Bounds on weighted discreteˇCebyšev functional - ProbStat Forum

ProbStat Forum, Volume 06, January 2013, Pages 1–9 ISSN 0974-3235

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ˇ Bounds on weighted discrete Cebyˇ sev functional Dragana Jankova , Tibor K. Pog´ anyb a University b University

of Osijek, Department of Mathematics, Osijek, Croatia of Rijeka, Faculty of Maritime Studies, Rijeka, Croatia

ˇ Abstract. Two sets of inequalities are derived for the weighted discrete Cebyˇ sev functional, considering the probabilistic setting and also the Lipschitz class of functions.

1. Introduction and motivation ˇ The weighted discrete Cebyˇ sev functional is defined as the sum Tp (a, b) :=

n X

pi ai bi −

i=1

n X

pi ai

i=1

n X

pi bi ,

(1)

i=1

n where and the weight–vector p = (p1 , . . . , pn ) ∈ Rn+ , satisfying Pn a = (a1 , . . . , an ), b = (b1 , . . . , bn ) ∈ R −1 by pj , j = 1, n in the the celebrated Korkin’s identity i=1 pi = 1. Replacing the uniform weights n (Korkin, 1882, 1883), (Mitrinovi´c and Vasi´c, 1974, pp. 6–7) n n n 1 X  1 X  1X 1 aj bj = aj bj + 2 n j=1 n j=1 n j=1 n

X

(aj − ak )(bj − bk ) ,

1≤j 1, by using weighted H¨older inequality, again by the Korkin’s identity, we have n X Tp (ξ, η) ≤ 1 pj |xj − xk | pk |yj − yk | 2 j,k=1 !1/r !1/s n n X 1 X ≤ {pj |xj − xk |}r {pk |yj − yk |}s 2 j,k=1 j,k=1 !1/r !1/s n n X 1 X r−1 s−1 r s pj |xj − xk | pk |yj − yk | ≤ 2 j,k=1

(7)

j,k=1

and also we have n X Tp (ξ, η) ≤ 1 pj pk |xj − xk | |yj − yk | 2 j,k=1 !1/r !1/s n n X 1 X r s ≤ pj pk |xj − xk | pj pk |yj − yk | 2 j,k=1 j,k=1 !1/r !1/s n n X 1 X r s ≤ pj |xj − xk | pk |yj − yk | . 2 j,k=1

(8)

j,k=1

Now, from (7) and (8), we get immediately the asserted constant M . Now, let us recall the following result, which will be used in the sequel. Lemma 2.4 (Weighted P´ olya–Szeg˝ o type inequality for sums). (Pog´any, 2005, p. 118) Assume that 0 < m1 ≤ xj ≤ M1 , j = 1, n, 0 < m2 ≤ yk ≤ M2 , Pn Pm Also, let pij ≥ 0 and i=1 j=1 pij = 1. Then n X

qj x2j

j=1

m X k=1

1 rk yk2 ≤ 4

r

m1 m2 + M1 M2

r

M1 M 2 m1 m2

!2

k = 1, m .

n X m X

!2 pjk xj yk

j=1 k=1

where m X

pjk = qj ,

k=1

j = 1, n,

n X

pjk = rk ,

k = 1, m .

j=1

Inequality (9) becomes an equality if and only if m = n and κ=

M1 m2 n ∈ N0 , m1 M2 + M1 m2

xl1 = · · · xlκ = m1 , xlκ+1 = · · · = xln = M1 , yl1 = · · · ylκ = M2 , ylκ+1 = · · · = yln = m2 , lj ∈ {1, . . . , n}, pjk = 0, (j, k) ∈ {l1 , . . . , lκ } × {lκ+1 , . . . , ln } ∪ {lκ+1 , . . . , ln } × {l1 , . . . , lκ }. Now, we can state the next result.

(9)

Jankov and Pog´ any / ProbStat Forum, Volume 06, January 2013, Pages 1–9

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Theorem 2.5. Let r.v.s ξ, η be as defined above. Then, we have ( Tp (ξ, η)
0 on Ξ, if there exists an absolute constant L > 0 such that |f (x) − f (y)| ≤ L|x − y|α

x, y ∈ Ξ .

Here L is the Lipschitz constant and the class consisting of such functions we write LipL (α). Theorem 3.1. Let r, s, r−1 + s−1 = 1,P r > 1, be conjugated H¨ older exponents and let a = (a1 , . . . , an ), b = n (b1 , . . . , bn ) ∈ Rn and p ≥ 0 such that j=1 pj = 1. Also, assume that f ∈ LipLf (αf ) and g ∈ LipLg (αg ). Then |Tp (f (a), g(b))| ≤ Lf Lg |an − a1 |αf |bn − b1 |αg min {1, M1 } , where f (x) := (f (x1 ), · · · , f (xn )), x ∈ {a, b} and  M1 = n 

n X

1/r  1/s n X  prj  psj  .

j=1

(12)

j=1

Proof. As r, s, r > 1 are conjugated exponents, by virtue of the discrete weighted H¨older inequality from (2) it holds: n 1 X {pj pk }1/r+1/s |f (aj ) − f (ak )| |g(bj ) − g(bk )| 2 j,k=1 1/s  1/r  n n X X 1 pj pk |g(bj ) − g(bi )|s  . pj pk |f (aj ) − f (ak )|r   ≤  2

|Tp (f (a), g(b))| ≤

(13)

j,k=1

j,k=1

Because of f ∈ LipLf (αf ), g ∈ LipLg (αg ), having in mind that coordinates of a, b can be taken increasing without loss of generality, we estimate the right–hand–side of (13) in the following way:  1/r  1/s n n X Lf Lg  X |Tp (f (a), g(b))| ≤ pj pk |aj − ak |αf r   pj pk |bj − bk |αg s  2 j,k=1

j,k=1

 = Lf Lg |an − a1 |αf |bn − b1 |αg 

n X

1/r+1/s pj pk 

j,k=1

= Lf Lg |an − a1 |αf |bn − b1 |αg . Now, it is left to derive M1 . Again, by using discrete H¨older inequality with the same pair of conjugated

Jankov and Pog´ any / ProbStat Forum, Volume 06, January 2013, Pages 1–9

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parameters r, s and the definition of Lipschitz functions f and g we get: n 1 X {pj |f (aj ) − f (ak )|} · {pk |g(bj ) − g(bk )|} 2 j,k=1  1/r  1/s n n X X 1 r s ≤  prj |f (aj ) − f (ak )|   psk |g(bj ) − g(bk )|  2

|Tp (f (a), g(b))| ≤

j,k=1

j,k=1

 ≤ Lf Lg |an − a1 |αf |bn − b1 |αg 

n X

j,k=1

1/r  prj 

n X



1/s psk 

j,k=1

 1/r  1/s n n X X  = Lf Lg |an − a1 |αf |bn − b1 |αg n  prj  psj  j=1

j=1

which completes the proof of the Theorem. Now, we discuss certain results which follow by the previous theorem. First, we would need the well– known fact that every Lipschitz function f : Ξ 7→ Υ is uniformly continuous, i.e. for every  > 0 there is a δ > 0 such that |f (x) − f (y)| <  for all points x, y ∈ Ξ such that |x − y| < δ. Corollary 3.2. Let a = (a1 , . . . , an ), b = (b1 , . . . , bn ) be two n–tuples coming from the metric space Ξ and let f, g : Ξ 7→ Υ be Lipschitz functions, f ∈ LipLf (αf ) and g ∈ LipLg (αg ). Then Tp (f (a), f (b)) is uniformly continuous. Proof. Desired result follows directly by Theorem 3.1, and uniform continuity of the functions f, g. It is enough to observe that for all max{|an − a1 |, |bn − b1 |} < δ it follows that |Tp (f (a), g(b))| < Lf Lg δ αf +αg min{1, M1 } =:  , where M1 is given with (12). Corollary 3.3. If f, g : Ξ 7→ Υ are injective Lipschitz functions between metric spaces, f ∈ LipLf (αf ) and g ∈ LipLg (αg ), corresponding metric d is discrete and a = (a1 , . . . , an ), b = (b1 , . . . , bn ) are two n–tuples from Ξ, then |Tp (f (a), g(b))| ≤ Lf Lg min{1, M1 } ,

(14)

where M1 is given with (12). Proof. Assertion (14) follows from Theorem 3.1 and definition of a discrete metric: ( 1, if x 6= y d(x, y) = . 0, if x = y So the asserted result. References Bogn´ ar, J. Mrs., Mogyor´ odi, J., Pr´ ekopa, A., R´ enyi, A., Sz´ asz, D. (2001) Probability Theory Problem Collection. Second edition. Typotex, Budapest. (In Hungarian). ˇ Cebyˇ sev, P. L. (1882) O pribliˇ zennyh vyraˇ zenijah odnih integralov ˇ cerez drugie. Soobˇs´ cenija i protokoly zasedani˘ı Matemmatiˇ ceskogo obˇ cestva pri Imperatorskom Har’kovskom Universitete No. 2, 93–98; (1948) Polnoe sobranie soˇ cineni˘ı P. L. ˇ Cebyˇ seva (1948), Moskva–Leningrad, 128–131.

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