TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
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1 TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity nd some of its generlistions for n time differentible functions.. Introduction The following result is known in the literture s Ostrowski s inequlity 3 p Theorem. Let f : b R be differentible mpping on b) with the property tht f t) M for ll t b). Then f x) b ) f t) dt b x +b 4 + b ) b ) M. The Ostrowski inequlity hs been generlised in number of different wys see 3 nd. Fink lso obtined the following result for n time differentible functions. Theorem. Let f ) t) be bsolutely continuous on b with f n) t) L p b) nd let.) F k x) : n k f k ) ) x ) k f k ) b) x b) k k! b then.) f x) + F k x) n x ) nq+ +b x) nq+ q b k... n ; n!b ) B q q + n ) q + ) f n) p for f L p b ; p > p + q ; ) n n n! mxx )n b x) n } b f n) ; for f L b ; x ) n+ +b x) n+ nn+)!b ) f n) ; for f L b ; Dte: My Mthemtics Subject Clssifiction. Primry 6D5; Secondry 6D0. Key words nd phrses. Trpezoidl inequlity Ostrowski s inequlity Integrl Inequlities.
2 S.S. DRAGOMIR AND A. SOFO where B α β) is Euler s Bet function.3) f n) p : f n) t) p dt ) p f n) : ess sup f n) t) <. p nd Remrk. The result bove on the infinite norm ws given by Milovnović nd Pečrić in 976 see 3 p. 468). Note tht for n nd the infinite norm Theorem reduces to Theorem. In the next section we develop n integrl equlity tht will permit us to obtin bounds for the error estimte in generlised trpezoid formul. The new results complement some of the erlier inequlities relted to the trpezoidl rule reported in. The Results The following generlistion of the trpezoid formul holds: Theorem 3. Let f : b R be mpping such tht its n ) th derivtive f ) is bsolutely continuous on b. Then we hve the equlity.) f ) + f b) n n! b ) n k) b ) k + k! } f k ) ) + ) k f k ) b) b t ) b t) b t) n + ) n t ) n f n) t) dt. Proof. We my use Fink s identity which sttes.) f x) + F k x) n b where t K t x) : t b n! b ) nd F k x) is defined by.). If in.) we put x then we obtin.3) f ) + F k ) n b if t x b if x t b )n n! b ) x t) K t x) f n) t) dt t ) b t) f n) t) dt
3 TRAPEZOIDAL TYPE INEQUALITIES 3 where F k ) )k n k) f k ) b) b ) k. k! Similrly if in.) we put x b we get.4) where f b) + F k b) n b n! b ) F k b) n k) f k ) ) b ) k. k! Adding.3) to.4) nd dividing by we hve f ) + f b) + F k ) + F k b)} n n! b ) t ) b t) b t) t ) f n) t) dt b b t) n + ) n t ) n f n) t) dt replcing F k ) nd F k b) we obtin identity.) hence the theorem is proved. Remrk..5) f ) + f b) ) For n we recpture the known identity b b t + b ) f t) dt. b) For n we deduce the equlity below which is lso well known in the literture.6) f ) + f b) b b ) b ) b b ) t t ) b t) f t) )) + b f t) dt. c) For n 3 we hve some extr terms involving the first derivtive t the end points nmely:.7) f ) + f b) + b f ) f b) b 6 b ) + b t ) b t) ) t f t) dt.
4 4 S.S. DRAGOMIR AND A. SOFO.8) d) Finlly for n 4 we hve the following f ) + f b) + b f ) f b) 8 b ) + f ) + f b) 48 b b t ) b t) b t) + t ) f 4) t) dt 48 b ) b t ) b t) t + b ) ) b + f 4) t) dt. 4 b ) The following inequlities cn now be stted. Theorem 4. Let f : b R be mpping such tht f ) is bsolutely continuous on b. Define T b n) } : f ) + f b) n k) b ) k f k ) ) + ) k f k ) b) + n k! then b.9) T b n) b ) + q n! B q q + n ) q + ) f n) p for p + q ; p > q > ; b ) n n )! + ) n n!) f n) n+)!n! for p q ; b ) n nn+)! f n) where B x y) is the Bet function nd f n) p f n) re s defined in.3). Proof. From.) nd the definition of T b n) we hve b.0) T b n) Y t) f n) t) dt n! b ) where.) Y t) : t ) b t) + ) n t ) b t). By Hölder s inequlity.) T b n) ) b q Y t) q dt n! b ) f n) t) p dt ) p p + q > p >. q
5 TRAPEZOIDAL TYPE INEQUALITIES 5 Now let us provide some upper bounds for the integrl: Y t) q dt. For rel α nd β nd q > we hve the elementry inequlity.3) α + β q q α q + β q ). Utilising.3) we obtin Y t) q dt q by the substitution w t b Also t ) q b t) )q + t ) )q b t) q dt q b ) nq+ B q + n ) q + ) B α β) B β α) Y t) q dt so from.) we deduce ) q nd the symmetry of the Bet function T b x) b )+ q n! 0 t α t) β dt; α β > 0. b ) n+ q B q q + n ) q + ) B q f q + n ) q + ) n) nd the first prt of the theorem is proved. For the Eucliden norm p q ) we cn compute exctly Y t) dt t ) b t) + ) n t ) b t) dt t ) b t) n + ) n t ) n b t) n + t ) n b t) ) dt b ) n+ B 3 n ) + ) n B n + n + ) b )n+ n )! + ) n n!). n + )! From.) T b n) b )n n )! + ) n n!) f n) n + )!n! nd the second prt of the theorem is proved. For the third prt of the theorem observe tht Y t) dt t ) b t) + ) n t ) b t) dt t ) b t) dt + b )n+ n n + ). p t ) b t) dt
6 6 S.S. DRAGOMIR AND A. SOFO From.) T b n) b )n f n) n n + )! the third prt of the theorem is proved hence Theorem 4 is proved. For the norm we cn lso delinete the following theorem. Theorem 5. Let f : b R be mpping such tht f ) is bsolutely continuous on b. i) For n even let n k k then } mx t )b t ) k +t ) k b t ) f k) t k)!b ) for k > ; T b k) b 8 f for k ; f 4) for k ; b ) where t +b is solution of the polynomil eqution b t) k t ) k + k ) t ) k b t) t ) b t) k 0. ii) For n odd let n k + k 0 mx τ T b k + ) τ )b τ) k τ ) k b τ) k+)!b ) f for k 0 n } f k) for k where τ +b is solution of the polynomil eqution t ) k + b t) k k t ) b t) k + t ) k b t) 0. The following two lemms will be useful in the proof of Theorem 5. Lemm. Let b > k is n integer nd t b. Define M t k) : t ) b t) k + t ) k b t) then M t k) hs exctly two zeros s function of t on b. Proof. Observe tht M t k) t ) t b) t ) k + t b) k. Since for t b) t ) k + t b) k > 0 hence M t k) 0 hs the only rel solutions t nd t b. Lemm. Let b > k is n integer nd t b. Define P t k) t ) b t) k + t ) k b t) then P t k) hs exctly three zeros in b.
7 TRAPEZOIDAL TYPE INEQUALITIES 7 Proof. We hve Observe tht Since nd P t k) t ) t b) t b) k + t b) k. P k) P b k) 0. t ) k + t b) k t + b) t ) k + + b t) k t ) k + t ) k 3 b t) + t ) b t) k 3 + b t) k > 0 for t b hence the third solution of P t k) 0 for t b is t +b. Proof of Theorem 5. i) Consider the cse of n even let n k k nd denote Y : sup t ) b t) k + t ) k b t). t b For M t k) defined bove simple clcultions show tht M t k) b t) k t ) k + k ) t ) k b t) t ) b t) k M t k) k ) t ) k + b t) k + k ) k ) t ) b t) k 3 + t ) k 3 b t) nd ) ) k + b b M k) M b k) 0; M k > 0 ) M k) b ) k > 0; M b k) b ) k + b < 0 M k 0 ) ) k + b b M k k ) k ) > 0 for k >. The locl extrem for the function M k) re the rel numbers t +b tht re solutions of the polynomil eqution b t) k t ) k + k ) t ) k b t) t ) b t) k 0. Therefore by Lemm } t Y mx t ) b t ) k + t ) k b t ) hence T b k) mx t for k >. t ) b t ) k + t ) k b t ) k)! b ) > 0 } f k)
8 8 S.S. DRAGOMIR AND A. SOFO For the two specil cses k n ) we hve T b ) b 8 f ) nd for k n 4) b )3 T b 4) f 4). 384 ii) When n is odd let n k + k 0 nd denote Z : sup t ) b t) k t ) k b t). t b With P t k) defined bove in Lemm we hve P t k) t ) k + b t) k k t ) b t) k + t ) k b t) ) + b P k) P b k) P k 0 P k) P b k) b ) k > 0 ) ) k + b b P k k) < 0 for k so there exists t lest one point τ ) +b such tht P τ k) : τ ) b τ) k τ ) k b τ) > 0. One cn relise tht the locl extrem for P k) re the rel numbers τ +b tht re solutions of the polynomil eqution t ) k + b t) k k t ) b t) k + t ) k b t) 0. Now by Lemm hence Z mx τ T b k) mx τ For the trivil cse k 0 n ) we hve } τ ) b τ) k τ ) k b τ) > 0 hence Theorem 5 is proved. τ ) b τ) k τ ) k b τ) k + )! b ) T b ) f 3. Some Exmples } f k+). In this section we give some exmples tht highlight Theorems 4 nd 5. i) For n 4 let T b 4) : f ) + f b) + b 8 + f ) f b) b ) 48 f ) + f b) b
9 TRAPEZOIDAL TYPE INEQUALITIES 9 then T b 4) b ) 3+ q 4 B q q + 3q + ) f 4) p for f L p b p q > b ) b ) b ) p + q ; f 4) for f L b ; f 4) for f L b ; f 4) for f L b. ii) For n 5 let T b 5) : then T b 5) f ) + f b) + b ) f ) + f b)) b ) f ) f b)) 0 + b )3 f ) f b)) 40 b b ) 4+ q 0 B q q + 4q + ) f 5) p for f L p b p q > b ) 9 70 b ) ) 770 p + q ; f 5) for f L b ; f 5) for f L b ; 9 6) 0 6 5b ) For n 5 k solving τ ) 4 + τ b) we obtin for which τ + b f 5) for f L b ; τ ) τ b) 3 + τ ) 3 τ b) b ) τ 0 P τ ) 9 6 ) b ) b is the required mximum in Theorem 5 ii). Therefore T b 5) 9 6 ) b ) f 5). 0
10 0 S.S. DRAGOMIR AND A. SOFO iii) For n 6 let f ) + f b) b ) T b 6) : + f ) f b)) 6 b ) + f ) + f b )3 b)) + f ) f b)) 4 44 b ) )4 + f 4) ) + f 4) b) 440 b then T b 6) b ) 5+ q 70 B q q + 5q + ) f 6) p for f L p b p q > b ) 440 b ) For n 6 k 3 stisfies 5 ) 00 p + q ; f 6) for f L b ; f 6) for f L b ; 5 0 4)b ) t + b b t ) 5 t ) f 6) for f L b b ) 6 + b t ) 4 b t ) t ) b t ) 4 0 nd ) b ) M t 3) 7 which provides the required mximum in Theorem 5 i). Therefore ) b ) T b 6) f 6) References S.S. DRAGOMIR nd Th.M RASSIAS Ostrowski Type Inequlities nd Their Applictions in Numericl Integrtion Acdemic Publishers Dordrecht 00. A.M. FINK Bounds on the devition of function from its verges Czechoslovk Mth. J. 4 7) 99) D.S. MITRINOVIĆ J.E. PEČARIĆ nd A.M. FINK Inequlities for Functions nd their Integrls nd Derivtives Kluwer Acdemic Publishers Dordrecht 99. School of Computer Science nd Mthemtics Victori University of Technology PO Box 448 Melbourne City Victori 800 Austrli. E-mil ddress: sever.drgomir@vu.edu.u URL: E-mil ddress: nthony.sofo@vu.edu.u URL:
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