S. S. Dragomir. 2, we have the inequality. b a

Transcription

1 Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely continuous functions nd pplictions for composite udrture rules nd for pdf s re provided Introduction In Guessb nd Schmeisser hve proved mong others the following compnion of Ostrowski s ineulity Theorem Let f : b R be such tht f t f s M t s k for ny t s b with k 0 ie f Lip M k Then for ech x +b we hve the ineulity f x + f + b x k+ x k+ + + b x k+ k M k + This ineulity is shrp for ech dmissible x Eulity is obtined if nd only if f = ±Mf + c with c R nd x t k for t x; 3 f t = t x k for x t + b ; f + b t for + b t b Received Mrch Mthemtics Subject Clssifiction: Primry 6D5; Secondry A55 Key words nd phrses: Ostrowski s ineulity trpezoid rule midpoint rule

2 S S Drgomir We remrk tht for k = ie f Lip M since x + + b x = x 3+b 8 + then we hve the ineulity f x + f + b x x 3+b 8 + M for ny x +b The constnt 8 is best possible in in the sense tht it cnnot be replced by smller constnt We must lso observe tht the best ineulity in is obtined for x = +3b giving the trpezoid type ineulity 5 f 3+b + f +3b M 8 The constnt 8 is shrp in 5 in the sense mentioned bove For recent monogrph devoted to Ostrowski type ineulities see In this pper we improve the bove results nd lso provide other bounds for bsolutely continuous functions whose derivtives belong to the Lebesgue spces L p b p Some nturl pplictions re lso provided Some integrl ineulities The following identity holds Lemm Assume tht f : b R is n bsolutely continuous function on b Then we hve the eulity

3 Ostrowski s ineulity 5 f x + f + b x = x for ny x +b t f t dt + +b x x + t + b +b x f t dt t b f t dt Proof Using the integrtion by prts formul for Lebesgue integrls we hve x +b x t + b x + b = f + b x t f t dt = f x x f t dt x x f x x + b +b x x nd t b f t dt = x f + b x +b x +b x Summing the bove eulities we deduce the desired identity Remrk The identity ws obtined in Lemm 3 for the cse of piecewise continuously differentible functions on b The following result holds Theorem Let f : b R be n bsolutely continuous function on b Then we hve the ineulity b f x + f + b x x t f t dt +b x + t + b x f t dt + b t f t dt +b x := M x

4 6 S S Drgomir for ny x +b If f L b then we hve the ineulities 3 M x x x 3+b b x + x f x+b x + x f b ; f +b xb α x α x +b + α f β x + f β x+b x + f β β +b xb if α > α + β = ; x x +b mx f x + f x+b x + f +b xb ; α for ny x +b The ineulity the first ineulity in 3 nd the constnt 8 re shrp Proof The ineulity follows by Lemm on tking the modulus nd using its properties If f L b then +b x x x t f t dt t + b +b x x f t dt + b b t f t dt x f x x f x+b x +b xb

5 nd the first ineulity in 3 is proved Denote M x := x + x Ostrowski s ineulity 7 + b x + x f x+b x +b xb for x +b Firstly observe tht { M x mx f x f x+b x f +b xb } x + b x + x + = f b 8 + x 3 + b nd the first ineulity in 3 is proved Using Hölder s ineulity for α > α + β { x M α x + x + b = we lso hve α + x α } α f β x + β x+b x + β β +b xb giving the second ineulity in 3 Finlly we lso observe tht { x M x mx x + b } f + x f + x+b x f +b xb The shrpness of the ineulities mentioned follows from Theorem for k = We omit the detils Remrk If in Theorem we choose x = then we get f + f b b with s shrp constnt see for exmple p5

6 8 S S Drgomir 5 If in the sme theorem we now choose x = +b then we get + b f f 8 +b + +b b b with the constnts 8 nd being shrp This result ws obtined in 3 It is nturl to consider the following corollry Corollry With the ssumptions in Theorem one hs the ineulity: 6 f 3+b + f +3b 8 f b The constnt 8 is best possible in the sense tht it cnnot be replced by smller constnt The cse when f theorem L p b p > is embodied in the following Theorem 3 Let f : b R be n bsolutely continuous function on b so tht f L p b p > If M x is s defined in then we hve the bounds: 7 M x x b + xp + x+b xp x x + f +b xbp

7 Ostrowski s ineulity 9 + x b mx + +b x b + + } { f xp f x+b xp f +b xbp ; x b α+ α + α +b x b α+ α α f β xp + f β x+b xp + f β β +b xbp if α > α + β = ; mx { x + +b x b + } b f xp + f x+b xp + f +b xbp ; for ny x +b Proof Using Hölder s integrl ineulity for p > p + hve = we x t f t dt x = x + + t dt f xp xp +b x = x +b x x t + b f t dt t + b dt f x+b xp +b x + + x+b xp

8 0 S S Drgomir nd +b x b t f t dt = +b x x + + b t dt f +b xbp +b xbp Summing the bove ineulities we deduce the first bound in 7 The lst prt my be proved in similr fshion to the one in Theorem nd we omit the detils Remrk 3 If in 7 we choose α = β = p p + then we get the ineulity = p > 8 M x + x + +b + x + f bp for ny x +b Remrk If in Theorem 3 we choose x = then we get the trpezoid ineulity 9 f + f b f bp + The constnt is best possible in the sense tht it cnnot be replced by smller constnt see for exmple p Indeed if we ssume tht 9 holds with constnt C > 0 insted of ie 0 f + f b f bp C +

9 Ostrowski s ineulity then for the function f : b R f x = k x +b k > 0 we hve f + f b nd by 0 we deduce k = k = k bp = k p ; k C k + giving C + Letting + we deduce C nd the shrpness of the constnt is proved Remrk 5 If in Theorem 3 we choose x = +b then we get the midpoint ineulity + b f f + +b p + +b bp + bp p > p + = In both ineulities the constnt is shrp in the sense tht it cnnot be replced by smller constnt To show this fct ssume tht holds with C D > 0 ie + b f C f + +b p + f +b bp D + bp

10 S S Drgomir For the function f : b R f x = k x +b k > 0 we hve + b b k f = 0 = f +b p + f +b bp = bp = p k; p k = p k nd then by we deduce k k C + k D + giving C D + for ny > Letting + we deduce C D nd the shrpness of the constnts in re proved The following result is useful in providing the best udrture rule in the clss for pproximting the integrl of n bsolutely continuous function whose derivtive is in L p b Corollry Assume tht f : b R is n bsolutely continuous function so tht f L p b p > Then one hs the ineulity 3 f 3+b where p + = + f +3b + f bp The constnt is the best possible in the sense tht it cnnot be replced by smller constnt Proof The ineulity follows by Theorem 3 nd Remrk 3 on choosing x = 3+b To prove the shrpness of the constnt ssume tht 3 holds with constnt E > 0 ie f 3+b + f +3b E f + bp

11 Ostrowski s ineulity 3 Consider the function f : b R x 3 + b if x +b f x = x + 3b if x +b b Then f is bsolutely continuous nd f L p b p > We lso hve f 3 + b + f + 3b = 0 = 8 bp = p nd then by we obtin: 8 E + giving E + 8 for ny > ie E nd the corollry is proved If one is interested in obtining bounds in terms of the norm for the derivtive then the following result my be useful Theorem Assume tht the function f : b R is bsolutely continuous on b If M x is s in eution then we hve the bounds x 5 M x f x + +b x f x+b x + x f +b xb

12 S S Drgomir + x 3+b f b x α +b α α x + f β x + f β x+b x + f β +b xb β if α > α + β = x + b 3 mx f x f x+b x f +b xb The proof is s in Theorem nd we omit it Remrk 6 By the use of Theorem 3 for x = we get the trpezoid ineulity see for exmple p55 6 f + f b b If in 5 we lso choose x = +b then we get the mid point ineulity see for exmple p b f b The following corollry lso holds Corollry 3 With the ssumption in Theorem 3 one hs the ineulity: 8 f 3+b + f +3b f b

13 3 A composite udrture formul Ostrowski s ineulity 5 We use the following ineulities obtined in the previous section: f 3+b + f +3b 3 8 f b if f L b ; + f bp if f L p b p > p + = ; f b if f L b Let I n : = x 0 < x < < x n < x n = b be division of the intervl b nd h i := x i+ x i i = 0 n nd ν I n := mx {h i i = 0 n } Consider the composite udrture rule 3 Q n I n f := n 3xi + x i+ f The following result holds + f xi + 3x i+ h i Theorem 5 Let f : b R be n bsolutely continuous function on b Then we hve 33 = Q n I n f + R n I n f where Q n I n f is defined by formul 3 nd the reminder stisfies the estimtes 3 8 f b n R n I n f n f + bp h + i f b ν I n h i if f L b ; if f L p b p > p + = ;

14 6 S S Drgomir Proof Applying ineulity 3 on the intervls x i x i+ we my stte tht 35 xi+ x i f 3xi + x i+ 8 h i f xi x i+ ; + + f xi + 3x i+ h i h + i f xi x i+ p p > p + = ; h i f xi x i+ ; for ech i {0 n } Summing the ineulity 35 over i from 0 to n nd using the generlized tringle ineulity we get 36 8 n R n I n f + h i f xi x i+ ; n h + i f xi x i+ p p > p + = ; n h i f xi x i+ Now we observe tht n h i xi x i+ n b h i

15 Ostrowski s ineulity 7 Using Hölder s discrete ineulity we my write tht n h + i Also we note tht n xi x i+ p h = = n + n i n xi+ h + i n h + i x i f bp p p x i x i+ p dt f t p dt n h i f n xi x i+ mx {h i} f 0in xi x i+ = ν I n b p Conseuently by the use of 36 we deduce the desired result 3 For the prticulr cse where the division I n is euidistnt ie I n : x i = + i i = 0 n n we my consider the udrture rule: 37 Q n f := n n { f + i + n + f + i + 3 The following corollry will be more useful in prctice n } Corollry With the ssumption of Theorem 5 we hve 38 = Q n f + R n f

16 8 S S Drgomir where Q n f is defined by 37 nd the reminder R n f stisfies the estimte: 8 f b ; n 39 R n I n f f + bp + ; n f b n Applictions for PDF s Summrizing some of the results in Section we my stte tht for f : b R n bsolutely continuous function we hve the ineulity b g x + g + b x x 3+b 8 + g b if g L b ; + + x + +b + x + x 3+b g bp if p > p + = nd g L p b ; g b for ll x +b Now let X be rndom vrible tking vlues in the finite intervl b with the probbility density function f : b 0 nd with the cumultive distribution function F x = Pr X x = x The following result holds

17 Ostrowski s ineulity 9 Theorem 6 With the bove ssumptions we hve the ineulity b E X F x + F + b x x 3+b 8 + f b if f L b ; + + x + +b + x + x 3+b f bp if p > p + = nd f L p b ; for ny x +b where E X is the expecttion of X Proof Follows by on choosing g = F nd tking into ccount tht E X = In prticulr we hve: tdf t = b F t dt Corollry 5 With the bove ssumptions we hve b + 3b F + F b E X 8 f b if f L b ; + f bp if p > p + = nd f L p b ; References A Guessb nd G Schmeisser Shrp integrl ineulities of the Hermite- Hdmrd type J Approx Theory

18 30 S S Drgomir S S Drgomir nd Th M Rssis Ed Ostrowski type ineulities nd pplictions in numericl integrtion Kluwer Acdemic Publishers Dordrecht/Boston/London 00 3 S S Drgomir A refinement of Ostrowski s ineulity for bsolutely continuous functions whose derivtives belong to L nd pplictions Liberts Mth School of Communictions nd Informtics Victori University of Technology PO Box 8 MCMC 800 Victori Austrli E-mil: sever@mtildvueduu