CHEBYSHEV INEQUALITIES AND SELF-DUAL CONES

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1 CHEBYSHEV INEQUALITIES AND SELF-DUAL CONES ZDZISŁAW OTACHEL Dept. of Applied Mathematics and Computer Science University of Life Sciences in Lublin Akademicka 13, Lublin, Poland Received: 20 December, 2008 Accepted: 12 April, 2009 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26D15, 26D20, 15A39. Key words: Abstract: Chebyshev type inequality, Convex cone, Dual cone, Orthoprojector. 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. Page 1 of 31

2 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 Page 2 of 31

3 1. Introduction and summary Let V be a real vector space provided with an inner product,. For fixed x V and y, z V the inequality (1.1) x, y x, z y, z x, x 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 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. Page 3 of 31

4 2. Preliminaries In this note V is a real Hilbert space with an inner product,. 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 : v, C 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 dual V1 C stands for V 1 dual C and means the relative dual of C with respect to a closed subspace V 1 of V. If for a closed convex cone D the identity dual V1 D = D holds, then D is called a self-dual cone w.r.t. V 1. 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. Page 4 of 31

5 3. Projection Inequality From now on we make the following assumptions: P is an idempotent and symmetric operator (orthoprojector) defined on V, V = V 1 + V 2, where V 1 is the range of P and V 2 is its orthogonal complement, i.e. V 1 = P V and V 2 = (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 y, P z 0. If y = z, then (PI) holds for any orthoprojector P taking the form P z 2 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 + V 2 ii) P z dual C iii) z dual P C. Proof. Since i), the inequality (PI) holds for every y C. Thus 0 y, P z = P y, z. Therefore 0 C, P z = P C, z. 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 x, V 1 = 0 are arbitrary have y, P z = c, P z + x, P z = c, P z 0. By a similar Page 5 of 31

6 argument, if z dual P C then y C + V 2 implies that P y P C. It leads to y, P z = P y, z 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 {f ν } be an orthogonal system in V. If P is the orthoprojector onto the subspace orthogonal to span{f ν }, i.e. P = id,f ν f ν f 2 ν, then we ν obtain the classic Bessel inequality z 2 ν z, f ν 2 f ν 2. Example 3.2 (Chebyshev type inequalities). Let x V be a fixed nonzero vector. Set P = id,x x. It is clear that P is the orthoprojector onto the subspace orthogonal x 2 to x. In the case where the inequality (PI) becomes a Chebyshev type inequality (1.1): x, z y, x y, z x 2. In the space V = R n under x = (1,..., 1), inequality (1.1) transforms into the Chebyshev sum inequality (or (CHSI) for short): n n n z i n y i z i. i=1 y i i=1 Consider the space V = L 2 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): ydµ zdµ yzdµ. i=1 Page 6 of 31

7 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) a, gb a, b 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 R n and the cone of nonincreasing vectors. Let C V be a convex cone. Every cone of the form C + V 2 has the representation: (3.1) C + V 2 = P C + V 2. Therefore, on studying the projection inequality (PI), according to Theorem 3.1, it is sufficient to consider convex cones of the form C = D + V 2, where D is a convex cone in V 1. The following proposition is a simple consequence of Theorem 3.1. Proposition 3.2. Let D V 1 be a convex cone. For y, z V the following conditions are equivalent. Page 7 of 31 i) (PI) holds for all y D + V 2 ii) z dual D.

8 Let D V 1 be a convex cone. Then V 2 dual D. This implies that P dual D = V 1 dual D. Applying (3.1) to dual D + V 2 = dual D, we get (3.2) dual V1 D + V 2 = dual D. According to the above equation and the last proposition, we need to find for (PI) such cones D for which D dual V1 D are as wide as possible. Proposition 3.3. The inequality (PI) holds for y, z D+V 2, where D is an arbitrary self-dual cone w.r.t. V 1. Proof. By assumption, D V 1, hence (3.2) gives dual D = D + V 2. Proposition 3.2 implies that (PI) holds for y, z (D + V 2 ) dual D = D + V 2. If D is a self-dual cone w.r.t. V 1 then D + V 2 is a maximal cone for (PI) in the following sense. Proposition 3.4. Let D be a self-dual cone w.r.t. V 1 with D +V 2 C, where C V is a convex cone. If (PI) holds for y, z C then C = D + V 2. Proof. Since V 2 C, (3.1) yields C = P C + V 2. By Proposition 3.2, (PI) holds for y, z (P C + V 2 ) dual P C. The assumption that (PI) holds for y, z C gives P C + V 2 dual P C. Since D + V 2 C, D = P (D + V 2 ) P C. From this we have dual P C dual D = D + V 2, by (3.2), because dual V1 D = D. Combining these inclusions we can see that C = P C + V 2 D + V 2. The converse inclusion holds by the hypothesis, and thus the proof is complete. Page 8 of 31 Let G P denote the set of all unitary operators acting on V with gv 2 = V 2. Notice that G P is a group of operators. The inequality (PI) is invariant with respect to G P.

9 Theorem 3.5. For fixed g G P 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 gv 2 = V 2. This is equivalent to g V 2 = V 2, where g is the adjoint operator of g. We first show that gv 1 V 1. Suppose, contrary to our claim, that there exists a u V 1 with the property gu = v 1 + v 2, v i V i, i = 1, 2, v 2 0. We have: Hence: u 2 = gu 2 = v 1 + v 2 2 = v v 2 2 (g unitary, v 1 v 2 ), u g v 2 2 = g(u g v 2 ) 2 = gu v 2 2 = v 1 2 (since g unitary, g g = id), u g v 2 2 = u 2 + g v 2 2 = u 2 + v 2 2 (u g v 2, g unitary). u 2 = v v 2 2 v 1 2 = u 2 + v 2 2 v 2 2 = 0 v 2 = 0, a contradiction. This completes the proof of gv 1 V 1. Note that g V 1 V 1, too. This implies that V 1 gv 1. Therefore Page 9 of 31 (3.3) gv 1 = V 1.

10 Now, let z V be arbitrary. We have z = z 1 + z 2, where z i V i, i = 1, 2. For an orthoprojector P onto V 1 we get: gp z = gp (z 1 + z 2 ) = gz 1 = P (gz 1 + gz 2 ) = P gz, because gz 1 V 1 by (3.3) and gz 2 V 2 by assumption. Thus (3.4) P g = gp. By (3.4), This proves required equivalence. gy, P gz = gy, gp z = g gy, P z = y, P z. A simple consequence of the above theorem is: Remark 1. For a convex cone C V and g 0 G P the following statements are equivalent. i) (PI) holds for y, z C ii) (PI) holds for y, z g 0 C. In the remainder of this section we assume that V is a real separable Hilbert space. Let {f ν } be an orthogonal basis of V 1, i.e. { > 0, η = ν f η, f ν = 0, η ν, for integers η, ν. Under the above assumption, the projection P z takes the form: (3.5) P z = ν z, f ν f ν 2 f ν. Page 10 of 31

11 From this, for y, z V we have y, P z = ν y, f ν z, f ν f ν 2. Therefore the following remark is evident. Remark 2. Let {f ν } be an orthogonal basis of V 1. For y, z V the inequality (PI) holds if and only if Set (3.6) D = { ν y, f ν z, f ν f ν 2 0. x V : x = ν α ν f ν, α ν 0 Clearly, D is a closed convex cone generated by the system {f ν }. The scalars α ν = x,f ν f ν are the Fourier coefficients of x w.r.t. the orthogonal system {f 2 ν }. Moreover, D is a self-dual cone w.r.t. V 1. By Proposition 3.3 we get Corollary 3.6. If {f ν } is an orthogonal basis of V 1, then (PI) holds for y, z D+V 2, where D is defined by (3.6). Let Ξ denote the set of all sequences ξ = (ξ 1, ξ 2,... ) with ξ 2 ν = 1, ν = 1, 2,.... For given ξ, let us define the operator g ξ on V as follows: }. Page 11 of 31 g ξ x = x P x + ν ξ ν x, f ν f ν 2 f ν.

12 This operator is an isometry, because g ξ x 2 = x 2 P x 2 + ξ 2 x, f ν 2 ν f ν ν 2 = x 2 P x 2 + P x 2 = x 2, by (3.5) and obvious orthogonality x P x ν ξ ν x, f ν f ν 2 f ν. If x V 2, then x, f ν = 0 for all ν. Hence P x = 0 = ν reason (3.7) g ξ x = x, x V 2. We write (3.8) G = {g ξ : ξ Ξ}. ξ ν x,f ν f ν 2 f ν. For this Page 12 of 31 We will show that G is a group of operators. It is evident that: (3.9) g ξ = id, for ξ = (1, 1,... ). Let ζ, ξ, γ Ξ. We have: g ζ f ν = f ν P f ν + η fν, f η ζ η f η f 2 η = ζ ν fν,

13 because P f ν = f ν, ν = 1, 2,.... From this, by x P x V 2 and (3.7) we get: Thus g ζ g ξ x = g ζ (x P x + ν = g ζ (x P x) + ν = x P x + ν ξ ν x, f ν f ν 2 f ν) ξ ν x, f ν f ν 2 g ζf ν ζ ν ξ ν x, f ν f ν 2 f ν. (3.10) g ζ g ξ = g ζ ξ = g ξ g ζ, where ζ ξ = (ζ 1 ξ 1, ζ 2 ξ 2,... ). This clearly gives: (3.11) g ζ (g ξ g γ ) = g ζ ξ γ = (g ξ g ζ )g γ and g ξ g ξ = g ξ ξ = id, which is equivalent to (3.12) (g ξ ) 1 = g ξ. Since g ξ is an isometry and invertible, Page 13 of 31 (3.13) g ξ unitary, 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 V 2. As a consequence, G G P.

14 Given any x V, we define ξ x = (ξ x,1, ξ x,2,... ) by { 1, x, fν 0 (3.14) ξ x,ν = 1, x, f ν < 0. It is clear that ξ x,ν x, f ν = x, f ν. Hence where x P x V 2 and ν g ξx x = x P x + ν x,f ν f ν f 2 ν D. Therefore (3.15) g ξx x D + V 2. Assertion (3.15) is simply the statement that (3.16) (Gx) C, x V x, f ν f ν 2 f ν, with C = D + V 2. 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 G P and for every cone C = P C + V 2, P C {0}. Fix v V. Clearly, v = v 1 + v 2, v i V i, i = 1, 2. If v 1 = 0 then v G P v V 2 C, i.e. (3.16) holds. Assume that 0 v 1 and note that there exists 0 u 1 P C. Let us construct the two orthogonal bases {e ν } and {f ν } of V 1 with e 1 = v 1 and f 1 = u 1. Set u = v 1 u 1 u 1 + v 2 and (3.17) g = id P + ν, e ν e ν f ν f ν. Page 14 of 31

15 Observe that u C, gv = u and g is the identity operator on V 2. Now, we prove that g is unitary. Firstly, we note that for any x V gx 2 = x 2 P x 2 + ν x, e ν 2 e ν 2 = x 2, because P x 2 = ν y V. We have Set x,e ν 2 e ν 2. Our next goal is to show that gv = V. To do this, fix y = y P y + ν x = y P y + ν y, f ν f ν y, f ν f ν f ν f ν. e ν e ν. It is easily seen that gx = y. So, g is unitary. Finally, g is a unitary operator on V with gv 2 = V 2 and gv = u. It gives u G P 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 + V 2, where P C is nontrivial. Let G be a subgroup of G 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 a g G such that gy, gz C. If G = G P, 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 Page 15 of 31

16 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 + V 2, P C {0} and let G be a subgroup of G P 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 + V 2. 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 + V 2 if and only if y, f ν z, f ν 0, ν. Page 16 of 31 Proof. If y, z are G-synchronous, then there exists a ξ such that g ξ y, g ξ z D + V 2. Hence ξ ν y, f ν 0 and ξ ν z, f ν 0 for all ν. Multiplying the above inequalities side by side we obtain 0 ξ 2 ν y, f ν z, f ν = y, f ν z, f ν for every ν.

17 Conversely, suppose that y, f ν z, f ν 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 {f ν } be an orthogonal basis of V 1. Set C = D + V 2, 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) y, g 0 f ν z, g 0 f ν 0, ν, then (PI) holds, where g 0 G P is fixed. Proof. The first part, i) ii). It is a consequence of Corollary 3.6 and Proposition 3.7. Conversely, if ii), then P y, P z 0. Firstly, suppose that P 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. Since z P z V 2, g(z P z) V 2, because gv 2 = V 2. Hence Page 17 of 31 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.

18 Next, assume that P y and P z are linearly independent. Let us construct an orthogonal basis {e ν } of V 1 with e 1 = P y, e 2 = P z P z, P y P y P y 2 and let g G P be defined by (3.17). There is no difficulty to showing that gz = z P z }{{} gy = y P y }{{} + z,p y e 1 f 1 }{{} V 2 0, by (PI) V 2 + P y f 1 f 1 C, }{{} D f 1 + P z 2 P y 2 P z,p y 2 P y 2 e 2 f 2 }{{} 0, by Cauchy Schwarz ineq. f 2 } {{ } D Therefore, y and z are synchronous as required. Now, let us note that (3.18) is equivalent to g 0y, f ν g 0z, f ν 0, ν. C. By Proposition 3.8, g 0y and g 0z are G-synchronous w.r.t. C. Hence there exists a g G such that gg 0y, gg 0 C. Since gg 0 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. Page 18 of 31

19 4. Applications to the Chebyshev Sum Inequality Throughout this section, V = R n with the standard inner product,. Let {s i } be the basis of R n, where s i = (1,..., 1, 0,..., 0), i = 1,..., n. The symbols }{{} i V 1 and V 2 stand for the subspace orthogonal to s n and its orthogonal complement, respectively, i.e. { } V 1 = (x 1,..., x n ) : i x i = 0, V 2 = span{s n }. Let P be the orthoprojector onto V 1, i.e. P = id,sn s n n. 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 = (x 1,..., x n ) : x 1 x 2 x n } is generated by {s 1,..., s n, s n }. On the other side, is a generator of { dual C = {(1, 1, 0,..., 0), (0, 1, 1, 0,..., 0), (0,..., 0, 1, 1)} x = (x 1,..., x n ) : n x i = 0, i=1 Set e i = np s i, i = 1,..., n 1. Clearly, } k x i 0, k = 1,..., n 1. i=1 Page 19 of 31 (4.1) e i = ns i is n = (n i,..., n i, i,..., i), i = 1,..., n 1. }{{}}{{} i n i

20 Write D = cone{e i }. Clearly, P C = D and V 2 C. Hence by (3.1), C = D + V 2. Applying Proposition 3.2, we conclude that (PI) holds for y, z (D+V 2 ) 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 {e i, i = 1,..., n 1} constitutes a basis of V 1. Observe that e i, e j = 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: { q1 = e 1, (4.2) ( q k+1 = n k ek+1 n k 1e n n k k), 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, n k, 1,..., 1), k = 1,..., n 1. }{{}}{{} k 1 n k Page 20 of 31

21 Let us denote K = D + V 2, where D stands for the cone{q k }. The convex cone D is self-dual w.r.t. V 1. According to Proposition 3.3 we can assert that (CHSI) holds for y, z K. Let g 0 (x 1,..., x n ) = ( x n,..., x 1 ). Clearly, g 0 G P. By Remark 1, (CHSI) holds for y, z g 0 K. Have: g 0 K = g 0 ( D + V 2 ) = g 0 D + V2 = cone{g 0 q k } + V 2. Define f k = g 0 q n k, k = 1,..., n 1. Since g 0 G P, g 0 is unitary and g 0 V 1 = V 1, by (3.3). Hence, {f k } is an orthogonal basis of V 1. Observe (4.4) f k = (1,..., 1, k, 0,..., 0), }{{} k = 1,..., n 1. k Write M = cone{f k } + V 2. By Remark 1, it is evident that (CHSI) holds for y, z M. Proposition 4.1. For x = (x 1,..., x n ) R n { } n 1 n x K the sequence x i is nonincreasing, n k + 1 i=k k=1 { } n 1 k x M the sequence x i is nonincreasing. k i=1 k=1 Proof. We prove only the first equivalence. The second one uses a similar procedure. Page 21 of 31

22 By (3.2), K = dual D, because D is self-dual w.r.t. V 1. Hence by (4.3) we can assert that x K is equivalent to n (n k)x k x i, k = 1,..., n 1. i=k+1 Adding to both of sides (n k) n i=k+1 x i and dividing by (n k)(n k + 1), we obtain ( n ) ( n ) 1 x i 1 x i, k = 1,..., n 1. n k + 1 n k 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 g 0 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 G P 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 G P, 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 = (x 1,..., x n ), y = (y 1,..., y n ) R n are said to be similarly ordered if (4.5) (x i x j )(y i y j ) 0, i,j. Page 22 of 31

23 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{f k }+V 2 where {f k } is an arbitrarily chosen orthogonal basis of V 1. Moreover, G-synchronicity gives (CHSI), where G is the group (3.8) acting on R n. For this reason, the specification of Theorem 3.9 can be as follows. Let {f k } 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 y, Uf k z, Uf k 0, k = 1,..., n 1, where U is a fixed unitary operation with Us n = s n or Us n = s n, 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 = (y 1,..., y n ) and z = (z 1,..., z n ) are G-synchronous w.r.t. M if and only if [ k ] [ k ] y i ky k+1 z i kz k+1 0, k = 1,..., n 1. i=1 i=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. Consider R n, n > 3. Page 23 of 31

24 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 = f 1 + f 2, z = f 2 + f 3, where f i are defined by (4.4). The vectors y and z are G-synchronous w.r.t. M, because y, z M, so (CHSI) holds. On the other hand y = (2, 0, 2, 0,..., 0), z = (2, 2, 1, 3, 0,..., 0) are not similarly ordered by (4.5), because (y 3 y 4)(z 3 z 4) = 2( 1 + 3) < 0. Page 24 of 31

25 5. Applications to the Chebyshev Integral Inequality Set V = L 2 as in Example 3.2. The characteristic function of the measurable set A [0, 1] is denoted by I A. Additionally we will write e s = I [0,s], 0 s 1. The symbol V 1 stands for the subspace orthogonal to V 2 = span{e 1 }, i.e. V 1 = { x L 2 : xdµ = 0 }. By Example 3.2, it is known that for the orthoprojector P onto V 1 (PI) transforms into the Chebyshev integral inequality (CHII). Let C L 2 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 ({e s : 0 s 1} { e 1 }), dual C = cone{i Π I Π+ε : ε > 0, Π, Π + ε [0, 1]}, where Π stands for an interval. The Haar system: (5.2) χ 0 0 = e 1 2 n/2 2k 2, t < 2k 1 2 n+1 2 n+1 χ k n(t) = 2 n/2, 0, otherwise n = 0, 1,..., 2k 1 t < 2k 2 n+1 2 n+1 k = 1, 2,..., 2 n forms an orthonormal basis of L 2. In particular, H = {χ k n : n = 0, 1,..., k = 1,..., 2 n } is an orthonormal basis of V 1. Let D = cone H. Page 25 of 31

26 The cone D is self-dual w.r.t. V 1, so by (3.2) we have: (5.3) dual D = D + V 2. 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 + V 2. From (5.4) and Corollary 3.6 it follows that Corollary 5.1. (CHII) holds for y, z D + V 2. The cone D + V 2 contains the cone of all nonincreasing µ a.e. functions in L 2. It is easily seen that the cone D +V 2 contains functions which are not nonincreasing µ a.e. Let G be the group (3.8) acting on L 2 with the Haar system. Employing the G-synchronicity relation w.r.t. D + V 2, by Theorem 3.9 we get: Corollary 5.2. (CHII) holds for y, z L 2 if only (5.5) y, χ z, χ 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. Page 26 of 31 (5.6) [y(s) y(t)] [z(s) z(t)]] 0, for all 0 s, t 1 (see e.g. [6, pp ]). 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 L 2.

27 Example 5.1. In L 2 let y = χ χ 2 2, z = χ χ 3 2, where χ j i are defined by (5.2). The vectors y and z are G-synchronous w.r.t. D + V 2, because they are in D. On the other hand y(s) = 2, 0 s 1 8 y(t) = 0, 4 8 t 5 8, z(s) = 0, 0 s 1 8 z(t) = 2, 4 8 t 5 8 From this, [y(s) y(t)] [z(s) z(t)]] = [2 0][0 2] 0 for any 0 s 1 and 8 4 t 5. Thus y and z are not similarly ordered. 8 8 Now, we recall that a function y L 2 is nonincreasing (nondecreasing, monotone) in mean if the function s 1 s ydµ, is nonincreasing (nondecreasing, monotone). s 0 Differentiating 1 s ydµ we can easy obtain that y is nonincreasing in mean if s 0 and only if 1 s 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 ]). Johnson in [2] gave a more general condition. Namely, if [ 1 s ] [ 1 s ] (5.7) ydµ y(s) zdµ z(s) 0, 0<s<1 s s then (CHII) holds for y and z. Remark 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 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).. Page 27 of 31

28 Proof. An easy verification shows that: Ad. 1) χ k n 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 f, χ 2 1 < 0. Ad. 3) Set y = χ 2 1, z = χ 3 2. For 5 < s < 6 have: 8 8 [ 1 s s 0 ] [ 1 ydµ y(s) s s 0 ] 2/2 zdµ z(s) = s 3/2 s < 0. The set of all L 2 -functions nonincreasing in mean constitutes a convex cone. It will be denoted by M. Let M 0 be the class of all step functions of the form Proposition 5.3. g s,t = I [0,s) s t s I [s,t], 0 < s < t < 1. M = dual M 0, P M = dual V1 M 0 = cone M 0, M = cone M 0 + V 2. Proof. By definition, f M if and only if 1 s s 0 fdµ 1 t t After equivalent transformations we obtain s 0 fdµ s t s 0 t s fdµ for all 0 < s < t < 1. fdµ for all 0 < s < t < 1. Page 28 of 31

29 This is simply f dual M 0, so the first equation holds. To show the second equation, note that M 0 M V 1. Hence dual(m V 1 ) dual M 0 = M, by the first equation. It follows that V 1 dual(m V 1 ) M V 1, i.e. (5.8) dual V1 (M V 1 ) M V 1. Fix f M V 1 and let g M V 1 be arbitrary. For such f and g (CHII) holds and takes the form: fgdµ fdµ gdµ = 0 0 = 0, i.e. f dual V1 (M V 1 ). Therefore 0 (5.9) M V 1 dual V1 (M V 1 ). 0 Since M = dual M 0, dual M = cone M 0. Now, observe that V 2 M. This implies by (3.1) that M = P M + V 2. Furthermore, in this situation P M = M V 1. The above gives Hence dual M = dual(m V 1 + V 2 ) = V 1 dual(m V 1 ) = dual V1 (M V 1 ). (5.10) dual V1 (M V 1 ) = cone M 0. 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. 0 Page 29 of 31

30 The second equation of the above propositions immediately gives: Remark 5. The convex cone of all L 2 -functions nonincreasing in mean with integral equal to 0 is self-dual w.r.t. V 1. Taking C = M in Theorem 3.1, by Proposition 5.3 we easily obtain: Corollary 5.4. If fdµ gdµ fgdµ holds for all functions f L 2 monotone in mean, then g L 2 is also monotone in mean. Page 30 of 31

31 References [1] M. BIERNACKI, Sur une inégalité entre les intégrales due á Tchébyscheff, Ann. Univ. Mariae Curie-Skłodowska, A5 (1951), [2] R. JOHNSON, Chebyshev s inequality for functions whose averages are monotone, J. Math. Anal. Appl., 172 (1993), [3] M. NIEZGODA, On the Chebyshev functional, Math. Inequal. Appl., 10(3) (2007), [4] M. NIEZGODA, Bifractional inequalities and convex cones, Discrete Math., 306(2) (2006), [5] Z. OTACHEL, Spectral orders and isotone functionals, Linear Algebra Appl., 252 (1997), [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) Page 31 of 31