Journal of Inequalities in Pure and Applied Mathematics

Transcription

1 Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006. A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Electronics Technologicl Eductionl Institute of Lmi Greece. EMil: kechrin@teilm.gr Deprtment of Electronics Technologicl Eductionl Institute of Lmi Greece. nd Deprtment of Informtics with Applictions to Biomedicine University of Centrl Greece Greece EMil: ssimkis@teilm.gr Received 01 April, 005; ccepted 10 My, 006. Communicted by: P. Cerone Abstrct Home Pge Go Bck c 000 Victori University ISSN electronic):

2 Abstrct Generliztions of the clssicl nd perturbed trpezoid inequlities re developed using new men vlue theorem for the reminder in Tylor s formul. The resulting inequlities for N-times differentible mppings re shrp. 000 Mthemtics Subject Clssifiction: 6D15. Key words: Clssicl trpezoid inequlity, Perturbed trpezoid inequlity, Men vlue theorem, Generliztions. We thnk Prof. P. Cerone for his constructive nd helpful suggestions. 1 Introduction Men Vlue Theorem Generl Integrl Inequlities Generlized Clssicl Trpezoid Inequlities Generlized Perturbed Trpezoid Inequlities References Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

3 1. Introduction In the literture on numericl integrtion, see for exmple [1], [13], the following estimtion is well known s the trpezoid inequlity: f b) + f ) 1 f x) dx b ) b sup f x), 1 x,b) where the mpping f : [, b] R is twice differentible on the intervl, b), with the second derivtive bounded on, b). In [3] N. Brnett nd S. Drgomir proved n inequlity for n time differentible functions which for n = 1 tkes the following form: f ) + f b) 1 b f x) dx b Γ γ), 8 where f : [, b] R is n bsolutely continuous mpping on [, b] such tht < γ f x) Γ <, x, b). In [15] N. Ujević reproved the bove result vi generliztion of Ostrowski s inequlity. For more results on the trpezoid inequlity nd their pplictions we refer to [4], [9], [11], [1]. In [10] S. Drgomir et l. obtined the following perturbed trpezoid inequlity involving the Grüss inequlity: 1 b f x) dx f b) + f ) + f b) f )) b ) Γ γ ) b ), Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 3 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

4 where f is twice differentible on the intervl, b), with the second derivtive bounded on, b), nd γ := inf x,b) f x), Γ =: sup x,b) f x). In [6] P. Cerone nd S. Drgomir improved the bove inequlity replcing the constnt 1 by nd in [8] X. Cheng nd J. Sun replced the constnt by For more results concerning the perturbed trpezoid inequlity we refer to the ppers of N. Brnett nd S. Drgomir [1], [], s well s, to the pper of N. Ujević [14]. In [5] P. Cerone nd S. Drgomir obtined some generl three-point integrl inequlities for n times differentible functions, involving two functions α, β : [, b] [, b] such tht α x) x nd β x) x for ll x [, b]. As specil cses for α x) := x, β x) := x) trpezoid type inequlities for n times differentible functions result. For more trpezoid-type inequlities involving n times differentible functions we refer to [6], [7], [16]. In this pper we stte men vlue Theorem for the reminder in Tylor s formul. We then develop shrp generl integrl inequlity for n times differentible mppings involving rel prmeter. Three generliztions of the clssicl trpezoid inequlity nd two generliztions of the perturbed trpezoid inequlity re obtined. The resulting inequlities for n times differentible mppings re shrp. Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 4 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

5 . Men Vlue Theorem For convenience we set R n f;, b) := f b) n b ) i f i) ). i! We prove the following men vlue Theorem for the reminder in Tylor s formul: Theorem.1. Let f, g C n [, b] such tht f n+1), g n+1) re integrble nd bounded on, b). Assume tht g n+1) x) > 0 for ll x, b). Then for ny t [, b] nd ny positive vlued mppings α, β : [, b] R, the following estimtion holds:.1) m α t) R n f; t, b) + 1) n+1 β t) R n f; t, ) α t) R n g; t, b) + 1) n+1 β t) R n g; t, ) M, i=0 f where m := inf n+1) x) x,b), M := sup f n+1) x) g n+1) x) x,b). g n+1) x) Proof. Since g n+1), α, β re positive vlued functions on, b), we clerly hve tht for ll t [, b] the following inequlity holds: α t) t b x) n g n+1) x) dx + β t) t x ) n g n+1) x) dx > 0, which, by using the Tylor s formul with n integrl reminder, cn be rewritten in the following form:.) α t) R n g; t, b) + 1) n+1 β t) R n g; t, ) > 0. Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 5 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

6 Moreover, we hve ) f α t) b x) n g n+1) n+1) x) x) t g n+1) x) m dx t ) f + β t) x ) n g n+1) n+1) x) x) g n+1) x) m 0, or equivlently.3) α t) t b x) n f n+1) x) dx + 1) n+1 β t) x) n f n+1) x) dx t m α t) b x) n g n+1) x) dx t ) + 1) n+1 β t) x) n g n+1) x) dx. Using the Tylor s formul with n integrl reminder,.3) cn be rewritten in the following form:.4) α t) R n f; t, b) + 1) n+1 β t) R n f; t, ) Dividing.4) by.) we get t m α t) R n g; t, b) + 1) n+1 β t) R n g; t, ) )..5) m α t) R n f; t, b) + 1) n+1 β t) R n f; t, ) α t) R n g; t, b) + 1) n+1 β t) R n g; t, ). Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 6 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

7 On the other hnd, we hve α t) or equivlently t b x) n g n+1) x) M f n+1) x) g n+1) x) + β t) t ) dx x ) n g n+1) x) M f ) n+1) x) 0. g n+1) x).6) α t) R n f; t, b) + 1) n+1 β t) R n f; t, ) Dividing.6) by.) we get.7) M α t) R n g; t, b) + 1) n+1 β t) R n g; t, ) ). α t) R n f; t, b) + 1) n+1 β t) R n f; t, ) α t) R n g; t, b) + 1) n+1 β t) R n g; t, ) M. Combining.5) with.7) we get.1). Theorem.. Let f, g C n [, b] such tht f n+1), g n+1) re integrble nd bounded on, b). Assume tht g n+1) x) > 0 for ll x, b). Then for ny t [, b] nd ny integrble nd positive vluted mppings α, β : [, b] R +, the following estimtion holds: α t) Rn f; t, b) + 1) n+1 β t) R n f; t, ) ) dt.8) m where m, M re s in Theorem.1. α t) Rn g; t, b) + 1) n+1 β t) R n g; t, ) ) dt M, Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 7 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

8 Proof. Integrting.),.4),.6) in Theorem.1 over [, b] we get.9) α t) Rn g; t, b) + 1) n+1 β t) R n g; t, ) ) dt > 0, nd.10) m α t) Rn g; t, b) + 1) n+1 β t) R n g; t, ) ) dt M α t) Rn f; t, b) + 1) n+1 β t) R n f; t, ) ) dt α t) Rn g; t, b) + 1) n+1 β t) R n g; t, ) ) dt. Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Dividing.10) by.9) we get.8). Title Pge Go Bck Pge 8 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

9 3. Generl Integrl Inequlities For convenience we denote γ n f) := inf f n) x), x,b) Γ n f) := sup f n) x). x,b) For our purpose we shll use Theorems.1 nd., s well s, n identity: Lemm 3.1. Let f : [, b] R be mpping such tht f n) is integrble on [, b]. Then for ny positive number ρ the following identity holds: 3.1) 1 b ) ρrn f; x, b) + 1) n+1 R n f; x, ) ) dx = n + 1) ρ + 1) n+1) f x) dx + ρf b) + 1) n+1 f ) b n 1 + n k) 1)n+k+1 f k) b) + ρf k) ) b ) k+1. k + 1)! k=0 Proof. Using the nlyticl form of the reminder in Tylor s formul we hve 3.) 1 b ) ρrn f; x, b) + 1) n+1 R n f; x, ) ) dx = ρf b) + 1) n+1 f ) 1 n b ) k=0 ρ b x) k + 1) n+1 x) k f k) x) dx k! Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 9 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

10 where I k := = ρf b) + 1) n+1 f ) 1 b ) n I k, k=0 ρ b x) k + 1) n+1 x) k f k) x) dx, k = 0, 1,..., n). k! For k 1, using integrtion by prts we obtin 3.3) I k I k 1 = 1)n+k f k 1) b) + ρf k 1) ) k! b ) k. Further, the following identity holds: n n 3.4) I k = n + 1) I 0 + n + 1 k) I k I k 1 ). k=0 k=1 Combining 3.) with 3.4) nd 3.3) we get 3.5) 1 b ) ρrn f; x, b) + 1) n+1 R n f; x, ) ) dt = ρf b) + 1) n+1 f ) n + 1) ρ + 1) n+1) + n k=1 b n + 1 k) 1)n+k f k 1) b) + ρf k 1) ) k! Replcing k by k + 1 in 3.5), we get 3.1). f x) dx b ) k. Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 10 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

11 Theorem 3.. Let f C n [, b] such tht f n+1) is integrble nd bounded on, b). Then for ny positive number ρ the following estimtion holds: 3.6) 1 + ρ) b ) n+1 n + )! n + 1) γ n+1 f) ρ + 1)n+1 b ) + n 1 k=0 f x) dx + ρf b) + 1)n+1 f ) n + 1) n k) 1) n+k+1 f k) b) + ρf k) ) b ) k n + 1) k + 1)! 1 + ρ) b )n+1 n + )! n + 1) Γ n+1 f), The inequlities in 3.6) re shrp. Proof. Choosing g x) = x n+1, α x) = ρ, β x) = 1 in.1) in Theorem.1, nd then using the identity R n g;, x) = x ) n+1 we get 3.7) ρ b t) n+1 + 1) n+1 t) n+1 γ n+1 f) n + 1)! ρr n f; t, b) + 1) n+1 R n f; t, ) ρ b t)n+1 + 1) n+1 t) n+1 Γ n+1 f), n + 1)! Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 11 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

12 for ll t [, b]. Integrting 3.7) with respect to t from to b we hve 3.8) 1 + ρ) b )n+1 γ n+1 f) n + )! 1 b ρrn f; t, b) + 1) n+1 R n f; t, ) ) dt 1 + ρ) b )n+1 Γ n+1 f). n + )! Setting 3.1) Lemm 3.1) in 3.8) nd dividing the resulting estimtion by n + 1), we get 3.6). Moreover, choosing f x) = x n+1 in 3.6), the equlity holds. Therefore the inequlities in 3.6) re shrp. Remrk 1. Applying Theorem 3. for n = 1 we get immeditely the clssicl trpezoid inequlity: 3.9) b ) γ f) 1 f b) + f ) b ) Γ f), 1 1 b f x) dx where f : [, b] R is continuously differentible on [, b] nd twice differentible on, b), with the second derivtive f integrble nd bounded on, b). Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 1 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

13 Remrk. Theorem 3. for n = becomes the following form: 1 + ρ) b ) 3 7 γ 3 f) 1 ρ b ) f x) dx + + f b) + ρf ) b ) ρ) b )3 Γ 3 f), 7 ρ 1) f ) ρ) f b) 3 where ρ R +, f C [, b] nd such tht f is bounded nd integrble on, b). Theorem 3.3. Let f, g be two mppings s in Theorem.. Then for ny ρ R + the following estimtion holds: 3.10) m I n f; ρ,, b) I n g; ρ,, b) M, f where m := inf n+1) x) x,b), M := sup f n+1) x) g n+1) x) x,b), nd g n+1) x) I n f; ρ,, b) := ρ + 1)n+1 b ) + n 1 k=0 f x) dx + ρf b) + 1)n+1 f ) n + 1) n k) 1) n+k+1 f k) b) + ρf k) ) b ) k. n + 1) k + 1)! Proof. Setting α x) = ρ, β x) = 1 in.1) of Theorem.1, nd using the identity 3.1) in Lemm 3.1 we get 3.9). Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 13 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

14 4. Generlized Clssicl Trpezoid Inequlities Using the inequlity 3.6) in Theorem 3. we obtin two generliztions of the clssicl trpezoid inequlity, which will be used in the lst section. Moreover, combining both generliztions we obtin third generliztion of the clssicl trpezoid inequlity. Theorem 4.1. Let f C n [, b] such tht f n+1) is integrble nd bounded on, b). Suppose n is odd. Then the following estimtion holds: 4.1) 1 n + )! n + 1) b )n+1 γ n+1 f) 1 b + n 1 k=1 n k) n + 1) The inequlities in 4.1) re shrp. f b) + f ) f x) dx + ) b ) k f k) ) + 1) k f k) b) k + 1)! 1 n + )! n + 1) b )n+1 Γ n+1 f). Proof. From 3.6) in Theorem 3. by ρ = 1, obviously we get 4.1). Theorem 4.. Let f C n [, b] such tht f n+1) is integrble nd bounded on Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 14 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

15 , b). Suppose n is odd. Then we hve 4.) b ) n+1 n n + 3)! γ n+1 f) 1 b + n 1 k=0 f x) dx n k) n b )n+1 n n + 3)! Γ n+1 f). The inequlities in 4.) re shrp. ) 1) k f k) b) + f k) ) b ) k k + )! Proof. Let m := n + 1. Then m is n even integer. Consider the mpping F : [, b] R, defined vi F x) := x f t) dt. Then we clerly hve tht F C m [, b] nd F m+1) is integrble nd bounded on, b). Now, pplying inequlity 3.6) in Theorem 3. to F by choosing ρ = 1, we redily get b ) m+1 m + )! m + 1) γ m+1 F ) m 1) F b) F )) m + 1) + m 1 k=1 m k) 1) k+1 F k) b) + F k) ) b ) k m + 1) k + 1)! Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 15 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

16 b )m+1 m + )! m + 1) Γ m+1 F ), or equivlently, b ) m+1 m + )! m 1) γ m f) m 1 m m 1 k=1 f x) dx m k) m + 1 b )m+1 m + )! m + 1) Γ m f). 1) k+1 f k 1) b) + f k 1) ) m+1 Multiplying the previous inequlity by we hve b ) n+1 n + 3)!n γ n+1 f) 1 b + n k=1 f x) dx n + 1 k) n m 1)b ) k + 1)! ) b ) k, nd then using m = n + 1 ) 1) k+1 f k 1) b) + f k 1) ) b ) k 1 k + 1)! Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 16 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

17 b )n+1 n + 3)!n Γ n+1 f), nd replcing k by k + 1we get 4.). Moreover, choosing f x) = x n+1 in 4.), the equlity holds. So, the inequlities in 4.) re shrp. Remrk 3. Applying Theorem 4. for n = 1 we gin obtin the clssicl trpezoid inequlity 3.9) in Remrk 1. 1 Remrk 4. A simple clcultion yields < for ny n > 1. nn+3)! n+)!n+1) Thus inequlity 4.) in Theorem 4. is better thn 4.1) in Theorem 4.1. Nevertheless inequlity 4.1) is useful, becuse suitble combintions of 4.1), 4.) led to some interesting results, s for exmple in the following theorem. Theorem 4.3. Let n be n odd integer such tht n 3. Let f C n [, b] such tht f n 1) is integrble nd bounded on, b). Then the following inequlities hold 4.3) 1 b ) n n + 3)! n ) n 1) n + 1) γ n 1 f) n n + 3) Γ n 1 f)) 1n n + 1) f x) dx n ) n 1) n k ) n 3 k=0 1) k f k) b) + f k) ) n k + ) n n + 1) ) b ) k+1 k + 4)! 1 b ) n n + 3)! n ) n 1) n + 1) Γ n 1 f) n n + 3) γ n 1 f)). Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 17 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

18 The inequlities in 4.3) re shrp. Proof. We set the mpping F : [, b] R by 4.4) F x) := x t f s) dsdt. Then we hve tht F C n [, b] nd F n+1) is bounded nd integrble on, b). Applying the inequlities 4.) in Theorem 4. nd 4.1) in Theorem 4.1 to F we respectively get the following inequlities: 4.5) nd 4.6) b ) n+1 n + 3)!n γ n+1 F ) 1 b + n 1 k=1 F x) dx + n k) n b )n+1 n + 3)!n Γ n+1 F ), b ) n+1 n + )! n + 1) γ n+1 F ) 1 b F ) + F b) 1) k F k) b) + F k) ) F x) dx + k + )! F b) + F ) ) b ) k Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 18 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

19 + n 1 k=1 n k) n + 1) b )n+1 n + )! n + 1) Γ n+1 F ). ) b ) k F k) ) + 1) k F k) b) k + 1)! Multiplying 4.6) by 1) nd dding the resulting estimtion with 4.5), we get b ) n+1 n + )! n n + 3) γ n+1 F ) 1 ) 4.7) n + 1 Γ n+1 F ) ) n 1 nk n k) 1) k F k) b) + F k) ) b ) k 1 n n + 1) k + )! k=1 b )n+1 n + )! n n + 3) Γ n+1 F ) 1 ) n + 1 γ n+1 F ). Dividing the lst estimtion with b ) nd splitting the first term of the sum we hve b ) n n + )! n n + 3) γ n+1 F ) 1 ) 4.8) n + 1 Γ n+1 F ) n ) n 1) F b) F )) n 1 k= 1n n + 1) nk n k) n n + 1) ) 1) k F k) b) + F k) ) b ) k 1 k + )! Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 19 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

20 b )n n + )! n n + 3) Γ n+1 F ) 1 ) n + 1 γ n+1 F ). Finlly, setting 4.4) in 4.7) nd multiplying the resulting estimtion by 1nn+1) n )n 1) we get 1 b ) n n + 3)! n ) n 1) n + 1) γ n 1 f) n n + 3) Γ n 1 f)) 1n n + 1) f x) dx n ) n 1) n 1 k= nk n k) n n + 1) 1) k f k ) b) + f k ) ) k + )! ) b ) k 1 1 b ) n n + 3)! n ) n 1) n + 1) Γ n 1 f) n n + 3) γ n 1 f)), nd replcing k by k + the inequlities in 4.3) re obtined. Moreover, choosing f x) = x n 1 in 4.3), the equlity holds. So, the inequlities in 4.3) re shrp. Applying Theorem 4.3 for n = 3 we immeditely obtin the following result: Corollry 4.4. Let f C 1 [, b] such tht f is integrble nd bounded on Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 0 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

21 , b). Then, 4.9) b ) 60 4γ f) 9Γ f)) 1 b b ) 60 f x) dx f ) + f b) 4Γ f) 9γ f)). Remrk 5. Let f be s in Corollry 4.4. If γ f) > 4 9 Γ f) then from 4.8) we get the following inequlity: 1 b f x) dx < f ) + f b). Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 1 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

22 5. Generlized Perturbed Trpezoid Inequlities In this section, using the results of the two previous sections, severl perturbed trpezoid inequlities re obtined involving n times differentible functions. Theorem 5.1. Let f C n [, b] such tht f n+1) is integrble nd bounded on, b). Then the following estimtions re vlid: 5.1) 5.) b ) n+1 n + )! n + 1) γ n+1 f) 1)n b ) + n 1 k=1 f x) dx + 1)n+1 f ) + nf b)) n + 1) n k) 1) n+k+1 f k) b) b ) k n + 1) k + 1)! b )n+1 n + )! n + 1) Γ n+1 f), b ) n+1 n + )! n + 1) γ n+1 f) 1 b ) + n 1 k=1 f x) dx + nf ) + f b) n + 1 n k) f k) ) b )k n + 1) k + 1)! b )n+1 n + )! n + 1) Γ n+1 f). Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

23 Further, if n is n even positive integer, then 5.3) 1 b ) n 1 + k=1 f x) dx + f ) + f b) n k) f k) ) + 1) k f k) b) b ) k n + 1) k + 1)! The inequlities in 5.1) nd 5.) re shrp. b ) n+1 n + )! n + 1) Γ n+1 f) γ n+1 f)). Proof. Tking the limit of 3.6) in Theorem 3. s ρ 0 we obtin 5.1). Further for ρ > 1, dividing 3.6) by ρ + 1) n+1) nd then obtining the limit from the resulting estimtion s ρ we get 5.). Now, let n be n even integer. Then multiplying 5.) by 1), dding the resulting inequlity with 5.1) nd finlly multiplying the obtined estimtion by 1 ) we esily get 5.3). Remrk 6. Applying Theorem 5.1 for n = we obtin the following inequlities: b ) 3 γ 3 f) 7 1 b ) b )3 Γ 3 f), 7 f x) dx f ) + f b) 3 + f b) 6 b ) Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 3 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

24 b ) 3 7 γ 3 f) 1 b ) b )3 Γ 3 f), 7 f x) dx + f ) + f b) 3 + f ) 6 b ) 5.4) 1 b ) f x) dx f ) + f b) + f b) f ) 1 b )3 144 b ) Γ 3 f) γ 3 f)), where f C [, b] nd is such tht f is bounded nd integrble on [, b]. Therefore, inequlity 5.4) cn be regrded s Grüss type generliztion of the perturbed trpezoid inequlity. Theorem 5.. Let f C n [, b] such tht f n+1) is integrble nd bounded on, b). Suppose n is odd nd greter thn 1. Then the following estimtion holds: 5.5) b ) n+1 n + 3) γ n+1 f) n + 1) Γ n+1 f)) n + 3)! n + 1) n ) 1 b n k=1 f b) + f ) f x) dx n k) n 1 k) n ) n + 1) ) f k) ) + 1) k f k) b) b ) k k + )! Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 4 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

25 b )n+1 n + 3) Γ n+1 f) n + 1) γ n+1 f)). n + 3)! n + 1) n ) The inequlities in 5.5) re shrp. Proof. Multiplying 4.) in Theorem 4. by n nd 4.1) in Theorem 4.1 n by nd then dding the resulting estimtions we see tht the lst term n of the sum in the intermedite prt of the obtined inequlity is vnishing, nd so, fter some lgebr, we get 5.5). Finlly, choosing f x) := x n+1 in 5.5), simple clcultion verifies tht the equlities hold. Therefore, the inequlities in 5.5) re shrp. Applying Theorem 5. for n = 3 we get immeditely the following result. Corollry 5.3. Let f C 3 [, b] such tht f 4) is integrble nd bounded on, b). Then the following estimtion holds: 5.6) 1 70 b )4 3γ 4 f) Γ 4 f)) 1 b f x) dx f b) + f ) 1 70 b )4 3Γ 4 f) γ 4 f)). The inequlities in 5.6) re shrp. + f b) f )) b ) 1 Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 5 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

26 References [1] N.S. BARNETT AND S.S. DRAGOMIR, On the perturbed trpezoid formul, Tmkng J. Mth., 33) 00), [] N.S. BARNETT AND S.S. DRAGOMIR, A perturbed trpezoid inequlity in terms of the third derivtive nd pplictions, RGMIA Reserch Report Collection, 4) 001), Art. 6. [3] N.S. BARNETT AND S.S. DRAGOMIR, Applictions of Ostrowski s version of the Grüss inequlity for trpezoid type rules, Tmkng J. Mth., 37) 006), [4] C. BUSE, S.S. DRAGOMIR, J. ROUMELIOTIS AND A. SOFO, Generlized trpezoid type inequlities for vector-vlued functions nd pplictions, Mth. Ineq. & Appl., 53) 00), [5] P. CERONE AND S.S. DRAGOMIR, Three point identities nd inequlities for n time differentible functions, SUT Journl of Mthemtics, 36) 000), [6] P. CERONE AND S.S. DRAGOMIR, Trpezoidl type rules from n inequlities point of view, Anlytic Computtionl Methods in Applied Mthemtics, G. Anstssiou Ed.), CRC press, N.Y., 000, [7] P. CERONE, S.S. DRAGOMIR, J. ROUMELIOTIS AND J. SUNDE, A new generliztion of the trpezoid formul for n time differentible mppings nd pplictions, Demonstrtio Mthemtic, 334) 000), Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 6 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006

27 [8] XIAO-LIANG CHENG AND JIE SUN, A note on the perturbed trpezoid inequlity, J. Inequl. Pure nd Appl. Mth., 3) 00), Art. 9. [ON- LINE: [9] S.S. DRAGOMIR, A generlised trpezoid type inequlity for convex functions, Est Asin J. Mth., 01) 004), [10] S.S. DRAGOMIR, P. CERONE AND A. SOFO, Some remrks on the trpezoid rule in numericl integrtion, Indin J. Pure Appl. Mth., 315) 000), [11] S.S. DRAGOMIR AND A. MCANDREW, On trpezoid inequlity vi Grüss type result nd pplictions, Tmkng J. Mth., 313) 000), [1] S. S. DRAGOMIR AND Th.M. RASSIAS, Ostrowski Inequlities nd Applictions in Numericl Integrtion, Kluwer Acdemic, Dordrecht, 00. [13] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequlities for Functions nd their Integrls nd Derivtives, Kluwer Acdemic, Dordrecht, [14] N. UJEVIĆ, On perturbed mid-point nd trpezoid inequlities nd pplictions, Kyungpook Mth. J., ), [15] N. UJEVIĆ, A generliztion of Ostrowski s inequlity nd pplictions in numericl integrtion, Appl. Mth. Lett., ), [16] N. UJEVIĆ, Error inequlities for generlized trpezoid rule, Appl. Mth. Lett., ), Generliztions of the Trpezoid Inequlities Bsed on New Men Vlue Theorem for the Reminder in Tylor s Formul A.I. Kechriniotis nd N.D. Assimkis Title Pge Go Bck Pge 7 of 7 J. Ineq. Pure nd Appl. Mth. 73) Art. 90, 006