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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

© 2012 Dragomir; licensee Springer

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M1">View MathML</a> is called a convex cone in X provided the following conditions hold:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a> imply <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M3">View MathML</a>;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M5">View MathML</a> imply <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M6">View MathML</a>.

A functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7">View MathML</a> is called superadditive (subadditive) on C if

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M8">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>

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

(iv) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M10">View MathML</a> for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4">View MathML</a>.

The functional h is s-positive homogeneous on C for a given <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M12">View MathML</a> if

(v) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M13">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M5">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4">View MathML</a>.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M16">View MathML</a>, we simply call it positive homogeneous.

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

Theorem 1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7">View MathML</a>be a nonnegative, superadditive ands-positive homogeneous functional onC. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19">View MathML</a>are such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M22">View MathML</a>

(1.1)

Now, consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M23">View MathML</a> an additive and strictly positive functional on C which is also positive homogeneous on C, i.e.,

(vi) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M24">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M25">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4">View MathML</a>.

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

Theorem 2LetCbe a convex cone in the linear spaceXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive functional onC. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive (subadditive) functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M29">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M30">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M31">View MathML</a>), then the functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M32">View MathML</a>

(1.2)

is superadditive (subadditive) onC.

Theorem 3LetCbe a convex cone in the linear spaceXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive functional onC. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M35">View MathML</a>, then the functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M36">View MathML</a>

(1.3)

is subadditive onC.

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

Theorem 4Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7">View MathML</a>be a nonnegative, superadditive ands-positive homogeneous functional onCandvbe an additive, strictly positive and positive homogeneous functional onC. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M29">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19">View MathML</a>are such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M43">View MathML</a>

(1.4)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M44">View MathML</a>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<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive functional onC.

(i) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive (subadditive) functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is concave (convex) and monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M49">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M50">View MathML</a>

(2.1)

is superadditive (subadditive) onC.

(ii) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive (subadditive) functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is convex (concave) and monotonic nonincreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a>is subadditive (superadditive) onC.

Proof (i) Assume that h is superadditive and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a> is concave and monotonic nondecreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M57">View MathML</a>

and since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M58">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>, by the monotonicity of Φ, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M60">View MathML</a>

(2.2)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>.

Now, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a> is concave,

(2.3)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>.

Utilizing (2.2) and (2.3), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M65">View MathML</a>

(2.4)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>, which shows that the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a> is superadditive on C.

Now, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M7">View MathML</a> is a subadditive functional on C and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69">View MathML</a> is convex and monotonic nondecreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the inequalities (2.2), (2.3) and (2.4) hold with the reverse sign for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>, which shows that the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a> 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<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive and positive homogeneous functional onC. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>and assume that there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M75">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21">View MathML</a>.

(a) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive and positive homogeneous functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69">View MathML</a>is concave and monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M81">View MathML</a>

(2.5)

(aa) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a subadditive and positive homogeneous functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69">View MathML</a>is concave and monotonic nonincreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then (2.5) is valid as well.

Proof We observe that if v and h are positive homogeneous functionals, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a> is also a positive homogeneous functional, and by the quasilinearity theorem above, it follows that in both cases <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a> is a superadditive functional on C. By applying Theorem 1 for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M16">View MathML</a>, we deduce the desired result. □

Remark 1 (Monotonicity property)

Let C be a convex cone in the linear space X. We say, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M88">View MathML</a>, that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M89">View MathML</a> (x is greater than y relative to the cone C) if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M90">View MathML</a>. Now, observe that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M89">View MathML</a>, then under the assumptions of Corollary 1, by (2.5), we have that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M93">View MathML</a>, which is a monotonicity property for the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a>.

There are various possibilities to build such functionals. For instance, for the finite families of functionals <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M95">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M96">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M97">View MathML</a> (I is a finite family of indices) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a> is concave/convex and monotonic, then the composite functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M99">View MathML</a>: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M100">View MathML</a> defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M101">View MathML</a>

(2.6)

has the same properties as the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a>.

If, for a given cone C, we consider the Cartesian product<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M103">View MathML</a> and define, for the vector <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M104">View MathML</a>, the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M105">View MathML</a> given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M106">View MathML</a>

(2.7)

where v and h defined on C are as above, then we observe that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M107">View MathML</a> has the same properties as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M54">View MathML</a>.

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

Proposition 1LetCbe a convex cone in the linear spaceXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive functional onC.

(i) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M110">View MathML</a>is a superadditive functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M111">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M112">View MathML</a>: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M113">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M114">View MathML</a>

(2.8)

is subadditive onC. In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M115">View MathML</a>is subadditive.

(ii) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M117">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M118">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M119">View MathML</a>

(2.9)

is superadditive onC. In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M120">View MathML</a>is superadditive.

(iii) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a subadditive functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M122">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M123">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M124">View MathML</a>

(2.10)

is subadditive onC. In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M125">View MathML</a>is subadditive.

Proof Follows from Theorem 5 for the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M126">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M127">View MathML</a> which is convex and decreasing for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M128">View MathML</a>, concave and increasing for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M129">View MathML</a> and convex and increasing for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M130">View MathML</a>. The details are omitted. □

The following boundedness property also holds.

Corollary 2LetCbe a convex cone in the linear spaceXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive and positive homogeneous functional onC. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>and assume that there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M75">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M136">View MathML</a>is a superadditive and positive homogeneous functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M117">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M138">View MathML</a>

(2.11)

In particular,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M139">View MathML</a>

(2.12)

Proposition 2LetCbe a convex cone in the linear spaceXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive functional onC.

(i) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a subadditive functional onC, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M142">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M143">View MathML</a>

(2.13)

is subadditive onCprovided<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M25">View MathML</a>.

(ii) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive functional onC, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M146">View MathML</a>is also subadditive onCwhen<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M147">View MathML</a>.

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

Remark 2 Similar composite functionals can be considered for the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a> defined as follows:

For instance, if we consider the composite functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M150">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M151">View MathML</a> is a superadditive functional on C, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a> 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M153">View MathML</a> is superadditive on C. Moreover, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a> is an additive and positive homogeneous functional on C, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a> is a superadditive and positive homogeneous functional on C and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a> such that there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M157">View MathML</a> with the property that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M160">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M161">View MathML</a>

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M162">View MathML</a>, I is an interval of real numbers for which lnΨ is convex.

We observe that such functions satisfy the elementary inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M163">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M164">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M165">View MathML</a>. 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<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive functional onC.

(i) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive (subadditive) functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M168">View MathML</a>is log-concave (log-convex) and monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M170">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M171">View MathML</a>

(2.14)

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M172">View MathML</a>

(2.15)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>.

(ii) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive (subadditive) functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M168">View MathML</a>is log-convex (log-concave) and monotonic nonincreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M177">View MathML</a>is submultiplicative (supemultiplicative) on C.

Proof We observe that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M178">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M4">View MathML</a>.

Applying now the quasilinearity theorem for the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M180">View MathML</a>, we deduce the desired result.

The details are omitted. □

Corollary 3 (Exponential boundedness)

LetCbe a convex cone in the linear spaceXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive and positive homogeneous functional onC. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M2">View MathML</a>and assume that there exist<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M75">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M20">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M21">View MathML</a>.

(a) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a superadditive and positive homogeneous functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M187">View MathML</a>is log-concave and monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M189">View MathML</a>

(2.16)

(aa) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a subadditive and positive homogeneous functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M187">View MathML</a>is log-concave and monotonic nonincreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, 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:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M193">View MathML</a>

(2.17)

for which we assume that the coefficients <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M194">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M195">View MathML</a> and the series is uniformly convergent for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M196">View MathML</a>.

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M197">View MathML</a>

and the lambda function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M198">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M196">View MathML</a>.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M200">View MathML</a> is the von Mangoldt function, where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M201">View MathML</a>

then [[4], p.3]:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M202">View MathML</a>

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M203">View MathML</a> is the number of divisors of n, we have [[4], p.35] the following relationships with the zeta function:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M204">View MathML</a>

and [[4], p.36]

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M205">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M206">View MathML</a> 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<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M207">View MathML</a>.

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

Proposition 3LetCbe a convex cone in the linear spaceXand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M27">View MathML</a>be an additive functional onC. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M28">View MathML</a>is a subadditive functional onCand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M210">View MathML</a>is defined by (2.17), then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M211">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M212">View MathML</a>

(2.18)

is submultiplicative onC.

Proof We observe that the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M213">View MathML</a> is well defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M214">View MathML</a> 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M215">View MathML</a> be a convex mapping. Here we consider the following well-known form of Jensen’s discrete inequality:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M216">View MathML</a>

(3.1)

where I denotes a finite subset of the set ℕ of natural numbers, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M218">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M97">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M220">View MathML</a>.

Let us fix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M221">View MathML</a> (the class of finite parts of ℕ) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223">View MathML</a>). Now, consider the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M224">View MathML</a> given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M225">View MathML</a>

(3.2)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M226">View MathML</a> and f is convex on C.

We observe that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a> is a convex cone and the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M228">View MathML</a> is nonnegative and positive homogeneous on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

Lemma 2 ([5])

The functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M230">View MathML</a>is a superadditive functional on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

For a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>, define the following functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M233">View MathML</a>:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M234">View MathML</a>

(3.3)

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

Proposition 4If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is concave and monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M233">View MathML</a>defined by (3.3) is superadditive on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is convex and monotonic nonincreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M241">View MathML</a>is subadditive on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

Proof Consider the functionals <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M243">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M244">View MathML</a>. We observe that v is additive, h is superadditive and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M245">View MathML</a>

Applying Theorem 5, we deduce the desired result. □

Corollary 4If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19">View MathML</a>are such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M248">View MathML</a>, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M249">View MathML</a>for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M97">View MathML</a>, then

(3.4)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M69">View MathML</a>concave and monotonic nondecreasing function on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>.

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

On utilizing Proposition 1, statement (ii), we observe that the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M254">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M117">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M256">View MathML</a>

(3.5)

is superadditive and monotonic nondecreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19">View MathML</a> are such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M260">View MathML</a> then

(3.6)

Now, if we consider the following composite functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M262">View MathML</a> given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M263">View MathML</a>

(3.7)

then by utilizing Remark 2 we conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M264">View MathML</a> is superadditive and monotonic nondecreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M265">View MathML</a>.

Moreover, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M19">View MathML</a> are such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M260">View MathML</a> then

(3.8)

It is also well known that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M215">View MathML</a> is a strictly convex mapping on C and, for a given sequence of vectors <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223">View MathML</a>), there exist at least two distinct indices k and j in I so that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M273">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M274">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M275">View MathML</a>.

In this situation, for the function f and the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223">View MathML</a>), we can define the functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M278">View MathML</a>

(3.9)

that is well defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>. Utilizing the statement (i) from Proposition 1, we conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M280">View MathML</a> is a subadditive functional on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M281">View MathML</a>.

We know that the hyperbolic cotangent function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M282">View MathML</a> is decreasing and convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M283">View MathML</a>. If we consider the composite functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M284">View MathML</a>

for a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M215">View MathML</a> that is strictly convex on C and for a given sequence of vectors <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M217">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M223">View MathML</a>) for which there exist at least two distinct indices k and j in I so that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M273">View MathML</a>, then we observe that this functional is well defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M289">View MathML</a>, and by the statement (ii) of Theorem 5, we conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M290">View MathML</a> is also a subadditive functional on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M265">View MathML</a>.

3.2 Applications for Hölder’s inequality

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M292">View MathML</a> be a normed space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M221">View MathML</a>. We define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M294">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M295">View MathML</a>

We consider for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M296">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M297">View MathML</a> the functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M298">View MathML</a>

The following result has been proved in [1].

Lemma 3For any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M300">View MathML</a>

(3.10)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M301">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M302">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M296">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M297">View MathML</a>.

Remark 3 The same result can be stated if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M305">View MathML</a> is a normed algebra and the functional H is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M306">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M307">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M308">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M296">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M297">View MathML</a>.

For a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>, define the following functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M312">View MathML</a>:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M313">View MathML</a>

(3.11)

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

Proposition 5If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is concave and monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M312">View MathML</a>defined by (3.11) is superadditive on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is convex and monotonic nonincreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M320">View MathML</a>is subadditive on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

By choosing various examples of concave and monotonic nondecreasing or convex and monotonic nonincreasing functions Φ on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, the reader can provide various examples of superadditive or subadditive functionals on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>. The details are omitted.

3.3 Applications for Minkowski’s inequality

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M292">View MathML</a> be a normed space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M221">View MathML</a>. We define the functional

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M326">View MathML</a>

(3.12)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M308">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M328">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M329">View MathML</a>.

The following result concerning the superadditivity of the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M330">View MathML</a> holds [1].

Lemma 4For any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M246">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M332">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M329">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M328">View MathML</a>.

For a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>, define the following functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M336">View MathML</a>:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M337">View MathML</a>

(3.13)

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

Proposition 6If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is concave and monotonic nondecreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M336">View MathML</a>defined by (3.13) is superadditive on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M47">View MathML</a>is convex and monotonic nonincreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, then the composite functional<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M344">View MathML</a>is subadditive on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>.

We notice that, by choosing various examples of concave and monotonic nondecreasing or convex and monotonic nonincreasing functions Φ on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M48">View MathML</a>, the reader can provide various examples of superadditive or subadditive functionals on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/276/mathml/M227">View MathML</a>. 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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