- Research Article
- Open Access

# Schur-Convexity of Averages of Convex Functions

- Vera Čuljak
^{1}, - Iva Franjić
^{2}, - Roqia Ghulam
^{3}Email author and - Josip Pečarić
^{4}

**2011**:581918

https://doi.org/10.1155/2011/581918

© Vera Čuljak et al. 2011

**Received: **12 November 2010

**Accepted: **11 January 2011

**Published: **18 January 2011

## Abstract

The object is to give an overview of the study of Schur-convexity of various means and functions and to contribute to the subject with some new results. First, Schur-convexity of the generalized integral and weighted integral quasiarithmetic mean is studied. Relation to some already published results is established, and some applications of the extended result are given. Furthermore, Schur-convexity of functions connected to the Hermite-Hadamard inequality is investigated. Finally, some results on convexity and Schur-convexity involving divided difference are considered.

## Keywords

## 1. Introduction

The property of Schur-convexity and Schur-concavity has invoked the interest of many researchers and numerous papers have been dedicated to the investigation of it. The object of this paper is to present an overview of the results related to the study of Schur-convexity of various means and functions, in particular, those connected with the Hermite-Hadamard inequality. Moreover, we contribute to the subject with some new results.

First, let us recall the definition of Schur-convexity. It generalizes the definition of the convex and concave function via the notion of majorization.

Definition 1.1.

where denotes the th largest component in .

Function is said to be Schur-concave on if is Schur-convex.

Note that every function that is convex and symmetric is also Schur-convex.

One of the references which will be of particular interest in this paper is [1]. The authors were inspired by some inequalities concerning gamma and digamma function and proved the following result for the integral arithmetic mean.

Theorem A1.

is Schur-convex (Schur-concave) on if and only if is convex (concave) on .

Few years later, Wulbert, in [2], proved that the integral arithmetic mean defined in (1.3) is convex on if is convex on . Zhang and Chu, in [3], rediscovered (without referring to and citing Wulbert's result) that the necessary and sufficient condition for the convexity of the integral arithmetic mean is for to be convex on . Note that the necessity is obvious. Namely, if is convex, then it is also Schur-convex since it is symmetric. Theorem A1 then implies the convexity of function .

Later, in [4], the Schur-convexity of the weighted integral arithmetic mean was proved.

Theorem A2.

holds (reverses) for all , in .

In the same reference, the authors left an open problem: under which conditions does (1.5) hold?

The monotonicity of the function defined in (1.4) was studied in [5].

Theorem A3.

Let be a continuous function on and let be a positive continuous weight on . Then, the function defined in (1.4) is increasing (decreasing) on if is increasing (decreasing) on .

In the following sections, Schur-convexity of the generalized integral and weighted integral quasiarithmetic mean is studied. Relation to some already published results is established. Further, a new proof of sufficiency in Theorem A1, which is also a new proof of Wulbert's result from [2], that is, Zhang and Chu's result from [3], is presented. Some applications of this extended result are given. Furthermore, Schur-convexity of various functions connected to the Hermite-Hadamard inequality is investigated. Finally, some results on convexity and Schur-convexity involving divided difference are considered.

To complete the Introduction, we state three very interesting lemmas related to Schur-convexity. They are needed later for the proofs of our results. All three can be found in both [6, 7]. The first one gives a useful characterization of Schur-convexity.

Lemma A1.

for all , , . Function is Schur concave if and only if the reversed inequality sign holds.

Lemma A2.

Let , and be defined as , where .

(1)If is convex (concave) and is increasing and Schur-convex (Schur-concave), then is Schur-convex (Schur-concave).

(2)If is concave (convex) and is decreasing and Schur-convex (Schur-concave), then is Schur-convex (Schur-concave).

Lemma A3.

Let , and be defined as , where .

(1)If each of is Schur-convex and is increasing (decreasing), then is Schur-convex (Schur-concave).

(2)If each of is Schur-concave and is increasing (decreasing), then is Schur-concave (Schur-convex).

## 2. Generalizations

For a special choice of functions , we can obtain various integral means. For example,

The next result discovers the property of Schur-convexity of the generalized integral quasiarithmetic means.

Theorem 2.1.

the following hold:

(i) is Schur-convex on if is convex on and is increasing on or if is concave on and is decreasing on ,

(ii) is Schur-concave on if is convex on and is decreasing on or if is concave on and is increasing on .

Proof.

is Schur-convex (Schur-concave) if and only if is convex (concave). Now, from Lemma A3 applied for , the statement follows.

Remark 2.2.

is Schur-convex for and Schur-concave for . This was also obtained in [1] as a consequence of Theorem A1.

Theorem 2.3.

the following hold:

(i) is Schur-convex on if is increasing, and and are convex or if is decreasing and is convex and is concave,

(ii) is Schur-concave on if is decreasing and is concave and is convex or if is increasing, and and are concave.

Proof.

is increasing (decreasing) and Schur-convex (Schur-concave) on if is increasing (decreasing) and convex (concave) on .

is

(a)Schur-convex if is convex and is convex and is increasing or if is concave and is convex and is decreasing,

(b)Schur-concave if is concave and is concave and is increasing or if is convex and is concave and is decreasing.

Finally, we apply Lemma A3 to in order to conclude that is

(a^{'})Schur-convex if
is Schur-convex and
is increasing or if
is Schur-concave and
is decreasing,

(b^{'})Schur-concave if
is Schur-convex and
is decreasing or if
is Schur-concave and
is increasing.

Combining (a), (b), (a^{'}), and (b^{'}) completes the proof.

Note that under an additional assumption that is strictly increasing, we have . Thus, using the same idea as in the proof of Theorem 2.3, an analogous result can easily be given for the mean .

Theorem 2.4.

Let and be real continuous strictly monotone functions on . Then, for the mean defined in (2.13), the following hold:

(i) is Schur-convex on if is increasing and is increasing and convex and is convex or if is increasing and is decreasing and concave and is convex or if is decreasing and is decreasing and concave and is concave or if is decreasing and is increasing and convex and is concave,

(ii) is Schur-concave on if is decreasing and is decreasing and convex and is convex or if is decreasing and is increasing and concave and is convex or if is increasing and is increasing and concave and is concave or if is increasing and is decreasing and convex and is concave.

### 2.1. Application of Theorem A1 for the Extended Mean Values

As a special case, the identric mean of order and the logarithmic mean of order are recaptured. Namely, and .

On the other hand, note that the generalized weighted quasiarithmetic mean defined in (2.1) is a generalization of the extended means. Namely, for .

Many properties of extended mean values have been considered in [10]. It was shown that are continuous on and symmetric with respect to both and , and and .

Schur-convexity of the extended mean values with respect to and was considered in [4, 5, 11].

Sándor in [12] (and also Qi et al. in [11]) proved the Schur-convexity of the extended mean values with respect to , using Theorem A1 and the integral representation .

Shi et al. in [5], using Theorem A1 and Lemma A3 obtained the following condition for the Schur-convexity of the extended mean values with respect to .

Theorem A4.

(i)if or , then the extended mean values are Schur-convex with respect to ,

(ii)if , then the extended mean values are Schur-concave with respect to .

Remark 2.5.

Chu and Zhang in [13] established the necessary and sufficient conditions for the extended mean values to be Schur-convex (Schur-concave) with respect to , for fixed .

Theorem A5.

(i)the extended mean values are Schur-convex with respect to if and only if ,

(ii)the extended mean values are Schur-concave with respect to if and only if .

We remark that the above result does not cover the case , that is, the case of the identric mean of order . Monotonicity and Schur-concavity of the identric mean with respect to and for fixed was discussed in [14], using the hyperbolic composite function.

Theorem A6.

## 3. Convexity

The following result is an extension of Wulbert's result from [2].

Theorem 3.1.

Let be a continuous function on an interval with a nonempty interior. If is convex on , then the integral arithmetic mean defined in (1.3) is convex on .

Proof.

Corollary 3.2.

Generalized logarithmic mean defined by (2.16) is convex for and concave for .

Proof.

Remark 3.3.

Remark 3.4.

The inequality (3.1) is strict if is a strictly convex function unless .

### 3.1. Applications

We recall the following definitions and remarks (see, e.g., [15]).

Definition 3.5.

for every and every , such that , .

Definition 3.6.

*,*where is an interval in

*,*is said to be convex if is convex, or equivalently, if for all and all , we have

Remark 3.7.

If
is exponentially convex, then
is a
*-* convex function.

Now, we will give some applications of (3.1).

Theorem 3.8.

where is given by (3.9). Then, the following hold:

(i)the function is continuous on ,

(iii)the function is exponentially convex on ,

(iv)if , the function is -convex on ,

Proof.

Analogous to the proof of Theorem 2.2 from [15].

Following the steps of the proofs of Theorems 2.4 and 2.5 given in [15], we can prove the following two mean value theorems.

Theorem 3.9.

Theorem 3.10.

provided that denominator on right-hand side is nonzero.

Remark 3.11.

*Let*

*,*

*where*

*,*

*are nonnegative real weights such that*

*and*

*,*

*in*

*. If the inverse of*

*exists, then various kinds of means can be defined by ( 3.14 ). Namely,*

Theorem 3.12.

Proof.

It follows the steps of the proof of Theorem 4.2 given in [15].

Remark 3.13.

*As a special case for*
*,*
*, we recapture the discrete version of the results obtained in*[16]*.*

## 4. Hermite-Hadamard Inequality

In [17], it was shown that is convex if and only if at least one of the inequalities in (4.1) is valid.

An interesting fact is that the original proof of Theorem A1 was given using the second Hermite-Hadamard inequality and the first one follows from the same theorem.

The same inequality was rediscovered later in [19] through an elementary analytic proof.

### 4.1. Application of Theorem A1 for a Function Connected with Hadamard Inequality

where and with , and showed convexity of if is convex function on .

Yang and Hong, in [22] (see also [21, page 147]) considered a similar function. Shi, in [23], found a similar result as Theorem A1 for the function .

Theorem A7.

the following hold:

(i)for , if is convex on , then is Schur-convex on ,

(ii)for , if is concave on , then is Schur-concave on .

In [24], we obtained Schur-convexity of the ebišev functional. In note [25], our first aim was to give another similar result to Theorem A1.

Theorem A8.

the following hold:

(i)for such that , , if is convex on , then is Schur-convex on ,

(ii)for such that , , if is concave on , then is Schur-concave on .

Shi, in [23], found a similar result as Theorem A1 for this function .

Theorem A9.

### 4.2. Schur-Convexity of Hermite-Hadamard Differences

In [27], the property of Schur-convexity of the difference between the middle part and the left-hand side of the Hermite-Hadamard inequality (4.1), and the difference between the right-hand side and the middle part of the same inequality, was investigated. The following theorems were proved.

Theorem A10.

is Schur-convex (concave) on if and only if is convex (concave) on .

Theorem A11.

is Schur-convex (concave) on if and only if is convex (concave) on .

First, we state a simple consequence of Theorems A1, A10, and A11.

Corollary 4.1.

Let be a continuous function. Then, the following statements are equivalent:

(ii) is Schur-convex (Schur-concave),

(iii) is Schur-convex (Schur-concave),

(iv) is Schur-convex (Schur-concave),

where is defined as in (1.4), as in (4.9) and as in (4.10).

Remark 4.2.

which, after applying Lemma A1, is another proof of (ii)*⇔*(iii) in Corollary 4.1.

Identity (4.13) enables us to give a new proof of sufficiency in Theorem A11.

Proof.

Since by assumption is convex (concave), Lemma A1 yields that is Schur-convex (Schur-concave).

Remark 4.3.

Note that with an additional assumption that
*,* since(*27*)is valid for all
*,* from (4.15) necessity in Theorem A11 follows as well.

With the help of identity (4.16), we can present the following.

Theorem 4.4.

is Schur-convex (Schur-concave).

If and is Schur-convex (Schur-concave), then is convex (concave).

Proof.

If is convex (concave), from Lemma A1, it follows that is Schur-convex (Schur-concave).

and this is valid for all . Since by assumption is Schur-convex (Schur-concave), from Lemma A1, it follows that is convex (concave).

Remark 4.5.

which is exactly (4.2). Since in Theorem 4.4 we have shown that is Schur-convex if is convex, this is in fact a new proof of (4.2).

## 5. Convexity and Schur-Convexity of Divided Differences

In this final section, we turn our attention towards divided differences. Let us first recall the definition.

Definition 5.1.

Notion closely related to divided differences is -convexity.

Definition 5.2.

If the inequality is reversed, then is said to be -concave on .

For more details on divided differences and -convexity, see [7].

In [30], Zwick proved the following theorem.

Theorem A12.

is a convex function of for all and all such that , .

This theorem is a generalization of a result from [31], where only 3-convex functions were considered. An additional generalization was given by Farwig and Zwick in [32].

Theorem A13.

holds for all such that , which is a generalization of (5.4).

Note that the divided difference is a permutation symmetric function. Thus, the following theorem follows from Theorem A13 and a result on majorization inequalities. It was obtained in [33] by Pečari and Zwick.

Theorem A14.

that is, function defined in (5.5) is Schur-convex.

Many more results involving divided differences were obtained, among others the multivariate analogues, all of which can be found in [7].

About a decade later, Merkle in [34] presented the following.

Theorem A15.

Then, the conditions (A)–(E) are equivalent and the conditions (A)–(E^{'}) are equivalent, where

and

Thus, it becomes clear that the statements (A)*⇔*(E) and (A^{'})*⇔*(E^{'}) are in fact an alternative statement of Theorem A1. Furthermore, implications (A)*⇒*(E) and (A^{'})*⇒*(E^{'}) are a special case of Theorem A14, while (A)*⇒*(D) and (A^{'})*⇒*(D^{'}) are a special case of Theorem A13.

Moreover, note that (B) and (C), that is, (B^{'}) and (C^{'}), are in fact the Hermite-Hadamard inequalities and we have already commented on their relation with Theorem A1—one side is used in the proof and the other is a consequence of the theorem.

Implications (D)*⇒*(E) and (D^{'})*⇒*(E^{'}) are trivial, since
is symmetric.

Furthermore, the statements (A)*⇔*(D) and (A^{'})*⇔*(D^{'}) are an alternative statement of Zhang and Chu's result from [3] and the necessity part recaptures Wulbert's result from [2] and the result from our Theorem 3.1.

### 5.1. Applications of Schur-Convexity of Divided Differences

are obtained as special cases of this new mean. In [36], necessary conditions under which Gini means (5.11) are Schur-convex and Schur-concave were given. In the short note [37], Witkowski completed this result with the proof of sufficiency of those conditions.

Using this form and Theorem A15, he proved the following.

Theorem A16.

The following conditions are equivalent:

(a)for all and all , is convex (concave) in and ,

(b)for all and all , is Schur-convex (Schur-concave) in and ,

(d)for all and all , is concave (convex) in and ,

(e)for all and all , is Schur-concave (Schur-convex) in and ,

Witkowski was able to apply all the results obtained for the two-parameter means, in particular Theorem A16, for this new family of means. The one of special interest to us is the following.

Theorem A17.

If , the following conditions are equivalent:

(a)for all and all , is convex (concave) in and ,

(b)for all and all , is Schur-convex (Schur-concave) in and ,

(c) increases (decreases) for ,

(d)for all and all , is concave (convex) in and ,

(e)for all and all , is Schur-concave (Schur-convex) in and ,

(f) decreases (increases) for .

If , then the conditions and reverse.

Note that the same four-parameter family of means was the object of interest to Yang in [39]. He gave conditions under which are increasing (decreasing) and logarithmically convex (logarithmically concave). Necessary and sufficient conditions for to be increasing (decreasing) were, however, given in [38].

## Declarations

### Acknowledgments

The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants nos. 117-1170889-0888 (for V. Čuljak and J. Pečarić) and 058-1170889-1050 (for I. Franjić).

## Authors’ Affiliations

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