Schur-convexity, Schur-geometric and Schur-harmonic convexity for a composite function of complete symmetric function.

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of the Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity for a composite function of the complete symmetric function.

Background

Throughout the article, denotes the set of real numbers, denotes n-tuple (n-dimensional real vectors), the set of vectors can be written as

In particular, the notations and denote and , respectively.

The following complete symmetric function is an important class of symmetric functions.

For , the complete symmetric function is defined aswhere are non-negative integers.

It has been investigated by many mathematicians and there are many interesting results in the literature.

Guan (2006) discussed the Schur-convexity of and proved that is increasing and Schur-convex on . Subsequently, Chu et al. (2011) proved that is Schur-geometrically convex and harmonically convex on .

Recently, Sun et al. (2014) studied the Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity of the following composite function of

Using the Lemma 1, Lemma 2 and Lemma 3 in second section, they proved as follows: Theorem A, Theorem B and Theorem C, respectively.

Theorem A

Forand,

(i) is increasing infor alliand Schur-convex onfor eachrfixed;

(ii) ifris even integer (or odd integer, respectively), thenis Schur-convex (or Schur-concave, respectively) on, and it is decreasing (or increasing, respectively) infor all.

Theorem B

Forand,

(i) is Schur-geometrically convex on;

(ii) ifris even integer (or odd integer, respectively), thenis Schur-geometrically convex (or Schur-geometrically concave, respectively) on.

Theorem C

Forand,

(i) is Schur-harmonically convex on;

(ii) ifris even integer (or odd integer, respectively), thenis Schur-harmonically convex (or Schur-harmonically concave, respectively) on.

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we will provide much simpler proofs of the above results.

Definitions and lemmas

For convenience, we recall some definitions as follows.

Definition 1

Let and .

(i) means for all .

(ii) Let , : is said to be increasing if implies . is said to be decreasing if and only if is increasing.

Definition 2

Let and .

is said to be majorized by (in symbols ) if for and , where and are rearrangements of and in a descending order.

Let , : is said to be a Schur-convex function on if on implies The function is said to be Schur-concave on if and only if is a Schur-convex function on .

Definition 3

Let and .

(i) is said to be a convex set if , implies .

(ii) Let be a convex set. A function : is said to be convex on if for all , and all . The function is said to be concave on if and only if is a convex function on .

Definition 4

(i) A set is called symmetric, if implies for every permutation matrix P.

(ii) A function is called symmetric if for every permutation matrix P, for all .

Lemma 1

(Schur-convex function decision theorem) (Marshall et al. 2011, p. 84) Letbe symmetric convex set with nonempty interior.is the interior of. The functionis continuous onand continuously differentiable on. Thenis aif and only ifis symmetric onandholds for any.

The first systematical study of the functions preserving the ordering of majorization was made by Issai Schur in 1923. In Schur’s honor, such functions are said to be “Schur-convex”. It has many important applications in analytic inequalities, combinatorial optimization, quantum physics, information theory, and other related fields. See Marshall et al. (2011), Rovenţa (2010), Čuljak et al. (2011), Zhang and Shi (2014).

Definition 5

Let , and .

(Zhang 2004, p. 64) is called a geometrically convex set if for all , and , such that .

(Zhang 2004, p. 107) The function : is said to be a Schur-geometrically convex function on , for any , if implies . The function is said to be a Schur-geometrically concave function on if and only if is a Schur-geometrically convex function on .

By Definition 5, the following is obvious.

Proposition 1

Let, and letThenis a Schur-geometrically convex (or Schur-geometrically concave, respectively) function onif and only ifis a Schur-convex (or Schur-concave, respectively) function on.

Lemma 2

(Schur-geometrically convex function decision theorem) (Zhang 2004, p.108) Letbe a symmetric and geometrically convex set with a nonempty interior. Letbe continuous onand differentiable in. Ifis symmetric onandholds for any, thenis a Schur-geometrically convex (or Schur-geometrically concave, respectively) function.

The Schur-geometric convexity was proposed by Zhang (2004), and was investigated by Chu et al. (2008), Guan (2007), Sun et al. (2009), and so on. We also note that some authors use the term “Schur multiplicative convexity”.

In 2009, Chu (Chu et al. (2011), Chu and Sun (2010), Chu and Lv (2009)) introduced the notion of Schur-harmonically convex function.

Definition 6

Chu and Sun (2010) Let , and .

(i) A set is said to be harmonically convex if for every .

(ii) A function is said to be Schur-harmonically convex on , for any , if implies . A function is said to be a Schur-harmonically concave function on if and only if is a Schur-harmonically convex function on .

By Definition6, the following is obvious.

Proposition 2

Letbe a set, and let. Thenis a Schur-harmonically convex (or Schur-harmonically concave, respectively) function onif and only ifis a Schur-convex (or Schur-concave, respectively) function on.

Lemma 3

(Schur-harmonically convex function decision theorem) (Chu and Sun 2010) Letbe a symmetric and harmonically convex set with inner points and letbe a continuous symmetric function which is differentiable on. Thenis Schur-harmonically convex (or Schur-harmonically concave, respectively) onif and only if

Lemma 4

Ifris even integer (or odd integer, respectively), thenis decreasing and Schur-convex (or increasing and Schur-concave, respectively) on.

Proof

Notice thati.e.

If r is even integer, then . For , if , then and , but is Schur-convex in , so that , i.e. , this shows that is Schur-convex in . If , then , but is increasing in , so that , i.e. , this shows that is decreasing in .

If r is odd integer, then . For , if , then and , but is Schur-convex in , so that , i.e. , this shows that is Schur-concave in . If , then , but is increasing in , so that , i.e. , this shows that is increasing in .

Lemma 5

(Marshall et al. 2011, p. 91; Wang 1990, p. 64–65) Let the set ,,and.

(i) Ifis increasing and Schur-convex andfis increasing and convex, thenis increasing and Schur-convex.

(ii) Ifis decreasing and Schur-convex andfis increasing and concave, thenis decreasing and Schur-convex.

(iii) Ifis increasing and Schur-concave andfis increasing and concave, thenis increasing and Schur-concave.

(iv) Ifis decreasing and Schur-convex andfis decreasing and concave, thenis increasing and Schur-convex.

(v) Ifis increasing and Schur-concave andfis decreasing and concave, thenis decreasing and Schur-concave.

Lemma 6

Let the set. The functionis differentiable.

(i) Ifis increasing and Schur-convex, thenis Schur-geometrically convex.

(ii) Ifis decreasing and Schur-concave, thenis Schur-geometrically concave.

We only give the proof of Lemma 6 (i) in detail. Similar argument leads to the proof of Lemma 6 (ii).

For and , we have

Since is Schur-convex on , by Lemma 1, we haveNotice that and is increasing, we have , and , so that , by Lemma 2, it follows that is Schur-geometrically convex on .

Lemma 7

Let the set. The functionis differentiable.

(i) Ifis increasing and Schur-convex, thenis Schur-harmonically convex.

(ii) Ifis decreasing and Schur-concave, thenis Schur-harmonically concave.

We only give the proof of Lemma 7 (ii) in detail. Similar argument leads to the proof of Lemma 7 (i).

For and , we have

Since is Schur-concave on , by Lemma 1, we haveNotice that is decreasing and is increasing, we have and , so that , by Lemma 3, it follows that is Schur-harmonically concave on .

Simple proof of theorems

Proof of Theorem A

Let . Directly calculating yields and , it is to see that g is increasing and convex on (0, 1) and g is increasing and concave on .

Since is increasing and Schur-convex in , from Lemma 5 (i) it follows that is increasing and Schur-convex in , and then by continuity of on , it follows that is increasing and Schur-convex on .

If r is even integer, then from Lemma 4, we known that is decreasing and Schur-convex, moreover g is increasing and concave on . By Lemma 5 (ii), it follows that is decreasing and Schur-convex.

If r is odd integer, then from Lemma 4, we known that is increasing and Schur-concave, moreover g is increasing and concave on . By Lemma 5 (iii), it follows that is increasing and Schur-concave.

The proof of Theorem A is completed.

Proof of Theorem B

From Theorem A (i) and Lemma 6 (i), it follows that Theorem B (i) holds.

Considing

Let . Then on . Directly calculating yields and , it is to see that h is increasing and concave on . From Lemma 4 and Lemma 5 (ii) (or (iii), respectively), it follows that if r is even integer (or odd integer, respectively), then is Schur-convex (or Schur-concave, respectively) on . And then, by Proposition 1, Theorem B (ii) holds.

The proof of Theorem B is completed.

Proof of Theorem C

From Theorem A (i) and Lemma 7 (i), it follows that Theorem C (i) holds.

Considing

Let . Then on (0, 1). Directly calculating yields and , it is to see that p is decreasing and concave on (0, 1). From Lemma 4 and Lemma 5 (iv) (or (v), respectively), it follows that if r is even integer (or odd integer, respectively), then is Schur-convex (or Schur-concave, respectively) on (0, 1). And then, by Proposition 2, Theorem C (ii) holds.

The proof of Theorem C is completed.

Conclusions

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of Theorem A, B, C.