Journal of Function Spaces

Volume 2015, Article ID 919470, 8 pages

http://dx.doi.org/10.1155/2015/919470

## Functions Like Convex Functions

Mechanical Engineering Faculty in Slavonski Brod, University of Osijek, Trg Ivane Brlić Mažuranić 2, 35000 Slavonski Brod, Croatia

Received 30 July 2014; Accepted 6 October 2014

Academic Editor: Janusz Matkowski

Copyright © 2015 Zlatko Pavić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper deals with convex sets, functions satisfying the global convexity property, and positive linear functionals. Jensen's type inequalities can be obtained by using convex combinations with the common center. Following the idea of the common center, the functional forms of Jensen's inequality are considered in this paper.

#### 1. Introduction

Introduction is intended to be a brief overview of the concept of convexity and affinity. Let be a real linear space. Let be points and let be coefficients. Their binomial combinationis convex if and ifIf , then the point itself is called the combination center.

A set is convex if it contains all binomial convex combinations of its points. The convex hull of the set is the smallest convex set containing , and it consists of all binomial convex combinations of points of .

Let be a convex set. A function is convex if the inequalityholds for all binomial convex combinations of pairs of points .

Requiring only the condition in (2) for coefficients and requiring the equality in (3), we get a characterization of the affinity.

Implementing mathematical induction, we can prove that all of the above applies to -membered combinations for any positive integer . In that case, the inequality in (3) is the famous Jensen’s inequality obtained in [1]. Numerous papers have been written on Jensen’s inequality; different types and variants can be found in [2, 3].

#### 2. Positive Linear Functionals and Convex Sets of Functions

Let be a nonempty set, and let be a subspace of the linear space of all real functions on the domain . We assume that contains the unit function defined by for every .

Let be an interval, and let be the set containing all functions with the image in . Then, is convex set in the space . The same is true for convex sets of Euclidean spaces. Let be a convex set, and let be the set containing all function -tuples with the image in . Then, is convex set in the space .

A linear functional is positive (nonnegative) if for every nonnegative function , and is unital (normalized) if . If , then for every unital positive functional the number is in the closed interval of real numbers containing the image of . Through the paper, the space of all linear functionals on the space will be denoted with .

Let be an affine function, that is, the function of the form where and are real constants. If are functions and if are positive functionals providing the unit equalitythenRespecting the requirement of unit equality in (4), the sum could be called the functional convex combination. In the case , the functional must be unital by the unit equality in (4).

In 1931, Jessen stated the functional form of Jensen’s inequality for convex functions of one variable; see [4]. Adapted to our purposes, that statement is as follows.

Theorem A. *Let be a closed interval, and let be a function.**Then, a unital positive functional ensures the inclusion**and satisfies the inequality**for every continuous convex function providing that .**If is concave, then the reverse inequality is valid in (7). If is affine, then the equality is valid in (7).*

The interval must be closed, otherwise it could happen that . The function must be continuous, otherwise it could happen that the inequality in (7) does not apply. Such boundary cases are presented in [5].

In 1937, McShane extended the functional form of Jensen’s inequality to convex functions of several variables. He has covered the generalization in two steps, calling them the geometric (the inclusion in (8)) and analytic (the inequality in (9)) formulation of Jensen’s inequality; see [6, Theorems 1 and 2]. Summarized in a theorem, that generalization is as follows.

Theorem B. *Let be a closed convex set, and let be a function.**Then, a unital positive functional ensures the inclusion**and satisfies the inequality**for every continuous convex function providing that .**If is concave, then the reverse inequality is valid in (9). If is affine, then the equality is valid in (9).*

#### 3. Main Results

##### 3.1. Functions of One Variable

The main result of this subsection is Theorem 1 relying on the idea of a convex function graph and its secant line. Using functions that are more general than convex functions and positive linear functionals, we obtain the functional Jensen’s type inequalities.

Through the paper, we will use an interval and a bounded closed subinterval with endpoints .

Every number can be uniquely presented as the binomial affine combinationwhich is convex if and only if the number belongs to the interval . Let be a function, and let be the function of the line passing through the points and of the graph of . Applying the affinity of the function to the combination in (10), we obtain its equationThe consequence of the representations in (10) and (11) is the fact that every convex function satisfies the inequalityand the reverse inequality

In the following consideration, we use continuous functions satisfying the inequalities in (12)-(13).

Theorem 1. *Let be a closed interval, let be a bounded closed subinterval, and let and be functions.**Then, a pair of unital positive functionals such that**satisfies the inequality**for every continuous function satisfying (12)-(13) and providing that .*

*Proof. *The number belongs to the interval by the inclusion in (6). Using the features of the function and applying the affinity of the function , we getbecause for every .

It is obvious that a continuous convex function satisfies Theorem 1 for every subinterval with endpoints . The function used in Theorem 1 is shown in Figure 1. Such a function satisfies only the global property of convexity on the sets and .