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.

Introduction is intended to be a brief overview of the concept of convexity and affinity. Let

A set

Let

Requiring only the condition in (

Implementing mathematical induction, we can prove that all of the above applies to

Let

Let

A linear functional

Let

In 1931, Jessen stated the functional form of Jensen’s inequality for convex functions of one variable; see [

Let

Then, a unital positive functional

If

The interval

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 (

Let

Then, a unital positive functional

If

The main result of this subsection is Theorem

Through the paper, we will use an interval

Every number

In the following consideration, we use continuous functions satisfying the inequalities in (

Let

Then, a pair of unital positive functionals

The number

It is obvious that a continuous convex function

A continuous function satisfying (

Involving the binomial convex combination

The functions used in Theorem

Let

Then, a unital positive functional

Putting

Now, we give a characterization of continuous convex functions by using unital positive functionals.

Let

Let us prove the sufficiency. Let

We want to transfer the results of the previous subsection to higher dimensions. The main result in this subsection is Theorem

Let

It should be noted that convex functions of two variables do not generally satisfy (

We take the convex function

The valuation of functions

The generalization of Theorem

Let

Then, a pair of unital positive functionals

The proof is similar to that of Theorem

The previous lemma suggests how the results of the previous subsection can be transferred to higher dimensions.

Let

Let

Let

Then, a pair of unital positive functionals

Relying on the hyperplane function

Including the

The following functional form of Jensen’s inequality is true for functions of several variables.

Let

Then, a unital positive functional

Continuous convex functions of several variables can be characterized by unital positive functionals in the following way. The dimension of a convex set is defined as the dimension of its affine hull.

Let

To prove the sufficiency, we take a convex combination

Functions investigated in Subsection

The next generalization of Theorem

Let

Then, a pair of collections of positive functionals

Now, we present a way of introducing the functional quasiarithmetic means. Let

In applications of the function convexity, we use a pair of strictly monotone continuous functions

Instead of the convexity of

Let

If

We take

Using the equality

All the cases of the above theorem are as follows.

Let

If either

If either

A special case of the quasiarithmetic means in (

To apply Theorem

Let

If

The proof follows from Theorem

Our aim is to use Theorem

Let

If the above convex combinations have the common center

We take the set

Let

Let

Applying the functional

Proposition

We take the convex function

Then, we have

More details on the behavior of a convex function of two variables on the triangle and outside the triangle can be found in [

The integral analogy of the concept of convex combination is the concept of barycenter. Let

The application of Theorem

Let

If the above sets have the common

The proof is similar to that of Proposition

If

Implementing convex combinations to the integral method, one may derive the following version of the Hermite-Hadamard inequality for convex functions on simplexes.

Let

If the convex combination center and the

More on the important and interesting Hermite-Hadamard’s inequality, including historical facts about its name, can be found in [

The author declares that there is no conflict of interests regarding the publication of this paper.

This work has been fully supported by Mechanical Engineering Faculty in Slavonski Brod and Croatian Science Foundation under Project 5435. The author thanks Velimir Pavić (graphic designer at Školska knjiga Zagreb) who has graphically prepared Figure