Abstract
The aim of this paper is to present the fundamental inequalities for convex functions on Euclidean spaces. The work is based on the geometry of the simplest convex sets and properties of convex functions. Some obtained inequalities are applied to demonstrate a natural way of generalizing the Hermite-Hadamard inequality.
1. Introduction
Let be a real linear (vector) space, points (vectors), and coefficients (scalars). The combination belongs to the linear subspace as the smallest linear space that contains all , and it is called the linear combination. If , the combination in (1) belongs to the affine hull = aff as the smallest translated linear space that contains all , and it is called the affine combination. If and all , the combination in (1) belongs to the convex hull as the smallest convex point set that contains all , and it is called the convex combination. The point itself is called the center of the observed combination.
A function is affine if the equality holds for all affine combinations in (1), and is convex if the inequality holds for all convex combinations in (1). The formulas may be referred to for and then by applying the mathematical induction extended to any positive integer . The important Jensen’s inequality in (3) was just so proven in [1].
Jensen’s inequality was extended to special affine combinations and their centers in [2], generalizing Mercer’s variant of Jensen’s inequality obtained in [3]. The connection between discrete and integral forms was observed in [4]; the integral forms were studied in [5], and the Jensen type inequalities for -class functions were investigated in [6]. Different variants and forms can be found in [7, 8].
2. Convex Functions on the Line
Known results are presented in this section, in a way that can be generalized. Convex combinations of the line segment and affine combinations of the convex cone represent the backbone of the work. Theorem 3 is the most important in terms of scope of its content.
If are different numbers, then every number can be uniquely presented as the affine combination where The above binomial combination is convex if and only if the number belongs to the interval . Given the function , let be the function of the line passing through the points and of the graph of . Using the affinity of , we get
Let be the convex cone (half-line) with the vertex at spanned by containing binomial affine combinations , where ; that is, Convex cone is defined similarly. That means that if , then and . The consequence of the representations in (4) and (6) is the well-known characterization of convex functions with one variable, as specified in the following lemma.
Lemma 1. If are the line segment endpoints, then every convex function satisfies the inequality and the reverse inequality
Using combinations with , the inequality in (9) takes the form
Example 2. Let , and be positive real numbers.
Substituting the values of the convex power function with the exponent in the inequality in (10), we get the inequality
and the reverse inequality by substituting the values of the concave power function with the exponent .
Substituting the values of the concave logarithmic function in the inequality in (10) and rearranging them, it follows the inequality
Combining the application of Lemma 1 and Jensen’s inequality to the convex combinations , it follows if all and if all or all .
Convex combinations with the common center are considered in the following theorem.
Theorem 3. Let be the line segment endpoints. Let be a convex combination of the points , and let be a convex combination of the points .
If the center equality
is valid, then the inequality
holds for every convex function .
Proof. The first inequality in (16) is the Jensen inequality. The last inequality follows from the series of inequalities derived by applying the inequality in (8) to and the inequality in (9) to .
The geometric insight to the inequality in (16) presented in Figure 1 shows that the point is below the point and the point is below the point
Corollary 4. Let be the line segment endpoints. Let be a convex combination of the points , and let be the convex combination such that
Then the inequality
holds for every convex function .
3. Main Results: Convex Functions on the Plane
We assume that is the real vector space with the standard coordinate addition and the scalar multiplication .
If , , and are the planar points that do not belong to one line, then every point can be presented by the unique affine combination: where The above trinomial combination is convex if and only if point belongs to the triangle . Given the function , let be the function of the plane passing through the points , , and of the graph of . Because of the affinity of , it follows
Let be the convex cone with the vertex at spanned by the vectors and containing trinomial affine combinations , where ; that is, Cones and are defined in the same way, and all tree cones can be seen in Figure 2.
Using the cones, we achieve the generalization of Lemma 1 to convex functions on the plane as follows.
Lemma 5. If are the triangle vertices, then every convex function satisfies the inequality and the reverse inequality
Proof. If , the combination in (22) is convex, and the inequality in (26) follows from the convexity of and the equality in (24).
If , say and , we can represent the point as the binomial affine combination:
Applying the inequality in (10) to the combination in (29), then using the convexity of and the affinity of , we get the series of inequalities:
which includes the inequality in (27).
The area outside the triangle and outside the cones (the white area in Figure 2) cannot be generally covered with one inequality. Such area does not exist in the previous one-dimensional case.
Applying Lemma 5 and Jensen’s inequality to the planar convex combinations , we get the inequality if all and the inequality if all , or all , or all .
The following are planar convex combinations with the common center.
Theorem 6. Let be the triangle vertices. Let be a convex combination of the points , and let be a convex combination of the points .
If the center equality
is valid, then the inequality
holds for every convex function .
Proof. We employ the proof of Theorem 3 using instead of .
If (33) is valid, and if all points and are the triangle vertices, then the equality holds. The reverse statement is true if the function is strictly convex.
Corollary 7. Let be the triangle vertices. Let be a convex combination of the points , and let be the convex combination such that
Then the inequality
holds for every convex function .
4. Generalization to Higher Dimensions
Let be the points so that the vectors be linearly independent. In this case, the convex hull is called the -simplex in with the vertices . All of the simplex vertices cannot belong to the same hyperplane in . Every point can be presented by the unique affine combination: where the coefficients are determined generalizing the coefficients in (23). The combination in (38) is convex if and only if the point belongs to the -simplex .
Given the function , let be the function of the hyperplane (in ) passing through the points of the graph of . Therefore
Let () be the convex cone with the vertex at spanned by the vectors for containing ()-membered affine combinations , where all ; that is,
Lemma 8. If are the -simplex vertices, then every convex function satisfies the inequality and the reverse inequality
Proof. The proof is similar to that of Lemma 5. We sketch the arguments briefly as follows.
To prove (42), we firstly apply Jensen’s inequality to the convex combination and then use (40).
To prove (43) for other than , we firstly implement the inequality in (10) to the binomial affine combination:
where and and then apply Jensen’s inequality to the convex combination of , thus obtaining
as the desired inequality.
Using the inequality in (45) with power and logarithmic functions, the inequalities of Example 2 can be generalized as follows.
Example 9. Let for be the -simplex vertices with all coordinates . Let be a point with all coordinates , where .
Including the values of the convex power sum function
in the inequality in (45), we get the inequality
and the reverse inequality by including the values of the concave power sum function with all .
Including the values of the concave logarithmic sum function
in the inequality in (45) and rearranging them, it follows the inequality
Applying Lemma 8 and Jensen’s inequality to the convex combinations , we get the inequality if all and the inequality if all belong to the same cone .
Relying on Lemma 8, we reach the conclusion written in the next theorem.
Theorem 10. Let be the -simplex vertices. Let be a convex combination of the points , and let be a convex combination of the points .
If the center equality
is valid, then the inequality
holds for every convex function .
Corollary 11. Let be the -simplex vertices. Let be a convex combination of the points , and let be the convex combination such that
Then the inequality
holds for every convex function .
5. Application to the Hermite-Hadamard Inequality
Applying the integral method with the convex combinations to the inequalities obtained in the theorems, we get their integral forms. Using the Jensen type inequalities, we briefly demonstrate the generalization of the Hermite-Hadamard inequality (for essentials on this inequality see [9] or [10]).
Let be a convex function. Given the positive integer , we make the partition , where all subsegments have the same length , and the adjacent subsegments have a common endpoint. If we take subsegment centers , then we have the convex combination equality: Applying the inequality in (21) to the above convex combination, it follows and letting to infinity, we obtain the classic Hermite-Hadamard inequality:
The transition to the planar case can be done using Corollary 7. Let be a convex function, where is the triangle with vertices , , and . Given the positive integer , we use the partition with congruent subtriangles having a common edge or endpoint if they are adjacent. So, the area of each subtriangle is equal to . If we take subtriangle centers , then we have the convex combination equality: Applying the inequality in (37) to the above convex combination, it follows and letting to infinity, we obtain the planar Hermite-Hadamard inequality:
The transition to any dimension suggests Corollary 11 using the -simplex . Applying the previous procedure to , we reach the conclusion that every convex function satisfies the inequality
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work of the first author has been fully supported by Mechanical Engineering Faculty in Slavonski Brod. The work of the second author has been supported by the Foundation of Scientific Research Project of Fujian Province Education Department of China under Grant JK2012049. The authors thank Velimir Pavić (graphic designer at Školska knjiga Zagreb) who has graphically prepared Figures 1 and 2.