Improvement of Ostrowski Integral Type Inequalities with Application

Transcription

1 Filomt 30:6 06), 56 DOI 098/FIL606Q Published by Fculty of Sciences nd Mthemtics, University of Niš, Serbi Avilble t: Improvement of Ostrowski Integrl Type Ineulities with Appliction Ather Qyyum,b, Ibrhim Fye, Muhmmd Shoib c Deprtment of Fundmentl nd Applied Sciences, Universiti Teknologi PETRONAS, 360 Bndr Seri Iskndr, Perk Drul Ridzun, Mlysi b Deprtment of Mthemtics, University of Hil, Sudi Arbi c Higher Colleges of Technology Abu Dhbi Mens College, PO Bo 5035, Abu Dhbi, United Arb Emirtes Abstrct The im of this pper is to estblish new ineulities which re more generlized thn the ineulities of Drgomir, Wng nd Cerone The current rticle lso obtins bounds for the devition of function from combintion of integrl mens over the end intervls covering the entire intervl A vriety of erlier results re recptured s specil cses of the ineulities obtined Some new perturbed results nd ppliction for cumultive distribution function re lso discussed Introduction In the lst few decdes, the field of mthemticl ineulities hs proved to be n etensively pplicble field Integrl ineulities ply n importnt role in severl brnches of mthemtics nd sttistics with reference to its pplictions The elementry ineulities re proved to be helpful in the development of mny other brnches of mthemtics Ostrowski [9 proved his fmous ineulity in 938 which, becuse of its pplictions in numericl nlysis, ttrcted lot of reserchers in the pst few yers [3-[7 For recent results nd generliztions concerning Ostrowski s ineulity see [3-[6 The first generliztion of Ostrowski s ineulity ws given by Milovnović nd Pecrić in [8 Further generliztions of Ostrowski s ineulity were given by Qyyum nd Hussin in [7 nd Qyyum etl in [8 Cheng gve shrp version of the ineulity derived in [ Cerone [, nd Drgomir nd Wng [3-[7 generlized the Ostrowski s ineulity for L, L p nd L norms In this work, we define new mpping which help refine the results of Cerone [, nd Drgomir nd Wng [3-[7 nd lso provide new results with wide rnging pplictions We lso derive some perturbed results by using Grüss nd Čebyŝev ineulities The obtined ineulities re of supreme importnce becuse they hve immedite pplictions in numericl integrtion, probbility theory, informtion theory nd integrl opertor theory etc In the lst, we will pply our ineulities to cumultive distribution functions 00 Mthemtics Subject Clssifiction Primry 6D5, 6D0; Secondry 65X Keywords Ostrowski ineulity, Numericl integrtion, Grüss nd Čebyŝev ineulities Received: 5 April 0; Accepted: 09 September 0 Communicted by Drgn S Djordjević Emil ddresses: theryyum@gmilcom Ather Qyyum), ibrhim_fye@petronscommy Ibrhim Fye), sfridi@gmilcom Muhmmd Shoib)

2 Preliminries A Qyyum et l / Filomt 30:6 06), 56 Let the functionl S f ;, b ) represent the devition of f ) from its integrl men over [, b nd be defined by S f ;, b ) f ) M f ;, b ) ) nd M f ;, b ) b f ) d Ostrowski [9 proved the following interesting integrl ineulity: Theorem Let f : [, b R be continuous on [, b nd differentible on, b), whose derivtive f :, b) R is bounded on, b), ie f sup t [,b f t) < then S f ;, b ) ) b + + b ) M b, 3) for ll [, b nd In this pper, we will use the usul L p norms defined for function k s follows: k : ess sup k t) t [,b k p : k t) p dt p, p < Drgomir nd Wng [3-[6 proved 3) nd other vrints for f L p [, b for p nd the Lebesgue norms mking use of peno kernel pproch nd Montgomery s identity [ Montgomery s identity sttes tht for n bsolutely continuous mppings f : [, b R f ) b where the kernel p: [, b R is given by t if t b P, t) t b if < t b f t)dt + P, t) f t)dt, ) b If we ssume tht f L [, b nd f ess t [,b f t) then M in 3) my be replced by f Drgomir nd Wng [3-[6 obtined the following ineulity by using P, t) nd n integrtion by prts rgument S f ;, b ) 5) ) b [ b [ b [ b b ) ) + +b f, f L [, b ) + +b) + + f p, f L p [, b, p >, p + + +b f, f L [, b,

3 where f : [, b R is bsolutely continuous on [, b Cerone [, proved the following ineulity: A Qyyum et l / Filomt 30:6 06), 56 3 Theorem Let f : [, b R be n bsolutely continuous mpping Define then τ ;, ) : f ) [ ) ) M f ;, + M f ;, b + τ ;, ) 6) 7) [ ) + b ) f, f +) L [, b +)+) [ ) + b ), f L p [, b, p >, p ) f, f L [, b f p Qyyum et l [3 lso proved Ostrowski s type integrl ineulities Lemm 3 Denote by P, ) : [, b R the kernel is given by P, t) : +)) t ), t +)b) t b), < t b 8) Then, τ ;, ) + ) [ ) b ) f ) f ) + [ ) ) M f ;, + M f ;, b + [ ) + b ) f 6+), f L [, b 9) +) [ ) + + b ) + f p +), f L p [, b, p >, p + ) + b ) + ) b ) ) f +), f L [, b Motivted by the result of Cerone [ nd Drgomir [3-[6, we will present new ineulities which will be the etended nd generlized form of Cerone [ nd Drgomir nd Wng [3-[6

4 A Qyyum et l / Filomt 30:6 06), 56 3 Min Results Before stting the min result, we need to estblish the following lemm Lemm 3 Let kernel is given by P, t) f : [, b R be n bsolutely continuous mpping Let P, ) : [, b R, the peno type + + b [ ) t + h b, t [ ) 0) t b h b, < t b, for ll [ + h b, b h b nd h [0,, where, R re non negtive nd not both zero, then the identity: holds P, t) f t)dt + + h + + ) b Proof From 0), we hve f ) + f t) dt + b ) + h b ) f ) ) b h b b f b) b ) f t) dt P, t) f t)dt b t + h b )) f t)dt t b h b )) f t)dt Using integrtion by prts, we get P, t) f t)dt + h b )) + f ) + h b + + b hb f b) b h b )) f ) f ) f t) dt f t) dt Combining like terms nd using lgebric mnipultion, we get the reuired identity given in ) We now stte nd prove our min result

5 A Qyyum et l / Filomt 30:6 06), 56 5 Theorem 3 Let f : [, b R be bsolutely continuous mpping Using ), we define τ ;, ) P, t) f t)dt ) : [ + + h ) b h b f ) + ) b ) b f b) [ M f ;, ) + M f ;, b ), b h b ) f ) where M f ;, b ) is the integrl men defined in ), then τ ;, ) 3) ) + [ ) + h b + + b) b + [ ) b h b +b +) f, f L [, b )) + ) + ) + h b h b b )) + ) + + b) + h b h b +) +) f p, f L p [, b, p >, p + ) + h b + ) + h b f L [, b [ b)+) )b) [ )b) )b) f +), for ll [ + h b, b h b nd h [0, Proof Tking the modulus of ) nd using ), we obtin τ ;, ) P, t) f t)dt By using the definition of L norm, we get τ ;, ) f P, t) dt P, t) f t) dt )

6 A Qyyum et l / Filomt 30:6 06), 56 6 Now P, t) dt + t + h b ) dt + + b t b h b ) dt Agin using integrtion by prts nd some lgebric mnipultion, we get P, t) dt Hence the first ineulity τ ;, ) b ) + [ + h b b ) + [ b h b ) + ) + [ ) + h b + ) +b + b b ) + [ ) b h b +b f + is obtined Now using Hölder s integrl ineulity nd definition of L p norm, from ) we get τ ;, ) b f p P, t) dt Now ) + ) P, t) dt t + h b ) dt + b ) t b h b )) dt Using integrtion by prts nd some lgebric mnipultion, we get ) + + ) P, t) dt )) + ) + ) + h b h b b )) + ) + + b) + h b h b

7 A Qyyum et l / Filomt 30:6 06), 56 7 The second ineulity τ ;, ) ) + + ) )) + ) + ) + h b h b b )) + ) + + b) + h b h b f p is obtined Finlly, using definition of L norm nd P, t), we hve from ) τ ;, ) sup P, t) f t [,b where ) + sup P, t) t [,b ) + h b + ) + h b [ b)+) )b) [ )b) )b) f This completes the proof of the theorem Remrk 33 If we put h 0, in 3), we get 7) If we put nd h 0, in 3), we get 5) This shows tht the results of Cerone nd Drgomir re our specil cses Remrk 3 By substituting h nd in ) nd 3), we get ) b f ) + f b) f t) dt b ) ) + [ +b + + b) b + [ +b +b f, ) + ) + ) +b b + +b b) ) + ) + b [ b) )b) + b +) [ )b) ) f )b) f p,

8 A Qyyum et l / Filomt 30:6 06), 56 8 Remrk 35 If we substitute h +b, nd in ) nd 3), we get nother result: f ) + ) b 8 f ) b f ) + f b) b ) f t) dt b + b ) + ) + b ) + ) + b b)+) f p f Similrly, for different vlues of h, we cn obtin vriety of results Remrk 36 It should be noted tht from ) nd ) + ) τ ;, ) S f ;, ) + S f ;, b ) 5) From 3), we obtin + ) τ ;, ) 6) ) + [ ) + h b + f,[, + b) b + [ ) b h b +b f,[,b, ) + b) + h b b + h b )) + ) + h b )) + ) + h b [ ) + h b f,[, + b +) +) f p,[, f p,[,b, [ ) b h b f,[,b

9 A Qyyum et l / Filomt 30:6 06), 56 9 Tht is, + ) τ ;, ) 7) ) + [ ) + h b + + b) b + [ ) b h b +b f, ) + b) + h b b + h b )) + ) + h b )) + ) + h b +) f p, [ ) + h b + b [ ) ) b h b f Remrk 37 We my write M f ;, ) + M f ;, b ) M f ;, ) + f u) du b f u) du M f ;, ) f u) du + f u) du b b + σ ) ) M f ;, ) + σ ) M f ;, b ), where b b σ ) 8) Thus, from ), we get τ ;, ) [ + h b ) + b + h ) b + f ) + ) b f b) [ ) + σ ) M f ;, ) + b h b + σ ) M f ;, b ) ) f ) 9) so tht for fied [, b, M f ;, b ) is lso fied

10 A Qyyum et l / Filomt 30:6 06), Corollry 38 If we tke nd +b in ) nd 3), we get ) + b h) f + h b) 6 + [ + h b ) b f ) + f b) b ) 3+b + b) b 6 + [ ) b h b +3b f t) dt f 0) b b) h)) + ) + h b + b b) h)) + ) + h b +) f p h) f Some Perturbed Results In 88, Čebyŝev [0 gve the following ineulity: T f, ) b ) f, ) where f, : [, b R re bsolutely continuous bounded functions T f, ) b b f ) ) d f ) d b b b M f, ;, b ) M f ;, b ) M ;, b ) ) d ) In 935, Grüss [ proved the following ineulity: b f ) ) d b f ) d b ) d ) ) Φ ϕ Γ γ, 3) provided tht f nd re two integrble functions on [, b nd stisfy the condition ϕ f ) Φ nd γ ) Γ for ll [, b ) The constnt is the best possible We will obtin the perturbed version of the results of Theorem 3, by using Grüss type results involving the Čebyŝev functionl T f, ) M f, ;, b ) M f ;, b ) M ;, b ), 5) where M is the integrl men defined in )

11 A Qyyum et l / Filomt 30:6 06), 56 5 Theorem Let f : [, b R be n bsolutely continuous mpping nd 0, 0, + ) 0, then τ ;, ) R + )S [ b ) N ) f b S ) Γ γ b ) λ where, τ ;, ) is s given by ), λ Φ ϕ nd R + h b )) h b ) 7) + b h b ) b h b )), 6) S f b) f ), b N ) ) b R + ) b ) ) [ + h b ) [ h b ) )) 3 ) 3 + h b ) 3 )) 3 b h b 8) Proof Associting f t) with P, t) nd t) with f t), nd using 5), we obtin the following T P, ), f );, b ) M P, ), f );, b ) M P, ) ;, b) M f );, b ) Now using identity ), we obtin b ) T P, ), f );, b ) τ ;, ) b ) M P, ) ;, b) S 9) Now from ) nd ), we get b ) M P, ) ;, b) P, t)dt 30) b + ) t + h b )) dt t b h b )) dt + + ) b R + ) + h b )) ) h b h b ) b h b ))

12 A Qyyum et l / Filomt 30:6 06), 56 5 By combining 30) with 8), the left hnd side of 6) cn esily be obtined Let f : [, b R nd f : [, b R be integrble on [, b, then [ T f, ) ) ) T f, f T, f, L [, b ) 3) ) Γ γ ) ) T f, f γ ) Γ, t [, b ) ) ) Φ ϕ Γ γ ϕ f ) Φ, t [, b Note tht 0 T f );, b, f );, b ) [ M f ) ;, b ) M f );, b ) f t) dt b f t) dt b f b S ) Γ γ, where γ f t) Γ, t [, b Now, for the bounds on 9), we hve to determine 3) T P, ) ;, b), P, ) ;, b)) nd ϕ P, ) Φ Using the definition of P, t) nd from 0), we hve T P, ) ;, b), P, ) ;, b)) M P, ) ;, b ) M P, ) ;, b) 33) From 30), we obtin nd M P, ) ;, b) M P, ) ;, b ) R + ) b ), ) + ) ) + + b ) b ) b t + h b )) dt t b h b )) dt ) [ + h b ) [ h b Thus, substituting the bove results into 33), we obtin )) 3 ) 3 + h b ) 3 )) 3 b h b 0 N ) T P, ) ;, b) P, ) ;, b)), 3) where N ) is given eplicitly by 8) Combining 9), 3) nd 33) gives the first ineulity in 3), nd the first ineulity in 6) Now utilizing ineulity 3) produces the second result in 6) Further it cn be seen from the definition of P, t) in 0), tht for, 0 Φ sup P, t) nd ϕ inf P, t), t [,b t [,b

13 where [ Φ sup + h b ), + b hb ) t [,b A Qyyum et l / Filomt 30:6 06), nd ϕ inf t [,b + hb [, b h b )) b 5 An Appliction to the Cumultive Distribution Function Let X be rndom vrible tking vlues in the finite intervl [, b with Cumultive Distributive Function F ) P r X ) f u) du, where f is the probbility density function In prticulr, f u) du The following theorem holds Theorem 5 Let X nd F be s bove, then [ b ) + h b ) ) b h b +h ) b b ) f ) + ) f b) ) f ) 35) [ b ) ) F ) ) ) + [ ) + h b + b ) ) + b) b + [ ) b h b +b f, b ) ) )) + ) + ) + h b + h b h ) + )) + + b b) + b h b +) f p, b ) ) ) + + h b + ) h b [ )b) )b) [ )+b) )b) f

14 Proof From ), we hve where A Qyyum et l / Filomt 30:6 06), 56 5 τ ;, ) : [ + h b ) + b + h ) b + b f b) ) f ) [ ) ) M f ;, + M f ;, b + [ + h b ) + b + h ) b + b f b) ) f ) I b h b b h b ) ) f ) f ) I [ ) ) M f ;, + M f ;, b f t) dt + f t) dt + f t) dt + b b b f t) dt f t) dt f t) dt f t) dt + f t) dt f t) dt By using the definition of Cumultive Distributive Function nd Probbility Density Function, we get I [ + F ) Thus, τ ;, ) becomes τ ;, ) : [ + + h ) b + + [ F ) b F ) + + h b f ) + b F ) + b ) b ) b f b) b b h b Using 3) nd vlue of τ ;, ) from the bove eution, we get reuired result 35) ) f ) Putting in Theorem 5 gives the following result

15 A Qyyum et l / Filomt 30:6 06), Corollry 5 Let X be rndom vrible, F ) Cumultive Distributive Function nd f is Probbility Density Function, then [ b ) ) ) + h b ) b h b f ) + h ) b ) f b) + b ) f ) 36) b ) ) +b ) F ) ) ) + [ + h b ) + + b) b + [ ) b h b +b f, b ) ) )) + ) + ) + h b + h b h ) + )) + + b b) + b h b [ b ) ) + h b b )b) + h b) )b) ) f +) f p, Remrk 53 The bove result llow the pproimtion of F ) in terms of f ) The pproimtion of R ) F ) could lso be obtined by simple substitution R ) is of importnce in relibility theory where f ) is the Probbility Density Function of filure We put 0 in 35), ssuming tht 0 to obtin [ b ) ) + h b f ) +h ) 37) b b ) f ) b ) F ) b ) ) ) + [ + h b ) ) + f, b ) ) b ) ) [ ) + h b + h b + h b [ )b) )b) [ )+b) )b) ) + ) + ) h b f +) f p, 6 Conclusion Cerone [, obtined bounds for the devition of function from combintion of integrl mens over the end intervls covering the entire intervl nd pplied these results to pproimte the cumultive distribution function in terms of the probbility density function On similr lines, we estblish new ineulities, which re more generlized s compred to the ineulities developed in [, [3-[6 The pproch tht we used not only generlized the results of [ nd [3-[6 but lso gve some other interesting ineulities s specil cses Approimtion of the cumultive distribution function is lso provided

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