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Journal of Inequalities in Pure and Applied Mathematics ON GRÜSS TYPE INEQUALITIES OF DRAGOMIR AND FEDOTOV J.E. PEČARIĆ AND B. TEPEŠ Faculty of Textile Technology University of Zagreb, Pirottijeva 6, 1000 Zagreb, Croatia. EMail: pecaric@mahazu.hazu.hr URL: http://mahazu.hazu.hr/DepMPCS/indexJP.html Faculty of Philosophy, I. Lučića 3, 1000 Zagreb, Croatia. volume 4, issue 5, article 91, 2003. Received 20 May, 2003; accepted 18 September, 2003. Communicated by: S.S. Dragomir Abstract Contents JJ J II I Home Page Go Back Close c Victoria University 2000 ISSN (electronic): 1443-5756 065-03 Quit Abstract Weighted versions of Grüss type inequalities of Dragomir and Fedotov are given. Some related results are also obtained. 2000 Mathematics Subject Classification: 26D15. Key words: Grüss type inequalities. On Grüss Type Inequalities of Dragomir and Fedotov Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 3 6 J.E. Pečarić and B. Tepeš Title Page Contents JJ J II I Go Back Close Quit Page 2 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au 1. Introduction In 1935, G. Grüss proved the following inequality: (1.1) Z b Z b Z b 1 1 1 f (x) g (x) dx − f (x) dx · g (x) dx b − a b−a a b−a a a 1 6 (Φ − ϕ) (Γ − γ) , 4 provided that f and g are two integrable functions on [a, b] satisfying the condition (1.2) ϕ 6 f (x) 6 Φ and γ 6 g(x) 6 Γ for all x ∈ [a, b] . The constant 1 4 is best possible and is achieved for a+b . f (x) = g (x) = sgn x − 2 The following result of Grüss type was proved by S.S. Dragomir and I. Fedotov [1]: Theorem 1.1. Let f, u : [a, b] → R be such that u is L−Lipschitzian on [a, b], i.e., (1.3) |u (x) − u (y)| 6 L |x − y| for all x ∈ [a, b] , On Grüss Type Inequalities of Dragomir and Fedotov J.E. Pečarić and B. Tepeš Title Page Contents JJ J II I Go Back Close Quit Page 3 of 13 f is Riemann integrable on [a, b] and there exist the real numbers m, M so that J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 (1.4) m 6 f (x) 6 M for all x ∈ [a, b] . http://jipam.vu.edu.au Then we have the inequality Z b Z 1 u (b) − u (a) b (1.5) f (x) du (x) − f (t) dt 6 L (M − m) (b − a) , b−a 2 a a and the constant one. 1 2 is sharp, in the sense that it cannot be replaced by a smaller The following result of Grüss’ type was proved by S.S. Dragomir and I. Fedotov [2]: Theorem 1.2. Let f, u : [a, b] → R be such that u is L−lipschitzian Wb on [a, b], and f is a function of bounded variation on [a, b]. Denote by a f the total variation of f on [a, b]. Then the following inequality holds: (1.6) Z b Z b b _ 1 f (b) − f (a) u (x) df (x) − · u (x) dx 6 L (b − a) f. b−a 2 a a a The constant 21 is sharp, in the sense that it cannot be replaced by a smaller one. Remark 1.1. For other related results see [3]. Let us also state that the weighted version of (1.1) is well known, that is we have with condition (1.2) the following generalization of (1.1): On Grüss Type Inequalities of Dragomir and Fedotov J.E. Pečarić and B. Tepeš Title Page Contents JJ J II I Go Back Close Quit (1.7) 1 |D (f, g; w)| 6 (Φ − ϕ) (Γ − γ) , 4 where Page 4 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 D (f, g; w) = A (f, g; w) − A (f ; w) A (g; w) , http://jipam.vu.edu.au and Rb A (f ; w) = a w (x) f (x) dx . Rb w (x) dx a So, in this paper we shall show that corresponding weighted versions of (1.5) and (1.6) are also valid. Some related results will be also given. On Grüss Type Inequalities of Dragomir and Fedotov J.E. Pečarić and B. Tepeš Title Page Contents JJ J II I Go Back Close Quit Page 5 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au 2. Results Theorem 2.1. Let f, u : [a, b] → R be such that f is Riemann integrable on [a, b] and u is L−Lipschitzian on [a, b], i.e. (1.3) holds true. If w : [a, b] → R is a positive weight function, then Z b (2.1) |T (f, u; w)| 6 L w (x) |f (x) − A (f ; w)| dx, a where On Grüss Type Inequalities of Dragomir and Fedotov Z (2.2) T (f, u; w) = a b w (x) f (x) du (x) Z b Z b 1 w (x) du (x) w (x) f (x) dx. − Rb w (x) dx a a a Moreover, if there exist the real numbers m, M such that (1.4) is valid, then Z b L (2.3) |T (f, u; w)| 6 (M − m) w (x) dx. 2 a J.E. Pečarić and B. Tepeš Title Page Contents JJ J II I Go Back Close Proof. As in [1], we have Z b |T (f, u; w)| = w (x) [f (x) − A (f ; w)] du (x) a Z b 6L w (x) |f (x) − A (f ; w)| dx. a Quit Page 6 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au That is, (2.1) is valid. Furthermore, from an application of Cauchy’s inequality we have: Z (2.4) |T (f, u; w)| 6 L b Z 2 w (x) (f (x) − A (f ; w)) dx w (x) dx a b 12 , a from where we obtain (2.5) 1 2 Z b |T (f, u; w)| 6 L · (D (f, f ; w)) · w (x) dx. a From (1.7) for g ≡ f we get: (2.6) J.E. Pečarić and B. Tepeš 1 (D (f, f ; w)) 2 6 1 (Φ − ϕ) . 2 Now, (2.4) and (2.5) give (2.3). Now, we shall prove the following result. Theorem 2.2. Let f : [a, b] → R be M −Lipschitzian on [a, b] and u : [a, b] → R be L−Lipschitzian on [a, b]. If w : [a, b] → R is a positive weight function, then RbRb w (x) w (x) |x − y| dxdy (2.7) |T (f, u; w)| 6 L · M · a a . Rb w (y) dy a Proof. It follows from (2.1) On Grüss Type Inequalities of Dragomir and Fedotov Title Page Contents JJ J II I Go Back Close Quit Page 7 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au R b w (y) (f (x) − f (y)) dy a |T (f, u; w)| 6 L · w (x) dx Rb w (y) dy a a R Z b b w (y) |f (x) − f (y)| dy a 6L· w (x) dx Rb w (y) dy a a RbRb w (x) w (x) |x − y| dxdy 6L·M · a a . Rb w (y) dy a b Z On Grüss Type Inequalities of Dragomir and Fedotov If in the previous result we set w (x) ≡ 1, then we can obtain the following corollary: Corollary 2.3. Let f and u be as in Theorem 2.2, then, Z b Z b L · M · (b − a)2 u (b) − u (a) 6 f (x) du (x) − f (t) dt . b−a 3 a a Proof. The proof follows by the fact that Z bZ b Z b Z b |x − y| dxdy = |x − y| dx dy a a a a Z b Z y Z b = (y − x) dx + (x − y) dx dy a 1 = 2 a Z a b y 1 (y − a)2 + (b − y)2 dy = (b − a)3 . 3 J.E. Pečarić and B. Tepeš Title Page Contents JJ J II I Go Back Close Quit Page 8 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au Theorem 2.4. Let f, u : [a, b] → R be such that u is L−Lipschitzian on [a, b], and f is a function of bounded variation on [a, b]. If w : [a, b] → R is a positive weight function, then the following inequality holds: |T (u, f ; w)| 6 M L b _ g 6 WML a b _ f, a T (u, f ; w) is defined by (2.2), g : [a, b] → R is the function g (x) = Rwhere x w (t) df (t), a ) (R b Rb w (t) (t − a) dt w (t) (b − t) dt a , a Rb W = supx∈[a,b] w (x) , M = max , Rb w (t) dt w (t) dt a a and Wb a g and J.E. Pečarić and B. Tepeš Title Page Wb a f denote the total variation of g and f on [a, b], respectively. Proof. We have T (u, f ; w) Z b = w (x) u (x) df (x) − R b a Rb Contents JJ J 1 Z b Z w (x) df (x) w (x) dx a ! b w (t) u (t) dt = w (x) u (x) − a R b df (x) w (t) dt a a ! Z b Rb w (t) (u (x) − u (t)) dt a = w (x) df (x). Rb w (t) dt a a a Z On Grüss Type Inequalities of Dragomir and Fedotov b w (x) u (x) dx a II I Go Back Close Quit Page 9 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au Using the fact that u is L−Lipschitzian on [a, b], we can state that: Z ! b R b w (t) (u (x) − u (t)) dt a w (x) df (x) |T (u, f ; w)| = Rb a w (t) dt a Z ! Z x b R b w (t) (u (x) − u (t)) dt a d w (t) df (t) = Rb a w (t) dt a a ! Rb b Z x w (t) |x − t| dt _ a 6 Lsupx∈[a,b] w (t) df (t) Rb w (t) dt a a a = ML b _ On Grüss Type Inequalities of Dragomir and Fedotov J.E. Pečarić and B. Tepeš g. a Title Page The constant M has the value Contents Rb M = supx∈[a,b] a w (t) |x − t| dt Rb w (t) dt a ! . If we denote a new function y(x) as: Z Close w (t) |x − t| dt Quit a Z x Z w (t) (x − t) dt + = a II I Go Back b y (x) = JJ J b w (t) (t − x) dt, Page 10 of 13 x J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au then the first derivative of this function is: Z x Z x Z b Z b dy d = x w (t) tdt − tw (t) dt + w (t) tdt − x w (t) dt dx dx a a x x Z x Z b w (t) dt + w (x) x − w (x) x − w (x) x − = a Z x Z w (t) dt − = w (t) dt + w (x) x x a b On Grüss Type Inequalities of Dragomir and Fedotov w (t) dt; x J.E. Pečarić and B. Tepeš and the second derivative is: d2 y = w (x) + w (x) = 2w (x) > 0. dx2 Title Page Contents Obviously f is a convex function, so we have: Rb M = supx∈[a,b] = supx∈[a,b] a y(x) Rb a (R b = max a w (t) |x − t| dt Rb w (t) dt a JJ J ! Go Back ! Close Quit w (t) dt w (t) (b − t) dt , Rb w (t) dt a II I Rb a w (t) (t − a) dt Rb w (t) dt a Page 11 of 13 ) . J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au That is: Z ! b R b w (t) (u (x) − u (t)) dt a |T (u, f ; w)| = w (x) df (x) Rb a w (t) dt a Z ! b R b w (t) (u (x) − u (t)) dt a = w (x) df (x) Rb a w (t) dt a Z b Rb w (t) |u (x) − u (t)| dt a 6 w (x) |df (x)| Rb w (t) dt a a ! b Rb _ w (t) |x − t| dt a 6 supx∈[a,b] w (x) Lsupx∈[a,b] f Rb w (t) dt a a = WML b _ On Grüss Type Inequalities of Dragomir and Fedotov J.E. Pečarić and B. Tepeš Title Page f. Contents a JJ J II I Go Back Close Quit Page 12 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au References [1] S.S. DRAGOMIR AND I. FEDOTOV, An inequality of Grüss type for Riemann-Stieltjes integral and applications for special means, Tamkang J. of Math., 29(4) (1998), 287–292. [2] S.S. DRAGOMIR AND I. FEDOTOV, A Grüss type inequality for mapping of bounded variation and applications to numerical analysis integral and applications for special means, RGMIA Res. Rep. Coll., 2(4) (1999). [ONLINE: http://rgmia.vu.edu.au/v2n4.html]. [3] S.S. DRAGOMIR, New inequalities of Grüss type for the Stieltjes integral, RGMIA Res. Rep. Coll., 5(4) (2002), Article 3. [ONLINE: http: //rgmia.vu.edu.au/v5n4.html]. On Grüss Type Inequalities of Dragomir and Fedotov J.E. Pečarić and B. Tepeš Title Page Contents JJ J II I Go Back Close Quit Page 13 of 13 J. Ineq. Pure and Appl. Math. 4(5) Art. 91, 2003 http://jipam.vu.edu.au