NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS MOHAMMAD ALOMARI A MASLINA DARUS A AND SEVER S DRAGOMIR B Abstrct In terms of the first derivtive some ineulities of Simpson s type bsed on s-convexity nd concvity re introduced Best Midpoint type ineulities re given Error estimtes for specil mens nd some numericl udrture rules re lso obtined Introduction Suppose f : b R is four times continuously differentible mpping on b) nd f 4) := f 4) x) < The following ineulity ) sup x b) f ) + f b) + b + f 3 ) f x) dx f 4) ) 4 88 holds nd it is well known in the literture s Simpson s ineulity It is well known tht if the mpping f is neither four times differentible nor its fourth derivtive f 4) bounded on b) then we cnnot pply the clssicl Simpson udrture formul In recent yers mny uthors hve estblished error estimtions for the Simpson s ineulity; for refinements counterprts generliztions nd new Simpson-type ineulities see 3 nd 9 4 Drgomir in 8 pointed out some recent developments on Simpson s ineulity for which the reminder is expressed in terms of lower derivtives thn the fourth Some of the importnt results re presented below Theorem Suppose f : b R is differentible mpping whose derivtive is continuous on b) nd f L b Then the following ineulity ) f ) + f b) + b ) + f ) f x) dx f 3 3 holds where f = x) dx The bound of ) for L-Lipschitzin mppings ws given in 8 by 5 36L ) Also the following ineulity ws obtined in 8 Dte: September 9 Mthemtics Subject Clssifiction Primry 6D5 Secondry 6D Key words nd phrses Simpson s ineulity Midpoint ineulity s-convex function corresponding uthor
ALOMARI DARUS AND DRAGOMIR Theorem Let f : b R be n bsolutely continuous mpping on b whose derivtive belongs to L p b Then we hve the ineulity: ) f ) + f b) + b 3) + f f x) dx 3 + + ) f 6 3 + ) p where p ) + ) = p > In 5 some ineulities of Hermite-Hdmrd type for differentible convex mppings were presented s follows: Theorem 3 Let f : I R R be differentible mpping on I where b I with < b If is convex on b then the following ineulity holds: b ) + b 4) f x) dx f ) + b) 8 A more generl result relted to 4) ws estblished in 6 8 In 8 Hudzik nd Mligrnd considered mong others the clss of functions which re s-convex in the second sense This clss is defined in the following wy: function f : R + R where R + = ) is sid to be s-convex in the second sense if f αx + βy) α s f x) + β s f y) for ll x y ) α β with α + β = nd for some fixed s This clss of s-convex functions is usully denoted by K s It cn be esily seen tht for s = s-convexity reduces to the ordinry convexity of functions defined on ) In 3 Drgomir nd Fitzptrick proved vrint of Hdmrd s ineulity which holds for s convex functions in the second sense: Theorem 4 Suppose tht f : ) ) is n s convex function in the second sense where s ) nd let b ) < b If f L b then the following ineulities hold: ) + b 5) s f f x) dx f ) + f b) s + The constnt k = s+ is the best possible in the second ineulity in 5) The bove ineulities re shrp For recent results nd generliztions concerning Hdmrd s ineulity see nd 4 8 The im of this pper is to estblish Simpson type ineulities bsed on s- convexity nd concvity Using these results we cn estimte the errorf) in the Simpson s formul without going through its higher derivtives which my not exist not be bounded or my be hrd to find Ineulities of Simpson type for s Convex In order to prove our min theorems we need the following lemm:
SIMPSON S INEQUALITIES 3 Lemm Let f : I R R be n bsolutely continuous mpping on I where b I with < b Then the following eulity holds: ) where f ) + 4f 6 Proof We note tht I = = / + b p t) f tb + t) ) dt ) + f b) = ) f x) dx t 6 t ) p t) = t 5 6 t p t) f tb + t) ) dt t ) f tb + t) ) dt + t 5 ) f tb + t) ) dt 6 / 6 Integrting by prts we get I = = t 6 ) / f tb + t) ) / + t 5 ) f tb + t) ) 6 / ) + b + f b) f ) + 4f 6 ) f tb + t) ) dt / Setting x = tb + t) nd dx = )dt we obtin ) I = 6 f ) + 4f + b which gives the desired representtion ) ) + f b) f tb + t) ) dt f tb + t) ) dt f x)dt The next theorem gives new refinement of the Simpson ineulity for s-convex functions Theorem 5 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is s-convex on b for some fixed s then the following ineulity holds: ) 6 f ) + 4f + b ) + f b) f x) dx ) 6 s 9 ) s + 5) s+ 6 s + 3s 8 s + 3s + ) ) + b)
4 ALOMARI DARUS AND DRAGOMIR Proof From Lemm nd since f is s-convex we hve ) + b f ) + 4f + f b) f x) dx 6 ) s t) f tb + t) ) dt / ) t 6) tb + t) ) dt + ) t 5 / 6) tb + t) ) dt / ) t t 6) s b) + t) s ) ) dt + ) t 5 t / 6) s b) + t) s ) ) dt /6 ) = ) 6 t t s b) + t) s ) ) dt / + ) t ) t s b) + t) s ) ) dt /6 6 5/6 ) 5 + ) / 6 t t s b) + t) s ) ) dt + ) t 5 ) t s b) + t) s ) ) dt 6 5/6 = ) 6 s 9 ) s + 5) s+ 6 s + 3s 8 s + 3s + ) which completes the proof ) + b) Therefore we cn deduce the following result for convex functions Corollry Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is convex on b then the following ineulity holds: 3) 6 f ) + 4f + b ) + f b) f x) dx 5 ) 7 ) + b) Remrk We note tht the obtined midpoint ineulity 3) is better thn the ineulity ) A best upper bound for the midpoint ineulity in terms of first derivtive my be stted s follows:
SIMPSON S INEQUALITIES 5 Corollry In Theorem 5 if f ) = f ) +b = f b) then we hve 4) ) + b f x) dx f ) 6 s 9 ) s + 5) s+ 6 s + 3s 8 s + 3s + ) ) + b) Corollry 3 In Corollry setting s = we hve 5) ) + b 5 ) f x) dx f ) + b) 7 Remrk We note tht the obtined midpoint ineulity 5) is better thn the ineulity 4) The corresponding version of the Simpson s ineulity for powers in terms of the first derivtive is incorported in the following result: Theorem 6 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If p/p ) is s-convex on b for some fixed s nd p > then the following ineulity holds: 6) 6 f ) + 4f + b ) + f b) ) + p+ p ) 6 p+ p + ) s + ) f x) dx ) + + b f + b + f ) ) ) + b) ) where p + = Proof From Lemm using the well known Hölder integrl ineulity we hve ) + b f ) + 4f + f b) f x) dx 6 ) p t) f tb + t) ) dt / ) t 6) tb + t) ) dt + ) t 5 6) tb + t) ) dt /
6 ALOMARI DARUS AND DRAGOMIR / ) t 6 + ) t 5 6 = ) /6 + ) / 6 t 5/6 / ) ) p p ) / dt tb + t) ) dt ) p dt+ / ) ) p p dt tb + t) ) dt / /6 t 6) p dt ) p / ) p 5 6 t dt + t 5 p dt 5/6 6) tb + t) ) dt / ) ) p Since f is s convex by 5) we hve / 7) tb + t) ) dt ) + ) f +b s + nd 8) 6 f ) + 4f / ) tb + t) ) f +b + b) dt s + + b ) + f b) ) + p+ p ) 6 p+ p + ) s + ) which completes the proof f x) dx ) + + b f + b + f ) tb + t) ) dt ) ) ) + b) ) Corollry 4 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If p/p ) is convex on b for some fixed p > then the following ineulity holds: 9) 6 f ) + 4f where p + = + b ) + f b) ) + p+ ) p 6 p+ p + ) f x) dx ) + + b f + b + f ) ) ) + b) ) )
SIMPSON S INEQUALITIES 7 Corollry 5 In Theorem 6 if in ddition ) = b) = then ) 6 f ) + 4f where p + = + b ) + f b) f x) dx ) + p+ s + ) 6 p+ p + ) ) p f + b The corresponding version of the midpoint ineulity for powers in terms of the first derivtive is observed in the following result: Corollry 6 In Theorem 6 if f ) = f ) +b = f b) then we hve b ) + b ) f x) dx f ) + p+ p ) ) 6 p+ ) + + b ) p + ) s + ) f ) + b ) + f + b) Another version of the Simpson ineulity for powers in terms of the first derivtive is obtined s follows: Theorem 7 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is s-convex on b for some fixed s nd then the following ineulity holds: ) 6 f ) + 4f ) 6 s + 3s + ) + b ) ) + f b) f x) dx ) 5 {3 s ) s) + 3s s) + 3 s) b) 7 + 5 s+ 3 s s 6s s) s) + 6s 4 ) ) + 3 s) s) + 3s s) + 3 s) ) + 5 s+ 3 s s 6s s) s) + 6s 4 b) ) Proof Suppose tht From Lemm nd using the well known power men ineulity we hve ) + b f ) + 4f + f b) f x) dx 6 ) s t) f tb + t) ) dt }
8 ALOMARI DARUS AND DRAGOMIR / ) t 6) tb + t) ) dt + ) t 5 6) tb + t) ) dt / / ) t dt 6) + ) / ) / t 6) tb + t) ) dt ) t 5 dt 6) / ) t 5 6) tb + t) ) dt Since is s-convex therefore we hve / t 6) tb + t) ) dt /6 ) t 6 t s b) + t) s ) ) dt / + t t 6) s b) + t) s ) ) dt nd /6 = 3 s ) s) + 3s s) + 3 s ) 36 s + 3s + ) / 5/6 b) + 5s+ 3 s s 6s s ) s ) + 6s 4 36 s + 3s + ) t 5 6) tb + t) ) dt ) 5 t / 6 t s b) + t) s ) ) dt + t 5 t 6) s b) + t) s ) ) dt 5/6 = 3 s ) s) + 3s s) + 3 s ) 36 s + 3s + ) Also we note tht ) + 5s+ 3 s s 6s s ) s ) + 6s 4 36 s + 3s + ) / t dt = 6) / t 5 ) dt = 5 6 7 ) b) Combining ll the bove ineulities gives the reuired result which completes the proof Theorem 8 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is concve on b for some fixed )
SIMPSON S INEQUALITIES 9 then the following ineulity holds: ) + b 3) f ) + 4f + f b) f x) dx 6 ) ) 5 ) 9b + 6 f + 6b + 9 7 9 f 9 Proof First we note tht by the concvity of nd the power-men ineulity we hve αx + α) y) α x) + α) y) Hence αx + α) y) α x) + α) y) so is lso concve Accordingly by Lemm nd the Jensen integrl ineulity we hve / t 4) 6 f tb + t) ) dt / t / ) 6 dt) f t 6 tb + t) dt / t 6 dt = 5 ) 9b + 6 7 f 9 nd 5) t 5 / 6 f tb + t) ) dt t 5 ) t / 6 dt f 5 / 6 tb + t) dt t 5 / 6 dt = 5 ) 6b + 9 7 f 9 Therefore ) + b f ) + 4f + f b) 6 which completes the proof 5 ) 7 f x) dx ) 9b + 6 f + 6b + 9 9 f 9 ) Theorem 9 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is concve on b for some fixed > then the following ineulity holds: 6) 6 f ) + 4f + b ) + f b) ) ) + ) f 3b + 4 f x) dx ) + b + 3 f 4 )
ALOMARI DARUS AND DRAGOMIR Proof From Lemm we hve 6 f ) + 4f ) + b + f b) / ) Using the Hölder ineulity for > nd p = nd ) / ) f x) dx t 6 tb + t) ) dt + ) t 5 6 tb + t) ) dt / we obtin t 6 tb + t) ) dt / ) t ) / 6 dt tb + t) ) dt / t 5 6 tb + t) ) dt ) t 5 ) 6 dt tb + t) ) dt / / ) ) It is esy to check tht / t 6 dt = t 5 6 dt = / 6 ) ) + Since is concve on b we cn use Jensen s integrl ineulity to obtin / tb + t) ) dt = / t tb + t) ) dt / / ) t dt) f tb + t) ) dt / t dt = / f tb + t) ) dt) = ) b + 3 f 4 Anlogously / tb + t) ) dt ) 3b + f 4
SIMPSON S INEQUALITIES Combining ll the obtined ineulities we get ) + b f ) + 4f + f b) f x) dx 6 ) ) ) ) ) ) 6 3b + + f + b + 3 4 f 4 ) ) ) ) ) + 3b + f + b + 3 4 f 4 which completes the proof Theorem Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is s concve on b for some fixed s nd > then the following ineulity holds: 7) 6 f ) + 4f + b ) s )/ ) + f b) 6 f x) dx ) ) + ) 3 + b f + + 3b f ) Proof We proceed similrly s in the proof of Theorem 9 by using 5) insted of Jensen s integrl ineulity for concve functions For s concve we hve / ) tb + t) ) 3 + b dt s f nd / ) tb + t) ) + 3b dt s f so tht ) + b f ) + 4f + f b) 6 ) s )/ which completes the proof 6 f x) dx ) ) + ) 3 + b f + + 3b f ) Remrk 3 ) In Theorems 7 if f ) = f ) +b = f b) one cn obtin new ineulities of midpoint type However the detils re left to the interested reder ) All of the bove ineulities obviously hold for convex functions Simply choose s = in ech of the results to obtin the desired ones
ALOMARI DARUS AND DRAGOMIR 3 Applictions to Specil Mens Let s nd u v w R We define function f : ) R s u t = ; f t) = vt s + w t > If v nd w u then f K s see 3) Hence for u = w = v = we hve f : b R f t) = t s f K s In 3 the following result is given: Let f : I R + be non decresing nd s convex function on I nd g : J I I be non negtive convex function on J then f g is s convex on I A simple conseuence of the previous result my be stted s follows: Corollry 7 Let g : I I ) be non negtive convex function on I then g s x) is s convex on ) < s < For rbitrry rel numbers α β α β) we consider the following mens: ) The rithmetic men: A = A α β) := α + β α β R; ) The logrithmic men: L = L α β) := α β R α β; ln b ln 3) The generlized log-men: β p+ α p+ p L p = L p α β) := p R\ { } α β R α β p + ) β α) It is well known tht L p is monotonic nondecresing over p R with L := L nd L := I In prticulr we hve the following ineulity L A In the following some new ineulities re derived for the bove mens ) Consider f : b R < < b) fx) = x s s Then f x) dx = L s s b) f ) + f b) = A s b s ) ) + b f = A s b) ) Using the ineulity ) we obtin 3 A s b s ) + 3 As b) L s s b) s ) 6 s 9 ) s + 5) s+ 6 s + 3s 8 s + 3s + ) For instnce if s = then we get A b) L b) 5 ) 36 s + b s
SIMPSON S INEQUALITIES 3 b) Using the ineulity 4) we hve A s b) L s s b) s ) 6 s 9 ) s + 5) s+ 6 s + 3s 8 s + 3s + ) For instnce if s = then we obtin s + b s A b) L b) 5 ) 7 c) Using the ineulity 6) we get 3 A s b s ) + 3 As b) L s s b) ) + p+ p s ) s 6 p+ + A s b) ) p + ) s + ) + A s b) + b s ) where p > nd p + = For instnce if s = then we hve ) + p+ p A b) L b) ) 6 p+ p > p + ) ) Consider f : b ) R < < b) fx) = x K s s by Corollry 7) s Then f x) dx = L s s b) f ) + f b) = A s b s) ) + b f = A s b) ) Using the ineulity ) we obtin 3 A s b s) + 3 A s b) L s s b) s ) 6 s 9 ) s + 5) s+ 6 s + 3s 8 s + 3s + ) s + b s For instnce if s = then we get 3 A b ) + 3 A b) L b) 5 36 ) + b b) Using the ineulity 4) we hve A s b) L s s b) s ) 6 s 9 ) s + 5) s+ 6 s + 3s 8 s + 3s + ) s + b s
4 ALOMARI DARUS AND DRAGOMIR For instnce if s = then we obtin A b) L b) 5 7 ) + b c) Using the ineulity 6) we get 3 A s b s) + 3 A s b) L s s b) + p+ ) 6 p+ p + ) ) p s s + ) s + A s b) ) A + s b) + b s ) where p > nd p + = For instnce if s = then we hve 3 A b ) + 3 A b) L b) ) + p+ p ) 6 p+ + A b) ) p + ) A + b) + b ) p > 4 Applictions to Some Numericl Qudrture Rules Using the results of Section we now provide some pplictions for numericl udrture rules Nmely we will consider the Simpson nd Midpoint rules 4 Applictions to Simpson s Formul Let d be division of the intervl b ie d : = x < x < < x n < x n = b h i = x i+ x i )/ nd consider the Simpson s formul 4) S f d) = n i= f x i ) + 4f x i + h i ) + f x i+ ) 6 x i+ x i ) It is well known tht if the mpping f : b R is differentible such tht f 4) x) exists on b) nd M = mx x b) f 4) x) < then 4) I = f x) dx = S f d) + E S f d) where the pproximtion error E S f d) of the integrl I by Simpson s formul S f d) stisfies 43) E S f d) K n x i+ x i ) 5 9 It is cler tht if the mpping f is not four times differentible or the fourth derivtive is not bounded on b) then 4) cnnot be pplied In the following we give mny different estimtions for the reminder term E S f d) in terms of the first derivtive i=
SIMPSON S INEQUALITIES 5 Proposition Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is convex on b then in 4) for every division d of b the following holds: E S f d) 5 n x i+ x i ) x i ) + x i+ ) 7 i= Proof Applying Corollry on the subintervls x i x i+ i = n ) of the division d we get ) ) x i+ x i ) xi + x xi+ i+ f x i ) + 4f + f x i+ ) f x) dx 3 x i 5 x i+ x i ) x i ) + x i+ ) 7 Summing over i from to n nd tking into ccount tht is convex we deduce by the tringle ineulity tht S f d) f x) dx 5 n x i+ x i ) x i ) + x i+ ) 7 which completes the proof i= Proposition Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If p/p ) is convex on b p > then in 4) for every division d of b the following holds: ) E S f d) + p+ p 6 p+ p + ) n x i+ x i ) x i ) + f xi + x i+ i= + f xi + x i+ ) ) ) + x i+ ) ) Proof The proof is similr to tht of Proposition using the proof of Corollry 4 Proposition 3 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is concve on b for some fixed then in 4) for every division d of b the following holds: E S f d) 5 n ) ) x i+ x i ) f 9xi+ + 6x i + 7 9 f 6xi+ + 9x i 9 i= Proof The proof is similr to tht of Proposition using the proof of Theorem 8 Proposition 4 Let f : I ) R be differentible mpping on I such tht f L b where b I with < b If is concve on b for some
6 ALOMARI DARUS AND DRAGOMIR fixed > then in 4) for every division d of b the following holds: ) ) E S f d) + n ) x i+ x i ) f 3xi+ + x i + 4 4 i= f xi+ + 3x i ) Proof The proof is similr to tht of Proposition using the proof of Theorem 9 4 Applictions to the Midpoint Formul Let d be division of the intervl b ie d : = x < x < < x n < x n = b nd consider the midpoint formul n ) xi + x i+ 44) M f d) = x i+ x i ) f i= It is well known tht if the mpping f : b R is differentible such tht f x) exists on b) nd K = sup x b) x) < then 45) I = f x) dx = M f d) + E M f d) where the pproximtion error E M f d) of the integrl I by the midpoint formul M f d) stisfies 46) E M f d) K n x i+ x i ) 3 4 In the following we propose some new estimtes for the reminder term E M f d) in terms of the first derivtive which re better thn the estimtions of 8 Proposition 5 Let f : I R R be differentible mpping on I b I with < b If is convex on b then in 45) for every division d of b the following holds: i= E M f d) 5 n x i+ x i ) x i ) + x i+ ) 7 i= Proof Applying Corollry 3 on the subintervls x i x i+ i = n ) of the division d we get ) x xi + x xi+ i+ i+ x i ) f f x) dx x i 5 x i+ x i ) x i ) + x i+ ) 7 Summing over i from to n nd tking into ccount tht is convex we deduce tht E M f d) 5 n x i+ x i ) x i ) + x i+ ) 7 which completes the proof i=
SIMPSON S INEQUALITIES 7 Proposition 6 Let f : I R R be differentible mpping on I b I with < b If p/p ) is convex on b p > then in 45) for every division d of b the following holds: ) E M f d) + p+ p 6 p+ p + ) n x i+ x i ) x i ) + f xi + x i+ i= + f xi + x i+ ) ) ) + x i+ ) ) Proof The proof is similr to tht of Proposition 5 using Corollry 6 Acknowledgement The first uthor cknowledges the finncil support of Universiti Kebngsn Mlysi Fculty of Science nd Technology UKM GUP TMK 7 7) References M Alomri M Drus nd SS Drgomir Ineulities of Hermite-Hdmrd s type for functions whose derivtives bsolute vlues re usi-convex RGMIA 9) Supp No 4 M Alomri et l Refinements of Hdmrd-type ineulities for usi-convex functions with pplictions to trpezoidl formul nd to specil mens Comp Mth Appl 9) doi:6/jcmw98 3 NS Brnett P Cerone SS Drgomir MR Pinheiro nd A Sofo Ostrowski type ineulities for functions whose modulus of derivtives re convex nd pplictions RGMIA Res Rep Coll 5) ) Article 4 P Cerone nd SS Drgomir Ostrowski type ineulities for functions whose derivtives stisfy certin convexity ssumptions Demonstrtio Mth 37 4) no 99-38 5 SS Drgomir On Simpson s udrture formul for mppings of bounded vrition nd pplictions Tmkng J Mthemtics 3 999) 53-58 6 SS Drgomir On Simpson s udrture formul for Lipschitzin mppings nd pplictions Soochow J Mthemtics 5 999) 75-8 7 SS Drgomir On Simpson s udrture formul for differentible mppings whose derivtives belong to L p spces nd pplictions J KSIAM 998) 57-65 8 SS Drgomir RP Agrwl nd P Cerone On Simpson s ineulity nd pplictions J of Ineul Appl 5 ) 533-579 9 SS Drgomir JE Pečrić nd S Wng The unified tretment of trpezoid Simpson nd Ostrowski type ineulities for monotonic mppings nd pplictions J of Ineul Appl 3 ) 6-7 SS Drgomir nd Th M Rssis Eds) Ostrowski Type Ineulities nd Applictions in Numericl Integrtion Kluwer Acdemic Publishers Dordrecht/Boston/London SS Drgomir nd S Fitzptrick The Hdmrd s ineulity for s-convex functions in the second sense Demonstrtio Mth 3 4) 999) 687-696 I Fedotov nd SS Drgomir An ineulity of Ostrowski type nd its pplictions for Simpson s rule nd specil mens Preprint RGMIA Res Rep Coll 999) 3-3 H Hudzik nd L Mligrnd Some remrks on s-convex functions Aeutiones Mth 48 994) - 4 US Kirmci et l Hdmrd-type ineulities for s-convex functions Appl Mth Comp 93 7) 6 35 5 US Kirmci Ineulities for differentible mppings nd pplictions to specil mens of rel numbers to midpoint formul Appl Mth Comp 47 4) 37 46
8 ALOMARI DARUS AND DRAGOMIR 6 US Kirmci nd ME Özdemir On some ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to midpoint formul Appl Mth Comp 53 4) 36 368 7 ME Özdemir A theorem on mppings with bounded derivtives with pplictions to udrture rules nd mens Appl Mth Comp 38 3) 45 434 8 CEM Perce nd J Pečrić Ineulities for differentible mppings with ppliction to specil mens nd udrture formul Appl Mth Lett 3 ) 5 55 9 CEM Perce J Pečrić N Ujević nd S Vrošnec Generliztions of some ineulities of Ostrowski-Grüss type Mth Ine Appl 3 ) ) 5 34 J Pečrić nd S Vrošnec Simpson s formul for functions whose derivtives belong to L p spces Appl Mth Lett 4 ) 3-35 NUjević Shrp ineulities of Simpson type nd Ostrowski type Comp Mth Appl 48 4) 45-5 NUjević Two shrp ineulities of Simpson type nd pplictions Georgin Mth J ) 4) 87 94 3 NUjević A generliztion of the modified Simpson s rule nd error bounds ANZIAM J 47 5) E E3 4 N Ujević New error bounds for the Simpson s udrture rule nd pplictions Comp Mth Appl 53 7) 64-7 A School Of Mthemticl Sciences Universiti Kebngsn Mlysi UKM Bngi 436 Selngor Mlysi E-mil ddress: mwomth@gmilcom E-mil ddress: mslin@ukmmy B Mthemtics School of Engineering & Science Victori University PO Box 448 Melbourne City MC 8 Austrli E-mil ddress: severdrgomir@vueduu URL: http://wwwstffvueduu/rgmi/drgomir/