CHEBYSHEV INEQUALITIES AND SELF-DUAL CONES

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Chebyshev Inequalities and Self-Dual Cones Zdzisław Otachel vol. 10, iss. 2, art. 54, 2009

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CHEBYSHEV INEQUALITIES AND SELF-DUAL CONES

ZDZISŁAW OTACHEL

Dept. of Applied Mathematics and Computer Science University of Life Sciences in Lublin

Akademicka 13, 20-950 Lublin, Poland EMail:zdzislaw.otachel@up.lublin.pl

Received: 20 December, 2008

Accepted: 12 April, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26D15, 26D20, 15A39.

Key words: Chebyshev type inequality, Convex cone, Dual cone, Orthoprojector.

Abstract: The aim of this note is to give a general framework for Chebyshev inequalities and other classic inequalities. Some applications to Chebyshev inequalities are made. In addition, the relations of similar ordering, monotonicity in mean and synchronicity of vectors are discussed.

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1. Introduction and summary

LetV be a real vector space provided with an inner producth·,·i.For fixedx ∈ V andy, z ∈V the inequality

(1.1) hx, yi hx, zi ≤ hy, zi hx, xi

is called a Chebyshev type inequality.

A general method for finding vectors satisfying the above inequality is given by Niezgoda in [4]. The same author in [3] proved a projection inequality for the Eaton system, obtaining a Chebyshev type inequality as a particular case for orthoprojectors of rank one. Furthermore, the relation of synchronicity with respect to the Eaton system is introduced there. It generalizes commonly known relations of similarly ordered vectors (cf. for example, [6, chap. 7.1]).

This paper is organized as follows. Section 2 contains basic notions related to convex cones. In Section 3 a projection inequality in an abstract Hilbert space is studied. The framework covers the projection inequality for the Eaton system, Chebyshev sum and integral inequalities and others, see Examples 3.1 – 3.3. We modify and extend the applicability of the relation of synchronicity to vector spaces with infinite bases. The results are applied to the Chebyshev sum inequality in Sec- tion4and the Chebyshev integral inequality in Section5.

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2. Preliminaries

In this noteV is a real Hilbert space with an inner producth·,·i. A convex cone is a nonempty setD ⊂ V such thatαD+βD ⊂Dfor all nonnegative scalars αand β. The closure of the convex cone of all nonnegative finite combinations inH ⊂ V is denoted byconeH. Similarly,spanH denotes the closure of the subspace of all finite combinations inH. The dual cone of a subsetC ⊂V is defined as follows

dualC ={v ∈V :hv, Ci ≥0}.

It is known, that the dual cone ofCis a closed convex cone and dualC = dual(coneC).

If for a subset G ⊂ V, a closed convex cone C is equal to coneG, then we say thatC is generated by Gor Gis a generator of C. The inclusionA ⊂ B implies dualB ⊂dualA. IfCandDare convex cones, then

dual(C+D) = dualC∩dualD.

The dual cone of a subspaceW is equal to its orthogonal complementW^⊥.If a set Cis a closed convex cone, then

dual dualC =C,

(cf. [5, lemma 2.1]). The symbol dual_V₁C stands for V₁ ∩dualC and means the relative dual ofC with respect to a closed subspaceV₁ ofV. If for a closed convex cone D the identity dual_V₁D = D holds, then D is called a self-dual cone w.r.t.

V1. For example, the convex cone generated by an orthogonal system of vectors is self-dual w.r.t. the subspace spanned by this system.

In other cases the standard mathematical notation is used.

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3. Projection Inequality

From now on we make the following assumptions:P is an idempotent and symmet- ric operator (orthoprojector) defined onV,V =V₁+V₂,whereV₁ is the range ofP andV₂ is its orthogonal complement, i.e. V₁ =P V andV₂ = (P V)^⊥.The identity operator is denoted byid.All subspaces and convex cones of a real Hilbert spaceV are assumed to be closed.

Fory, z ∈V we will consider a projection inequality (briefly (PI)) of the form hy, P zi ≥0.

Ify = z, then (PI) holds for any orthoprojector P taking the formkP zk² ≥ 0. A general method of solution of (PI) is established by our following theorem (cf. [4, Theorem 3.1]).

Theorem 3.1. For vectorsy, z ∈ V and a convex cone C ⊂ V the following state- ments are mutually equivalent.

i) (PI) holds for ally∈C+V₂ ii) P z ∈dualC

iii) z ∈dualP C.

Proof. Since i), the inequality (PI) holds for everyy∈C. Thus 0≤ hy, P zi=hP y, zi.

Therefore0≤ hC, P zi=hP C, zi.HenceP z ∈dualCandz ∈dualP C.It proves that i)⇒ii), iii).

Conversely, if P z ∈ dualC then for y = c+x, where c ∈ C and hx, V₁i = 0 are arbitrary have hy, P zi = hc, P zi +hx, P zi = hc, P zi ≥ 0. By a similar

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argument, if z ∈ dualP C then y ∈ C +V₂ implies that P y ∈ P C. It leads to hy, P zi = hP y, zi ≥ 0. From this we conclude that ii),iii) ⇒i), which completes the proof.

Example 3.1 (Bessel inequality). For an orthoprojectorP the inequality (PI) holds provided thaty = z.Let {f_ν}be an orthogonal system in V. If P is the orthoprojector onto the subspace orthogonal tospan{f_ν}, i.e. P = id−P

ν h·,fνi

kfνk²f_ν, then we obtain the classic Bessel inequality

kzk² ≥X

ν

hz, f_νi² kf_νk² .

Example 3.2 (Chebyshev type inequalities). Letx∈V be a fixed nonzero vector. Set P = id−_kxk^h·,xi2x. It is clear thatP is the orthoprojector onto the subspace orthogonal tox.In the case where the inequality (PI) becomes a Chebyshev type inequality (1.1):

hx, zi hy, xi ≤ hy, zi kxk².

In the space V = Rⁿ underx = (1, . . . ,1), inequality (1.1) transforms into the Chebyshev sum inequality (or (CHSI) for short):

n

X

i=1

y_i

n

X

i=1

z_i ≤n

n

X

i=1

y_iz_i.

Consider the spaceV =L²of all 2-nd power integrable functions with respect to the Lebesgue measureµon the unit interval[0,1]. Forx≡ 1inequality (1.1) takes the form of a Chebeshev integral inequality (or (CHII) for short):

Z ydµ

Z

zdµ≤ Z

yzdµ.

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Example 3.3 (Projection inequality for Eaton systems). LetGbe a closed subgroup of the orthogonal group acting onV, dimV < ∞,and C ⊂ V be a closed convex cone. Let us assume:

i) for each vectora∈V there existg ∈Gandb∈C satisfyinga=gb, ii) ha, gbi ≤ ha, bifor alla, b∈Candg ∈G.

IfP is the orthoprojector onto a subspace orthogonal to{a∈V :Ga=a}, then the inequality (PI) holds, provided thaty, z ∈C,(cf. [3, Theorem 2.1]).

The triplet(V, G, C)fulfiling the conditions i)-ii) is said to be an Eaton system, (see e.g. [3] and the references given therein). The main example of this structure is the permutation group acting onRⁿand the cone of nonincreasing vectors.

LetC⊂V be a convex cone. Every cone of the formC+V₂has the representation:

(3.1) C+V₂ =P C+V₂.

Therefore, on studying the projection inequality (PI), according to Theorem3.1, it is sufficient to consider convex cones of the form C = D +V₂, where D is a convex cone inV₁. The following proposition is a simple consequence of Theorem 3.1.

Proposition 3.2. LetD ⊂ V₁ be a convex cone. Fory, z ∈ V the following condi- tions are equivalent.

i) (PI) holds for ally∈D+V2

ii) z ∈dualD.

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LetD⊂V₁ be a convex cone. ThenV₂ ⊂dualD. This implies thatP dualD= V₁∩dualD.Applying (3.1) todualD+V₂ = dualD,we get

(3.2) dual_V₁D+V₂ = dualD.

According to the above equation and the last proposition, we need to find for (PI) such conesDfor whichD∩dual_V₁Dare as wide as possible.

Proposition 3.3. The inequality (PI) holds fory, z ∈D+V₂, whereDis an arbitrary self-dual cone w.r.t.V₁.

Proof. By assumption, D ⊂ V1, hence (3.2) givesdualD = D+V2. Proposition 3.2implies that (PI) holds fory, z ∈(D+V2)∩dualD=D+V2.

If Dis a self-dual cone w.r.t. V₁ thenD+V₂ is a maximal cone for (PI) in the following sense.

Proposition 3.4. LetDbe a self-dual cone w.r.t.V₁withD+V₂ ⊂C, whereC ⊂V is a convex cone.

If (PI) holds fory, z ∈CthenC =D+V₂.

Proof. SinceV₂ ⊂ C, (3.1) yieldsC = P C +V₂.By Proposition 3.2, (PI) holds fory, z ∈(P C +V₂)∩dualP C.The assumption that (PI) holds fory, z ∈C gives P C +V₂ ⊂ dualP C.SinceD+V₂ ⊂ C, D = P(D+V₂) ⊂ P C. From this we havedualP C ⊂ dualD = D+V₂, by (3.2), becausedual_V₁D = D.Combining these inclusions we can see thatC =P C +V2 ⊂D+V2.

The converse inclusion holds by the hypothesis, and thus the proof is complete.

LetG_P denote the set of all unitary operators acting onV withgV₂ =V₂. Notice thatG_P is a group of operators. The inequality (PI) is invariant with respect toG_P.

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Theorem 3.5. For fixedg ∈G_P the following statements are equivalent.

i) (PI) holds fory, z ii) (PI) holds forgy, gz.

Proof. Assume thatgis a unitary operator satisfyinggV₂ =V₂.This is equivalent to g^∗V₂ =V₂, whereg^∗is the adjoint operator ofg. We first show thatgV₁ ⊂V₁.

Suppose, contrary to our claim, that there exists a u ∈ V₁ with the property gu=v₁+v₂, v_i ∈V_i, i = 1,2, v₂ 6= 0.We have:

kuk² =kguk² =kv₁+v₂k² =kv₁k²+kv₂k² (g−unitary, v₁ ⊥v₂),

ku−g^∗v₂k² =kg(u−g^∗v₂)k²

=kgu−v2k²

=kv₁k² (sinceg −unitary, g^∗g = id),

ku−g^∗v₂k² =kuk²+kg^∗v₂k²

=kuk²+kv2k² (u⊥g^∗v2, g^∗−unitary).

Hence:







kuk² =kv₁k²+kv₂k² kv₁k² =kuk²+kv₂k²

⇒ kv₂k² = 0⇒v₂ = 0,

a contradiction. This completes the proof ofgV₁ ⊂V₁.

Note thatg^∗V₁ ⊂V₁, too. This implies thatV₁ ⊂gV₁. Therefore

(3.3) gV1 =V1.

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Now, letz ∈V be arbitrary. We havez = z₁+z₂,wherez_i ∈ V_i, i= 1,2.For an orthoprojectorP ontoV₁ we get:

gP z=gP(z1+z2) =gz1 =P(gz1+gz2) =P gz, becausegz₁ ∈V₁by (3.3) andgz₂ ∈V₂ by assumption. Thus

(3.4) P g =gP.

By (3.4),

hgy, P gzi=hgy, gP zi=hg^∗gy, P zi=hy, P zi. This proves required equivalence.

A simple consequence of the above theorem is:

Remark 1. For a convex cone C ⊂ V and g₀ ∈ G_P the following statements are equivalent.

i) (PI) holds fory, z ∈C ii) (PI) holds fory, z ∈g₀C.

In the remainder of this section we assume thatV is a real separable Hilbert space.

Let{f_ν}be an orthogonal basis ofV₁,i.e.

hf_η, f_νi

>0, η =ν

= 0, η 6=ν, for integersη, ν.

Under the above assumption, the projectionP ztakes the form:

(3.5) P z=X

ν

hz, fνi kf_νk² f_ν.

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From this, fory, z ∈V we have hy, P zi=X

ν

hy, f_νi hz, f_νi kf_νk² .

Therefore the following remark is evident.

Remark 2. Let{f_ν}be an orthogonal basis ofV₁. Fory, z ∈V the inequality (PI) holds if and only if

X

ν

hy, f_νi hz, f_νi kf_νk² ≥0.

Set

(3.6) D=

(

x∈V :x=X

ν

ανfν, αν ≥0 )

.

Clearly,Dis a closed convex cone generated by the system{f_ν}. The scalarsα_ν =

hx,f_νi

kfνk² are the Fourier coefficients ofxw.r.t. the orthogonal system{f_ν}.Moreover, Dis a self-dual cone w.r.t. V₁. By Proposition3.3we get

Corollary 3.6. If{f_ν}is an orthogonal basis ofV₁, then (PI) holds fory, z ∈D+V₂, whereDis defined by (3.6).

LetΞdenote the set of all sequencesξ = (ξ₁, ξ₂, . . .)withξ²_ν = 1, ν = 1,2, . . . . For givenξ, let us define the operatorg_ξ onV as follows:

g_ξx=x−P x+X

ν

ξ_νhx, f_νi kf_νk² f_ν.

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This operator is an isometry, because

kg_ξxk² =kxk²− kP xk²+X

ν

ξ_ν²hx, f_νi² kfνk²

=kxk²− kP xk²+kP xk² =kxk², by (3.5) and obvious orthogonality

x−P x⊥X

ν

ξν

hx, f_νi kf_νk² fν.

If x ∈ V₂, then hx, f_νi = 0 for all ν. Hence P x = 0 = P

ν

ξ_ν^hx,f_kf ^νⁱ

νk²f_ν. For this reason

(3.7) gξx=x, x∈V2.

We write

(3.8) G={g_ξ :ξ ∈Ξ}.

We will show thatGis a group of operators. It is evident that:

(3.9) g_ξ = id, forξ = (1,1, . . .).

Letζ, ξ, γ ∈Ξ. We have:

g_ζf_ν =f_ν −P f_ν +X

η

ζ_ηhf ν, f_ηi

kfηk² f_η =ζ_νf ν,

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becauseP f_ν =f_ν, ν = 1,2, . . . .From this, byx−P x∈V₂and (3.7) we get:

g_ζg_ξx=g_ζ(x−P x+X

ν

ξ_νhx, f_νi kfνk² f_ν)

=g_ζ(x−P x) +X

ν

ξ_νhx, f_νi kfνk² g_ζf_ν

=x−P x+X

ν

ζ_νξ_νhx, f_νi kfνk² f_ν.

Thus

(3.10) g_ζg_ξ =gζ·ξ=g_ξg_ζ, whereζ·ξ = (ζ₁ξ₁, ζ₂ξ₂, . . .).This clearly gives:

(3.11) g_ζ(g_ξg_γ) =gζ·ξ·γ = (g_ξg_ζ)g_γ and

gξgξ =gξ·ξ = id, which is equivalent to

(3.12) (g_ξ)⁻¹ =g_ξ.

Sinceg_ξis an isometry and invertible,

(3.13) g_ξ−unitary, ∀ξ∈Ξ.

By (3.13), (3.7), (3.9) – (3.12) we can assert thatGis an Abelian group of unitary operators that are identities onV₂.As a consequence,G⊂G_P.

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Given anyx∈V, we defineξ_x = (ξ_x,1, ξ_x,2, . . .)by

(3.14) ξ_x,ν =

( 1, hx, f_νi ≥0

−1, hx, f_νi<0.

It is clear thatξx,νhx, fνi=|hx, fνi|. Hence gξxx=x−P x+X

ν

|hx, f_νi|

kf_νk² fν,

wherex−P x∈V₂andP

ν

|hx,fνi|

kfνk² f_ν ∈D.Therefore

(3.15) g_ξ_xx∈D+V₂.

Assertion (3.15) is simply the statement that

(3.16) (Gx)∩C 6=∅, ∀_x∈V

withC = D+V₂. This condition ensures that the sum of the cones gC, where g runs overG, covers the whole spaceV. Now, we show that (3.16) holds forG_P and for every coneC =P C+V₂, P C 6={0}.

Fix v ∈ V. Clearly, v = v₁ + v₂, v_i ∈ V_i, i = 1,2. If v₁ = 0 then v ∈ G_Pv ⊂ V₂ ⊂ C, i.e. (3.16) holds. Assume that 0 6= v₁ and note that there exists 06= u₁ ∈P C. Let us construct the two orthogonal bases{e_ν}and{f_ν}ofV₁ with e₁ =v₁ andf₁ =u₁. Setu=kv₁k_ku^u¹

1k +v₂ and

(3.17) g = id−P +X

ν

h·, e_νi ke_νkkf_νkf_ν.

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Observe thatu∈C,gv=uandgis the identity operator onV₂. Now, we prove that gis unitary. Firstly, we note that for anyx∈V

kgxk² =kxk²− kP xk²+X

ν

hx, e_νi²

ke_νk² =kxk²,

becausekP xk² =P

ν

hx,eνi²

ke_νk² . Our next goal is to show thatgV = V.To do this, fix y∈V.We have

y=y−P y+X

ν

hy, f_νi kf_νk

f_ν kf_νk. Set

x=y−P y+X

ν

hy, f_νi kfνk

e_ν keνk. It is easily seen thatgx=y. So,gis unitary.

Finally, g is a unitary operator on V with gV₂ = V₂ andgv = u. It givesu ∈ G_Pv∩C, as desired.

We are now in a position to introduce a notion of synchronicity of vectors for (PI).

For an orthoprojectorP letCbe a convex cone which admits the representation C =P C +V₂,

whereP C is nontrivial. LetGbe a subgroup ofG_P with the property (3.16).

The two vectors y, z ∈ V are said to be G-synchronous (with respect to C) if there exists ag ∈Gsuch thatgy, gz ∈C.IfG=GP, then we simply say thatyand zare synchronous.

The definition is motivated by [3, sec. 2]. It generalizes the notion of synchronicity with respect to Eaton systems. Obviously,G-synchronicity forces synchronicity

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under fixedC. In the sequel, for special cones we show that synchronicity is equivalent to (PI) butG-synchronicity is a sufficient condition for (PI).

According to Theorem3.5, by the notion of synchronicity, it is possible to extend (PI) beyond a coneCif only (PI) holds for vectors inC.

Proposition 3.7. LetC ⊂V be a convex cone withC =P C+V₂, P C 6={0}and letGbe a subgroup ofG_P with property (3.16).

The following statements are equivalent.

i) (PI) holds fory, z ∈C

ii) (PI) holds for the vectorsyandzwhich areG-synchronous w.r.t. C.

Proof. i)⇒ii). Assume y andz are G-synchronous w.r.t. C. There exists g ∈ G withgy, gz ∈ C. Since i), (PI) holds forgy, gz. By Theorem 3.5we conclude that (PI) holds foryandz.

The converse implication is evident becausey, z ∈Care of courseG-synchronous.

Now, we are able to give an equivalent condition for G-synchronicity. Simulta- neously, the condition is sufficient for synchronicity w.r.t. D+V₂.

Proposition 3.8. LetGbe the group defined by (3.8) and letDbe the cone defined by (3.6).

The vectorsy, z ∈V areG-synchronous w.r.t.D+V₂ if and only if hy, fνi hz, fνi ≥0, ∀ν.

Proof. Ify, z areG-synchronous, then there exists aξsuch thatg_ξy, g_ξz ∈ D+V₂. Henceξ_νhy, f_νi ≥ 0andξ_νhz, f_νi ≥0for allν.Multiplying the above inequalities side by side we obtain0≤ξ_ν²hy, f_νi hz, f_νi=hy, f_νi hz, f_νifor everyν.

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Conversely, suppose that hy, f_νi hz, f_νi ≥ 0 for every ν. In this situation, the sequences defined foryandzby (3.14) are equal. HenceyandzareG-synchronous by (3.15).

Summarizing the above considerations we give sufficient and necessary conditions for (PI) to hold.

Theorem 3.9. Let{f_ν}be an orthogonal basis ofV₁. SetC =D+V₂, whereDis defined by (3.6). The following statements are equivalent.

i) yandz are synchronous w.r.t.C ii) (PI) holds foryandz.

In particular, if

(3.18) hy, g₀f_νi hz, g₀f_νi ≥0, ∀ν, then (PI) holds, whereg₀ ∈G_P is fixed.

Proof. The first part, i)⇒ii). It is a consequence of Corollary3.6 and Proposition 3.7.

Conversely, if ii), thenhP y, P zi ≥0. Firstly, suppose thatP z =αP y. Clearly, α ≥ 0. By (3.16), which holds for G_P and C, there exists a g ∈ G_P such that gy ∈ C. Hence P gy ∈ P C = D. By (3.4), gP y ∈ D. Since α ≥ 0, αgP y ∈ D.

Sincez−P z ∈V2,g(z−P z)∈V2, becausegV2 =V2. Hence gz =gP z+g(z−P z) =αgP y+g(z−P z)∈C.

Thereforegy, gz ∈C, i.e.yandz are synchronous.

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Next, assume that P y andP z are linearly independent. Let us construct an orthogonal basis{e_ν}ofV₁ with

e₁ =P y, e₂ =P z− hP z, P yi kP yk² P y

and letg ∈G_P be defined by (3.17). There is no difficulty to showing that gy =y−P y

| {z }

∈V₂

+kP yk kf₁k f₁

| {z }

∈D

∈C,

gz =z−P z

| {z }

∈V2

+ _ke^{hz,P yi}

1kkf1k

| {z }

≥0,by (PI)

f₁+ ^{kP zk}_{kP yk}²^{kP yk}2ke²^{−hP z,P yi}2kkf2k ²

| {z }

≥0,by Cauchy−Schwarz ineq.

f₂

| {z }

∈D

∈C.

Therefore,yandzare synchronous as required.

Now, let us note that (3.18) is equivalent to

hg₀^∗y, f_νi hg₀^∗z, f_νi ≥0, ∀ν.

By Proposition3.8, g₀^∗y andg^∗₀z areG-synchronous w.r.t. C. Hence there exists a g ∈ G such thatgg^∗₀y, gg₀^∗ ∈ C. Sincegg₀^∗ ∈ G_P, y and z are synchronous w.r.t.

C. For this reason (PI) holds, by the first part of this proposition. The proof is complete.

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4. Applications to the Chebyshev Sum Inequality

Throughout this section, V = Rⁿ with the standard inner product h·,·i. Let {s_i} be the basis of Rⁿ, where s_i = (1, . . . ,1

| {z }

i

,0, . . . ,0), i = 1, . . . , n. The symbols

V₁ andV₂ stand for the subspace orthogonal to s_n and its orthogonal complement, respectively, i.e.

V1 = (

(x1, . . . , xn) :X

i

xi = 0 )

, V2 = span{s_n}.

Let P be the orthoprojector onto V₁, i.e. P = id−^h·,s_nⁿⁱs_n. In this situation, by Example3.2, (PI) becomes the Chebyshev sum inequality (CHSI).

It is known that the convex cone of nonincreasing vectors C ={x= (x₁, . . . , x_n) :x₁ ≥x₂ ≥ · · · ≥x_n} is generated by{s₁, . . . , s_n,−s_n}.On the other side,

{(1,−1,0, . . . ,0), (0,1,−1,0, . . . ,0), (0, . . . ,0,1,−1)}

is a generator of dualC=

(

x= (x₁, . . . , x_n) :

n

X

i=1

x_i = 0,

k

X

i=1

x_i ≥0, k = 1, . . . , n−1 )

.

Sete_i =nP s_i, i= 1, . . . , n−1. Clearly, (4.1) ei =nsi−isn = (n−i, . . . , n−i

| {z }

i

,−i, . . . ,−i

| {z }

n−i

), i= 1, . . . , n−1.

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Write

D= cone{e_i}.

Clearly,P C =DandV2 ⊂C. Hence by (3.1), C =D+V₂.

Applying Proposition3.2, we conclude that (PI) holds fory, z ∈(D+V₂)∩dualD= C ∩ dualD. With the aid of generators we can check that D ⊂ dualC. Hence C = dual dualC ⊂dualD.

By the above considerations, for arbitraryy, z ∈ C,the inequality (CHSI) holds.

This is a classic Chebyshev result.

The system{e_i, i= 1, . . . , n−1}constitutes a basis ofV₁.Observe that hei, eji=i(n−j)n, i≤j, i, j= 1, . . . , n−1.

Hence, easy computations lead to

e_k+1−n−k−1 n−k e_k, e_i

= 0, i= 1, . . . , k; k = 1, . . . , n−2.

From this, the Gram-Schmidt orthogonalization gives the orthogonal system{q_i}for the basis{e_i}as follows:

(4.2)

( q₁ =e₁,

qk+1 = ^n−k_n ek+1−^n−k−1_n−k ek

, k= 1, . . . , n−2.

According to (4.1) and (4.2) we obtain the explicit form of the orthogonal basis {q_i}

(4.3) q_k = (0, . . . ,0

| {z }

k−1

, n−k,−1, . . . ,−1

| {z }

n−k

), k= 1, . . . , n−1.

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Let us denote

K =De +V₂,

whereDe stands for thecone{q_k}. The convex coneDe is self-dual w.r.t.V₁. According to Proposition3.3we can assert that (CHSI) holds fory, z ∈K. Letg0(x1, . . . , xn) = (−xn, . . . ,−x1).Clearly,g0 ∈GP. By Remark1, (CHSI) holds fory, z ∈g0K. Have:

g₀K =g₀(De +V₂) =g₀De +V₂ = cone{g₀q_k}+V₂.

Definef_k=g₀qn−k, k = 1, . . . , n−1.Sinceg₀ ∈G_P,g₀is unitary andg₀V₁ =V₁, by (3.3). Hence,{f_k}is an orthogonal basis ofV₁. Observe

(4.4) fk= (1, . . . ,1

| {z }

k

,−k,0, . . . ,0), k = 1, . . . , n−1.

Write

M = cone{f_k}+V₂. By Remark1, it is evident that (CHSI) holds fory, z ∈M. Proposition 4.1. Forx= (x1, . . . , xn)∈Rⁿ

x∈K ⇐⇒ the sequence

( 1 n−k+ 1

n

X

i=k

x_i )n

k=1

is nonincreasing,

x∈M ⇐⇒ the sequence (1

k

X

i=1

x_i )ⁿ

k=1

is nonincreasing.

Proof. We prove only the first equivalence. The second one uses a similar procedure.

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By (3.2), K = dualD, becausee De is self-dual w.r.t. V₁. Hence by (4.3) we can assert thatx∈K is equivalent to

(n−k)x_k≥

n

X

i=k+1

x_i, k = 1, . . . , n−1.

Adding to both of sides(n−k)Pn

i=k+1xiand dividing by(n−k)(n−k+ 1),we obtain

1 n−k+ 1

n

X

i=k

x_i

!

≥ 1 n−k

n

X

i=k+1

x_i

!

, k = 1, . . . , n−1.

This is equivalent to our claim.

By the above proposition, we can see thatC ⊂ K andC ⊂ M. The coneM is said to be a cone of vectors nonincreasing in mean. It is easily seen that (CHSI) holds fory, z ∈ −K and fory, z ∈ −M (for e.g., by takingC = K, M and substituting

−idintog₀ in Remark1). The statement that (CHSI) holds for vectors monotonic in mean is due to Biernacki, see [1].

The remainder of this section will be devoted to (CHSI) for synchronous vectors.

We will consider relations between synchronicity and similar ordering.

HereG_P is the group of all orthogonal matrices such that the sum of the entries of each row and column is equal to1or−1. The group of alln×npermutation matrices is a subgroup ofG_P,which together with the coneCfulfil (3.16). The permutation group synchronicity w.r.t. C is simply the relation "to be similarly ordered". It implies synchronicity w.r.t. every cone which containsC, e.g. M orK.

The two vectorsx= (x₁, . . . , x_n), y = (y₁, . . . , y_n)∈Rⁿare said to be similarly ordered if

(4.5) (xi−xj)(yi−yj)≥0, ∀i,j.

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The assertion that (CHSI) holds for similarly ordered vectors is a consequence of Proposition3.7.

Theorem3.9states that (CHSI) is equivalent to synchronicity w.r.t.cone{f_k}+V₂ where{f_k}is an arbitrarily chosen orthogonal basis ofV₁. Moreover,G-synchronicity gives (CHSI), whereGis the group (3.8) acting onRⁿ. For this reason, the specifi- cation of Theorem3.9can be as follows.

Let{f_k}be defined by (4.4) andGby (3.8) in compliance with the basis.

Corollary 4.2. (CHSI) holds fory, z if and only if yand z are synchronous w.r.t.

M.

In particular, (CHSI) is satisfied byyandzsuch that

hy, U fki hz, U fki ≥0, k = 1, . . . , n−1,

where U is a fixed unitary operation with U s_n = s_n or U s_n = −s_n, i.e. U is represented by an orthogonal matrix whose rows and columns sum up to1or to−1.

By Proposition3.8we have:

Remark 3. The vectorsy = (y₁, . . . , y_n) and z = (z₁, . . . , z_n) areG-synchronous w.r.t. M if and only if

" _k X

i=1

y_i−ky_k+1

# " _k X

i=1

z_i−kz_k+1

#

≥0, k= 1, . . . , n−1.

Relations of similar ordering and G-synchronicity w.r.t. M are not comparable, i.e. there exist similarly ordered vectors which are not synchronous and there exist synchronous vectors that are not similarly ordered. On the other hand, both relations imply synchronicity w.r.t.M and as a consequence, (CHSI) holds.

Example 4.1. ConsiderRⁿ, n >3.

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For0 < α <1 < β < n−1sety = (0, . . . ,0,1−n,−α), z = (0, . . . ,0,1− n,−β).According to (4.5) and Remark3 the vectorsy andz are similarly ordered and are notG-synchronous, but they are synchronous w.r.t. M, so (CHSI) holds.

Now, sety⁰ =f₁+f₂,z⁰ =f₂+f₃,wheref_i are defined by (4.4). The vectorsy⁰ andz⁰ areG-synchronous w.r.t. M, becausey⁰, z⁰ ∈M, so (CHSI) holds.

On the other handy⁰ = (2,0,−2,0, . . . ,0), z⁰ = (2,2,−1,−3,0, . . . ,0)are not similarly ordered by (4.5), because(y⁰₃−y₄⁰)(z₃⁰ −z₄⁰) = −2(−1 + 3)<0.

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5. Applications to the Chebyshev Integral Inequality

Set V = L² as in Example 3.2. The characteristic function of the measurable set A ⊂ [0,1] is denoted byI_A. Additionally we will write e_s = I_[0,s], 0 ≤ s ≤ 1.

The symbol V₁ stands for the subspace orthogonal to V₂ = span{e₁}, i.e. V₁ = x∈L² :R

xdµ= 0 .By Example 3.2, it is known that for the orthoprojector P ontoV1 (PI) transforms into the Chebyshev integral inequality (CHII). Let C ⊂ L² be the closed convex cone of all nonincreasingµ a.e.functions. It is known (see [5, Theorem 3.1 and 3.3]) that:

C = cone ({e_s: 0≤s≤1} ∪ {−e₁}), (5.1)

dualC = cone{I_Π−I_Π+ε:ε >0,Π,Π +ε ⊂[0,1]}, whereΠstands for an interval.

The Haar system:

χ⁰₀ =e₁ (5.2)

χ^k_n(t) =











2^n/2, ^2k−2₂n+1 ≤t < ^2k−1₂n+1

−2^n/2, ^2k−1₂n+1 ≤t < ₂n+1^2k

0, otherwise n = 0,1, . . . , k= 1,2, . . . ,2ⁿ

forms an orthonormal basis of L². In particular, H = {χ^k_n : n = 0,1, . . . , k = 1, . . . ,2ⁿ}is an orthonormal basis ofV₁.

Let

D= coneH.

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The coneDis self-dual w.r.t.V₁,so by (3.2) we have:

(5.3) dualD=D+V₂.

By (5.1), observe that H ⊂ dualC, hence C = dual dualC ⊂ dualH = dualD.Combining this with (5.3), we obtain

(5.4) C ⊂D+V₂.

From (5.4) and Corollary3.6it follows that Corollary 5.1. (CHII) holds fory, z ∈D+V2.

The coneD+V2contains the cone of all nonincreasingµ a.e.functions inL². It is easily seen that the coneD+V₂contains functions which are not nonincreas- ingµ a.e.

Let G be the group (3.8) acting on L² with the Haar system. Employing the G-synchronicity relation w.r.t.D+V₂, by Theorem3.9we get:

Corollary 5.2. (CHII) holds fory, z ∈L² if only

(5.5) hy, χi hz, χi ≥0, ∀χ∈H.

We next discuss the relation between the condition (5.5) and the known sufficient conditions for (CHII). One of these is the condition thatyandzare similarly ordered, i.e.

(5.6) [y(s)−y(t)] [z(s)−z(t)]]≥0, for all0≤s, t ≤1

(see e.g. [6, pp. 198-199]). Now, we show by an example that theG-synchronicity condition (5.5) is not stronger than the condition of similar ordering (5.6) inL².

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Example 5.1. InL² let y = χ¹₂+χ²₂, z = χ²₂+χ³₂,whereχ^j_i are defined by (5.2).

The vectorsyandz areG-synchronous w.r.t. D+V₂, because they are inD.

On the other hand

y(s) = 2, 0≤s ≤ ¹₈ y(t) = 0, ⁴₈ ≤t≤ ⁵₈

,

z(s) = 0, 0≤s≤ ¹₈ z(t) = 2, ⁴₈ ≤t≤ ⁵₈

.

From this,[y(s)−y(t)] [z(s)−z(t)]] = [2−0][0−2] ≤0for any 0≤ s ≤ ¹₈ and

4

8 ≤t≤ ⁵₈.Thusyandzare not similarly ordered.

Now, we recall that a function y ∈ L² is nonincreasing (nondecreasing, monotone) in mean if the functions7→ ¹_sRs

0 ydµ,is nonincreasing (nondecreasing, monotone).

Differentiating ¹_sRs

0 ydµ we can easy obtain that y is nonincreasing in mean if and only if ¹_sRs

0 ydµ≥y(s), µ a.e.

It is known that (CHII) holds foryandzwhich are monotone in mean in the same direction (see [1], cf. also [6, pp. 198-199]). Johnson in [2] gave a more general condition. Namely, if

(5.7)

1 s

Z s 0

ydµ−y(s) 1 s

Z s 0

zdµ−z(s)

≥0, ∀0<s<1

then (CHII) holds foryandz.

Remark 4.

1. There exist functions inconeHwhich are not nonincreasing in mean.

2. There exist functions nonincreasing in mean which are not inconeH.

3. There exist functions inconeHfor which (5.7) does not hold, i.e. the condition (5.5) is not stronger than (5.7).

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Proof. An easy verification shows that:

Ad. 1)χ^k_n∈ H, k >1are not nonincreasing in mean.

Ad. 2) Set f = I_[0,1/2) −2I_[1/2,3/4). f is nonincreasing in mean and is not in coneHbecausehf, χ²₁i<0.

Ad. 3) Sety=χ²₁, z=χ³₂. For ⁵₈ < s < ⁶₈ have:

1 s

Z s 0

ydµ−y(s) 1 s

Z s 0

zdµ−z(s)

=−

√2/2 s · 3/2

s <0.

The set of all L²-functions nonincreasing in mean constitutes a convex cone. It will be denoted byM. LetM₀ be the class of all step functions of the form

g_s,t =I_[0,s)− s

t−sI_[s,t], 0< s < t <1.

Proposition 5.3.

M = dualM₀,

P M = dualV1M0 = coneM0, M = coneM₀+V₂.

Proof. By definition,f ∈M if and only if 1

s Z s

0

f dµ≥ 1 t

Z t 0

f dµ for all0< s < t <1.

After equivalent transformations we obtain Z s

0

f dµ ≥ s t−s

Z t s

f dµ for all0< s < t <1.

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This is simplyf ∈dualM₀, so the first equation holds.

To show the second equation, note thatM₀ ⊂M ∩V₁. Hence dual(M ∩V1)⊂dualM0 =M,

by the first equation. It follows thatV₁∩dual(M ∩V₁)⊂M∩V₁, i.e.

(5.8) dual_V₁(M ∩V₁)⊂M ∩V₁.

Fixf ∈M ∩V₁ and letg ∈M ∩V₁ be arbitrary. For suchf andg(CHII) holds and takes the form:

Z 1 0

f gdµ≥ Z 1

0

f dµ· Z 1

0

gdµ= 0·0 = 0,

i.e.f ∈dual_V₁(M ∩V₁). Therefore

(5.9) M ∩V₁ ⊂dual_V₁(M∩V₁).

Since M = dualM0, dualM = coneM0. Now, observe thatV2 ⊂ M. This implies by (3.1) thatM =P M+V₂. Furthermore, in this situationP M =M ∩V₁. The above gives

dualM = dual(M∩V1+V2)

=V₁∩dual(M ∩V₁) = dual_V₁(M ∩V₁).

Hence

(5.10) dualV1(M ∩V1) = coneM0.

Combining (5.8), (5.9) and (5.10) we obtain the required equations.

The third equation is a consequence of the second one. The proof is complete.

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The second equation of the above propositions immediately gives:

Remark 5. The convex cone of allL²-functions nonincreasing in mean with integral equal to0is self-dual w.r.t.V₁.

TakingC =M in Theorem3.1, by Proposition5.3we easily obtain:

Corollary 5.4. IfR f dµR

gdµ ≤ R

f gdµholds for all functionsf ∈ L² monotone in mean, theng ∈L² is also monotone in mean.

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References

[1] M. BIERNACKI, Sur une inégalité entre les intégrales due á Tchébyscheff, Ann.

Univ. Mariae Curie-Skłodowska, A5 (1951), 23–29.

[2] R. JOHNSON, Chebyshev’s inequality for functions whose averages are mono- tone, J. Math. Anal. Appl., 172 (1993), 221–232.

[3] M. NIEZGODA, On the Chebyshev functional, Math. Inequal. Appl., 10(3) (2007), 535–546.

[4] M. NIEZGODA, Bifractional inequalities and convex cones, Discrete Math., 306(2) (2006), 231–243.

[5] Z. OTACHEL, Spectral orders and isotone functionals, Linear Algebra Appl., 252 (1997), 159–172.

[6] J.E. PE ˇCARI ´C, F. PROSCHANANDY.L. TONG, Convex Functions, Partial Or- derings, and Statistical Applications, Mathematics in Science and Engineering, Vol. 187, Academic Press, Inc. (1992)