Journal of Inequalities in Pure and Applied Mathematics A GENERALIZATION OF THE PRE-GRÜSS INEQUALITY AND APPLICATIONS TO SOME QUADRATURE FORMULAE volume 3, issue 1, article 13, 2002. NENAD UJEVIĆ Department of Mathematics University of Split Teslina 12/III 21000 Split, Croatia. EMail: ujevic@pmfst.hr Received 03 May, 2001; accepted 26 October, 2001. Communicated by: P. Cerone Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 038-01 Quit Abstract A generalization of the pre-Grüss inequality is presented. It is applied to estimations of remainders of some quadrature formulas. 2000 Mathematics Subject Classification: 26D10, 41A55. Key words: Pre-Grüss inequality, Generalization, Quadrature formulae. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 3 6 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Quit Page 2 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au 1. Introduction In recent years a number of authors have written about generalizations of Ostrowski’s inequality. For example, this topic is considered in [1], [2], [5], [7], [9] and [12]. In this way some new types of inequalities are formed, such as inequalities of Ostrowski-Grüss type, inequalities of Ostrowski-Chebyshev type, etc. An important role in forming these inequalities is played by the preGrüss inequality. This paper develops a new approach to the topic obtaining better results than the approach using the pre-Grüss inequality. It presents new, improved versions of the mid-point and trapezoidal inequality. The mid-point inequality is considered in [1], [2], [3], [7] and [9], while the trapezoidal inequality is considered in [4], [5], [7] and [9]. In [11] we can find the pre-Grüss inequality: (1.1) T (f, g)2 ≤ T (f, f )T (g, g), where f, g ∈ L2 (a, b) and T (f, g) is the Chebyshev functional: Z b Z b Z b 1 1 (1.2) T (f, g) = f (t)g(t)dt − f (t)dt g(t)dt. b−a a (b − a)2 a a If there exist constants γ, δ, Γ, ∆ ∈ R such that δ ≤ f (t) ≤ ∆ and γ ≤ g(t) ≤ Γ, t ∈ [a, b] then, using (1.1), we get the Grüss inequality: (1.3) (∆ − δ)(Γ − γ) |T (f, g)| ≤ . 4 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Quit Page 3 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au Specially, we have T (f, f ) ≤ (1.4) (∆ − δ)2 . 4 Using the above inequalities we get the following inequalities: Z b a+b (1.5) (b − a) − f (t)dt f 2 a " 2 # 12 f (b) − f (a) (b − a)2 1 2 √ ≤ kf 0 k2 − b − a b−a 2 3 ≤ (b − a)2 √ (Γ − γ) 4 3 where f : [a, b] → R is an absolutely continuous function whose derivative f 0 ∈ L2 (a, b) and γ ≤ f 0 (t) ≤ Γ, t ∈ [a, b] . As usual, k·k2 is the norm in L2 (a, b). Further, Z b f (a) + f (b) (1.6) (b − a) − f (t)dt 2 a " 2 # 12 2 1 f (b) − f (a) (b − a) 2 √ ≤ kf 0 k2 − b−a b−a 2 3 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Quit 2 ≤ (b − a) √ (Γ − γ) 4 3 Page 4 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au and (1.7) Z b f (a) + 2f a+b + f (b) 2 (b − a) − f (t)dt 4 a " 2 # 12 f (b) − f (a) (b − a)2 1 2 √ kf 0 k2 − ≤ b−a b−a 4 3 ≤ (b − a)2 √ (Γ − γ) 8 3 where the function f satisfies the above conditions. The inequalities (1.5)-(1.7) are considered (and proved) in [2], [9] and [12]. In this paper we generalize (1.1). We use the generalization to improve the above inequalities. A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Quit Page 5 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au 2. Main Results Lemma 2.1. Let f, g, Ψi ∈ L2 (a, b), i = 0, 1, 2, ..., n, where Ψ0i = Ψi (t)/ kΨi k2 are orthonormal functions. If Sn (f, g) is defined by Z b Z b n Z b X 0 Sn (f, g) = f (t)g(t)dt − f (s)Ψi (s)ds g(s)Ψ0i (s)ds a i=0 a then we have a 1 1 |Sn (f, g)| ≤ Sn (f, f ) 2 Sn (g, g) 2 . The proof follows by the known inequality holding in inner product spaces (H, h·, ·i) 2 n X hx, li i hli , yi hx, yi − i=0 ! ! n n X X kyk2 − ≤ kxk2 − |hx, li i|2 |hli , yi|2 , i=0 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents i=0 where x, y ∈ H and {li }i=0,n is an orthonormal family in H, i.e., (li , lj ) = δij for i, j ∈ {0, . . . , n}. √ We here use only the case n = 1. We choose Ψ00 (t) = 1/ b − a, Ψ1 (t) = Ψ(t) and denote S1 (g, h) = SΨ (g, h) such that Z b Z b Z b 1 (2.1) SΨ (g, h) = g(t)h(t)dt − g(t)dt h(t)dt b−a a a a Z b Z b − g(t)Ψ0 (t)dt h(t)Ψ0 (t)dt a JJ J II I Go Back Close Quit Page 6 of 20 a J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au where g, h, Ψ ∈ L2 (a, b), Ψ0 (t) = Ψ(t)/ kΨk2 and Z b (2.2) Ψ(t)dt = 0. a Lemma 2.2. With the above notations we have 1 1 |SΨ (g, h)| ≤ SΨ (g, g) 2 SΨ (h, h) 2 . (2.3) It is obvious that Z (2.4) SΨ (g, h) = (b − a)T (g, h) − b Z g(t)Ψ0 (t)dt a b h(t)Ψ0 (t)dt a so that SΨ (g, h) is a generalization of the Chebyshev functional. We also define the functions: , t ∈ a, a+b t − 2a+b 3 2 (2.5) Φ(t) = t − a+2b , t ∈ a+b ,b 3 2 and (2.6) χ(t) = t− 5a+b , 6 t ∈ a, a+b 2 t − a+5b , t ∈ a+b ,b . 6 2 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Quit It is not difficult to verify that Z b Z b (2.7) Φ(t)dt = χ(t)dt = 0 Page 7 of 20 a J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 a http://jipam.vu.edu.au and kΦk22 = kχk22 = (2.8) (b − a)3 . 36 We define (2.9) Φ0 (t) = Φ(t) χ(t) , χ0 (t) = . kΦk2 kχk2 Integrating by parts, we have Z b (2.10) Q(f ; a, b) = Φ0 (t)f 0 (t)dt a Z b a+b 3 2 =√ f (a) + f + f (b) − f (t)dt 2 b−a a b−a A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page and Contents (2.11) P (f ; a, b) Z b = χ0 (t)f 0 (t)dt a Z b 6 1 a+b + f (b) − f (t)dt . =√ f (a) + 4f 2 b−a a b−a Remark 2.1. It is obvious that JJ J II I Go Back Close Quit Z (2.12) SΨ (g, g) = (b − a)T (g, g) − 2 b g(t)Ψ0 (t)dt ≤ (b − a)T (g, g). Page 8 of 20 a J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au Theorem 2.3. (Mid-point inequality) Let I ⊂ R be a closed interval and a, b ∈ Int I, a < b. If f : I → R is an absolutely continuous function whose derivative f 0 ∈ L2 (a, b) then we have Z b (b − a) 32 a+b √ C1 , (b − a) − f (t)dt ≤ (2.13) f 2 2 3 a where ( (2.14) C1 = 2 kf 0 k2 [f (b) − f (a)]2 − − [Q(f ; a, b)]2 b−a and Q(f ; a, b) is defined by (2.10). ) 12 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Proof. We define Title Page (2.15) p(t) = t − a, t ∈ a, a+b 2 t − b, t ∈ a+b , b . 2 Then we have p(t)dt = 0 (2.16) a II I Close Quit and (2.17) JJ J Go Back b Z Contents kpk22 Z = a b (b − a)3 p(t) dt = . 12 Page 9 of 20 2 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au We now calculate Z b (2.18) p(t)Φ(t)dt a Z = a a+b 2 2a + b (t − a) t − 3 Integrating by parts, we have Z b Z 0 (2.19) p(t)f (t)dt = a+b 2 Z b a + 2b dt + (t − b) t − a+b 3 2 0 = f dt = 0. b Z 0 (t − a)f (t)dt + (t − b)f (t)dt a+b 2 a a a+b 2 Z (b − a) − A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae b f (t)dt. Nenad Ujević a Using (2.16), (2.18) and (2.19) we get Z b Z b Z b 1 0 0 (2.20) SΦ (p, f ) = p(t)dt p(t)f (t)dt − f 0 (t)dt b−a a a a Z b Z b − f 0 (t)Φ0 (t)dt p(t)Φ0 (t)dt a a Z b a+b (b − a) − f (t)dt. = f 2 a From (2.20) and (2.3) it follows that Z b a + b f ≤ SΦ (f 0 , f 0 ) 12 SΦ (p, p) 12 . (2.21) (b − a) − f (t)dt 2 a Title Page Contents JJ J II I Go Back Close Quit Page 10 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au From (2.16)-(2.18) we get (2.22) 1 SΦ (p, p) = − b−a 3 (b − a) = . 12 kpk22 Z b 2 Z b 2 p(t)dt − p(t)Φ0 (t)dt a a We also have C12 (2.23) 0 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae 0 = SΦ (f , f ). From (2.21)-(2.23) we easily find that (2.13) holds. Nenad Ujević Remark 2.2. It is not difficult to see that (2.13) is better than the first estimation in (1.5). Theorem 2.4. (Trapezoidal inequality) Under the assumptions of Theorem 2.3 we have Z b (b − a) 32 f (a) + f (b) ≤ √ C2 , (2.24) (b − a) − f (t)dt 2 2 3 a Contents JJ J II I Go Back where ( (2.25) Title Page C2 = 2 2 kf 0 k2 − [f (b) − f (a)] − [P (f ; a, b)]2 b−a ) 12 Close Quit Page 11 of 20 and P (f ; a, b) is defined by (2.11). J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au Proof. Let p(t) be defined by (2.15). We calculate b Z p(t)χ(t)dt (2.26) a Z a+b 2 = a = 5a + b (t − a) t − 6 Z b a + 5b dt + (t − b) t − a+b 6 2 dt (b − a)3 . 24 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Integrating by parts, we have Z b (2.27) f 0 (t)χ(t)dt| Nenad Ujević a a+b 2 Z b a + 5b 5a + b 0 f 0 (t)dt t− f (t)dt + t− = a+b 6 6 a 2 Z b f (a) + 4f a+b + f (b) 2 = (b − a) − f (t)dt. 6 a Z Using (2.16), (2.19), (2.26), (2.27) and (2.8) we get Title Page Contents JJ J II I Go Back (2.28) Sχ (f 0 , p) = Z a b p(t)f 0 (t)dt − 1 b−a Z b f 0 (t)dt Z b p(t)dt a a Z b Z b − p(t)χ0 (t)dt f 0 (t)χ0 (t)dt a Close Quit Page 12 of 20 a J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au a+b 2 Z b (b − a) − f (t)dt a " # Z b ) + f (b) 3 f (a) + 4f ( a+b 2 (b − a) − − f (t)dt 2 6 a Z 1 f (a) + f (b) 1 b = − (b − a) + f (t)dt. 2 2 2 a =f From (2.3) and (2.28) it follows that Z b f (a) + f (b) 1 1 (2.29) (b − a) − f (t)dt ≤ 2Sχ (f 0 , f 0 ) 2 Sχ (p, p) 2 . 2 a A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević We have (2.30) 1 Sχ (p, p) = − b−a 3 (b − a) = 48 kpk22 Z b 2 Z b 2 p(t)dt − p(t)χ0 (t)dt a a and (2.31) Title Page Contents JJ J II I Go Back C22 = Sχ (f 0 , f 0 ). From (2.29)-(2.31) we easily get (2.24). Remark 2.3. We see that (2.24) is better than the first estimation in (1.6). Close Quit Page 13 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au We now consider a simple quadrature rule of the form (2.32) f (a) + 2f a+b 2 + f (b) Z b (b − a) − f (t)dt 4 a Z b 1 a+b f (a) + f (b) = f + (b − a) − f (t)dt = R(f ). 2 2 2 a It is not difficult to see that (2.32) is a convex combination of the mid-point quadrature rule and the trapezoidal quadrature rule. In [5] it is shown that (2.32) has a better estimation of error than the well-known Simpson quadrature rule (when we estimate the error in terms of the first derivative f 0 of integrand f ). We here have a similar case. Theorem 2.5. Under the assumptions of Theorem 2.3 we have Z b f (a) + 2f a+b + f (b) (b − a) 32 2 √ C3 , (2.33) (b − a) − f (t)dt ≤ 4 4 3 a where " 2 (2.34) C3 = kf 0 k2 − [f (b) − f (a)]2 b−a 2 # 12 1 a+b − f (a) − 2f + f (b) . b−a 2 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Quit Page 14 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au Proof. We define η(t) = (2.35) 1, t ∈ a, a+b 2 −1, t ∈ η0 (t) = (2.36) a+b ,b 2 , η(t) . kηk2 A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae We easily find that Z (2.37) b Nenad Ujević η(t)dt = 0, kηk22 = b − a. a Title Page Let p(t) be defined by (2.15). Then we have b Z Z p(t)η(t)dt = (2.38) a a+b 2 Z (t − a)dt − (t − b)dt a+b 2 a = Contents b (b − a)2 . 4 JJ J II I Go Back Close We also have Z b 0 f (t)η(t)dt = −f (a) + 2f (2.39) a a+b 2 Quit − f (b). Page 15 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au From (2.37)-(2.39) we get Z b Z b Z b 1 0 0 0 (2.40) Sη (f , p) = p(t)f (t)dt − f (t)dt p(t)dt b−a a a a Z b Z b − p(t)η0 (t)dt f 0 (t)η0 (t)dt a a Z b a+b =f (b − a) − f (t)dt 2 a a+b b−a − −f (a) + 2f − f (b) 4 2 Z b + f (b) f (a) + 2f a+b 2 (b − a) − f (t)dt. = 4 a A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević From (2.3) and (2.40) it follows that Z b f (a) + 2f a+b + f (b) 2 (2.41) (b − a) − f (t)dt 4 a Title Page Contents 1 1 ≤ Sη (f 0 , f 0 ) 2 Sη (p, p) 2 . II I Go Back We now calculate (2.42) JJ J 1 Sη (p, p) = − b−a 3 (b − a) = . 48 kpk22 Z a b 2 Z b 2 p(t)dt − p(t)η0 (t)dt Close Quit a Page 16 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au We also have (2.43) C32 = Sη (f 0 , f 0 ). From (2.41)-(2.43) we easily get (2.33). Remark 2.4. It is not difficult to see that (2.33) is better than the first estimation in (1.7). Finally, in [12] we can find the next inequality Z b f (a) + 4f a+b + f (b) (b − a)2 2 (2.44) (b − a) − f (t)dt ≤ (Γ − γ), 6 12 a where f : I → R, (I ⊂ R is an open interval, a < b, a, b ∈ I) is a differentiable function, f 0 is integrable and there exist constants γ, Γ ∈ R such that γ ≤ f 0 (t) ≤ Γ, t ∈ [a, b]. Inequality (2.44) is a variant of the Simpson’s inequality. On the other hand, we have Z b f (a) + 2f a+b + f (b) (b − a)2 2 √ (Γ − γ). (2.45) (b − a) − f (t)dt ≤ 4 8 3 a A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Inequality (2.45) follows from (2.33), since (2.46) Sη (f 0 , f 0 ) ≤ (b − a) Quit Γ−γ 2 2 Page 17 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au and (2.46) follows from (2.4) and (1.4). Form (2.44) and (2.45) we see that the simple 3-point quadrature rule (2.32) has a better estimation of error than the well-known 3-point Simpson quadrature rule. Note that the estimations are expressed in terms of the first derivative f 0 of integrand. Finally, the following remark is valid. Remark 2.5. The considered case n = 1 illustrates how to apply Lemma 2.1 to quadrature formulas. It is also shown that the derived results are better than some recently obtained results. We can use Lemma 2.1 to derive further improvements of the obtained results. However, in such a case we must require Z b g(t)Ψ0i (t)dt = 0, i = 0, 1, 2, ..., n. A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević a n Thus, the construction of such a finite sequence {Ψ0i }0 can be complicated. However, if we really need better error bounds, without taking into account possible complications, then we can apply the procedure described in this section. Title Page Contents JJ J II I Go Back Close Quit Page 18 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au References [1] G.A. ANASTASSIOU, Ostrowski type inequalities, Proc. Amer. Math. Soc., 123(12) (1995), 3775–3781. [2] N.S. BARNETT, S.S. DRAGOMIR AND A. SOFO, Better bounds for an inequality of the Ostrowski type with applications, RGMIA Research Report Collection, 3(1) (2000), Article 11. [3] P. CERONE AND S.S. DRAGOMIR, Midpoint-type rules from an inequalities point of view, Handbook of Analytic-Computational Methods in Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York, (2000), 135–200. [4] P. CERONE AND S.S. DRAGOMIR, Trapezoidal-type rules from an inequalities point of view, Handbook of Analytic-Computational Methods in Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York, (2000), 65–134. [5] S.S. DRAGOMIR, P. CERONE AND J. ROUMELIOTIS, A new generalization of Ostrowski integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, Appl. Math. Lett., 13 (2000), 19–25. [6] S.S. DRAGOMIR AND T.C. PEACHY, New estimation of the remainder in the trapezoidal formula with applications, Stud. Univ. Babeş-Bolyai Math., XLV (4), (2000), 31–42. [7] S.S. DRAGOMIR AND S. WANG, An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some special A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević Title Page Contents JJ J II I Go Back Close Quit Page 19 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au means and some numerical quadrature rules, Comput. Math. Appl., 33 (1997), 15–20. [8] A. GHIZZETTI AND A. OSSICINI, Quadrature Formulae, Birkhaüser Verlag, Basel/Stuttgart, 1970. [9] M. MATIĆ, J. PEČARIĆ AND N. UJEVIĆ, Improvement and further generalization of some inequalities of Ostrowski-Grüss type, Comput. Math. Appl., 39 (2000), 161–179. [10] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Acad. Publ., Dordrecht/Boston/Lancaster/Tokyo, 1991. A Generalization of the Pre-Grüss Inequality and Applications to some Quadrature Formulae Nenad Ujević [11] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Acad. Publ., Dordrecht/Boston/Lancaster/Tokyo, 1993. [12] C.E.M. PEARCE, J. PEČARIĆ, N. UJEVIĆ AND S. VAROŠANEC, Generalizations of some inequalities of Ostrowski-Grüss type, Math. Inequal. Appl., 3(1) (2000), 25–34. Title Page Contents JJ J II I Go Back Close Quit Page 20 of 20 J. Ineq. Pure and Appl. Math. 3(1) Art. 13, 2002 http://jipam.vu.edu.au