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 inequality1 Introduction
The problem of studying the quasilinearity properties of functionals associated with some celebrated inequalities such as the Jensen, CauchyBunyakowskySchwarz, 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
(i)
(ii)
A functional
(iii)
and nonnegative (strictly positive) on C if, obviously, it satisfies
(iv)
The functional h is spositive homogeneous on C for a given
(v)
If
In [1], the following result has been obtained.
Theorem 1Let
Now, consider
(vi)
In [2] we obtained further results concerning the quasilinearity of some composite functionals.
Theorem 2LetCbe a convex cone in the linear spaceXand
is superadditive (subadditive) onC.
Theorem 3LetCbe a convex cone in the linear spaceXand
is subadditive onC.
Another result similar to Theorem 1 has been obtained in [2] as well, namely
Theorem 4Let
where
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
(i) If
is superadditive (subadditive) onC.
(ii) If
Proof (i) Assume that h is superadditive and
and since
for any
Now, since
for any
Utilizing (2.2) and (2.3), we get
for any
Now, if
(ii) Follows in a similar manner and the details are omitted. □
Corollary 1 (Boundedness property)
LetCbe a convex cone in the linear spaceXand
(a) If
(aa) If
Proof We observe that if v and h are positive homogeneous functionals, then
Remark 1 (Monotonicity property)
Let C be a convex cone in the linear space X. We say, for
There are various possibilities to build such functionals. For instance, for the finite
families of functionals
has the same properties as the functional
If, for a given cone C, we consider the Cartesian product
where v and h defined on C are as above, then we observe that
There are some natural examples of composite functionals that are embodied in the propositions below.
Proposition 1LetCbe a convex cone in the linear spaceXand
(i) If
is subadditive onC. In particular,
(ii) If
is superadditive onC. In particular,
(iii) If
is subadditive onC. In particular,
Proof Follows from Theorem 5 for the function
The following boundedness property also holds.
Corollary 2LetCbe a convex cone in the linear spaceXand
In particular,
Proposition 2LetCbe a convex cone in the linear spaceXand
(i) If
is subadditive onCprovided
(ii) If
The proof follows from Theorem 5. The details are omitted.
Remark 2 Similar composite functionals can be considered for the functions
For instance, if we consider the composite functional
where
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 logconvex functions, namely functions
We observe that such functions satisfy the elementary inequality
for any
Theorem 6 (Quasimultiplicity theorem)
LetCbe a convex cone in the linear spaceXand
(i) If
is supermultiplicative (submultiplicative) onC, i.e., we recall that
for any
(ii) If
Proof We observe that
for any
Applying now the quasilinearity theorem for the functions
The details are omitted. □
Corollary 3 (Exponential boundedness)
LetCbe a convex cone in the linear spaceXand
(a) If
(aa) If
There are numerous examples of logconvex (logconcave) functions of interest that can provide some nice examples.
Following [3], we consider the following Dirichlet series:
for which we assume that the coefficients
It is obvious that in this class we can find the zeta function
and the lambda function
where
If
then [[4], p.3]:
If
and [[4], p.36]
where
We use the following result, see [3]
Lemma 1The functionψdefined by (2.17) is nonincreasing and logconvex on
Utilizing the quasimultiplicity theorem and this lemma, we can state the following result as well.
Proposition 3LetCbe a convex cone in the linear spaceXand
is submultiplicative onC.
Proof We observe that the function
3 Applications
3.1 Applications for Jensen’s inequality
Let C be a convex subset of the real linear space X and let
where I denotes a finite subset of the set ℕ of natural numbers,
Let us fix
where
We observe that
Lemma 2 ([5])
The functional
For a function
By the use of Theorem 5, we can state the following proposition.
Proposition 4If
If
Proof Consider the functionals
Applying Theorem 5, we deduce the desired result. □
Corollary 4If
for any
The proof follows from Corollary 1 and the details are omitted.
On utilizing Proposition 1, statement (ii), we observe that the functional
is superadditive and monotonic nondecreasing on
If
Now, if we consider the following composite functional
then by utilizing Remark 2 we conclude that
Moreover, if
It is also well known that if
for any
In this situation, for the function f and the sequence
that is well defined on
We know that the hyperbolic cotangent function
for a function
3.2 Applications for Hölder’s inequality
Let
and
We consider for
The following result has been proved in [1].
Lemma 3For any
where
Remark 3 The same result can be stated if
where
For a function
By the use of Theorem 5, we can state the following proposition.
Proposition 5If
If
By choosing various examples of concave and monotonic nondecreasing or convex and
monotonic nonincreasing functions Φ on
3.3 Applications for Minkowski’s inequality
Let
where
The following result concerning the superadditivity of the functional
Lemma 4For any
where
For a function
By the use of Theorem 5, we can state the following proposition.
Proposition 6If
If
We notice that, by choosing various examples of concave and monotonic nondecreasing
or convex and monotonic nonincreasing functions Φ on
Remark 4 For other examples of superadditive (subadditive) functionals that can provide interesting inequalities similar to the ones outlined above, we refer to [69] and [1012].
Competing interests
The author declares that he has no competing interests.
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