An identity due to P. Cerone for the Čebyšev functional is extended

for Stieltjes integrals. A sharp inequality and its application in approximating

Stieltjes integrals are also given

Dragomir, Sever S

English

An identity due to P. Cerone for the Čebyšev functional is extended\ud
for Stieltjes integrals. A sharp inequality and its application in approximating\ud
Stieltjes integrals are also given

Name not available

A Generalisation of Cerone's Identity and Applications

Victoria University Eprints Repository

A GENERALISATION OF CERONE’S IDENTITY ANDAPPLICATIONSS.S. DRAGOMIRAbstract. An identity due to P. Cerone for the Cˇebysˇev functional is ex-tended for Stieltjes integrals. A sharp inequality and its application in ap-proximating Stieltjes integrals are also given.1. IntroductionIn 2001, P. Cerone [1] established the following identity for the Cˇebysˇev func-tional:T (f, g) :=1b− a∫ baf (t) g (t) dt− 1b− a∫ baf (t) dt · 1b− a∫ bag (t) dt(1.1)=1(b− a)2∫ ba[(t− a)∫ btg (s) ds− (b− t)∫ tag (s) ds]df (t) ,provided f is of bounded variation on [a, b] and g is continuous on [a, b] . He proved(1.1) on utilising the auxiliary function Ψ : [a, b]→ R,(1.2) Ψ (t) := (t− a)∫ btg (s) ds− (b− t)∫ tag (s) dsand integrating by parts in the Stieltjes integral∫ baΨ(t) df (t) , which exists, sincef is of bounded variation and Ψ is differentiable on (a, b) .One may observe that the result remains valid if one assumes that g is Lebesgueintegrable on [a, b] and f is of bounded variation. This follows by the fact that, inthis case Ψ becomes absolutely continuous on [a, b], the Stieltjes integral∫ baΨ(t) df (t)still exists and the argument will follow as in [1].The weighted version of this inequality has been obtained in the same paper [1]and can be stated as:T (f, g; p) :=1∫ bap (s) ds∫ bap (t) f (t) g (t) dt(1.3)− 1∫ bap (s) ds∫ bap (t) f (t) dt · 1∫ bap (s) ds∫ bap (t) g (t) dtDate: 23 May, 2005.2000 Mathematics Subject Classification. Primary 26D15, 26D10.Key words and phrases. Integral inequalities, Stieltjes integrals.12 S.S. DRAGOMIR=1(∫ bap (s) ds)2 ∫ ba[∫ tap (s) ds∫ btp (s) g (s) ds−∫ btp (s) ds∫ tap (s) g (s) ds]df (t) ,provided f is of bounded variation on [a, b] and p, g are continuous on [a, b] with∫ bap (s) ds > 0. The same remark for the extension of the identity in the case thatp, g are Lebesgue integrable on [a, b] so that pg is also integrable, may apply.The above two identities have been applied in [1] to obtain some interestingnew bounds for the Cˇebysˇev functionals T (f, g) and T (f, g; p) from which we onlymention the following:(1.4) |T (f, g)|≤ 1(b− a)2 ×supt∈[a,b]|Ψ(t)|∨ba (f) ;L∫ ba|Ψ(t)| dt for f L− Lipschitzian;∫ ba|Ψ(t)| df (t) for f monotonic nondecreasing,where∨ba (f) is the total variation of f on [a, b] , Ψ(t) is given by (1.2) , and(1.5) |T (f, g; p)|≤ 1(∫ bap (s) ds)2×supt∈[a,b]|Ψp (t)|∨ba (f) ;L∫ ba|Ψp (t)| dt if f is L− Lipschitzian;∫ ba|Ψp (t)| df (t) for f monotonically nondecreasing,where in this case the wighted auxiliary mapping Ψp is defined as Ψp : [a, b]→ R,Ψp (t) :=∫ tap (s) ds∫ btp (s) g (s) ds−∫ btp (s) ds∫ tap (s) g (s) ds.For other inequalities and applications for moments, see [1].For further results, see the follow up paper [2] where various lower and otherupper bounds were established.2. A Related FunctionalIn [4], the authors have considered the following functional(2.1) D (f ;u) :=∫ baf (x) du (x)− [u (b)− u (a)] · 1b− a∫ baf (t) dt,provided that the Stieltjes integral∫ baf (x) du (x) exists.This functional palys an important role in approximating the Stieltjes integral∫ baf (x) du (x) in terms of the Riemann integral∫ baf (t) dt and the divided diferenceof the integrator u. Therefore, further bounds on D (f ;u) will generate a flow ofdifferent error estimates for the approximation of the Stieltjes integral that plays anA GENERALISATION OF CERONE’S IDENTITY 3important role in various fields of Analysis, Numerical Analysis, Integral OperatorTheory, Probability & Statistics and other fields of Modern Mathematics.In [4], the following result in estimating the above functional D (f ;u) has beenobtained:(2.2) |D (f ;u)| ≤ 12L (M −m) (b− a) ,provided u is L−Lipschitzian and f is Riemann integrable and with the propertythat there exists the constants m,M ∈ R such that(2.3) m ≤ f (x) ≤M for any x ∈ [a, b] .The constant 12 is best possible in (2.2) in the sense that it cannot be replaced bya smaller quantity.If one assumes that u is of bounded variation and f is K−Lipschitzian, thenD (f, u) satisfies the inequality [5](2.4) |D (f ;u)| ≤ 12K (b− a)b∨a(u) .Here the constant 12 is also best possible.The above inequalities have been used in [4] and [5] for obtaining inequalitiesbetween special means and on estimating the error in approximating the Stieltjesintegral∫ baf (x) du (x) in terms of the Riemann integral for the function f and thedivided difference of u.Now, for the function u : [a, b] → R, consider the following auxiliary mappingsΦ,Γ and ∆ [3]:Φ (t) :=(t− a)u (b) + (b− t)u (a)b− a − u (t) , t ∈ [a, b] ,Γ (t) := (t− a) [u (b)− u (t)]− (b− t) [u (t)− u (a)] , t ∈ [a, b] ,∆(t) := [u; b, t]− [u; t, a] , t ∈ (a, b) ,where [u;α, β] is the divided difference of u in α, β, i.e.,[u;α, β] :=u (α)− u (β)α− β .The following representation of D (f, u) may be stated.Theorem 1. Let f, u : [a, b] → R be such that the Stieltjes integral ∫ baf (t) du (t)and the Riemann integral∫ baf (t) dt exist. ThenD (f, u) =∫ baΦ (t) df (t) =1b− a∫ baΓ (t) df (t)(2.5)=1b− a∫ ba(t− a) (b− t)∆ (t) df (t) .4 S.S. DRAGOMIRProof. Since∫ baf (t) du (t) exists, hence∫ baΦ (t) df (t) also exists, and the integra-tion by parts formula for Stieltjes integrals gives that∫ baΦ (t) df (t) =∫ ba[(t− a)u (b) + (b− t)u (a)b− a − u (t)]df (t)=[(t− a)u (b) + (b− t)u (a)b− a − u (t)]f (t)∣∣∣∣ba−∫ baf (t) d[(t− a)u (b) + (b− t)u (a)b− a − u (t)]= −∫ baf (t)[u (b)− u (a)b− a dt− du (t)]= D (f, u) ,proving the required identity.Remark 1. The identity (2.5) has been established in [3]. There were some typo-graphical errors in [3] that have been corrected above.Remark 2. If u is an integral, i.e., u (t) =∫ tag (s) ds, t ∈ [a, b] , thenΦ (t) =t− ab− a∫ bag (s) ds−∫ tag (s) ds,Γ (t) = (t− a)∫ btg (s) ds− (b− t)∫ tag (s) ds,∆(t) =∫ btg (s) dsb− t −∫ tag (s) dst− a ,and then, from (2.5), one may recapture Cerone’s identity (1.1) for the Cˇebysˇevfunctional T (f, g) .Since it well known that u is an integral if and only if u is absolutely continuous,and in this case g (s) = u′ (s) for s ∈ [a, b] , hence (2.5) is indeed a proper generali-sation of (1.1) holding for larger classes of functions than the absolutely continuousfunctions.Remark 3. If one chooses u : [a, b]→ R,u (t) =∫ tap (s) g (s) ds∫ bap (s) ds, t ∈ [a, b] ,where p, g are Lebesgue integrable with pg is also integrable and∫ bap (s) ds 6= 0, thenthe identity (2.5) produces the representation:E (f, g; p) :=∫ bap (s) f (s) g (s) ds∫ bap (s) ds−∫ bap (s) g (s) ds∫ bap (s) ds· 1b− a∫ baf (t) dt(2.6)=∫ baΦp (t) df (t) =1b− a∫ baΓp (t) df (t)=1b− a∫ ba(t− a) (b− t)∆p (t) df (t) ,A GENERALISATION OF CERONE’S IDENTITY 5whereΦp (t) :=t− ab− a ·∫ bap (s) g (s) ds∫ bap (s) ds−∫ tap (s) g (s) ds∫ bap (s) ds,Γp (t) := (t− a)∫ btp (s) g (s) ds∫ bap (s) ds− (b− t)∫ tap (s) g (s) ds∫ bap (s) dsand∆p (t) :=∫ btp (s) g (s) ds(b− t) ∫ bap (s) ds−∫ tap (s) g (s) ds(t− a) ∫ bap (s) ds.One must observe that the identity (2.6) is not the same as Cerone’s identity forweighted integrals (1.3).For recent inequalities related to D (f ;u) for various pairs of functions (f, u) ,see [3, pp. 112-118].3. A Bound for f of Bounded Variation and u ContinuousIt is known that if u is continuous on [a, b] and f is of bounded variation on [a, b] ,then the Stieltjes integral∫ baf (t) du (t) exists. This integral may exists even forlarger clases of integrators f , for instance, piecewise continuous functions for whichthe discontinuities of the integrand f do not overlap with those of the integrator u.The following result may be stated:Theorem 2. Let f : [a, b]→ R be of bounded variation on [a, b] and u : [a, b]→ Rsuch that there exist the constants γ,Γ ∈ R with:(3.1) γ ≤ u (t) ≤ Γ for any t ∈ [a, b]and the Stieltjes integral∫ baf (t) du (t) exists. Then(3.2) |D (f ;u)| ≤ (Γ− γ)b∨a(f) .The multiplicative constant 1 in front of Γ − γ cannot be replaced by a smallerquantity.Proof. By (1.1), we obviously have:γ (b− t) ≤ (b− t)u (a) ≤ (b− t) Γ,γ (t− a) ≤ (t− a)u (b) ≤ (t− a) Γ,− (b− a) Γ ≤ − (b− a)u (t) ≤ − (b− a) γ,which gives by addition and division with b− a that− (Γ− γ) ≤ (b− t)u (a) + (t− a)u (b)b− a − u (t) ≤ Γ− γ,showing that |Φ (t)| ≤ Γ− γ for any t ∈ [a, b] .Taking into account that for ϕ bounded and ψ of bounded variation on [a, b] onehas ∣∣∣∣∣∫ baϕ (t) dψ (t)∣∣∣∣∣ ≤ supt∈[a,b] |ϕ (t)|b∨a(ψ) ,6 S.S. DRAGOMIRprovided the Stieltjes integral exists, we have by (2.5) that|D (f ;u)| ≤ supt∈[a,b]|φ (t)|b∨a(f) ≤ (Γ− γ)b∨a(f) ,proving the required inequality (3.2).Now, for the sharpness of the inequality.Assume that there exists a c > 0 such that(3.3) |D (f ;u)| ≤ c (Γ− γ)b∨a(f) ,where u and f are as in the hypothesis of the theorem.Consider u, f : [a, b]→ R withu (t) =12(t− a+ b2)2, f (t) = sgn(t− a+ b2), t ∈ [a, b] .Then u is continuous, f is of bounded variation, the integral∫ baf (t) du (t) existsandb∨a(f) = 2,∫ baf (t) dt = 0,Γ = supt∈[a,b]u (t) =(b− a)28, γ = inft∈[a,b]u (t) = 0,∫ baf (t) du (t) =∫ basgn(t− a+ b2)(t− a+ b2)dt=∫ ba∣∣∣∣t− a+ b2∣∣∣∣ dt = (b− a)24 .Substituting into (3.3) we get (b−a)24 ≤ c(b−a)24 , which implies that c ≥ 1.Corollary 1. Let f : [a, b] → R be of bounded variation and u : [a, b] → R contin-uous on [a, b] . Then:(3.4) |D (f ;u)| ≤[maxt∈[a,b]u (t)− mint∈[a,b]u (t)] b∨a(f) .The inequality (3.4) is sharp.If we consider the Cˇebysˇev functional T (f, g) , then we can state the followingcorollary as well:Corollary 2. Let f : [a, b] → R be of bounded variation and g : [a, b] → R aLebesgue integrable function such that there exists the constants m and M with(3.5) m ≤ g (s) ≤M for a.e. s ∈ [a, b] .Then(3.6) |T (f, g)| ≤ (b− a) (M −m)b∨a(f) .A GENERALISATION OF CERONE’S IDENTITY 7Proof. We choose u (t) :=∫ tag (s) ds which is continuous on [a, b] and satisfies theinequality (3.1) with γ = (b− a)m and Γ = (b− a)M and apply Theorem 2.Remark 4. If we assume that for the Lebesgue integrable function g,∫ ·ag (s) dssatisfies the conditionγ ≤∫ tag (s) ds ≤ Γ for any t ∈ [a, b] ,then|T (f, g)| ≤ (Γ− γ)b∨a(f)and the inequality is sharp. The equality case is realised for g (t) = t − a+b2 andf (t) = sgn(t− a+b2), t ∈ [a, b] .It is an open problem wether or not the bound in (3.6) is sharp.Remark 5. If p, g ∈ L [a, b] so that pg ∈ L [a, b] and ∫ bap (s) ds 6= 0 and there existsthe constants δ,∆ so thatδ ≤∫ tap (s) g (s) ds∫ bap (s) ds≤ ∆for any t ∈ [a, b] , then|E (f, g; p)| ≤ (∆− δ)b∨a(f) .The last inequality is sharp.4. Application for Approximating the Stieltjes IntegralLet us consider the partition of the interval [a, b] given byIn : a = t0 < t1 < · · · < tn−1 < tn = b.Denote v (In) := max {hi|i = 0, . . . , n− 1}, where hi := ti+1 − ti, i = 0, . . . , n − 1.If u : [a, b]→ R is continuous on [a, b] and if we defineMi := supt∈[ti,ti+1]u (t) , mi := inft∈[ti,ti+1]u (t)andv (u, In) := max0≤i≤n−1(Mi −mi) ,then, obviously, by the continuity of u on [a, b] , for any ε ≥ 0, there exists a δ > 0and a division In with norm v (In) < δ such that v (u, In) < ε.Consider now the quadrature rule(4.1) Sn (f, u, In) :=n−1∑i=0[u (ti+1)− u (ti)]ti+1 − ti ·∫ ti+1tif (t) dt,provided u is continuous on [a, b] and f is of bounded variation on [a, b] .We may state the following result in approximating the Stieltjes integral:8 S.S. DRAGOMIRTheorem 3. Let f, u : [a, b] → R be such that f is of bounded variation on [a, b]and u is continuous on [a, b] . Then for any division In as above,(4.2)∫ baf (t) du (t) = Sn (f, u, In) +Rn (f, u, In) ,where the remainder Rn (f, u, In) satisfies the estimate:(4.3) |Rn (f, u, In)| ≤ v (u, In)b∨a(f) .Proof. Applying Theorem 2 on the intervals [ti, ti+1] , i = 0, . . . , n − 1, we havesucessively:|Rn (f, u, In)| =∣∣∣∣∣n−1∑i=0∫ ti+1tif (t) du (t)− u (ti+1)− u (ti)ti+1 − ti∫ ti+1tif (t) dt∣∣∣∣∣≤n−1∑i=0∣∣∣∣∫ ti+1tif (t) du (t)− u (ti+1)− u (ti)ti+1 − ti∫ ti+1tif (t) dt∣∣∣∣≤n−1∑i=0(Mi −mi)ti+1∨ti(f) ≤ v (u, In)b∨a(f)and the estimate (4.3) is obtained.References[1] P. CERONE, On an identity for the Chebychev functional and some ramifications, J. Inequal.Pure & Appl. Math., 3(1) (2002), Article 2.[2] P. CERONE and S.S. DRAGOMIR, New upper and lower bounds for the Cˇebysˇev functional,J. Inequal. Pure & Appl. Math., 3(5) (2002), Article 77.[3] S.S. DRAGOMIR, Inequalities of Gru¨ss type for the Stieltjes integral and applications, Kragu-jevac J. Math., 26 (2004), 89-112.[4] S.S. DRAGOMIR and I. FEDOTOV, An inequality of Gru¨ss type for Riemann-Stieltjes inte-gral and applications for special means, Tamkang J. Math., 29(4) (1998), 287-292.[5] S.S. DRAGOMIR and I. FEDOTOV, A Gru¨ss type inequality for mappings of bounded varia-tion and applications to numerical analysis, Non. Funct. Anal. & Appl., 6(3) (2001), 425-437.School of Computer Science and Mathematics, Victoria University of Technology,PO Box 14428, Melbourne City, VIC 8001, Australia.E-mail address: sever.dragomir@vu.edu.auURL: http://rgmia.vu.edu.au/dragomir

A Generalisation of Cerone's Identity and Applications

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