S. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:

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1 FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy 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) b k+ x ) k+ + + b x) k+ k M k + ) b ) Received Jnury Mthemtics Subject Clssifiction Primry 6D5; Secondry A55

2 S S Drgomir This ineulity is shrp for ech dmissible x Eulity is obtined if nd only if f = ±Mf + c with c R nd 3) x t) k for t x f t) = t x) k for x t + b) f + b t) for + b) t b We remrk tht for k = ie f Lip M since x ) + + b x) b ) = x 3+b 8 + b ) b ) then we hve the ineulity ) f x) + f + b x) b x 3+b 8 + b ) b ) 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 ) b b ) 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

3 Some Compnions of Ostrowski s Ineulity 3 The following identity holds Some Integrl Ineulities Lemm Assume tht f : b R is n bsolutely continuous function on b Then we hve the eulity ) f x) + f + b x) b + b +b x x = b t + b ) f t) dt + b x +b x t ) f t) dt t b) f t) dt for ny x +b Proof Using the integrtion by prts formul for Lebesgue integrls we hve nd +b x x +b x x t ) f t) dt = f x) x ) x t + b ) ) + b f t) dt = f + b x) x f x) x + b ) +b x x t b) f t) dt = x ) f + 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

4 S S Drgomir Theorem Let f : b R be n bsolutely continuous function on b Then we hve the ineulity ) b f x) + f + b x) b x t ) f t) dt b +b x + t + b x f t) dt + := M x) for ny x +b If f L b then we hve the ineulities 3) +b x b t) f t) dt M x) x ) + b b x + x) f x+b x x ) + +b xb x 3+b 8 + b ) b ) f b α ) x α x +b + b b ) α f β x + f β x+b x + f β β +b xb b ) if α > α + β = ) ) x x +b mx b b f x + f x+b x + f +b xb b ) α for ny x +b The ineulity ) the first ineulity in 3) nd the constnt /8 re shrp

5 Some Compnions of Ostrowski s Ineulity 5 Proof The ineulity ) follows by Lemm on tking the modulus nd using it properties If f L b then +b x x x t ) f t) dt x ) x t + b f t) + b dt x) f x+b x +b x b t) f t) dt x ) nd the first ineulity in 3) is proved Denote M x) := x ) +b xb f + b x + x) f x+b x x ) + +b xb for x +b Firstly observe tht { M x) mx f x f x+b x f +b xb } x ) ) + b + x + = b 8 b ) + x ) ) x 3 + b nd the first ineulity in 3) is proved Using Hölder s ineulity for α > α + β = we lso hve { x ) M α x) + x + b giving the second ineulity in 3) ) α + x ) α } α f β x + β x+b x + β β +b xb

6 6 S S Drgomir 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 b ) f b with / s shrp constnt see for exmple p 5) If in the sme theorem we now choose x = +b then we get ) + b f b 5) 8 b ) f +b + f +b b ) f b with the constnts /8 nd / being shrp This result ws obtined in 3 b It is nturl to consider the following corollry Corollry With the ssumptions in Theorem one hs the ineulity: f ) 3+b + f +3b ) b 8 b ) 6) b The constnt /8 is best possible in the sense tht it cnnot be replced by smller constnt The cse when f L p b p > is embodied in the following theorem Theorem 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

7 hve the bounds: Some Compnions of Ostrowski s Ineulity 7 7) M x) x + ) b x b mx + ) + +b + x b x b ) + f xp +b x b ) + f x+b xp ) + f +b xbp ) + b ) + } { f xp f x+b xp f +b xbp b ) + ) x b ) α+ α + α +b x b ) α+ α α f β xp + f β x+b xp + f β β +b xbp b ) if α > α + β = mx { x ) + +b x b ) + } b f xp + f x+b xp + f +b xbp b ) for ny x +b Proof Using Hölder s integrl ineulity for p > p + = we hve x t ) f t) x dt t ) dt ) xp = x )+ + ) f xp +b x x t + b f t) dt = +b x x t + b dt) f x+b xp +b x ) + + ) x+b xp

8 8 S S Drgomir nd +b x b t) f t) dt = +b x x )+ + ) b t) dt) f +b xbp f +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 the ineulity 8) M x) + ) If in 7) we choose α = β = p p + for ny x +b Remrk ineulity 9) ) ) + + x +b x + b b = p > then we get b ) f bp If in Theorem we choose x = then we get the trpezoid f ) + f b) b 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) b b ) f bp C + ) then for the function f : b R f x) = k x +b k > 0 we hve f ) + f b) = k b b = k b bp = k b ) p ;

9 Some Compnions of Ostrowski s Ineulity 9 nd by 0) we deduce k b ) k b ) C k b ) + ) / giving C +)/ Letting + we deduce C / nd the shrpness of the constnt is proved Remrk 5 If in Theorem we choose x = + b)/ then we get the midpoint ineulity ) + b f ) b b ) + ) b ) + ) f +b p + f +b bp f 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 b C b ) + ) D b ) + ) f +b p + f +b bp bp For the function f : b R f x) = k x +b ) + b b f = 0 = b f +b p + f +b bp = b bp = b ) p k; k > 0 we hve k b ) ) p k = b ) p k

10 0 S S Drgomir nd then by ) we deduce k b ) k b ) k b ) C 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 f ) 3+b + f +3b ) b b ) 3) + ) bp where p + = The constnt / is the best possible in the sense tht it cnnot be replced by smller constnt Proof The ineulity follows by Theorem 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 ) b b ) E + ) Consider the function f : b R x 3 + b if x +b f x) = x + 3b if x +b b bp Then f is bsolutely continuous nd f L p b p > We lso hve ) ) 3 + b + 3b b f + f = 0 = b b 8

11 Some Compnions of Ostrowski s Ineulity nd then by ) we obtin: bp = b ) /p b 8 b ) 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 3 Assume tht the function f : b R is bsolutely continuous on b If M x) is s in eution ) then we hve the bounds 5) ) x f M x) b x + + +b x 3+b b x b ) f x+b x + x b ) f +b xb f b ) α α ) x α +b x + b b f β x + f β x+b x + f β +b xb β if α > α + β = x + b 3 mx f b 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 for x = we get the trpezoid ineulity see for exmple p 55) f ) + f b) 6) b f b

12 S S Drgomir If in 5) we lso choose x = + b)/ then we get the mid point ineulity see for exmple p 56) ) + b f 7) b f b The following corollry lso holds With the ssumption in Theorem one hs the ineul- Corollry 3 ity: f ) 3+b + f +3b ) 8) b b 3 A Composite Qudrture Formul We use the following ineulities obtined in the previous section: f ) 3+b + f +3b ) 3) b 8 b ) f b if f L b ; 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 Q n I n f) := n ) ) 3xi + x i+ xi + 3x i+ 3) f + f h i The following result holds Theorem 3 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)

13 Some Compnions of Ostrowski s Ineulity 3 where Q n I n f) is defined by formul 3) nd the reminder stisfies the estimtes 8 f n b h i if f L b ; n ) R n I n f) f + ) bp h + 3) i if f L p b f b ν I n ) p > p + = ; Proof Applying ineulity 3) on the intervls x i x i+ we my stte tht xi+ ) ) 3xi + x i+ xi + 3x i+ 35) f + f h i x i 8 h i f xi x i+ h + + ) i f xi x i+ p p > p + = ; for ech i {0 n } h i f xi x i+ ; Summing the ineulity 35) over i from 0 to n nd using the generlised tringle ineulity we get n h i 8 f xi x i+ n R n I n f) h + 36) + ) i f xi x i+ p p > p + = ; Now we observe tht n h i n h i f xi x i+ f xi x i+ f n b h i

14 S S Drgomir Using Hölder s discrete ineulity we my write tht n h + i Also we note tht xi x i+ p = = n 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 ) p n h i f xi x i+ mx n n {h i } xi x i+ = ν I n) b 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 = + ib )/n i = 0 n we my consider the udrture rule: 37) Q n f) := b n n { f + i + n ) b ) + f + i + 3 The following corollry will be more useful in prctice n ) } b ) Corollry 3 With the ssumption of Theorem 3 we hve 38) = Q n f) + R n f) where Q n f) is defined by 37) nd the reminder R n f) stisfies the estimte: 8 f b ) b n R n I n f) f b ) + 39) + ) bp n f b ) b n

15 Some Compnions of Ostrowski s Ineulity 5 Applictions for PDF s Summrising 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) b ) x 3+b 8 + b ) g b b if g L b ; ) ) x + +b + x + b ) + ) g b b bp + x 3+b b for ll x + b)/ if p > p + = nd g L p b ; g 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 Theorem With the bove ssumptions we hve the ineulity b E X) ) F x) + F + b x) b ) x 3+b 8 + b ) f b b if f L b ; ) ) x + +b + x + b ) + ) f b b bp + x 3+b b if p > p + = nd f L p b ; for ny x + b)/ where E X) is the expecttion of X

16 6 S S Drgomir Proof Follows by ) on choosing g = F nd tking into ccount tht E X) = In prticulr we hve: t df t) = b F t) dt Corollry With the bove ssumptions we hve ) ) 3 + b + 3b F + F b E X) 3) b 8 b ) f b if f L b ; b ) f + ) bp if p > p + = nd f L p b ; R E F E R E N C E S A Guessb nd G Schmeisser: Shrp integrl ineulities of the Hermite-Hdmrd type J Approx Theory 5 00) SS Drgomir nd ThM Rssis ed): Ostrowski Type Ineulities nd Applictions in Numericl Integrtion Kluwer Acdemic Publishers Dordrecht Boston London 00 3 SS Drgomir: A refinement of Ostrowski s ineulity for bsolutely continuous functions whose derivtives belong to L nd pplictions Liberts Mthemtic 00) 9 6 School of Computer Science & Mthemtics Victori University of Technology PO Box 8 MCMC 800 Victori Austrli e-mil: sever@mtildvueduu Home pge: