• Research Article
• Open Access

# Jensen's Inequality for Convex-Concave Antisymmetric Functions and Applications

Journal of Inequalities and Applications20082008:185089

https://doi.org/10.1155/2008/185089

• Accepted: 11 August 2008
• Published:

## Abstract

The weighted Jensen inequality for convex-concave antisymmetric functions is proved and some applications are given.

## Keywords

• Convex Function
• Related Result
• Classical Case
• Main Motivation
• Simple Proof

## 1. Introduction

where is a convex function, is interval in , and . Recall that a function is convex if

holds for every and every (see [1, Chapter 2]).

The natural problem in this context is to deduce Jensen-type inequality weakening some of the above assumptions. The classical case is the case of Jensen-convex (or mid-convex) functions. A function is Jensen-convex if
holds for every . It is clear that every convex function is Jensen-convex. To see that the class of convex functions is a proper subclass of Jensen-convex functions, see [2, page 96]. Jensen's inequality for Jensen-convex functions states that if is a Jensen-convex function, then

where . For the proof, see [2, page 71] or [1, page 53].

A class of functions which is between the class of convex functions and the class of Jensen-convex functions is the class of Wright-convex functions. A function is Wright-convex if

holds for every , where (see [1, page 7]).

The following theorem was the main motivation for this paper (see  and [1, pages 55-56]).

Theorem 1.1.

Let be Wright-convex on and . If and for , then (1.4) is valid.

Another way of weakening the assumptions for (1.1) is relaxing the assumption of positivity of weights . The most important result in this direction is the Jensen-Steffensen inequality (see, e.g., [1, page 57]) which states that (1.1) holds also if and , where .

The main purpose of this paper is to prove the weighted version of Theorem 1.1. For some related results, see [4, 5]. In Section 3, to illustrate the applicability of this result, we give a generalization of the famous Ky-Fan inequality.

## 2. Main Results

Theorem 2.1.

Let be a convex function on and for every . If , and for , then (1.1) holds.

Proof.

Without loss of generality, we can suppose that . So, is an odd function. First we consider the case . If , then we have the known case of Jensen inequality for convex functions. Thus, we will assume that and . The equation of the straight line through points is
Since is convex on and , it follows that

Since the function is convex on and , by Galvani's theorem it follows that the function is increasing on . Therefore, from and we have ; so (2.4) holds.

Now, for an arbitrary , we have

so the proof is complete.

Remark 2.2.

Remark 2.3.

Neither condition , nor condition , can be removed from the assumptions of Theorem 2.1. To see this, consider the function on . That the first condition cannot be removed can be seen by considering , and . That the second condition cannot be removed can be seen by considering , and . In both cases, (1.1) does not hold.

Remark 2.4.

Using Jensen and Jensen-Steffensen inequalities, it is easy to prove the following inequalities (see also [6, 7]):

where is a convex function on , and for . If is concave, the reverse inequalities hold in (2.7).

Now, suppose the conditions in Theorem 2.1 are fulfilled except that the function satisfies . It is immediate (consider the function ) that inequality (1.1) still holds. Using , the inequality (1.1) gives
so the left-hand side of inequality (2.7) is valid also in this case. On the other hand, if (so ), the previous inequality can be written as

which is the reverse of the right-hand side inequality of (2.7); so the concavity properties of the function are prevailing in this case.

## 3. Applications

In the following corollary, we give a simple proof of a known generalization of the Levinson inequality (see  and [1, pages 71-72]).

Recall that a function is 3-convex if for , and , where denotes third-order divided difference of . It is easy to prove, using properties of divided differences or using classical case of the Levinson inequality, that if is a 3-convex function, then the function is convex on (see [1, pages 71-72]).

Corollary 3.1.

Let be a 3-convex function; , and
for . Then,

Proof.

It is a simple consequence of Theorem 2.1 and the above-mentioned fact that is convex on .

Remark 3.2.

In fact, the following improvement of inequality (3.2) is valid:
where and are the weighted geometric and arithmetic means, respectively, defined by

where (see [6, page 295]).

In the following corollary, we give an improvement of the Ky-Fan inequality.

Corollary 3.3.

Let , and . If and , then

Proof.

Set and in (3.3). It follows that

which by obvious rearrangement implies (3.6).

## Declarations

### Acknowledgments

The research of J. Pečarić and I. Perić was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 117-1170889-0888 (J. Pečarić) and 058-1170889-1050 (I. Perić). S. Hussain and J. Pečarić also acknowledge with thanks the facilities provided to them by Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan. The authors also thank the careful referee for helpful suggestions which have improved the final version of this paper.

## Authors’ Affiliations

(1)
Abdus Salam School of Mathematical Sciences, GC University Lahore, Gulberge, Lahore, 54660, Pakistan
(2)
Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
(3)
Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

## References 