Filomt 30:3 06, 360 36 DOI 0.9/FIL6360Q Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://www.pmf.ni.c.rs/filomt A Compnion of Ostrowski Type Integrl Inequlity Using 5-Step Kernel with Some Applictions Ather Qyyum,, Muhmmd Shoi c, Irhim Fye Deprtment of Fundmentl nd Applied Sciences, Universiti Teknologi PETRONAS, 360 Bndr Seri Iskndr, Perk Drul Ridzun, Mlysi Deprtment of Mthemtics, University of H il, P.O. Box 0, H il 5, Sudi Ari c Higher Colleges of Technology Au Dhi Mens College, P.O. Box 5035, Au Dhi, United Ar Emirtes Astrct. The im of this pper is to estlish new version of Ostrowski s type integrl inequlity. The results re otined y using new type of kernel with five sections. Applictions to composite qudrture rule nd to Cumultive Distriutive Functions re considered.. Introduction In 93, Ostrowski 0] estlished the following interesting integrl inequlity. Theorem.. Let f :, ] R e continuous on, ] nd differentile on,, whose derivtive f :, R is ounded on,, i.e f = sup f t < t,] then f x f tdt + x + f. This inequlity hs powerful pplictions in numericl integrtion, proility nd optimiztion theory, sttistics, nd integrl opertor theory. The integrl inequlity tht estlishes connection etween the integrl of the product of two functions nd the product of the integrls is known in literture s Grüss inequlity 9], which is given elow. Theorem.. Let f, :, ] R e integrle functions such tht ϕ f x Φ nd γ x Γ, for some constnts ϕ, Φ, γ, Γ nd x, ]. Then 00 Mthemtics Suject Clssifiction. Primry 6D5; Secondry A55, A0, 65C50 Keywords. keywords, Ostrowski inequlity, Grüss nd Čeyŝev inequlities Received: 0 Octoer 0; Accepted: 0 Septemer 05 Communicted y Drgn S. Djordjević Emil ddresses: therqyyum@gmil.com Ather Qyyum, sfridi@gmil.com Muhmmd Shoi, irhim_fye@petrons.com.my Irhim Fye
A. Qyyum et l. / Filomt 30:3 06, 360 36 360 f x xdx f xdx. xdx Φ ϕγ γ. In 5], Guess nd Schmeisser proved the following Ostrowski s inequlity: Let f :, ] R stisfy the Lipschitz condition i.e, ] f t f s M t s. Then for ll x, +, we hve f x + x f tdt + x 3+ M. In, the point x = 3+ yields the following trpezoid type inequlity. f 3+ +3 f tdt M. Some generliztion of ostrowski type inequlities re lso done in ]-7]. In 3], Drgomir proved the inequlities for mppings of ounded vrition. In ], Brnett et. l proved some Ostrowski nd generlized trpezoid inequlity. Drgomir ] nd Liu 6] estlished some compnions of ostrowski type integrl inequlities. Alomri ] proved the following inequlity: Let f :, ] R e differentile mpping on,. If f L, ] nd γ f t Γ, for ll t, ], then the inequlity f x + x Γ γ. Recently Liu 7], used 3-step kernel to prove some ostrowski type inequlities. He hs demonstrted improvement in pproximtion errors. More recently Qyyum et. l ]-] proved some ostrowski type inequlities for L norm, L norm nd L p norm. In ll the references mentioned, uthors proved their results y using kernels with two or three sections. In this pper we introduce new kernel which hs five sections tht further generlize vrious results. By using this specil type of kernel, one cn otin different type of useful nd interesting results. We will derive our inequlities using Grüss inequlity, Cuchy inequlity nd Diz-Metclf inequlity. Finlly, some otined inequlities will then e pplied for qudrture formul nd for cumultive distriutive function.. Min Results Before we stte nd prove our min theorem, we need to prove the following lemm. Lemm.. Let us define the kernel Px, t s: Px, t = t, t, +x ], t 3+, t +x, x], t +, t x, + x], t +3 + x, t + x, ], t, t + x, ], 3
for ll x ], +, the following identity holds: A. Qyyum et l. / Filomt 30:3 06, 360 36 3603 Px, t f t dt = f x + x + x Proof. From 3, using integrtion y prts, we get required identity. We now give our min theorem. ] + x... When f L, ] :... Cse.: Theorem.. Let f :, ] R e differentile on,. If f L, ] nd γ f t Γ, for ll t, ], then the inequlity ] + x + x f x + x holds for ll x ], +. Proof. As we know tht for ll t, ] nd x ], +, we hve x 3 + Px, t x. Applying Grüss-Inequlity 9] to the mppings Px,. nd f., we otin Px, t f tdt Px, tdt for ll x ], +. It is strightforwrd exercise to show tht nd f tdt 6 Γ γ 5 6 Γ γ, 6 Px, tdt = 0 7 f tdt = f f. Hence using 6-, we get our required result 5. Our otined result 5, further generlizes the results given in ]-3], nd 7]. importnce of the ove otined result 5, we will now discuss some corollries. To emphsize the
A. Qyyum et l. / Filomt 30:3 06, 360 36 360 Corollry.3. Let f is defined s in Theorem., nd, dditionlly, if f x = f + x, then we hve ] + x + x f x 6 Γ γ, for ll x ], +. For instnce; choose x =, we hve 3 f 6 Γ γ nd choose x = +, we hve + f ] 3 + + 3 Corollry.. If we sustitute x =, in 5, we get f Corollry.5. If we sustitute x = +, in 5, we get ] + 3 + + 3 f Corollry.6. If we sustitute x = 3+, in 5, we get 3 + + 3 7 + + 7 f Corollry.7. If we sustitute x = +3, in 5, we get + 3 3 + 5 + 3 3 + 5 f 6 Γ γ. 6 Γ γ. 9 By using 3, we cn prove nother interesting theorem. ] ] 6 Γ γ. 0 6 Γ γ. 6 Γ γ.... Cse.: Theorem.. Let f : I R R e differentile mpping on I 0, the interior of the intervl I, nd let, I with <. If f L, ] nd γ f t Γ for ll x, ], then the following inequlity holds for ll x ], +. ] + x + x f x + x 3 3 + x + x x 5 ] Γ + γ.
A. Qyyum et l. / Filomt 30:3 06, 360 36 3605 Proof. Let then where C = Γ + γ Px, t f t dt C Px, tdt = Px, t f t C ] dt = ] + x + x f x + x Px, tdt = 0. On the other hnd, we hve Px, t f t C ] dt.mx f t C Px, t dt. t,] Since mx f t C Γ + γ t,] 5 nd Px, t dt = From -6, we get 3. Corollry.9. If we sustitute x =, in 3, we get ] f x + x x + + x 3 +. 6 + ] Γ + γ. 7 Corollry.0. If we sustitute x = +, in 3, we get ] + 3 + + 3 f + ] Γ + γ. 6
A. Qyyum et l. / Filomt 30:3 06, 360 36 3606 Corollry.. If we sustitute x = 3+, in 3, we get ] 3 + + 3 7 + + 7 f 3 + + ] Γ + γ. Corollry.. If we sustitute x = +3, in 3, we get ] + 3 3 + 5 + 3 3 + 5 f + + + 3 ] Γ + γ. 6 By using 3, we cn prove nother interesting theorem. 9 0..3. Cse.c: Theorem.3. Let f :, ] R e differentile mpping in,. If f L, ] nd γ f x Γ, then we hve nd + x f x + x ] + x ] + x + x f x + x for ll x ; + ], where f f Ω = mx Px, t, S =, γ = inf t,] f t, Γ = sup f t. t,] t,] Proof. As we know tht We denote Px, t f tdt = + x f x + x R n x = Px, t f tdt Px, tdt. If C R is n ritrry constnt, then we hve R n x = f t C Px, t Ω. S γ Ω. Γ S, f tdt 3 ] + x Px, tdt. f tdt. Px, sds dt. 5
Since Px, t Furthermore, we hve R n x mx t,] nd A. Qyyum et l. / Filomt 30:3 06, 360 36 3607 Px, sdt dt = 0. Px, t 0 f t C dt mx Px, t = Ω. t,] 6 We lso hve ] f t γ dt = S γ. 7 f t Γ dt = Γ S. By using 7,, 3, 6, 7 nd, we get nd... Cse.: When f L, ] Theorem.. Let f :, ] R e n solutely continuous mpping in, with f L, ]. Then, we hve ] + x + x f x + x 9 σ f 3 + + 3 3x x + 30x ] for ll x ], +, where σ f = f f f = f S. Proof. Let R n s defined s in. Then from 3, we get R n x = ] + x + x f x + x If we choose C = f s ds
A. Qyyum et l. / Filomt 30:3 06, 360 36 360 in 5 nd using the cuchy inequlity, we get R n x f t f s ds Px, t Px, sds dt f t f s ds dt Px, t Px, sds dt The shrpness of the constnt 6]. σ f 3 + + 3 3x x + 30x ]. in 9 cn e otined for x = or x = + which is lredy proven in Corollry.5. If we sustitute x =, in 9, we get f σ f + ] 30 Corollry.6. If we sustitute x = +, in 9, we get ] + 3 + + 3 f Corollry.7. If we sustitute x = 3+, in 9, we get 3 + + 3 7 + + 7 f Corollry.. If we sustitute x = +3, in 9, we get + 3 3 + 5 + 3 3 + 5 f We cn stte ostrowski inequlity in n other wy lso:.3. Cse.3: When f L, ] 0 + σ f 3. 3 ] ] 5 σ f. 3 6 σ f. 33 6 Theorem.9. Let f :, ] R e twice continuously differentile mpping in, with f L, ]. Then ] + x + x f x + x 3 3 + + 3 3x x + 30x ] 3 f, π for ll x ], +.
Proof. Let R n x e defined y. From 3, A. Qyyum et l. / Filomt 30:3 06, 360 36 3609 R n x = ] + x + x f x + x If we choose C = f + in 5 nd use the Cuchy Inequlity, we get R n x + f t f Px, t + f t f dt We cn use the Diz-Metclf inequlity ] or ], to get + f t f dt f π. We lso hve Px, t Px, sds dt = Therefore, using the ove reltions, we otin 3. Corollry.0. If we sustitute x =, in 3, we get f = Px, sds dt. Px, t Px, sds dt Px, t dt 35 3 + + 3 3x x + 30x. + ]] 3 Corollry.. If we sustitute x = +, in 3, we get ] + 3 + + 3 f π. f. 36 3 π f. 37 Corollry.. If we sustitute x = 3+, in 3, we get 3 + + 3 7 + + 7 f π 5 + 5 6 f. 6 ] 3
A. Qyyum et l. / Filomt 30:3 06, 360 36 360 Corollry.3. If we sustitute x = +3, in 3, we get ] + 3 3 + 5 + 3 3 + 5 f π 6 f. 39 3. An ppliction to Composite Qudrture Rules Let I n : = x 0 < x < x <... < x n < x n = e division of the intervl, ], ξ i, + ] i = 0,,..., n ; sequence of intermedite points h i = + i = 0,,...n. We hve the following qudrture formul: Theorem 3.. Let f : I R R e differentile mpping on I 0, the interior of the intervl I, nd let, I with <. If f L, ] nd γ f t Γ for ll x, ], then we hve the following qudrture formul: f tdt = A f, I n + R f, I n, where A f, I n = n 3xi + + h i f nd reminder stisfies the estimtion R f, I n 6 n Γ γ h i, xi + 3+ for ll ξ i, + ], where h i := +, i =,,...n. 7xi + + ] xi + 7+ Proof. Apply on the intervl, + ], ξ i, + ], where h i := + i =,,...n, to get R f, I n = + f tdt n h i f 3xi ++ Summing over i from 0 to n, we get R f, I n = = x n i+ f tdt f tdt n n From, it follows tht R f, I n = f tdt n 6 h i Γ γ. which completes the proof. xi +3+ h i f 3xi ++ h i f 3xi ++ h i f 3xi ++ 7xi ++ xi +3+ xi +3+ xi +3+ ] xi +7+. 7xi ++ 7xi ++ 7xi ++ ] xi +7+ ] xi +7+. xi +7+ ] 0
A. Qyyum et l. / Filomt 30:3 06, 360 36 36 Theorem 3.. Let h i = + = h = n i = 0,,..., n nd let f :, ] R e n solutely continuous mpping in, with f L, ]. Then we hve f xdx = A f, I n + R f, I n, nd reminder stisfies the estimtion R f, I n 3 σ f Proof. Applying 3 to the intervl, + ], then we get ] h xi + + 3xi, + xi, 3+ f h 3 0x i + + x i+ + + f t f xi+ f dt h for i = 0,,..., n. Now summing over i from 0 to n, using the tringle inequlity nd Cuchy inequlity twice, we get h n ] xi + + 3xi, + xi, 3x i+ f h n + 0x i + + x i t f xi+ f dt 3 h h 3 h 3 = n f n n f xi+ f n 0x i + + x i+ f 0x x 3 i i + + xi+ σ f. f f Theorem 3.3. Let h i = + = h = n i = 0,,,, n nd let f : ; ] R e twice continuously differentile mpping in, with f L, ]. Then we hve f xdx = A f, I n + R f, I n, where the reminder stisfies the estimtion R f, I n 6 3 πn 5 f.
Proof. Applying 39 to the intervl, + ], we get h f xi + 3+ 3xi + + A. Qyyum et l. / Filomt 30:3 06, 360 36 36 5xi + 3+ 3xi + 5+ ] + π 6 h3 + f t dt for i = 0,,..., n. Now summing over i from 0 to n, using the tringle inequlity nd Cuchy inequlity twice, we get h n f π xi + 3+ 6 h3 n π = 3xi + + x n i+ f t dt 6 h3 3 6 πn 5 x n i+ f t dt f. 5xi + 3+ 3xi + 5+ ],. An Appliction to Cumultive Distriution Function Let X e rndom vrile tking vlues in the finite intervl, ] with the proility density function f :, ] 0, ] nd cumultive distriutive function F x = Pr X x = x. 3 F = Pr X = f u du =. Theorem.. With the ssumptions of Theorem., we hve the following inequlity which holds ] + x + x F x + F + x + F + F E X 6 Γ γ, 5 for ll x ], +. Where E X is the expecttion of X. Proof. In the proof of Theorem., let f = F nd using the fct tht E X = tdf t = F t dt. Further detils re left to the interested reders.
A. Qyyum et l. / Filomt 30:3 06, 360 36 363 Theorem.. With the ssumptions of Theorem., we hve the following inequlity which holds ] + x + x F x + F + x + F + F E X x + x x + + x 3 + Γ + γ, 6 for ll x ], +, where E X is the expecttion of X. Proof. Applying 3 nd on 3 nd using the sme conditions tht we used in Theorem., we get the required inequlity. Corollry.3. Under the ssumptions of Theorem., if we put x = in 5, then we get F + F E X 6 Γ γ. Corollry.. Under the ssumptions of Theorem., if we put x = + in 5, then we get ] + 3 + + 3 F + F + F E X 6 Γ γ. Corollry.5. Under the ssumptions of Theorem., if we put x = 3+ in 5, then we get ] 3 + + 3 7 + + 7 F + F + F + F E X 6 Γ γ. References ] M.W. Alomri, A compnion of Ostrowski s inequlity with pplictions, Trnsylvnin Journl of Mthemtics nd Mechnics, 3 0, no., 9. ] N. S. Brnett, S. S. Drgomir nd I. Gomm, A compnion for the Ostrowski nd the generlised trpezoid inequlities, Mthemticl nd Computer Modelling, 50 009, 79-7. 3] S. S. Drgomir, A compnion of Ostrowski s inequlity for functions of ounded vrition nd pplictions, RGMIA Preprint, Vol. 5 Supp. 00 rticle No.. ] S. S. Drgomir, Some compnions of Ostrowski s inequlity for solutely continuous functions nd pplictions, Bulletin of the Koren Mthemticl Society., 005, No., 3-30. 5] A. Guess nd G. Schmeisser, Shrp integrl inequlities of the Hermite-Hdmrd type, Journl of Approximtion Theory, 5 00, no., 60. 6] Z. Liu, Some compnions of n ostrowski type inequlity nd pplictions, journl of inequlities in pure nd pplied mthemtics, vol. 0, iss., rt. 5, 009. 7] W. Liu, New Bounds for the Compnion of Ostrowski s Inequlity nd Applictions, Filomt : 0, 67 7. ] D. S. Mitrinović, J. E. Pecrić nd A. M. Fink, Inequlities involving functions nd their integrls nd derivtives, Mthemtics nd its Applictions Est Europen Series, 53, Kluwer Acd. Pul., Dordrecht, 99. 9] D. S. Mitrinović, J. E. Pecrić nd A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Pulishers, Dordrecht, 993. 0] A. Ostrowski, Üer die Asolutweichung einer differentienren Funktionen von ihren Integrlimittelwert, Comment. Mth. Hel. 0 93, 6-7. ] N. Ujević, New ounds for the first inequlity of Ostrowski-Gruss type nd pplictions, Computers nd Mthemtics with Applictions, 6 003, no. -3, 7. ] G. A. Anstssio, Frctionl Representtion Formule Under Initil Conditions nd Frctionl Ostrowski Type Inequlities, Demonstrtio Mthemtic, 05, no. 3, 357 37. 3] W. Liu, A. Tun, Dimond-lph weighted Ostrowski nd Gruss type inequlities on time scles, Applied Mthemtics nd Computtion, 70 05 5-60. ] P. Cerone, S. S. Drgomir, E. Kikinty, Jensen Ostrowski type inequlities nd pplictions for f-divergence mesures, Applied Mthemtics nd Computtion, 66 05, 30 35. 5] W. Liu, Ostrowski type frctionl integrl inequlities for MT-convex functions, Miskolc Mthemticl Notes, 6 05, 9-56.
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