Schur-Convexity of Averages of Convex Functions

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.

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.

Functionis said to be Schur-convex onif

(11)

for every , such that , that is, such that

(12)

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.

Let be a continuous function on an interval with a nonempty interior. Then,

(13)

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.

Let be a continuous function on and let be a positive continuous weight on . Then, the function

(14)

is Schur-convex (Schur-concave) on if and only if the inequality

(15)

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.

Let and let be a continuous symmetric function. If is differentiable on , then is Schur-convex on if and only if

(16)

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

Let be a real positive Lebesgue integrable function on , a real Lebesgue integrable function on , and a real continuous strictly monotone function defined on , the range of . The generalized weighted quasiarithmetic mean of function with respect to weight function is given by

(21)

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

(i)for on , we get the classical quasiarithmetic integral mean of a function

(22)

(ii)for on , we get the classical weighted quasiarithmetic integral mean

(23)

(iii)for on , we get the weighted arithmetic integral mean

(24)

(iv)for on , we obtain the weighted power integral mean of order

(25)

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

Theorem 2.1.

Let be a real Lebesgue integrable function defined on the interval , with range . Let be a real continuous strictly monotone function on . Then, for the generalized integral quasiarithmetic mean of function defined as

(26)

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.

Applying Theorem A1 for function yields that

(27)

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

Remark 2.2.

Applying this theorem forandshows that the generalized logarithmic mean defined foras

(28)

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

Theorem 2.3.

Let be a real continuous strictly monotone function on and be a differentiable and strictly increasing function on . Then, for the generalized weighted integral quasiarithmetic mean defined by

(29)

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.

Applying Theorem A1 and Lemma A3 (for for function , we conclude that

(210)

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

Using Lemma A2, we now deduce that

(211)

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.

Using substitution , we can rewrite

(212)

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.

In [8], a new symmetric mean was defined for two strictly monotone functions and on as

(213)

If we change the variable , we have

(214)

Further, by substitution , we obtain

(215)

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

For and , extended mean values were defined in [9] by Stolarsky as follows:

(216)

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.

For fixed ,

(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.

As a special case forandin Theorem 2.3, we recapture the result from Theorem A4 for the extended mean values

(217)

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.

For fixed ,

(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.

For fixed ,

(i) is increasing with respect to ,

(ii)if, then is Schur-concave with respect to .

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 .

Furthermore, for , and nonnegative real weights , such that , the following hold:

(31)

(32)

where and .

Proof.

Using the discrete Jensen inequality for the convex function , we have the following conclusion:

(33)

So, function is convex on .

Using the Hermite-Hadamard inequality for the convex function , we can extend inequality (3.1) on the left and on the right hand side as follows:

(34)

Corollary 3.2.

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

Proof.

Apply Theorem 3.1 for .

Remark 3.3.

Theorem 3.1 is a generalization of the discrete Jensen inequality. For, , the inequality

(35)

recaptures the Jensen inequality

(36)

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.

A function is exponentially convex if it is continuous and

(37)

for every and every , such that , .

Definition 3.6.

A function, whereis an interval in, is said to be convex ifis convex, or equivalently, if for alland all, we have

(38)

Remark 3.7.

Ifis exponentially convex, thenis a- convex function.

Consider a family of functions , from [15], defined as

(39)

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

Theorem 3.8.

Let , , let , be nonnegative real weights such that and . Let us define function

(310)

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

(i)the function is continuous on ,

(ii)for each and matrix is positive semidefinite. Particularly,

(311)

(iii)the function is exponentially convex on ,

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

(v)for , such that , one has

(312)

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.

Let be any compact interval, , , where , are nonnegative real weights such that and . If , then there exists such that

(313)

Theorem 3.10.

Let be any compact interval and as in Theorem 3.9. If such that does not vanish for any value of , then there exists such that

(314)

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,

(315)

Moreover, we can define three-parameter means as in [15]

(316)

where, including all the limit cases,

(317)

and the weighted power mean of is denoted as

(318)

All the limiting cases of (3.16) are given as follows:

(319)

Theorem 3.12.

Let , , where , are nonnegative real weights such that and . If are such that , then the following inequality is valid:

(320)

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

Let us recall the Hermite-Hadamard inequality: if is a convex function on and such that , then the following double inequality holds:

(41)

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.

A very interesting inequality closely connected with the Hermite-Hadamard inequality was given in [18]. Namely, it was shown by a simple geometric argument that for a convex function , the following is valid:

(42)

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

Dragomir et al. in [20] (see also [21, page 108]) considered a function , connected to Hadamard's inequality, given by

(43)

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.

Let be an interval with a nonempty interior and be a continuous function on . For function defined on as

(44)

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.

Let be an interval with a nonempty interior. Let be a continuous function on and a continuous function on . Let be a function defined by

(45)

For a function defined on as

(46)

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 .

Another function defined by a double integral in connection with the Hermite-Hadamard inequalities is considered in [26]

(47)

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

Theorem A9.

Let be an interval with a nonempty interior, a continuous function on , and . If is convex (concave) on , the function defined on as

(48)

is Schur-convex (Schur-concave) on .

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.

Suppose is an open interval and is a continuous function. Function

(49)

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

Theorem A11.

Suppose is an open interval and is a continuous function. Function

(410)

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:

(i) is convex (concave),

(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.

It is not difficult to verify that

(411)

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

In [28], the following identity was derived: if is such that is absolutely continuous for some ,, and , then

(412)

Applying identity (4.12) for , then choosing, respectively, and , adding up two thus obtained identities, and finally dividing by two procures

(413)

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

Proof.

We have

(414)

Using (4.13), we see that in fact

(415)

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.

Identity similar to (4.13) can be found in [29]: if is twice differentiable, then the following identity is valid:

(416)

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

Theorem 4.4.

If is a convex (concave) function, then the function

(417)

is Schur-convex (Schur-concave).

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

Proof.

Using (4.16), we deduce

(418)

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

Now, assume in addition that . Applying the integral mean value theorem yields that there exists such that

(419)

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.

If is Schur-convex, since , one has

(420)

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.

Let. A th-order divided difference of at distinct pointsis defined recursively by

(51)

Notion closely related to divided differences is -convexity.

Definition 5.2.

A function is said to be -convex on ,, if and only if for all choices ofdistinct points in,

(52)

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.

Let be -convex on . Then, the function

(53)

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

Therefore, for and , , where is the domain of , Jensen's inequality yields

(54)

where .

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.

Let be -convex on . Then,

(55)

is a convex function of the vector . Consequently,

(56)

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.

Let be an -convex function on . If and , then

(57)

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.

Let be differentiable on and continuous on . Define

(58)

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

(A) is convex on ,

(B) for all ,

(C) for all ,

(D) is convex on ,

(E) is Schur-convex on

and

(A^') is concave on ,

(B^') for all ,

(C^') for all ,

(D^') is concave on ,

(E^') is Schur-concave on .

First, note that function defined in (5.8) is the 1st-order divided difference of function . Also,

(59)

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

In [35], Yang introduced the following mean: let be a symmetric and positively homogeneous function (i.e., such that for , ), satisfying . For , the two-parameter family generated by is defined as

(510)

Note that the extended mean vales and the Gini means

(511)

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.

In a series of papers, Yang investigated various properties of the mean , such as monotonicity and logarithmic convexity. In [38], Witkowski continued his research by extending his results, giving simplified proofs and other conditions equivalent to monotonicity and convexity of . In order to do this, he introduced the function: , so as to present in the form

(512)

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 ,

(c) is convex (concave) for ,

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

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

(f) is concave (convex) for .

If is positively homogeneous, then so are for every and so the four-parameter family can be created by

(513)

Since

(514)

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].