ON SIMPSON S INEQUALITY AND APPLICATIONS. 1. Introduction The following inequality is well known in the literature as Simpson s inequality : 2 1 f (4)

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1 ON SIMPSON S INEQUALITY AND APPLICATIONS SS DRAGOMIR, RP AGARWAL, AND P CERONE Abstrct New neultes of Smpson type nd ther pplcton to udrture formule n Numercl Anlyss re gven Introducton The followng neulty s well known n the lterture s Smpson s neulty : f x dx b f + f b + b + f f 4 b 5, 880 where the mppng f :, b R s ssumed to be four tmes contnuously dfferentble on the ntervl, b nd for the fourth dervtve to be bounded on, b, tht s, f 4 : sup f 4 x < x,b Now, f we ssume tht I n : x 0 < x < < x < x n b s prtton of the ntervl, b nd f s s bove, then we hve the clsscl Smpson s udrture formul: f x dx A S f, I n + R S f, I n where A S f, I n s the Smpson rule A S f, I n : f x + f x + h + x + x + f h nd the remnder term R S f, I n stsfes the estmte R S f, I n 880 f 4 4 where h : x + x for 0,, n When we hve n eudstnt prttonng of, b gven by 5 I n : x : + b n, 0,, n; Dte: Aprl, Mthemtcs Subject Clssfcton Prmry D 5, D 0; Secondry 4 A 55, 4 A 99 Key words nd phrses Smpson s Ineulty, Qudrture Formule h 5

2 SS DRAGOMIR, RP AGARWAL, AND P CERONE then we hve the formul where 7 A S,n f : b n f x dx A S,n f + R S,n f b + f n f + b n nd the remnder stsfes the estmton 8 + b n + + f + b n, R S,n f b n 4 f 4 + For some other ntegrl neultes see the recent book nd the ppers -4 nd 5-7 The mn purpose of ths survey pper s to pont out some very recent developments on Smpson s neulty for whch the remnder s expressed n terms of lower dervtves thn the fourth It s well known tht f the mppng f s nether four tmes dfferentble nor s the fourth dervtve f 4 bounded on, b, then we cnnot pply the clsscl Smpson udrture formul, whch, ctully, s one of the most used udrture formule n prctcl pplctons The frst secton of our pper dels wth n upper bound for the remnder n Smpson s neulty for the clss of functons of bounded vrton The second secton provdes some estmtes for the remnder when f s Lpschtzn mppng whle the thrd secton s concerned wth the sme problem for bsolutely contnuous mppngs whose dervtves re n the Lebesgue spces L p, b The fourth secton s devoted to the pplcton of celebrted result due to Grüss to estmtng the remnder n Smpson udrture rule n terms of the supremum nd nfmum of the frst dervtve The ffth secton dels wth generl convex combnton of trpezod nd nteror pont udrture formul from whch, n prtculr, we cn obtn the clsscl Smpson rule The lst secton contns some results relted to Smpson, trpezod nd md pont formule for monotonc mppngs nd some pplctons for probblty dstrbuton functons Lst, but not lest, we would lke to menton tht every secton contns specl subsecton n whch the theoretcl results re ppled for the specl mens of two postve numbers: dentrc men, logrthmc men, p-logrthmc men etcnd provdes mprovements nd relted results to the clsscl seuence of neultes H G L I A, where H, G, L, I nd A re defned n the seuel

3 SIMPSON S INEQUALITY Smpson s Ineulty for Mppngs of Bounded Vrton Smpson s Ineulty The followng result holds Theorem Let f :, b R be mppng of bounded vrton on, b Then we hve the neulty f x dx b b b f, f + f b + f + b where b f denotes the totl vrton of f on the ntervl, b The constnt s the best possble Proof hve: where Usng the ntegrton by prts formul for Remnn-Steltjes ntegrl we s x df x b s x : f + f b + f + b x 5+b, x, +b x +5b, x +b, b f x dx, Indeed, +b s x df x x 5 + b x 5 + b b f x df x + x + 5b +b +b + x + 5b f + f b + b + f f x df x b +b f x dx, f x dx b s seuence of d- nd the dentty s proved Now, ssume tht n : x n 0 < x n < < x n < xn n vsons wth ν n 0 s n, where ν n : mx {0,,} nd ξ n x n, x n + x n + xn If p :, b R s contnuous on, b nd v :, b R s

4 4 SS DRAGOMIR, RP AGARWAL, AND P CERONE of bounded vrton on, b, then p x dv x lm ν n 0 lm ν n 0 mx x,b p p p x sup n ξ n ξ n v v v b mx p x v x,b x n + x n + x n + v v v x n x n x n Applyng the neulty for p x s x nd v x f x we get b 4 s x df x mx s x f x,b Tkng nto ccount the fct tht the mppng s s monotonc nondecresng on the ntervls, +b nd +b, b nd nd we deduce tht s b, + b s 0 b, + b s b s b b, mx s x b x,b Now, usng the neulty 4 nd the dentty we deduce the desred result Now, for the best constnt Assume tht the followng neulty holds f x dx b f + f b + f wth constnt C > 0 Let us choose the mppng f :, b R gven by f x, +b f x f x +b + b b C b f +b, b

5 SIMPSON S INEQUALITY 5 Then we hve nd f x dx b b f + f b + f b f 4 b + b 4 b Now, usng the bove neulty, we get 4C b 4 b whch mples tht C nd then s the best possble constnt n It s nturl to consder the followng corollry whch follows from dentty Corollry Suppose tht f :, b R s dfferentble mppng whose dervtve s contnuous on, b nd f : Then we hve the neulty 5 f x dx b f b f x dx < f + f b + f + b The followng corollry for Smpson s composte formul holds: Corollry Let f :, b R be mppng of bounded vrton on, b nd I h prtton of, b Then we hve the Smpson s udrture formul nd the remnder term R S f, I h stsfes the estmte: where γ h : mx {h 0,, n } R S f, I h b γ h f, The cse of eudstnt prttonng s emboded n the followng corollry: Corollry Let I n be n eudstnt prttonng of, b nd f be s n Theorem Then we hve the formul nd the remnder stsfes the estmte: 7 R S,n f b n b f Remrk If we wnt to pproxmte the ntegrl f x dx by Smpson s formul A S,n f wth n ccurcy less tht ε > 0, we need t lest n ε N ponts for the dvson I n, where b n ε : ε b f + nd r denotes the nteger prt of r R

6 SS DRAGOMIR, RP AGARWAL, AND P CERONE Comments If the mppng f :, b R s nether four tme dfferentble nor the fourth dervtve s bounded on, b, then we cnnot pply the clsscl estmton n Smpson s formul usng the fourth dervtve But f we ssume tht f s of bounded vrton, then we cn use nsted the formul We gve here clss of mppngs whch re of bounded vrton but whch hve the fourth dervtve unbounded on the gven ntervl Let f p :, b R, f p x : x p where p, 4 Then obvously nd f p x : p x p, x, b f p 4 p p p p x x 4 p, x, b It s cler tht f p s of bounded vrton nd but lm x + f p 4 x + b f b p <, Applctons for Specl mens Let us recll the followng mens: The rthmetc men A A, b : + b,, b 0; The geometrc men The hrmonc men 4 The logrthmc men 5 The dentrc men G G, b : b,, b 0; H H, b : +,, b > 0; b L L, b : I I, b : e b,, b > 0, b; ln b ln b b b,, b > 0, b; The p-logrthmc men b p+ p+ p L p L p, b :, p R\ {, 0},, b > 0, b p + b It s well known tht L p s monotonc nondecresng over p R wth L : L nd L 0 : I In prtculr, we hve the followng neultes H G L I A Usng Theorem, some new neultes re derved for the bove mens

7 SIMPSON S INEQUALITY 7 Let f :, b R 0 < < b, f x x p, p R\ {, 0} Then nd f x dx L p, b, b f + f b A p, b p, + b f A p, b f p b L p p, p R\ {, 0, } Usng the neulty 5 we get Lp p, b A p, b p Ap, b p Let f :, b R 0 < < b, f x x Then nd Lp f x dx L, b, b f + f b H, b, + b f A, b f Usng the neulty 5 we get AH AL HL b G, b b G LHA Let f :, b R 0 < < b, f x ln x Then nd f x dx ln I, b, b f + f b ln G, b, + b f ln A, b f Usng the neulty 5 we obtn ln I G A b L, b b L p b

8 8 SS DRAGOMIR, RP AGARWAL, AND P CERONE Smpson s Ineulty for Lpschtzn Mppngs Smpson s Ineulty The followng result holds : Theorem Let f :, b R be n L Lpschtzn mppng on, b Then we hve the neulty f x dx b f + f b + b + f 5 L b Proof Usng the ntegrton by prts formul for Remnn-Steltjes ntegrl we hve see lso the proof of Theorem tht s x df x b f + f b + f + b f x dx where s x : x 5+b, x, +b x +5b, x +b, b b s seuence of d- Now, ssume tht n : x n 0 < x n < < x n < xn n vsons wth ν n 0 s n, where ν n : mx {0,,} nd ξ n x n, x n + R s L-Lpschtzn on, b, then x n + xn If p :, b R s Remnn ntegrble on, b nd v :, b p x dv x lm p ν n 0 lm ν n 0 ξ n pξ n v x n + v x n + xn L lm pξ n x n + ν n 0 xn L p x dx x n v x n + v x n + xn x n Applyng the neulty for p x s x nd v x f x we get 4 s x df x L s x dx

9 SIMPSON S INEQUALITY 9 Let us compute s x dx +b 5+b +5b + +b x 5 + b dx b x dx + + 5b +b x dx + x + 5b dx x 5 + b +b 5+b +5b x + 5b dx dx 5 b Now, usng the neulty 4 nd the dentty we deduce the desred result Corollry 4 Suppose tht f :, b R s dfferentble mppng whose dervtve s contnuous on, b Then we hve the neulty f x dx b f + f b + b 5 + f 5 f b The followng corollry for Smpson s composte formul holds: Corollry 5 Let f :, b R be n L Lpschtzn mppng on, b nd I h prtton of, b Then we hve the Smpson s udrture formul nd the remnder term R S f, I h stsfes the estmton: R S f, I h 5 L h The cse of eudstnt prttonng s emboded n the followng corollry: Corollry Let I n be n eudstnt prttonng of, b nd f be s n Theorem Then we hve the formul nd the remnder stsfes the estmton: 7 R S,n f 5 L n b Remrk If we wnt to pproxmte the ntegrl f x dx by Smpson s formul A S,n f wth n ccurcy less tht ε > 0, we need t lest n ε N ponts for the dvson I n, where 5 n ε : L b + ε nd r denotes the nteger prt of r R Comments If the mppng f :, b R s nether four tme dfferentble nor the fourth dervtve s bounded on, b, then we cn not pply the clsscl estmton n Smpson s formul usng the fourth dervtve But f we ssume tht f s Lpschtzn, then we cn use nsted the formul We gve here clss of mppngs whch re lpschtzn but hvng the fourth dervtve unbounded on the gven ntervl

10 0 SS DRAGOMIR, RP AGARWAL, AND P CERONE nd Let f p :, b R, f p x : x p where p, 4 Then obvously f p x : p x p, x, b f p 4 p p p p x x 4 p, x, b It s cler tht f p s Lpschtzn wth the constnt but lm x + f p 4 x + L p b p <, Applctons for Specl Mens Usng Theorem, we now pont out some new neultes for the specl mens defned n the prevous secton Let f :, b R 0 < < b, f x x p, p R\ {, 0} Then pb p f p f δ p, b : p p f p, \ {, 0} Usng the neulty 5 we get L p p, b A p, b p Ap, b 5 5 δ p, b b Let f :, b R 0 < < b, f x x Then Usng the neulty 5 we get f HA LA LH 5 b LAH Let f :, b R 0 < < b, f x ln x Then f Usng the neulty 5 we get ln I 5 b G A 4 Smpson s Ineulty n Terms of the p-norm 4 Smpson s neulty The followng result holds 4: Theorem Let f :, b R be n bsolutely contnuous mppng on, b whose dervtve belongs to L p, b Then we hve the neulty f x dx b f + f b + b 4 + f + + b + f + p, where p +, p >

11 SIMPSON S INEQUALITY Proof Usng the ntegrton by prts formul for bsolutely contnuous mppngs, we hve: 4 s x f x dx b f + f b + f + b f x dx where s x : x 5+b, x, +b x +5b, x +b, b Indeed, +b s x f x dx x 5 + b x 5 + b b f x f x dx + x + 5b +b +b + x + 5b f + f b + b + f f x f x dx b +b f x dx, f x dx nd the dentty s proved Applyng Hölder s ntegrl neulty we obtn 4 s x f x dx s x dx f p

12 SS DRAGOMIR, RP AGARWAL, AND P CERONE Let us compute s x dx +b x 5 + b dx + x + 5b +b dx 5+b +b 5 + b x dx + x 5 + b dx 5+b +5b + 5b b + x dx + x + 5b dx +b +5b 5+b b x + x 5 + b b + + 5b x + x + 5b + b +b +5b b + b b b + + b + + b + 5b b + + +b 5+b Now, usng the neulty 4 nd the dentty 4 we deduce the desred result 4 The followng corollry for Smpson s composte formul holds: Corollry 7 Let f nd I h be s bove Then we hve Smpson s rule nd the remnder R S f, I h stsfes the estmte 44 R S f, I h f p h + Proof Apply Theorem on the ntervl x, x + 0,, n to obtn x+ x f x dx h f x + f x + x + x + + f x+ f t p p dt + + f + p h + x

13 SIMPSON S INEQUALITY Summng the bove neultes over from 0 to, usng the generlzed trngle neulty nd Hölder s dscrete neulty, we get R S f, I h x+ f x dx h x nd the corollry s proved h + f x + f x + x + x + + f x+ x h + f p h + f t p p dt, x+ x p p f t p p dt The cse of eudstnt prttonng s emboded n the followng corollry: Corollry 8 Let f be s bove nd f I n s n eudstnt prttonng of, b, then we hve the estmte R S,n f + + b + f n + p Remrk If we wnt to pproxmte the ntegrl f x dx by Smpson s formul A S,n f wth n ccurcy less tht ε > 0, we need t lest n ε N ponts for the dvson I n, where + + n ε : b + f ε + p + nd r denotes the nteger prt of r R Comments If the mppng f :, b R s nether four tme dfferentble nor the fourth dervtve s bounded on, b, then we cn not pply the clsscl estmton n Smpson s formul usng the fourth dervtve But f we ssume tht f L p, b, then we cn use the formul 44 nsted We gve here clss of mppngs whose frst dervtves belong to L p, b but hvng the fourth dervtves unbounded on the gven ntervl Let f s :, b R, f s x : x s where s, 4 Then obvously f s x : s x s, x, b nd f s 4 s s s s x x 4 s, x, b It s cler tht lm x + f s 4 x +, but f s p s b s + p s p+ p <

14 4 SS DRAGOMIR, RP AGARWAL, AND P CERONE 4 Applctons for Specl Mens See Secton for the defnton of the mens Let f :, b R 0 < < b, f x x s, s R\ {, 0} Then nd b f x dx L s s, b, + b f A s, b, f + f b A s, b s f p s L s s p b p Usng the neulty 4 we get Ls s, b As, b s As, b + + s L s + s p, b b where p +, p > Let f :, b R 0 < < b, f x x Then nd b f x dx L, b, f + b A, b, f + f b H, b f p L p, b b p Usng the neulty 4 we get HA LA LH + AHL + L p + b p Let f :, b R 0 < < b, f x ln x Then nd b f x dx ln I, b, + b f ln A, b, f + f b ln A, b f p L p, b b p

15 SIMPSON S INEQUALITY 5 Usng the neulty 4, we obtn ln I + + L G / A / p, b b + 5 Grüss Ineulty for the Smpson Formul 5 Some Prelmnry results The followng ntegrl neulty whch estblshes connecton between the ntegrl of the product of two functons nd the product of the ntegrls of the two functons s well known n lterture s Grüss neulty 5, p 9: Theorem 4 Let f, g :, b R be two ntegrble functons such tht ϕ f x Φ nd γ g x Γ for ll x, b; ϕ, Φ, γ nd Γ re constnts Then we hve the neulty b f x g x dx b 5 f x dx g x dx b b b φ ϕ Γ γ, 4 nd the neulty s shrp n the sense tht the constnt 4 cn not be replced by smller one In 98, Ostrowsk cf, for exmple, p 48 proved the followng neulty whch gves n pproxmton of the ntegrl b f t dt s follows: Theorem 5 Let f :, b R be dfferentble mppng on, b whose dervtve f :, b R s bounded on, b, e, f : sup t,b f t dt < T hen: 5 f x b for ll x, b f t dt 4 + x +b b b f, In the recent pper, SS Drgomr nd S Wng proved the followng verson of Ostrowsk s neulty by usng the Grüss neulty 5 Theorem Let f : I R R be dfferentble mppng n the nteror of I nd let, b nti wth < b If f L, b nd γ f x Γ for ll x, b, then we hve the followng neulty: f x f b f 5 f t dt b b for ll x, b b Γ γ, 4 x + b

16 SS DRAGOMIR, RP AGARWAL, AND P CERONE They lso ppled ths result for specl mens nd n Numercl Integrton obtnng some udrture formule generlzng the md-pont udrture rule nd the trpezod rule Note tht the error bounds they obtned re n terms of the frst dervtve whch re prtculrly useful n the cse when f does not exst or s very lrge t some ponts n, b For other relted results see the ppers 7-7 In ths secton of our pper we gve generlzton of the bove neulty whch contns s prtculr cse the clsscl Smpson formul Applcton for specl mens nd n Numercl Integrton re lso gven 5 An Integrl Ineulty of Grüss Type For ny rel numbers A, B, let us consder the functon { t + A p t p x t t b + B f t x x < t b It s cler tht p x hs the followng propertes It hs the jump t pont t x nd p x B A b dp x t dt + p x δ t x b Let M x : sup t,b p x t nd m x : nf t,b p x t Then the dfference M x m x cn be evluted s follows : For B A 0, we hve M x m x p x For B A > 0, the followng three cses re possble If 0 B A b, then x + b for x + B A ; M x m x p x for + B A < x b B A ; x for b B A < x b If b < B A b, then x + b for x < b B A ; M x m x B A for b B A x < + B A ; x for + B A x b If B A > b, then M x m x p x The followng neulty of Ostrowsk type holds Theorem 7 Let f :, b R be dfferentble mppng on, b whose dervtve stsfes the ssumpton 54 γ f t Γ for ll t, b,

17 SIMPSON S INEQUALITY 7 where γ, Γ re gven rel numbers Then we hve the neulty: 55 C A f + b B + A f x + B C f b where 4 Γ γ M x m x b, C x : x x + A x b x b + B, b nd A, B, M x nd m x re s bove, x, b Proof 5 Usng the Grüss neulty 5, we cn stte tht b p x t f f b f t dt b b b 4 Γ γ M x m x, for ll x, b Integrtng the frst term by prts we obtn: 57 Also, s p x t f t dt Bf b Af p x t dt f t dt + p x f x p x t dt x x + A x b x b + B, then 5 gves the neulty: Bf b Af b 4 Γ γ M x m x, f t dt + p x f x whch s clerly euvlent wth the desred result 55 C x f t dt f b f b Remrk 4 Settng n 55, A B 0 nd tkng nto ccount, by the property b, tht M x m x b, we obtn the neulty 5 by Drgomr nd Wng The followng corollry s nterestng: Corollry 9 Let A, B rel numbers so tht 0 B A b If f s s bove, then we hve the neulty 58 B A f + b B A f Γ γ b B + A b 4 Proof Consder x +b Then, from 55, + b x b, + B A f b f t dt

18 8 SS DRAGOMIR, RP AGARWAL, AND P CERONE nd x b b C x A + B, x + B A, b B A By property b we hve M x m x b B A Applyng Theorem 7 for x +b, we get esly 58 Remrk 5 If we choose n the bove corollry B A b, then we get f + f b + b b 59 + f b f t dt 8 Γ γ b, whch s the rthmetc men of the md-pont nd trpezod formule Remrk If we choose n 58 B A, then we get the md-pont neulty + b b 50 b f f t dt Γ γ b 4 dscovered by SS Drgomr nd S Wng n the pper see Corollry Remrk 7 If we choose n 58 B A b, then we obtn the celebrted Smpson s formul + b b 5 f + 4f + f b f t dt Γ γ b, for whch we hve n estmton n terms of the frst dervtve not s n the clsscl cse n whch the forth dervtve s reured s follows: + b b 5 f + 4f + f b f t dt f b 5 The method of evluton of the error for the Smpson rule consdered bove cn be ppled for ny udrture formul of Newton - Cotes type For exmple, to get the nlogous evluton of the error for the Newton-Cotes rule of order t s suffcent to replce the functon p x t n by the functon t A f t + h; p x t : t +b + A+B f + h < t b h; t b B f b h < t b; where B A b 4, h b

19 SIMPSON S INEQUALITY 9 5 Applctons for Specl Mens See Secton for the defnton of the mens Consder the mppng f x x p p >, x > 0 Then Γ γ b p L p p for, b R wth 0 < < b Conseuently, we hve the neulty Ap, b + A p, b p L p p, b b p L p p Consder the mppng f x x, x > 0 Then Γ γ b b A, b b G 4, b for 0 < < b Conseuently we hve the neulty: A, b + H, b L, b b A, b G 4, b whch s euvlent to HL + AL AH b A HL G 4 Consder the mppng f x ln x, x > 0 Then we hve Γ γ b G for, b R wth 0 < < b Conseuently, we hve the neulty ln A + ln G ln I whch s euvlent to ln A G I b G, b G 54 Estmton of Error Bounds n Smpson s Rule The followng theorem holds Theorem 8 Let f :, b R be dfferentble mppng on, b whose dervtve stsfes the condton γ f t Γ for ll t, b ; where γ, Γ re gven rel numbers Then we hve 5 f t dt S n I n, f + R n I n, f

20 0 SS DRAGOMIR, RP AGARWAL, AND P CERONE where 54 I h s the prtton gven by S n I n, f h f x + 4f x + h + f x +, o I n : x 0 < x < < x < x n b h : x + x, 0,, n nd the remnder term R n I n, f stsfes the estmton: R n I n, f Γ γ 55 h Proof Let us set n 5 x, b x +, h x + x nd x + h x + x + where 0,, n Then we hve the estmton: h f x + 4f x + h + f x + x + for ll 0,, n After summng nd usng the trngle neulty, we obtn h b f x + 4f x + h + f x + Γ γ h, whch proves the reured estmton x f t dt Γ γ h, f t dt Corollry 0 Under the bove ssumptons nd f we put f : sup t,b f t <, then we hve the followng estmton of the remnder term n Smpson s formul R n I n, f 4 f 5 h The clsscl error estmtes bsed on the Tylor expnson for the Smpson s rule nvolve the forth dervtve f 4 In the cse tht f 4 does not exsts or s very lrge t some ponts n, b, the clsscl estmtes cn not be ppled, nd thus 55 nd 5 provde lterntve error estmtes for the Smpson s rule A Convex Combnton The followng generlzton of Ostrowsk s neulty holds 9:

21 SIMPSON S INEQUALITY Theorem 9 Let f :, b R be bsolutely contnuous on, b, nd whose dervtve f :, b R s bounded on, b Denote f : ess sup t,b f x < Then 4 b δ + δ + f + f b f t dt f x δ + for ll δ 0, nd + δ b x b δ b x + b Proof Let us defne the mppng p :, b R gven by t + δ b, t, x p x, t : t b δ b, t x, b Integrtng by prts, we hve: x δ b p x, t f t dt t + δ b f t dt + f + f b x + δ f x δ b f t b δ b f t dt f t dt On the other hnd, p x, t f t dt p x, t f t dt f p x, t dt x f t + δ b b dt + t b δ b dt : f L Now, let us observe tht r p t dt p t dt + r x t dt p + r 4 p r + for ll r, p, such tht p r Usng the prevous dentty, we hve tht x t + δ b dt 4 x + + δ b + x r + p

22 SS DRAGOMIR, RP AGARWAL, AND P CERONE nd Then we get L x + b x b 4 t b δ b dt x 4 b x + b δ b x + b δ + δ + nd the theorem s thus proved + δ b x + b x b x + δ b Remrk 8 If we choose n, δ 0, we get Ostrowsk s neulty b If we choose n, δ nd x +b we get the trpezod neulty: f + f b f t dt b 4 b f Corollry Under the bove ssumptons, we hve the neulty: f t dt f + f b f x + b 8 b + x + b f for ll x b+, nd, n prtculr, the followng mxture of the trpezod 4, +b 4 neulty nd md-pont neulty: 4 f t dt f 8 b f + b + f + f b b Fnlly, we lso hve the followng generlzton of Smpson s neulty: Corollry Under the bove ssumptons, we hve b f t dt f + 4f x + f b b 5 b + x + b f for ll x b+5 4, +5b 4, nd, n prtculr, the Smpson s neulty f t dt + b 5 f + 4f + f b b 5 b f

23 SIMPSON S INEQUALITY Applctons n Numercl Integrton The followng pproxmton of the ntegrl f x dx holds 9 Theorem 0 Let f :, b R be n bsolutely contnuous mppng on, b whose dervtve s bounded on, b If I n : x 0 < x < < x < x n b s prtton of, b nd h : x + x, 0,, n, then we hve: where 7 f x dx A δ I n, ξ, δ, f + R δ I n, ξ, δ, f A δ I n, ξ, δ, f δ f ξ h + δ f x + f x + δ 0,, x + δ h ξ x + δ h, 0,, n ; nd the remnder term stsfes the estmton: 8 R δ I n, ξ, δ, f f δ + δ h + ξ 4 x + x + h, Proof Applyng Theorem 9 on the ntervl x, x +, 0,, n we get h δ f ξ + f x + f x + x+ δ f x dx x δ + δ h 4 + ξ x + x + f for ll δ 0, nd ξ x, x +, 0,, n Summng over from 0 to n nd usng the trngle neulty we get the estmton 8 Remrk 9 If we choose δ 0, then we get the udrture formul 9 f x dx A T I n, ξ, f + R T I n, ξ, f where A T I n, ξ, f s the Remnn s sum, e, A T I n, ξ, f : f ξ h, ξ x, x +, 0,, n ; nd the remnder term stsfes the estmte see lso 8: R T I n, ξ, f f h 4 + ξ x + x + 0 b If we choose δ, then we get the trpezod formul f x dx A T I n, f + R T I n, f

24 4 SS DRAGOMIR, RP AGARWAL, AND P CERONE where A T I n, f s the trpezodl rule A T I n, f f x + f x + nd the remnder terms stsfes the estmton R T I n, f f h 4 Corollry Under the bove ssumptons we hve where h f x dx B T I n, ξ, f + Q T I n, ξ, f B T I n, ξ, f x+ + x ξ 4 f ξ h +, x + x + 4 f x + f x + h,, nd the remnder term stsfes the estmton Q T I n, ξ, f f h + ξ 8 x + x + 4 In prtculr, we hve 5 where B T I n, f f x dx B T I n, f + Q T I n, f x + x + f nd Q T I n, f stsfes the estmton: Q T I n, f f 8 h + h f x + f x + h Fnlly, we hve the followng generlzton of Smpson s neulty whose remnder term s estmted by the use of the frst dervtve only Corollry 4 Under the bove ssumptons we hve: 7 where f x dx S T I n, ξ, f + W T I n, ξ, f S T I n, ξ, f f ξ h + f x + f x + h, x+ + 5x ξ, x + 5x +,

25 SIMPSON S INEQUALITY 5 nd the remnder term W T I n, ξ, f stsfes the bound: W T I n, ξ, f f 5 h + ξ x + x + 8 nd, n prtculr, the Smpson s rule: 9 where f x dx S T I n, f + W T I n, f S T I n, f x + x + f h + f x + f x + h nd the remnder term stsfes the estmton: 0 W T I n, f 5 f h Applctons for Specl Mens Now, let us reconsder the neulty n the followng euvlent form: f + f b δ f x + δ f t dt b δ + δ x +b b + f 4 b for ll δ 0, nd x, b such tht b b + δ x b δ Consder the mppng f : 0, 0,, f x x p, p R\ {, 0} Then, for 0 < < b, we hve { f p b p f p > p p f p, \ {, 0}, nd then, by, we deduce tht: δ x p + δ A p, b p L p p, b { δ + δ b + 4 x A b } δ p, b where { p b δ p, b : p f p > p p f p, \ {, 0} nd δ 0,, x + δ b, b δ b Consder the mppng f : 0, 0,, f x x nd 0 < < b We hve: f

26 SS DRAGOMIR, RP AGARWAL, AND P CERONE nd then by, we deduce, for ll δ 0,, nd +δ b x b δ b tht: δ δl + Lxδ xδ xδl δ + δ b + 4 x A b Consder the mppng f : 0, R, f x ln x nd 0 < < b We hve: f, nd then, by, we deduce tht x δ ln G δ δ + δ b + I 4 x A, b for ll δ 0,, nd x + δ b, b δ b 7 A Generlzton for Monotonc Mppngs In 0, SS Drgomr estblshed the followng Ostrowsk type neulty for monotonc mppngs Theorem Let f :, b R be monotonc nondecresng mppng on, b Then for ll x, b, we hve the neulty: f x f x dx b x + b f x + b sgnt xftdt x f x f + b x f b f x b + x +b f b f b All the neultes re shrp nd the constnt s the best possble one In ths secton we shll obtn generlzton of ths result whch lso contns the trpezod nd Smpson type neultes The followng result holds 8:

27 SIMPSON S INEQUALITY 7 Theorem Let f :, b R be monotonc nondecresng mppng on, b nd t, t, t, b be such tht t < t < Then 7 f x dx t f + b t f b + t t ft b t f b + t t t ft t f + b t f b ft + t t ft ft + t t ft ft + t ft ft mx {t, t t, t t, b t } f b f T x f x dx where sgnt x, for x, t T x sgnt x, for x t, b Proof Usng ntegrton by prts formul for Remnn-Steltjes ntegrl we hve s x df x t f + b t f b + t t ft f x d x where Indeed, x t, x, t s x x t, x t, b s x df x t x t df x + x t f x t t x t df x + x t ft b t f x d x t f + b t f b + t t ft Assume tht A n : x n 0 < x n < < x n < xn n dvsons wth νa n 0 s n, where νa n : ξ n x n, x n + f x dx b s seuence of mx x n,, xn nd If p :, b R s contnuous mppng on, b nd v s

28 8 SS DRAGOMIR, RP AGARWAL, AND P CERONE monotonc nondecresng on, b, then 7 p x dv x lm pξ n vx n + νan vxn lm pξ n vx n + vxn νan lm pξ n vx n + νan vxn p x dv x Applyng the neulty 7 for p x s x nd v x f x, x, b we cn stte: s x df x t s x df x t xdf x + t xf x t + t xf x t t + t t + t x t df x + t t t t t xdf x + t f x dx + x t f x t t f x dx + t b f x dx + x t f x t + f x dx t f + t t ft t t ft + b t f b + T x f x dx t t x t df x whch s the frst neulty n 7 If f :, b R s monotonc nondecresng n, b, we cn lso stte t f x dx ft t, t t f x dx ft t t,

29 SIMPSON S INEQUALITY 9 nd So, We hve t t t f x dx ft t t, T x f x dx t f x dx ft b t t t f x dx f x dx + f x dx f x dx t t t ft t ft t t + ft t t ft b t t f + t t ft t t ft b t f b + T x f x dx t f + t t ft t t ft + b t f b + t ft t t ft + t t ft b t ft t ft f + t t ft ft + t t ft ft + b t f b ft mx {t, t t, t t, b t } f b f The theorem s thus proved Remrk 0 For t 0, t x, t b, generlzed trpezod neulty s obtned nd we get Theorem from the bove Theorem For t t t x Theorem becomes: Corollry 5 Let f be defned s n Theorem Then 7 ftdt x f + b xf b b xf b x f + sgnx tftdt b x f b f x + x f x f b + x + b f b f All the neultes n 7 re shrp nd the constnt s the best possble

30 0 SS DRAGOMIR, RP AGARWAL, AND P CERONE Proof We only need to prove tht the constnt the mppng f 0 :, b R gven by s the best possble one Choose 0, f x, b, f 0 x, f x b Then, f 0 s monotonc nondecresng on, b, nd for x we hve ftdt x f + b xf b b xf b x f + sgnt xftdt b x f b f x + x f x f b C b + x + b f b f C + b whch prove the shrpness of the frst two neultes nd the fct tht C cnnot be less thn For x +b we get the trpezod neulty Corollry Let f :, b R be monotonc nondecresng mppng on, b Then 74 f + f b ftdt b b b f b f sgn t + b ftdt b f b f The constnt fctor s the best n both neultes

31 SIMPSON S INEQUALITY Corollry 7 Let f be s n Theorem 9 nd p, R + wth p > Then f x dx b f + f b + p + b f p + where b f b f + p + p + b f b f + p pb + b f p + p + { mx, p T x T x f x dx f } b f b f p + sgn p+b p+ x p + b p +, f x, +b sgn pb+ p+ x, f x +b, b Proof Set n Theorem : t p+b p+, t +b, t +pb p+ Remrk Of specl nterest s the cse p 5 nd where we get from Corollry 7 the followng result of Smpson type; f x dx f + f b + b b + f b f b f + b f b f + f T x f x dx 5b + f 5 + b b f b f, where T x { sgn 5+b x x, +b, sgn +5b x, x +b, b Remrk For p we get Corollry from Corollry 7 7 An Ineulty for the Cumultve Dstrbuton Functon Let X be rndom vrble tkng vlues n the fnte ntervl, b, wth cumultve dstrbutons functon F x PrX x The followng result from 0 cn be obtned from Theorem

32 SS DRAGOMIR, RP AGARWAL, AND P CERONE Theorem Let X nd F be s bove Then we hve the neultes 75 b E x PrX x b x + b PrX x + b b x PrX x + x PrX x b x +b + b for ll x, b All the neultes n 75 re shrp nd the constnt Now we shll prove the followng result sgn t x F tdt s the best possble Theorem 4 Let X nd F be s bove Then we hve the neultes 7 E x x b x + sgn x t F tdt b x PrX x + x PrX x b + x + b for ll x, b All the neultes n 7 re shrp nd the constnt s the best possble Proof Apply Corollry 5 for the monotonc nondecresng mppng ft : F t, t, b to get 77 F tdt x F + b xf b b xf b + x F + sgn x t F tdt b xf b F x + x F x F b + x + b F b F nd s F 0, F b

33 SIMPSON S INEQUALITY by the ntegrton by prts formul for Remnn-Steltjes ntegrls b E x tdf t tf t b F tdt Tht s, bf b F b F tdt F tdt b E x F tdt The neultes 77 gve the desred estmton 7 Corollry 8 Let X be rndom vrble tkng vlues n the fnte ntervl, b, wth cumultve dstrbuton functon F x PrX x nd the expectton E x Then we hve the neulty The constnt E x + b b b s the best n both neultes References sgn t + b F tdt b DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Ineultes for Functons nd Ther Integrls nd Dervtves, Kluwer Acdemc Publshers, Dordrecht, 994 SS DRAGOMIR, On Smpson s udrture formul for mppngs of bounded vrton nd pplctons, Tmkng J of Mthemtcs, n press SS DRAGOMIR, On Smpson s udrture formul for Lpschtzn mppngs nd pplctons, Soochow J of Mthemtcs, n press 4 SS DRAGOMIR, On Smpson s udrture formul for dfferentble mppngs whose dervtves belong to L p spces nd pplctons, J KSIAM, 998, DS MITRINOVIĆ, JE PEČARIĆ nd AM FINK, Clsscl nd New Ineultes n Anlyss, Kluwer Acdemc Publshers, Dordrecht, 99 SS DRAGOMIR nd S WANG, An neulty of Ostrowsk-Grüss type nd ts pplctons to the estmton of error bounds for some specl mens nd for some numercl udrture rules, Computers Mth Applc, 997, SS DRAGOMIR nd S WANG, A new neulty of Ostrowsk s type n L norm nd pplctons to some specl mens nd to some numercl udrture rules, Tmkng J of Mthemtcs, 8997, SS DRAGOMIR nd S WANG, Applctons of Ostrowsk s neulty to the estmton of error bounds for some specl mens nd some numercl udrture rules, Appl Mth Lett, 998, SS DRAGOMIR nd S WANG, A new neulty of Ostrowsk s type n L p norm, Indn J of Mthemtcs, n press 0 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An neulty of Ostrowsk-Grüss type for twce dfferentble mppngs nd pplctons n numercl ntegrton, Kyungpook Mth J, n press

34 4 SS DRAGOMIR, RP AGARWAL, AND P CERONE NS BARNETT nd SS DRAGOMIR, An neulty of Ostrowsk s type for cumultve dstrbuton functons, Kyungpook Mth J, n press SS DRAGOMIR, NS BARNETT nd P CERONE, An Ostrowsk type neulty for double ntegrls n terms of L p -norms nd pplctons n numercl ntegrton, Rev D Anly Num Theor Approx, n press T PEACHEY, A MC ANDREW nd SS DRAGOMIR, The best constnt n n eulty of Ostrowsk type, Tmkng J of Mth, n press 4 SS DRAGOMIR, YJ CHO nd SS KIM, Some remrks on Mlnovć-Pečrć neulty nd pplctons for specl mens nd numercl ntegrton, Tmkng J of Mth, ccepted 5 SS DRAGOMIR, RP AGARWAL nd NS BARNETT, Ineultes for Bet nd Gmm functons v some clsscl nd new ntegrl neultes, Journl of Ineultes nd Applctons, ccepted SS DRAGOMIR, A generlzton of Ostrowsk ntegrl neulty for mppngs of bounded vrton nd pplctons n numercl ntegrton, Bull Austrln Mth Soc, ccepted 7 SS DRAGOMIR,NS BARNETT nd S WANG, An Ostrowsk type neulty for rndom vrble whose probblty densty functon belongs to L p, b, p >, Mthemtcl Ineultes nd Applctons, ccepted 8 SS DRAGOMIR nd NS BARNETT, An Ostrowsk type neulty for mppngs whose second dervtves re bounded nd pplctons, The Journl of the Indn Mthemtcl Socety, ccepted 9 SS DRAGOMIR, P CERONE nd J ROUMELIOTIS, A new generlzton of Ostrowsk s ntegrl neulty for mppngs whose dervtves re bounded nd pplctons n numercl ntegrton nd for specl mens, Appl Mth Lett, ccepted 0 SS DRAGOMIR, Ostrowsk s neulty for monotonc mppngs nd pplctons, J KSIAM, ccepted I FEDOTOV nd SS DRAGOMIR, An neulty of Ostrowsk type nd ts pplctons for Smpson s rule nd specl mens, Preprnt, RGMIA Res Rep Coll, 999, -0 rgm SS DRAGOMIR, On the Ostrowsk s ntegrl neulty for mppngs wth bounded vrton nd pplctons, Preprnt, RGMIA Reserch Report Collecton, 999, -70 SS DRAGOMIR, On the Ostrowsk s ntegrl neulty for Lpschtzn mppngs wth bounded vrton, Preprnt, RGMIA Reserch Report Collecton, 999, SS DRAGOMIR, P CERONE, J ROUMELIOTIS nd S WANG, A weghted verson of Ostrowsk neulty for mppngs of Hölder type nd pplctons n numercl nlyss, Preprnt, RGMIA Reserch Report Collecton, 999, P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, On Ostrowsk type neulty for mppngs whose second dervtves belong to L p, b nd pplctons, Preprnt, RGMIA Reserch Report Collecton, 998, 4-50 NS BARNETT nd SS DRAGOMIR, An Ostrowsk type neulty for double ntegrls nd pplctons for embouchure formule, Preprnt, RGMIA Reserch Report Collecton, 998, - 7 NS BARNETT nd SS DRAGOMIR, An Ostrowsk s type neulty for rndom vrble whose probblty densty functon belongs to L, b, Preprnt, RGMIA Reserch Report Collecton, 998, - 8 P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An neulty of Ostrowsk type for mppngs whose second dervtves re bounded nd pplctons, Preprnt, RGMIA Reserch Report Collecton, 998, P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, Some Ostrowsk type neultes for n tme dfferentble mppngs nd pplctons, Preprnt, RGMIA Reserch Report Collecton, 998, J ROUMELIOTIS, P CERONE nd SS DRAGOMIR, An Ostrowsk type neulty for weghted mppngs wth bounded second dervtves, Preprnt, RGMIA Reserch Report Collecton, 998, P CERONE, SS DRAGOMIR nd J ROUMELIOTIS, An neulty of Ostrowsk type for mppngs whose second dervtves belong to L, b nd pplctons, Preprnt, RGMIA Reserch Report Collecton, 998, 5-58

35 SIMPSON S INEQUALITY 5 SS DRAGOMIR, A generlzton of Ostrowsk ntegrl neulty for mppngs whose dervtves belong to L nd pplctons n numercl ntegrton, submtted P CERONE nd SS DRAGOMIR, On weghted generlzton of Iyengr type neultes nvolvng bounded frst dervtve, submtted 4 P CERONE nd SS DRAGOMIR, Lobtto type udrture rules for functons wth bounded dervtve, submtted 5 SS DRAGOMIR, NS BARNETT nd P CERONE, An N-dmensonl verson of Ostrowsk s neulty for mppngs of the Hölder type, submtted SS DRAGOMIR, A generlzton of Ostrowsk ntegrl neulty for mppngs whose dervtves belong to L, b nd pplctons n numercl ntegrton, submtted 7 SS DRAGOMIR, A generlzton of Ostrowsk ntegrl neulty for mppngs whose dervtves belong to L p, b nd pplctons n numercl ntegrton, submtted 8 SS DRAGOMIR, JE PEČARIĆ nd S WANG, The unfed tretment of trpezod, Smpson nd Ostrowsk type neultes for monotonc mppngs nd pplctons, n preprton SS Drgomr School of Communctons nd Informtcs, Vctor Unversty of Technology, PO Box 448, MC Melbourne Cty, 800 Vctor, Austrl E-ml ddress, SS Drgomr: sever@mtldvueduu URL: Current ddress, RP Agrwl: Deprtment of Mthemtcs, Ntonl Unversty of Sngpore, 0 Kent Rdge Crescent, Sngpore 90 E-ml ddress, RP Agrwl: mtrvp@leonsnussg PCerone School of Communctons nd Informtcs, Vctor Unversty of Technology, PO Box 448, MC Melbourne Cty, 800 Vctor, Austrl URL: