NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a

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1 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

2 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 ) f ) 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:

3 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 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:

5 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 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) ) )

7 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) 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 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 = ) 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 )

9 SIMPSON S INEQUALITIES 9 then the following ineulity holds: ) + b 3) f ) + 4f + f b) f x) dx 6 ) ) 5 ) 9b + 6 f + 6b 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 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 f 9 Therefore ) + b f ) + 4f + f b) 6 which completes the proof 5 ) 7 f x) dx ) 9b + 6 f + 6b 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 )

10 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

11 SIMPSON S INEQUALITIES Combining ll the obtined ineulities we get ) + b f ) + 4f + f b) f x) dx 6 ) ) ) ) ) ) 6 3b + + f + b f 4 ) ) ) ) ) + 3b + f + b 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

12 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

13 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

14 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=

15 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 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

16 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 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=

17 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 SS Drgomir On Simpson s udrture formul for mppings of bounded vrition nd pplictions Tmkng J Mthemtics 3 999) SS Drgomir On Simpson s udrture formul for Lipschitzin mppings nd pplictions Soochow J Mthemtics 5 999) SS Drgomir On Simpson s udrture formul for differentible mppings whose derivtives belong to L p spces nd pplictions J KSIAM 998) SS Drgomir RP Agrwl nd P Cerone On Simpson s ineulity nd pplictions J of Ineul Appl 5 ) 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) 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 ) - 4 US Kirmci et l Hdmrd-type ineulities for s-convex functions Appl Mth Comp 93 7) US Kirmci Ineulities for differentible mppings nd pplictions to specil mens of rel numbers to midpoint formul Appl Mth Comp 47 4) 37 46

18 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) ME Özdemir A theorem on mppings with bounded derivtives with pplictions to udrture rules nd mens Appl Mth Comp 38 3) CEM Perce nd J Pečrić Ineulities for differentible mppings with ppliction to specil mens nd udrture formul Appl Mth Lett 3 ) 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) 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: