CHEBYSHEV INEQUALITIES AND SELF-DUAL
CONES
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
Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
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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
3
2
Preliminaries
4
3
Projection Inequality
5
4
Applications to the Chebyshev Sum Inequality
19
5
Applications to the Chebyshev Integral Inequality
25
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
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1.
Introduction and summary
Let V be a real vector space provided with an inner product h·, ·i . For fixed x ∈ V
and y, 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 Section 4 and the Chebyshev integral inequality in Section 5.
Chebyshev Inequalities and
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vol. 10, iss. 2, art. 54, 2009
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2.
Preliminaries
In this note V is a real Hilbert space with an inner product h·, ·i. A convex cone is
a nonempty set D ⊂ V such that αD + βD ⊂ D for all nonnegative scalars α and
β. The closure of the convex cone of all nonnegative finite combinations in H ⊂ V
is denoted by cone H. Similarly, span H denotes the closure of the subspace of all
finite combinations in H. The dual cone of a subset C ⊂ V is defined as follows
dual C = {v ∈ V : hv, Ci ≥ 0}.
It is known, that the dual cone of C is a closed convex cone and
dual C = dual(cone C).
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
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If for a subset G ⊂ V , a closed convex cone C is equal to cone G, then we say
that C is generated by G or G is a generator of C. The inclusion A ⊂ B implies
dual B ⊂ dual A. If C and D are convex cones, then
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dual(C + D) = dual C ∩ dual D.
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The dual cone of a subspace W is equal to its orthogonal complement W ⊥ . If a set
C is a closed convex cone, then
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dual dual C = C,
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(cf. [5, lemma 2.1]). The symbol dualV1 C stands for V1 ∩ dual C and means the
relative dual of C with respect to a closed subspace V1 of V . If for a closed convex
cone D the identity dualV1 D = 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 symmetric operator (orthoprojector) defined on V , V = V1 + V2 , where V1 is the range of P
and V2 is its orthogonal complement, i.e. V1 = P V and V2 = (P V )⊥ . The identity
operator is denoted by id . All subspaces and convex cones of a real Hilbert space V
are assumed to be closed.
For y, z ∈ V we will consider a projection inequality (briefly (PI)) of the form
Chebyshev Inequalities and
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Zdzisław Otachel
hy, P zi ≥ 0.
If y = z, then (PI) holds for any orthoprojector P taking the form kP zk2 ≥ 0. A
general method of solution of (PI) is established by our following theorem (cf. [4,
Theorem 3.1]).
vol. 10, iss. 2, art. 54, 2009
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Theorem 3.1. For vectors y, z ∈ V and a convex cone C ⊂ V the following statements are mutually equivalent.
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i) (PI) holds for all y ∈ C + V2
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ii) P z ∈ dual C
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iii) z ∈ dual P C.
Proof. Since i), the inequality (PI) holds for every y ∈ C. Thus
0 ≤ hy, P zi = hP y, zi .
Therefore 0 ≤ hC, P zi = hP C, zi . Hence P z ∈ dual C and z ∈ dual P C. It proves
that i) ⇒ ii), iii).
Conversely, if P z ∈ dual C then for y = c + x, where c ∈ C and hx, V1 i =
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 ∈ dual P 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 orthoprojector P the inequality (PI) holds
provided that y = z. Let {fν } be an orthogonal system in V . If P is the orthoproP h·,fν i
jector onto the subspace orthogonal to span{fν }, i.e. P = id − kf
2 fν , then we
νk
ν
obtain the classic Bessel inequality
Zdzisław Otachel
kzk2 ≥
X hz, fν i2
ν
kfν
k2
.
Example 3.2 (Chebyshev type inequalities). Let x ∈ V be a fixed nonzero vector. Set
h·,xi
P = id − kxk
2 x. It is clear that P is the orthoprojector onto the subspace orthogonal
to x. 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 under x = (1, . . . , 1), inequality (1.1) transforms into the
Chebyshev sum inequality (or (CHSI) for short):
n
X
i=1
yi
Chebyshev Inequalities and
Self-Dual Cones
n
X
i=1
zi ≤ n
n
X
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yi zi .
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i=1
Consider the space V = L2 of all 2-nd power integrable functions with respect to
the Lebesgue measure µ on the unit interval [0, 1]. For x ≡ 1 inequality (1.1) takes
the form of a Chebeshev integral inequality (or (CHII) for short):
Z
Z
Z
ydµ zdµ ≤ yzdµ.
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Example 3.3 (Projection inequality for Eaton systems). Let G be a closed subgroup
of the orthogonal group acting on V, dim V < ∞, and C ⊂ V be a closed convex
cone. Let us assume:
i) for each vector a ∈ V there exist g ∈ G and b ∈ C satisfying a = gb,
ii) ha, gbi ≤ ha, bi for all a, b ∈ C and g ∈ G.
If P is the orthoprojector onto a subspace orthogonal to {a ∈ V : Ga = a}, then
the inequality (PI) holds, provided that y, 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 on Rn and the cone of nonincreasing vectors.
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
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Let C ⊂ V be a convex cone. Every cone of the form C + V2 has the representation:
(3.1)
C + V2 = P C + V2 .
Therefore, on studying the projection inequality (PI), according to Theorem 3.1,
it is sufficient to consider convex cones of the form C = D + V2 , where D is a
convex cone in V1 . The following proposition is a simple consequence of Theorem
3.1.
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Proposition 3.2. Let D ⊂ V1 be a convex cone. For y, z ∈ V the following conditions are equivalent.
i) (PI) holds for all y ∈ D + V2
ii) z ∈ dual D.
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Let D ⊂ V1 be a convex cone. Then V2 ⊂ dual D. This implies that P dual D =
V1 ∩ dual D. Applying (3.1) to dual D + V2 = dual D, we get
(3.2)
dualV1 D + V2 = dual D.
According to the above equation and the last proposition, we need to find for (PI)
such cones D for which D ∩ dualV1 D are as wide as possible.
Proposition 3.3. The inequality (PI) holds for y, z ∈ D+V2 , where D is an arbitrary
self-dual cone w.r.t. V1 .
Chebyshev Inequalities and
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vol. 10, iss. 2, art. 54, 2009
Proof. By assumption, D ⊂ V1 , hence (3.2) gives dual D = D + V2 . Proposition
3.2 implies that (PI) holds for y, z ∈ (D + V2 ) ∩ dual D = D + V2 .
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If D is a self-dual cone w.r.t. V1 then D + V2 is a maximal cone for (PI) in the
following sense.
Proposition 3.4. Let D be a self-dual cone w.r.t. V1 with D + V2 ⊂ C, where C ⊂ V
is a convex cone.
If (PI) holds for y, z ∈ C then C = D + V2 .
Proof. Since V2 ⊂ C, (3.1) yields C = P C + V2 . By Proposition 3.2, (PI) holds
for y, z ∈ (P C + V2 ) ∩ dual P C. The assumption that (PI) holds for y, z ∈ C gives
P C + V2 ⊂ dual P C. Since D + V2 ⊂ C, D = P (D + V2 ) ⊂ P C. From this we
have dual P C ⊂ dual D = D + V2 , by (3.2), because dualV1 D = D. Combining
these inclusions we can see that C = P C + V2 ⊂ D + V2 .
The converse inclusion holds by the hypothesis, and thus the proof is complete.
Let GP denote the set of all unitary operators acting on V with gV2 = V2 . Notice
that GP is a group of operators. The inequality (PI) is invariant with respect to GP .
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Theorem 3.5. For fixed g ∈ GP the following statements are equivalent.
i) (PI) holds for y, z
ii) (PI) holds for gy, gz.
Proof. Assume that g is a unitary operator satisfying gV2 = V2 . This is equivalent to
g ∗ V2 = V2 , where g ∗ is the adjoint operator of g. We first show that gV1 ⊂ V1 .
Suppose, contrary to our claim, that there exists a u ∈ V1 with the property
gu = v1 + v2 , vi ∈ Vi , i = 1, 2, v2 6= 0. We have:
Chebyshev Inequalities and
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vol. 10, iss. 2, art. 54, 2009
kuk2 = kguk2 = kv1 + v2 k2 = kv1 k2 + kv2 k2
(g − unitary, v1 ⊥ v2 ),
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ku − g ∗ v2 k2 = kg(u − g ∗ v2 )k2
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= kgu − v2 k2
= kv1 k2
∗
2
2
(since g − unitary, g ∗ g = id),
∗
2
ku − g v2 k = kuk + kg v2 k
2
= kuk + kv2 k
Hence:
2

 kuk2 = kv1 k2 + kv2 k2
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∗
∗
(u ⊥ g v2 , g − unitary).
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⇒ kv2 k2 = 0 ⇒ v2 = 0,
 kv k2 = kuk2 + kv k2
1
2
a contradiction. This completes the proof of gV1 ⊂ V1 .
Note that g ∗ V1 ⊂ V1 , too. This implies that V1 ⊂ gV1 . Therefore
(3.3)
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gV1 = V1 .
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Now, let z ∈ V be arbitrary. We have z = z1 + z2 , where zi ∈ Vi , i = 1, 2. For
an orthoprojector P onto V1 we get:
gP z = gP (z1 + z2 ) = gz1 = P (gz1 + gz2 ) = P gz,
because gz1 ∈ V1 by (3.3) and gz2 ∈ V2 by assumption. Thus
P g = gP.
(3.4)
By (3.4),
hgy, P gzi = hgy, gP zi = hg ∗ gy, P zi = hy, P zi .
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
This proves required equivalence.
A simple consequence of the above theorem is:
Remark 1. For a convex cone C ⊂ V and g0 ∈ GP the following statements are
equivalent.
i) (PI) holds for y, z ∈ C
ii) (PI) holds for y, z ∈ g0 C.
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In the remainder of this section we assume that V is a real separable Hilbert space.
Let {fν } be an orthogonal basis of V1 , i.e.
> 0, η = ν
hfη , fν i
= 0, η 6= ν,
for integers η, ν.
Under the above assumption, the projection P z takes the form:
X hz, fν i
(3.5)
Pz =
fν .
kfν k2
ν
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From this, for y, z ∈ V we have
hy, P zi =
X hy, fν i hz, fν i
kfν k2
ν
.
Therefore the following remark is evident.
Remark 2. Let {fν } be an orthogonal basis of V1 .
For y, z ∈ V the inequality (PI) holds if and only if
Chebyshev Inequalities and
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X hy, fν i hz, fν i
ν
kfν k2
vol. 10, iss. 2, art. 54, 2009
≥ 0.
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Set
(
(3.6)
D=
x∈V :x=
)
X
αν fν , αν ≥ 0 .
ν
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Clearly, D is a closed convex cone generated by the system {fν }. The scalars αν =
hx,fν i
are the Fourier coefficients of x w.r.t. the orthogonal system {fν }. Moreover,
kfν k2
D is a self-dual cone w.r.t. V1 . By Proposition 3.3 we get
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Corollary 3.6. If {fν } is an orthogonal basis of V1 , then (PI) holds for y, z ∈ D+V2 ,
where D is defined by (3.6).
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Let Ξ denote the set of all sequences ξ = (ξ1 , ξ2 , . . . ) with ξν2 = 1, ν = 1, 2, . . . .
For given ξ, let us define the operator gξ on V as follows:
gξ x = x − P x +
X
ν
ξν
hx, fν i
fν .
kfν k2
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This operator is an isometry, because
2
2
2
2
2
kgξ xk = kxk − kP xk +
X
ν
2
2 hx, fν i
ξν
kfν k2
= kxk − kP xk + kP xk2 = kxk2 ,
by (3.5) and obvious orthogonality
Chebyshev Inequalities and
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x − Px ⊥
X
ξν
ν
hx, fν i
fν .
kfν k2
If x ∈ V2 , then hx, fν i = 0 for all ν. Hence P x = 0 =
reason
gξ x = x,
(3.7)
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P
ν
νi
ξν hx,f
f . For this
kfν k2 ν
x ∈ V2 .
We write
G = {gξ : ξ ∈ Ξ}.
(3.8)
We will show that G is a group of operators. It is evident that:
gξ = id,
(3.9)
for ξ = (1, 1, . . . ).
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Let ζ, ξ, γ ∈ Ξ. We have:
gζ fν = fν − P fν +
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X
η
ζη
hf ν, fη i
fη = ζν f ν,
kfη k2
because P fν = fν , ν = 1, 2, . . . . From this, by x − P x ∈ V2 and (3.7) we get:
gζ gξ x = gζ (x − P x +
X
ξν
ν
= gζ (x − P x) +
X
hx, fν i
fν )
kfν k2
ξν
ν
= x − Px +
X
ν
hx, fν i
gζ fν
kfν k2
hx, fν i
ζν ξν
fν .
kfν k2
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vol. 10, iss. 2, art. 54, 2009
Thus
(3.10)
gζ gξ = gζ·ξ = gξ gζ ,
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where ζ · ξ = (ζ1 ξ1 , ζ2 ξ2 , . . . ). This clearly gives:
(3.11)
gζ (gξ gγ ) = gζ·ξ·γ = (gξ gζ )gγ
and
gξ gξ = gξ·ξ = id,
(gξ )−1 = gξ .
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gξ − unitary,
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Since gξ is an isometry and invertible,
(3.13)
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which is equivalent to
(3.12)
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∀ξ∈Ξ .
By (3.13), (3.7), (3.9) – (3.12) we can assert that G is an Abelian group of unitary
operators that are identities on V2 . As a consequence, G ⊂ GP .
Given any x ∈ V , we define ξx = (ξx,1 , ξx,2 , . . . ) by
(
1, hx, fν i ≥ 0
(3.14)
ξx,ν =
−1, hx, fν i < 0.
It is clear that ξx,ν hx, fν i = |hx, fν i|. Hence
gξx x = x − P x +
X |hx, fν i|
ν
where x − P x ∈ V2 and
P |hx,fν i|
ν
kfν k2
kfν k2
fν ,
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fν ∈ D. Therefore
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(3.15)
gξx x ∈ D + V2 .
Assertion (3.15) is simply the statement that
(3.16)
(Gx) ∩ C 6= ∅,
∀x∈V
with C = D + V2 . This condition ensures that the sum of the cones gC, where g
runs over G, covers the whole space V . Now, we show that (3.16) holds for GP and
for every cone C = P C + V2 , P C 6= {0}.
Fix v ∈ V . Clearly, v = v1 + v2 , vi ∈ Vi , i = 1, 2. If v1 = 0 then v ∈
GP v ⊂ V2 ⊂ C, i.e. (3.16) holds. Assume that 0 6= v1 and note that there exists
0 6= u1 ∈ P C. Let us construct the two orthogonal bases {eν } and {fν } of V1 with
e1 = v1 and f1 = u1 . Set u = kv1 k kuu11 k + v2 and
(3.17)
g = id −P +
X h·, eν i
fν .
ke
kkf
k
ν
ν
ν
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Observe that u ∈ C, gv = u and g is the identity operator on V2 . Now, we prove that
g is unitary. Firstly, we note that for any x ∈ V
kgxk2 = kxk2 − kP xk2 +
X hx, eν i2
ν
because kP xk2 =
y ∈ V. We have
P hx,eν i2
ν
keν k2
keν k2
= kxk2 ,
. Our next goal is to show that gV = V. To do this, fix
Zdzisław Otachel
X hy, fν i fν
y = y − Py +
.
kf
k
kf
k
ν
ν
ν
Set
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X hy, fν i eν
x = y − Py +
.
kf
k
ke
k
ν
ν
ν
It is easily seen that gx = y. So, g is unitary.
Finally, g is a unitary operator on V with gV2 = V2 and gv = u. It gives u ∈
GP v ∩ C, as desired.
We are now in a position to introduce a notion of synchronicity of vectors for (PI).
For an orthoprojector P let C be a convex cone which admits the representation
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C = P C + V2 ,
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where P C is nontrivial. Let G be a subgroup of GP with the property (3.16).
The two vectors y, z ∈ V are said to be G-synchronous (with respect to C) if
there exists a g ∈ G such that gy, gz ∈ C. If G = GP , then we simply say that y and
z are 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 fixed C. In the sequel, for special cones we show that synchronicity is equivalent to (PI) but G-synchronicity is a sufficient condition for (PI).
According to Theorem 3.5, by the notion of synchronicity, it is possible to extend
(PI) beyond a cone C if only (PI) holds for vectors in C.
Proposition 3.7. Let C ⊂ V be a convex cone with C = P C + V2 , P C 6= {0} and
let G be a subgroup of GP with property (3.16).
The following statements are equivalent.
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i) (PI) holds for y, z ∈ C
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ii) (PI) holds for the vectors y and z which are G-synchronous w.r.t. C.
Proof. i) ⇒ ii). Assume y and z are G-synchronous w.r.t. C. There exists g ∈ G
with gy, gz ∈ C. Since i), (PI) holds for gy, gz. By Theorem 3.5 we conclude that
(PI) holds for y and z.
The converse implication is evident because y, z ∈ C are of course G-synchronous.
Now, we are able to give an equivalent condition for G-synchronicity. Simultaneously, the condition is sufficient for synchronicity w.r.t. D + V2 .
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Proposition 3.8. Let G be the group defined by (3.8) and let D be the cone defined
by (3.6).
The vectors y, z ∈ V are G-synchronous w.r.t. D + V2 if and only if
hy, fν i hz, fν i ≥ 0,
∀ν .
Proof. If y, z are G-synchronous, then there exists a ξ such that gξ y, gξ z ∈ D + V2 .
Hence ξν hy, fν i ≥ 0 and ξν hz, fν i ≥ 0 for all ν. Multiplying the above inequalities
side by side we obtain 0 ≤ ξν2 hy, fν i hz, fν i = hy, fν i hz, fν i for every ν.
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Conversely, suppose that hy, fν i hz, fν i ≥ 0 for every ν. In this situation, the
sequences defined for y and z by (3.14) are equal. Hence y and z are G-synchronous
by (3.15).
Summarizing the above considerations we give sufficient and necessary conditions for (PI) to hold.
Theorem 3.9. Let {fν } be an orthogonal basis of V1 . Set C = D + V2 , where D is
defined by (3.6). The following statements are equivalent.
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i) y and z are synchronous w.r.t. C
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Zdzisław Otachel
ii) (PI) holds for y and z.
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In particular, if
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(3.18)
hy, g0 fν i hz, g0 fν i ≥ 0,
∀ν,
then (PI) holds, where g0 ∈ GP is fixed.
Proof. The first part, i)⇒ii). It is a consequence of Corollary 3.6 and Proposition
3.7.
Conversely, if ii), then hP y, P zi ≥ 0. Firstly, suppose that P z = αP y. Clearly,
α ≥ 0. By (3.16), which holds for GP and C, there exists a g ∈ GP such that
gy ∈ C. Hence P gy ∈ P C = D. By (3.4), gP y ∈ D. Since α ≥ 0, αgP y ∈ D.
Since z − P z ∈ V2 , g(z − P z) ∈ V2 , because gV2 = V2 . Hence
gz = gP z + g(z − P z) = αgP y + g(z − P z) ∈ C.
Therefore gy, gz ∈ C, i.e. y and z are synchronous.
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Next, assume that P y and P z are linearly independent. Let us construct an orthogonal basis {eν } of V1 with
e1 = P y,
e2 = P z −
hP z, P yi
Py
kP yk2
and let g ∈ GP be defined by (3.17). There is no difficulty to showing that
kP yk
gy = y − P y +
f ∈ C,
| {z } kf1 k 1
| {z }
∈V2
Chebyshev Inequalities and
Self-Dual Cones
Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
∈D
yi
gz = z| −{zP z} + kehz,P
f1 +
1 kkf1 k
|
{z
}
∈V
2
≥0, by (PI)
|
kP zk2 kP yk2 −hP z,P yi2
kP yk2 ke2 kkf2 k
|
{z
f2 ∈ C.
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}
Contents
≥0, by Cauchy−Schwarz ineq.
{z
}
∈D
Therefore, y and z are synchronous as required.
Now, let us note that (3.18) is equivalent to
hg0∗ y, fν i hg0∗ z, fν i ≥ 0,
∀ν.
By Proposition 3.8, g0∗ y and g0∗ z are G-synchronous w.r.t. C. Hence there exists a
g ∈ G such that gg0∗ y, gg0∗ ∈ C. Since gg0∗ ∈ GP , 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 {si }
be the basis of Rn , where si = (1, . . . , 1, 0, . . . , 0), i = 1, . . . , n. The symbols
| {z }
i
V1 and V2 stand for the subspace orthogonal to sn and its orthogonal complement,
respectively, i.e.
(
)
X
V1 = (x1 , . . . , xn ) :
xi = 0 , V2 = span{sn }.
Chebyshev Inequalities and
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vol. 10, iss. 2, art. 54, 2009
i
Let P be the orthoprojector onto V1 , i.e. P = id − h·,snn i sn . In this situation, by
Example 3.2, (PI) becomes the Chebyshev sum inequality (CHSI).
It is known that the convex cone of nonincreasing vectors
C = {x = (x1 , . . . , xn ) : x1 ≥ x2 ≥ · · · ≥ xn }
is generated by {s1 , . . . , sn , −sn }. On the other side,
dual C =
x = (x1 , . . . , xn ) :
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n
X
xi = 0,
i=1
k
X
)
xi ≥ 0, k = 1, . . . , n − 1 .
i=1
Set ei = nP si , i = 1, . . . , n − 1. Clearly,
(4.1)
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{(1, −1, 0, . . . , 0), (0, 1, −1, 0, . . . , 0), (0, . . . , 0, 1, −1)}
is a generator of
(
Title Page
ei = nsi − isn = (n − i, . . . , n − i, −i, . . . , −i),
{z
} | {z }
|
i
n−i
i = 1, . . . , n − 1.
Close
Write
D = cone{ei }.
Clearly, P C = D and V2 ⊂ C. Hence by (3.1),
C = D + V2 .
Applying Proposition 3.2, we conclude that (PI) holds for y, z ∈ (D+V2 )∩dual D =
C ∩ dual D. With the aid of generators we can check that D ⊂ dual C. Hence
C = dual dual C ⊂ dual D.
By the above considerations, for arbitrary y, z ∈ C, the inequality (CHSI) holds.
This is a classic Chebyshev result.
The system {ei , i = 1, . . . , n − 1} constitutes a basis of V1 . Observe that
Chebyshev Inequalities and
Self-Dual Cones
Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
Title Page
hei , ej i = i(n − j)n,
i ≤ j, i, j = 1, . . . , n − 1.
Contents
Hence, easy computations lead to
n−k−1
ek+1 −
ek , ei = 0,
n−k
i = 1, . . . , k; k = 1, . . . , n − 2.
From this, the Gram-Schmidt orthogonalization gives the orthogonal system {qi } for
the basis {ei } as follows:
(
q1 = e1 ,
(4.2)
qk+1 = n−k
ek+1 − n−k−1
ek , k = 1, . . . , n − 2.
n
n−k
According to (4.1) and (4.2) we obtain the explicit form of the orthogonal basis
{qi }
(4.3)
qk = (0, . . . , 0, n − k, −1, . . . , −1),
| {z }
| {z }
k−1
n−k
k = 1, . . . , n − 1.
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Let us denote
e + V2 ,
K=D
e stands for the cone{qk }. The convex cone D
e is self-dual w.r.t. V1 .
where D
According to Proposition 3.3 we can assert that (CHSI) holds for y, z ∈ K.
Let g0 (x1 , . . . , xn ) = (−xn , . . . , −x1 ). Clearly, g0 ∈ GP . By Remark 1, (CHSI)
holds for y, z ∈ g0 K. Have:
e + V2 ) = g0 D
e + V2 = cone{g0 qk } + V2 .
g0 K = g0 (D
Define fk = g0 qn−k , k = 1, . . . , n − 1. Since g0 ∈ GP , g0 is unitary and g0 V1 = V1 ,
by (3.3). Hence, {fk } is an orthogonal basis of V1 . Observe
(4.4)
fk = (1, . . . , 1, −k, 0, . . . , 0),
| {z }
k = 1, . . . , n − 1.
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
Title Page
Contents
k
Write
M = cone{fk } + V2 .
By Remark 1, it is evident that (CHSI) holds for y, z ∈ M .
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n
Proposition 4.1. For x = (x1 , . . . , xn ) ∈ R
(
)n
n
X
1
x ∈ K ⇐⇒ the sequence
xi
is nonincreasing,
n − k + 1 i=k
k=1
)n
( k
X
1
xi
is nonincreasing.
x ∈ M ⇐⇒ the sequence
k i=1
k=1
Proof. We prove only the first equivalence. The second one uses a similar procedure.
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e because D
e is self-dual w.r.t. V1 . Hence by (4.3) we can
By (3.2), K = dual D,
assert that x ∈ K is equivalent to
(n − k)xk ≥
n
X
xi ,
k = 1, . . . , n − 1.
i=k+1
Pn
Adding to both of sides (n − k) i=k+1 xi and dividing by (n − k)(n − k + 1), we
obtain
!
!
n
n
X
X
1
1
xi ≥
xi , k = 1, . . . , n − 1.
n − k + 1 i=k
n − k i=k+1
This is equivalent to our claim.
(xi − xj )(yi − yj ) ≥ 0,
Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
Title Page
By the above proposition, we can see that C ⊂ K and C ⊂ M . The cone M is
said to be a cone of vectors nonincreasing in mean. It is easily seen that (CHSI) holds
for y, z ∈ −K and for y, z ∈ −M (for e.g., by taking C = K, M and substituting
− id into g0 in Remark 1). 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.
Here GP is the group of all orthogonal matrices such that the sum of the entries of
each row and column is equal to 1 or −1. The group of all n×n permutation matrices
is a subgroup of GP , which together with the cone C fulfil (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 contains C, e.g. M or K.
The two vectors x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn are said to be similarly
ordered if
(4.5)
Chebyshev Inequalities and
Self-Dual Cones
∀i,j .
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The assertion that (CHSI) holds for similarly ordered vectors is a consequence of
Proposition 3.7.
Theorem 3.9 states that (CHSI) is equivalent to synchronicity w.r.t. cone{fk }+V2
where {fk } is an arbitrarily chosen orthogonal basis of V1 . Moreover, G-synchronicity
gives (CHSI), where G is the group (3.8) acting on Rn . For this reason, the specification of Theorem 3.9 can be as follows.
Let {fk } be defined by (4.4) and G by (3.8) in compliance with the basis.
Corollary 4.2. (CHSI) holds for y, z if and only if y and z are synchronous w.r.t.
M.
In particular, (CHSI) is satisfied by y and z such that
hy, U fk i hz, U fk i ≥ 0,
k = 1, . . . , n − 1,
where U is a fixed unitary operation with U sn = sn or U sn = −sn , i.e. U is
represented by an orthogonal matrix whose rows and columns sum up to 1 or to −1.
By Proposition 3.8 we have:
Remark 3. The vectors y = (y1 , . . . , yn ) and z = (z1 , . . . , zn ) are G-synchronous
w.r.t. M if and only if
" k
#" k
#
X
X
yi − kyk+1
zi − kzk+1 ≥ 0, k = 1, . . . , n − 1.
i=1
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
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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. Consider Rn , n > 3.
For 0 < α < 1 < β < n − 1 set y = (0, . . . , 0, 1 − n, −α), z = (0, . . . , 0, 1 −
n, −β). According to (4.5) and Remark 3 the vectors y and z are similarly ordered
and are not G-synchronous, but they are synchronous w.r.t. M , so (CHSI) holds.
Now, set y 0 = f1 + f2 , z 0 = f2 + f3 , where fi are defined by (4.4). The vectors y 0
and z 0 are G-synchronous w.r.t. M , because y 0 , z 0 ∈ M , so (CHSI) holds.
On the other hand y 0 = (2, 0, −2, 0, . . . , 0), z 0 = (2, 2, −1, −3, 0, . . . , 0) are not
similarly ordered by (4.5), because (y30 − y40 )(z30 − z40 ) = −2(−1 + 3) < 0.
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
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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 by IA . Additionally we will write es = I[0,s] , 0 ≤ s ≤ 1.
The
symbol
R V1 stands for the subspace orthogonal to V2 = span{e1 }, i.e. V1 =
2
x ∈ L : xdµ = 0 . By Example 3.2, it is known that for the orthoprojector P
onto V1 (PI) transforms into the Chebyshev integral inequality (CHII). Let C ⊂ L2
be the closed convex cone of all nonincreasing µ a.e. functions. It is known (see [5,
Theorem 3.1 and 3.3]) that:
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
(5.1)
C = cone ({es : 0 ≤ s ≤ 1} ∪ {−e1 }) ,
dual C = cone{IΠ − IΠ+ε : ε > 0, Π, Π + ε ⊂ [0, 1]},
where Π stands for an interval.
The Haar system:
(5.2)
Contents
χ00 = e1

2k−2

2n/2 ,
≤t<

2n+1


χkn (t) =
−2n/2 , 2k−1
≤t<
2n+1



 0,
otherwise
n = 0, 1, . . . ,
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2k−1
2n+1
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2k
2n+1
k = 1, 2, . . . , 2n
forms an orthonormal basis of L2 . In particular, H = {χkn : n = 0, 1, . . . , k =
1, . . . , 2n } is an orthonormal basis of V1 .
Let
D = cone H.
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The cone D is self-dual w.r.t. V1 , so by (3.2) we have:
(5.3)
dual D = D + V2 .
By (5.1), observe that H ⊂ dual C, hence C = dual dual C ⊂ dual H =
dual D. Combining this with (5.3), we obtain
(5.4)
C ⊂ D + V2 .
Chebyshev Inequalities and
Self-Dual Cones
From (5.4) and Corollary 3.6 it follows that
Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
Corollary 5.1. (CHII) holds for y, z ∈ D + V2 .
The cone D + V2 contains the cone of all nonincreasing µ a.e. functions in L2 .
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It is easily seen that the cone D + V2 contains functions which are not nonincreasing µ 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 Theorem 3.9 we get:
Corollary 5.2. (CHII) holds for y, z ∈ L2 if only
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(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 that y and z are similarly ordered,
i.e.
(5.6)
[y(s) − y(t)] [z(s) − z(t)]] ≥ 0,
for all 0 ≤ s, t ≤ 1
(see e.g. [6, pp. 198-199]). Now, we show by an example that the G-synchronicity
condition (5.5) is not stronger than the condition of similar ordering (5.6) in L2 .
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Example 5.1. In L2 let y = χ12 + χ22 , z = χ22 + χ32 , where χji are defined by (5.2).
The vectors y and z are G-synchronous w.r.t. D + V2 , because they are in D.
On the other hand
y(s) = 2,
0≤s≤
1
8
y(t) = 0,
4
8
5
8
≤t≤
z(s) = 0,
0≤s≤
1
8
z(t) = 2,
4
8
5
8
,
≤t≤
.
From this, [y(s) − y(t)] [z(s) − z(t)]] = [2 − 0][0 − 2] ≤ 0 for any 0 ≤ s ≤ 81 and
4
≤ t ≤ 58 . Thus y and z are not similarly ordered.
8
Now, we recall that a function yR ∈ L2 is nonincreasing (nondecreasing, monos
tone) in mean if the function s 7→ 1s 0 ydµ, is nonincreasing (nondecreasing, monotone).
R
1 s
ydµ we can easy obtain that y is nonincreasing in mean if
Differentiating
s 0
R
1 s
and only if s 0 ydµ ≥ y(s), µ a.e.
It is known that (CHII) holds for y and z which 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
Z s
Z s
1
1
ydµ − y(s)
zdµ − z(s) ≥ 0, ∀0<s<1
(5.7)
s 0
s 0
then (CHII) holds for y and z.
Remark 4.
1. There exist functions in cone H which are not nonincreasing in mean.
2. There exist functions nonincreasing in mean which are not in cone H.
3. There exist functions in cone H for which (5.7) does not hold, i.e. the condition
(5.5) is not stronger than (5.7).
Chebyshev Inequalities and
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vol. 10, iss. 2, art. 54, 2009
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Proof. An easy verification shows that:
Ad. 1) χkn ∈ H, k > 1 are 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
cone H because hf, χ21 i < 0.
Ad. 3) Set y = χ21 , z = χ32 . For 85 < s < 68 have:
√
Z s
Z s
1
2/2 3/2
1
·
< 0.
ydµ − y(s)
zdµ − z(s) = −
s 0
s 0
s
s
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
2
The set of all L -functions nonincreasing in mean constitutes a convex cone. It
will be denoted by M . Let M0 be the class of all step functions of the form
s
gs,t = I[0,s) −
I[s,t] , 0 < s < t < 1.
t−s
Proposition 5.3.
M = dual M0 ,
P M = dualV1 M0 = cone M0 ,
M = cone M0 + V2 .
Proof. By definition, f ∈ M if and only if
Z
Z
1 s
1 t
f dµ ≥
f dµ for all 0 < s < t < 1.
s 0
t 0
After equivalent transformations we obtain
Z s
Z t
s
f dµ ≥
f dµ for all 0 < s < t < 1.
t−s s
0
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This is simply f ∈ dual M0 , so the first equation holds.
To show the second equation, note that M0 ⊂ M ∩ V1 . Hence
dual(M ∩ V1 ) ⊂ dual M0 = M,
by the first equation. It follows that V1 ∩ dual(M ∩ V1 ) ⊂ M ∩ V1 , i.e.
dualV1 (M ∩ V1 ) ⊂ M ∩ V1 .
(5.8)
Fix f ∈ M ∩ V1 and let g ∈ M ∩ V1 be arbitrary. For such f and g (CHII) holds
and takes the form:
Z 1
Z 1
Z 1
f gdµ ≥
f dµ ·
gdµ = 0 · 0 = 0,
0
0
0
Chebyshev Inequalities and
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Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
Title Page
i.e. f ∈ dualV1 (M ∩ V1 ). Therefore
Contents
M ∩ V1 ⊂ dualV1 (M ∩ V1 ).
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Since M = dual M0 , dual M = cone M0 . Now, observe that V2 ⊂ M . This
implies by (3.1) that M = P M + V2 . Furthermore, in this situation P M = M ∩ V1 .
The above gives
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(5.9)
dual M = dual(M ∩ V1 + V2 )
= V1 ∩ dual(M ∩ V1 ) = dualV1 (M ∩ V1 ).
Hence
(5.10)
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dualV1 (M ∩ V1 ) = cone M0 .
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.
The second equation of the above propositions immediately gives:
Remark 5. The convex cone of all L2 -functions nonincreasing in mean with integral
equal to 0 is self-dual w.r.t. V1 .
Taking C = M in Theorem 3.1, by Proposition 5.3 we easily obtain:
R
R
R
Corollary 5.4. If f dµ gdµ ≤ f gdµ holds for all functions f ∈ L2 monotone
in mean, then g ∈ L2 is also monotone in mean.
Chebyshev Inequalities and
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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 monotone, J. Math. Anal. Appl., 172 (1993), 221–232.
[3] M. NIEZGODA, On the Chebyshev functional, Math. Inequal. Appl., 10(3)
(2007), 535–546.
Chebyshev Inequalities and
Self-Dual Cones
Zdzisław Otachel
vol. 10, iss. 2, art. 54, 2009
[4] M. NIEZGODA, Bifractional inequalities and convex cones, Discrete
Math., 306(2) (2006), 231–243.
Title Page
[5] Z. OTACHEL, Spectral orders and isotone functionals, Linear Algebra
Appl., 252 (1997), 159–172.
[6] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering,
Vol. 187, Academic Press, Inc. (1992)
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