Functions Like Convex Functions

holds for all binomial convex combinations αa + βb of pairs of points a, b ∈ C. Requiring only the condition in (2) for coefficients and requiring the equality in (3), we get a characterization of the affinity. Implementingmathematical induction, we can prove that all of the above applies to n-membered combinations for any positive integer n. 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].


Introduction
Introduction is intended to be a brief overview of the concept of convexity and affinity.Let X be a real linear space.Let ,  ∈ X be points and let ,  ∈ R be coefficients.Their binomial combination is convex if ,  ≥ 0 and if If  = +, then the point  itself is called the combination center.
A set S ⊆ X is convex if it contains all binomial convex combinations of its points.The convex hull convS of the set S is the smallest convex set containing S, and it consists of all binomial convex combinations of points of S.
Let C ⊆ X be a convex set.A function  : C → R is convex if the inequality  ( + ) ≤  () +  () holds for all binomial convex combinations  +  of pairs of points ,  ∈ C.
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].

Positive Linear Functionals and Convex Sets of Functions
Let X be a nonempty set, and let X be a subspace of the linear space of all real functions on the domain X.We assume that X contains the unit function 1 defined by 1() = 1 for every  ∈ X.
Let I ⊆ R be an interval, and let X I be the set containing all functions  ∈ X with the image in I.Then, X I is convex set in the space X.The same is true for convex sets of Euclidean spaces.Let C ⊆ R  be a convex set, and let (X  ) C be the set containing all function -tuples  = ( 1 , . . .,   ) ∈ X  with the image in C.Then, (X  ) C is convex set in the space X  .
A linear functional  : X → R is positive (nonnegative) if () ≥ 0 for every nonnegative function  ∈ X, and  is unital (normalized) if (1) = 1.If  ∈ X, 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 X will be denoted with L(X).
Let  : R → R be an affine function, that is, the function of the form () :=  +  where  and  are real constants.

Journal of Function Spaces
If  1 , . . .,   ∈ X are functions and if  1 , . . .,   ∈ L(X) are positive functionals providing the unit equality then ( (  )) . ( Respecting the requirement of unit equality in (4), the sum ∑  =1   (  ) could be called the functional convex combination.In the case  = 1, the functional  =  1 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 I ⊆ R be a closed interval, and let  ∈ X I be a function.
Then, a unital positive functional  ∈ L(X) ensures the inclusion  () ∈ I and satisfies the inequality  ( ()) ≤  ( ()) for every continuous convex function  : If  is concave, then the reverse inequality is valid in (7).If  is affine, then the equality is valid in (7).
The interval I must be closed, otherwise it could happen that () ∉ I.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 C ⊆ R  be a closed convex set, and let  = ( 1 , . . .,   ) ∈ (X  ) C be a function.
Then, a unital positive functional  ∈ L(X) ensures the inclusion and satisfies the inequality for every continuous convex function  : If  is concave, then the reverse inequality is valid in (9).If  is affine, then the equality is valid in (9).

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 I ⊆ R and a bounded closed subinterval [, ] ⊆ I with endpoints  < .
Every number  ∈ R can be uniquely presented as the binomial affine combination which is convex if and only if the number  belongs to the interval [, ].Let  : I → R be a function, and let  line {,} : R → R be the function of the line passing through the points (, ()) and (, ()) of the graph of .Applying the affinity of the function  line {,} to the combination in (10), we obtain its equation The consequence of the representations in ( 10) and ( 11) is the fact that every convex function  : I → R satisfies the inequality and the reverse inequality In the following consideration, we use continuous functions satisfying the inequalities in (12)-(13).
Theorem 1.Let I ⊆ R be a closed interval, let [, ] ⊆ I be a bounded closed subinterval, and let  ∈ X [,] and ℎ ∈ X I\(,) be functions.
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  line {,} , we get because  line {,} (ℎ()) ≤ (ℎ()) for every  ∈ X.
It is obvious that a continuous convex function  : I → R satisfies Theorem 1 for every subinterval [, ] ⊆ I 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 I \ (, ).
Involving the binomial convex combination  +  with the equality in (14) by assuming that and inserting the term () + () in ( 16) via the double equality which is true because  line {,} ( + ) = () + (), we achieve the double inequality  ( ()) ≤  () +  () ≤  ( (ℎ)) . ( The functions used in Theorem 1 satisfy the functional form of Jensen's inequality in the following case. Corollary 2. Let I ⊆ R be a closed interval, let [, ] ⊆ I be a bounded closed subinterval, and let ℎ ∈ X I\(,) be a function.
Then, a unital positive functional  ∈ L(X) such that satisfies the inequality for every continuous function satisfying (12)-( 13) and providing that (ℎ) ∈ X.
Proof.Putting  +  = (ℎ), it follows that by the right inequality in (19).Now, we give a characterization of continuous convex functions by using unital positive functionals.

Proposition 3. Let I ⊆ R be a closed interval. A continuous function 𝑓 : I → R is convex if and only if it satisfies the inequality
for every pair of interval endpoints ,  ∈ I, every function  ∈ X [,] such that () ∈ X, and every unital positive functional  ∈ L(X).
Proof.Let us prove the sufficiency.Let  :=  +  be a convex combination of points ,  ∈ I where  < .We take the constant function  = 1 in X [,] (actually () = via (23), we get the convexity inequality in (3).

Functions of Several Variables.
We want to transfer the results of the previous subsection to higher dimensions.The main result in this subsection is Theorem 6 generalizing Theorem 1 to functions of several variables.Let C ⊆ R 2 be a convex set, let △ ⊆ C be a triangle with vertices , , and , and let △  be its interior.In the following observation, we assume that  : C → R is a continuous function satisfying the inequality  () ≤  plane {,,} () for  ∈ △ (25) and the reverse inequality where  plane {,,} is the function of the plane passing through the corresponding points of the graph of .
It should be noted that convex functions of two variables do not generally satisfy (26).The next example confirms this claim.
The valuation of functions  and  The generalization of Theorem 1 to two dimensions is as follows.
Proof.The proof is similar to that of Theorem 1.Using the triangle vertices , , and , we apply the plane function  plane {,,} instead of the line function  line {,} .
The previous lemma suggests how the results of the previous subsection can be transferred to higher dimensions.
Let  1 , . . .,  +1 ∈ R  be points.Their convex hull is the -simplex in R  if the points  1 −  +1 , . . .,   −  +1 are linearly independent.Let C ⊆ R  be a convex set, and let S ⊆ C be a -simplex with vertices  1 , . . .,  +1 .In the consideration that follows, we use a function  : C → R satisfying the inequality and the reverse inequality where  hyperplane { 1 ,..., +1 } is the function of the hyperplane passing through the corresponding points of the graph of .Theorem 6.Let C ⊆ R  be a closed convex set, let S ⊆ C be a -simplex, and let g = ( 1 , . . .,   ) ∈ (X  ) S and ℎ = (ℎ 1 , . . ., ℎ  ) ∈ (X  ) C\S  be functions.

Applications to Functional Quasiarithmetic Means
Functions investigated in Subsection 3.1 can be included to quasiarithmetic means by applying methods such as those for convex functions.The basic facts relating to quasiarithmetic and power means can be found in [7].For more details on different forms of quasiarithmetic and power means, as well as their refinements, see [8].