Research

# Quasilinearity of some functionals associated with monotonic convex functions

SS Dragomir

Author Affiliations

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

Journal of Inequalities and Applications 2012, 2012:276  doi:10.1186/1029-242X-2012-276

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/276

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

This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.

### 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 50 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.

Let X be a linear space. A subset C X is called a convex cone in X provided the following conditions hold:

(i) x , y C imply x + y C ;

(ii) x C , α 0 imply α x C .

A functional h : C R is called superadditive (subadditive) on C if

(iii) h ( x + y ) ( ) h ( x ) + h ( y ) for any x , y C

and nonnegative (strictly positive) on C if, obviously, it satisfies

(iv) h ( x ) ( > ) 0 for each x C .

The functional h is s-positive homogeneous on C for a given s > 0 if

(v) h ( α x ) = α s h ( x ) for any α 0 and x C .

If s = 1 , we simply call it positive homogeneous.

In [1], the following result has been obtained.

Theorem 1Let x , y C and h : C R be a nonnegative, superadditive ands-positive homogeneous functional onC. If M m 0 are such that x m y and M y x C , then

M s h ( y ) h ( x ) m s h ( y ) . (1.1)

Now, consider v : C R an additive and strictly positive functional on C which is also positive homogeneous on C, i.e.,

(vi) v ( α x ) = α v ( x ) for any α > 0 and x C .

In [2] we obtained further results concerning the quasilinearity of some composite functionals.

Theorem 2LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive functional onC. If h : C [ 0 , ) is a superadditive (subadditive) functional onCand p , q 1 ( 0 < p , q < 1 ), then the functional

Ψ p , q : C [ 0 , ) , Ψ p , q ( x ) = h q ( x ) v q ( 1 1 p ) ( x ) (1.2)

Theorem 3LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive functional onC. If h : C [ 0 , ) is a superadditive functional onCand 0 < p , q < 1 , then the functional

Φ p , q : C [ 0 , ) , Φ p , q ( x ) = v q ( 1 1 p ) ( x ) h q ( x ) (1.3)

Another result similar to Theorem 1 has been obtained in [2] as well, namely

Theorem 4Let x , y C , h : C R be a nonnegative, superadditive ands-positive homogeneous functional onCandvbe an additive, strictly positive and positive homogeneous functional onC. If p , q 1 and M m 0 are such that x m y , M y x C , then

M s q + q ( 1 1 p ) Ψ p , q ( y ) Ψ p , q ( x ) m s q + q ( 1 1 p ) Ψ p , q ( y ) , (1.4)

where Ψ p , q is defined by (1.2).

As shown in [1] and [2], the above results can be applied to obtain refinements of the Jensen, Hölder, Minkowski and Schwarz inequalities for weights satisfying certain conditions.

The main aim of the present paper is to study quasilinearity properties of other composite 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 5 (Quasilinearity theorem)

LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive functional onC.

(i) If h : C [ 0 , ) is a superadditive (subadditive) functional onCand Φ : [ 0 , ) R is concave (convex) and monotonic nondecreasing on [ 0 , ) , then the composite functional η Φ : C R defined by

η Φ ( x ) : = v ( x ) Φ ( h ( x ) v ( x ) ) (2.1)

(ii) If h : C [ 0 , ) is a superadditive (subadditive) functional onCand Φ : [ 0 , ) R is convex (concave) and monotonic nonincreasing on [ 0 , ) , then the composite functional η Φ is subadditive (superadditive) onC.

Proof (i) Assume that h is superadditive and Φ : [ 0 , ) R is concave and monotonic nondecreasing on [ 0 , ) . Then

h ( x + y ) h ( x ) + h ( y ) for any  x , y C ,

and since v ( x + y ) = v ( x ) + v ( y ) for any x , y C , by the monotonicity of Φ, we have

Φ ( h ( x + y ) v ( x + y ) ) = Φ ( h ( x + y ) v ( x ) + v ( y ) ) Φ ( h ( x ) + h ( y ) v ( x ) + v ( y ) ) = Φ ( v ( x ) h ( x ) v ( x ) + v ( y ) h ( y ) v ( y ) v ( x ) + v ( y ) ) (2.2)

for any x , y C .

Now, since Φ : [ 0 , ) R is concave,

(2.3)

for any x , y C .

Utilizing (2.2) and (2.3), we get

v ( x + y ) Φ ( h ( x + y ) v ( x + y ) ) v ( x ) Φ ( h ( x ) v ( x ) ) + v ( y ) Φ ( h ( y ) v ( y ) ) (2.4)

for any x , y C , which shows that the functional η Φ is superadditive on C.

Now, if h : C R is a subadditive functional on C and Φ : [ 0 , ) [ 0 , ) is convex and monotonic nondecreasing on [ 0 , ) , then the inequalities (2.2), (2.3) and (2.4) hold with the reverse sign for any x , y C , which shows that the functional η Φ is subadditive on C.

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

Corollary 1 (Boundedness property)

LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive and positive homogeneous functional onC. Let x , y C and assume that there exist M m > 0 such that x m y and M y x C .

(a) If h : C [ 0 , ) is a superadditive and positive homogeneous functional onCand Φ : [ 0 , ) [ 0 , ) is concave and monotonic nondecreasing on [ 0 , ) , then

M η Φ ( y ) η Φ ( x ) m η Φ ( y ) . (2.5)

(aa) If h : C [ 0 , ) is a subadditive and positive homogeneous functional onCand Φ : [ 0 , ) [ 0 , ) is concave and monotonic nonincreasing on [ 0 , ) , then (2.5) is valid as well.

Proof We observe that if v and h are positive homogeneous functionals, then η Φ is also a positive homogeneous functional, and by the quasilinearity theorem above, it follows that in both cases η Φ is a superadditive functional on C. By applying Theorem 1 for s = 1 , we deduce the desired result. □

Remark 1 (Monotonicity property)

Let C be a convex cone in the linear space X. We say, for x , y X , that x C y (x is greater than y relative to the cone C) if x y C . Now, observe that if x , y C and x C y , then under the assumptions of Corollary 1, by (2.5), we have that η Φ ( x ) η Φ ( y ) , which is a monotonicity property for the functional η Φ .

There are various possibilities to build such functionals. For instance, for the finite families of functionals v i : C ( 0 , ) and h i : C [ 0 , ) with i I (I is a finite family of indices) and Φ : [ 0 , ) R is concave/convex and monotonic, then the composite functional σ Φ : C R defined by

σ Φ ( x ) : = i I v i ( x ) Φ ( h i ( x ) v i ( x ) ) (2.6)

has the same properties as the functional η Φ .

If, for a given cone C, we consider the Cartesian product C n : = C × × C X n and define, for the vector x ¯ : = ( x 1 , , x n ) C n , the functional ϖ Φ : C n R given by

ϖ Φ ( x ¯ ) : = i = 1 n v ( x i ) Φ ( h ( x i ) v ( x i ) ) , (2.7)

where v and h defined on C are as above, then we observe that ϖ Φ has the same properties as η Φ .

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

Proposition 1LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive functional onC.

(i) If h : C ( 0 , ) is a superadditive functional onCand r > 0 , then the composite functional η r : C ( 0 , ) defined by

η r ( x ) : = [ v ( x ) ] 1 + r [ h ( x ) ] r (2.8)

is subadditive onC. In particular, η 1 ( x ) = v 2 ( x ) h ( x ) is subadditive.

(ii) If h : C [ 0 , ) is a superadditive functional onCand q ( 0 , 1 ) , then the composite functional η q : C [ 0 , ) defined by

η q ( x ) : = [ v ( x ) ] 1 q [ h ( x ) ] q (2.9)

is superadditive onC. In particular, η 1 / 2 ( x ) = v ( x ) h ( x ) is superadditive.

(iii) If h : C [ 0 , ) is a subadditive functional onCand p 1 , then the composite functional η p : C [ 0 , ) defined by

η p ( x ) : = [ h ( x ) ] p [ v ( x ) ] p 1 (2.10)

is subadditive onC. In particular, η 2 ( x ) = h 2 ( x ) v ( x ) is subadditive.

Proof Follows from Theorem 5 for the function Φ : ( 0 , ) ( 0 , ) , Φ ( t ) = t s which is convex and decreasing for s ( , 0 ) , concave and increasing for s ( 0 , 1 ) and convex and increasing for s [ 1 , ) . The details are omitted. □

The following boundedness property also holds.

Corollary 2LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive and positive homogeneous functional onC. Let x , y C and assume that there exist M m > 0 such that x m y and M y x C . If h : C [ 0 , ) is a superadditive and positive homogeneous functional onCand q ( 0 , 1 ) , then

M [ v ( y ) ] 1 q [ h ( y ) ] q [ v ( x ) ] 1 q [ h ( x ) ] q m [ v ( y ) ] 1 q [ h ( y ) ] q . (2.11)

In particular,

M 2 v ( y ) h ( y ) v ( x ) h ( x ) m 2 v ( y ) h ( y ) . (2.12)

Proposition 2LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive functional onC.

(i) If h : C [ 0 , ) is a subadditive functional onC, then the composite functional ε α : C ( 0 , ) defined by

ε α ( x ) : = v ( x ) exp ( α h ( x ) v ( x ) ) (2.13)

is subadditive onCprovided α > 0 .

(ii) If h : C [ 0 , ) is a superadditive functional onC, then the composite functional ε α is also subadditive onCwhen α < 0 .

The proof follows from Theorem 5. The details are omitted.

Remark 2 Similar composite functionals can be considered for the functions Φ : [ 0 , ) R defined as follows:

For instance, if we consider the composite functional

η arctan ( x ) : = v ( x ) arctan ( h ( x ) v ( x ) ) ,

where h : C [ 0 , ) is a superadditive functional on C, v : C ( 0 , ) is an additive functional on C and C is a convex cone in the linear space X, then by the quasilinearity theorem, we conclude that η arctan : C [ 0 , ) is superadditive on C. Moreover, if v : C ( 0 , ) is an additive and positive homogeneous functional on C, h : C [ 0 , ) is a superadditive and positive homogeneous functional on C and x , y C such that there exist M m > 0 with the property that x m y and M y x C , then

M v ( y ) arctan ( h ( y ) v ( y ) ) v ( x ) arctan ( h ( x ) v ( x ) ) m v ( y ) arctan ( h ( y ) v ( y ) ) .

The same properties hold for the composite functional generated by the hyperbolic tangent function, namely

η tanh ( 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 of log-convex functions, namely functions Ψ : I ( 0 , ) , I is an interval of real numbers for which lnΨ is convex.

We observe that such functions satisfy the elementary inequality

Ψ ( ( 1 t ) a + t b ) [ Ψ ( a ) ] 1 t [ Ψ ( b ) ] t

for any a , b I and t [ 0 , 1 ] . 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 6 (Quasimultiplicity theorem)

LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive functional onC.

(i) If h : C [ 0 , ) is a superadditive (subadditive) functional onCand Ϝ : [ 0 , ) ( 0 , ) is log-concave (log-convex) and monotonic nondecreasing on [ 0 , ) , then the composite functional ξ Ϝ : C ( 0 , ) defined by

ξ Ϝ ( x ) : = [ Ϝ ( h ( x ) v ( x ) ) ] v ( x ) (2.14)

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

ξ Ϝ ( x + y ) ( ) ξ Ϝ ( x ) ξ Ϝ ( y ) (2.15)

for any x , y C .

(ii) If h : C [ 0 , ) is a superadditive (subadditive) functional onCand Ϝ : [ 0 , ) ( 0 , ) is log-convex (log-concave) and monotonic nonincreasing on [ 0 , ) , then the composite functional η Ϝ is submultiplicative (supemultiplicative) on C.

Proof We observe that

log ξ Ϝ ( x ) = v ( x ) log [ Ϝ ( h ( x ) v ( x ) ) ] = η log ( Ϝ ) ( x )

for any x C .

Applying now the quasilinearity theorem for the functions log ( Ϝ ) , we deduce the desired result.

The details are omitted. □

Corollary 3 (Exponential boundedness)

LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive and positive homogeneous functional onC. Let x , y C and assume that there exist M m > 0 such that x m y and M y x C .

(a) If h : C [ 0 , ) is a superadditive and positive homogeneous functional onCand Ϝ : [ 0 , ) [ 1 , ) is log-concave and monotonic nondecreasing on [ 0 , ) , then

[ Ϝ ( h ( y ) v ( y ) ) ] M v ( y ) [ Ϝ ( h ( x ) v ( x ) ) ] v ( x ) [ Ϝ ( h ( y ) v ( y ) ) ] m v ( y ) . (2.16)

(aa) If h : C [ 0 , ) is a subadditive and positive homogeneous functional onCand Ϝ : [ 0 , ) [ 1 , ) is log-concave and monotonic nonincreasing on [ 0 , ) , then (2.16) is valid as well.

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

Following [3], we consider the following Dirichlet series:

ψ ( s ) : = n = 1 a n n s (2.17)

for which we assume that the coefficients a n 0 for n 1 and the series is uniformly convergent for s > 1 .

It is obvious that in this class we can find the zeta function

ζ ( s ) : = n = 1 1 n s

and the lambda function

λ ( s ) : = n = 0 1 ( 2 n + 1 ) s = ( 1 2 s ) ζ ( s ) ,

where s > 1 .

If Λ ( n ) is the von Mangoldt function, where

Λ ( n ) : = { log p , n = p k ( p  prime,  k 1 ) , 0 , otherwise ,

then [[4], p.3]:

ζ ( s ) ζ ( s ) = n = 1 Λ ( n ) n s , s > 1 .

If d ( n ) is the number of divisors of n, we have [[4], p.35] the following relationships with the zeta function:

ζ 2 ( s ) = n = 1 d ( n ) n s , ζ 3 ( s ) ζ ( 2 s ) = n = 1 d ( n 2 ) n s , ζ 4 ( s ) ζ ( 2 s ) = n = 1 d 2 ( n ) n s ,

and [[4], p.36]

ζ 2 ( s ) ζ ( 2 s ) = n = 1 2 ω ( n ) n s , s > 1 ,

where ω ( n ) is the number of distinct prime factors of n.

We use the following result, see [3]

Lemma 1The functionψdefined by (2.17) is nonincreasing and log-convex on ( 1 , ) .

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

Proposition 3LetCbe a convex cone in the linear spaceXand v : C ( 0 , ) be an additive functional onC. If h : C [ 0 , ) is a subadditive functional onCand ψ : ( 1 , ) ( 0 , ) is defined by (2.17), then the composite functional ξ ψ : C ( 0 , ) defined by

ξ ψ ( x ) : = [ ψ ( h ( x ) + v ( x ) v ( x ) ) ] v ( x ) (2.18)

is submultiplicative onC.

Proof We observe that the function Ϝ ( t ) : = ψ ( t + 1 ) is well defined on ( 0 , ) and is nonincreasing and log-convex on this interval. Applying Theorem 6, we deduce the desired result. □

### 3 Applications

#### 3.1 Applications for Jensen’s inequality

Let C be a convex subset of the real linear space X and let f : C R be a convex mapping. Here we consider the following well-known form of Jensen’s discrete inequality:

f ( 1 P I i I p i x i ) 1 P I i I p i f ( x i ) , (3.1)

where I denotes a finite subset of the set ℕ of natural numbers, x i C , p i 0 for i I and P I : = i I p i > 0 .

Let us fix I P f ( N ) (the class of finite parts of ℕ) and x i C ( i I ). Now, consider the functional J : S + ( I ) R given by

J I ( p ) : = i I p i f ( x i ) P I f ( 1 P I i I p i x i ) 0 , (3.2)

where S + ( I ) : = { p = ( p i ) i I | p i 0 , i I  and  P I > 0 } and f is convex on C.

We observe that S + ( I ) is a convex cone and the functional J I is nonnegative and positive homogeneous on S + ( I ) .

Lemma 2 ([5])

The functional J I ( ) is a superadditive functional on S + ( I ) .

For a function Φ : [ 0 , ) R , define the following functional ξ Φ , I : S + ( I ) R :

ξ Φ , I ( p ) : = P I Φ ( 1 P I i I p i f ( x i ) f ( 1 P I i I p i x i ) ) . (3.3)

By the use of Theorem 5, we can state the following proposition.

Proposition 4If Φ : [ 0 , ) R is concave and monotonic nondecreasing on [ 0 , ) , then the composite functional ξ Φ , I : S + ( I ) R defined by (3.3) is superadditive on S + ( I ) .

If Φ : [ 0 , ) R is convex and monotonic nonincreasing on [ 0 , ) , then the composite functional ξ Φ , I is subadditive on S + ( I ) .

Proof Consider the functionals v ( p ) : = P I and h ( p ) : = J I ( p ) . We observe that v is additive, h is superadditive and

η Φ ( p ) = v ( p ) Φ ( h ( p ) v ( p ) ) = ξ Φ , I ( p ) .

Applying Theorem 5, we deduce the desired result. □

Corollary 4If p , q S + ( I ) and M m 0 are such that M p q m p , i.e., M p i q i m p i for each i I , then

(3.4)

for any Φ : [ 0 , ) [ 0 , ) concave and monotonic nondecreasing function on [ 0 , ) .

The proof follows from Corollary 1 and the details are omitted.

On utilizing Proposition 1, statement (ii), we observe that the functional ξ q , I : S + ( I ) [ 0 , ) , where q ( 0 , 1 ) and

ξ q , I ( p ) : = P I ( 1 P I i I p i f ( x i ) f ( 1 P I i I p i x i ) ) q = ( P I q 1 i I p i f ( x i ) P I q f ( 1 P I i I p i x i ) ) q , (3.5)

is superadditive and monotonic nondecreasing on S + ( I ) .

If p , q S + ( I ) and M m 0 are such that M p q m p , then

(3.6)

Now, if we consider the following composite functional ξ arctan , I : S + ( I ) [ 0 , ) given by

ξ arctan , I ( p ) = P I arctan ( 1 P I i I p i f ( x i ) f ( 1 P I i I p i x i ) ) , (3.7)

then by utilizing Remark 2 we conclude that ξ arctan , I is superadditive and monotonic nondecreasing on S + ( I ) .

Moreover, if p , q S + ( I ) and M m 0 are such that M p q m p , then

(3.8)

It is also well known that if f : C R is a strictly convex mapping on C and, for a given sequence of vectors x i C ( i I ), there exist at least two distinct indices k and j in I so that x k x j , then

J I ( p ) : = i I p i f ( x i ) P I f ( 1 P I i I p i x i ) > 0

for any p S + ( I ) = { p = ( p i ) i I | p i 0 , i I  and  P I > 0 } .

In this situation, for the function f and the sequence x i C ( i I ), we can define the functional

η r , I ( p ) : = P I 1 + r [ i I p i f ( x i ) P I f ( 1 P I i I p i x i ) ] r (3.9)

that is well defined on S + ( I ) . Utilizing the statement (i) from Proposition 1, we conclude that η r , I ( ) is a subadditive functional on S + ( I ) .

We know that the hyperbolic cotangent function coth ( t ) : = e t + e t e t e t is decreasing and convex on ( 0 , ) . If we consider the composite functional

ξ coth , I ( p ) : = P I coth ( 1 P I i I p i f ( x i ) f ( 1 P I i I p i x i ) )

for a function f : C R that is strictly convex on C and for a given sequence of vectors x i C ( i I ) for which there exist at least two distinct indices k and j in I so that x k x j , then we observe that this functional is well defined on S + ( I ) , and by the statement (ii) of Theorem 5, we conclude that ξ coth , I ( ) is also a subadditive functional on S + ( I ) .

#### 3.2 Applications for Hölder’s inequality

Let ( X , ) be a normed space and I P f ( N ) . We define

E ( I ) : = { x = ( x j ) j I | x j X , j I }

and

K ( I ) : = { λ = ( λ j ) j I | λ j K , j I } .

We consider for γ , β > 1 , 1 γ + 1 β = 1 the functional

H I ( p , λ , x ; γ , β ) : = ( j I p j | λ j | γ ) 1 γ ( j I p j x j β ) 1 β j I p j λ j x j .

The following result has been proved in [1].

Lemma 3For any p , q S + ( I ) , we have

H I ( p + q , λ , x ; γ , β ) H I ( p , λ , x ; γ , β ) + H I ( q , λ , x ; γ , β ) , (3.10)

where x E ( I ) , λ K ( I ) and γ , β > 1 with 1 γ + 1 β = 1 .

Remark 3 The same result can be stated if ( B , ) is a normed algebra and the functional H is defined by

H I ( p , λ , x ; γ , β ) : = ( i I p i x i γ ) 1 γ ( i I p i y i β ) 1 β i I p i x i y i ,

where x = ( x i ) i I , y = ( y i ) i I B , p S + ( I ) and γ , β > 1 with 1 γ + 1 β = 1 .

For a function Φ : [ 0 , ) R , define the following functional ω Φ , I : S + ( I ) R :

ω Φ , I ( p ) : = P I Φ ( ( 1 P I j I p j | λ j | γ ) 1 γ ( 1 P I j I p j x j β ) 1 β 1 P I j I p j λ j x j ) . (3.11)

By the use of Theorem 5, we can state the following proposition.

Proposition 5If Φ : [ 0 , ) R is concave and monotonic nondecreasing on [ 0 , ) , then the composite functional ω Φ , I : S + ( I ) R defined by (3.11) is superadditive on S + ( I ) .

If Φ : [ 0 , ) R is convex and monotonic nonincreasing on [ 0 , ) , then the composite functional ω Φ , I is subadditive on S + ( I ) .

By choosing various examples of concave and monotonic nondecreasing or convex and monotonic nonincreasing functions Φ on [ 0 , ) , the reader can provide various examples of superadditive or subadditive functionals on S + ( I ) . The details are omitted.

#### 3.3 Applications for Minkowski’s inequality

Let ( X , ) be a normed space and I P f ( N ) . We define the functional

M I ( p , x , y ; δ ) = [ ( i I p i x i δ ) 1 δ + ( i I p i y i δ ) 1 δ ] δ i I p i x i + y i δ , (3.12)

where p S + ( I ) , δ 1 and x , y E ( I ) .

The following result concerning the superadditivity of the functional M I ( , x , y ; δ ) holds [1].

Lemma 4For any p , q S + ( I ) , we have

M I ( p + q , x , y ; δ ) M I ( p , x , y ; δ ) + M I ( q , x , y ; δ ) ,

where x , y E ( I ) and δ 1 .

For a function Φ : [ 0 , ) R , define the following functional ϰ Φ , I : S + ( I ) R :

ϰ Φ , I ( p ) : = P I Φ ( [ ( 1 P I i I p i x i δ ) 1 δ + ( 1 P I i I p i y i δ ) 1 δ ] δ 1 P I i I p i x i + y i δ ) . (3.13)

By the use of Theorem 5, we can state the following proposition.

Proposition 6If Φ : [ 0 , ) R is concave and monotonic nondecreasing on [ 0 , ) , then the composite functional ϰ Φ , I : S + ( I ) R defined by (3.13) is superadditive on S + ( I ) .

If Φ : [ 0 , ) R is convex and monotonic nonincreasing on [ 0 , ) , then the composite functional ϰ Φ , I is subadditive on S + ( I ) .

We notice that, by choosing various examples of concave and monotonic nondecreasing or convex and monotonic nonincreasing functions Φ on [ 0 , ) , the reader can provide various examples of superadditive or subadditive functionals on S + ( I ) . The details are omitted.

Remark 4 For other examples of superadditive (subadditive) functionals that can provide interesting inequalities similar to the ones outlined above, we refer to [6-9] and [10-12].

### Competing interests

The author declares that he has no competing interests.

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