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

Abstract

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

1. Introduction

The famous Jensen inequality states that

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

The following theorem was the main motivation for this paper (see [3] 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

(2.1)

Since is convex on and , it follows that

(2.2)

It is enough to prove that

(2.3)

which is obviously equivalent to the inequality

(2.4)

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

(2.5)

so the proof is complete.

Remark 2.2.

In fact, we have proved that

(2.6)

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]):

(2.7)

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

(2.8)

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

(2.9)

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 [8] 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

(3.1)

for . Then,

(3.2)

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:

(3.3)

A famous inequality due to Ky-Fan states that

(3.4)

where and are the weighted geometric and arithmetic means, respectively, defined by

(3.5)

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

(3.6)

Proof.

Set and in (3.3). It follows that

(3.7)

which by obvious rearrangement implies (3.6).

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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.

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Hussain, S., Pečarić, J. & Perić, I. Jensen's Inequality for Convex-Concave Antisymmetric Functions and Applications. J Inequal Appl 2008, 185089 (2008). https://doi.org/10.1155/2008/185089

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