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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 194830, 5 pages
http://dx.doi.org/10.1155/2013/194830
Research Article

The Applications of Functional Variants of Jensen's Inequality

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

Received 31 May 2013; Accepted 2 September 2013

Academic Editor: Eva A. Gallardo Gutiérrez

Copyright © 2013 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 is inspired by McShane's results on the functional form of Jensen's inequality for convex functions of several variables. The work is focused on applications and generalizations of this important result. At that, the generalizations of Jensen's inequality are obtained using the positive linear functionals.

1. Introduction

1.1. Vector Space of Real Valued Functions and Positive Linear Functionals

Given a nonempty set we consider a space of functions containing the unit function defined with for every . Let be a real vector space of real valued functions on containing the unit function. A linear functional is said to be positive (nonnegative) or monotone if for every nonnegative function . If , we say that the functional is unital or normalized.

1.2. Functional Forms of Jensen’s Inequality

In 1931, Jessen [1] formulated the functional form of Jensen’s inequality for convex functions of one variable.

Theorem A. Let be a closed interval. Let be a function such that for every . Let be a continuous convex function such that .
Then every unital positive linear functional satisfies the inequality

The interval must be closed; otherwise, it could happen that as mentioned in [2].

Example 1. Let , , and is defined by . Take the identity function for . Then and for every , but .
The function must be continuous; otherwise, it could happen that the inequality in (1) does not apply as noted in [3].

Example 2. Consider the previous example with . Take the convex function defined by and for . Then .

In 1937, McShane [4, Theorems ] 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 (Theorem B) and analytic (Theorem C) formulation of Jensen’s inequality.

Theorem B. Let be a closed convex set. Let be functions such that for every .
Then every unital positive linear functional satisfies the inclusion

Theorem C. Let be a closed convex set. Let be functions such that for every . Let be a continuous convex function such that .
Then every unital positive linear functional satisfies the inequality

If the function is concave, then the reverse inequality is valid in (3). The hyperplanes that contain the set were used in the proof of Theorem B. The epigraph of the function : and Theorem B were applied in the proof of Theorem C.

2. The Applications and Generalizations of McShane’s Result

This section presents general applications and generalizations of Theorem C. The main generalization is Theorem 8.

Using McShane’s inequality in (3) it is very easy to get the functional form of Hölder’s and Minkowski’s inequality for nonnegative functions.

Corollary 3 (the functional form of Hölder’s inequality). Let be nonnegative functions such that . Let be real numbers such that and all .
Then every unital positive linear functional satisfies the inequality

Proof. Applying the concave function , where all , together with the functions to the reverse inequality in (3), it follows that which is the required inequality.

Corollary 4 (the functional form of Minkowski’s inequality). Let be nonnegative functions for . Let be a real number such that and all .
Then every unital positive linear functional satisfies the inequality

Proof. Applying the concave function , where all , and the functions to the reverse inequality in (3), it follows that and rising to the power of gives the desired inequality.

The McShane inequality in (3) can be easily generalized by introducing the weighted function as follows.

Theorem 5. Let be a closed convex set. Let be functions such that for every . Let be either a nonnegative or nonpositive function such that all . Let be a continuous convex function such that .
Then every positive linear functional such that satisfies the inequality

Proof. We use the function vector space and the linear functional defined with . The functional is positive and unital. Since all and , the functional values and may be applied to the inequality in (3).

If is a vector space of real valued -integrable functions on , then using the inequality in (9) with the positive linear functional we get the following integral form of Jensen’s inequality for convex functions of several variables.

Corollary 6. Let be a measure on a closed convex set . Let be -measurable functions such that for every . Let be either a nonnegative or nonpositive -integrable function such that and all are -integrable.
Then every continuous convex function such that is -integrable satisfies the integral inequality

Choosing points for and applying the inequality in (9) with the positive linear functional projections , and weighted function , we have the following discrete form of Jensen’s inequality for convex functions of several variables.

Corollary 7. Let be a closed convex set. Let be points for . Let be either nonnegative or nonpositive coefficients such that .
Then every continuous convex function satisfies the discrete inequality

Now we use nonempty sets and associated function vector spaces containing the unit function.

Theorem 8. Let be a closed convex set. Let be functions such that for every for . Let be either nonnegative or nonpositive functions such that . Let be a continuous convex function such that .
Then every -tuple of positive linear functionals such that all satisfies the inequality

Proof. First we apply the inequality in (13) to the numbers and coefficients then the inequality in (9) to every : and so we get the inequality in (14).

The inequality in (14) covers all the inequalities in this section.

3. The Generalizations of Jensen’s Inequality Using Functionals

This section presents the actual results of the paper. The main results are the generalizations of Jensen’s inequality written in Theorem 12 for convex functions of one variable and Theorem 16 for convex functions of several variables.

3.1. Inequalities for Convex Functions with One Variable

In what follows, we use a bounded interval assuming . Every can be uniquely presented as the affine combination where

Lemma 9. Let be a point. Let and be coefficients of the sum .
Then the affine combination belongs to .

Proof. Since , it has to be for coefficients and taken from the formulas in (17). Then we have The coefficients in the square brackets are nonnegative with the sum equal to , so the observed expression belongs to .

Sufficient conditions on the coefficients in Lemma 9 are and . Applying these conditions, it follows that .

Theorem 10. Let be points for . Let and be coefficients of the sums .
Then the affine combination .

Proof. Since , the affine combination by Lemma 9.

Corollary 11. Let be a monotone function, and let be points for . Let and be coefficients of the sums .
Then the affine combination .

The convex hull of a set will be denoted by .

Theorem 12. Let be a monotone function, and let be points for . Let and be coefficients of the sums .
Then every continuous convex function satisfies the inequality

Proof. If is monotone, then the Jensen inequality in (1) should be applied to the function space generated with and , that is, and linear functional on defined by The functional is unital since and positive since the function is monotone.
If is not monotone, then we translate so that its minimum and express where with . The previous procedure should be applied to the monotone convex functions and .

Putting the identity function for , the inequality in (19) is reduced to If and , the above inequality represents the Jensen inequality. If and , it represents the Jensen-Mercer inequality.

3.2. Inequalities for Convex Functions with Several Variables

We assume that is the real vector space treating its points as vectors with the standard coordinate addition and scalar multiplication . The aim of this subsection is to generalize Theorem 12 to simplexes.

To begin let us show how Lemma 9 can be generalized to the triangle (). Take the three planar points , , and that do not belong to one line. Then any point is presented by the unique affine combination where

So, we suppose that is the triangle with vertices , , and .

Lemma 13. Let be a point. Let and be coefficients such that and .
Then the affine combination .

Proof. Let us show that the affine combination is actually the convex combination. Since the point , the convex combination holds with the coefficients , , and taken from the formulas in (25). Therefore, we have The coefficients in the square brackets are nonnegative with the sum equal to , so the observed combination is convex, and its center belongs to .

If are points such that the points are linearly independent, then the convex hull is called the -simplex with vertices . Geometrically speaking, all the simplex vertices cannot belong to the same hyperplane. If , then any point is presented by the unique affine combination where the numerators and denominators of the coefficients are

In what follows we assume that is the -simplex with vertices .

The generalization of Lemma 13 (in dimensions) to several points stands as follows.

Theorem 14. Let be points for . Let and be coefficients such that all and .
Then the affine combination

Corollary 15. Let be a monotone mapping (either nondecreasing in each variable or nonincreasing in each variable), and let be points for . Let and be coefficients such that all and .
Then the affine combination

Theorem 16. Let be a monotone mapping (either nondecreasing in each variable or non-increasing in each variable), and let be points for . Let and be coefficients such that all and .
Then every continuous convex function satisfies the inequality

Proof. Let us prove the case .
If is monotone (either nondecreasing in both variables or nonincreasing in both variables), then we apply the McShane inequality in (3) to the function space and linear functional on defined by The functional is unital since and positive since the function is monotone.
If is not monotone, then we translate so that its minimum and divide the triangle into parts considering the lines and . Then we express as the sum of the monotone convex functions, , and apply the previous procedure to these functions.
The proof goes in the same way for any integer .

Applying the identity mapping for to the inequality in (31), it reads as follows:

References

  1. B. Jessen, “Bemrkninger om konvekse Funktioner og Uligheder imellem Middelvrdier. I,” Matematisk Tidsskrift B, vol. 2, pp. 17–28, 1931.
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  3. I. Raşa, “A note on Jessen's inequality,” in Itinerant Seminar on Functional Equations, Approximation and Convexity, pp. 275–280, Universitatea Babes-Bolyai, Cluj-Napoca, Romania, 1988. View at Zentralblatt MATH · View at MathSciNet
  4. E. J. McShane, “Jensen's inequality,” Bulletin of the American Mathematical Society, vol. 43, no. 8, pp. 521–527, 1937. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet