Dragomir, Sever S

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A Companion of Ostrowski's Inequality for Functions of Bounded Variation and ApplicationsThis is the Published version of the following publicationDragomir, Sever S (2014) A Companion of Ostrowski's Inequality for Functionsof Bounded Variation and Applications. International Journal of Nonlinear Analysis and Applications, 5 (1). pp. 89-97. ISSN 2008-6822  The publisher’s official version can be found at https://ijnaa.semnan.ac.ir/article_118.htmlNote that access to this version may require subscription.Downloaded from VU Research Repository  https://vuir.vu.edu.au/40605/ Int. J. Nonlinear Anal. Appl. 5 (2014) No. 1, 89-97ISSN: 2008-6822 (electronic)http://www.ijnaa.semnan.ac.irA Companion of Ostrowski’s Inequality forFunctions of Bounded Variation and ApplicationsSever S. DragomiraaSchool of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050,South Africa.Dedicated to the Memory of Charalambos J. Papaioannou(Communicated by Th. M. Rassias)AbstractA companion of Ostrowski’s inequality for functions of bounded variation and applications are given.Keywords: Ostrowski’s Inequality, Trapezoid Rule, Midpoint Rule.2000 MSC: 26D15, 41A55.1. IntroductionIn [11], the author has proved the following inequality of Ostrowski type [24] for functions of boundedvariation.Theorem 1.1. Let f : [a, b]→ R be a function of bounded variation on [a, b] . Denote by∨ba (f) itstotal variation on [a, b] . Then, for any x ∈ [a, b], one has the inequality:∣∣∣∣f (x)− 1b− a∫ baf (t) dt∣∣∣∣ ≤[12+∣∣∣∣∣x− a+b2b− a∣∣∣∣∣]b∨a(f) . (1.1)The constant 12is best possible in the sense that it cannot be replaced by a smaller quantity.The above inequality (1.1) has as a remarkable particular case, the mid-point inequality, namely|f(a+b2)− 1b−a∫ baf (t) dt∣∣∣ ≤ 12 ∨ba (f) .Here 12is a best constant as well.The corresponding version for the generalized trapezoid inequality was obtained in [4].∗Corresponding authorEmail address: sever.dragomir@vu.edu.au (Sever S. Dragomir)Received: January 2013 Revised: January 201490 S. S. DragomirTheorem 1.2. With the assumptions in Theorem 1.1, one has the inequality|(x− a) f (a) + (b− x) f (b)b− a− 1b− a∫ baf (t) dt∣∣∣∣ ≤[12+∣∣∣∣∣x− a+b2b− a∣∣∣∣∣]b∨a(f) (1.2)for any x ∈ [a, b] .Here the constant 12is also best possible.The trapezoid inequality|f(a)+f(b)2− 1b−a∫ baf (t) dt∣∣∣ ≤ 12 ∨ba (f)is the best inequality one can derive from (1.2). Here the constant 12is also sharp.Recently, Guessab and Schmeisser [23], in the effort of incorporating together the mid-point andtrapezoid inequality, have proved amongst others, the following companion of Ostrowski’s inequality.Theorem 1.3. Assume that the function f : [a, b]→ R is of H − r−Hölder type with r ∈ (0, 1], i.e.,|f (t)− f (s)| ≤ H |t− s|r foranyt, s ∈ [a, b] . (1.3)Then, for each x ∈[a, a+b2], one has the inequality∣∣∣∣f (x) + f (a+ b− x)2 − 1b− a∫ baf (t) dt∣∣∣∣ ≤[2r+1 (x− a)r+1 + (a+ b− 2x)r+12r (r + 1) (b− a)]H. (1.4)This inequality is sharp for each admissible x. Equality is obtained if and only if f = ±Hf∗+ c, withc ∈ R andf∗ (t) =(x− t)r , fora ≤ t ≤ x(t− x)r , forx ≤ t ≤ 12(a+ b)f∗ (a+ b− t) , for 12 (a+ b) ≤ t ≤ b.(1.5)Remark 1.4. For r = 1, i.e., f is Lipschitzian with the constant L > 0, and since4(x− a)2 + (a+ b− 2x)24(b− a)=18+ 2(x− 3a+b4b− a)2 (b− a)then, by (1.4), we get the following companion of Ostrowski’s inequality for Lipschitzian functions∣∣∣∣f (x) + f (a+ b− x)2 − 1b− a∫ baf (t) dt∣∣∣∣ ≤18+ 2(x− 3a+b4b− a)2 (b− a)L, (1.6)for any x ∈[a, a+b2].The constant 18is best possible in (1.6) in the sense that it cannot be replaced by a smaller constant.By substituting x = 3a+b4into the above inequality, we obtain the following trapezoid type inequal-ity, which is the best in the class,∣∣∣∣∣f(3a+b4)+ f(a+3b4)2− 1b− a∫ baf (t) dt∣∣∣∣∣ ≤ 18 (b− a)L. (1.7)The constant 18here is also best possible in the above sense.A Companion of Ostrowski’s Inequality ...5 (2014) No. 1,89-97 91For a monograph devoted to Ostrowski type inequalities, see [18].For research papers on Ostrowski’s inequality see [1]-[17], [19]-[21] and [22].The main aim of this paper is to provide a sharp bound for the differencef (x) + f (a+ b− x)2− 1b− a∫ baf (t) dt,where f is assumed to be of bounded variation. Some applications for cumulative distributionfunction and quadrature rules are also given.2. Some Integral InequalitiesThe following identity holds.Lemma 2.1. Assume that the function f : [a, b] → R is of bounded variation on [a, b] . Then wehave the equality12[f (x) + f (a+ b− x)]− 1b− a∫ baf (t) dt (2.1)=1b− a[∫ xa(t− a) df (t) +∫ a+b−xx(t− a+ b2)df (t) +∫ ba+b−x(t− b) df (t)]for any x ∈[a, a+b2].Proof . Obviously, all the Riemann-Stieltjes integrals from the right hand side of (2.1) exist becausethe functions (· − a) ,(· − a+b2)and (· − b) are continuous on these intervals and f is of boundedvariation.Using the integration by parts formula for Riemann-Stieltjes integrals, we have, for any x ∈[a, a+b2], that ∫ xa(t− a) df (t) = f (x) (x− a)−∫ xaf (t) dt,∫ a+b−xx(t− a+ b2)df (t) = f (a+ b− x)(a+ b2− x)− f (x)(x− a+ b2)−∫ a+b−xxf (t) dtand ∫ ba+b−x(t− b) df (t) = (x− a) f (a+ b− x)−∫ ba+b−xf (t) dt.Summing the above equalities we deduce (2.1). Remark 2.2. A version of this identity for piecewise continuously differentiable functions has beenobtained in [23, Lemma 3.2].The following companion of Ostrowski’s inequality holds.Theorem 2.3. Assume that the function f : [a, b] → R is of bounded variation on [a, b] . Then wehave the inequalities:∣∣∣∣12 [f (x) + f (a+ b− x)]− 1b− a∫ baf (t) dt∣∣∣∣ (2.2)92 S. S. Dragomir≤ 1b−a[(x− a)∨xa (f) +(a+b2− x)∨a+b−xx (f) + (x− a)∨ba+b−x (f)]≤[14 +∣∣x− 3a+b4b− a∣∣]ba(f)[2 (x− ab− a)α +(a+b2− xb− a)α] 1α×[[x∨a(f)]β+[a+b−x∨x(f)]β+[b∨a+b−x(f)]β] 1β, if α > 1, 1α+ 1β= 1[x− a+ b−a2b− a]max{x∨a(f) ,a+b−x∨x(f) ,b∨a+b−x(f)}for any x ∈[a, a+b2],where∨dc (f) denotes the total variation of f on [c, d] . The constant14is best possible in the firstbranch of the second inequality in (2.2).Proof . We use the fact that for a continuous function p : [c, d] → R and a function v : [a, b] → Rof bounded variation, one has the inequality∣∣∣∣∫ dcp (t) dv (t)∣∣∣∣ ≤ supt∈[c,d]|p (t)|d∨c(v) . (2.3)Taking the modulus in (2.1) we have∣∣∣12 [f (x) + f (a+ b− x)]− 1b−a ∫ ba f (t) dt∣∣∣≤ 1b−a[∣∣∫ xa(t− a) df (t)∣∣+ ∣∣∣∫ a+b−xx (t− a+b2 ) df (t)∣∣∣+∣∣∣∫ ba+b−x (t− b) df (t)∣∣∣]≤ 1b−a[(x− a)x∨a(f) +(a+b2− x) a+b−x∨x(f) + (x− a)b∨a+b−x(f)]=: M (x)and the first inequality in (2.2) is obtained.Now, observe thatM(x) ≤ 1b−a max{x− a, a+b2− x} [ x∨a(f) +a+b−x∨x(f) +b∨a+b−x(f)]= 1b−a[14(b− a) +∣∣x− 3a+b4∣∣] b∨a(f)and the first branch in the second inequality in (2.2) is proved.Using Hölder’s discrete inequality we have (for α > 1, 1α+ 1β= 1) thatM(x) ≤ 1b−a[(x− a)α +(a+b2− x)α+ (x− a)α] 1α×[[x∨a(f)]β+[a+b−x∨x(f)]β+[b∨a+b−x(f)]β] 1βgiving the second branch in the second inequality.Finally, we haveM(x) ≤ 1b−a max{x∨a(f) ,a+b−x∨x(f) ,b∨a+b−x(f)}×[(x− a) +(a+b2− x)+ (x− a)],which is equivalent with the last inequality in (2.2).The sharpness of the constant 14in the first branch of the second inequality in (2.2) will be provedin a particular case later. A Companion of Ostrowski’s Inequality ...5 (2014) No. 1,89-97 93Corollary 2.4. With the assumptions in Theorem 2.3, one has the trapezoid inequality∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (t) dt∣∣∣∣ ≤ 12b∨a(f) . (2.4)The constant 12is best possible in (2.4).Proof . Follows from the first inequality in (2.2) on choosing x = a. For the sharpness of theconstant, assume that (2.4) holds with a constant A > 0, i.e.,∣∣∣∣f (a) + f (b)2 − 1b− a∫ baf (t) dt∣∣∣∣ ≤ A b∨a(f) . (2.5)If we choose f : [a, b]→ R withf (x) =1 if x = a,0 if x ∈ (a, b) ,1 if x = b,then f is of bounded variation on [a, b] andf (a) + f (b)2= 1,∫ baf (t) dt = 0, andb∨a(f) = 2,giving in (2.5) 1 ≤ 2A, thus A ≥ 12and the corollary is proved. Remark 2.5. The inequality (2.4) was first proved in a different manner in [8].Corollary 2.6. With the assumptions in Theorem 2.3, one has the midpoint inequality∣∣∣∣f (a+ b2)− 1b− a∫ baf (t) dt∣∣∣∣ ≤ 12b∨a(f) . (2.6)The constant 12is best possible in (2.6).Proof . Follows from the first inequality in (2.2) on choosing x = a+b2. For the sharpness of theconstant, assume that (2.6) holds with a constant B > 0, i.e.,∣∣∣∣f (a+ b2)− 1b− a∫ baf (t) dt∣∣∣∣ ≤ B b∨a(f) . (2.7)If we choose f : [a, b]→ R withf (x) =0 if x ∈[a, a+b2),1 if x = a+b2,0 if x ∈(a+b2, b],then f is of bounded variation on [a, b] , and f(a+b2)= 1,∫ baf (t) dt = 0, andb∨a(f) = 2, giving in(2.7), 1 ≤ 2B, thus B ≥ 12. 94 S. S. DragomirRemark 2.7. The inequality (2.6) was firstly proved in a different manner in [9].The best inequality we may get from Theorem 2.3 on using the bound provided by the first branchin the second inequality in (2.2) is incorporated in the following corollary.Corollary 2.8. With the assumptions in Theorem 2.3, one has the inequality:∣∣∣∣∣f(3a+b4)+ f(a+3b4)2− 1b− a∫ baf (t) dt∣∣∣∣∣ ≤ 14b∨a(f) . (2.8)The constant 14is best possible.Proof . Follows by Theorem 2.3 on choosing x = 3a+b4.To prove the sharpness of the constant 14, assume that (2.8) holds with a constant C > 0, i.e.,∣∣∣∣∣f(3a+b4)+ f(a+3b4)2− 1b− a∫ baf (t) dt∣∣∣∣∣ ≤ Cb∨a(f) . (2.9)Consider the function f : [a, b]→ R, given byf (x) =1 if x ∈{3a+b4, a+3b4},0 if x ∈ [a, b] \{3a+b4, a+3b4}.Then f is of bounded variation on [a, b] ,f(3a+b4)+ f(a+3b4)2= 1,∫ baf (t) dt = 0andb∨a(f) = 4,giving in (2.9) 4C ≥ 1, thus C ≥ 14.This example can be used to prove the sharpness of the constant 14in (2.2) as well. 3. Applications for P.D.F.’sLet X be a random variable taking values in the finite interval [a, b] , with the probability densityfunction f : [a, b] → [0,∞) and with the cumulative distribution function F (x) = Pr (X ≤ x) =∫ xaf (t) dt.We may state the following theorem.Theorem 3.1. With the above assumptions, we have the inequality∣∣∣∣12 [F (x) + F (a+ b− x)]− b− E (X)b− a∣∣∣∣ (3.1)≤ 1b−a{(2x− 3a+b4)[F (x)− F (a+ b− x)] + (x− a)}≤ 14+∣∣x− 3a+b4b− a∣∣ ,foranyx∈[a, a+b2], where E (X) denotes the expectation of X, namely E (X) =∫ batdF (t) .A Companion of Ostrowski’s Inequality ...5 (2014) No. 1,89-97 95Proof . If we apply Theorem 2.3 for F, which is monotonic nondecreasing, we get∣∣∣∣12 [F (x) + F (a+ b− x)]− 1b− a∫ baF (t) dt∣∣∣∣ (3.2)≤ 1b−a[(x− a)F (x) +(a+b2− x)(F (a+ b− x)− F (x))+ (x− a) (1− F (a+ b− x))]≤ 14+∣∣x− 3a+b4b− a∣∣ .SinceE (X) =∫ batdF (t) = b−∫ baF (t) dt,thenby(3.2)weget(3.1)andthetheoremisproved.In particular, we have:Corollary 3.2. With the above assumptions, we have:∣∣∣∣12[F(3a+ b4)+ F(a+ 3b4)]− b− E (X)b− a∣∣∣∣ ≤ 14 .4. A Composite Quadrature FormulaLet In : a = x0 < x1 < · · · < xn−1 < xn = b be a division of the interval [a, b] and hi := xi+1 − xi(i = 0, . . . , n− 1) and ν (In) := max {hi|i = 0, . . . , n− 1} .Consider the composite quadrature ruleQn (In, f) =12n−1∑i=0[f(3xi + xi+14)+ f(xi + 3xi+14)]hi. (4.1)The following result holds.Theorem 4.1. Let f : [a, b]→ R be a function of bounded variation on [a, b] . Then we have∫ baf (t) dt = Qn (In, f) +Rn (In, f) (4.2)where Qn (In, f) is defined in formula (4.1), and the remainder Rn (In, f) satisfies the estimate|Rn (In, f)| ≤14ν (In)b∨a(f) . (4.3)The constant 14is best possible.Proof . Applying Corollary 2.8 on the interval [xi, xi+1] we may state that∣∣∣∣∫ xi+1xif (t) dt− 12[f(3xi + xi+14)+ f(xi + 3xi+14)]hi∣∣∣∣ ≤ 14hixi+1∨xi(f) , (4.4)for any i ∈ {0, . . . , n− 1} .Summing the inequality (4.4) over i from 0 to n− 1, and using the generalized triangle inequalitywe get|Rn (In, f)| ≤14n−1∑i=0hixi+1∨xi(f) ≤ 14ν (In)n−1∑i=0xi+1∨xi(f) =14ν (In)b∨a(f) ,96 S. S. Dragomirand the proof is completed. For the particular case when the division In is equidistant, i.e.,In : xi = a+ i ·b− an, i = 0, . . . , n,we may consider the quadrature rule:Qn (f) :=b− a2nn−1∑i=0{f[a+(4i+ 14n)(b− a)]+ f[a+(4i+ 34n)(b− a)]}.The following corollary will be more useful in practice.Corollary 4.2. With the assumption of Theorem 4.1, we have∫ baf (t) dt = Qn (f) +Rn (f) , (4.5)where Qn (f) is defined by (4) and the remainder Rn (f) satisfies the estimate|Rn (f)| ≤14· b− anb∨a(f) . (4.6)The constant 14is sharp.Remark 4.3. If one is interested in finding the minimal number of points for the equidistant partitionIn so that the theoretical error in (4.6) is smaller that ε > 0, then this number nε is given bynε :=[14· b− aεb∨a(f)]+ 1, (4.7)where [a] denotes the integer part of the positive number a.References[1] G. A. Anastassiou, Univariate Ostrowski inequalities, revisited. Monatsh. Math., 135 (3) (2002) 175-189.[2] P. Cerone and S. S. Dragomir, Midpoint-type rules from an inequalities point of view, Ed. G. A. Anastassiou,Handbook of Analytic-Computational Methods in Applied Mathematics, CRC Press, New York, 135-200.[3] P. Cerone and S. S. Dragomir, New bounds for the three-point rule involving the Riemann-Stieltjes integrals, inAdvances in Statistics Combinatorics and Related Areas, C. Gulati, et al. (Eds.), World Science Publishing, 53-62(2002).[4] P. Cerone, S. S. Dragomir and C. E. M. Pearce, A generalised trapezoid inequality for functions of boundedvariation, Turkish J. Math., 24 (2) (2000) 147-163.[5] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time differentiable mappingsand applications, Demonstratio Mathematica, 32 (2) (1999) 697-712.[6] S. S. Dragomir, Ostrowski’s inequality for monotonous mappings and applications, J. KSIAM, 3 (1) (1999) 127-135.[7] S. S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Comp. Math.Appl., 38 (1999) 33-37.[8] S. S. Dragomir, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc.,60 (1) (1999) 495-508.[9] S. S. Dragomir, On the midpoint quadrature formula for mappings with bounded variation and applications,Kragujevac J. Math., 22 (2000) 13-18.A Companion of Ostrowski’s Inequality ...5 (2014) No. 1,89-97 97[10] S. S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000)477-485.[11] S. S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications,Math. Ineq. & Appl., 4 (1) (2001) 33-40.[12] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral∫ baf (t) du (t) where f is of Hölder typeand u is of bounded variation and applications, J. KSIAM, 5 (1) (2001) 35-45.[13] S. S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure & Appl. Math., 3 (5)(2002) Art. 68.[14] S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined onlinear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math., 3 (2) (2002) Article 31, 8pages.[15] S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense,16 (2) (2003) 373-382.[16] S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type, Springer Briefs in Mathematics. Springer,New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1.[17] S. S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang, A weighted version of Ostrowski inequality for mappingsof Hölder type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Romanie, 42 (90) (4) (1999)301-314.[18] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration,Kluwer Academic Publishers, Dordrechy/Boston/London, 2002.[19] S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in L1−norm and applications to some specialmeans and to some numerical quadrature rules, Tamkang J. of Math., 28 (1997) 239-244.[20] S. S. Dragomir and S. Wang, Applications of Ostrowski’s inequality to the estimation of error bounds for somespecial means and some numerical quadrature rules, Appl. Math. Lett., 11 (1998) 105-109.[21] S. S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in Lp−norm and applications to some specialmeans and to some numerical quadrature rules, Indian J. of Math., 40 (3) (1998) 245-304.[22] A. M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J., 42 (2) (1992) 298-310.[23] A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Th., 115(2002) 260-288.[24] A. Ostrowski, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Com-ment. Math. Hel, 10 (1938) 226-227.

A Companion of Ostrowski's Inequality for Functions of Bounded Variation and Applications

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A Companion of Ostrowski's Inequality for Functions of Bounded Variation and Applications

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