ON THE WEIGHTED OSTROWSKI INEQUALITY N.S. BARNETT AND S.S. DRAGOMIR School of Computer Science nd Mthemtics Victori University, PO Bo 14428 Melbourne City, VIC 8001, Austrli. EMil: {neil.brnett, sever.drgomir}@vu.edu.u Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Received: 14 My, 2007 Accepted: 30 September, 2007 Communicted by: B.G. Pchptte 2000 AMS Sub. Clss.: 26D15, 26D10. Key words: Abstrct: Ostrowski inequlity, Integrl inequlities, Absolutely continuous functions. On utilising n identity from [5, some weighted Ostrowski type inequlities re estblished. Pge 1 of 21 Go Bck
1 Introduction 3 2 Ostrowski Type Inequlities 6 3 Some Emples 17 Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 2 of 21 Go Bck
1. Introduction In [5, the uthors obtined the following generlistion of the weighted Montgomery identity: (1.1 f ( = 1 ϕ (1 w (t ϕ w (s ds f (t dt + 1 ϕ (1 P w,ϕ (, t f (t dt, where f : [, b R is n bsolutely continuous function, ϕ : [0, 1 R is differentible function with ϕ (0 = 0, ϕ (1 0 nd w : [, b [0, is probbility density function such tht the weighed Peno kernel (1.2 P w,ϕ (, t := ( t ϕ w (s ds, t, ( t ϕ w (s ds ϕ (1, < t b, is integrble for ny [, b. If ϕ (t = t, then (1.1 reduces to the weighted Montgomery identity obtined by Pečrić in [21: (1.3 f ( = w (t f (t dt + where the weighted Peno kernel P w is P w (, t f (t dt, Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 3 of 21 Go Bck (1.4 P w (, t := { t w (s ds, t, w (s ds, < t b. t
Finlly, the uniform distribution is used to provide the Montgomery identity [17, p. 565: (1.5 f ( = 1 b with P (, t := f (t dt + P (, t f (t dt, { t b if t, t b b if < t b, tht hs been etensively used to obtin Ostrowski type results, see for instnce the reserch ppers [3 [6, [7 [16, [19 [20, [22 nd the book [15. In the sme pper [5, on introducing the generlised Čebyšev functionl, (1.6 T ϕ (w, f, g := 1 ϕ (1 w (t dt f ( g ( d ( w ( ϕ w (t dt f ( d [ ( w ( ϕ w (t dt g ( d, w ( ϕ ( [ the uthors obtined the representtion: (1.7 T ϕ (w, f, g = 1 ( w ( ϕ w (t dt ϕ 2 (1 [ [ P w,ϕ (, t f (t dt P w,ϕ (, t g (t dt d Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 4 of 21 Go Bck
nd used it to obtin n upper bound for the bsolute vlue of the Čebyšev functionl in the cse where f, g, ϕ L [, b. This bound cn be stted s: (1.8 T ϕ (w, f, g 1 ϕ 2 (1 f g ϕ w ( H 2 ( d, where H ( := P w,ϕ (, t dt. The inequlity (1.8 provides generlistion of result obtined by Pchptte in [18. The min im of this pper is to obtin some weighted inequlities of the Ostrowski type by providing vrious upper bounds for the devition of f (, [, b, from the integrl men 1 ϕ (1 w (t ϕ w (s ds f (t dt, when f is bsolutely continuous, of bounded vrition or Lipschitzin on the intervl [, b. Some prticulr cses of interest re lso given. Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 5 of 21 Go Bck
2. Ostrowski Type Inequlities In order to stte some Ostrowski type inequlities, we consider the Lebesgue norms g [α,β, := ess sup g (t t [α,β nd [ β g [α,β,l := α 1 l g (t l dt, l [1, ; provided tht the integrl nd the supremum re finite. Theorem 2.1. Let ϕ : [0, 1 R be continuous on [0, 1, differentible on (0, 1 with the property tht ϕ (0 = 0 nd ϕ (1 0. If w : [, b R + is probbility density function, then for ny f : [, b R n bsolutely continuous function, we hve (2.1 f ( 1 w (t ϕ w (s ds f (t dt ϕ (1 b ( ϕ t w (s ds f (t dt + ϕ w (s ds ϕ (1 f (t dt for ny [, b. If nd H 1 ( := H 2 ( := ϕ ϕ w (s ds f (t dt w (s ds ϕ (1 f (t dt, Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 6 of 21 Go Bck
then (2.2 H 1 ( nd ϕ ( w (s ds [,, f [,,1 ; ϕ ( w (s ds [,,p f [,,q if p > 1, 1 p + 1 q = 1 nd f L q [, ; ϕ ( w (s ds [,,1 f [,, if f L [, ; (2.3 H 2 ( ( ϕ w (s ds ϕ (1 [,b, f [,b,1 ; for ny [, b. ϕ ( w (s ds ϕ (1 [,b,r f [,b,t if r > 1, 1 r + 1 t = 1 nd f L t [, b ; ϕ ( w (s ds ϕ (1 [,b,1 f [,b, if f L [, b Proof. Follows from the identity (1.1 on observing tht (2.4 f ( 1 w (t ϕ w (s ds f (t dt ϕ (1 Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 7 of 21 Go Bck
= [ ϕ w (s ds f (t dt + ϕ b [ ϕ w (s ds f (t dt + ϕ b ( ϕ t w (s ds f (t dt + ϕ w (s ds ϕ (1 w (s ds w (s ds for ny [, b, nd the first prt of (2.1 is proved. The bounds from (2.2 nd (2.3 follow by the Hölder inequlity. f (t dt ϕ (1 f (t dt ϕ (1 f (t dt Remrk 1. It is obvious tht, the bove theorem provides 9 possible upper bounds for the bsolute vlue of the devition of f ( from the integrl men, 1 b w (t ϕ w (s ds f (t dt ϕ (1 lthough they re not stted eplicitly. The bove result, which provides n Ostrowski type inequlity for the bsolutely continuous function f, cn be etended to the lrger clss of functions of bounded vrition s follows: Theorem 2.2. Let ϕ nd w be s in Theorem 2.1. If w is continuous on [, b nd f : [, b R is function of bounded vrition on [, b, then: f ( 1 (2.5 w (t ϕ w (s ds f (t dt ϕ (1 Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 8 of 21 Go Bck
[ 1 sup ϕ (1 t [, { 1 ϕ (1 m ϕ w (s ds + sup t [,b t [,b (f ϕ w (s ds ϕ (1 sup ϕ w (s ds, t [, } sup ϕ w (s ds ϕ (1 (f (f, where b (f denotes the totl vrition of f on [, b. Proof. We recll tht, if p : [α, β R is continuous on [α, β nd v : [α, β R is of bounded vrition, then the Riemnn-Stieltjes integrl β p (t dv (t eists nd α β β (2.6 p (t dv (t sup p (t (v. α t [α,β Since the functions ϕ ( w (s ds nd ϕ ( w (s ds ϕ (1 re continuous on [, nd [, b, respectively, the Riemnn-Stieltjes integrls [ ϕ w (s ds df (t nd ϕ w (s ds ϕ (1 df (t eist nd (2.7 ϕ w (s ds df (t sup t [, α ϕ w (s ds (f, Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 9 of 21 Go Bck
while (2.8 [ ϕ w (s ds ϕ (1 df (t sup ϕ w (s ds ϕ (1 t [,b (f. Integrting by prts in the Riemnn-Stieltjes integrl, we hve (2.9 ϕ w (s ds df (t [ = f (t ϕ w (s ds f (t d ϕ w (s ds ( = f ( ϕ w (s ds w (t ϕ w (s ds f (t dt nd (2.10 [ ϕ = [ ϕ [ = ϕ w (s ds ϕ (1 df (t w (s ds ϕ (1 f (t [ f (t d ϕ w (s ds w (s ds ϕ (1 f ( b ϕ (1 Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 10 of 21 Go Bck
w (t ϕ w (s ds f (t dt. If we dd (2.9 nd (2.10 we deduce the following identity of the Montgomery type for the Riemnn-Stieltjes integrl which is of interest in itself: (2.11 f ( = 1 w (t ϕ w (s ds f (t dt ϕ (1 + 1 ϕ w (s ds df (t ϕ (1 + 1 ϕ (1 [ ϕ w (s ds ϕ (1 df (t, for ny [, b. Now, by (2.11 nd (2.7 (2.8 we obtin the estimte: f ( 1 w (t ϕ w (s ds f (t dt ϕ (1 1 ϕ (1 ϕ w (s ds df (t + 1 [ ϕ (1 ϕ w (s ds ϕ (1 df (t 1 ( ϕ (1 sup t ϕ w (s ds (f t [, + 1 ϕ (1 sup t [,b ϕ w (s ds ϕ (1 which provides the first inequlity in (2.5. The lst prt of (2.5 is obvious. (f, [, b Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 11 of 21 Go Bck
The following prticulr cse is of interest for pplictions. Corollry 2.3. Assume tht f, ϕ, w re s in Theorem 2.2. In ddition, if ϕ is monotonic nondecresing on [0, 1, then f ( 1 (2.12 w (t ϕ w (s ds f (t dt ϕ (1 ϕ ( [ w (s ds (f + 1 ϕ ( w (s ds (f ϕ (1 ϕ (1 [ 1 2 + ϕ ( w (s ds 1 (f. ϕ (1 2 Proof. Follows by Theorem 2.2 on observing tht, if ϕ is monotonic nondecresing on [, b, then: ( t ( sup ϕ w (s ds = sup ϕ w (s ds = ϕ w (s ds t [, t [, nd sup t [,b ϕ w (s ds [ ϕ (1 = sup ϕ (1 ϕ t [,b = ϕ (1 inf ϕ w (s ds t [,b ( = ϕ (1 ϕ w (s ds. w (s ds Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 12 of 21 Go Bck
Corollry 2.4. With the ssumptions of Theorem 2.2 nd if K := sup t (0,1 ϕ (t <, then we hve the bounds: f ( 1 (2.13 w (t ϕ w (s ds f (t dt ϕ (1 [ 1 ϕ (1 K t sup w (s ds b (f + sup w (s ds (f t [, t [,b t { K ϕ (1 m t sup w (s ds, sup b } w (s ds (f. t [, t [,b Remrk 2. If w (s 0 for s [, b, then from (2.13 we get f ( 1 (2.14 w (t ϕ w (s ds f (t dt ϕ (1 [ K b w (s ds (f + w (s ds (f ϕ (1 K [ 1 b w (s ds + 1 ϕ (1 2 2 w (s ds w (s ds t (f. The following result, tht provides n Ostrowski type inequlity for L Lipschitzin functions, cn be stted s well. Theorem 2.5. Let ϕ nd w be s in Theorem 2.1. If w is continuous on [, b nd f : [, b R is n L 1 Lipschitzin function on [, nd L 2 Lipschitzin on Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 13 of 21 Go Bck
[, b, with [, b, then f ( 1 (2.15 w (t ϕ w (s ds f (t dt ϕ (1 1 [ L 1 ϕ (1 ϕ w (s ds dt + L 2 ϕ w (s ds ϕ (1 dt [ 1 m {L 1, L 2 } ϕ (1 ϕ w (s ds dt + ϕ w (s ds ϕ (1 dt. Proof. We recll tht, if p : [α, β R is L Lipschitzin nd v is Riemnn integrble, then the Riemnn-Stieltjes integrl β f (t du (t eists nd α β β (2.16 p (t dv (t L p (t dt. Now, if we pply the bove property to the integrls [ ϕ w (s ds df (t nd ϕ then we cn stte tht (2.17 ϕ α w (s ds α α df (t L 1 w (t ds ϕ (1 df (t, ϕ w (s ds dt Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 14 of 21 Go Bck
nd (2.18 [ ϕ w (s ds ϕ (1 df (t L 2 ϕ w (s ds ϕ (1 dt. By mking use of the identity (2.11, by (2.17 nd (2.18 we deduce the first prt of (2.15. The lst prt is obvious. The following prticulr cse is of interest s well. Corollry 2.6. With the ssumptions of Theorem 2.5 nd if K := sup t (0,1 ϕ (t <, then f ( 1 (2.19 w (t ϕ w (s ds f (t dt ϕ (1 K [ t L 1 ϕ (1 w (s ds dt + L b 2 w (s ds dt t K [ ϕ (1 m {L t 1, L 2 } w (s ds dt + b w (s ds dt. Remrk 3. If w : [, b R is nonnegtive weight, then t w (s ds, w (s ds t 0 for ech t [, b nd since w (s ds dt = w (s ds t w (t dt = w (t dt tw (t dt = t ( t w (t dt Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 15 of 21 Go Bck
nd ( t ( w (s ds dt = t = t b w (s ds w (t dt + then we get, from (2.19, the following result: f ( 1 (2.20 w (t ϕ ϕ (1 [ L 1 K ϕ (1 K ϕ (1 m {L 1, L 2 } + w (t dt tw (t dt = w (s ds ( t w (t dt + L 2 t w (t dt. (t w (t dt, f (t dt (t w (t dt Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 16 of 21 Go Bck
3. Some Emples The inequlity (2.12 is source of numerous prticulr inequlities tht cn be obtined by specifying the function ϕ : [0, 1 R which is continuous, differentible nd monotonic nondecresing with ϕ (0 = 0. For instnce, if we choose ϕ (t = t α, α > 0, then we get the inequlity: α 1 f ( α (3.1 w (t w (s ds f (t dt ( α w (s ds [ 1 2 + (f + ( α w (s ds 1 2 [ ( 1 w (s ds (f, α (f for ny [, b provided tht f is of bounded vrition on [, b, w (s 0 for ny s [, b nd the involved integrls eist. Another simple emple cn be given by choosing ϕ (t = ln (t + 1. In this sitution, we obtin the inequlity: [ (3.2 f ( 1 ln 2 t ln ( w (s ds + 1 ln 2 [ 1 2 + ln ( w (s ds + 1 ln 2 w (t f (t dt w (s ds + 1 [ (f + 1 2 1 ln ( (f, w (s ds + 1 ln 2 (f Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 17 of 21 Go Bck
for ny [, b provided tht f is of bounded vrition on [, b, w (s 0 for ny s [, b nd the involved integrls eist. Finlly, by choosing the function ϕ (t = ep(t 1, we obtin, from the inequlity (2.12, the following result s well: f ( 1 w (t ep w (s ds f (t dt e 1 ep ( w (s ds 1 (f + e ep ( w (s ds (f e 1 e 1 [ 1 2 + ep ( w (s ds 1 e 1 1 2 (f, for ny [, b, provided f is of bounded vrition on [, b nd the involved integrls eist. Weighted Ostrowski Inequlity N.S. Brnett nd S.S. Drgomir vol. 8, iss. 4, rt. 96, 2007 Title Pge Pge 18 of 21 Go Bck
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