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DRAGOMIR-AGARWAL TYPE INEQUALITIES FOR SEVERAL FAMILIES OF QUADRATURES I. FRANJIĆ J. PEČARIĆ Faculty of Food Technology and Biotechnology University of Zagreb Pierottijeva 6, 10000 Zagreb, Croatia EMail: ifranjic@pbf.hr Faculty of Textile Technology University of Zagreb Prilaz baruna Filipovića 28a 10000 Zagreb, Croatia EMail: pecaric@hazu.hr Received: 05 May, 2009 Accepted: 15 September, 2009 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26D15, 65D30, 65D32. Key words: Dragomir-Agarwal-type inequalities, k-convex functions, General 3-point, 4point and 5-point quadrature formulae, Corrected quadrature formulae. Abstract: Acknowledgements: Inequalities estimating the absolute value of the difference between the integral and the quadrature, i.e. the Dragomir-Agarwal-type inequalities, are given for the general 3, 4 and 5-point quadrature formulae, both classical and corrected. Beside values of the function in the chosen nodes, "corrected" quadrature formula includes values of the first derivative at the end points of the interval and has a higher accuracy than the adjoint classical quadrature formula. The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 058-1170889-1050 (first author) and 117-1170889-0888 (second author). Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 1 of 23 Go Back Full Screen Close Contents 1 Introduction 3 2 Preliminaries 5 3 Main Result 3.1 CASE α = Q3 and n = 3, 4 . 3.2 CASE α = CQ3 and n = 5, 6 3.3 CASE α = Q4 and n = 3, 4 . 3.4 CASE α = CQ4 and n = 5, 6 3.5 CASE α = Q5 and n = 5, 6 . 3.6 CASE α = CQ5 and n = 7, 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 12 13 15 16 17 17 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 2 of 23 Go Back Full Screen Close 1. Introduction The well-known Hermite-Hadamard inequality states that if f : [a, b] → R is a convex function, then Z b a+b 1 f (a) + f (b) (1.1) f ≤ f (t)dt ≤ . 2 b−a a 2 Dragomir-Agarwal Type Inequalities This pair of inequalities has been improved and extended in a number of ways. One of the directions estimated the difference between the middle and rightmost term in (1.1). For example, Dragomir and Agarwal presented the following result in [2]: suppose f : I ⊆ R → R is differentiable on I and |f 0 |q is convex on [a, b] for some q ≥ 1, where I is an open interval in R and a, b ∈ I (a < b). Then 1 Z b b − a |f 0 (a)|q + |f 0 (b)|q q f (a) + f (b) 1 . − f (t)dt ≤ 2 b−a a 4 2 Generalizations to higher-order convexity for this type of inequality were given in [3]. Related results for Euler-midpoint, Euler-twopoint, Euler-Simpson, dual EulerSimpson, Euler-Maclaurin, Euler-Simpson 3/8 and Euler-Boole formulae were given in [11]. Furthermore, related results for the general Euler 2-point formulae were given in [10], unifying the cases of Euler trapezoid, Euler midpoint and Eulertwopoint formulae. The aim of this paper is to give related results for the general 3, 4 and 5-point quadrature formulae, as well as for the corrected general 3, 4 and 5-point quadrature formulae. In addition to values of the function at the chosen nodes, "corrected" quadrature formulae include values of the first derivative at the end points of the interval and also have higher accuracy than adjoint classical quadrature formulae. They are sometimes called "quadratures with end corrections". I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 3 of 23 Go Back Full Screen Close Our first course of action was to obtain the quadrature formulae. This was done using the extended Euler formulae, in which Bernoulli polynomials play an important role. For the reader’s convenience, let us recall some basic properties of Bernoulli polynomials. Bernoulli polynomials Bk (t) are uniquely determined by Bk0 (x) = kBk−1 (x), Bk (t + 1) − Bk (t) = ktk−1 , k ≥ 0, B0 (t) = 1. For the kth Bernoulli polynomial we have Bk (1 − x) = (−1)k Bk (x), x ∈ R, k ≥ 1. The kth Bernoulli number Bk is defined by Bk = Bk (0). For k ≥ 2, we have Bk (1) = Bk (0) = Bk . Note that B2k−1 = 0, k ≥ 2 and B1 (1) = −B1 (0) = 1/2. Bk∗ (x) are periodic functions of period 1 defined by Bk∗ (x + 1) = Bk∗ (x), x ∈ R, and related to Bernoulli polynomials as Bk∗ (x) = Bk (x), 0 ≤ x < 1. For k ≥ 2, Bk∗ (t) is a continuous function, while B1∗ (x) is a discontinuous function with a jump of −1 at each integer. For further details on Bernoulli polynomials, see [1] and [9]. Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 4 of 23 Go Back Full Screen Close 2. Preliminaries General 3-point quadrature formulas were obtained in [5] and general corrected 3point quadrature formulas in [6]; general closed 4-point quadrature formulas were considered in [7] and finally, general closed 5-point quadrature formulas were derived in [8]. Namely, if f : [0, 1] → R is such that f (n−1) is continuous and of bounded variation on [0, 1] for some n ≥ 1, then we have Z Z 1 1 1 α α (2.1) f (t)dt − Qα (x) + Tn−1 (x) = Fn (x, t)df (n−1) (t), n! 0 0 for α = Q3, CQ3 and x ∈ [0, 1/2), for α = Q4, CQ4 and x ∈ (0, 1/2], and for α = Q5, CQ5 and x ∈ (0, 1/2), where f (x) + 24B2 (x)f 12 + f (1 − x) 1 , QQ3 (x) := Q x, , 1 − x = 6(1 − 2x)2 2 7f (x) − 480B4 (x)f 12 + 7f (1 − x) 1 QCQ3 (x) := QC x, , 1 − x = , 2 30(1 − 2x)2 (1 + 4x − 4x2 ) QQ4 (x) := Q(0, x, 1 − x, 1) −6B2 (x)f (0) + f (x) + f (1 − x) − 6B2 (x)f (1) , = 12x(1 − x) QCQ4 (x) := QC (0, x, 1 − x, 1) 30B4 (x)f (0) + f (x) + f (1 − x) + 30B4 (x)f (1) , = 60x2 (1 − x)2 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 5 of 23 Go Back Full Screen Close 1 QQ5 (x) := Q 0, x, , 1 − x, 1 2 h 1 = f (x) + f (1 − x) 60x(1 − x)(1 − 2x)2 − (10x2 − 10x + 1)(1 − 2x)2 (f (0) + f (1)) 1 2 +32x(1 − x)(5x − 5x + 1)f , 2 1 QCQ5 (x) := QC 0, x, , 1 − x, 1 2 1 = [f (x) + f (1 − x) 2 420x (1 − x)2 (1 − 2x)2 + (98x4 − 196x3 + 102x2 − 4x − 1)(1 − 2x)2 (f (0) + f (1)) + 64x2 (1 − x)2 (14x2 − 14x + 3)f (1/2) and b(n−1)/2c α Tn−1 (x) (2.2) = Fnα (x, t) = X k=1 α Gn (x, t) (2.3) (2.4) I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 6 of 23 1 Gα (x, 0) [f (2k−1) (1) − f (2k−1) (0)], (2k)! 2k − Gαn (x, 0), and finally, Bn∗ (x − t) + 24B2 (x) · Bn∗ 12 − t + Bn∗ (1 − x − t) = , 6(1 − 2x)2 ∗ ∗ 1 7B (x − t) − 480B (x) · B − t + 7Bn∗ (1 − x − t) 4 n n 2 GnCQ3 (x, t) = , 30(1 − 2x)2 (1 + 4x − 4x2 ) GQ3 n (x, t) Dragomir-Agarwal Type Inequalities Go Back Full Screen Close (2.5) (2.6) Bn∗ (x − t) − 12B2 (x) · Bn∗ (1 − t) + Bn∗ (1 − x − t) , 12x(1 − x) 60B4 (x) · Bn∗ (1 − t) + Bn∗ (x − t) + Bn∗ (1 − x − t) , GnCQ4 (x, t) = 60x2 (1 − x)2 GQ4 n (x, t) = 10x2 − 10x + 1 ∗ Bn (1 − t) 30x(x − 1) Bn∗ (x − t) + Bn∗ (1 − x − t) 8(5x2 − 5x + 1) ∗ 1 + + Bn −t , 60x(1 − x)(1 − 2x)2 15(1 − 2x)2 2 (2.7) GQ5 n (x, t) = 98x4 − 196x3 + 102x2 − 4x − 1 ∗ Bn (1 − t) 210x2 (1 − x)2 Bn∗ (x − t) + Bn∗ (1 − x − t) 16(14x2 − 14x + 3) ∗ 1 + + Bn −t . 420x2 (1 − x)2 (1 − 2x)2 105(1 − 2x)2 2 (2.8) GCQ5 (x, t) = n The following lemma was the key result for obtaining the results in [5], [6], [7] and [8], and we shall need it here as well. GQ3 2n−1 (x, t) Lemma 2.1. For x ∈ {0} ∪ [1/6, 1/2) and n ≥ 2, variable t on (0, 1/2). The sign of the function is determined by: has no zeros in (−1)n+1 GQ3 for x ∈ [1/6, 1/2) and (−1)n GQ3 2n−1 (x, t) > 0 2n−1 (0, t) > 0. √ For x ∈ 0, 1/2 − 15/10 ∪ [1/6, 1/2) and n ≥ 3, GCQ3 2n−1 (x, t) has no zeros in variable t on (0, 1/2). The sign of the function is determined by: √ (−1)n GCQ3 for x ∈ [0, 1/2 − 15/10], 2n−1 (x, t) > 0 (−1)n+1 GCQ3 2n−1 (x, t) > 0 for x ∈ [1/6, 1/2). Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 7 of 23 Go Back Full Screen Close √ For x ∈ (0, 1/2 − 3/6] ∪ [1/3, 1/2] and n ≥ 2, GQ4 2n−1 (x, t) has no zeros in variable t on (0, 1/2). The sign of the function is determined by: √ (−1)n+1 GQ4 for x ∈ (0, 1/2 − 3/6], 2n−1 (x, t) > 0 (−1)n GQ4 for x ∈ [1/3, 1/2]. 2n−1 (x, t) > 0 √ CQ4 For x ∈ (0, 1/2 − 5/10] ∪ [1/3, 1/2] and n ≥ 3, G2n−1 (x, t) has no zeros in variable t on (0, 1/2). The sign of the function is determined by: √ (−1)n+1 GCQ4 for x ∈ (0, 1/2 − 5/10], 2n−1 (x, t) > 0 CQ4 (−1)n G2n−1 (x, t) > 0 for x ∈ [1/3, 1/2]. √ For x ∈ (0, 1/2 − 15/10] ∪ [1/5, 1/2) and n ≥ 3, GQ5 2n−1 (x, t) has no zeros in variable t on (0, 1/2). The sign of the function is determined by: i √ (−1)n GQ5 (x, t) > 0 for x ∈ 0, 1/2 − 15/10 , 2n−1 n+1 GQ5 2n−1 (x, t) > 0 for x ∈ [1/5, 1/2) . √ √ For x ∈ (0, 1/2 − 21/14] ∪ [3/7 − 2/7, 1/2) and n ≥ 4, GCQ5 2n−1 (x, t) has no zeros in variable t on (0, 1/2). The sign of the function is determined by: i √ (−1)n GCQ5 (x, t) > 0 for x ∈ 0, 1/2 − 21/14 , 2n−1 h √ CQ5 (−1)n+1 G2n−1 (x, t) > 0 for x ∈ 3/7 − 2/7, 1/2 , (−1) CQ3 Q4 CQ4 where GQ3 2n−1 is as in (2.3), G2n−1 as in (2.4), G2n−1 as in (2.5), G2n−1 as in (2.6), CQ5 GQ5 2n−1 as in (2.7) and G2n−1 as in (2.8). Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 8 of 23 Go Back Full Screen Close Applying properties of Bernoulli polynomials, it easily follows that functions Gαn for α = Q3, CQ3, Q4, CQ4, Q5, CQ5 and n ≥ 1, have the following properties: (2.9) (2.10) Gαn (x, 1 − t) = (−1)n Gαn (x, t), t ∈ [0, 1], ∂ j Gαn (x, t) n! = (−1)j Gα (x, t), j = 1, 2, . . . , n, j ∂t (n − j)! n−j α also, that Gα2n−1 (x, 0) = 0 for n ≥ 1, and so F2n−1 (x, t) = Gα2n−1 (x, t). α These properties and Lemma 2.1 yield that functions F2n , defined by (2.2), are monotonous on (0, 1/2) and (1/2, 1), have constant sign on (0, 1), so the functions α |F2n (t)| attain their maximal value at t = 1/2. Finally, using (2.9) and (2.10), it is not hard to establish that under the assumptions of Lemma 2.1 we have: Z 1 Z 1 α α (2.11) |F2n (x, t)|dt = 2 t|F2n (x, t)|dt = |Gα2n (x, 0)|, 0 Z0 1 Z 1 1 α α (2.12) |G2n−1 (x, t)|dt = 2 t|Gα2n−1 (x, t)|dt = |F2n (x, 1/2)| . n 0 0 Dragomir-Agarwal Type Inequalities Now that we have stated all the previously obtained results which form a basis for the results of this paper, we proceed to the main result. Page 9 of 23 I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Go Back Full Screen Close 3. Main Result To shorten notation, we denote the left-hand side of (2.1) by R1 0 f (t)dt − ∆αn (x), i.e. α ∆αn (x) := Qα (x) − Tn−1 (x) for α = Q3, CQ3, Q4, CQ4, Q5, CQ5 and n ≥ 1. Theorem 3.1. Let f : [0, 1] → R be n-times differentiable. If |f (n) |p is convex for some p ≥ 1 and Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 • n ≥ 3 and Title Page 1. α = Q3 and x ∈ {0} ∪ [1/6, 1/2), √ 2. α = Q4 and x ∈ (0, 1/2 − 3/6] ∪ [1/3, 1/2], Contents • n ≥ 5 and √ 1. α = CQ3 and x ∈ [0, 1/2 − 15/10] ∪ [1/6, 1/2), √ 2. α = CQ4 and x ∈ (0, 1/2 − 5/10] ∪ [1/3, 1/2], √ 3. α = Q5 and x ∈ (0, 1/2 − 15/10] ∪ [1/5, 1/2), • n ≥ 7 and 1. α = CQ5 and x ∈ (0, 1/2 − JJ II J I Page 10 of 23 Go Back Full Screen √ 21/14] ∪ [3/7 − √ 2/7, 1/2), then we have Z 1 (n) 1 p (n) p p |f (0)| + |f (1)| (3.1) f (t)dt − ∆αn (x) ≤ Cα (n, x) · 2 0 Close while if, under same conditions, |f (n) | is concave, then Z 1 (n) 1 α , (3.2) f (t)dt − ∆n (x) ≤ Cα (n, x) · f 2 0 where α F2k x, 1 2 2 Cα (2k − 1, x) = (2k)! with functions and (2.8). α F2k and defined as in (2.2) and Gα2k Cα (2k, x) = 1 |Gα (x, 0)| (2k)! 2k as in (2.3), (2.4), (2.5), (2.6), (2.7) α Proof. First, recall that F2k−1 (x, t) = Gα2k−1 (x, t). Now, starting from (2.1), we apply Hölder’s and then Jensen’s inequality for the convex function |f (n) |p , to obtain Z 1 α n! f (t)dt − ∆n (x) 0 Z 1 ≤ |Fnα (x, t)| · |f (n) (t)|dt 0 Z 1 1− p1 |Fnα (x, t)|dt ≤ ≤ 1 |f 0 Z Z (n) p ((1 − t) · 0 + t · 1)| · 1 p |Fnα (x, t)|dt 0 1 1− p1 α |Fn (x, t)|dt × |f (n) (0)|p Z 1 (1 − t)|Fnα (x, t)|dt + |f (n) (1)|p 0 Inequality (3.1) now follows from (2.11) and (2.12). I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 11 of 23 Go Back Full Screen Close 0 Dragomir-Agarwal Type Inequalities Z 0 1 p1 t|Fnα (x, t)|dt . To prove (3.2), apply Jensen’s integral inequality to (2.1) to obtain Z 1 α n! f (t)dt − ∆n (x) 0 Z 1 ≤ |Fnα (x, t)| · |f (n) ((1 − t) · 0 + t · 1)|dt 0 ! R1 Z 1 α ((1 − t) · 0 + t · 1)|F (x, t)|dt n α (n) 0 ≤ |Fn (x, t)|dt · f . R1 |Fnα (x, t)|dt 0 0 Theorem 3.1 provides numerous interesting special cases. Particular choices of node will procure the Dragomir-Agarwal-type estimates for many classical quadrature formulas, as well as the adjoint corrected ones. 3.1. CASE α = Q3 and n = 3, 4 For x = 0, Theorem 3.1 gives Dragomir-Agarwal-type estimates for Simpson’s formula; for x = 1/4 it provides the estimates for the dual Simpson formula and for x = 1/6 for Maclaurin’s formula. These were already obtained in [11]; Simpson’s formula was also considered in [4]. √ For x = 1/2 − 3/6 (⇔ B2 (x) = 0), the following estimates are obtained for the Gauss 2-point formula: √ ! √ ! √ ! 3 − 3 1 3 − 3 1 3 − 3 ∆Q3 = f + f n 6 2 6 2 6 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 12 of 23 Go Back Full Screen Close and CQ3 CQ3 √ √ ! 9−4 3 3− 3 = 3, ≈ 1.2 · 10−3 , 6 1728 √ ! 3− 3 1 4, = ≈ 2.3 · 10−4 . 6 4320 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić 3.2. CASE α = CQ3 and n = 5, 6 For x = 0, the following estimates for the corrected Simpson’s formula are produced: 1 1 1 CQ3 ∆n (0) = 7f (0) + 16f + 7f (1) − [f 0 (1) − f 0 (0)] 30 2 60 and 1 CCQ3 (5, 0) = ≈ 8.68 · 10−6 , 115200 1 CCQ3 (6, 0) = ≈ 1.65 · 10−6 . 604800 For x = 1/6, the following estimates for the corrected Maclaurin’s formula are produced: 1 1 1 1 5 1 0 CQ3 ∆n = 27f + 26f + 27f + [f (1) − f 0 (0)] 6 80 6 2 6 240 vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 13 of 23 Go Back Full Screen Close and 1 1 CCQ3 5, = ≈ 1.45 · 10−6 , 6 691200 1 31 CCQ3 6, = ≈ 3.56 · 10−7 . 6 87091200 For x = 1/4, the following estimates for the corrected dual Simpson’s formula are produced: 1 1 3 1 1 1 0 CQ3 ∆n = 8f −f + 8f + [f (1) − f 0 (0)] 4 15 4 2 4 120 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page and Contents 1 1 CCQ3 5, = ≈ 8.68 · 10−6 , 4 115200 31 1 = ≈ 1.6 · 10−6 . CCQ3 6, 4 19353600 √ (x, 0) = 0), the following estimates for the For x = 1/2 − 15/10 (⇔ GCQ3 2 Gauss 3-point formula are produced: " √ ! √ ! √ !# 5 − 15 1 5 − 15 1 5 + 15 ∆CQ3 = 5f + 8f + 5f n 10 18 10 2 10 JJ II J I Page 14 of 23 Go Back Full Screen Close and √ 25 − 6 15 5 − 15 = CCQ3 5, ≈ 3.06 · 10−6 , 10 576000 √ ! 5 − 15 1 CCQ3 6, = ≈ 4.96 · 10−7 . 10 2016000 p √ . For x = x0 := 1/2 − 225 − 30 30 30 (⇔ B4 (x) = 0) (the case when the weight next to f (1/2) is annihilated), the following estimates for the corrected Gauss 2-point formula are produced: √ 1 1 5 − 30 0 CQ3 ∆n (x0 ) = f (x0 ) + f (1 − x0 ) − [f (1) − f 0 (0)] 2 2 60 √ ! and p √ ! 15 − 225 − 30 30 CCQ3 5, 30 p p √ √ √ 46 225 − 30 30 − 120 30 − 4 30 + 150 30 − 825 = ≈ 7.86 · 10−6 , 1728000 p √ ! √ 15 − 225 − 30 30 45 − 7 30 CCQ3 6, = ≈ 1.47 · 10−6 . 30 4536000 3.3. CASE α = Q4 and n = 3, 4 For x = 1/3, the estimates for the Simpson 3/8 formula from [11] are recaptured. Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 15 of 23 Go Back Full Screen Close 3.4. CASE α = CQ4 and n = 5, 6 For x = 1/3, the following estimates for the corrected Simpson 3/8 formula are produced: 1 1 2 1 1 0 CQ4 ∆n = 13f (0) + 27f + 27f + 13f (1) − [f (1)−f 0 (0)] 3 80 3 3 120 Dragomir-Agarwal Type Inequalities and I. Franjić and J. Pečarić 1 1 CCQ4 5, = ≈ 1.45 · 10−6 , 3 691200 1 1 CCQ4 6, = ≈ 3.67 · 10−7 . 3 2721600 √ (x, 0) = 0 , the following estimates for the For x = 1/2 − 5/10 ⇔ GCQ4 2 Lobatto 4-point formula are produced: " # √ ! √ ! 1 1 5 − 5 5 + 5 ∆CQ4 = f (0) + 5f + 5f + f (1) n 3 12 10 10 and CCQ4 CCQ4 √ ! √ 5− 5 5 5, = ≈ 3.88 · 10−6 , 10 576000 √ ! 5− 5 1 6, = ≈ 6.61 · 10−7 . 10 1512000 vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 16 of 23 Go Back Full Screen Close 3.5. CASE α = Q5 and n = 5, 6 For x = 1/4, the following estimates for Boole’s formula from [11] are recaptured. CASE α = CQ5 and n = 7, 8 √ For x = 1/2 − 21/14 ⇔ GCQ5 (x, 0) = 0 , the following estimates for the 2 Lobatto 5-point formula are produced: 1 CQ5 ∆n 4 " # √ ! √ ! 1 7 − 21 1 7 + 21 = 9f (0) + 49f + 64f + 49f + 9f (1) 180 14 2 14 3.6. and √ CCQ5 CCQ5 ! √ 12 21 − 49 = ≈ 4.74 · 10−9 , 1264435200 7 − 21 14 √ ! 7 − 21 1 8, = ≈ 7.03 · 10−10 . 14 1422489600 7, For x = 1/4, the following estimates for the corrected Boole’s formula are produced: 1 1 1 1 CQ5 ∆n = 217f (0) + 512f + 432f 4 1890 4 2 3 1 0 +512f + 217f (1) − [f (1) − f 0 (0)] 4 252 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 17 of 23 Go Back Full Screen Close and CCQ5 1 7, 4 = 17 ≈ 3.49 · 10−9 , 4877107200 1 1 = CCQ5 8, ≈ 6.15 · 10−10 . 4 1625702400 √ Further, for x = 1/2 − 7/14 (⇔ 14x2 − 14x + 3 = 0), which is the case when the weight next to f (1/2) is annihilated, the following estimates for the corrected Lobatto 4-point formula are produced: " √ ! √ ! 7− 7 1 7− 7 ∆CQ5 = 37f (0) + 98f n 14 14 270 # √ ! 1 0 7+ 7 + 37f (1) − [f (1) − f 0 (0)] +98f 14 180 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 18 of 23 and CCQ5 √ ! √ 7− 7 343 − 16 7 = ≈ 8.81 · 10−9 , 7, 14 34139750400 CCQ5 √ ! 7− 7 1 = ≈ 1.41 · 10−9 . 8, 14 711244800 Go Back Full Screen Close Finally, for 1 x = x0 := − 2 p √ 45 − 2 102 14 ⇔ 98x4 − 196x3 + 102x2 − 4x − 1 = 98 ! ! ! √ √ 1 1 102 102 x2 − x + =0 , x2 − x + − + 49 98 49 98 which is the case when the weight next to f (0) and f (1) is annihilated, the estimates for the corrected Gauss 3-point formula are produced: √ 1977 + 16 102 CQ5 ∆n (x0 ) = [f (x0 ) + f (1 − x0 )] 6930 √ √ 1 9 − 102 0 1488 − 16 102 + f − [f (1) − f 0 (0)] 3465 2 420 and p √ √ 24 60933 − 6014 102 − 49(87 − 8 102) CCQ5 (7, x0 ) = ≈ 8.12 · 10−9 , 3793305600 √ 43 − 3 102 ≈ 1.28 · 10−9 . CCQ5 (8, x0 ) = 9957427200 Remark 1. An interesting fact to point out is that out of all the 3-point quadrature formulae, Maclaurin’s formula gives the least estimate of error in Theorem 3.1; among Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 19 of 23 Go Back Full Screen Close the corrected 3-point quadrature formulae, the corrected Maclaurin’s formula has the same property. Further, among the closed 4-point quadrature formulas, the Simpson 3/8 formula gives the best estimate and the corrected Simpson 3/8 formula is the optimal corrected closed 4-point quadrature formula. Finally, the node x = 1/5 produces the closed 5-point √ quadrature formula with the best error estimate, while the node x = 3/7 − 2/7 produces the corrected closed 5-point quadrature formula with the same property. The proofs are similar to those in [5], [6], [7] and [8], respectively. In view of the previous remark, let us consider the case α = Q5, n = 5, 6 and x = 1/5. We have: 1 1 1 4 1 Q5 ∆n = 27f (0) + 125f + 128f + 125f + 27f (1) 5 432 5 2 5 and 1 1 CQ5 5, = ≈ 8.68 · 10−7 , 5 1152000 1 1 CQ5 6, = ≈ 1.98 · 10−7 . 5 5040000 √ Further, for α = CQ5, n = 7, 8 and x = x0 := 3/7 − 2/7 we obtain: ∆CQ5 (x0 ) = 0.10143 [f (0) + f (1)] + 0.259261 [f (x0 ) + f (1 − x0 )] n 1 + 0.278617 f + 3.07832 · 10−3 [f 0 (1) − f 0 (0)] 2 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 20 of 23 Go Back Full Screen Close and √ 27 − 16 2 ≈ 1.15 · 10−9 , CCQ5 (7, x0 ) = 3793305600 √ 11 − 6 2 CCQ5 (8, x0 ) = ≈ 2.53 · 10−10 . 9957427200 Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 21 of 23 Go Back Full Screen Close References [1] M. ABRAMOWITZ AND I.A. STEGUN (Eds), Handbook of Mathematical Functions with Formulae, Graphs and Mathematical Tables, National Bureau of Standards, Applied Math. Series 55, 4th printing, Washington, 1965. [2] S.S. DRAGOMIR AND R.P. AGARWAL, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoid formula, Appl. Mat. Lett., 11(5) (1998), 91–95. [3] Lj. DEDIĆ, C.E.M. PEARCE AND J. PEČARIĆ, Hadamard and DragomirAgarwal inequalities, higher-order convexity and the Euler formula, J. Korean Math. Soc., 38 (2001), 1235–1243. [4] Lj. DEDIĆ, C.E.M. PEARCE AND J. PEČARIĆ, The Euler formulae and convex functions, Math. Inequal. Appl., 3(2) (2000), 211–221. [5] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, Quadrature formulae of Gauss type based on Euler identitites, Math. Comput. Modelling, 45(3-4) (2007), 355–370. [6] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, General 3-point quadrature formulas of Euler type, (submitted for publication). [7] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, General closed 4-point quadrature formulae of Euler type, Math. Inequal. Appl., 12 (2009), 573–586. [8] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, On families of quadrature formulas based on Euler identities, (submitted for publication). [9] V.I. KRYLOV, Approximate Calculation of Integrals, Macmillan, New YorkLondon, 1962. Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 22 of 23 Go Back Full Screen Close [10] J. PEČARIĆ AND A. VUKELIĆ, Hadamard and Dragomir-Agarwal inequalities, the general Euler two point formulae and convex functions, Rad Hrvat. akad. znan. umjet., 491. Matematičke znanosti, 15 (2005), 139–152. [11] J. PEČARIĆ AND A. VUKELIĆ, On generalizations of Dragomir-Agarwal inequality via some Euler-type identitites, Bulletin de la Sociètè des Mathèmaticiens de R. Macèdonie, 26 (LII) (2002), 463–483. Dragomir-Agarwal Type Inequalities I. Franjić and J. Pečarić vol. 10, iss. 3, art. 65, 2009 Title Page Contents JJ II J I Page 23 of 23 Go Back Full Screen Close