Improvements of some Integral Inequalities of H. Gauchman involving Taylor s Remainder

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1 Divulgciones Mtemátics Vol. 11 No. 2(2003), pp Improvements of some Integrl Inequlities of H. Guchmn involving Tylor s Reminder Mejor de lguns Desigulddes Integrles de H. Guchmn que involucrn el Resto de Tylor Mohmed Akkouchi (mkkouchi@crmil.com) Déprtement de Mthémtiques Université Cdi Ayyd, Fculté des Sciences-Semlli Bd. du prince My. Adellh B.P Mrrkech, Morocco. Astrct In this pper we improve some integrl inequlities recently otined y H. Guchmn involving Tylor s reminder. Key words nd phrses: Tylor s reminder, Grüss inequlity, Inequlity of Cheng-Sun, differentile mppings. Resumen En este trjo se mejorn lguns desigulddes integrles recientemente otenids por H. Guchmn que involucrn restos de Tylor. Plrs y frses clve: Resto de Tylor, desiguldd de Grüss, desiguldd de Cheng-Sun, plicción diferencile. 1 Introduction nd reclls This pper is continution of two recent works of H. Guchmn (see [5] nd [6]). Its im is to improve some integrl inequlities otined y H. Guchmn in [6] involving Tylor s reminder. Our method is sed on the use of n inequlity of Grüss type recently otined y X. L. Cheng nd J. Sun in [2]. Received 2002/05/30. Accepted 2003/06/25. MSC (2000): 26D15.

2 116 Mohmed Akkouchi In the following, n will e non-negtive integer. We denote y R n,f (c, x) the nth Tylor s reminder of function f with center c, tht is n f (k) R n,f (c, x) = f(x) (x c) k. k! k=0 We recll the following lemm estlished in [6]. Lemm 1. Let f e function dfined on [, ]. Assume tht f C ([, ]). Then ( x) f (x) dx = (x ) R n,f (, x) dx (1) f (x) dx =( 1) R n,f (, x) dx (2) The following result contins n integrl inequlity which is well known in the literture s Grüss inequlity (cf., for exmple [8], p. 296), Theorem 2. Let I e n intervl of the rel line nd let F, G : I R e two integrle functions such tht m F (x) M nd ϕ G(x) Φ for ll x [, ] ; m, M, ϕ nd Φ re constnts. Then we hve the inequlity F (x)g(x) dx 1 F (x) dx. G(x) dx (M m)(φ ϕ) 4 (3) nd the inequlity is shrp in the sense tht the constnt 1 4 cn not e replced y smller one. Using (3), H. Guchmn hs proved (in [6]) the following result contining integrl inequlities involving Tylor s reminder. Theorem 3. Let f e function defined on [, ]. Assume tht f C ([, ]) nd m f () M for ech x [, ], where m nd M re constnts. Then R n,f (, x) dx f (n) () f (n) () ( ) ( )n+2 (M m), (4) 4 ( 1) R n,f (, x) dx f (n) () f (n) () ( ) ( )n+2 4 (M m). (5) The purpose of this pper is to provide some improvements to the inequlities (4) nd (5) ove. Divulgciones Mtemátics Vol. 11 No. 2(2003), pp

3 Improvements of some Integrl Inequlities of H. Guchmn The result Before we give the min result of this pper we need to recll the following vrint of the Grüss inequlity which is recently otined y X. L. Cheng nd J. Sun (see [2]). Theorem 4. Let F, G : [, ] R e two integrle functions such tht ϕ G(x) Φ for some rel constnts ϕ, Φ nd for ll x [, ], then F (x)g(x) dx 1 F (x) dx G(x) dx ( 1 2 F (x) 1 The min result now follows. ) F (y) dy dx (Φ ϕ) (6) Theorem 5. Let f e function defined on [, ]. Assume tht f C ([, ]) nd m f () M for ech x [, ], where m nd M re constnts. Then R n,f (, x) dx f (n) () f (n) () ( ) ( 1) ( )n+2 n!(n + 2) 2n+3 R n,f (, x) dx f (n) () f (n) () ( ) ( )n+2 n!(n + 2) 2n+3 (M m), (7) (M m). (8) Proof. (i) For ll x, [, ] we set F (x) = ( x) ()! nd G(x) = f () (x). Then y ssumption, F, G re integrle on [, ], with m G M. By using lemm 1 nd Cheng-Sun inequlity, we hve = R n,f (, x) dx f (n) () f (n) () ( ) ( x) f () (x) dx 1 f () ( x) (x) dx dx Divulgciones Mtemátics Vol. 11 No. 2(2003), pp

4 118 Mohmed Akkouchi ( 1 ( x) 2 For ll x in [, ], we set θ(x) = ( x) ( ) ) dx (M m) (9) ( ). It is esy to see tht θ is strictly decresing function from [, ] onto [θ(), θ()], where θ() = ( ) (n+2)! x n := nd θ() = ()( ) (n+2)!. Let us set (n + 2) 1 Then x n is the unique point where θ vnishes nd it is esy to show tht θ is nonnegtive on the intervl [, x n ] nd is negtive on the intervl [x n, ]. Therefore, we hve xn θ(x) dx = θ(x) dx θ(x) dx := I 1 I 2. x n By esy computtions, we get However x n = I 1 I 2 = (n + 2) 1 ( ) nd From (10) nd (11), we deduce tht θ(x) dx = 2( ) n+2 (n + 2) 1. ( x n ) ( x n) n+2. (10) ( x n ) n+2 ( )n+2 =. (11) (n + 2) n+2 ( 1 1 ) = n + 2 From (9) nd (12) we get the inequlity (7). (ii) In similr mnner, one could derive inequlity (8). 2( )n+2 n!(n + 2) 2n+3. (12) Remrk. (7) nd (8) re ctully improvements of (4) nd (5) since for every nturl numer n, we hve n + 1 (n + 2) 2n+3 < 1 4. Now we consider the cses when n = 0 or 1 in Theorem 5. Divulgciones Mtemátics Vol. 11 No. 2(2003), pp

5 Improvements of some Integrl Inequlities of H. Guchmn Corollry 6. Let f e function defined on [, ]. Assume tht f C 2 ([, ]) nd m f M for ech x [, ], where m nd M re constnts. Then f(x) dx f()( ) 2f () + f () ( ) 2 ( )3 6 9 (M m), 3 (13) f(x) dx f()( ) + 2f () + f () ( ) 2 ( )3 6 9 (M m), 3 (14) f(x) dx f() + f() 2 ( ) + f () f () ( ) 2 12 ( )3 9 3 (M m). (15) Proof. To otin (13) nd (14) we tke n = 1 in (7) nd (8) of Theorem 5. (15) is otined y tking hlf the sum of (13) nd (14). Corollry 7. Let f e function defined on [, ]. Assume tht f C 1 ([, ]) nd m f M for ech x [, ], where m nd M re constnts. Then f() + f() ( )2 f(x) dx ( ) (M m). (16) 2 8 Thus, we recpture the trpezoid inequlity which hs een otined y severl uthors (see the ppers [1,3,7]). References [1] X.-L. Cheng, Improvement of some Ostrowski-Grüss type inequlities, Comput. Mth. Appl., 42, (2001). [2] X.-L. Cheng, A note on the pertured trpezoid inequlity, RGMIA - report (2002), 1 4. URL: {urlhttp://sci.vut.edu.u/ rgmi [3] S. S. Drgomir nd S. Wng, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error ounds for some specil mens nd for some numericl qudrture rules, Comput. Mth. Appl., , (1997). Divulgciones Mtemátics Vol. 11 No. 2(2003), pp

6 120 Mohmed Akkouchi [4] I. Fedotov nd S. S. Drgomir, An inequlity of Ostrowski s type nd its pplictions for Simpson s rule in numericl integrtion nd for specil mens, Mth. Ineq. Appl., , (1999). [5] H. Guchmn, Some integrl inequlities involving Tylor s reminder I, to pper in J. Inequl. Pure nd Appl. Mth., 3 (2) (2002), Article 26. URL: [6] H. Guchmn, Some integrl inequlities involving Tylor s reminder II, to pper in J. Inequl. Pure nd Appl. Mth. [7] M. Mtić, J. E. Pečrić nd N. Ujević, Improvement nd further generliztion of some inequlities of Ostrowski-Grüss type, Comput. Mth. Appl., 39, (2000). [8] D. S. Mitrinović, J. E. Pečrić nd A. M. Fink, Inequlities for Functions nd their Integrls nd derivtives, Kluwer Acdemic, Dordrecht, Divulgciones Mtemátics Vol. 11 No. 2(2003), pp