Volume 10 (2009), Issue 2, Article 54, 15 pp.
CHEBYSHEV INEQUALITIES AND SELF-DUAL CONES
ZDZISŁAW OTACHEL
D EPARTMENT OF A PPLIED M ATHEMATICS AND C OMPUTER S CIENCE
U NIVERSITY OF L IFE S CIENCES IN L UBLIN
A KADEMICKA 13, 20-950 L UBLIN , P OLAND
zdzislaw.otachel@up.lublin.pl
Received 20 December, 2008; accepted 12 April, 2009
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Chebyshev type inequality, Convex cone, Dual cone, Orthoprojector.
2000 Mathematics Subject Classification. 26D15, 26D20, 15A39.
1. I NTRODUCTION 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.
001-09
2
Z DZISŁAW OTACHEL
2. P RELIMINARIES
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).
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
dual(C + D) = dual C ∩ dual D.
The dual cone of a subspace W is equal to its orthogonal complement W ⊥ . If a set C is a closed
convex cone, then
dual dual C = C,
(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.
3. P ROJECTION I NEQUALITY
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
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]).
Theorem 3.1. For vectors y, z ∈ V and a convex cone C ⊂ V the following statements are
mutually equivalent.
i): (PI) holds for all y ∈ C + V2
ii): P z ∈ dual C
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 argument, if z ∈ dual P C then
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
C HEBYSHEV I NEQUALITIES AND S ELF -D UAL C ONES
3
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 orthoprojector onto the subspace
P h·,fν i
orthogonal to span{fν }, i.e. P = id − kf
2 fν , then we obtain the classic Bessel inequality
νk
ν
kzk2 ≥
X hz, fν i2
ν
kfν k2
.
Example 3.2 (Chebyshev type inequalities). Let x ∈ V be a fixed nonzero vector. Set P =
h·,xi
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
n
X
i=1
zi ≤ n
n
X
yi zi .
i=1
2
Consider the space V = L 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.
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.
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.
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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
4
Z DZISŁAW OTACHEL
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 .
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 .
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 .
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:
kuk2 = kguk2 = kv1 + v2 k2 = kv1 k2 + kv2 k2
(g − unitary, v1 ⊥ v2 ),
ku − g ∗ v2 k2 = kg(u − g ∗ v2 )k2
= kgu − v2 k2
= kv1 k2
(since g − unitary, g ∗ g = id),
ku − g ∗ v2 k2 = kuk2 + kg ∗ v2 k2
= kuk2 + kv2 k2
Hence:
(u ⊥ g ∗ v2 , g ∗ − unitary).

 kuk2 = kv1 k2 + kv2 k2
⇒ kv2 k2 = 0 ⇒ v2 = 0,
 kv k2 = kuk2 + kv k2
1
2
a contradiction. This completes the proof of gV1 ⊂ V1 .
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
C HEBYSHEV I NEQUALITIES AND S ELF -D UAL C ONES
5
Note that g ∗ V1 ⊂ V1 , too. This implies that V1 ⊂ gV1 . Therefore
gV1 = V1 .
(3.3)
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 .
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.
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 =
f .
2 ν
kf
k
ν
ν
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
X hy, fν i hz, fν i
ν
kfν k2
≥ 0.
Set
(
(3.6)
D=
x∈V :x=
)
X
αν fν , αν ≥ 0 .
ν
νi
Clearly, D is a closed convex cone generated by the system {fν }. The scalars αν = hx,f
are
kfν k2
the Fourier coefficients of x w.r.t. the orthogonal system {fν }. Moreover, D is a self-dual cone
w.r.t. V1 . By Proposition 3.3 we get
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).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
6
Z DZISŁAW OTACHEL
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:
X hx, fν i
gξ x = x − P x +
ξν
f .
2 ν
kf
k
ν
ν
This operator is an isometry, because
kgξ xk2 = kxk2 − kP xk2 +
X
ξν2
ν
2
hx, fν i2
kfν k2
2
= kxk − kP xk + kP xk2 = kxk2 ,
by (3.5) and obvious orthogonality
hx, fν i
fν .
kfν k2
ν
P
νi
If x ∈ V2 , then hx, fν i = 0 for all ν. Hence P x = 0 = ξν hx,f
f . For this reason
kfν k2 ν
x − Px ⊥
X
ξν
ν
x ∈ V2 .
gξ x = x,
(3.7)
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, . . . ).
Let ζ, ξ, γ ∈ Ξ. We have:
gζ fν = fν − P fν +
X
ζη
η
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:
X hx, fν i
gζ gξ x = gζ (x − P x +
ξν
f )
2 ν
kf
k
ν
ν
X hx, fν i
= gζ (x − P x) +
ξν
gζ fν
kfν k2
ν
X
hx, fν i
= x − Px +
ζν ξν
fν .
kfν k2
ν
Thus
(3.10)
gζ gξ = gζ·ξ = gξ gζ ,
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,
which is equivalent to
(gξ )−1 = gξ .
(3.12)
Since gξ is an isometry and invertible,
(3.13)
gξ − unitary,
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
∀ξ∈Ξ .
http://jipam.vu.edu.au/
C HEBYSHEV I NEQUALITIES AND S ELF -D UAL C ONES
7
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 ,
fν ∈ D. Therefore
gξx x ∈ D + V2 .
(3.15)
Assertion (3.15) is simply the statement that
(Gx) ∩ C 6= ∅,
(3.16)
∀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
X h·, eν i
(3.17)
g = id −P +
fν .
keν kkfν k
ν
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
X hx, eν i2
2
2
2
kgxk = kxk − kP xk +
= kxk2 ,
2
ke
k
ν
ν
because kP xk2 =
P hx,eν i2
ν
keν k2
. Our next goal is to show that gV = V. To do this, fix y ∈ V. We
have
y = y − Py +
X hy, fν i fν
.
kfν k kfν k
ν
x = y − Py +
X hy, fν i eν
.
kfν k keν k
ν
Set
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
C = P C + V2 ,
where P C is nontrivial. Let G be a subgroup of GP with the property (3.16).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
8
Z DZISŁAW OTACHEL
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 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.
i): (PI) holds for y, z ∈ C
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 .
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.
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.
i): y and z are synchronous w.r.t. C
ii): (PI) holds for y and z.
In particular, if
(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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
C HEBYSHEV I NEQUALITIES AND S ELF -D UAL C ONES
9
Next, assume that P y and P z are linearly independent. Let us construct an orthogonal basis
{eν } of V1 with
hP z, P yi
Py
e1 = P y, e2 = P z −
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
∈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.
}
≥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.
4. A PPLICATIONS TO THE C HEBYSHEV S UM I NEQUALITY
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 V1 and V2 stand for the
| {z }
i
subspace orthogonal to sn and its orthogonal complement, respectively, i.e.
(
)
X
V1 = (x1 , . . . , xn ) :
xi = 0 , V2 = span{sn }.
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,
{(1, −1, 0, . . . , 0), (0, 1, −1, 0, . . . , 0), (0, . . . , 0, 1, −1)}
is a generator of
(
dual C =
x = (x1 , . . . , xn ) :
n
X
i=1
xi = 0,
k
X
)
xi ≥ 0, k = 1, . . . , n − 1 .
i=1
Set ei = nP si , i = 1, . . . , n − 1. Clearly,
(4.1)
ei = nsi − isn = (n − i, . . . , n − i, −i, . . . , −i),
|
{z
} | {z }
i
i = 1, . . . , n − 1.
n−i
Write
D = cone{ei }.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
10
Z DZISŁAW OTACHEL
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
hei , ej i = i(n − j)n,
i ≤ j, i, j = 1, . . . , n − 1.
Hence, easy computations lead to
n−k−1
ek , ei = 0,
ek+1 −
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 }
qk = (0, . . . , 0, n − k, −1, . . . , −1),
| {z }
| {z }
(4.3)
k−1
k = 1, . . . , n − 1.
n−k
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.
k
Write
M = cone{fk } + V2 .
By Remark 1, it is evident that (CHSI) holds for y, z ∈ M .
Proposition 4.1. For x = (x1 , . . . , xn ) ∈ Rn
(
)n
n
X
1
x ∈ K ⇐⇒ the sequence
xi
is nonincreasing,
n − k + 1 i=k
k=1
( k
)n
1X
x ∈ M ⇐⇒ the sequence
xi
is nonincreasing.
k i=1
k=1
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
C HEBYSHEV I NEQUALITIES AND S ELF -D UAL C ONES
11
Proof. We prove only the first equivalence. The second one uses a similar procedure.
e because D
e is self-dual w.r.t. V1 . Hence by (4.3) we can assert that
By (3.2), K = dual D,
x ∈ K is equivalent to
n
X
(n − k)xk ≥
xi ,
k = 1, . . . , n − 1.
i=k+1
Adding to both of sides (n − k)
1
n−k+1
Pn
xi and dividing by (n − k)(n − k + 1), we obtain
!
!
n
n
X
X
1
xi ≥
xi , k = 1, . . . , n − 1.
n − k i=k+1
i=k
i=k+1
This is equivalent to our claim.
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
(xi − xj )(yi − yj ) ≥ 0,
(4.5)
∀i,j .
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
i=1
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
12
Z DZISŁAW OTACHEL
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 Gsynchronous, 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.
5. A PPLICATIONS TO THE C HEBYSHEV I NTEGRAL I NEQUALITY
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]
V1 stands for
, 0 ≤2s ≤
R 1. The symbol
the subspace orthogonal to V2 = span{e1 }, i.e. V1 = 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:
(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)
χ00 = e1

2k−2

2n/2 ,
≤ t < 2k−1

2n+1
2n+1


2k
χkn (t) =
−2n/2 , 2k−1
≤ t < 2n+1
2n+1



 0,
otherwise
n = 0, 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.
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 .
From (5.4) and Corollary 3.6 it follows that
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 .
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
C HEBYSHEV I NEQUALITIES AND S ELF -D UAL C ONES
13
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
hy, χi hz, χi ≥ 0,
(5.5)
∀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 .
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 ≤
Thus y and z are not similarly ordered.
1
8
and
4
8
≤ t ≤ 58 .
Now, we recall thatRa function y ∈ L2 is nonincreasing (nondecreasing, monotone) in mean
s
if the function s 7→ R1s 0 ydµ, is nonincreasing (nondecreasing, monotone).
s
Differentiating 1s 0 ydµ we can easy obtain that y is nonincreasing in mean if and only if
R
s
1
ydµ ≥ y(s), µ a.e.
s 0
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
(5.7)
ydµ − y(s)
zdµ − z(s) ≥ 0, ∀0<s<1
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).
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
1
2/2 3/2
ydµ − y(s)
zdµ − z(s) = −
·
< 0.
s 0
s 0
s
s
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
14
Z DZISŁAW OTACHEL
The set of all L2 -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
I[s,t] , 0 < s < t < 1.
gs,t = I[0,s) −
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 t
Z s
s
f dµ ≥
f dµ for all 0 < s < t < 1.
t−s s
0
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
i.e. f ∈ dualV1 (M ∩ V1 ). Therefore
(5.9)
M ∩ V1 ⊂ dualV1 (M ∩ V1 ).
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
dual M = dual(M ∩ V1 + V2 )
= V1 ∩ dual(M ∩ V1 ) = dualV1 (M ∩ V1 ).
Hence
(5.10)
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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/
C HEBYSHEV I NEQUALITIES AND S ELF -D UAL C ONES
15
R EFERENCES
[1] M. BIERNACKI, Sur une inégalité entre les intégrales due á Tchébyscheff, Ann. Univ. Mariae CurieSkłodowska, 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.
[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. 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)
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 54, 15 pp.
http://jipam.vu.edu.au/