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volume 7, issue 3, article 79, 2006.

Received 03 March, 2006;

accepted 21 March, 2006.

Communicated by:I. Olkin

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Journal of Inequalities in Pure and Applied Mathematics

A VARIANCE ANALOG OF MAJORIZATION AND SOME ASSOCIATED INEQUALITIES

MICHAEL G. NEUBAUER AND WILLIAM WATKINS

Department of Mathematics California State University Northridge Northridge, CA 91330.

EMail:michael.neubauer@csun.edu EMail:bill.watkins@csun.edu URL:http://www.csun.edu/˜vcmth006

c

2000Victoria University ISSN (electronic): 1443-5756 064-06

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A Variance Analog of Majorization and Some Associated Inequalities

Michael G. Neubauer and William Watkins

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J. Ineq. Pure and Appl. Math. 7(3) Art. 79, 2006

Abstract

We introduce a partial order, variance majorization, onRⁿ, which is analogous to the majorization order. A new class of monotonicity inequalities, based on variance majorization and analogous to Schur convexity, is developed.

2000 Mathematics Subject Classification: Primary: 26D05, 26D07; Secondary:

15A42.

Key words: Inequality, Symmetric polynomial, Majorization, Schur convex, Variance.

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1. Introduction

Let (x₁, . . . , x_n)and (y₁, . . . , y_n)be two sequences of real numbers in nonincreasing order. The sequencexmajorizesyif

i

X

k=1

x_k≥

i

X

k=1

y_k,

for i = 1, . . . , n with equality for i = n. Majorization is a partial order on the set of nonincreasing sequences having the same sum and it plays a large role in the theory of inequalities dating back to the work of I. Schur [7]. In- deed a function F(x₁, . . . , x_n) of n real variables is said to be Schur convex if F(x) ≥ F(y) whenever the sequence x majorizes y. Marshall and Olkin [6] catalog many functions and results of this type with particular emphasis on statistical inequalities. As a simple example, take the product function F(x) = Qn

k=1x_k. If x, y are n-tuples of nonnegative real numbers and if x majorizes y, then F(x) ≤ F(y). That is, −F is a Schur convex function. In particular, ify = (¯x, . . . ,x)¯ thenxmajorizesy and therefore the product ofn nonnegative numbers with fixed mean is maximized when all of them are equal to the meanx. Another way to state this well-known elementary result is that¯ the product of a sequence x of nonnegative reals with fixed mean x¯ attains a maximum when the variance ofxis zero.

Now suppose that the variance ofxis also fixed. In this paper, we define a partial order (variance majorization) on the set of sequences x having a fixed mean and a fixed variance. We obtain a monotonicity result similar to the one above for sequences in which one is variance-majorized by the other. In particular the maximum value of the product of a sequence of nonnegative reals with

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fixed mean and fixed variance is attained when the sequence takes on only two values α < β and the multiplicity of β is 1. This simple consequence of the main theorem is known as Cohn’s Inequality [1]: ifx₁, . . . , x_nare nonnegative reals then

n

Y

k=1

x_k ≤αⁿ⁻¹β,

whereαandβare chosen so that the sequencesx= (x1, . . . , xn)and(α, . . . , α, β) have the same means and the same variances.

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2. Majorization and Variance Majorization

LetI(I^st) be the set of nondecreasing (strictly increasing) sequences inRⁿ: I={x∈Rⁿ :x1 ≤x2 ≤ · · · ≤xn}

I^st ={x∈Rⁿ :x₁ < x₂ <· · ·< x_n}.

The variance majorization order involves the variances of leading subsequences of x ∈ I. So let x[i] = (x₁, . . . , x_i)be the leading subsequence ofx for i = 1, . . . , n. Note thatx[i]consists of thei smallest components ofx. We denote the mean ofx[i]by

x[i] = (1/i)

i

X

k=1

x_k, and the variance ofx[i]by

Var(x[i]) = (1/i)

i

X

k=1

(x_k−x[i])².

2.1. Definitions of Variance Majorization and Majorization

Definition 2.1 (Variance Majorization). Let x = (x₁, . . . , x_n) and y = (y1, . . . , yn)be sequences of real numbers in Isuch thatx¯ = ¯y and Var(x) = Var(y). We say thatxis variance majorized byy(oryvariance majorizesx), if

Var(x[i])≤Var(y[i]), fori= 2, . . . n. We writex^vm≺ yory^vm x.

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For fixed mean m and variance v ≥ 0, variance majorization is a partial order on the set

S(m, v) = {x∈I: ¯x=m,Var[x] =v},

which is the intersection of Iwith the sphere in Rⁿ centered atm(1,1, . . . ,1) with radius √

nv, and the hyperplane throughm(1,1, . . . ,1)orthogonal to the vector(1,1, . . . ,1).

By contrast, majorization is a partial order of the set of nonincreasing sequences

D={z∈R:z1 ≥z2 ≥ · · · ≥zn}.

Definition 2.2 (Majorization). Let x = (x₁, . . . , x_n)andy = (y₁, . . . , y_n)be sequences inDsuch thatx¯= ¯y. We say thatxis majorized byy(orymajorizes x), if

x[i]≤y[i], fori= 1, . . . , n. In this case we writex^maj≺ y.

The definition of majorization is usually given in this equivalent form:

i

X

k=1

x_k≤

i

X

k=1

y_k, fori= 1, . . . , nwith equality fori=n.

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2.2. Least and Greatest Sequences with Respect to the Variance Majorization Order

Returning now to the variance majorization order, there is a least element and a greatest element inS(m, v)for eachmandv ≥0.

Lemma 2.1. Letmandv ≥0be real numbers and let x_min = (α₁, . . . , α₁, β₁) x_max= (α₂, β₂, . . . , β₂), where

α₁ =m−p

v/(n−1), β₁ =m+p

(n−1)v, α₂ =m−p

(n−1)v, β₂ =m+p

v/(n−1).

Thenx_min, x_max ∈S(m, v)and

x_min ^vm≺ x^vm≺ x_max, for allx∈S(m, v).

Figure1shows the Hasse diagram for the variance majorization partial order for all integral sequences of length six with sum 0 and sum of squares equal to 30. In this case,x_min = (−1,−1,−1,−1,−1,5)andx_max= (−5,1,1,1,1,1).

By contrast, the least and greatest elements inD∩ {x: ¯x=m}with respect to the majorization order are(¯x, . . . ,x)¯ and(n¯x,0, . . . ,0).

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http://jipam.vu.edu.au

Variance-majorization 5

8-1,-1,-1,-1,-1, 5<

8-2,-2,-1,-1, 2, 4< 8-2,-2,-2, 1, 1, 4<

8-3,-2, 0, 0, 1, 4<

8-2,-2,-2, 0, 3, 3<

8-3,-1,-1,-1, 3, 3<

8-4,-1, 0, 0, 2, 3<

8-3,-3, 1, 1, 1, 3< 8-3,-3, 0, 2, 2, 2<

8-4,-1,-1, 2, 2, 2< 8-4,-2, 1, 1, 2, 2<

8-5, 1, 1, 1, 1, 1<

Figure 1: Variance majorization partial order for integral sequences inS(0,6)

Figure 1: Variance majorization partial order for integral sequences inS(0,30).

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3. Variance Monotone Functions and Schur Convex Functions

LetI be a closed interval inRand letF(x₁, . . . , x_n)be a real-valued function defined onI∩Iⁿ.

Definition 3.1 (Variance Monotone). The function F is variance monotone increasing onI∩Iⁿif

x^vm≺ y =⇒ F(x)≤F(y),

for allx, y ∈ I∩Iⁿ. If−F is variance monotone increasing, we say thatF is variance monotone decreasing.

Definition 3.2 (Schur Convex). The functionF is Schur convex onD∩Iⁿif x^maj≺ y =⇒ F(x)≤F(y),

for allx, y ∈D∩Iⁿ.

3.1. The Main Result

The next theorem is the main result.

Theorem 3.1. Let I be a closed interval in Rⁿ, and let F(z₁, . . . , z_n) be a continuous, real-valued function onI∩Iⁿthat is differentiable on the interior ofI∩Iⁿwith gradient∇F(z) = (F1(z), . . . , Fn(z)). Suppose that

(3.1) F2(z)−F1(z)

z₂−z₁ ≥ F3(z)−F2(z)

z₃−z₂ ≥ · · · ≥ Fn(z)−Fn−1(z) z_n−zn−1

,

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for allz ∈I^st∩Iⁿ. ThenF is variance monotone increasing onI∩Iⁿ, that is x^vm≺ y =⇒ F(x)≤F(y).

for allx, y ∈I∩Iⁿ.

For functions of the formF(z) =φ(x₁) +· · ·+φ(x_n), Theorem3.1special- izes to the following corollary:

Corollary 3.2. Let φ(t) be a continuous, real-valued function on a closed in- tervalI such thatφis twice-differentiable on the interior ofI andφ⁰⁰ is nonin- creasing. Then the function

F(x₁, . . . , x_n) =φ(x₁) +· · ·+φ(x_n)

is variance monotone increasing on the set of nondecreasing sequences inIⁿ. That is,

x^vm≺ y =⇒ φ(x₁) +· · ·+φ(x_n)≤φ(y₁) +· · ·+φ(y_n), for allx, y ∈I∩Iⁿ.

It turns out thatS(m, v)⊂Iⁿwhen the interval I =h

m−p

(n−1)v, m+p

(n−1)v)i

(see Corollary 4.8). Thus the sequences xmax and xmin (described in Lemma 2.1) are in Iⁿ. So, if F is variance monotone increasing on I∩Iⁿ, then the maximum and minimum values ofF are attained atx_maxandx_min. This means we can boundF(x)by expressions involving only the mean and variance ofx:

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Corollary 3.3. Letmandv ≥0be real numbers and I =

h

m−p

(n−1)v, m+p

(n−1)v i

. LetF be a variance monotone increasing function onI∩Iⁿ. Then

F(α₁, . . . , α₁, β₁)≤F(x)≤F(α₂, β₂, . . . , β₂), for allx∈S(m, v), whereα₁, β₁, α₂, β₂are defined as in Lemma2.1.

3.2. Schur Convex Functions

By comparison, the following theorem by Schur is the result analogous to The- orem3.1for majorization. It plays the central role in the theory of majorization inequalities:

Theorem 3.4 ([7]). Let F(z) be a continuous, real-valued function onDthat is differentiable on in the interior ofD. Then

x^maj≺ y =⇒ F(x)≤F(y), for allx, y ∈Dif and only if

(3.2) F₁(z)≥F₂(z)≥ · · · ≥F_n(z), for allz in the interior ofD.

The result analogous to Corollary 3.2 for majorization is known as Kara- mata’s Theorem:

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Corollary 3.5 ([4]). Let φ be a continuous, real-valued function on a closed intervalI such thatφ is twice differentiable on the interior ofI andφ⁰⁰ is non- negative. Then

x^maj≺ y =⇒ φ(x₁) +· · ·+φ(x_n)≤φ(y₁) +· · ·+φ(y_n), for allx, y ∈D∩Iⁿ.

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4. Some Variance Monotone and Schur Convex Functions

In this section, we give a short list of some common functions that are monotone in both the regular majorization and variance majorization orders. And we give as corollaries some samples of the kinds of inequalities one can obtain from Corollary3.3.

4.1. Elementary Symmetric Functions

Let E_k(z)denote the kth elementary symmetric function of the sequencez = (z₁, . . . , z_n)∈Rⁿ:

E_k(z) = X

z_i₁z_i₂· · ·z_i_k,

where the sum is taken over all sets ofkindices with1≤i₁ <· · ·< i_k ≤n.

Theorem 4.1. LetE_kbe thekth elementary symmetric function. Then x^vm≺ y =⇒ Ek(y)≤Ek(x),

x^vm≺ y =⇒ Ek+1(x)

E_k(x) ≤ Ek+1(y) E_k(y) , for allx, y ∈I∩[0,∞)ⁿ.

The next corollary is obtained from Corollary3.3by evaluating the elementary symmetric functionE_katx_minandx_max.

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Corollary 4.2. Letx ∈ I∩[0,∞)ⁿ. ThenA(v, m, k) ≤E_k(x)≤ B(v, m, k), where

A(v, m, k) =E_k(x_max) = n

k m+

r v n−1

^k−1

m−(k−1) r v

n−1

B(v, m, k) =E_k(x_min) = n

k m−

r v n−1

^k−1

m+ (k−1) r v

n−1

The inequality analogous to Theorem4.1for (regular) majorization is given next:

Theorem 4.3 ([6, p. 80]).

x^maj≺ y =⇒ E_k(y)≤E_k(x) x^maj≺ y =⇒ Ek+1(y)

E_k(y) ≤ Ek+1(x) E_k(x) , for allx, y ∈D∩[0,∞)ⁿ.

4.2. Moment Functions

Letpbe a positive real number and letφ(t) = t^p. Thepth moment function of z ∈[0,∞)ⁿis given by

M_p(z) =z₁^p+· · ·+z^p_n.

The following results are applications of Corollaries3.5and3.2:

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Theorem 4.4. LetM_p be thepth moment function. Then

x^maj≺ y =⇒ M_p(x)≤M_p(y), forp∈(−∞,0]∪[1,∞) x^maj≺ y =⇒ M_p(y)≤M_p(x), forp∈[0,1],

for allx, y ∈D∩[0,∞)ⁿand

x^vm≺ y =⇒ Mp(x)≤Mp(y), forp∈(−∞,0]∪[1,2]

x^vm≺ y =⇒ M_p(y)≤M_p(x), forp∈[0,1]∪[2,∞), for allx, y ∈I∩[0,∞)ⁿ.

Again we obtain bounds on M_p(x), which depend only on the mean and variance ofx, from Corollary3.3:

Corollary 4.5. Letx∈I∩[0,∞)ⁿwith meanmare variancev. Let A(m, v, p)=M_p(x_min) =(n−1)

m−p v n−1

p

+

m+p

(n−1)vp

B(m, v, p)=M_p(x_max)=

m−p

(n−1)vp

+ (n−1)

m+p _v

n−1

p

. Then

A(m, v, p)≤M_p(x)≤B(m, v, p), forp∈(−∞,0)∪[1,2]

and

B(m, v, p)≤Mp(x)≤A(m, v, p), forp∈[0,1]∪[2,∞).

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4.3. Entropy Function

The entropy function is defined forx∈[0,∞)ⁿby

H(x) =−(x₁logx₁ +· · ·+x_nlogx_n).

Letting φ(t) = −tlogt, we have φ⁰⁰(t) = −1/t, which is nonpositive and increasing on [0,∞). Thus −φ satisfies the conditions of Corollaries 3.2 and 3.5. Thus we have the following inequalities:

Theorem 4.6. LetHbe the entropy function. Then x^vm≺ y =⇒ H(y)≤H(x), for allx, y ∈I∩[0,∞)ⁿ, and

x^maj≺ y =⇒ H(y)≤H(x), for allx, y ∈D∩[0,∞)ⁿ.

4.4. Coordinates of x

The smallest and the largest coordinates of a sequence are variance monotone decreasing.

Lemma 4.7. Letx, y ∈I. Then

x^vm≺ y =⇒ x₁ ≥y₁ andx_n≥y_n.

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We call this result a lemma because it is a part of the proof of Theorem3.1 rather than a consequence of it.

When combined with Corollary3.3, Lemma4.7gives bounds for the smallest and largest coordinates of a sequence inS(m, v)in terms ofmandv:

Corollary 4.8. Letx= (x₁, . . . , x_n)be a sequence inS(m, v). Then m−p

(n−1)v ≤ x₁ ≤ m−p

v/(n−1) m+p

v/(n−1) ≤ x_n ≤ m+p

(n−1)v.

Applying Corollary4.8to the eigenvalues of a symmetric matrix, we recover an equivalent form of an inequality of Wolkowicz and Styan [8, Theorem 2.1]

that bounds the maximum and minimum eigenvalues by expressions involving only the trace and Euclidean norm of the matrix:

Corollary 4.9. Let Gbe a symmetric matrix with eigenvaluesλ₁ ≤ · · · ≤ λ_n. Then

tr(G)

n + _n^√¹_n−1p

n||G||²−(tr(G))² ≤λ_n≤ tr(G)

n +

√n−1 n

pn||G||²−(tr(G))² tr(G)

n −

√n−1 n

pn||G||²−(tr(G))² ≤λ₁ ≤ tr(G)

n − _n^√¹_n−1p

n||G||²−(tr(G))². Corollary4.9 follows from the fact that the mean m and variance v of the eigenvalues can be expressed in terms of the trace and Euclidean norm||G||of

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Gas follows:

m= tr(G) n v = 1

n X

i

(λ_i−m)²

= 1 n

X

i

λ²_i −2mλ_i+m²

!

= 1

n(tr(G²)−nm²)

= 1

n² n||G||² −tr(G)² .

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5. Restricting S(m, v) to an Interval

Lemma 2.1 guarantees that there is a least and a greatest element in S(m, v) with respect to the variance majorization order. Now we restrict the setS(m, v) to an intervalI = [m−δ, m+β], withδ, β ≥0, containingm. From Corollary 4.8, we have

h

m−p

v/(n−1), m+p

v/(n−1)in

⊂S(m, v)⊂h

m−p

(n−1)v, m+p

(n−1)vin

. So if eitherδ, β <p

v/(n−1), thenS(m, v)∩Iⁿ=∅. However, ifS(m, v)∩ Iⁿis not empty, then it contains a least element, but it may not contain a greatest element.

Lemma 5.1. Let m and v ≥ 0 be real numbers. Let I be the interval I = [m−δ, m+β]such thatS(m, v)∩Iⁿis not empty. Then there exist unique real numbersm−δ≤α≤γ < m+βand an integer1≤j ≤n−1such that the sequence

x_min = (

j

z }| { α, . . . , α, γ,

n−j−1

z }| {

m+β, . . . , m+β)∈S(m, v)∩Iⁿ. Moreoverx_min ^vm≺ xfor allx∈S(m, v)∩Iⁿ.

Example 5.1. Let n = 5. The least element in S(0,1944/5) ∩ [−36,24]⁵ is (−18,−18,−12,24,24). There is no greatest element in S(0,1944/5)∩ [−36,24]⁵. See Section6.7.

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The situation for restricting the sequences to a closed interval is a little different for majorization. There is always a least element and a greatest in D∩[m, M]ⁿ. The sequence(¯x, . . . ,x)¯ ∈D∩[m, M]ⁿis the least element. The greatest element with respect to majorization takes at most three values for its coordinates, two of which are the end points of the closed interval. That is, the greatest element in for majorization inD∩[m, M]ⁿis of the form

(M, . . . , M, θ, m, . . . , m).

A discussion of restricting the majorization order to an interval is given in [5] and [6, page 132].

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6. Proofs

The main technique used in the proofs is to express the sequences inIas linear combinations of a special basis forRⁿ, the so-called Helmert basis.

6.1. Helmert basis

The purpose of this section is to describe the relationship between the coordinates of ann-tuplex∈Rⁿand the coordinates ofxwith respect to the so-called Helmert basis forRⁿ(see [6, p. 47] for a discussion of the Helmert basis). The Helmert basis forRⁿis defined as follows:

w^T₀ = 1

√n(1,1, . . . ,1) w^T₁ = 1

√2(−1,1,0, . . . ,0) w^T₂ = 1

√6(−1,−1,2,0, . . . ,0) ...

w^T_i = 1 pi(i+ 1)(

i

z }| {

−1, . . . ,−1, i,0, . . . ,0) ...

w^T_n−1 = 1

p(n−1)n(−1, . . . ,−1, n−1).

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It is clear that {w₀, w₁, . . . , wn−1}is an orthonormal basis forRⁿ. Thus every vectorx∈Rⁿis a linear combinationx=Pn−1

k=0akwk. The Helmert coefficient a₀ is determined by the mean x¯ of the sequence x. Specifically, a₀ = √

nx.¯ The other Helmert coefficients, a₁, . . . , an−1 are related to the variance, partial variances, and order of the coordinates ofx.

Lemma 6.1. Letxbe a sequence of real numbers withx=Pn−1

k=0a_kw_k. Then Var(x[i]) = 1

i(a²₁+· · ·+a²_i−1), fori= 2, . . . , n. In particular,

Var(x) = 1

n(a²₁+· · ·+a²_n−1).

Proof. Sincea_k=x·w_k,

i−1

X

k=1

a²_k=x

i−1

X

k=1

w^T_kw_k

! x^T

=x((I_i−(1/i)J_i)⊕0n−i)x^T

=

i

X

k=1

x²_k−(1/i)(

i

X

k=1

x_k)²

!

=iVar(x[i]).

Thei×iidentity matrix is denoted byI_i andJ_i denotes thei×imatrix all of whose entries are one. The fact thatPi−1

k=1w^T_kw_k =I_i−(1/i)J_ifollows from a simple inductive argument.

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Letx = P

a_iw_i. In the next definition and lemma, we give necessary and sufficient conditions on the sequence (a1, . . . , an−1) for the sequence x to be nondecreasing. (Clearly, the coefficienta₀does not influence the relative order- ing of the coordinates ofx.)

Definition 6.1 (Admissible Sequence). Letα= (α₁, . . . , αn−1)be a sequence of nonnegative real numbers. Thenαis admissible if

(6.1) (i−1)iαi−1 ≤i(i+ 1)α_i, for2≤i≤n−1.

Lemma 6.2. Letx= (x₁, . . . , x_n)be a vector inRⁿanda₀, . . . , an−1be scalars such thatx=Pn−1

i=0 aiwi. The following conditions are equivalent:

1. xis nondecreasing.

2. a_i ≥ 0, for i = 1, . . . , n−1, and the sequence a⁽²⁾ = (a²₁, . . . , a²_n−1)is admissible.

3. the kth component of ai−1w_i −a_iwi−1 is nonnegative for allk 6= i and nonpositive fork =i.

Proof. Let1≤i≤n−1. Thenx₂−x₁ =√

2a₁, and x_i+1−x_i = ia_i

pi(i+ 1) − (i−1)ai−1

p(i−1)i − a_i pi(i+ 1)

!

= (i+ 1)a_i

pi(i+ 1) − (i−1)ai−1

p(i−1)i

= 1 i

pi(i+ 1)a_i−p

(i−1)iai−1

, (6.2)

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fori ≥ 2. Thusxis nondecreasing if and only ifa_i ≥ 0, fori = 1, . . . , n−1 anda⁽²⁾is admissible. So Conditions1and2are equivalent.

Now let(a_i−1w_i−a_iw_i−1)_kbe thekcomponent ofa_i−1w_i−a_iw_i−1. Then

(6.3) (ai−1wi−aiwi−1)k =











−√^aⁱ⁻¹

i(i+1) +√^aⁱ

(i−1)i , ifk < i,

−√^aⁱ⁻¹

i(i+1) − ^(i−1)a√ ⁱ⁻¹

(i−1)i , ifk =i,

iai−1

√

i(i+1) , ifk =i+ 1,

0 , ifk > i+ 1.

To prove that Condition 2 implies Condition 3, suppose that a_i ≥ 0 for i = 1, . . . , n−1and thata⁽²⁾is admissible. It is clear from Equation (6.3) that (ai−1w_i−a_iwi−1)_k ≥0for allk 6=iand that(ai−1w_i−a_iwi−1)_i ≤0.

Conversely, suppose that Condition3holds. Withk = 3, i = 2in Equation (6.3), we geta1 ≥ 0. Withk = 1 < iwe get thata⁽²⁾ is admissible andai ≥ 0 for alli. Thus Condition2holds.

6.2. Proof of Theorem 3.1 and Lemma 4.7

Let F be a differentiable, real-valued function onI∩Iⁿ satisfying Inequality (3.1). Let x, y be nondecreasing sequences in Iⁿ such thatx¯ = ¯y, Var(x) = Var(y)andx^vm≺ y. Let

x=

n−1

X

k=0

a_kw_k, y=

n−1

X

k=0

b_kw_k,

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for scalarsa_k, b_k,k = 0, . . . , n−1, and let

a⁽²⁾ = (a²₁, . . . , a²_n−1), b⁽²⁾ = (b²₁, . . . , b²_n−1).

Sincex¯= ¯y, we havea₀ =b₀. By Lemma6.2,a_k, b_k ≥0fork = 1, . . . , n−1, a⁽²⁾andb⁽²⁾are admissible, and by Lemma6.1

(6.4)

i

X

k=1

a²_k≤

i

X

k=1

b²_k,

for1≤i≤n−1with equality in Inequality (6.4) fori=n−1since Var(x) = Var(y).

Next define a pathc(t) = (c₀, c(t)₁, . . . , c(t)n−1)fromatob byc₀ = a₀ = b₀, and

c(t)_k = q

(1−t)a²_k+tb²_k,

for t ∈ [0,1]and k = 1, . . . , n−1. Then c(t)⁽²⁾ = (1 −t)a⁽²⁾ +tb⁽²⁾, from which it follows thatc⁽²⁾is admissible and that

i

X

k=1

a²_k≤

i

X

k=1

c(t)²_k≤

i

X

k=1

b²_k, fori= 1, . . . , n−1.

Now define a pathz(t)fromxtoyby z(t) = a₀w₀+

n−1

X

k=1

c(t)_kw_k.

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Sincec(t)_k ≥0andc(t)⁽²⁾ is admissible,z(t)is a nondecreasing sequence.

Letj be the smallest index for whichaj 6= bj. Thenc(t)k = ak fork < j andc(t)_j >0fort >0. It is easy to verify thatc⁰(t)_k = (b²_k−a²_k)/c(t)_k(unless c(t)_k = 0). Thus the tangent vectorz⁰(t)is given by

z⁰(t) =

n−1

X

k=j

b²_k−a²_k c(t)_k w_k (6.5)

= (b²_j −a²_j) w_j

c_j − w_j+1 c_j+1

+ (b²_j+1+b²_j+2−a²_j+1−a²_j+2)

w_j+2 cj+2

−w_j+3 cj+3

+· · ·+ b²₁+b²₂+· · ·+b²_n−1

−a²₁−a²₂− · · · −a²_n−1

wn−2

cn−2

− wn−1

cn−1

,

for t > 0. It follows from Inequality (6.4) that z⁰(t) is a nonnegative linear combination of the vectors ^w_cⁱ⁻¹

i−1 − ^w_cⁱ

i, fori=j + 1, . . . , n−1.

We now show thatz(t)∈ Iⁿfor allt ∈ [0,1]. Indeed, both the first and the last coordinates ofz(t)are nonincreasing functions oft. Thus

y₁ =z(1)₁ ≤z(t)₁ ≤z(t)₂ ≤ · · · ≤z(t)_n≤z(0)_n=x_n.

Sincey₁, x_n ∈I,z(t)∈Iⁿand thusF(z(t))is defined for allt∈[0,1]. To see thatz(t)₁andz(t)_nare decreasing, we examine the first and last coordinates of

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the vectors ^w_cⁱ⁻¹

i−1 − ^w_cⁱ

i. The first coordinate is

−1 p(i−1)ici−1

+ 1

pi(i+ 1)c_i,

which is nonpositive sincec⁽²⁾ is an admissible sequence. Thusz⁰(t)₁ ≤ 0and z(t)₁ is nonincreasing int. This proves the first part of Lemma4.7.

The last coordinate of ^w_cⁱ⁻¹

i−1 −^w_cⁱ

i is zero unlessi=n−1and in that case it is

−(n−1) p(n−1)ncn−1

,

which is also nonpositive. Thus z(t)_n is nonincreasing. This proves the other part of Lemma4.7.

Finally to prove thatF(z(t))is an increasing function int, we show that dF

dt =∇F ·z⁰(t)≥0.

In view of Equation (6.5), it suffices to show that

∇F ·

wi−1

ci−1

− wi

c_i

≥0, fori=j + 1, . . . , n−1.

Sincew_kis orthogonal to the all-ones vectore,

∇F ·

wi−1

ci−1

−w_i c_i

= (∇F −F_ie)·

wi−1

ci−1

− w_i c_i

.

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For eachi= 1, . . . , n−1define a functionK_ionI^st∩Iⁿby K_i(z) = F_i+1(z)−F_i(z)

z_i+1−z_i . Now let i < j. Then ^F^j^(z)−F_z ⁱ^(z)

j−zi is a convex combination of K_k(z) for k = i, . . . , j−1:

Fj(z)−Fi(z) z_j −z_i =

j−1

X

k=i

zk+1−zk

z_j−z_i Kk(z).

Thus by Condition (3.1),

F_j(z)−F_i(z)

z_j−z_i ≤Ki(z) and so

F_j(z)−F_i(z)≤(z_j−z_i)K_i(z) for alli < j andz ∈I^st∩Iⁿ.

Sincec(t)⁽²⁾is an admissible sequence, Lemma6.2guarantees that all components of ^w_cⁱ⁻¹

i−1 − ^w_cⁱ

i are nonpositive except theith component. Of course the ith component of∇F −F_ieis zero. It follows that

(∇F −F_ie)·

wi−1

ci−1

− w_i c_i

≥K_i(z)(z−z_ke)·

wi−1

ci−1

− w_i c_i

=K_k(z)z·

wi−1

ci−1

− w_i c_i

= 0.

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The last equality holds becausez =Pn−1

k=j c_kw_kis clearly orthogonal to ^w_cⁱ⁻¹

i−1 −

wi

ci.

We have shown that ^dF_dt ≥ 0, forz ∈ I^st∩Iⁿ andt ∈ (0,1). ThusF(z(t)) is an increasing function oft. SoF(x) =F(z(0))≤F(z(1)) =F(y).

By the continuity ofF,F is variance monotone increasing onI∩Iⁿ.

6.3. Proof of Corollary 3.2

Let φ(t)be a continuous, real-valued function on a closed interval I such that φ(t)is twice-differentiable on the interior ofI andφ⁰⁰(t)is nonincreasing onI.

Let z be an increasing sequence inIⁿ. Let ibe an integer satisfyng 1 ≤ i ≤ n−1. By the mean-value theorem, there existsξ_iin the intervalz_i < ξ_i < z_i+1 such that

φ⁰(zi+1)−φ⁰(zi)

z_i+1−z_i =φ⁰⁰(ξi).

Inequality (3.1) follows sinceξ₁ ≤ ξ₂ ≤ · · · ≤ ξn−1, φ⁰⁰ is nonincreasing, and F_i(z) =φ⁰(z_i).

6.4. Proof of Theorem 4.1

Since E₂(z)andE₁(z)are constant on S(m, v), we assume that k ≥ 3. Leti be an integer satisfying 1 ≤ i ≤ n −1 and let E_k^(i,i+1) be the kth elementary symmetric polynomial of then−2variablesz1, z2, . . . , zi−1, zi+2, . . . , zn. Then

E_k(z) =E_k^(i,i+1)+ (z_i+z_i+1)E_k−1^(i,i+1)+z_iz_i+1E_k−2^(i,i+1).

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Thus ∂E_k(z)

∂z_i+1 − ∂E_k(z)

∂z_i =−(z_i+1−z_i)E_k−2^(i,i+1), and so

1 z_i+1−z_i

∂E_k(z)

∂z_i+1 − ∂E_k(z)

∂z_i

=−E_k−2^(i,i+1),

for all z ∈ I∩[0,∞]ⁿ. But the sequence z is nondecreasing so E_k−2^(i−1,i) ≥ E_k−2^(i,i+1) for i = 1, . . . , n − 1. It follows from Theorem 3.1 that −Ek(z) is variance monotone increasing. Thus E_k(z) is variance monotone decreasing.

This proves the first inequality in Theorem4.1.

To prove that the function F(z) = E_k+1(z)/E_k(z) is variance monotone increasing, we must show that Inequality (3.1) holds. It suffices to show that

F₂(z)−F₁(z)

z₂−z₁ ≥ F₃(z)−F₂(z) z₃−z₂ .

We writeE_kfor thekth elementary symmetric function ofz₁, . . . , z_nandE_k⁰ for thekth elementary symmetric function ofz4, . . . , zn. Then

(6.6) E_k=E_k⁰ + (z₁+z₂+z₃)E_k−1⁰ + (z₁z₂+z₁z₃+z₂z₃)E_k−2⁰ +z₁z₂z₃E_k−3⁰ . It follows that

F₁ = ∂

∂z₁

E_k+1 E_k

= 1

E_k²[EkE_k⁰ −Ek+1E_k−1⁰ + (z2+z3)(EkE_k−1⁰ −Ek+1E_k−2⁰ ) +z₂z₃(E_kE_k−2⁰ −E_k+1E_k−3⁰ )].

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Thus

F₂−F₁

z₂−z₁ =− 1 E_k²

E_kE_k−1⁰ −E_k+1E_k−2⁰ +z₃(E_k⁰E_k−2⁰ −E_k+1E_k−3⁰ ) . Similarly,

F₃−F₂

z₃−z₂ =− 1 E_k²

E_kE_k−1⁰ −E_k+1E_k−2⁰ +z₁(E_k⁰E_k−2⁰ −E_k+1E_k−3⁰ ) , so that

F₂−F₁

z₂−z₁ − F₃−F₂

z₃−z₂ = z₃−z₁

E_k² (E_kE_k−2⁰ −E_k+1E_k−3⁰ ).

Sincez3−z1andE_k²are positive, it remains to show thatEkE_k−2⁰ −Ek+1E_k−3⁰ is nonnegative, which can be rewritten using Equation (6.6) as

E_kE_k−2⁰ −E_k+1E_k−3⁰

= (E_k⁰E_k−2⁰ −E_k+1⁰ E_k−3⁰ ) + (z1+z2+z3)(E_k−1⁰ E_k−2⁰ −E_k⁰E_k−3⁰ ) + (z₁z₂+z₁z₃+z₂z₂)(E_k−2⁰ E_k−2⁰ −E_k−1⁰ E_k−3⁰ ).

Each of the expressions inE_r⁰ above are nonnegative. These weak inequalities follow from a simple counting argument. Or we can use an old result in Hardy, Littlewood and Pólya [3, p. 52]:

z ∈[0,∞)ⁿands > r =⇒ Es−1E_r > E_sEr−1, withr=k−2ands=k+ 1, k, k−1.