]^r (.)

that is well defined onS+(I). Utilizing the statement (i) from Proposition , we conclude thatηr,I(·) is asubadditive functionalonS+(I).

We know that the hyperbolic cotangent functioncoth(t) :=^e_et^t+e–e^–t–t is decreasing and con- vex on (,∞). If we consider the composite functional

ξcoth,I(p) :=PIcoth 

PI

i∈I

pif(xi) –f 

PI

i∈I

pixi

for a functionf :C→Rthat isstrictly convexonCand for a given sequence of vectors xi∈C(i∈I) for which there exist at least two distinct indiceskandjinIso thatxk=xj, then we observe that this functional is well defined onS+(I), and by the statement (ii) of Theorem , we conclude thatξcoth,I(·) is also asubadditive functionalonS+(I).

3.2 Applications for Hölder’s inequality

Let (X, · ) be a normed space andI∈Pf(N). We define E(I) :=

x= (xj)_j∈I|xj∈X,j∈I and

K(I) :=

λ= (λj)_j∈I|λj∈K,j∈I .

We consider forγ,β> , _γ^ +_β^ =  the functional

HI(p,λ,x;γ,β) :=

j∈I

pj|λj|^γ

_γ^

j∈I

pjxj^β _β^

–

j∈I

pjλjxj

.

The following result has been proved in [].

Lemma  For anyp,q∈S+(I),we have

HI(p+q,λ,x;γ,β)≥HI(p,λ,x;γ,β) +HI(q,λ,x;γ,β), (.) where x∈E(I),λ∈K(I)andγ,β> with_γ^ +_β^ = .

Remark  The same result can be stated if (B, · ) is a normed algebra and the functional His defined by

H_I(p,λ,x;γ,β) :=

i∈I

p_ix_i^γ

_γ^

i∈I

p_iy_i^β _β^

–

i∈I

p_ix_iy_i ,

wherex= (xi)_i∈I,y= (yi)_i∈I⊂B,p∈S+(I) andγ,β>  with _γ^ +_β^ = .

For a function: [,∞)→R, define the following functionalω,I:S₊(I)→R: ω,I(p) :=P_I

 PI

j∈I

p_j|λj|^γ _γ^

 PI

j∈I

p_jx_j^β _β^

– 

PI

j∈I

p_jλjx_j

. (.)

By the use of Theorem , we can state the following proposition.

Proposition  If: [,∞)→Ris concave and monotonic nondecreasing on[,∞),then the composite functionalω,I:S+(I)→Rdefined by(.)is superadditive on S+(I).

If: [,∞)→Ris convex and monotonic nonincreasing on[,∞),then the composite functionalω,Iis subadditive on S+(I).

By choosing various examples of concave and monotonic nondecreasing or convex and monotonic nonincreasing functionson [,∞), the reader can provide various examples of superadditive or subadditive functionals onS₊(I). The details are omitted.

3.3 Applications for Minkowski’s inequality

Let (X, · ) be a normed space andI∈Pf(N). We define the functional

MI(p,x,y;δ) =

i∈I

pixi^δ _δ^

+

i∈I

piyi^{δ }_δδ

–

i∈I

pixi+yi^δ, (.)

wherep∈S+(I),δ≥ andx,y∈E(I).

The following result concerning the superadditivity of the functional MI(·,x,y;δ) holds [].

Lemma  For anyp,q∈S₊(I),we have

MI(p+q,x,y;δ)≥MI(p,x,y;δ) +MI(q,x,y;δ), where x,y∈E(I)andδ≥.

For a function: [,∞)→R, define the following functionalκ,I:S+(I)→R:

κ,I(p) :=PI  PI

i∈I

pixi^{δ }_δ

+ 

PI

i∈I

piyi^{δ }_δδ

–  PI

i∈I

pixi+yi^δ

. (.)

By the use of Theorem , we can state the following proposition.

Proposition  If: [,∞)→Ris concave and monotonic nondecreasing on[,∞),then the composite functionalκ,I:S+(I)→Rdefined by(.)is superadditive on S+(I).

If: [,∞)→Ris convex and monotonic nonincreasing on[,∞),then the composite functionalκ,I is subadditive on S+(I).

We notice that, by choosing various examples of concave and monotonic nondecreas- ing or convex and monotonic nonincreasing functionson [,∞), the reader can pro- vide various examples of superadditive or subadditive functionals onS+(I). The details are omitted.

Remark  For other examples of superadditive (subadditive) functionals that can provide interesting inequalities similar to the ones outlined above, we refer to [–] and [–].

Competing interests

The author declares that he has no competing interests.

Received: 10 May 2012 Accepted: 8 November 2012 Published: 28 November 2012

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doi:10.1186/1029-242X-2012-276

Cite this article as:Dragomir:Quasilinearity of some functionals associated with monotonic convex functions.

Journal of Inequalities and Applications20122012:276.