Majorization involving the cyclic moving average.

We solve an open problem on some majorization inequalities involving the cyclic moving average.

Introduction

We first recall two definitions.

Definition 1.1

([1])

For fixed , let and be two n-tuples of real numbers.

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

Let . A function is said to be a Schur-convex function (shortly, an S-convex function) if

For example:

In 2006, I. Olkin, one of the authors of the book [1], wrote a letter to K. Z. Guan, referring to the following interesting question: is it true that However, a proof for remains elusive (see [1], p. 63).

In 2010, Shi [2] proved that (1) holds when , and , . In this paper, we prove that (1) holds for any and .

For any , let be the ordered component of the sequence . We denote

Lemmas and corollaries

For proving our main results, we need the following lemmas.

Lemma 2.1

Let and . Then

Proof

For any and , we have Note that , so we can induce that  □

if , that is, , then . It follows that .

if , that is, , then . Since , we have . So . It follows that .

. Therefore (2) holds.

From the proof of Lemma 2.1 it is easy to deduce the following:

Corollary 2.2

Let , , and let . For any , there exist and such that and

Lemma 2.3

Let , , and .

If then

If then

For , if then

If then

For , if then

For , if then

□

By Lemma 2.1 we have It follows that Thus we have By the left inequality of (3) we have Note that , so we have Therefore

By Lemma 2.1 we have By the right inequality of (4) we get Note that , so we have This means that

By (5) we get Note that and , so we have It follows that and Therefore (6) holds.

By (7) we get Since , we have It follows that

By (8) we have Note that and , so we have This means that

By (9) we get Note that and , so we have It follows that So we get Therefore Note that , so we have It follows that Therefore

Lemma 2.4

Let , , , and .

If , then

If , then

For , if , then

We only prove (i). Using a similar method, we can obtain (ii) and (iii).

By Lemma 2.1 we have By Corollary 2.2 we let where , , and . It is clear that

Next, we prove that and .

If , which means that the right-hand side of (12) includes , then by (11) the right-hand side of (12) should include and , so we have This is a contradiction with (13).

If , then by (13) we have . Together with (11), we get So the right-hand side of (12) must include , which means that . This is a contradiction with .

Therefore and . So (10) holds. □

Corollary 2.5

Let , , , and , and let

For , we have

For , we have

For and , must be one of the following two cases: or

If , by (14) we have By Lemma 2.3(i) we have and then by Lemma 2.4(i) we can induce that (15) holds.

If , then by (14) we have By Lemma 2.3(ii) we have and then by Lemma 2.4(i) we can induce that (16) holds.

By Corollary 2.2 we let where , , and . Then we have

Next, we prove that or .

If , then by Lemma (2.3)(iii) we have . So we get Thus the right-hand side of (17) includes , which means that . Therefore This is a contradiction with (18).

If , then by (18) we have . By Lemma 2.3(iii) we get . It follows that So the right-hand side of (17) must include . Therefore This is a contradiction with .

Thus or . □

In a similar way as in Corollary 2.5, we can prove the following corollaries.

Corollary 2.6

Let , , , and .

If , then must be one of the following two cases: or

If , then must be one of the following two cases: or

Corollary 2.7

Let , , , and , and let . If then we have:

if , then

if , then must be one of the following two cases: or

Lemma 2.8

Let , , , and , and let . If then

By a simple calculation we obtain By (19) we have It follows that So we have Note that Thus we can induce that This means that (20) holds. □

Lemma 2.9

Let , , , and , and let . If then

Note that By (21) we have It follows that So we can induce that Since we have This means that (22) holds. □

Main results

We are now in a position to prove our main results (1) in two cases: and .

Theorem 3.1

For any , we have

It is clear that (23) holds if . Next, let . Then we have

For , we prove that in the following two cases:  □

If , then

If , then So (23) holds.

Theorem 3.2

For any and , we have

It is clear that (24) holds for any , and for , .

Next, let and .

For any and , let , and let Next, we prove that in the following two cases:  □

If and , then by Corollary 2.7(i) and Lemma 2.8 we get

If and , then by Corollary 2.7(ii), Lemma 2.8, and Lemma 2.9 we get Note that so (24) holds.

Discussion

In the theory of majorizations, there are two key concepts, majorizing relations and Schur-convex functions. Majorizing relations are weaker ordered relations among vectors, and Shur-convex functions are an extension of classical convex functions. Combining these two objects is an effective method of constructing inequalities.

In the theory of majorization, there are two important and fundamental objects, establishing majorizing relations among vectors and finding various Schur-convex functions. Majorizing relations deeply characterize intrinsic connections among vectors, and combining a new majorizing relation with suitable Schur-convex functions can lead to various interesting inequalities; see [3-13].