Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using the Tylor-Lngrnge formul s well s generliztion of this one, we give some generliztions of the integrl midpoint inequlity s well of the Ostrowski inequlity for n -time differentible mppings. A new shrp generlized weighted Ostrowski type inequlity is given. Keywords : Generlized Ostrowski s inequlity, generlized midpoint inequlity, Tylor s formul.. Introduction Integrl inequlities hve be used extensively in most subjects involving mthe mticl nlysis. They re prticulry useful for pproximtion theory nd numericl nlysis in which estimtes of pproximtion errors re involved. In 938, Ostrowski see for exmple [], [5], [6]) proved the following integrl inequlity: f x) [ ] b +b f t)dt b x + ) 4 b ) b ) f,.) E-mil: nt.nstsiou@teilm.gr E-mil: kechrin@teilm.gr E-mil: bkotsos@teilm.gr Journl of Interdisciplinry Mthemtics Vol. 9 006), No., pp. 49 60 c Tru Publictions
50 K. S. ANASTASIOU, A. I. KECHRINIOTIS AND B. A. KOTSOS for ll x [, b], where the mpping f : [, b] R is differentible on, b), nd f is bounded on, b), which mens f := sup f t) < x,b). The constnt 4 is shrp in the sense tht it cn not be replced by smller one. On the Numericl Integrtion is lso importnt, the following inequlity see for exmple [3], [5], [6]) well known s the integrl midpoint inequlity: b + b f t)dt f ) b ) f,.) 4 where the mpping f : [, b] R is twice differentible on, b), nd f is bounded on, b). For some pplictions of Ostrowski s inequlity to some specil mens nd some numericl qudrture rules, we refer to the recent pper [4] by S. S. Drgomir nd S. Wng. Drgomir, Cerone nd Sof [3], using the celebrted Hermite- Hdmrd inequlity proved the following inequlity, which is better thn.): b ) γ 4 f t)dt f b ) + b Γ b ) 4,.3) where γ := inf f x), Γ := sup f x). x,b) x,b) In [], Cerone, Drgomir nd Roumeliotis considered nother midpoint type inequlity, s follows: Let f be s bove. Moreover, if f L, b), then we hve b ) + b f t)dt f b b 8 f,.4) where f = n f x) dx i.e is the norm of L, b). G. V. Milovnovic nd J. E. Pecric see for exmple [6, p. 468]), proved the following generliztion of Ostrowski s inequlity: Let f : [, b] R be n-times differentible function, n, nd such tht f n) <.
OSTROWSKI S INEQUALITY 5 Then f x) n n + k= for ll x [, b]. n k) k!n x ) k f k ) ) x b) k f k ) b) b ) b f x)dx f n) nn+)! x )n+ +b x) n+,.5) b ) In this pper, by using the Tylor s formul with Lngrnge s form of reminder, s well s generliztion of the Tylor s formul, we present generl weighted nlyticl integrl Ostrowski type inequlity involving n-time differentible mppings. Also, bsed on this inequlity, we prove shrp extension of the clssicl Ostrowski inequlity.) nd some new shrp generliztions of the inequlities.),.),.3) for n-time differentible mppings re obtined. Moreover, we give generlized midpoint-gruess type inequlity for n-time differentible mppings, which in specil cse yields n improvement of the inequlity.4). All these inequlities cn be used in the theory of Numericl Integrtion nd generlly in Mthemticl Anlysis. The pper is orgnized s follows: In section, we prove the generl weighted inequlity Theorem ) nd ll our results over Ostrowski s inequlity. The section 3 is devoted to obtin the generlized midpoint inequlities.. Generliztions of the Ostrowski inequlity Let us denote R n f ;, b) := f b) n b ) i f i) ) the Tylor s reminder. Moreover, for ny c, d R, we denote by {c, d}) the open intervl min{c, d}, mx{c, d}), nd by [{c, d}] the closed [min{c, d}, mx{c, d}]. For the proof of the following Theorem, we need generliztion of the Tylor s formul [, p. 3]: Theorem. If f n), g n) re continuous on [{, b}], f n+), g n+) exist on {, b}), nd g n+) x) = 0 for ny x {, b}), then there is number ξ {, b}), such tht R n f ;, b) R n g;, b) = f n+) ξ) g n+) ξ).
5 K. S. ANASTASIOU, A. I. KECHRINIOTIS AND B. A. KOTSOS Theorem. Let I R be closed intervl. Let, b, t be ny numbers in I such tht b >. Let f, g C n I). Suppose tht f, g re differentible of order n + on the interior I of I such tht g n+) > 0 or g n+) < 0 on I nd f n+) g n+) is bounded on I. Let w : [, b] R be n integrble nd positive vluted mpping. Then for ll x I, b), m where m := inf t I g holds for ll x I. wt) f t)dt n wt)gt)dt n f n+) n+), M := sup t I f i) x) g i) x) wt)t x) i dt M,.) wt)t x) i dt f n+) t) g n+). Specilly, if n is odd, then.) t) Proof. Let t be ny number in, b) with t = x. According to Theorem, there is one ξ {x, t}) such tht or R n f ; x, t) R n g; x, t) = f n+) ξ) g n+) ξ), f n+) t) inf t {x,t}) g n+) t) R n f ; x, t) R n g; x, t) sup f n+) t) t {x,t}) g n+) t), f nd since obviously m n+) y) inf y {x,t}) g n+) y) we get tht for ll t, x) x, b), sup f n+) y) y {x,t}) g n+) y) M, m R n f ; x, t) R n g; x, t) M..) Applying the Tylor-Lngrnge formul to g bout x, we hve tht for some σ {t, x}), R n g; x, t) = We distinguish two cses. t x)n+ g n+) σ)..3) FIRST CASE: n is odd. Then from.3), nd the ssumptions of this theorem, for ll x I {t}, follows signg n+) )R n g; x, t)wt) > 0..4)
OSTROWSKI S INEQUALITY 53 Combining.) with.4) we get m signg n+) )R n g; x, t)wt) signg n+) )R n f ; x, t)wt) M signg n+) )R n g; x, t)wt),.5) for ll x I {t}, nd since.5) is true t t = x, we hve tht.5) is true for ll t I. Now, integrting.5) over [, b], we obtin m signg n+) ) signg n+) ) wt)r n g; x, t)dt M signg n+) ) wt)r n f ; x, t)dt wt)r n g; x, t)dt..6) Integrting on both sides of.4) over [, b], we obtin signg n+) ) Consequently, from.6) we get or m m wt)r n g; x, t)dt > 0. wt)r n f ; x, t)dt M, wt)r n g; x, t)dt wt) f t) n wt)gt) n From.7) we get immeditely.). ) wt) t x)i f i) x)dt wt) t x)i g i) x)dt ) M..7) SECOND CASE: n is even. Then from.3), nd the ssumptions of this theorem we hve tht for ll x I, b), nd ll t, b), signt x) signg n+) )R n g; x, t)wt) > 0..8) Combining.) with.8), we obtin m signt x) signg n+) )R n g; x, t)wt) signt x) signg n+) )R n f ; x, t)wt) M signt x) signg n+) )R n g; x, t)wt)..9)
54 K. S. ANASTASIOU, A. I. KECHRINIOTIS AND B. A. KOTSOS Now from.9), working similrly s in the first cse, we obtin, tht for ll x I, b) the estimtion.) holds. Theorem 3. If f C n [, b]), f n+) is bounded on, b), then for ny x, b), it follows b x) f t)dt i+ x) i+ f i) x) i + )! b x)n+ + x ) n+ n + )! sup f n+) t),.0) t [,b] Specilly, if n is odd, then the inequlity.0) is shrp in the sense tht the constnt cnnot be replced by smller one. n+)! Proof. Let t be ny number in [, b] such tht t = x. Applying the Tylor- Lngrnge formul to f bout x we obtin f t) t x) i f i) x) = t x)n+ f n+) ξ), for some ξ min{t, x}, mx{t, x}), b). From this, we conclude, tht for ny t, b), f t) t x) i f i) t x) n+ x) sup f n+) t)..) t [,b] By integrting on both sides of.) over [, b], we get f t) t x) i f i) x) dt nd hence f t)dt f i) x) sup f n+) t) t [,b] t x) i dt sup f n+) t) t [,b] x x t) n+ dt + x t x) n+ dt, t x) n+ dt From this, we get immeditely the estimtion.0). For the shrpness of the inequlity.0) by n odd, let us choose f x) = x n+. ).
OSTROWSKI S INEQUALITY 55 Then we hve b x) n+ + x ) n+ n + )! = b x)n+ + x ) n+ n + ) sup f n+) t) t [,b] On the other hnd, we hve b x) f t)dt i+ x) i+ f i) x) i + )!..) = bn+ n+ b x) i+ x) i+ n+)n... n+ i)x n+ i n + i + )! = ) ) n + b x) b n+ i+ x n+ i n + i + i + )! ) ) n + x) n+ i+ x n+ i n + i + i + )! n+ ) n + b x) b n+ j x n+ j n + j j! j= n+ ) n + x) n+ i+ x n+ i n + j j! = i+= j j= = n + bn+ b x + x) n+ x n+ b x) n+ )) n + n+ x + x) n+ x n+ x) n+ )) = b x)n+ + x ) n+ n + )..3) From.) nd.3), follows f t)dt f i) x) b x)i+ x) i+ i + )! = b x)n+ + x ) n+ n + )! sup f n+) t)..4) t [,b] From.4), we conclude, tht the constnt is the best possible in n + )!.0).
56 K. S. ANASTASIOU, A. I. KECHRINIOTIS AND B. A. KOTSOS Remrk. Putting in.0) n = 0, we get f t) f t)dx b t ) + b t) f. b ) A simple clcultion yields t ) + b t) b ) = [ 4 ] +b t + ) b ) b ). Combining both reltions, the Ostrowski s inequlity.) is obtined. If n is odd, then the inequlity.0) cn be replced by better one: Theorem 4. Let n be n odd positive integer. Let f be mpping s in Theorem 3. Then b x) n+ + x ) n+ γ n+ n + )! f t)dt f i) x) b x)i+ x) i+ i + )! where γ n+ :=.5) is shrp. Γ n+ b x) n+ + x ) n+ n + )! inf f n+) x), Γ n+ := x,b).5) sup f n+) x). The inequlity x,b) Proof. We pply Theorem by choosing I = [, b], wt) =, gt) := t n+. Further, by pplying the Tylor-Lngrnge formul to g bout x, we get gt) t x) i g i) x) = t x) n+..6) Putting.6) in.) we obtin.5). Choosing f x) := x n+, the equlity in.5) holds, nd hence the shrpness of the inequlity is proved. In the following theorem n extension of Ostrowksi s inequlity.) is given. Theorem 5. Let f be differentible mpping with bounded f on the interior I of n intervl I R Then for ll, b I with b > nd ll x I, b),
OSTROWSKI S INEQUALITY 57 holds f x) b f t)dt b + x sup f x)..7) x I The inequlity.7) is shrp in the sense tht the constnt by smller one. cnnot be replced Proof. Applying Theorem for n = 0, wx) =, gx) = x, we obtin inf f x) x I f t)dt b ) f x) b xb ) sup x I f x)..8) From this esily, we get.7). The equlity in.7) is verified by ny first degree polynomil, nd hence the shrpness of inequlity is proved. Remrk. For ll x R, simple clcultion yields [ ] +b b + x x + ) 4 b ) b ), nd hence the Ostrowski s inequlity cn be extended in the following wy: Let f be differentible mpping with bounded f on the interior I of n open intervl I R Then for ll, b I with b > nd for ll x I, holds f x) b f t)dt [ 4 ] +b x + ) b ) b ) sup f x)..9) x I Since.8) holds, we conclude tht the inequlity.7) is better thn the restriction of inequlity.9) on I, b). 3. Generlized midpoint inequlities Now, we will prove two generlized midpoint integrl inequlities: Theorem 6. Let f C n [, b]), nd f n+) exists nd is bounded on, b). Then, if n is odd we hve γ n+ b ) n+ n+ n + )! f t)dt i even ) f i) + b b ) i+ i i + )! Γ n+ b ) n+ n+ n + )!, 3.)
58 K. S. ANASTASIOU, A. I. KECHRINIOTIS AND B. A. KOTSOS where γ n+ := inf f n+) x), Γ n+ := x,b) 3.) is shrp in the sense tht the constnt n+ n+)! smller one. sup f n+) x). The inequlity x,b) cnnot be replced by Proof. From.5) by x = +b, we get esily 3.). Similrly s in Theorem 4, we cn prove tht the equlity in 3.) holds by f x) := x n+, nd hence the shrpness of inequlity is verified. Remrk 3. For n =, from 3.) we get.3). Theorem 7. Let f be s in Theorem 6. If n is even, then holds ) f t)dt f i) + b b ) i+ i Γ n+ Γ n+ )b ) n+ i + )! n+. 3.) n + )! i even Proof. Let t be ny number in, b). Applying the Tylor-Lngrnge formul to f bout +b we R n f ; + b ) +b, t t = )n+ f n+) ξ), { for some ξ t, + b }). Tht is t +b )n+ inf f n+) x) R n f ; + b ) x {t +b }), t Moreover, we hve γ n+ inf x {t, +b If t > +b +b, then t γ n+ t +b )n+ }) f n+) x) sup +b t )n+ sup f n+) x). 3.3) x {t, +b }) x {t, +b }) f n+) x) Γ n+. 3.4) )n+ > 0, nd hence from 3.3), 3.4), we get R n f ; + b ), t t +b Γ )n+ n+. 3.5)
OSTROWSKI S INEQUALITY 59 Now, integrting 3.5) over [ +b, b], we obtin γ n+ t +b )n+ n + )! n+ +b f t)dt ) b ) i+ f i) +b i + )! i+ b ) n+ Γ n+. 3.6) n+ If t < +b +b, then since n is even we hve t )n+ < 0, nd hence from 3.3), 3.4) we result in: t +b Γ )n+ n+ R n f ; + b ), t t +b γ )n+ n+. 3.7) Now, integrting 3.7) over [, +b ], we obtin b ) n+ +b Γ n+ n + )! n+ Adding 3.6) nd 3.8), we obtin Γ n+ γ n+ )b ) n+ n + )! n+ tht is 3.). f t)dt ) i b ) i+ f i) +b i+ b ) n+ γ n+. 3.8) n+ f t)dt i even Γ n+ γ n+ )b ) n+ n + )! n+, Remrk 4. Applying the estimtion 3.) by n = 0 we obtin: ) b ) i+ f i) ) +b i + )! i Let f C[, b]) be mpping such tht f is bounded on, b). Then holds ) + b f f t)dt b Γ γ )b ), 8 where γ := inf f x), Γ := x,b) sup f x). Obviously, if f is convex x,b) then the inequlities.4), 3.9) re identicl. The dvntge by inequlity 3.9) is tht the ssumptions, f is twice differentible on, b), nd f L, b) re not necessry.
60 K. S. ANASTASIOU, A. I. KECHRINIOTIS AND B. A. KOTSOS References [] T. M. Apostol, Mthemticl Anlysis, nd edn., Addison-Wesley Publishing Compny, 6th printing, 98. [] P. Cerone, S. S. Drgomir nd J. Roumeliotis, An inequlity of Ostrowski type for mppings whose second derivtives belong to L, b) nd pplictions, RGMIA Reserch Report Collection, Vol. ) 998). [3] S. S. Drgomir, P. Cerone nd A. Sofo, Some remrks on the midpoint rule in numericl integrtion, RGMIA Reserch Report Collection, Vol. ) 998). [4] S. S. Drgomir nd S. Wng, Applictions of Ostrowski s inequlity to the estimtion of error bounds for some specil mens nd some numericl quntrture rules, Appl. Mth. Lett., Vol. 998), pp. 05 09. [5] D. S. Mitrinovic, J. E. Pecric nd A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic, Dortrecht, 993. [6] D. S. Mitrinovic, J. E. Pecric nd A. M. Fink, Inequlities for Functions nd their Integrls nd Derivtives, Kluwer Acdemic, Dortrecht, 994. Received Februry, 005 ; Revised July, 005