90 S.S. Drgomr nd (t b)du(t) =u()(b ) u(t)dt: If we dd the bove two equltes, we get (.) u()(b ) u(t)dt = p(; t)du(t) where p(; t) := for ll ; t [; b]:

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1 RGMIA Reserch Report Collecton, Vol., No. 1, ON THE OSTROWSKI INTEGRAL INEQUALITY FOR LIPSCHITZIAN MAPPINGS AND APPLICATIONS S.S. Drgomr Abstrct. A generlzton of Ostrowsk's nequlty for lpschtzn mppngs nd pplctons n Numercl Anlyss nd for Euler's Bet functon re gven. 1 Introducton The followng theorem contns the ntegrl nequlty whch s known n the lterture s Ostrowsk's nequlty [,p.469]. Theorem 1.1. Let f :[; b]! R be dfferentble mppng on (; b) whose dervtve s bounded on (; b) nd denote kf 0 k 1 = sup t(;b) jf 0 (t)j < 1: Then for ll [; b] we hve the nequlty Z f() 1 b # f(t)dt 1 +b b 4 + (b ) (b )kf 0 k 1: The constnt 1 4» " s shrp n the sense tht t cn not be replced by smller one. In ths pper we prove tht Ostrowsk's nequlty lso holds for lpschtzn mppngs nd pply t n obtnng Remnn's type qudrture formul for ths clss of mppngs. Applctons for Euler's Bet functon re lso gven. Ostrowsk's Inequlty For Lpschtzn Mppngs The followng nequlty for lpschtzn mppngs holds: Theorem.1. Let u :[; b]! R be nllpschtzn mppng on [; b];.e., ju() u(y)j»l j yj for ll ; y [; b]: Then we hve the nequlty (.1) u(t)dt u()(b ) for ll [; b] : The constnt 1 4 s the best possble one. " 1» L(b ) 4 + # +b (b ) Proof. Usng the ntegrton by prts formul for Remnn-Steltjes ntegrl we hve : Z Z (t )du(t) =u()( ) u(t)dt Dte. Jnury, Mthemtcs Subject Clsscton. Prmry 6 D 15; Secondry 6 D 99. Key words nd phrses. Ostrowsk's Inequlty,Numercl Integrton,Bet Mppng. 89

2 90 S.S. Drgomr nd (t b)du(t) =u()(b ) u(t)dt: If we dd the bove two equltes, we get (.) u()(b ) u(t)dt = p(; t)du(t) where p(; t) := for ll ; t [; b]: Now, ssume tht n ν( n)! 0sn!1; where ν( n) := ρ t f t [; ) t b f [; b] : = (n) 0 < (n) 1 <:::< (n) <(n) n m f0;:::;g ((n) +1 (n) s Remnn ntegrble on [; b] ndv :[; b]! R s L-lpschtzn on [; b]; then (.3) (.4) p()dv()» lm ν( n)!0» L lm ν( n)!0 = lm p(ο (n) ν( n)!0 ) (n) p(ο (n) ) +1 (n) (n) ; = b s sequence of dvsons wth ) nd ο (n) [ (n) ; (n) ]: If p :[; b]! R h p(ο (n) ) v( (n) +1 ) ) v((n) v +1 (n) (n) +1 v (n) (n) +1 (n) = L jp()j d: Applyng the nequlty (:3) for p(; t) s bove nd v() = u(); [; b]; we get = L Z 4 Zb jt j dt + p(; t)du(t)» L " = L (b ) 3 jp(; t)j dt jt bj dt5 L = # 1 +b 4 + (b ) ( ) +(b ) Λ ndthenby(:4), v the dentty (:), we deduce the desred nequlty (:1). Now, ssume tht the nequlty (:1) holds wth constnt C>0;.e., (.5) u(t)dt u()(b ) "» L (b ) C + for ll [; b]: Consder the mppng f :[; b]! R;f() = n (:5). Then + b for ll [; b]; nd then for = ; we get b» +b» C + (b ) C + 1 (b ) 4 whch mples tht C 1 nd the theorem s completely proved. 4 # +b (b ) +1 RGMIA Reserch Report Collecton, Vol., No. 1, 1999

3 Ostrowsk Integrl Inequlty for Lpschtzn Mppngs 91 The followng corollry holds: Corollry.. Let u :[; b]! R be sbove. Then we hve the nequlty: (.6) u(t)d u + b (b )» 1 4 L(b ) : Remrk.1. It s well known tht f f : [; b]! R s conve mppng on [; b], then Hermte- Hdmrd's nequlty holds (.7) f + b Z» 1 b b f()d» f()+f(b) : Now, f we ssume tht f : I ρ R! R s conve on I nd ; b Int(I); < b; then f 0 + s monotonous nondecresng on [; b] nd by Theorem.1 we get (.8) Z 0» 1 b f()d f b + b» 1 4 f +(b)(b 0 ) whch gves counterprt for the rst membershp of Hdmrd's nequlty. 3 A Qudrture Formul of Remnn Type Let I n : = 0 < 1 <:::< < n = b be dvson of the ntervl [; b] nd ο [ ; +1] ( =0;:::;n 1)sequenceontermedte ponts for I n: Construct the Remnn sums R n(f; I n;ο)= where h := +1 : We hve the followng qudrture formul f(ο )h Theorem 3.1. Let f :[; b]! R be nllpschtzn mppng on [; b] nd I n;ο ( =0;:::;n 1) be s bove. Then we hve the Remnn qudrture formul (3.1) f()d = R n(f; I n;ο)+w n(f; I n;ο) where the remnder stses the estmton (3.) for ll ο ( =0;:::;n 1) s bove. The constnt 1 4 s shrp n (3:). jw n(f; I n;ο)j» 1 4 L h + L» 1 L Proof. Apply Theorem.1 on the ntervl [ ; +1] toget h ο + +1 (3.3) Z +1 f()d f(ο )h» 1» L 4 h + ο + +1 : RGMIA Reserch Report Collecton, Vol., No. 1, 1999

4 9 S.S. Drgomr Summng over from 0 to n 1 nd usng the generlzed trngle nequlty weget Now, s jw n(f; I n;ο)j»» L Z +1» 1 4 h + ο f()d f(ο )h + +1 : ο » 4 h for ll ο [ ; +1]( =0; :::; n 1) the second prt of (3:) s lso proved. Note tht the best estmton we cn get from (3:) s tht one for whch ο = + +1 obtnng the followng mdpont formul: Corollry 3.. Let f; I n be sbove. Then we hve the mdpont rule f()d = M n(f; I n)+s n(f; I n) where M n(f; I n)= + +1 f h nd the remnder S n(f; I n) stses the estmton js n(f; I n)j» 1 4 L h : Remrk 3.1. If we ssume tht f :[; b]! R s dfferentble on (; b) nd whose dervtve f 0 s bounded on (; b) we cn put nsted of L the nnty norm kf 0 k 1 obtnng the estmton due to Drgomr-Wng from the pper [1]. Consder the mppng Bet for rel numbers 4 Applctons for Euler's Bet Mppng B(p; q) := nd the mppng e p;q(t) :=t p1 (1 t) q1 ;t [0; 1]: We hve for p; q > 1tht If t t 0 = p1 p+q h 0; Z 1 0 t p1 (1 t) q1 dt; p; q > 0 e 0 p;q(t) =e p1;q1(t)[p 1 (p + q )t] : p1 p+q ; 1 then e 0 p;q(t) < 0 whch shows tht for p1 then e 0 p+q p;q(t) > 0 nd f t wehve mmum for ep;q nd then: sup e p;q(t) =e p;q(t 0)= (p 1)p1 (q 1) q1 ; t[0;1] (p + q ) p+q p; q > 1: RGMIA Reserch Report Collecton, Vol., No. 1, 1999

5 Ostrowsk Integrl Inequlty for Lpschtzn Mppngs 93 Consequently 0 e p;q(t) (p ) p (q ) q» m (p + q 4) p+q4 jp 1 (p + q )tj t[0;1] for ll t [0; 1] nd then (4.1) e 0 p;q = m fp 1;q 1g (p )p (q ) q (p + q 4) p+q4 ; p; q > 1» m fp 1;q 1g (p )p (q ) q The followng nequlty for Bet mppng holds (p + q 4) p+q4 p; q > : Proposton 4.1. Let p; q > nd [0; 1]: Then we hve the nequlty B(p; q) p1 (1 ) q1 (4.) "» mfp 1;q 1g (p )p (q ) q 1 # (p + q 4) p+q : The proof follows by Theorem.1 ppled for the mppng e p;q nd tkng nto ccounttht e 0 stses the nequlty (4:1). Corollry 4.. Let p; q > : Then we hve the nequlty B(p; q) 1 p+q» 1 4 mfp 1;q 1g (p )p (q ) q (p + q 4) p+q4 : Now, f we pply Theorem 3.1 for the mppng e p;q we get the followng ppromton of Bet mppng n terms of Remnn sums. Proposton 4.3. Let I n : = 0 < 1 <:::< < n = b be dvson of the ntervl [; b];ο [ ; +1] ; ( =0;:::;n 1) sequence of ntermedte ponts for I n nd p; q > : Then we hve the formul B(p; q) = where theremnder T n(p; q) stses the estmton ο p1 (1 ο ) q1 h + T n(p; q) jt n(p; q)j»mfp 1;q 1g (p )p (q ) q " 1 h + 4 ο (p + q 4) p+q4 + # +1» 1 mfp 1;q 1g (p )p (q ) q (p + q 4) p+q4 Prtculrly, f we choose bove ο = + +1 ( =0;:::;n 1) then we get the ppromton where B(p; q) = h : 1 p+q ( + +1) p1 ( +1) q1 + V n (p; q) jv n(p; q)j» 1 4 mfp 1;q 1g (p )p (q ) q (p + q 4) p+q4 h : p;q 1 RGMIA Reserch Report Collecton, Vol., No. 1, 1999

6 94 S.S. Drgomr References [1] S.S. DRAGOMIR nd S. WANG, Applctons of Ostrowsk's nequlty to the estmton of error bounds for some specl mens nd for some numercl qudrture rules, Appl. Mth. Lett., 11(1)(1998), [] D.S. MITRINOVIĆ, J.E. PE»CARIĆ nd A.M. FINK, Inequltes for Functons nd ther Integrls nd Dervtves, Kluwer Acdemc Publshers, School of Communctons nd Informtcs, Vctor Unversty of Technology, PO Bo1448, Melbourne Cty MC, Vctor 8001, Austrl. E-ml ddress: RGMIA Reserch Report Collecton, Vol., No. 1, 1999