Some new bounds for Čebyšev functional for sequences of vectors in normed linear spaces are pointed out

Dragomir, Sever S

English

Victoria University Eprints Repository

BOUNDING THE CˇEBYSˇEV FUNCTIONAL FOR SEQUENCESOF VECTORS IN NORMED LINEAR SPACESS.S. DRAGOMIRAbstract. Some new bounds for Cˇebysˇev functional for sequences of vectorsin normed linear spaces are pointed out.1. IntroductionConsider the Cˇebysˇev functional defined for p = (p1, ..., pn) ∈ Rn,α = (α1, ..., αn) ∈Kn (K = R or C) and x = (x1, ..., xn) ∈ Xn, where X is a linear space over the realor complex number field K:(1.1) Tn (p;α,x) := Pnn∑i=1piαixi −n∑i=1piαi ·n∑i=1pixi,where Pn :=∑ni=1 pi.The following Gru¨ss type inequalities for sequences in normed linear spaces hold.Theorem 1. Let (X, ‖.‖) be a normed linear space over the real or complex numberfield K, α = (α1, ..., αn) ∈ Kn,p = (p1, ..., pn) ∈ Rn+ with∑ni=1 pi = 1 and x =(x1, ..., xn) ∈ Xn. Then one has the inequalities‖Tn (p;α,x)‖(1.2)≤[∑ni=1 i2pi − (∑ni=1 ipi)2]max1≤j≤n−1 |∆αj |max1≤j≤n−1 ‖∆xj‖ , [1];12∑ni=1 pi (1− pi)∑n−1j=1 |∆αj |∑n−1j=1 ‖∆xj‖ , [3];∑1≤i<j≤n pipj (j − i)(∑n−1j=1 |∆αj |p)1/p (∑n−1j=1 ‖∆xj‖q)1/q,p > 1, 1p +1q = 1, [2].The constant 1 in the first branch, 12 in the second branch and 1 in the third branchare best possible in the sense that they cannot be replaced by smaller constants.The following particular inequalities for unweighted means hold as well, whereTn (α,x) is defined as follows:Tn (α,x) :=1nn∑i=1αixi − 1nn∑i=1ai · 1nn∑i=1xi.Date: March 13, 2003.1991 Mathematics Subject Classification. Primary 26D15; Secondary 26D10.Key words and phrases. Cˇebysˇev functional, Gru¨ss Type Inequalities .12 S.S. DRAGOMIRCorollary 1. With the assumptions of Theorem 1 for X,α and x, we have‖Tn (α,x)‖(1.3)≤112(n2 − 1)max1≤j≤n−1 |∆αj |max1≤j≤n−1 ‖∆xj‖ , [1];12 ·(1− 1n)∑n−1j=1 |∆αj |∑n−1j=1 ‖∆xj‖ , [3];16n2−1n(∑n−1j=1 |∆αj |p)1/p (∑n−1j=1 ‖∆xj‖q)1/q,p > 1, 1p +1q = 1, [2].Here the constants 112 ,12 and16 are best possible in the sense that they cannot bereplaced by smaller constants.For applications to estimate the p-moments of guessing mappings, see [1]. Forapplications in approximating the discrete Fourier transform, the discrete Mellintransform as well as some applications for polynomials and Lipschitzian mappings,see [2] and [3].For classical results related the Cˇebysˇev functional, see [4], [5], [6], [7], [8], [10]and [12]. For more recent results, see [12], [13], [14], [15], [9] and [11].2. The IdentitiesThe first result is embodied in the followingTheorem 2. Let p = (p1, ..., pn) ,a = (a1, ..., an) be n-tuples of real or complexnumbers and x = (x1, ..., xn) an n-tuple of vectors in the linear space X. If wedefinePi : =i∑k=1pk, P¯i := Pn − Pi, i ∈ {1, ..., n− 1} ,Ai (p) : =i∑k=1pkak, A¯i (p) := An (p)−Ai (p) , i ∈ {1, ..., n− 1} ,then we have the identityTn (p;a,x) =n−1∑i=1det(Pi PnAi (p) An (p))·∆xi(2.1)= Pnn−1∑i=1Pi(An (p)Pn− Ai (p)Pi)·∆xi (if Pi 6= 0, i ∈ {1, ..., n})=n−1∑i=1PiP¯i(A¯i (p)P¯i− Ai (p)Pi)·∆xi(if Pi, P¯i 6= 0, i ∈ {1, ..., n− 1});where ∆xi := xi+1 − xi (i ∈ {1, ..., n− 1}) is the forward difference.Proof. We use the following well known summation by parts formula(2.2)q−1∑l=pdl∆vl = dlvl|qp −q−1∑l=pvl+1∆dl,CˇEBYSˇEV FUNCTIONAL 3where dl are real or complex numbers, and vl are vectors in a linear space, l = p, ..., q(q > p; p, q are natural numbers) .If we choose in (2.2), p = 1, q = n, di = PiAn (p) − PnAi (p) and vi = xi(i ∈ {1, ..., n− 1}) , then we getn−1∑i=1(PiAn (p)− PnAi (p)) ·∆xi= [PiAn (p)− PnAi (p)] · xi|n1 −n−1∑i=1∆(PiAn (p)− PnAi (p)) · xi+1= [PnAn (p)− PnAn (p)] · xn − [P1An (p)− PnA1 (p)] · x1−n−1∑i=1[Pi+1An (p)− PnAi+1 (p)− PiAn (p) + PnAi (p)] · xi+1= Pnp1a1x1 − p1An (p)x1 −n−1∑i=1(pi+1An (p)− Pnpi+1ai+1) · xi+1= Pnp1a1x1 − p1An (p)x1 −An (p)n−1∑i=1pi+1xi+1 + Pnn−1∑i=1pi+1ai+1xi+1= Pnn∑i=1piaixi −n∑i=1piai ·n∑i=1pixi= Tn (p;a,x) ,which produce the first identity in (2.1) .The second and the third identities are obvious and we omit the details. Before we prove the second result, we need the following lemma providing anidentity that is interesting in itself as well.Lemma 1. Let p = (p1, ..., pn) and a = (a1, ..., an) be n-tuples of real or complexnumbers. Then we have the equality(2.3) det(Pi PnAi (p) An (p))=n−1∑j=1Pmin{i,j}P¯max{i,j} ·∆aj ,for each i ∈ {1, ..., n− 1} .Proof. Define, for i ∈ {1, ..., n− 1} ,K (i) :=n−1∑j=1Pmin{i,j}P¯max{i,j} ·∆aj .4 S.S. DRAGOMIRWe haveK (i) =i∑j=1Pmin{i,j}P¯max{i,j} ·∆aj +n−1∑j=i+1Pmin{i,j}P¯max{i,j} ·∆aj(2.4)=i∑j=1PjP¯i ·∆aj +n−1∑j=i+1PiP¯j ·∆aj= P¯ii∑j=1Pj ·∆aj + Pin−1∑j=i+1P¯j ·∆aj .Using the summation by parts formula, we havei∑j=1Pj ·∆aj = Pj · aj |i+11 −i∑j=1(Pj+1 − Pj) · aj+1(2.5)= Pi+1ai+1 − p1a1 −i∑j=1pj+1 · aj+1= Pi+1ai+1 −i+1∑j=1pj · ajandn−1∑j=i+1P¯j ·∆aj = P¯j · aj∣∣ni+1−n−1∑j=i+1(P¯j+1 − P¯j) · aj+1(2.6)= P¯nan − P¯i+1ai+1 −n−1∑j=i+1(Pn − Pj+1 − Pn + Pj) · aj+1= −P¯i+1ai+1 +n−1∑j=i+1pj+1 · aj+1.Using (2.5) and (2.6) we haveK (i) = P¯iPi+1ai+1 − i+1∑j=1pj · aj+ Pi n−1∑j=i+1pj+1 · aj+1 − P¯i+1ai+1= P¯iPi+1ai+1 − PiP¯i+1ai+1 − P¯ii+1∑j=1pj · aj + Pin−1∑j=i+1pj+1 · aj+1= [(Pn − Pi)Pi+1 − Pi (Pn − Pi+1)] ai+1+Pin−1∑j=i+1pj+1 · aj+1 − P¯ii+1∑j=1pj · ajCˇEBYSˇEV FUNCTIONAL 5= Pnpi+1ai+1 + Pin−1∑j=i+1pj+1 · aj+1 − P¯ii+1∑j=1pj · aj=(Pi + P¯i)pi+1ai+1 + Pin−1∑j=i+1pj+1 · aj+1 − P¯ii+1∑j=1pj · aj= Pin−1∑j=i+1pj · aj − P¯ii∑j=1pj · aj = PiA¯i (p)− P¯iAi (p)= det(Pi PnAi (p) An (p));and the identity is proved. We are able now to state and prove the second identity for the Cˇebysˇev functionalTheorem 3. With the assumptions of Theorem 2, we have the equality(2.7) Tn (p;a,x) =n−1∑i=1n−1∑j=1Pmin{i,j}P¯max{i,j} ·∆aj ·∆xi.The proof is obvious by Theorem 2 and Lemma 1.Remark 1. The identity (2.7), for n-tuples of real numbers, was stated without aproof in paper [12]. It also may be found for the same sequences in [9, p. 281],again without a proof. In the second place mentioned above there is a misprint forthe index of P¯ which, instead of max {i, j}+ 1, should be max {i, j}.3. Some New InequalitiesThe following result holdsTheorem 4. Let (X, ‖.‖) be a normed linear space over the real or complex numberfield K, a = (a1, ..., an) ∈ Kn,p = (p1, ..., pn) ∈ Rn and x = (x1, ..., xn) ∈ Xn. Thenone has the inequalities‖Tn (p;a,x)‖(3.1)≤max1≤i≤n−1∣∣∣∣det( Pi PnAi (p) An (p))∣∣∣∣ ·∑n−1j=1 ‖∆xj‖ ;(∑n−1i=1∣∣∣∣det( Pi PnAi (p) An (p))∣∣∣∣q)1/q · (∑n−1j=1 ‖∆xj‖p)1/pfor p > 1, 1p +1q = 1;∑n−1i=1∣∣∣∣det( Pi PnAi (p) An (p))∣∣∣∣ ·max1≤j≤n−1 ‖∆xj‖ .All the inequalities in (3.1) are sharp in the sense that the constants 1 cannot bereplaced by smaller constants.Proof. Using the first identity in (2.1), we have‖Tn (p;a,x)‖ ≤n∑i=1∣∣∣∣det( Pi PnAi (p) An (p))∣∣∣∣ ‖∆xi‖ .6 S.S. DRAGOMIRUsing Ho¨lder’s inequality, we deduce the desired result (3.1) .Let prove, for instance, that the constant 1 in the second inequality is bestpossible.Assume, for C > 0, we have that(3.2) ‖Tn (p;a,x)‖ ≤ C(n−1∑i=1∣∣∣∣det( Pi PnAi (p) An (p))∣∣∣∣q)1/qn−1∑j=1‖∆xj‖p1/pfor p > 1, 1p +1q = 1, n ≥ 2.If we choose n = 2, then we getT2 (p;a,x) = p1p2 (a2 − a1) (x2 − x1) .Also, for n = 2,(n−1∑i=1∣∣∣∣det( Pi PnAi (p) An (p))∣∣∣∣q)1/q= |p1p2| |a2 − a1|and n−1∑j=1‖∆xj‖p1/p = ‖x2 − x1‖ .Then by (3.2), holding for n = 2, p1, p2 > 0, a1 6= a2, x2 6= x1, we deduce C ≥ 1,proving that 1 is the best possible constant in that inequality. The following corollary for the uniform distribution of the probability p holds.Corollary 2. With the assumptions of Theorem 4 for a and x, we have the in-equalities‖Tn (a,x)‖≤ 1n2×max1≤i≤n−1∣∣∣∣det( i n∑ik=1 ak∑nk=1 ak)∣∣∣∣ ·∑n−1j=1 ‖∆xj‖ ;(∑n−1i=1∣∣∣∣det( i n∑ik=1 ak∑nk=1 ak)∣∣∣∣q)1/q · (∑n−1j=1 ‖∆xj‖p)1/pfor p > 1, 1p +1q = 1;∑n−1i=1∣∣∣∣det( i n∑ik=1 ak∑nk=1 ak)∣∣∣∣ ·max1≤j≤n−1 ‖∆xj‖ .The following result may be stated as well.CˇEBYSˇEV FUNCTIONAL 7Theorem 5. With the assumptions of Theorem 4 and if Pi 6= 0 (i = 1, ..., n) , thenwe have the inequalities‖Tn (p;a,x)‖(3.3)≤ |Pn| ×max1≤i≤n−1∣∣∣An(p)Pn − Ai(p)Pi ∣∣∣ ·∑n−1i=1 |Pi| ‖∆xi‖ ;(∑n−1i=1 |Pi|∣∣∣An(p)Pn − Ai(p)Pi ∣∣∣q)1/q · (∑n−1i=1 |Pi| ‖∆xi‖)1/pfor p > 1, 1p +1q = 1;∑n−1i=1 |Pi|∣∣∣An(p)Pn − Ai(p)Pi ∣∣∣ ·max1≤i≤n−1 ‖∆xi‖ .All the inequalities in (3.3) are sharp in the sense that the constant 1 cannot bereplaced by a smaller constant.Proof. Follows by the second identity in (2.1) and taking into account that‖Tn (p;a,x)‖ ≤ |Pn|n−1∑i=1∣∣∣∣An (p)Pn − Ai (p)Pi∣∣∣∣ · |Pi| ‖∆xi‖ .Using Ho¨lder’s weighted inequality, we easily deduce (3.3) .The sharpness of the constant may be shown in a similar manner. We omit thedetails. The following corollary containing the unweighted inequalities holds.Corollary 3. With the above assumptions for a and x one, has‖Tn (a,x)‖(3.4)≤ 1n×max1≤i≤n−1∣∣∣ 1n∑nk=1 ak − 1i ∑ik=1 ak∣∣∣ ·∑n−1i=1 i ‖∆xi‖ ;(∑n−1i=1 i∣∣∣ 1n∑nk=1 ak − 1i ∑ik=1 ak∣∣∣q)1/q · (∑n−1i=1 i ‖∆xi‖p)1/pfor p > 1, 1p +1q = 1;∑n−1i=1 i∣∣∣ 1n∑nk=1 ak − 1i ∑ik=1 ak∣∣∣ ·max1≤i≤n−1 ‖∆xi‖ .The inequalities in (3.4) are sharp in the sense mentioned above.Another type of inequalities may be stated if one uses the third identity in (2.1).8 S.S. DRAGOMIRTheorem 6. With the assumptions in Theorem 4 and if Pi, P i 6= 0, i ∈ {1, ..., n− 1} ,then we have the inequalities‖Tn (p;a,x)‖(3.5)≤max1≤i≤n−1∣∣∣Ai(p)P i− Ai(p)Pi∣∣∣ ·∑n−1i=1 |Pi| ∣∣P i∣∣ ‖∆xi‖ ;(∑n−1i=1 |Pi|∣∣P i∣∣ ∣∣∣Ai(p)P i − Ai(p)Pi ∣∣∣q)1/q · (∑n−1i=1 |Pi| ∣∣P i∣∣ ‖∆xi‖p)1/pfor p > 1, 1p +1q = 1;∑n−1i=1 |Pi|∣∣P i∣∣ ∣∣∣Ai(p)P i − Ai(p)Pi ∣∣∣ ·max1≤i≤n−1 ‖∆xi‖ .In particular, if pi = 1n , i ∈ {1, ..., n} , then we have‖Tn (a,x)‖(3.6)≤ 1n2·max1≤i≤n−1∣∣∣ 1n−i∑nk=i+1 ak − 1i ∑ik=1 ak∣∣∣ ·∑n−1i=1 i (n− i) ‖∆xi‖ ;(∑n−1i=1 i (n− i)∣∣∣ 1n−i∑nk=i+1 ak − 1i ∑ik=1 ak∣∣∣q)1/q×(∑n−1i=1 i (n− i) ‖∆xi‖p)1/pfor p > 1, 1p +1q = 1;∑n−1i=1 i (n− i)∣∣∣ 1n−i∑nk=i+1 ak − 1i ∑ik=1 ak∣∣∣ ·max1≤i≤n−1 ‖∆xi‖ .The inequalities in (3.5)and (3.6) are sharp in the above mentioned sense.A different approach may be considered if one uses the representation in termsof double sums for the Cˇebysˇev functional provided by the Theorem 3.The following result holds.Theorem 7. With the assumptions in Theorem 4, we have the inequalities‖Tn (p;a,x)‖(3.7)≤max1≤i,j≤n−1{∣∣Pmin{i,j}∣∣ ∣∣P¯max{i,j}∣∣} ·∑n−1i=1 |∆ai|∑n−1i=1 ‖∆xi‖ ;(∑n−1i=1∑n−1j=1∣∣Pmin{i,j}∣∣q ∣∣P¯max{i,j}∣∣q)1/q×(∑n−1i=1 |∆ai|p)1/p (∑n−1i=1 ‖∆xi‖p)1/pfor p > 1, 1p +1q = 1;∑n−1i=1∑n−1j=1∣∣Pmin{i,j}∣∣ ∣∣P¯max{i,j}∣∣×max1≤i≤n−1 |∆ai|max1≤i≤n−1 ‖∆xi‖ .The inequalities are sharp in the sense mentioned above.The proof follows by the identity (2.7) on using Ho¨lder’s inequality for doublesums and we omit the details.CˇEBYSˇEV FUNCTIONAL 9Now, definek∞ := max1≤i,j≤n−1{min {i, j}n(1− max {i, j}n)}, n ≥ 2.Using the elementary inequalityab ≤ 14(a+ b)2 , a, b ∈ R;we deducemin {i, j} · (n−max {i, j}) ≤ 14(n+min {i, j} −max {i, j})2=14(n− |i− j|)2 , 1 ≤ i, j ≤ n− 1.Consequently, we observe thatk∞ ≤ 14n2 max1≤i,j≤n−1{(n− |i− j|)2}=14.We may state now the following corollary of Theorem 7.Corollary 4. Let (X, ‖.‖) be a normed linear space, a = (a1, ..., an) ∈ Kn andx = (x1, ..., xn) ∈ Xn. Then we have the inequality(3.8) ‖Tn (a,x)‖ ≤ k∞n−1∑i=1|∆ai|n−1∑i=1‖∆xi‖ ≤ 14n−1∑i=1|∆ai|n−1∑i=1‖∆xi‖ .The constant 14 cannot be replaced in general by a smaller constant.Remark 2. The inequality (3.8) is better than the second inequality in Corollary1.Consider now, for q > 1, the numberkq :=1n2n−1∑i=1n−1∑j=1[min {i, j} · (n−max {i, j})]q1/q .We observe, by the symmetry of the terms under the sums symbol, we have thatkq =1n22 ∑1≤i<j≤n−1iq (n− j)q +n−1∑i=1iq (n− i)q1/q ,that may be computed exactly if q = 2 or another natural number.Since, as above,[min {i, j} · (n−max {i, j})]q ≤ 14q(n− |i− j|)2qwe deducekq ≤ 14n2n−1∑i=1n−1∑j=1(n− |i− j|)2q1/q≤ 14n2[(n− 1)2 n2q]1/q=14(n− 1)2/q .Consequently, we may state the following corollary as well.10 S.S. DRAGOMIRCorollary 5. With the assumption in Corollary 4, we have the inequalities‖Tn (a,x)‖ ≤ kq(n−1∑i=1|∆ai|p)1/p(n−1∑i=1‖∆xi‖p)1/p≤ 14(n− 1)2/q(n−1∑i=1|∆ai|p)1/p(n−1∑i=1‖∆xi‖p)1/p;provided p > 1, 1p+1q = 1. The constant14 cannot be replaced in general by a smallerconstant.Finally, if we denotek1 :=1n2n−1∑i=1n−1∑j=1[min {i, j} · (n−max {i, j})] ,then we observe, for u =(1n , ...,1n), e = (1, 2, ..., n) , thatk1 = Tn (u; e, e) =1nn∑i=1i2 −(1nn∑i=1i)2=112(n2 − 1) ,and by Theorem 7, we deduce the inequality‖Tn (a,x)‖ ≤ 112(n2 − 1) max1≤j≤n−1|∆aj | max1≤j≤n−1‖∆xj‖ .Note that, the above inequality, has been discovered with a different method in [1].The constant 112 is best possible.References[1] Dragomir, S.S., Booth G.L. (2000), Gru¨ss-Lupas¸ type inequality and its applications for theestimation of p-moments of guessing mappings, Math. Comm., 5, 117-126.[2] Dragomir, S.S. (2002), Another Gru¨ss type inequality for sequences of vectors in normedlinear spaces and applications, J. Comp. Anal. & Appl., 4(2), 155-172.[3] Dragomir, S.S. (2002), A Gru¨ss type inequality for sequences of vectors in normedlinear spaces and applications, RGMIA Res. Rep. Coll., 5(2). Article 9, [ON LINEhttp://rgmia.vu.edu.au/v5n2.html][4] Cˇebysˇev, P.L. (1882), O priblizˇennyh vyrazˇenijah odnih integralov cˇerez drugie. Soobsˇc´enijai protokoly zasedani˘ı Matemmaticˇeskogo obcˇestva pri Imperatorskom Har’kovskom Univer-sitete No. 2, 93–98; Polnoe sobranie socˇineni˘ı P. L. Cˇebysˇeva. Moskva–Leningrad, 1948a,128-131.[5] Cˇebysˇev, P.L. (1883), Ob odnom rjade, dostavljajusˇc´em predel’nye velicˇiny integralov prirazlozˇenii podintegral’no˘ı funkcii na mnozˇeteli. Prilozˇeni k 57 tomu Zapisok Imp. Akad.Nauk, No. 4; Polnoe sobranie socˇineni˘ı P. L. Cˇebysˇeva. Moskva–Leningrad, 1948b, 157-169.[6] Hardy, G.H., Littlewood, J.E., and Po´lya, G. (1934, 1952), Inequalities, 1st Ed. and 2nd Ed.Cambridge University Press, Cambridge, England.[7] Biernacki, M. (1951), Sur une ine´galite´ entre les inte´grales due a` Tchebyscheff. Ann. Univ.Mariae Curie-Sklodowska A5, 23-29.[8] Sonin, N. Ja. (1898), O nekotoryh neravenstvah, ostnosjasˇcˇihsja k opredelennym integralam,Zap. Imp. Akad. Nauk po Fiziko-Matem. Otd. 6, 1-54.[9] Mitrinovic´ D.S., Pecˇaric´ J.E. and Fink A.M. (1993), Classical and New Inequalities in Anal-ysis, Kluwer Academic, Dordrecht.[10] Korkine, A.N. (1883), Sur une the´ore`me de M. Tchebychef, C.R. Acad. Sci. Paris, 96, 326-327.[11] Dragomir S.S. (in preparation) A survey on Cˇebysˇev and Gru¨ss type discrete inequalities.CˇEBYSˇEV FUNCTIONAL 11[12] Mitrinovic´ D.S. and Pecˇaric´, J.E. (1991), On an identity of D.Z. Djokovic´, Prilozi Mak.Akad.Nauk. Umj. (Skopje), 12(1), 21-22.[13] Dragomir, S.S. and Pecˇaric´, J.E., (1989), Refinements of some inequalities for isotonic linearfunctionals, L’Anal. Num. The´or de L’Approx. 18 (1), 61-65.[14] Dragomir, S.S. (1990), On some improvements of Cˇebysˇev’s inequality for sequences andintegrals, Studia Univ. Babes¸-Bolyai, Mathematica, XXXV (4), 35-40.[15] Pecˇaric´, J.E. and Dragomir, S.S. (1990), Some remarks on Cˇebysˇev’s inequality, L’Anal. Num.The´or de L’Approx. 19 (1), 58-65.School of Computer Science & Mathematics, Victoria University, Melbourne, Aus-traliaE-mail address: sever@matilda.vu.edu.auURL: http://rgmia.vu.edu.au/SSDragomirWeb.html

(in preparation) A survey on ˇ Cebyˇ sev and Gr¨ uss type discrete inequalities.ˇ

A Gr¨ uss type inequality for sequences of vectors in normed linear spaces and applications,

Another Gr¨ uss type inequality for sequences of vectors in normed linear spaces and applications,

Classical and New Inequalities in Analysis,

Gr¨ uss-Lupa¸ s type inequality and its applications for the estimation of p-moments of guessing mappings,

O nekotoryh neravenstvah, ostnosjaˇ sˇ cihsja k opredelennym integralam,

O pribliˇ zennyh vyraˇ zenijah odnih integralov ˇ cerez drugie. Soobˇ s´ cenija i protokoly zasedani˘ ı

Ob odnom rjade, dostavljajuˇ s´ cem predel’nye veliˇ ciny integralov pri razloˇ zenii podintegral’no˘ ı funkcii na mnoˇ zeteli.

On an identity of D.Z.

On some improvements of ˇ Cebyˇ sev’s inequality for sequences and integrals,

Reﬁnements of some inequalities for isotonic linear functionals,

Some remarks on ˇ Cebyˇ sev’s inequality,

Sur une in´ egalit´ e entre les int´ egrales due ` a Tchebyscheﬀ.

Sur une th´ eor`

Bounding the Čebyšev Functional for Sequences of Vectors in Normed Linear Spaces

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Bounding the Čebyšev Functional for Sequences of Vectors in Normed Linear Spaces

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