Revista Colombiana de Matemáticas Volumen 41 (2007), páginas 1 13
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1 Revist Colombin de Mtemátics Volumen 4 7, págins 3 Ostrowski, Grüss, Čebyšev type inequlities for functions whose second derivtives belong to Lp,b nd whose modulus of second derivtives re convex Arif Rfiq COMSATS Institute of Informtion Technology, Pkistná Frooq Ahmd Bhuddin Zkriy University, Pkistná Abstrct. Ostrowski, Grüss, Čebyšev type inequlities involving functions whose second derivtives belong to L p, b nd whose modulus of second derivtives re convex re estblished. The results provide better bounds thn those currently vilble in the literture. Keywords. Ostrowski Grüss-Čebyšev inequlities, modulus of second derivtive convex, convex function. Mthemtics Subject Clssifiction. Primry: 65C. Secondry: 65A. Resumen. Se estblecen desigulddes de tipo Ostrowski, Grüss, Čebyšev que comprenden funciones cuys segunds derivds pertenecen L p, b y cuyos módulos de segunds derivds son convexos. Los resultdos obtenidos proporcionn mejores cots que ls ctulmente disponibles en l litertur.. Introduction In 938, A. M. Ostrowski [6] proved the following: Theorem.. Let f : [, b] R be continuous on [, b] nd differentible on, b whose derivtive f :, b R is bounded on, b i.e., f x M <
2 ARIF RAFIQ & FAROOQ AHMAD, then f x b for ll x [, b], where M is constnt. f t dt x b 4 b M,. b For two bsolutely continuous functions f, g : [, b] R, consider the functionl, T f, g = f x g x dx f x dx g x dx, b b b provided, the involved integrls exist. In 88, P. L. Čebyšev [7] proved tht, if f, g L [, b], then, In 934, G. Grüss [7] showed tht. T f, g b f g..3 T f, g M m N n,.4 4 provided m, M, n nd N re rel numbers stisfying the conditions, < m f x M <, < n g x N <, for ll x [, b]. Pchptte in [] proved the following results. Theorem.. Let f : [, b] R be bsolutely continuous on [, b]. If f, g re convex on [, b] nd f, g L [, b], then, S f, g [ gx f x f fx g x g ] [ x b ] 4 b b 4,.5 for ll x [, b]. Corollry.. Under the ssumptions of theorem., we hve the mid point inequlity, [ b b b S M f, g g f f 6 ] b b f g g..6
3 OSTROWSKI, GRÜSS, ČEBYŠEV TYPE INEQUALITIES... 3 Theorem.3. Let f : [, b] R be bsolutely continuous on [, b]. If f, g re convex on [, b] nd f, g L [, b], then, T f, g for ll x [, b], where Ex = x b x. b 4 b [ gx f x f fx g x g ] Exdx,.7 Corollry.. Under the ssumptions of theorem.3, we hve the mid point inequlity, [ T M f, g b b b 6 g f f ] b b f g g..8 Theorem.4. Let f : [, b] R be bsolutely continuous on [, b]. If f, g re convex on [, b] nd f, g L [, b], then, T f, g 4 b 3 for ll x [, b], where Ex = x b x. [ f x f g x g ] E xdx,.9 Corollry.3. Under the ssumptions of theorem.4, we hve the mid point inequlity [ b b T M f, g f f 64 ] b g g.. During the pst few yers, mny reserchers hve given considerble ttention to the bove inequlities nd vrious generliztions, extensions nd vrints of them hve ppered in the literture, see [ ], nd the references cited therein. Motivted by results given in [8 ], we estblish here some inequlities similr to those given by Ostrowski, Grüss nd Čebyšev involving functions whose derivtives belong to L p, b spce nd whose modulus of second derivtives re convex. The nlysis used in the proofs is elementry nd bsed on integrl identities proved in [ ].
4 4 ARIF RAFIQ & FAROOQ AHMAD. Sttement of results Let I be suitble intervl of the rel line R. A function f : I R is clled convex if fλx λy λfx λfy, for ll x, y I nd λ [, ] see []. We need the following identities proved by Mir et l. in [5]: f x = ftdt x b f x b x t λ f λ x λt dλ dt, b for ll x [, b], where f : I R is n bsolutely continuous function on [, b] nd λ [, ]. We use the following nottion to simplify the detils of presenttion, S f, g = fxgx fx b x b gtdt gx fxg x gxf x. ftdt At the mid-point we denote this by S M f, g, noting tht the lst term on the RHS vnishes. T f, g = Sf, gdx = fxgxdx b b b b b = b b fxdx x b fxgxdx gtdt gxdx ftdt fxg x gxf x dx b t the mid-point we denote this by T M f, g. fxdx b x b fxg x gxf x dx, gxdx
5 OSTROWSKI, GRÜSS, ČEBYŠEV TYPE INEQUALITIES... 5 nd S f, g = fxgx x b fx b fxg x gxf x gtdt gx x b f xg x x b f x b b b ftdt b ftdt gtdt g x b b gtdt, ftdt T f, g = Sf, gdx = fxgxdx b b b x b fxg x gxf x dx b b f x x b gxdx gtdt g x f xg xdx b b 3 ftdt dx, b x b nd t the mid-point we denote this by T f, g. We lso use [ ] /q b x q x q Qx =. q fxdx We define. p s the usul Lebesgue norm on L p [, b],; in other words, h p /p = dt ht p for h Lp [, b] nd p, q >, p q =.
6 6 ARIF RAFIQ & FAROOQ AHMAD The following theorem dels with Ostrowski type inequlities involving two functions. Theorem.. Let f : [, b] R be bsolutely continuous on [, b]. If f, g re convex on [, b] nd f, g L p [, b], then, S f, g Qx [ b /p fx g x gx f x b ] gx f p fx g p,. for ll x [, b]. Proof. From the hypothesis of theorem., the following identities hold: f x = ftdt x b f x b x t λ f λ x λt dλ dt,. b g x = gtdt b b x t x b g x λ g λ x λt dλ dt,.3 for ll x [, b]. Multiplying both sides of. nd.3 by gx nd fx respectively, dding the resulting identities nd rewriting, we hve, b Sf, g = gx x t λ f λ x λt dλ dt b fx x t λ g λ x λt dλ dt..4 Since f, g re convex on [, b] then, from.4, nd using properties of modulus, we hve,
7 OSTROWSKI, GRÜSS, ČEBYŠEV TYPE INEQUALITIES... 7 Sf, g b gx x t f x fx b fx = Qx b x t g x gx λ dλ f t λ dλ g t x t f x f t dt x t [ g x g t ] dt Qx [ gx f x b /p f b p ] fx g x b /p g p [ b /p fx g x gx f x gx f p fx g p ]. λ λ dλ dt λ λ dλ dt We therefore hve the desired inequlity.. Corollry.. Under the ssumptions of theorem., we hve the mid point inequlity, S M f, g [ b q b q q g f p b f g p b /p b b ] f g b b g f..5 Remrk.. As we know tht in the bove inequlity q > nd so q q >, then clerly bounds obtined in.5 re t lest 9 times better thn the bounds obtined in.6. The Grüss type inequlities re embodied in the following theorem.
8 8 ARIF RAFIQ & FAROOQ AHMAD Theorem.. Let f : [, b] R be bsolutely continuous on [, b]. If f, g re convex on [, b] nd f, g L p [, b], then, T f, g b for ll x [, b]. [ b /p fx g x gx f x gx f p fx g p ] Qxdx,.6 Proof. From the proof of theorem., we hve, Sf, g = b gx fx x t x t λ f λ x λt dλ dt λ g λ x λt dλ dt..7 Integrting.7 with respect to x over [, b] nd dividing by b, we get, T f, g = b b gx fx x t x t λ f λ x λt dλ dt λ g λ x λt dλ dt dx. Since f, g re convex on [, b] nd using the properties of modulus, we hve, T f, g b b gx f t fx λ λ dλ dt x t f x x t g x λ dλ λ dλ g t λ λ dλ dt dx
9 OSTROWSKI, GRÜSS, ČEBYŠEV TYPE INEQUALITIES... 9 b b gx f x x t q dt fx g x /q x t q dt x t q dt /p f t p dt x t q dt /q /q g t p dt p dt /p dx /q = [ b b /p fx g x gx f x ] gx f p fx g p Qxdx. /p p dt /p Hence we get desired inequlity.6. Corollry.. Under the ssumptions of theorem., we hve the mid point inequlity, T M f, g [ b q b q q g f p b f g p b /p b b ] f g b b g f..8 Remrk.. As we know tht in the bove inequlity q > nd so q q >, then clerly bounds obtined in.8 re t lest 9 times better thn the bounds obtined in.8. The next theorem contins Čebyšev type inequlities.
10 ARIF RAFIQ & FAROOQ AHMAD Theorem.3. Let f : [, b] R be bsolutely continuous on [, b]. If f, g re convex on [, b] nd f, g L p [, b], then, T f, g 36 b 3 {[ ] gx f x b /p f p [ ]} fx g x b /p g p Q xdx.9 for ll x [, b]. Proof. From the hypothesis of theorem.3 the identities. nd.3 hold. Multiplying both sides of these by ech other, we hve: implying f x b g x b = b b ftdt x b f x x t fxgx x b gtdt x b x t g x λ f λ x λt dλ dt λ g λ x λt dλ dt, [fxg x gxf x] fx gtdt gx b x b f x b b ftdt b ftdt gtdt g x b gtdt x b f xg x ftdt
11 OSTROWSKI, GRÜSS, ČEBYŠEV TYPE INEQUALITIES... = x t b x t b which gives Sf, g = x t b x t b nd, consequently, we obtin T f, g b b 3 x t = 36 b 3 [ fx λ f λ x λt dλ dt x t λ g λ x λt dλ dt λ f λ x λt dλ dt λ g λ x λt dλ dt. λ f x λ λ f t dλ dt [ ] λ g x λ λ g t {[ ] gx f x b /p f p g x b /p g p ]} Q xdx. dλ dt dx This completes the proof. Corollry.3. Under the ssumptions of theorem.3, we hve the mid-point inequlity,
12 ARIF RAFIQ & FAROOQ AHMAD b 4q T M f, g 36 4q q [ b /p b b f g b g b g f p b f b f g p ].. Remrk.3. As we know tht in the bove inequlity q > nd so 4q q > 44, then clerly bounds obtined in. re t lest 8 times better thn the bounds obtined in.. References [] N. S. Brnett, P. Cerone, S. S. Drgomir, M. R. Pinheiro & A. Sofo, Ostrowski type inequlities for functions whose modulus of derivtives re convex nd pplictions, RGMIA Res. Rep. Collec. 5, 9 3. [] P. Cerone & S. S. Drgomir, Ostrowski type inequlities for functions whose derivtives stisfy certin convexity ssumptions, Demonstrtio Mth. 37 4, [3] S. S. Drgomir & Th. M. Rssis Eds., Ostrowski Type Inequlities nd Applictions in Numericl Integrtion, Kluwer Acdemic Publishers, Dordrect,. [4] S. S. Drgomir & A. Sofo, Ostrowski type inequlities for functions whose derivtives re convex, Proceeding of the 4th Interntionl Conference on Modelling nd Simultion, November. Victori University, Melbourne Austrsli. RGMIA Res. Rep. Collec. 5 Supp., Art. 3. [5] N. A. Mir, A. Rfiq & M. Rizwn, Ostrowski Grüss Čebyšev type inequlities for functions whose modulus of second derivtives re convex, submitted. [6] D. S. Mitrinovic, J. E. Pecric & A. M. Fink, Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluver Acdemic Publishers, Dordrecht, 99. [7] D. S. Mitrinovic, J. E. Pecric & A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, Dordrect, 993. [8] B. G. Pchptte, A note on integrl inequlities involving two log-convex functions, Mth. Inequl. Appl , [9] B. G. Pchptte, A note on Z Hdmrd type integrl inequlities involving severl log-convex functions, Tmkng J. Mth. 36 5, [] B. G. Pchptte, Mthemticl Inequlities, North-Hollnd Mthemticl Librry, Vol. 67 Elsvier, 5. [] B. G. Pchptte, On Ostrowski-Grüss-Čebyšev type inequlities for functions whose modulus of derivtives re convex, JIPAM 6 5 4, 4. [] J. E. Pecric, F. Proschn & Y. L. Tng, Convex Functions, Prtil Orderings nd Sttisticl Applictions, Acdemicx Press, New York, 99.
13 OSTROWSKI, GRÜSS, ČEBYŠEV TYPE INEQUALITIES... 3 Recibido en gosto de 6. Aceptdo en mrzo de 7 Deprtment of Mthemtics COMSATS Institute of Informtion Technology, Plot 3, H-8/ Islmbd 44, Pkistn e-mil: rfiq@comsts.edu.pk Centre for Advnced Studies in Pure nd Applied Mthemtics Bhuddin Zkriy University Multn 68, Pkistn e-mil: frooqgujr@gmil.com
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