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.


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.
Let X be a linear space.A subset C ⊆ X is called a convex cone in X provided the following conditions hold: 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) ≥ (>)  for each x ∈ C.
The functional h is s-positive homogeneous on C for a given s >  if (v) h(αx) = α s h(x) for any α ≥  and x ∈ C. If s = , we simply call it positive homogeneous.
In [], the following result has been obtained.Now, consider v : C → R an additive and strictly positive functional on C which is also positive homogeneous on C, i.e., (vi) v(αx) = αv(x) for any α >  and x ∈ C.
In [] we obtained further results concerning the quasilinearity of some composite functionals.
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 Theorem  Let C be a convex cone in the linear space X and v : Another result similar to Theorem  has been obtained in [] as well, namely Theorem  Let x, y ∈ C, h : C → R be a nonnegative, superadditive and s-positive homogeneous 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 xmy, Myx ∈ C, then where p,q is 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 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.

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. http://www.journalofinequalitiesandapplications.com/content/2012/1/276 and monotonic nonincreasing on [, ∞), then the composite functional η is subadditive (superadditive) on C. Proof (i) Assume that h is superadditive and : [, ∞) → R is concave and monotonic nondecreasing on [, ∞).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 , we have for any x, y ∈ C. Utilizing (.) and (.), we get for any x, y ∈ C, which shows that the functional η is superadditive on C.
) is convex and monotonic nondecreasing on [, ∞), then the inequalities (.), (.) and (.) hold with the reverse sign for any x, y ∈ C, which shows that the functional η is subadditive on C.
(ii) Follows in a similar manner and the details are omitted.
Proof We observe that if v and h are positive homogeneous functionals, then η is also a positive homogeneous functional, and by the quasilinearity theorem above, it follows that in both cases η is a superadditive functional on C. By applying Theorem  for s = , we deduce the desired result.
Remark  (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 xy ∈ C. Now, observe that if x, y ∈ C and x ≥ C y, then under the assumptions of Corollary , by (.), we have that η (x) ≥ η (y), which is a monotonicity property for the functional η .
There are various possibilities to build such functionals.For instance, for the finite families of functionals has the same properties as the functional η .If, for a given cone C, we consider the Cartesian product C n := C × • • • × C ⊂ X n and define, for the vector x := (x  , . . ., x n ) ∈ C n , the functional : where v and h defined on C are as above, then we observe that has the same properties as η .
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
) is a superadditive functional on C and q ∈ (, ), then the composite functional η q : C → [, ∞) defined by Proof Follows from Theorem  for the function : (, ∞) → (, ∞), (t) = t s which is convex and decreasing for s ∈ (-∞, ), concave and increasing for s ∈ (, ) and convex and increasing for s ∈ [, ∞).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 xmy and My - ) is a superadditive and positive homogeneous functional on C and q ∈ (, ), then In particular, Proposition  Let C be a convex cone in the linear space X and v : C → (, ∞) be an additive functional on C.
The proof follows from Theorem .The details are omitted.
For instance, if we consider the composite functional ) 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

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 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 → (, ∞), I is an interval of real numbers for which ln is convex.
We observe that such functions satisfy the elementary inequality Theorem  (Quasimultiplicity theorem) Let C be a convex cone in the linear space X and v : C → (, ∞) be an additive functional on C.
is supermultiplicative (submultiplicative) on C, i.e., we recall that for any x ∈ C. Applying now the quasilinearity theorem for the functions log( ), 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 xmy and My - ) is a superadditive and positive homogeneous functional on C and for which we assume that the coefficients a n ≥  for n ≥  and the series is uniformly convergent for s > .
It is obvious that in this class we can find the zeta function and the lambda function where s > .
If (n) is the von Mangoldt function, where If d(n) is the number of divisors of n, we have [, p.] the following relationships with the zeta function: and [, p.] where ω(n) is the number of distinct prime factors of n.

We use the following result, see []
Lemma  The function ψ defined 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 additive functional on C. If h
is submultiplicative on C. http://www.journalofinequalitiesandapplications.com/content/2012/1/276 Proof We observe that the function (t) := ψ(t + ) is well defined on (, ∞) and is nonincreasing and log-convex on this interval.Applying Theorem , we deduce the desired result.

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: where I denotes a finite subset of the set N of natural numbers, x i ∈ C, p i ≥  for i ∈ I and 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 where S + (I) := {p = (p i ) i∈I |p i ≥ , i ∈ I and P I > } 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  ([]) The functional J I (•) is a superadditive functional on S + (I).
For a function : [, ∞) → R, define the following functional ξ ,I : S + (I) → R: By the use of Theorem , we can state the following proposition.
Proof Consider the functionals v(p) := P I and h(p) := J I (p).We observe that v is additive, h is superadditive and Applying Theorem , we deduce the desired result.
The proof follows from Corollary  and the details are omitted.On utilizing Proposition , statement (ii), we observe that the functional ξ q,I : S + (I) → [, ∞), where q ∈ (, ) and ξ q,I (p) : is superadditive and monotonic nondecreasing on S + (I).If p, q ∈ S + (I) and M ≥ m ≥  are such that Mp ≥ q ≥ mp, then Now, if we consider the following composite functional ξ arctan,I : S + (I) → [, ∞) given by ξ arctan,I (p) = P I arctan then by utilizing Remark  we conclude that ξ arctan,I is superadditive and monotonic nondecreasing on S + (I).Moreover, if p, q ∈ S + (I) and M ≥ m ≥  are such that Mp ≥ q ≥ mp, then 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 , we conclude that ξ coth,I (•) is also a subadditive functional on S + (I).

Applications for Hölder's inequality
Let (X, • ) be a normed space and I ∈ P f (N).We define The following result has been proved in [].
Lemma  For any p, q ∈ S + (I), we have Remark  The same result can be stated if (B, • ) is a normed algebra and the functional H is defined by

Applications for Minkowski's inequality
Let (X, • ) be a normed space and I ∈ P f (N).We define the functional Lemma  For 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 ≥ .

Theorem
Let x, y ∈ C and h : C → R be a nonnegative, superadditive and s-positive homogeneous functional on C. If M ≥ m ≥  are such that xmy and Myx ∈ C, then M s h(y) ≥ h(x) ≥ m s h(y).(.) ∞) is an additive and positive homogeneous functional on C, h : C → [, ∞) is a superadditive and positive homogeneous functional on C and x, y ∈ C such that there exist M ≥ m >  with the property that xmy and Myx ∈ C, then
For a function : [, ∞) → R, define the following functional κ ,I : S + (I) → R: κ ,I (p) := P I  p i x i + y i δ .(.)By the use of Theorem , we can state the following proposition.
) http://www.journalofinequalitiesandapplications.com/content/2012/1/276It 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 ∈S + (I) = {p = (p i ) i∈I |p i ≥ , i ∈ I and P I > }.In this situation, for the function f and the sequence x i ∈ C (i ∈ I), we can define the functional + (I).Utilizing the statement (i) from Proposition , 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 (, ∞).If we consider the composite functional