Quasilinearity of some functionals associated with monotonic convex functions

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.

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 is called a convex cone in X provided the following conditions hold:

(i) imply ;

(ii) , imply .

A functional is called superadditive (subadditive) on C if

(iii) for any

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

(iv) for each .

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

(v) for any and .

If , we simply call it positive homogeneous.

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

Theorem 1Letandbe a nonnegative, superadditive ands-positive homogeneous functional onC. Ifare such thatand, then

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

(vi) for any and .

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

Theorem 2LetCbe a convex cone in the linear spaceXandbe an additive functional onC. Ifis a superadditive (subadditive) functional onCand (, ), then the functional

is superadditive (subadditive) onC.

Theorem 3LetCbe a convex cone in the linear spaceXandbe an additive functional onC. Ifis a superadditive functional onCand, then the functional

is subadditive onC.

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

Theorem 4Let, be a nonnegative, superadditive ands-positive homogeneous functional onCandvbe an additive, strictly positive and positive homogeneous functional onC. Ifandare such that, , then

whereis 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.

We start with the following general result.

Theorem 5 (Quasilinearity theorem)

LetCbe a convex cone in the linear spaceXandbe an additive functional onC.

(i) Ifis a superadditive (subadditive) functional onCandis concave (convex) and monotonic nondecreasing on, then the composite functionaldefined by

is superadditive (subadditive) onC.

(ii) Ifis a superadditive (subadditive) functional onCandis convex (concave) and monotonic nonincreasing on, then the composite functionalis subadditive (superadditive) onC.

Proof (i) Assume that h is superadditive and is concave and monotonic nondecreasing on . Then

and since for any , by the monotonicity of Φ, we have

for any .

Now, since is concave,

for any .

Utilizing (2.2) and (2.3), we get

for any , which shows that the functional is superadditive on C.

Now, if is a subadditive functional on C and is convex and monotonic nondecreasing on , then the inequalities (2.2), (2.3) and (2.4) hold with the reverse sign for any , which shows that the functional 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 spaceXandbe an additive and positive homogeneous functional onC. Letand assume that there existsuch thatand.

(a) Ifis a superadditive and positive homogeneous functional onCandis concave and monotonic nondecreasing on, then

(aa) Ifis a subadditive and positive homogeneous functional onCandis concave and monotonic nonincreasing on, then (2.5) is valid as well.

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

Remark 1 (Monotonicity property)

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

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

has the same properties as the functional .

If, for a given cone C, we consider the Cartesian product and define, for the vector , the functional given by

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

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

Proposition 1LetCbe a convex cone in the linear spaceXandbe an additive functional onC.

(i) Ifis a superadditive functional onCand, then the composite functional: defined by

is subadditive onC. In particular, is subadditive.

(ii) Ifis a superadditive functional onCand, then the composite functionaldefined by

is superadditive onC. In particular, is superadditive.

(iii) Ifis a subadditive functional onCand, then the composite functionaldefined by

is subadditive onC. In particular, is subadditive.

Proof Follows from Theorem 5 for the function , which is convex and decreasing for , concave and increasing for and convex and increasing for . The details are omitted. □

The following boundedness property also holds.

Corollary 2LetCbe a convex cone in the linear spaceXandbe an additive and positive homogeneous functional onC. Letand assume that there existsuch thatand. Ifis a superadditive and positive homogeneous functional onCand, then

In particular,

Proposition 2LetCbe a convex cone in the linear spaceXandbe an additive functional onC.

(i) Ifis a subadditive functional onC, then the composite functionaldefined by

is subadditive onCprovided.

(ii) Ifis a superadditive functional onC, then the composite functionalis also subadditive onCwhen.

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

Remark 2 Similar composite functionals can be considered for the functions defined as follows:

For instance, if we consider the composite functional

where is a superadditive functional on C, 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 is superadditive on C. Moreover, if is an additive and positive homogeneous functional on C, is a superadditive and positive homogeneous functional on C and such that there exist with the property that and , then

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

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 , I is an interval of real numbers for which lnΨ is convex.

We observe that such functions satisfy the elementary inequality

for any and . 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 spaceXandbe an additive functional onC.

(i) Ifis a superadditive (subadditive) functional onCandis log-concave (log-convex) and monotonic nondecreasing on, then the composite functionaldefined by

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

for any.

(ii) Ifis a superadditive (subadditive) functional onCandis log-convex (log-concave) and monotonic nonincreasing on, then the composite functionalis submultiplicative (supemultiplicative) on C.

Proof We observe that

for any .

Applying now the quasilinearity theorem for the functions , we deduce the desired result.

The details are omitted. □

Corollary 3 (Exponential boundedness)

LetCbe a convex cone in the linear spaceXandbe an additive and positive homogeneous functional onC. Letand assume that there existsuch thatand.

(a) Ifis a superadditive and positive homogeneous functional onCandis log-concave and monotonic nondecreasing on, then

(aa) Ifis a subadditive and positive homogeneous functional onCandis log-concave and monotonic nonincreasing on, 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:

for which we assume that the coefficients for and the series is uniformly convergent for .

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

and the lambda function

where .

If is the von Mangoldt function, where

then [[4], p.3]:

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

and [[4], p.36]

where 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.

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

Proposition 3LetCbe a convex cone in the linear spaceXandbe an additive functional onC. Ifis a subadditive functional onCandis defined by (2.17), then the composite functionaldefined by

is submultiplicative onC.

Proof We observe that the function is well defined on and is nonincreasing and log-convex on this interval. Applying Theorem 6, we deduce the desired result. □

3.1 Applications for Jensen’s inequality

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

where I denotes a finite subset of the set ℕ of natural numbers, , for and .

Let us fix (the class of finite parts of ℕ) and (). Now, consider the functional given by

where and f is convex on C.

We observe that is a convex cone and the functional is nonnegative and positive homogeneous on .

Lemma 2 ([5])

The functionalis a superadditive functional on.

For a function , define the following functional :

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

Proposition 4Ifis concave and monotonic nondecreasing on, then the composite functionaldefined by (3.3) is superadditive on.

Ifis convex and monotonic nonincreasing on, then the composite functionalis subadditive on.

Proof Consider the functionals and . We observe that v is additive, h is superadditive and

Applying Theorem 5, we deduce the desired result. □

Corollary 4Ifandare such that, i.e., for each, then

for anyconcave and monotonic nondecreasing function on.

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

On utilizing Proposition 1, statement (ii), we observe that the functional , where and

is superadditive and monotonic nondecreasing on .

If and are such that then

Now, if we consider the following composite functional given by

then by utilizing Remark 2 we conclude that is superadditive and monotonic nondecreasing on .

Moreover, if and are such that then

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

for any .

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

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

We know that the hyperbolic cotangent function is decreasing and convex on . If we consider the composite functional

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

3.2 Applications for Hölder’s inequality

Let be a normed space and . We define

and

We consider for , the functional

The following result has been proved in [1].

Lemma 3For any, we have

where, andwith.

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

where , and with .

For a function , define the following functional :

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

Proposition 5Ifis concave and monotonic nondecreasing on, then the composite functionaldefined by (3.11) is superadditive on.

Ifis convex and monotonic nonincreasing on, then the composite functionalis subadditive on.

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

3.3 Applications for Minkowski’s inequality

Let be a normed space and . We define the functional

where , and .

The following result concerning the superadditivity of the functional holds [1].

Lemma 4For any, we have

whereand.

For a function , define the following functional :

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

Proposition 6Ifis concave and monotonic nondecreasing on, then the composite functionaldefined by (3.13) is superadditive on.

Ifis convex and monotonic nonincreasing on, then the composite functionalis subadditive on.

We notice that, by choosing various examples of concave and monotonic nondecreasing or convex and monotonic nonincreasing functions Φ on , the reader can provide various examples of superadditive or subadditive functionals on . 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].

The author declares that he has no competing interests.

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