[[abstract]]The aim of the present article is to establish two new discrete Ostrowski and Gr"uss type inequalities involving functions and their first second forward differences.[[notice]]補正完畢[[journaltype]]國

Hwang, Dah-yan

The aim of the present article is to establish two new discrete Ostrowski and Gr"uss type inequalities involving functions and their first second forward differences.補正完畢國

Tamkang University Institutional Repository

TAMKANG JOURNAL OF MATHEMATICSVolume 35, Number 1, Spring 2004ON OSTROWSKI AND GRU¨SS TYPE DISCRETE INEQUALITIES FORSECOND FORWARD DIFFERENCESDAH-YAN HWANG∗ AND GOU-SHENG YANGAbstract. The aim of the present article is to establish two new discrete Ostrowski and Gru¨sstype inequalities involving functions and their first second forward differences.1. IntroductionIn [10, p.468], Ostrowski proved the following interesting inequality.Let f : [a, b]→ R be continuous on [a, b] and differentiable on (a, b) whose derivativeis bounded on (a, b) i.e. ‖f ′‖∞ := supt∈(a,b) |f′(t)| <∞. Then∣∣∣∣∣f(x)− 1b− a∫ baf(t)dt∣∣∣∣∣ ≤[14+(x− a+b2)2(b− a)2](b − a)‖f ′‖∞ (1.1)for all x ∈ [a, b].In [1], P. Ceroue, S. S. Dragomir and J. Roumelioties proved the following Ostrowskitype inequality with twice differentiable mapping.Let f : [a, b] → R be continuous and twice differentiable mapping on (a, b) whosesecond derivate are bounded on (a, b), i.e. ‖f ′′‖∞ := supt∈(a,b) |f′′(t)| < ∞. Then wehave the inequality:∣∣∣∣∣f(x)− 1b− a∫ baf(t)dt−(x−a+ b2)f ′(x)∣∣∣∣∣≤[124(b− a)2 +12(x−a+ b2)2]‖f ′′‖∞ ≤(b− a)26‖f ′′‖∞ (1.2)for all x ∈ [a, b].Another celebrated inequality that gives estimation for the integral of a product interms of the product of integrals, is Gru¨ss inequality [9, p.296].Received December 16, 2002.2000 Mathematics Subject Classification. 26D15.Key words and phrases. Ostrowski inequality, Gru¨ss inequality.∗The first author thanks the Kuang Wu Institute of Technology for its generous support duringthe preparation of the manuscripts.8788 DAH-YAN HWANG AND GOU-SHENG YANGIf f and g are two integrable functions on [a, b] and φ, Φ, γ and Γ are constants suchthatφ ≤ f(x) ≤ Φ, γ ≤ g(x) ≤ Γfor all x ∈ [a, b], then∣∣∣∣∣ 1b− a∫ baf(x)g(x)dx −1b− a∫ baf(x)dx ·1b− a∫ bag(x)dx∣∣∣∣∣ ≤ (Φ− φ)(Γ − γ)4 . (1.3)In [14], B. G. Pachpatte proved the following Gru¨ss type inequality twice differentiablemapping.Let f , g : [a, b] → R be continuous on [a, b] and twice differentiable on (a, b), whosesecond derivatives are bounded on (a, b), i.e.‖f ′′‖∞ := supt∈(a,b)|f ′′(t)| <∞ and ‖g′′‖∞ := supt∈(a,b)|g′′(t)| <∞.Then∣∣∣∣∣ 1b−a∫ baf(x)g(x)dx−1b−a∫ baf(x)dx ·1b−a∫ bag(x)dx−12(b−a)∫ ba(x−a+b2)(fg)′(x)dx∣∣∣∣∣≤12(b− a)2∫ ba(‖f ′′‖∞|g(x)|+ ‖g′′‖∞|f(x)|)B(x)dx (1.4)where B(x) =∫ ba|K(x, t)|dt.for x ∈ [a, b], in whichK(x, t) =(t− a)22if t ∈ [a, x](b− a)22if t ∈ (x, b].In the past years, a large number of papers have appeared in the literature that dealwith various extensions and applications of inequalities (1.1), (1.2), (1.3) and (1.4), see[2-14] and the reference given therein. Recently, in [15], B. G. Pachpatte established twonew discrete types of the inequality (1.1) and (1.3) involving functions and their firstorder forward differences. The main purpose of the present note is to establish two newdiscrete types of the inequality (1.2) and (1.4) involving functions and their first andsecond forward differences.2. Statement of ResultsIn what followings, R and N denote the set of real numbers and natural numbers,respectively, and Na,b = {a, a + 1, . . . , a + n = b} for a ∈ R, n ∈ N . For any functionOSTROWSKI AND GRU¨SS TYPE DISCRETE INEQUALITIES 89u(t), t ∈ Na,b, we define the operator ∆ by ∆u(t) = u(t+ 1)− u(t) and ∆2 by ∆2u(t) =∆u(t+ 1)−∆u(t). We use the usual convention that empty sum is taken to be zero.The first result reads as follows.Theorem 1. Let f , g be real-valued functions defined on Na,b+1 for which ∆f(t),∆g(t), ∆2f(t), ∆2g(t) exist and |∆2f(t)| ≤ A, |∆2g(t)| ≤ B on Na,b+1. Then∣∣∣∣∣f(t+ 1)g(t+ 1)− 12(b− a)[g(t+ 1)b−1∑s=af(s+ 2) + f(t+ 1)b−1∑s=ag(s+ 2)]−(t− a+b2)2[g(t+ 1)∆f(t) + f(t+ 1)∆g(t)]+14(b− a)[(f(b + 1)− f(a+ 1)) · g(t+ 1) + (g(b+ 1)− g(a+ 1) · f(t+ 1))]∣∣∣∣∣≤12(b− a)[Ag(t+ 1) +Bf(t+ 1)] ·H(t) (2.1)and∣∣∣∣∣f(t+1)− 1b−ab−1∑s=af(s+2)−(t−a+b2)∆f(t)+f(b+1)−f(a+1)2(b−a)∣∣∣∣∣≤ Ab−a ·H(t) (2.2)for n ∈ Na,b, whereH(t) =b−1∑s=a|r(t, s)|, (2.3)in whichr(t, s) =(s− a)22if s ∈ [a, t− 1](b − s)22if s ∈ [t, b](2.4)for all t, s ∈ Na,b and A, B are nonnegative constants.Theorem 2. Let f , g, ∆f , ∆g, ∆2f , ∆2g, A and B be as in Theorem 1. Then∣∣∣∣∣ 1b− ab−1∑t=af(t+ 1)g(t+ 1)−12(b− a)2[(b−1∑t=ag(t+ 1))(b−1∑t=af(t+ 2))+(b−1∑t=af(t+ 1))(b−1∑t=ag(t+ 2))]−12(b− a)b−1∑t=a[(t−a+ b2)(g(t+ 1)∆f(t) + f(t+ 1)∆g(t))]90 DAH-YAN HWANG AND GOU-SHENG YANG+14(b− a)2[(f(b + 1)− f(a+ 1))b−1∑t=ag(t+ 1) + (g(b + 1)− g(a+ 1))b−1∑t=af(t+ 1)] ∣∣∣∣∣≤12(b− a)2b−1∑t=a[(A|g(t+ 1)|+B|f(t+ 1)|) ·H(t)] (2.5)3. Proof of Theorem 1Using the summation by parts formula,β−1∑s=αu(s)∆v(s) = [u(β)v(β) − u(α)v(α)] −β−1∑s=αv(s+ 1)∆u(s),we haveβ−1∑s=αu(s)∆2v(s)= [u(β)∆v(β) − u(α)∆v(α)] −β−1∑s=α∆u(s)∆v(s + 1)= u(β)∆v(β) − u(α)∆(α) −∆u(β)v(β + 1) + ∆u(α)v(α + 1) +β−1∑s=αv(s+ 2)∆2u(s)where α, β ∈ Na,b and u, v are real-valued functions defined on Na,b+1.Let u(s) = (s−a)22 , v(s) = f(s), β = t, α = a. Then u(a) = 0, ∆u(s) = (s − a +12 )and ∆2u(s) = 1, so that for t ∈ Na,b, we havet−1∑s=a(s−a)22·∆2f(s)=(t−a)22·∆f(t)−(n−a+12)·f(t+1)+12f(a+1)+t−1∑s=af(s+2). (3.1)Similarly, we haveb−1∑s=t(b−s)22·∆2f(s)=−(b−t)22·∆f(t)−12f(b+1)+(n−b+12)·f(t+1)+b−1∑s=tf(s+2). (3.2)Adding (3.1), (3.2) and using (2.4), we getb−1∑s=ar(t, s)∆2f(s)= (b− a)(t−a+ b2)∆f(t)− (b − a)f(t+ 1) +f(a+ 1)− f(b+ 1)2(b− a)+b−1∑s=af(s+ 2)i.e.OSTROWSKI AND GRU¨SS TYPE DISCRETE INEQUALITIES 91f(t+1)−1b−ab−1∑s=af(s+2)−(t−a+b2)∆f(t)+f(b+1)−f(a+1)2(b− a)=−1b−ab−1∑s=ar(t, s)∆2f(s),(3.3)for all t ∈ Na,b.Similarly, we hvaeg(t+1)−1b−ab−1∑s=ag(s+2)−(t−a+b2)∆g(t)+g(b+1)−g(a+1)2(b−a)=−1b−ab−1∑s=ar(t, s)∆2g(s),(3.4)for all t ∈ Na,b.Multiplying (3.3) by g(t + 1) and (3.4) by f(t + 1), t ∈ Na,b, adding the resultingidentities and rewriting we getf(t+ 1)g(t+ 1)−12(b− a)g(t+ 1)b−1∑s=af(s+ 2)−12(b− a)f(t+ 1)b−1∑s=ag(s+ 2)−12(t−a+ b2)g(t+ 1)∆f(t)−12(t−a+ b2)f(t+ 1)∆g(t)f(b+ 1)− f(a+ 1)4(b− a)· g(t+ 1) +g(b+ 1)− g(a+ 1)4(b− a)· f(t+ 1)=−12(b− a)g(t+ 1)b−1∑s=ar(t, s)∆2f(s) +−12(b− a)f(t+ 1)b−1∑s=ar(t, s)∆2g(s) (3.5)From (3.5), we have∣∣∣∣∣f(t+ 1)g(t+ 1)− 12(b− a)[g(t+ 1)b−1∑s=af(s+ 2) + f(t+ 1)b−1∑s=ag(s+ 2)]−(t− a+b2)2[g(t+ 1)∆f(t) + f(t+ 1)∆g(t)]+14(b− a)[(f(b+ 1)− f(a+ 1))g(t+ 1) + (g(b+ 1)− g(a+ 1))f(t+ 1)]∣∣∣∣∣≤12(b− a)[A|g(t+ 1)|+B|f(t+ 1)|] ·b−1∑s=a|r(t, s)|=12(b− a)[A|g(t+ 1)|+B|f(t+ 1)|] ·H(t)which is the required inequality in (2.1).The inequality (2.2) follows immediately from (3.3). The proof is complete.92 DAH-YAN HWANG AND GOU-SHENG YANG4. Proof of Theorem 2From the hypotheses, as in the proof of Theorem 1, the identities (3.3), (3.4) and(3.5) hold. Summing both sides of (3.5) over n from a to b − 1, and rewriting we getb−1∑t=af(t+ 1)g(t+ 1)−12(b− a)[(b−1∑t=ag(t+ 1))(b−1∑t=af(t+ 2))+(b−1∑t=af(t+ 1))(b−1∑t=ag(t+ 2))]−12b−1∑t=a[(t−a+ b2)(g(t+ 1)∆f(t) + f(t+ 1)∆g(t))]+14(b− a)[(f(b+ 1)− f(a+ 1))b−1∑t=ag(t+ 1) + (g(b+ 1)− g(a+ 1))b−1∑t=af(t+ 1)]=−12(b−a)[(b−1∑t=ag(t+1)b−1∑s=ar(t, s)∆2f(s))+(b−1∑t=af(t+1)b−1∑s=ar(t, s)∆2g(s))]. (4.1)From (4.1) and using the properties of modulus, we observe that∣∣∣∣∣ 1b− ab−1∑t=af(t+ 1)g(t+ 1)−12(b− a)2[(b−1∑t=ag(t+ 1))(b−1∑t=af(t+ 2))+(b−1∑t=af(t+ 1))(b−1∑t=ag(t+ 2))]−12(b− a)b−1∑t=a[(t−a+ b2)(g(t+ 1)∆f(t) + f(t+ 1)∆g(t))]+14(b− a)2[(f(b+ 1)− f(a+ 1))b−1∑t=ag(t+ 1) + (g(b+ 1)− g(a+ 1))b−1∑t=af(t+ 1)] ∣∣∣∣∣≤12(b−a)2[b−1∑t=a(|g(t+1)|b−1∑s=a|r(t, s)| · |∆2f(s)|)+b−1∑t=a(|f(t+1)|b−1∑s=a|r(t, s)| · |∆2g(s)|)]≤12(b− a)2[b−1∑t=a(|g(t+ 1)| ·H(t) · A) +b−1∑s=a(|f(t+ 1)|H(t) · B)]≤12(b− a)2b−1∑t=a[(Ag(t+ 1)|+Bf(t+ 1)) ·H(t)].This proves the inequality (2.5). The proof is complete.OSTROWSKI AND GRU¨SS TYPE DISCRETE INEQUALITIES 93References[1] P. Cerone, S. S. Dragomir and J. Roumeliotis, An inequality of Ostrowski type for mappingswhose second derivatives are bounded and applications, RGMIA, Research Report Collection1(1998), 33-39.[2] P. Cerone, S. S. Dragomir and J. Roumeliotis, An Ostrowski type inequality for mappingswhose second derivatives belong to Lp(A,B) and applications, RGMIA, Research ReportCollection 1(1998), 41-49.[3] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-timediffernetiable mappings and applications, RGMIA, Research Report Collection 1(1998), 51-65.[4] P. Cerone, S. S. Dragomir and J. Roumeliotis, An Ostrowski type inequalities for weightedmappings with bounded second derivatives, RGMIA, Research Report Collection 1(1998),95-104.[5] P. Cerone, S. S. Dragomir and J. Roumeliotis, An inequality of Ostrowski – Gru¨ss type fortwice differentiable mappings and applications in numerical integration, RGMIA, ResearchReport Collection 1(1998), article 8.[6] P. Cerone, S. S. Dragomir and J. Roumeliotis, An Ostrowski type inequality for mappingswhose second derivatives are bounded and applications, RGMIA, Research Report Collection1(1998), 69-77.[7] S. S. Dragomir and N. S. Barnett, An Ostrowski type inequality for mapping whose secondderivatives are bounded and applications, RGMIA, Research Report Collection 1(1998),article 9.[8] S. S. Dragomir and A. Sofo, An integral inequality for twice-differentiable mappings andapplications, RGMIA, Research Report Collection 2(1999), Article 9.[9] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis,Kluwer Academic Publishers, Dordrecht, 1993.[10] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities for Functions and Their Inte-grals and Derivatives for Functions and Their Integrals and Derivatives, Kluwer AcademicPublishers, Dordrecht, 1994.[11] B. G. Pachpatte, On a new Ostrowski type inequality in two independent variables, TamkangJ. Math. 32(2001), 45-49.[12] B. G. Pachpatte, A note on Ostrowski type inequalities, Demonstratio Mathematica 35(2002), 27-30.[13] B. G. Pachpatte, On Gru¨ss type inequalities for double integrals, J. Math. Anal. Appl.267(2002), 454-459.[14] B. G. Pachpatte, On Trapezoid and Gru¨ss like integral inequalities, Tamkang J. Math. (toappear)[15] B. G. Pachpatte, A note Ostrowski and Gru¨ss type discrete inequalities, Tamkang J. Math.(to appear)Center for General Education, Kuang Wu Institute of Technology, Peito, Taipei, Taiwan, 11271,R.O.C.E-mail: dyhuang@kwit.edu.twDepartment of Mathematics, Tamkang University, Tamsui, Taiwan 25137, R.O.C.

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On ostrowski and grüss type discrete inequalities for second forward differences

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On ostrowski and grüss type discrete inequalities for second forward differences

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