The purpose of this paper is to use a variant of the Grüss inequality to derive two new sharp Ostrowski-Grüss type inequalities related to a perturbed trapezoidal type rule and a perturbed generalized interior point rule, respectively, which provide improvements of some previous results in the literatures

Liu, Zheng

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXVIII (2013) – Fasc. II, pp. 3–14doi: 10.4418/2013.68.2.1TWO NEW SHARP OSTROWSKI-GRU¨SS TYPEINEQUALITIESZHENG LIUThe purpose of this paper is to use a variant of the Gru¨ss inequalityto derive two new sharp Ostrowski-Gru¨ss type inequalities related to aperturbed trapezoidal type rule and a perturbed generalized interior pointrule, respectively, which provide improvements of some previous resultsin the literatures.1. IntroductionIn 1935, G.Gru¨ss proved the following integral inequality which gives an ap-proximation for the integral of the product of two functions in terms of theproduct of the integrals of the two functions (see for example [11,p.296]).Theorem 1.1. Let h,g : [a,b]→ R be two integrable functions such that φ ≤h(t) ≤ Φ and γ ≤ g(t) ≤ Γ for all t ∈ [a,b], where φ , Φ, γ and Γ are realnumbers. Then we haveEntrato in redazione: 24 luglio 2012AMS 2010 Subject Classification: 26D15.Keywords: Gru¨ss inequality, Ostrowski-Gru¨ss inequality, Absolutely continuous, Sharp bound.4 ZHENG LIU|T (h,g)| := | 1b−a∫ bah(t)g(t)dt− 1b−a∫ bah(t)dt · 1b−a∫ bag(t)dt|≤ 14(Φ−φ)(Γ− γ), (1)and the inequality is sharp, in the sense that the constant 14 can not be replacedby a smaller one.It is clear that the constant 14 is achieved forh(t) = g(t) = sgn(t− a+b2).From then on, (1) is well known in the literature as the Gru¨ss inequality.A premature Gru¨ss inequality originated from the work of Gru¨ss (see also[11, p. 296]). It is embodied in the following theorem and was also consideredand applied for the first time in the paper [10, Theorem 5] by M. Matic´, J.Pecˇaric´ and N. Ujevic´ in 2000.Theorem 1.2. Let h,g : [a,b]→ R be integrable functions such that hg is alsointegrable, and γ ≤ g(t)≤ Γ for all t ∈ [a,b], where γ,Γ ∈R are constants.Then|T (h,g)| ≤ 12√T (h,h)(Γ− γ). (2)In 2002, almost at the same time, by using similar, somewhat complicatedmethods, X. L. Cheng and J. Sun in [6, Theorem 1.1] as well as M. Matic´ in [9,Theorem 3] have proved the following variant of the Gru¨ss inequality respec-tively.Theorem 1.3. Let h,g : [a,b]→ R be two integrable functions such that γ ≤g(t)≤ Γ for some constants γ , Γ for all t ∈ [a,b]. Then∣∣∣∣∫ ba h(t)g(t)dt− 1b−a∫ bah(t)dt∫ bag(t)dt∣∣∣∣≤ 12(∫ ba|h(t)− 1b−a∫ bah(y)dy|dt)(Γ− γ). (3)Moreover, Matic´ has proved that there exists a function g for which theequality in (3) is attained, Cerone and Dragomir have proved in [4, Theorem 3]TWO NEW SHARP OSTROWSKI-GRU¨SS TYPE INEQUALITIES 5that the constant 12 in (3) cannot be replaced by a smaller one. In [8, p.122],the author has provided a simple proof of Theorem 1.3 with the sharpness ofinequality (3) in the sense that we can choose the function g such that eitherg(x) ={Γ, if h(x)− 1b−a∫ ba h(y)dy≥ 0,γ , if h(x)− 1b−a∫ ba h(y)dy < 0org(x) ={γ , if h(x)− 1b−a∫ ba h(y)dy≥ 0,Γ, if h(x)− 1b−a∫ ba h(y)dy < 0to attain the equality in (3).The result stated in Theorem 1.3 is of particular interest and very useful inthe case when∫ ba∣∣∣h(t)− 1b−a ∫ ba h(y)dy∣∣∣ dt can be evaluated exactly.In [4, (2.19)], we can see that Theorem 1.2 improves Theorem 1.1 and The-orem 1.3 improves Theorem 1.2.From [2, Theorem 10] and [3, Theorem 13], we see that applying the pre-mature Gru¨ss inequality (2) has derived two Ostrowski-Gru¨ss type inequalitiesrelated to a perturbed trapezoidal type rule and a perturbed generalized interiorpoint rule as follows:Theorem 1.4. Let f : [a,b]→ R be a mapping such that the derivative f (n−1)(n ≥ 1) is absolutely continuous on [a,b]. Assume that there exist constantsγ,Γ ∈ R such that γ ≤ f (n)(t) ≤ Γ a.e. on [a,b]. Then for all x ∈ [a,b] , thefollowing inequality holds∣∣∣∣∣∫ baf (t)dt−n−1∑k=01(k+1)![(x−a)k+1 f (k)(a)+(−1)k(b− x)k+1 f (k)(b)]−(x−a)n+1+(−1)n(b− x)n+1(n+1)![f (n−1)(b)− f (n−1)(a)b−a]∣∣∣∣∣≤ Γ− γ2(n+1)!√2n+1{n2(b−a)[(x−a)2n+1+(b− x)2n+1]+ (2n+1)(x−a)(b− x)[(x−a)n− (x−b)n]2} 12 .(4)Theorem 1.5. Let f : [a,b]→ R be a mapping such that the derivative f (n−1)(n ≥ 1) is absolutely continuous on [a,b]. Assume that there exist constantsγ,Γ ∈ R such that γ ≤ f (n)(t) ≤ Γ a.e. on [a,b]. Then for all x ∈ [a,b] , thefollowing inequality holds6 ZHENG LIU|∫ ba f (t)dt−∑n−1k=0 1(k+1)! [(b− x)k+1+(−1)k(x−a)k+1] f (k)(x)− (b−x)n+1+(−1)n(x−a)n+1(n+1)! [ f(n−1)(b)− f (n−1)(a)b−a ]|≤ Γ−γ2(n+1)!√2n+1{n2(b−a)[(x−a)2n+1+(b− x)2n+1]+(2n+1)(x−a)(b− x)[(x−a)n− (x−b)n]2} 12 .(5)The purpose of this paper is to provide, using the variant of the Gru¨ss in-equality, two new sharp Ostrowski-Gru¨ss type inequalities related to a perturbedtrapezoidal type rule and a perturbed generalized interior point rule, which giveimprovements of the above inequalities (4) and (5), respectively. We need thefollowing two lemmas:Lemma 1.6. [2, Theorem 7] Let f : [a,b]→ R be a mapping such that thederivative f (n−1)(n≥ 1) is absolutely continuous on [a,b]. Then for all x∈ [a,b]we have the identity:∫ baf (t)dt =n−1∑k=01(k+1)![(x−a)k+1 f (k)(a)+(−1)k(b− x)k+1 f (k)(b)]+1n!∫ ba(x− t)n f (n)(t)dt. (6)Lemma 1.7. [3, Theorem 7] Let f : [a,b]→ R be a mapping such that thederivative f (n−1)(n≥ 1) is absolutely continuous on [a,b]. Then for all x∈ [a,b]we have the identity:∫ baf (t)dt =n−1∑k=01(k+1)![(b− x)k+1+(−1)k(x−a)k+1] f (k)(x)+(−1)n∫ baKn(x, t) f (n)(t)dt, (7)where the kernel Kn : [a,b]2→ R is given byKn(x, t) :={(t−a)nn! , if t ∈ [a,x],(t−b)nn! , if t ∈ (x,b],(8)and x ∈ [a,b].TWO NEW SHARP OSTROWSKI-GRU¨SS TYPE INEQUALITIES 72. The resultsIn what follows, we will use the notationsHx =1b−a∫ ba(x− t)n dt = (x−a)n+1− (x−b)n+1(n+1)(b−a)andKx =(x−a)(x−b)n+1+(b− x)(x−a)n+1b−a .Theorem 2.1. Let f : [a,b]→ R be a mapping such that the derivative f (n−1)(n ≥ 1) is absolutely continuous on [a,b]. Assume that there exist constantsγ,Γ ∈ R such that γ ≤ f (n)(t) ≤ Γ a.e. on [a,b]. Then for all x ∈ [a,b], thefollowing inequality holds:∣∣∣∣∣∫ baf (t)dt−n−1∑k=01(k+1)![(x−a)k+1 f (k)(a)+(−1)k(b− x)k+1 f (k)(b)]−Hx[ f(n−1)(b)− f (n−1)(a)]n!∣∣∣∣∣≤ Γ− γ(n+1)!G(a,b,x;n) (9)whereG(a,b,x;n) =nHx n√Hx+Kx, n oddnHx n√Hx−Kx, n even and x ∈ [a,ξ ]2nHx n√Hx, n even and x ∈ (ξ ,η)nHx n√Hx+Kx, n even and x ∈ [η ,b](10)where ξ and η are the real roots of the equations(ξ −a)n−Hξ = 0 (11)and(η−b)n−Hη = 0 (12)respectively and a < ξ < a+b2 < η < b. The inequality (9) with (10)-(12) issharp.Proof. For all x ∈ [a,b], it is clear that by applying (6) and (3) we can derive8 ZHENG LIU∣∣∣∣∣∫ baf (t)dt−n−1∑k=01(k+1)![(x−a)k+1 f (k)(a)+(−1)k(b− x)k+1 f (k)(b)]−Hx[ f(n−1)(b)− f (n−1)(a)]n!∣∣∣∣∣=1n!∣∣∣∣∫ ba (x− t)n f (n)(t)dt−Hx∫ baf (n)(t)dt∣∣∣∣≤ Γ− γ2n!∫ ba|(x− t)n−Hx|dt. (13)For brevity, putp(x, t) = (x− t)n−Hx, (x, t) ∈ [a,b]× [a,b]. (14)Then∂ p(x, t)∂ t=−n(x− t)n−1. (15)If n is odd, we see from (15) that for any fixed x ∈ [a,b], p(x, t) is a strictlydecreasing continuous function for t ∈ [a,b], and from (14) we havep(x,a) = (x−a)n[1− x−a(n+1)(b−a)]+(x−b)n+1(n+1)(b−a) > 0,p(x,b) = (x−b)n[1− b− x(n+1)(b−a)]− (x−a)n+1(n+1)(b−a) < 0.So, by the intermediate value theorem we can conclude that for any fixed x ∈[a,b], p(x, t) has unique zero t1 = x− n√Hx in (a,b). Thus the last integral in(13) is equal to∫ ba|p(x, t)|dt =∫ t1ap(x, t)dt−∫ bt1p(x, t)dt =2n+1(nHx n√Hx+Kx). (16)If n is even, we see from (15) that for any fixed x ∈ [a,b], p(x, t) is strictlydecreasing for t ∈ (a,x) and strictly increasing for t ∈ (x,b). From (14), we havep(x,a) = (x−a)n− (x−a)n+1+(b− x)n+1(n+1)(b−a) ,TWO NEW SHARP OSTROWSKI-GRU¨SS TYPE INEQUALITIES 9p(x,b) = (x−b)n− (x−a)n+1+(b− x)n+1(n+1)(b−a) ,and then∂ p(x,a)∂x= (x−a)n−1[n− x−ab−a ]+(b− x)nb−a > 0,∂ p(x,b)∂x= (x−b)n−1[n− b− xb−a ]−(x−a)nb−a < 0.Thus we see that p(x,a) is strictly increasing for x ∈ [a,b] and p(x,b) is strictlydecreasing for x ∈ [a,b], andp(a,a) = p(b,b) =−(b−a)nn+1< 0,p(a+b2,a) = p(a+b2,b) =nn+1(b−a2)n > 0.So, by the intermediate value theorem we can conclude that there exist uniqueξ ∈ (a, a+b2 ) and unique η ∈ (a+b2 ,b) such that p(ξ ,a) = 0 and p(η ,b) = 0,i. e. ξ and η are the real roots of equations (11) and (12) respectively anda< ξ < a+b2 <η < b. If x∈ [a,ξ ], then p(x,a)≤ 0 and p(x,b)> 0. If x∈ (ξ ,η),then p(x,a)> 0 and p(x,b)> 0. If x ∈ [η ,b], then p(x,a)> 0 and p(x,b)≤ 0.Therefore, there are three possible cases to be determined.(i) In case x ∈ [a,ξ ], p(x, t) ≤ 0 for t ∈ [a,x] and p(x, t) has a zero t2 =x+ n√Hx in (x,b). We have∫ ba|p(x, t)|dt =−∫ t2ap(x, t)dt+∫ bt2p(x, t)dt =2n+1(nHx n√Hx−Kx). (17)(ii) In case x ∈ (ξ ,η), p(x, t) has a zero t1 = x− n√Hx in (a,x)and a zerot2 = x+ n√Hxin (x,b). We have∫ ba|p(x, t)|dt =∫ t1ap(x, t)dt−∫ t2t1p(x, t)dt+∫ bt2p(x, t)dt=4nn+1Hx n√Hx. (18)(iii) In case x∈ [η ,b], p(x, t) has a zero t1 = x− n√Hx in (a,x) and p(x, t)≤ 0for t ∈ [x,b]. We have10 ZHENG LIU∫ ba|p(x− t)|dt =∫ t1ap(x, t)dt−∫ bt1p(x, t)dt =2n+1(nHx n√Hx+Kx). (19)Consequently, the inequality (9) with (10)-(12) follows from (13) and (16)-(19). The function f for which the equality in (9) is attained is in fact such thatf (n) is equal to Γ (or γ) when p(x, t)≥ 0 and equal to γ (or Γ) when p(x, t)< 0.The proof is completed.Remark 2.2. It is not difficult to find that the sharp inequality (9) with (10)-(12)provides an improvement of the inequality (4).Remark 2.3. If we take x = a+b2 , we can obtain the following sharp perturbedtrapezoid inequality:∣∣∣∣∫ ba f (t)dt− b−a2 [ f (a)+ f (b)]−n−1∑k=11(k+1)!(b−a2)k+1[ f (k)(a)+(−1)k f (k)(b)]−1+(−1)n(n+1)!(b−a2)n+1f (n−1)(b)− f (n−1)(a)b−a∣∣∣∣∣≤ Γ− γ(n+1)![1− (−1)n2+n+(−1)nn(n+1) n√n+1](b−a2)n+1.Theorem 2.4. Let f : [a,b]→ R be a mapping such that the derivative f (n−1)(n ≥ 1) is absolutely continuous on [a,b]. Assume that there exist constantsγ,Γ ∈ R such that γ ≤ f (n)(t) ≤ Γ a.e. on [a,b]. Then for all x ∈ [a,b], thefollowing inequality holds:∣∣∣∣∣∫ baf (t)dt−n−1∑k=01(k+1)![(b− x)k+1+(−1)k(x−a)k+1] f (k)(x)−Hx[ f(n−1)(b)− f (n−1)(a)]n!∣∣∣∣∣≤ Γ− γ(n+1)!G(a,b,x;n) (20)where G(a,b,x;n) is as defined in (10)-(12). The inequality (20) with (10)-(12)is sharp.TWO NEW SHARP OSTROWSKI-GRU¨SS TYPE INEQUALITIES 11Proof. For all x∈ [a,b], it is clear that by applying (7), (8) and (3) we can derive∣∣∣∣∣∫ baf (t)dt−n−1∑k=01(k+1)![(b− x)k+1+(−1)k(x−a)k+1] f (k)(x)−Hx[ f(n−1)(b)− f (n−1)(a)]n!∣∣∣∣∣=∣∣∣∣∫ ba Kn(x, t) f (n)(t)dt− 1b−a∫ baKn(x, t)dt∫ baf (n)(t)dt∣∣∣∣≤ Γ− γ2∫ ba∣∣∣∣Kn(x, t)− 1b−a∫ baKn(x,y)dy∣∣∣∣ dt=Γ− γ2n![∫ xa|(t−a)n−Hx| dt+∫ bx|(t−b)n−Hx| dt].By substituting t1 = a+ x− t and t2 = b+ x− t, we get∫ xa|(t−a)n−Hx|dt+∫ bx|(t−b)n−Hx|dt=∫ xa|(x− t1)n−Hx|dt1+∫ bx|(x− t2)n−Hx|dt2=∫ ba|(x− t)n−Hx|dt,and so the proof is reduced to the proof of Theorem 2.1.Remark 2.5. It is not difficult to find that the inequality (20) with (10)-(12)provides an improvement of the inequality (5).Remark 2.6. If we take x = a+b2 , we can obtain the following sharp perturbedmidpoint inequality:∣∣∣∣∣∫ baf (t)dt− (b−a) f (a+b2)−n−1∑k=11+(−1)k(k+1)!(b−a2)k+1 f (k)(a+b2)−1+(−1)n(n+1)!(b−a2)n+1f (n−1)(b)− f (n−1)(a)b−a∣∣∣∣∣≤ Γ− γ(n+1)![1− (−1)n2+n+(−1)nn(n+1) n√n+1](b−a2)n+1.12 ZHENG LIURemark 2.7. Setting x = a and x = b in (9) and (20) yields the same left andright rectangle inequalities∣∣∣∣∣∫ baf (t)dt− (b−a) f (a)−n−1∑k=1(b−a)k+1(k+1)!f (k)(a)−(b−a)n+1(n+1)!f (n−1)(b)− f (n−1)(a)b−a∣∣∣∣∣≤ Γ− γ(n+1)!n(b−a)n+1(n+1) n√n+1and∣∣∣∣∣∫ baf (t)dt− (b−a) f (b)−n−1∑k=1(−1)k(b−a)k+1(k+1)!f (k)(b)−(−1)n(b−a)n+1(n+1)!f (n−1)(b)− f (n−1)(a)b−a∣∣∣∣∣≤ Γ− γ(n+1)!n(b−a)n+1(n+1) n√n+1respectively.Remark 2.8. If we take n = 2 in (20), then we recapture the following sharpinequality∣∣∣∣ f (x)− (x− a+b2 ) f ′(x)+ [ 124(b−a)2+ 12(x− a+b2 )2] f ′(b)− f ′(a)b−a− 1b−a∫ baf (t)dt∣∣∣∣≤ (Γ2− γ2)G(a,b,x)where G(a,b,x) ==13(b−a) [(x−a)(a+b2 − x)(b− x)+( 112(b−a)2+(x− a+b2 )2)32 ], a≤ x≤ 13(2a+b),23(b−a) [112(b−a)2+(x− a+b2 )2]32 , 13(2a+b)< x <13(a+2b)13(b−a) [(x−a)(x− a+b2 )(b− x)+( 112(b−a)2+(x− a+b2 )2)32 ], 13(a+2b)≤ x≤ b,TWO NEW SHARP OSTROWSKI-GRU¨SS TYPE INEQUALITIES 13which has been proved in [5, Theorem 1.5] and [7, Theorem 2] in different ways.Finally, it should be noticed that the bounds of inequality (9) in Theorem 2.1and inequality (20) in Theorem 2.4 are the same which has verified and extendedthe results in [1].Acknowledgment.The author wishes to thank the referee for his helpful comments and informationon reference [1] which has been unaware.REFERENCES[1] P. Cerone, On relationships between Ostrowski, trapezoidal and Chebychev iden-tites and inequalities, Soochow J. Math. 28 (3) (2002), 311–328.[2] P. Cerone - S. S. Dragomir, Trapezoidal type rules from an inequalities point ofview, Handbook of Analytic-Computational Methods in Applied Mathematics,CRC Press N.Y. 2000, 65–134.[3] P. Cerone - S. S. Dragomir, Midpoint type rules from an inequalities point of view,Handbook of Analytic-Computational Methods in Applied Mathematics, CRCPress N.Y. 2000, 135–200.[4] P. Cerone - S. S. Dragomir, A refinement of the Gru¨ss inequality and applications,RGMIA Research Report Collection 5 (2002) Article 14. Tamkang J. Math. 38(1) (2007), 37–49.[5] X. L. Cheng, Improvement of some Ostrowski-Gru¨ss inequalities, Computer Math.Appl. 42 (2001), 109–114.[6] X. L. Cheng - J. Sun, A note on the perturbed trapezoid inequality, Jour-nal of Inequalities in Pure and Applied Mathematics, 3 (2) (2002), Article 29(http://jipam.vu.edu.au/).[7] Z. Liu, Some sharp Ostrowski-Gru¨ss type inequalities, Univ. Beograd. Publ. Elek-trotehn. Fak. Ser. Mat. 18 (2006), 14–21.[8] Z. Liu, Notes on a Gru¨ss type inequality and its application, Vietnam Journal ofMathematics, 2 (35)(2007), 121–128.[9] M. Matic´, Improvement of some estimations related to the remainder in gener-alized Taylor’s formula, Mathematical Inequalities & Applications 5 (4) (2002),637–648.[10] M. Matic´ - J. Pecˇaric´ - N. Ujevic´, Improvement and further generalization of in-equalities of Ostrowski-Gru¨ss type, Computer Math. Appl. 39 (2000), 161–175.14 ZHENG LIU[11] D. S. Mitrinovic´ - J. Pecˇaric´ - A. M. Fink, Classical and new inequalities in anal-ysis, Kluwer Academic Publishers, Dordrecht, 1993.ZHENG LIUInstitute of Applied Mathematics, School of ScienceUniversity of Science and Technology LiaoningAnshan 114051, Liaoning, Chinae-mail: lewzheng@163.net

Two New Sharp Ostrowski-Grüss Type Inequalities

Zheng Liu

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Two New Sharp Ostrowski-Grüss Type Inequalities

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