Some inequalities related to the perturbed trapezoid formula are given. An application for the expectation of a random variable is also pointed out

Barnett, Neil S

Dragomir, Sever S

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ON THE PERTURBED TRAPEZOID FORMULAN.S. BARNETT AND S.S. DRAGOMIRAbstract. Some inequalities related to the perturbed trapezoid formula aregiven. An application for the expectation of a random variable is also pointedout.1. IntroductionIn [3], the authors have pointed out the following trapezoid inequality in termsof the p−norms of the second derivative.Theorem 1. Let f : [a, b] → R be a twice differentiable function on (a, b). Thenwe have the estimate∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)]∣∣∣∣∣(1.1)≤‖f ′′‖∞12 (b− a)3 if f ′′ ∈ L∞ [a, b] ;12 ‖f ′′‖p [B (q + 1, q + 1)]1q (b− a)2+ 1q , if f ′′ ∈ Lp [a, b] ,p > 1, 1p +1q = 1;‖f ′′‖18 (b− a)2 if f ′′ ∈ L1 [a, b] ;where‖f ′′‖∞ : = supt∈[a,b]|f ′′ (t)| ,‖f ′′‖1 : =∫ ba|f ′′ (t)| dtand‖f ′′‖p :=(∫ ba|f ′′ (t)|p dt) 1p, p > 1and B is the Beta function, that is,B (r, s) :=∫ 10tr−1 (1− t)s−1 dt, r, s > 0.Using Gru¨ss’ integral inequality, the following perturbed trapezoid inequality interms of the upper and lower bounds of the second derivative, may be stated (see[3]):Date: February 8, 2001.1991 Mathematics Subject Classification. Primary 26D15; Secondary 26D10, 26D99, 41A55.Key words and phrases. Perturbed Trapezoid Formula.12 N.S. BARNETT AND S.S. DRAGOMIRTheorem 2. Let f : [a, b] → R be a twice differentiable function on (a, b) andassume that(1.2) γ := infx∈(a,b)f ′′ (x) > −∞ and Γ := supx∈(a,b)f ′′ (x) <∞.Then we have the estimation∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)212[f ′ (b)− f ′ (a)]∣∣∣∣∣(1.3)≤ 132(Γ− γ) (b− a)3 .In [2], by the use of a finer argument based on the pre-Gru¨ss inequality, theauthors have improved (1.3) as follows.Theorem 3. If f is as in Theorem 2, then∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)212[f ′ (b)− f ′ (a)]∣∣∣∣∣(1.4)≤ 124√5(b− a)3 (Γ− γ) ,where γ and Γ are given in (1.2).Remark 1. Atkinson [1] defines the quadrature rulePT (f ; a, b) :=b− a2[f (a) + f (b)]− (b− a)212[f ′ (b)− f ′ (a)]as a corrected trapezoidal rule and obtains it using an asymptotic error estimateapproach which does not provide an expression for the error bound.In this paper we point out different bounds for the corrected trapezoidal rule. Anatural application for the expectation of a random variable is also given.2. The ResultsWe have the following representation result.Lemma 1. Let f : [a, b] → R be a differentiable function so that f ′ is absolutelycontinuous on [a, b]. Then we have the representation:∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b](2.1)=12∫ ba(x− a) (b− x) {[f ′; a, b]− f ′′ (x)} dx,where[f ′; a, b] :=f ′ (b)− f ′ (a)b− ais the divided difference.Proof. By twice applying the integration by parts formula, we may state (see forexample [1]) that(2.2)∫ baf (x) dx− b− a2[f (a) + f (b)] = −12∫ ba(x− a) (b− x) f ′′ (x) dx.PERTURBED TRAPEZOID FORMULA 3On the other hand, by the simple identity:1b− a∫ bah (x) g (x) dx− 1b− a∫ bah (x) dx · 1b− a∫ bag (x) dx(2.3)=1b− a∫ bah (x)[g (x)− 1b− a∫ bag (y) dy]dx,we may state that∫ ba(x− a) (b− x) f ′′ (x) dx−∫ ba(x− a) (b− x) dx ·∫ baf ′′ (x) dx=∫ ba(x− a) (b− x) [f ′′ (x) dx− [f ′; a, b]] dx,which is clearly equivalent to:∫ ba(x− a) (b− x) f ′′ (x) dx(2.4)=(b− a)26[f ′ (b)− f ′ (a)] +∫ ba(x− a) (b− x) [f ′′ (x) dx− [f ′; a, b]] dx.Combining (2.2) with (2.4), we deduce (2.1).Using the above representation, we may state the following result on the errorof the perturbed trapezoid formula:Theorem 4. With the assumptions of Lemma 1, we have∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.5)≤(b−a)312 ‖f ′′ − [f ′; a, b]‖∞ if f ′′ ∈ L∞ [a, b] ;12 [B (q + 1, q + 1)]1q (b− a)2+ 1q ‖f ′′ − [f ′; a, b]‖p , if f ′′ ∈ Lp [a, b] ,p > 1, 1p +1q = 1;18 (b− a)2 ‖f ′′ − [f ′; a, b]‖1 if f ′′ ∈ L1 [a, b] ,where B is the Beta function and‖g‖∞ : = ess sup |g (x)|‖g‖p : =(∫ ba|g (x)|s dx) 1s, s ≥ 1.Proof. Using Lemma 1, we have∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.6)≤ 12∫ ba(x− a) (b− x) |f ′′ (x) dx− [f ′; a, b]| dx =:M.4 N.S. BARNETT AND S.S. DRAGOMIRIt is obvious thatM ≤ 12‖f ′′ − [f ′; a, b]‖∞ ·∫ ba(x− a) (b− x) dx=(b− a)312‖f ′′ − [f ′; a, b]‖∞and the first part of (2.5) is proved.Using Ho¨lder’s integral inequality, we have for p > 1, 1p +1q = 1, that(2.7) M ≤ 12(∫ ba(x− a)q (b− x)q dx) 1q(∫ ba|f ′′ (x)− [f ′; a, b]|p dx) 1p.Now, using the transformation x = (1− t) a+ tb, t ∈ [0, 1], we get(x− a)q (b− x)q = (b− a)2q tq (1− t)q ,dx = (b− a) dtand thus ∫ ba(x− a)q (b− x)q dx = (b− a)2q+1∫ 10tq (1− t)q dt= (b− a)2q+1B (q + 1, q + 1) .Using (2.7) we deduce the second part of (2.5).Finally, asM ≤ 12supx∈[a,b]{(x− a) (b− x)}∫ ba|f ′′ (x)− [f ′; a, b]| dx=(b− a)28‖f ′′ − [f ′; a, b]‖1the theorem is completely proved.The following corollary concerning the Euclidean norm is useful in practice.Corollary 1. If f : [a, b]→ R is such that f ′′ ∈ L2 [a, b], then we have the inequal-ity: ∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.8)≤ (b− a)32√30∣∣∣∣ 1b− a ‖f ′′‖22 − [f ′; a, b]2∣∣∣∣ 12 .Proof. Choosing in (2.5) p = q = 2, we get∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.9)≤ 12[B (3, 3)]12 (b− a)2+ 1q ‖f ′′ − [f ′; a, b]‖2 .However,B (3, 3) =130,PERTURBED TRAPEZOID FORMULA 5and‖f ′′ − [f ′; a, b]‖2=[∫ ba(f ′′ (x)− [f ′; a, b])2 dx] 12=[∫ ba(f ′′ (x))2 dx− 2∫ baf ′′ (x) [f ′; a, b] dx+ (b− a) [f ′; a, b]2] 12=(‖f ′′‖22 − 2 (b− a) [f ′; a, b]2 + (b− a) [f ′; a, b]2) 12=√b− a(1b− a ‖f′′‖22 − [f ′; a, b]2) 12,then, by (2.9) we get (2.8).Remark 2. (1) The Gru¨ss integral inequality for a function g : [a, b]→ R with−∞ < m ≤ g (x) ≤ M < ∞ for almost every x ∈ [a, b] states that (see forexample [4, p. 296])(2.10) 0 ≤ 1b− a ‖g‖22 −(1b− a∫ bag (x) dx)2≤ 14(M −m)2 .Applying (2.10) for the mapping f ′′ under the assumption that γ ≤ f ′′ (x) ≤Γ for a.e. x ∈ [a, b], we deduce(1b− a ‖f′′‖2 − [f ′; a, b]2) 12≤ 12(Γ− γ)and then, by (2.8), we deduce∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.11)≤ (b− a)34√30(Γ− γ) .which is not as good as the result in (1.3).(2) Chebychev’s inequality for a differentiable function g : [a, b] → R, withg′ ∈ L∞ [a, b] states that (see [4, p. 297])(2.12) 0 ≤ 1b− a ‖g‖22 −(1b− a∫ bag (x) dx)2≤ 112(b− a)2 ‖g′‖∞ .Applying (2.12) for the mapping f ′′ under the assumption that f ′′′ ∈ L∞ [a, b],we deduce by (2.8) that∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.13)≤ (b− a)4 ‖f ′′′‖∞12√10.6 N.S. BARNETT AND S.S. DRAGOMIR(3) Lupas¸’s inequality for a differentiable function f with f ′′′ ∈ L2 [a, b] statesthat (see [4, p. 301])(2.14) 0 ≤ 1b− a ‖g‖22 −(1b− a∫ bag (x) dx)2≤ b− api2‖g′‖22 .Applying (2.14) for the mapping f ′′ under the assumption that f ′′′ ∈ L2 [a, b],we deduce, by (2.8), that∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.15)≤ (b− a)72 ‖f ′′′‖22pi√30.The following lemma of representation also holds.Lemma 2. Let f : [a, b]→ R be a differentiable function such that f ′ is absolutelycontinuous on [a, b]. We have the representation:-∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b](2.16)=12∫ ba(x− a+ b2)2[f ′′ (x)− [f ′; a, b]] dx,where [f ′; a, b] is the divided difference.Proof. The identity (2.16) may be proven directly.A simpler proof uses Lemma 1 as follows.Since(x− a) (b− x) = a2 + b22−(x− a+ b2)2and12∫ ba(x− a) (b− x) {[f ′; a, b]− f ′′ (x)} dx=12∫ ba[a2 + b22−(x− a+ b2)2]{[f ′; a, b]− f ′′ (x)} dx=12∫ baa2 + b22{[f ′; a, b]− f ′′ (x)} dx− 12∫ ba(x− a+ b2)2{[f ′; a, b]− f ′′ (x)} dx=12∫ ba(x− a+ b2)2{f ′′ (x)− [f ′; a, b]} dx,then by (2.1) we deduce (2.16).The following result also holds.PERTURBED TRAPEZOID FORMULA 7Theorem 5. With the assumptions of Lemma 2, we have the inequality∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.17)≤(b− a)324‖f ′′ − [f ′; a, b]‖∞ if f ′′ ∈ L∞ [a, b] ;(b− a)2+ 1q8 (2q + 1)1q‖f ′′ − [f ′; a, b]‖p , if f ′′ ∈ Lp [a, b] ,p > 1, 1p +1q = 1;(b− a)28‖f ′′ − [f ′; a, b]‖1 .Proof. Using Lemma 2, we have:-∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.18)≤ 12∫ ba(x− a+ b2)2|f ′′ (x)− [f ′; a, b]| dx =: N.It is obvious thatN ≤ 12‖f ′′ − [f ′; a, b]‖∞ ·∫ ba(x− a+ b2)2dx=(b− a)324‖f ′′ − [f ′; a, b]‖∞ .Using Ho¨lder’s integral inequality, we have for p > 1, 1p +1q = 1, that(2.19) N ≤(∫ ba∣∣∣∣x− a+ b2∣∣∣∣2q dx) 1q(∫ ba|f ′′ (x)− [f ′; a, b]|p dx) 1p.However, ∫ ba∣∣∣∣x− a+ b2∣∣∣∣2q dx = 2 ∫ ba+b2(x− a+ b2)2qdx =(b− a)2q4q (2q + 1)and then, by (2.19), we deduce the second part of (2.17).Finally, assupx∈[a,b]∣∣∣∣x− a+ b2∣∣∣∣2 = (b− a)24 ,thenN ≤ (b− a)28‖f ′′ − [f ′; a, b]‖1 ,proving the last part of (2.17).Remark 3. It is obvious that the first inequality in (2.17) is better than the similarone in (2.5), while the last ones are identical.8 N.S. BARNETT AND S.S. DRAGOMIRRemark 4. A computer simulation for the function 12 [B (q + 1, q + 1)]1q , 18 ·1(2q+1)1qshows that the latter is smaller for any q > 1, but we do not have ananalytic proof of this fact. We conjecture that the second inequality in (2.7) isbetter than the second inequality in (2.5) for every p > 1, 1p +1q = 1.For p = q = 2, we get the following particular case for the euclidean norm:Corollary 2. If f : [a, b]→ R is such that f ′′ ∈ L2 [a, b], then∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.20)≤ (b− a)38√5[1b− a ‖f′′‖2 − [f ′; a, b]2] 12.Remark 5. We note that (2.20) is a better result that the corresponding one in(2.8), and thus, we may note the following better results following via Gru¨ss typeinequalities.If f ′′ is such that γ ≤ f ′′ ≤ Γ for a.e. x ∈ [a, b], then by Gru¨ss’ (2.10), we have∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.21)≤ (b− a)3 (Γ− γ)16√5.If f ′′′ ∈ L∞ [a, b], then by the Chebychev inequality (2.12), we have∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.22)≤ (b− a)4 ‖f ′′′‖∞16√15.Finally, if f ′′′ ∈ L2 [a, b], then by the Lupas¸ inequality (2.15), we have:∣∣∣∣∣∫ baf (x) dx− b− a2[f (a) + f (b)] +(b− a)312[f ′; a, b]∣∣∣∣∣(2.23)≤ (b− a)72 ‖f ′′′‖28pi√5.3. Applications for ExpectationLet X be a random variable having the p.d.f., f : [a, b]→ R and the cumulativedistribution function F : [a, b]→ [0, 1], i.e.,F (x) =∫ xaf (t) dt, x ∈ [a, b]The following result holds.PERTURBED TRAPEZOID FORMULA 9Theorem 6. With the above assumptions and if the p.d.f. is absolutely continuouson [a, b], then we have the inequality:∣∣∣∣∣E (X)− a+ b2 − (b− a)212 [f (b)− f (a)]∣∣∣∣∣(3.1)≤(b−a)324 ‖f ′ − [f ; a, b]‖∞ if f ′ ∈ L∞ [a, b](b−a)2+1q8(2q+1)1q‖f ′ − [f ; a, b]‖p if f ′ ∈ Lp [a, b] ,p > 1, 1p +1q = 1;(b−a)28 ‖f ′ − [f ; a, b]‖1 .Proof. Applying Theorem 5 for the c.d.f., F , we may write that∣∣∣∣∣∫ baF (t) dt− F (a) + F (b)2(b− a) + (b− a)212[f (b)− f (a)]∣∣∣∣∣(3.2)≤(b−a)324 ‖f ′ − [f ; a, b]‖∞(b−a)2+1q8(2q+1)1q‖f ′ − [f ; a, b]‖p(b−a)28 ‖f ′ − [f ; a, b]‖1 .However, F (a) = 0, F (b) = 1 and∫ baF (t) dt = b− E (X) ,and then, by (3.2) we deduce the desired inequality (3.1).References[1] K.E. ATKINSON, An Introduction to Numerical Analysis, Wiley and Sons, Second Edition,1989.[2] P. CERONE and S. S. DRAGOMIR, Trapezoidal type rules from an inequalities point of view,Handbook of Analytic-Computational Methods in Applied Mathematics, G.A. Anastassiou(Ed), CRC Press, New York (2000), 65-134.[3] S.S. DRAGOMIR, P. CERONE and A. SOFO, Some remarks on the trapezoid rule in numericalintegration, Indian J. of Pure and Appl. Math., 31(5) (2000), 475-494. Preprint: RGMIA Res.Rep. Coll., 2(5) (1999), Article 1. [ONLINE] http://rgmia.vu.edu.au/v2n5.html[4] D.S. MITRINOVIC´, J.E. PECˇARIC´ and A.M. FINK, Classical and New Inequalities in Anal-ysis, Kluwer Academic Publishers, 1993.School of Communications and Informatics, Victoria University of Technology, POBox 14428, Melbourne City MC, Victoria 8001, Australia.E-mail address: neil@sci.vu.edu.auURL: http://sci.vu.edu.au/staff/neilb.htmlE-mail address: sever@matilda.vu.edu.auURL: http://rgmia.vu.edu.au/SSDragomirWeb.html

On the Perturbed Trapezoid Formula

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