RGMIA Research Report Collection, Vol. 1, No. 1, SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIA

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1 ttp//sci.vut.edu.u/rgmi/reports.tml SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIABLE MAPPINGS AND APPLICATIONS P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS Astrct. Some generliztions of te Ostrowski inequlity for n-time differentile mppings re given. Applictions in Numericl Integrtion nd for power series expnsions re lso presented. Introduction In 93, Ostrowski (see for exmple [3, p.46]) proved te following integrl inequlity Let f I R! R e dierentile mpping on I ( I is te interior of I), nd let I wit < If f 0 ( )! R is ounded on ( ), i.e., kf 0 k sup jf 0 (t)j < ten we ve te inequlity t( ) f (x) f (t) dt x ( ) ( ) kf 0 k for ll x [ ] Te constnt 4 is srp in te sense tt it cn not e replced y smller one. For pplictions of Ostrowski's inequlity to some specil mens nd some numericl qudrture rules, we refer te reder to te recent pper [] y S.S. Drgomir nd S. Wng. In 976, G.V. Milovnovic nd J.E. Pecric (seefor exmple [3, p. 46]), proved te following generliztion of Ostrowski's result Let f [ ]! R e n n-times dierentile function, n nd suc tt sup t( ) n f(x)+ k f (n) (t) < Ten n k k!! Z f (k) ()(x ) k f (k) ()(x ) k f (t) dt Dte. Novemer, Mtemtics Suject Clssiction. Primry 6D5 Secondry 4A55. Key words nd prses. Ostrowski Integrl Inequlity, Numericl Integrtion, Tylor's Expnsions. 53

2 54 Cerone, Drgomir nd Roumeliotis n(n +)! (x )n+ +( x) n+ for ll x [ ] In [], P. Cerone, S.S. Drgomir nd J. Roumeliotis proved te following Ostrowski type inequlity for twice dierentile mppings Let f [ ]! R e twice dierentile mpping on ( ) nd f 00 ( )! R is ounded, i.e., kf 00 k inequlity f (x) 4 ( ) + sup t( ) f (t) dt x + jf 00 (t)j < Ten we ve te f 0 (x) x + kf 00 ( ) k kf 00 k 6 for ll x [ ] In tis pper we estlis noter generliztion of Ostrowski inequlity for n-time dierentile mppings wic nturlly generlizes te result from [] nd pply it in Numericl Integrtion nd for power series expnsions of functions on n intervl. Te following lemm olds Integrl identities Lemm.. Let f [ ]! R e mpping suc tt f () is solutely continuous on [ ]. Ten for ll x [ ] we ve te identity Z ( x) k+ +() k (x ) k+ (.) f (t) dt f (k) (x) (k +)! +() n K n (x t) f (n) (t) dt te kernel K n [ ]! R is given y (.) K n (x t) >< > (t ) n (t ) n if t [ x] if t (x ] x [ ] nd n is nturl numer, n

3 Ostrowski for n-time Dierentile Mppings 55 Proof. Te proof is y mtemticl induction. For n we ve to prove te equlity (.3) f (t) dt ( ) f (x) K (x t) Integrting y prts,we ve < K (x t) f () (t) dt Zx (t ) f (t)j x Z x (x ) f (x)+( x) f (x) K (x t) f () (t) dt t if t [ x] t if t (x ] (t ) f 0 (t) dt + f (t) dt +(t ) f (t)j x x (t ) f 0 (t) dt Z x f (t) dt f (t) dt ( ) f (x) f (t) dt nd te identity (3) is proved. Assume tt () olds for n ndletusprove itforn + Tt is, we ve toprove te equlity Z n ( x) k+ +() k (x ) k+ (.4) f (t) dt f (k) (x) (k +)!, oviously, We ve +() n+ K n+ (t) >< > K n+ (x t) f (n+) (t) dt (t ) n+ (n +)! (t ) n+ (n +)! if t [ x] if t (x ] K n+ (x t) f (n+) (t) dt Z x (t ) n+ (n + )! f (n+) (t) dt + x (t ) n+ f (n+) (t) dt (n +)!

4 56 Cerone, Drgomir nd Roumeliotis nd integrting y prts gives K n+ (x t) f (n+) (t) dt (t )n+ (n +)! f (n) (t) x Z x (t ) n f (n) (t) dt + (t )n+ (n + )! f (n) (t) x x (t ) n f (n) (t) dt Tt is Z (x )n+ +() n+ ( x) n+ f (n) (x) K n (x t) f (n) (t) dt (n + )! K n (x t) f (n) (t) dt (x )n+ +() n+ ( x) n+ f (n) (x) (n +)! K n+ (x t) f (n+) (t) dt Now, using te mtemticl induction ypotesis, we get Z ( x) k+ +() k (x ) k+ f (t) dt f (k) (x) (k +)! Z + ( x)n+ +() n (x ) n+ f (n) (x) () n K n+ (x t) f (n+) (t) dt (n +)! n ( x) k+ +() k (x ) k+ (k +)! f (k) (x) +() n+ K n+ (x t) f (n+) (t) dt Tt is, te identity (4) nd te teorem is tus proved.

5 Ostrowski for n-time Dierentile Mppings 57 Corollry.. Wit te ove ssumptions, we ve te representtion Z +() k ( ) k+ f (t) dt (k + )! k+ f (k) + (.5) M n (t) +() n >< > (t ) n (t ) n M n (t) f (n) (t) dt if t + if t + Te proof follows y Lemm. y coosing x + Corollry.3. Wit te ove ssumptions, we ve te representtion (.6) f (t) dt ( ) k+ (k + )! f (k) ()+() k f (k) () + T n (t) f (n) (t) dt T n (t) ( t) n +() n (t ) n t [ ] Proof. Firstly, coose in () x to get f (t) dt ( ) k+ (k +)! f (k) ()+() n (t ) n f (n) (t) dt ( ) k+ (k +)! f (k) ()+ Also, if we put x in () we get f (t) dt ( t) n f (n) (t) dt Z () k ( ) k+ f (k) ()+() n (k +)! (t ) n f (n) (t) dt

6 5 Cerone, Drgomir nd Roumeliotis Summing te ove two identities nd dividing y,we get f (t) dt n ( ) k+ (k +)! f (k) ()+() k f (k) () nd te corollry is proved. + T n (t) f (n) (t) dt Te following Tylor-like formul wit integrl reminder lso olds Corollry.4. Let g [ y]! R e mpping suc tt g (n) is solutely continuous on [ y] Ten for ll x [ y] we ve te identity (y x) k+ +() k (x ) k+i (.7) g (y) g ()+ g (k+) (x) (k +)! Z y +() n K n (y t) g (n+) (t) dt Te proof is ovious y Lemm. coosing f g 0 nd y Remrk.. If we coose n in (), we get te identity (.) f (t) dt ( ) f (x) K (x t) f 0 (t) dt for ll x [ ] K (x t) < t if t [ x] t if t (x ] wic is te identity employed ys.s. Drgomir nd S. Wng to prove n Ostrowski type inequlity in pper []. (.9) If in (5) we coose n, ten we get + f (t) dt ( ) f M (t) < t t if t + if t + M (t) f 0 (t) dt

7 Ostrowski for n-time Dierentile Mppings 59 wic gives te mid-point type inequlity useful in Numericl Anlysis. Also, if we put n in(6), we get te trpezoid identity (.0) f (t) dt f ()+f () ( )+ T (t) f 0 (t) dt T (t) + t t [ ] Finlly, if in te Tylor-like formul (7) we putn we get x [ y] g (y) g ()+(y ) g 0 (x) Z y K (y t) g () (t) dt Remrk.. If we coose n in (), we get te identity (.) Z f (t) dt ( ) f (x) x + f 0 (x)+ K (x t) f 00 (t) dt K (x t) >< > (t ) (t ) if t [ x] if t (x ] nd x [ ] wic is te identity employed yp.cerone, S.S. Drgomir nd J. Roumeliotis to prove some Ostrowski type inequlities for twice dierentile mppings in te pper [] (.) If in (5) we coose n,tenweget + f (t) dt ( ) f + M (t) f 00 (t) dt M (t) >< > (t ) (t ) wic is te clssicl mid-point identity. if t + if t +

8 60 Cerone, Drgomir nd Roumeliotis Also, if we put n in(6) we get te identity (.3) f (t) dt f ()+f () ( )+ ( ) f 0 () f 0 () (.4) + T (t) f 00 (t) dt T (t) ( t) +(t ) t [ ] Finlly, ifwe put n in(7), we get g (y) g ()+(y ) g 0 (x) (y ) x + y g 00 (x) + Z y K is s ove nd x y K (y t) g (3) (t) dt 3 Some integrl inequlities for kk norm Te following teorem olds Teorem 3.. Let f [ ]! R e mpping suc tt f () is solutely continuous on [ ] nd f (n) L [ ] Ten for ll x [ ] we ve te inequlity Z ( x) k+ +() k (x ) k+ (3.) f (t) dt f (k) (x) (k +)! (x ) n+ +( x) n+i f (n) ( ) n+ (n +)! (n +)! f (n) sup t[ ] Proof. Using te identity () we ve f (t) dt f (n) (t) < ( x) k+ +() k (x ) k+ (k +)! f (k) (x)

9 Ostrowski for n-time Dierentile Mppings 6 K n (x t) f (n) (t) dt Z 4 x f (n) (t ) n dt + x jk n (x t)j dt ( t) n dt (x ) n+ +( x) n+i (n + )! nd te rst prt of inequlity (3) is proved. To prove te second inequlity in(3) we oserve tt 3 5 (3.) (x ) n+ +( x) n+ ( ) n+ for ll x [ ] Tking into ccount te fct tt te mpping n [ ]! R n (x) (x ) n+ +( x) n+ s te property + inf n (x) n x[ ] ( )n+ n ten te est inequlity we cn get from (3) is te one for wic x + In tis wy, we cn stte te following corollry. Ten we ve te in- Corollry 3.. Assume tt f is s in Teorem 3.. equlity (3.3) f (t) dt +() k (k +)! ( ) n+ n (n +)! ( ) k+ k+ f (k) + Anoter result wic generlizes te trpezoid inequlity, is te following one Corollry 3.3. Wit te ove ssumptions, we ve te inequlity (3.4) f (t) dt ( )n+ (n +)! ( ) k+ (k + )! >< > f (k) ()+() k f (k) () if n r r+ r if n r +

10 6 Cerone, Drgomir nd Roumeliotis Proof. Using te identity (6), we get (3.5) f (t) dt If n r ten ( ) k+ (k + )! T n (t) f (n) (t) dt Z jt n (t)j dt (r)! f (k) ()+() k f (k) () f (n) jt n (t)j dt ( t) r +(t ) r dt ( (r)! ) r+ + (r +) ( )r+ (r +) ( )r+ (r + )! For n r+ put r+ (t) ( t) r+ (t ) r+ t [ ] Oserve tt r+ (t) 0it + nd r+ (t) > 0ift [ + )nd r+ (t) < 0 if t ( + ] Ten jt r+ (t)j dt + Z ( t) r+ (t ) r+i dt + + (t ) r+ ( t) r+i dt ( )r+ r + 4 r+ r + ( ) r+ ( )r+ ( )r+ r + r r + r ( )r+ r + r+ r Using (34) we get te desired inequlity (33) Te following inequlity in terms of kk norm for te Tylor like expnsion (7) lso olds

11 Ostrowski for n-time Dierentile Mppings 63 Corollry 3.4. Let g e sincorollry.4. Ten we ve te inequlity (3.6) for ll x y g (y) g () (y x) k+ +() k (x ) k+i (k +)! g (k+) (x) g (n+) (y x) n+ +(x ) n+i g (n+) (y ) n+ (n + )! (n +)! Remrk 3.. It is well known tt for te clssicl Tylor expnsion round we ve te inequlity (3.7) n g (y) (y ) k g (k) () k! g (n+) (n +)! (y ) n+ for ll y It is cler now tt te ove pproximtion (36) round te ritrry point x [ y] provides etter pproximtion for te mpping g t te point y tn te clssicl Tylor expnsion round te point. If in (36) we coose x +y ten we get n +() ki (y ) k g (y) g () k! k g (k) + y (3.) k g (n+) (n +)! n (y )n+ Te ove inequlity (3) sows tt for g C [ ] te series +() ki (y ) k+ g ()+ (k +)! k+ g (k+) + y converges more rpidly to g (y) tn te usul one (y ) k g (k) () k! wic comes from Tylor's expnsion. Furter, it sould e noted tt (3) only involves odd derivtives of g Remrk 3.. If in te inequlity (3) we coose n we get f (t) dt ( ) f (x) (x ) +( x) kf 0 k

12 64 Cerone, Drgomir nd Roumeliotis As simple clcultion sows tt (x ) +( x) i 4 ( ) + consequently we otin te Ostrowski inequlity (3.9) for ll x [ ] f (t) dt ( ) f (x) x + x ( ) ( ) kf 0 k If in (3) we put n,we get te mid-point inequlity + (3.0) f (t) dt f ( ) 4 ( ) kf 0 k (3.) (3.) From te inequlity (34), forn we get te trpezoid inequlity f (t) dt Also, from (36)we deduce for ll x y f ()+f () jg (y) g () (y ) g 0 (x)j ( ) ( ) kf 0 k x +y 4 + (y ) kg 00 k Remrk 3.3. If in te inequlity (3) we coose n ten we get f (t) dt ( ) f (x)+ x + ( ) f 0 (x) for ll x [ ] Now, oserve tt 6 (x ) 3 +( x) 3 ( ) (x ) 3 +( x) 3i kf 00 k ( ) ( ) (x ) +( x) (x )( x) (x + x) 3(x )( x) ( ) +3 x ( + ) x + i ( ) ( ) +3 ( ) +3 x + x + i i

13 Ostrowski for n-time Dierentile Mppings 65 ndtenwe recpture te result otined in [], nmely (3.3) f (t) dt ( ) f (x)+ x + ( ) f 0 (x) 4 + x + ( ) ( ) 3 kf 00 k If we putn in(33) we get te clssicl mid-point inequlity + (3.4) f (t) dt ( ) f (3.5) 4 ( )3 kf 00 k Now, if in (34) we putn ten we get te inequlity f (t) dt f ()+f () ( ) ( ) f 0 () f 0 () ( )3 6 kf 00 k Finlly, ifweputn in(36), tenwe get te inequlity g (y) g () (y ) g0 (x)+(y ) x + y 4 + x +y (y ) (y ) 3 kg 000 k g 00 (x) 4 Applictions for numericl integrtion Consider te prtition I m x 0 <x <<x m <x m of te intervl [ ] nd te intermedite points 0 m j [x j x j+ ] j 0 m Dene te formul F m k (f I m ) m k+ xj+ j +() k k+ i j x j f (k) j j0 (k + )! wic cn e regrded s perturtion of Riemnn's sum (f I m ) j x j+ x j j 0 m Te following teorem olds. m j0 f j j

14 66 Cerone, Drgomir nd Roumeliotis Teorem 4.. Let f [ ]! R e mpping suc tt f () is solutely continuous on [ ] nd I m prtitioning of [ ] s ove. Ten we ve te qudrture formul (4.) f (x) dx F m k (f I m )+R m k (f I m ) F m k is dened ove nd te reminder R m k stises te estimtion (4.) jr m k (f I m )j for ll s ove. f (n) (n + )! m j0 f (n) m (n +)! j x j n+ + xj+ j n+ i j0 n+ j Proof. Apply Teorem 3. on te intervl [x j x j+ ]toget Z x j+ xj f (t) dt k+ xj+ 4 j +() k k+ i3 j x j 5 f (k) (k + )! j n+ n+ i j x j + xj+ (n +)! j (n +)! n+ j for ll j f0 m g Summing over j from 0 to m nd using te generlized tringle inequlity, we deduce te desired estimtion (4) As n interesting prticulr cse, we cn consider te following pertured mid-point formul m +() k k+ j M m k (f I m ) (k +)! k+ f (k) xj + x j+ j0 wic in eect involves only even k. We stte te following result concerning te estimtion of te reminder term. Corollry 4.. Let f nd I m esinteorem 4.. Ten we ve (4.3) f (t) dt M m k (f I m )+R m k (f I m )

15 Ostrowski for n-time Dierentile Mppings 67 nd te reminder term R m k stises te estimtion (4.4) jr m k (f I m )j f (n) m n (n +)! j0 n+ j We cn consider te following pertured version of te trpezoid formul T m k (f I m ) m j0 k+ j (k +)! f (k) (x j )+() k f (k) (x j+ ) By te use of Corollry 3.3, we ve te following pproximtion of te integrl R f (t) dt in terms of T m k (f I m ) Corollry 4.3. Let f nd I m esinteorem 4.. Ten we ve (4.5) f (t) dt T m k (f I m )+ ~ Rm k (f I m ) nd te reminder ~ Rm k (f I m ) stises te inequlity R ~ m k (f I m ) C n >< > C n (n +)! if n r m j0 r+ r if n r + n+ j Remrk 4.. ). If we coose n in te ove qudrture formule (4) nd (43), we recpture some results from te pper []. ). If we put n, ten y te ove Teorem 4. nd Corollry 4., we recover some results from te pper []. We omit te detils. 5 Applictions for some prticulr mppings ) Consider g R! R g(x) e x Ten g (n) (x) e x n N nd g (n+) sup Using inequlity (36) we ve (5.) t[ y] g (n+) (t) e y (y x) k+ +() k (x ) k+i ey e e x (k +)!

16 6 Cerone, Drgomir nd Roumeliotis e y (y x) n+ +(x ) n+i e y (y )n+ (n +)! (n +)! for ll x y Prticulrly, ifwecoose 0 ten we get i (y x) k+ +() k x k+ (5.) ey e x (k +)! (5.3) e y (y x) n+ + n+i e y x (n + )! (n +)! yn+ Moreover, if we coose x y ten we get ey e y +() k (k +)! for ll y 0 ) Consider g (0 )! R, g (x) lnx Ten yk+ k+ ey y n+ n (n + )! nd g (n) (x) () (n )! x n n x>0 g (n+) sup t[ y] ()n t n+ n+ >0 Using te inequlity (36) we cn stte ln y ln (y x) k+ +() k (x ) k+ (k +)! ()k k! x k+ (n +)! n+ (y x) n+ +(x ) n+i (y )n+ (n +)!n+ wic is equivlent to (5.4) y ln k + (x )k+ +() k (y x) k+ x k+ (y x)n+ +(x ) n+ (n +) n+ (n +) n+ (y )n+ Now, if we coose in (54) y z + x w + z w 0 ten we get (5.5) ln (z +) k + wk+ +() k (z w) k+ (w +) k+

17 Ostrowski for n-time Dierentile Mppings 69 (z w)n+ + w n+ n + (n +) zn+ Finlly, ifwecoose in (54) y u x w wit u w> ten we ve ln u k + (w ) k+ +() k (u w) k+ w k+ (u w)n+ +(w ) m+ n + (u )n+ (n +) References [] P. CERONE, S.S. DRAGOMIR nd J. ROUMELIOTIS, An inequlity of Ostrowski type for mppings wose second derivtives re ounded nd pplictions, sumitted. [] S.S. DRAGOMIR nd S. WANG, Applictions of Ostrowski's inequlity to te estimtion of error ounds for some specil mens nd some numericl qudrture rules, Appl. Mt. Lett., (99), [3] D.S. MITRINOVIC, J.E. PECARIC nd A.M. FINK, Inequlities for Functions nd Teir Integrls nd Derivtives, Kluwer Acdemic, Dordrect, 994. Scool of Communictions nd Informtics Victori University of Tecnology, PO Box 44 MCMC Melourne, Victori 00, Austrli. E-mil ddress fpc, sever, jonrg@mtild.vut.edu.u