Journal of Inequalities in Pure and Applied Mathematics

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1 Journl of Inequlities in Pure nd Applied Mthemtics Volume, Issue, Article, 00 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS SCHOOL OF COMMUNICATIONS AND INFORMATICS, PO BOX 448, MELBOURNE CITY MC 800, VICTORIA, AUSTRALIA neil@mtildvueduu pc@mtildvueduu sever@mtildvueduu johnr@mtildvueduu Received 7 Jnury, 000; ccepted 6 June, 000 Communicted by CEM Perce ABSTRACT Some inequlities for the dispersion of rndom vrible whose pdf is defined on finite intervl nd pplictions re given Key words nd phrses: Rndom vrible, Expecttion, Vrince, Dispersion, Grüss Inequlity, Chebychev s Inequlity, Lupş Inequlity 000 Mthemtics Subject Clssifiction 60E5, 6D5 INTRODUCTION In this note we obtin some inequlities for the dispersion of continuous rndom vrible X hving the probbility density function pdf) f defined on finite intervl, b Tools used include: Korkine s identity, which plys centrl role in the proof of Chebychev s integrl inequlity for synchronous mppings 4, Hölder s weighted inequlity for double integrls nd n integrl identity connecting the vrince σ X) nd the expecttion E X) Perturbed results re lso obtined by using Grüss, Chebyshev nd Lupş inequlities In Section 4, results from n identity involving double integrl re obtined for vriety of norms SOME INEQUALITIES FOR DISPERSION Let f :, b R R + be the pdf of the rndom vrible X nd E X) : ISSN electronic): c 00 Victori University All rights reserved tf t) dt

2 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS its expecttion nd σ X) t E X)) f t) dt t f t) dt E X) its dispersion or stndrd devition The following theorem holds Theorem With the bove ssumptions, we hve b ) f 6, provided f L,, b ; + b ) q ) 0 σ X) f p, provided f L p, b Proof Korkine s identity 4, is ) p t) dt q+)q+) q b ) p t) g t) h t) dt nd p >, p + q ; p t) g t) dt p t) h t) dt p t) p s) g t) g s)) h t) h s)) dtds, which holds for the mesurble mppings p, g, h :, b R for which the integrls involved in ) exist nd re finite Choose in ) p t) f t), g t) h t) t E X), t, b to get ) σ X) It is obvious tht 4) f t) f s) t s) dtds f t) f s) t s) dtds sup t,s),b f t) f s) t s) dtds b )4 f 6 nd then, by ), we obtin the first prt of ) For the second prt, we pply Hölder s integrl inequlity for double integrls to obtin ) f t) f s) t s) dtds f p t) f p p b s) dtds t s) q dtds f p b ) q+ q, q + ) q + ) where p > nd +, nd the second inequlity in ) is proved p q For the lst prt, observe tht s f t) f s) t s) dtds f t) f s) dtds sup t s) t,s),b f t) dt f t) f s) dtds b ) f s) ds ) q J Inequl Pure nd Appl Mth, ) Art, 00

3 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE Using finer rgument, the lst inequlity in ) cn be improved s follows Theorem Under the bove ssumptions, we hve 5) 0 σ X) b ) Proof We use the following Grüss type inequlity: 6) 0 p t) g t) dt p t) dt ) p t) g t) dt p t) dt 4 M m), provided tht p, g re mesurble on, b nd ll the integrls in 6) exist nd re finite, p t) dt > 0 nd m g M e on, b For proof of this inequlity see 9 Choose in 6), p t) f t), g t) t E X), t, b Observe tht in this cse m E X), M b E X) nd then, by 6) we deduce 5) Remrk The sme conclusion cn be obtined for the choice p t) f t) nd g t) t, t, b The following result holds Theorem 4 Let X be rndom vrible hving the pdf given by f :, b R R + Then for ny x, b we hve the inequlity: 7) σ X) + x E X)) b ) b ) + x +b b x) q+ +x ) q+ b Proof We observe tht q+ + x +b ) ) f, provided f L, b ; q f p, provided f L p, b, p >, nd p + q ; 8) nd s x t) f t) dt x xt + t ) f t) dt x xe X) + t f t) dt 9) σ X) we get, by 8) nd 9), 0) x E X) + σ X) which is of interest in itself too t f t) dt E X), x t) f t) dt, J Inequl Pure nd Appl Mth, ) Art, 00

4 4 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS We observe tht x t) f t) dt ess sup f t) t,b x t) dt b x) + x ) f b ) b ) f + x + b ) nd the first inequlity in 7) is proved For the second inequlity, observe tht by Hölder s integrl inequlity, ) x t) f t) dt f p p b t) dt x t) q dt nd the second inequlity in 7) is estblished Finlly, observe tht, nd the theorem is proved ) q b x) q+ + x ) q+ q f p, q + x t) f t) dt sup x t) f t) dt t,b mx { x ), b x) } The following corollries re esily deduced Corollry 5 With the bove ssumptions, we hve ) 0 σ X) b ) b ) + E X) +b b EX)) q+ +EX) ) q+ q+ mx {x, b x}) b + x + b ), ) f, provided f L, b ; q f p, if f L p, b, p > nd p + q ; b + E X) +b Remrk 6 The lst inequlity in ) is worse thn the inequlity 5), obtined by technique bsed on Grüss inequlity The best inequlity we cn get from 7) is tht one for which x +b, nd this pplies for ll the bounds since b ) min x,b + x + b ) b x) q+ + x ) q+ min x,b q + b ) b )q+ q q + ),, J Inequl Pure nd Appl Mth, ) Art, 00

5 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 5 nd b min + x,b x + b Consequently, we cn stte the following corollry s well b Corollry 7 With the bove ssumptions, we hve the inequlity: 0 σ X) + E X) + b ) b ) f, provided f L, b ; b ) q+ f p, if f L p, b, p >, 4q+) q b ) 4 nd p + q ; Remrk 8 From the lst inequlity in ), we obtin ) 0 σ X) b E X)) E X) ) 4 b ), which is n improvement on 5) PERTURBED RESULTS USING GRÜSS TYPE INEQUALITIES In 95, G Grüss see for exmple 6) proved the following integrl inequlity which gives n pproximtion for the integrl of product in terms of the product of the integrls Theorem Let h, g :, b R be two integrble mppings such tht φ h x) Φ nd γ g x) Γ for ll x, b, where φ, Φ, γ, Γ re rel numbers Then, ) T h, g) Φ φ) Γ γ), 4 where ) T h, g) b h x) g x) dx b h x) dx b g x) dx nd the inequlity is shrp, in the sense tht the constnt cnnot be replced by smller 4 one For simple proof of this s well s for extensions, generlistions, discrete vrints nd other ssocited mteril, see 5, nd - where further references re given A premture Grüss inequlity is embodied in the following theorem which ws proved in It provides shrper bound thn the bove Grüss inequlity Theorem Let h, g be integrble functions defined on, b nd let d g t) D Then ) T h, g) D d where T h, g) is s defined in ) T h, h), Theorem will now be used to provide perturbed rule involving the vrince nd men of pdf J Inequl Pure nd Appl Mth, ) Art, 00

6 6 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Perturbed Results Using Premture Inequlities In this subsection we develop some perturbed results Theorem Let X be rndom vrible hving the pdf given by f :, b R R + Then for ny x, b nd m f x) M we hve the inequlity 4) P V x) : σ X) + x E X)) M m M m) 6) T h, h) b Now, b ) 45 b ) 45 b b ) x + b ) ) + 5 x + b ) Proof Applying the premture Grüss result ) by ssociting g t) with f t) nd h t) x t), gives, from )-) 5) x t) f t) dt x t) dt f t) dt b b ) M m T h, h), where from ) b x t) 4 dt x t) dt b 7) nd b x t) dt x ) + b x) b ) b b x t) 4 dt x )5 + b x) 5 5 b ) ) + x + b ) giving, for 6), x ) 5 + b x) 5 x ) + b x) 8) 45T h, h) 9 5 b b Let A x nd B b x in 8) to give ) A 5 + B 5 A + B 45T h, h) 9 5 A + B A + B 9 A 4 A B + A B AB + B 4 5 A AB + B ) 4A 7AB + 4B ) A + B) A ) ) + B A B + 5 A + B) Using the fcts tht A + B b nd A B x + b) gives b ) b ) 9) T h, h) + 5 x + b ) 45 J Inequl Pure nd Appl Mth, ) Art, 00

7 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 7 nd from 7) b giving 0) x t) dt A + B A + B) b x t) dt A A AB + B ) + B + b ) + x + b ) ) A B, Hence, from 5), 9) 0) nd 0), the first inequlity in 4) results The corsest uniform bound is obtined by tking x t either end point Thus the theorem is completely proved Remrk 4 The best inequlity obtinble from 4) is t x +b giving ) σ X) + E X) + b b ) M m b ) 5 The result ) is tighter bound thn tht obtined in the first inequlity of ) since 0 < M m < f For symmetric pdf E X) +b nd so the bove results would give bounds on the vrince The following results hold if the pdf f x) is differentible, tht is, for f x) bsolutely continuous Theorem 5 Let the conditions on Theorem be stisfied Further, suppose tht f is differentible nd is such tht f : sup f t) < t,b Then ) P V x) b f I x), where P V x) is given by the left hnd side of 4) nd, b ) b ) ) I x) + 5 x + b ) 45 Proof Let h, g :, b R be bsolutely continuous nd h, g be bounded Then Chebychev s inequlity holds see ) T h, g) b ) sup h t) sup g t) t,b t,b Mtić, Pečrić nd Ujević using premture Grüss type rgument proved tht 4) T h, g) b ) sup g t) T h, h) t,b Associting f ) with g ) nd x ) with h ) in ) gives, from 5) nd 9), I x) b ) T h, h), which simplifies to ) nd the theorem is proved J Inequl Pure nd Appl Mth, ) Art, 00

8 8 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Theorem 6 Let the conditions of Theorem be stisfied Further, suppose tht f is loclly bsolutely continuous on, b) nd let f L, b) Then 5) P V x) b π f I x), where P V x) is the left hnd side of 4) nd I x) is s given in ) Proof The following result ws obtined by Lupş see ) For h, g :, b) R loclly bsolutely continuous on, b) nd h, g L, b), then where T h, g) b ) π h g, ) k : k t) for k L, b) b Mtić, Pečrić nd Ujević further show tht 6) T h, g) b π g T h, h) Associting f ) with g ) nd x ) with h in 6) gives 5), where I x) is s found in ), since from 5) nd 9), I x) b ) T h, h) Alternte Grüss Type Results for Inequlities Involving the Vrince Let 7) S h x)) h x) M h) where 8) M h) b Then from ), h u) du 9) T h, g) M hg) M h) M g) Drgomir nd McAndrew 9 hve shown, tht 0) T h, g) T S h), S g)) nd proceeded to obtin bounds for trpezoidl rule Identity 0) is now pplied to obtin bounds for the vrince Theorem 7 Let X be rndom vrible hving the pdf f :, b R R + Then for ny x, b the following inequlity holds, nmely, ) P V x) 8 ν x) f ) if f L, b, b where P V x) is s defined by the left hnd side of 4), nd ν ν x) b Proof Using identity 0), ssocite with h ), x ) nd f ) with g ) Then ) + ) x +b ) x t) f t) dt M x ) ) x t) M x ) ) f t) dt, b J Inequl Pure nd Appl Mth, ) Art, 00

9 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 9 where from 8), M x ) ) b x t) dt x ) + b x) b ) nd so ) M x ) ) ) b + x + b ) Further, from 7), nd so, on using ) 4) S x ) ) x t) S x ) ) x t) M x ) ) b ) x + b ) Now, from ) nd using 0), ) nd 4), the following identity is obtined b 5) σ X) + x E X) ) + x + b ) where S ) is s given by 4) Tking the modulus of 5) gives 6) P V x) S x t) ) f t) ) dt b S x t) ) f t) b ) dt, Observe tht under different ssumptions with regrd to the norms of the pdf f x) we my obtin vriety of bounds For f L, b then 7) P V x) f ) b S x t) ) dt Now, let 8) S x t) ) t x) ν t X ) t X + ), where 9) ν M x ) ) x ) + b x) b ) nd b ) + x + b ), 0) X x ν, X + x + ν Then, ) H t) S x t) ) t dt x) ν dt t x) ν t + k J Inequl Pure nd Appl Mth, ) Art, 00

10 0 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS nd so from ) nd using 8) - 9) gives, ) S x t) ) dt H X ) H ) H X + ) H X ) + H b) H X + ) H X ) H X + ) + H b) H ) { ν } ν X ν + b x) ν X + + ν b + ν ν + b x) + x ) ν b ) x ) + ν 8 ν Thus, substituting into 7), 6) nd using 9) redily produces the result ) nd the theorem is proved Remrk 8 Other bounds my be obtined for f L p, b, p however obtining explicit expressions for these bounds is somewht intricte nd will not be considered further here They involve the clcultion of sup t,b for f L, b nd t x) ν mx { x ) ν, ν, b x) ν } t x) ν q dt for f L p, b, p + q, p >, where ν is given by 9) 4 SOME INEQUALITIES FOR ABSOLUTELY CONTINUOUS PDFS We strt with the following lemm which is interesting in itself Lemm 4 Let X be rndom vrible whose probbility density function f :, b R + is bsolutely continuous on, b Then we hve the identity 4) σ X) + E X) x b ) + x + b ) + b ) q where the kernel p :, b R is given by s, if s t b, p t, s) : s b, if t < s b, for ll x, b Proof We use the identity see 0)) 4) σ X) + E X) x for ll x, b x t) f t) dt t x) p t, s) f s) dsdt, J Inequl Pure nd Appl Mth, ) Art, 00

11 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE On the other hnd, we know tht see for exmple for simple proof using integrtion by prts) 4) f t) b for ll t, b Substituting 4) in 4) we obtin 44) σ X) + E X) x t x) b b Tking into ccount the fct tht f s) ds + b f s) ds + b x ) + b x) + b x ) + b x) b ) + p t, s) f s) ds p t, s) f s) ds dt t x) p t, s) f s) dsdt x + b ), x, b, then, by 44) we deduce the desired result 4) The following inequlity for PDFs which re bsolutely continuous nd hve the derivtives essentilly bounded holds Theorem 4 If f :, b R + is bsolutely continuous on, b nd f L, b, ie, f : ess sup f t) <, then we hve the inequlity: t,b 45) σ X) + E X) x for ll x, b Proof Using Lemm 4, we hve σ X) + E X) x b ) b ) x + b b b f b b ) ) b ) 0 x + b ) + x + b ) f t x) p t, s) f s) dsdt t x) p t, s) f s) dsdt t x) p t, s) dsdt J Inequl Pure nd Appl Mth, ) Art, 00

12 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS We hve I : I + I b Let A x, B b x then nd Now, I + I b t x) p t, s) dsdt t t x) s ) ds + t b t x) t ) + b t) dt t x) t ) dt + I I b nd the theorem is proved 0 t x) t ) dt b ) 0 b ) b ) b ) u Au + A ) u du b s) ds dt t x) b t) dt A A b ) + 5 b ) t x) b t) dt b ) u Bu + B ) u du B B b ) + 5 b ) A + B 4 A + B) b ) + 5 b ) b ) + x + b ) b ) 0 b ) + x + b ) 0 The best inequlity we cn get from 45) is embodied in the following corollry Corollry 4 If f is s in Theorem 4, then we hve 46) σ X) + E X) + b b ) b )4 f 0 We now nlyze the cse where f is Lebesgue p integrble mpping with p, ) J Inequl Pure nd Appl Mth, ) Art, 00

13 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE Remrk 44 The results of Theorem 4 my be compred with those of Theorem 5 It my be shown tht both bounds re convex nd symmetric bout x +b Further, the bound given by the premture Chebychev pproch, nmely from )-) is tighter thn tht obtined by the current pproch 45) which my be shown from the following Let these bounds be described by B p nd B c so tht, neglecting the common terms nd where B p b 5 B c Y b b ) 00 ) + 5Y + Y, x + b ) It my be shown through some strightforwrd lgebr tht B c B p > 0 for ll x, b so tht B c > B p The current development does however hve the dvntge tht the identity 4) is stisfied, thus llowing bounds for L p, b, p rther thn the infinity norm Theorem 45 If f :, b R + is bsolutely continuous on, b nd f L p, ie, then we hve the inequlity 47) f p : ) f t) p p dt σ X) + E X) x b ) f p b ) p q + ) q x + b x ) q+ B <, p, ) ) b, q +, q + x + b x) q+ B ) ) b, q +, q + b x for ll x, b, when p + q nd B,, ) is the qusi incomplete Euler s Bet mpping: B z; α, β) : z 0 u ) α u β du, α, β > 0, z Proof Using Lemm 4, we hve, s in Theorem 4, tht 48) σ X) + E X) x b ) x + b ) b t x) p t, s) f s) dsdt J Inequl Pure nd Appl Mth, ) Art, 00

14 4 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Using Hölder s integrl inequlity for double integrls, we hve 49) t x) p t, s) f s) dsdt where p >, + p q We hve to compute the integrl 40) Define D : q + 4) E : b ) p f p t x) q p t, s) q dsdt ) f s) p p b dsdt t t x) q s ) q ds + t t x) q t ) q+ + b t) q+ dt q + t x) q t ) q+ dt + ) t x) q p t, s) q q dsdt t x) q p t, s) q dsdt b s) q ds t x) q t ) q+ dt dt ) q t x) q b t) q+ dt If we consider the chnge of vrible t u) + ux, we hve t implies u 0 nd t b implies u b, dt x ) du nd then x 4) Define E x 0 u) + ux x q u) + ux x ) du x ) q+ x u ) q u q+ du 0 ) b x ) q+ B, q +, q + x 4) F : t x) q b t) q+ dt If we consider the chnge of vrible t v) b + vx, we hve t b implies v 0, nd t implies v b, dt x b) dv nd then 44) F b x 0 b b x v) b + vx x q b v) b vx q+ x b) dv b x) q+ b x v ) q v q+ dv 0 ) b b x) q+ B, q +, q + b x, J Inequl Pure nd Appl Mth, ) Art, 00

15 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 5 Now, using the inequlities 48)-49) nd the reltions 40)-44), since D E + F ), q+ we deduce the desired estimte 47) The following corollry is nturl to be considered Corollry 46 Let f be s in Theorem 45 Then, we hve the inequlity: 45) σ X) + E X) + b b ) f p b ) + q q + ) q + q B q +, q + ) + Ψ q +, q + ) q, where p + q, p > nd B, ) is Euler s Bet mpping nd Ψ α, β) : 0 uα u + ) β du, α, β > 0 Proof In 47) put x +b The left side is cler Now B, q +, q + ) The right hnd side of 47) is thus: f b ) q+ q p b ) p q + ) q 0 0 u ) q u q+ du u ) q u q+ du + u ) q u q+ du B q +, q + ) + Ψ q +, q + ) B q +, q + ) + Ψ q +, q + ) q f p b ) + q q + ) q + q B q +, q + ) + Ψ q +, q + ) q nd the corollry is proved Finlly, if f is bsolutely continuous, f L, b nd f f t) dt, then we cn stte the following theorem Theorem 47 If the pdf, f :, b R + is bsolutely continuous on, b, then 46) σ X) + E X) x for ll x, b b ) x + b ) f b ) b ) + x + b J Inequl Pure nd Appl Mth, ) Art, 00

16 6 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS Proof As bove, we cn stte tht σ X) + E X) x b ) where nd the theorem is proved b x + b ) t x) p t, s) f s) dsdt sup t x) p t, s) t,s),b b f G G : sup t,s),b t x) p t, s) b ) sup t x) t,b b ) mx x, b x) b ) b ) + x + b, f s) dsdt It is cler tht the best inequlity we cn get from 46) is the one when x +b, giving the following corollry Corollry 48 With the ssumptions of Theorem 47, we hve: 47) σ X) + E X) + b b ) b ) f 4 REFERENCES P CERONE AND SS DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted, RGMIA Res Rep Coll, 4) 999), Article 8 ONLINE P CERONE AND SS DRAGOMIR, Trpezoidl type rules from n inequlities point of view, Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics, GA Anstssiou Ed), CRC Press, New York 000), 65 4 P CERONE AND SS DRAGOMIR, Midpoint type rules from n inequlities point of view, Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics, GA Anstssiou Ed), CRC Press, New York 000), P CERONE, SS DRAGOMIR AND J ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions, Est Asin Mth J, 5) 999), 9 Preprint RGMIA Res Rep Coll, ) 998), Article 4, 998 ONLINE 5 P CERONE, SS DRAGOMIR AND J ROUMELIOTIS, An inequlity of Ostrowski- Grüss type for twice differentible mppings nd pplictions, Kyungpook Mth J, 9) 999), 4 Preprint RGMIA Res Rep Coll, ) 998), Article 8, 998 ONLINE J Inequl Pure nd Appl Mth, ) Art, 00

17 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE 7 6 P CERONE, SS DRAGOMIR AND J ROUMELIOTIS, An Ostrowski type inequlity for mppings whose second derivtives belong to L p, b) nd pplictions, Preprint RGMIA Res Rep Coll, ) 998), Article 5 ONLINE 7 P CERONE, SS DRAGOMIR AND J ROUMELIOTIS, On Ostrowski type for mppings whose second derivtives belong to L, b) nd pplictions, Honm Mth J, ) 999), 7 7 Preprint RGMIA Res Rep Coll, ) 998), Article 7 ONLINE 8 P CERONE, SS DRAGOMIR AND J ROUMELIOTIS, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth, ) 999), Preprint RGMIA Res Rep Coll, ) 998), 5 66 ONLINE 9 P CERONE, SS DRAGOMIR, J ROUMELIOTIS AND J SUNDE, A new generliztion of the trpezoid formul for n-time differentible mppings nd pplictions, Demonstrtio Mth, 4) 000), RGMIA Res Rep Coll, 5) 999), Article 7 ONLINE 0 SS DRAGOMIR, Grüss type integrl inequlity for mppings of r-hölder s type nd pplictions for trpezoid formul, Tmkng J Mth, ) 000), 4 47 SS DRAGOMIR, A Tylor like formul nd ppliction in numericl integrtion, submitted SS DRAGOMIR, Grüss inequlity in inner product spces, Austrl Mth Soc Gz, 6) 999), SS DRAGOMIR, New estimtion of the reminder in Tylor s formul using Grüss type inequlities nd pplictions, Mth Inequl Appl, ) 999), SS DRAGOMIR, Some integrl inequlities of Grüss type, Indin J of Pure nd Appl Mth, 4) 000), SS DRAGOMIR AND N S BARNETT, An Ostrowski type inequlity for mppings whose second derivtives re bounded nd pplictions, J Indin Mth Soc, 66-4) 999), 7 45 Preprint RGMIA Res Rep Coll, ) 998), Article 9 ONLINE 6 SS DRAGOMIR, P CERONE AND A SOFO, Some remrks on the midpoint rule in numericl integrtion, Studi Mth Bbeş-Bolyi Univ, in press) 7 SS DRAGOMIR, P CERONE AND A SOFO, Some remrks on the trpezoid rule in numericl integrtion, Indin J Pure Appl Mth, 5) 000), Preprint: RGMIA Res Rep Coll, 5) 999), Article ONLINE 8 SS DRAGOMIR, YJ CHO AND SS KIM, Some remrks on the Milovnović-Pečrić Inequlity nd in Applictions for specil mens nd numericl integrtion, Tmkng J Mth, 0) 999), 0 9 SS DRAGOMIR AND A McANDREW, On Trpezoid inequlity vi Grüss type result nd pplictions, Tmkng J Mth, ) 000), 9 0 RGMIA Res Rep Coll, ) 999), Article 6 ONLINE 0 SS DRAGOMIR, JE PEČARIĆ AND S WANG, The unified tretment of trpezoid, Simpson nd Ostrowski type inequlity for monotonic mppings nd pplictions, Mth nd Comp Modelling, 000), 6 70 Preprint: RGMIA Res Rep Coll, 4) 999), Article ONLINE SS DRAGOMIR AND A SOFO, An integrl inequlity for twice differentible mppings nd pplictions, Preprint: RGMIA Res Rep Coll, ) 999), Article 9 ONLINE J Inequl Pure nd Appl Mth, ) Art, 00

18 8 NS BARNETT, P CERONE, SS DRAGOMIR, AND J ROUMELIOTIS SS DRAGOMIR AND S WANG, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules, Comput Mth Appl, 997), 5 M MATIĆ, JE PEČARIĆ AND N UJEVIĆ, On New estimtion of the reminder in Generlised Tylor s Formul, Mth Inequl Appl, ) 999), DS MITRINOVIĆ, JE PEČARIĆ AND AM FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, 99 5 DS MITRINOVIĆ, JE PEČARIĆ AND AM FINK, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, JE PEČARIĆ, F PROSCHAN AND YL TONG, Convex Functions, Prtil Orderings, nd Sttisticl Applictions, Acdemic Press, 99 J Inequl Pure nd Appl Mth, ) Art, 00