Research Article On New Inequalities via Riemann-Liouville Fractional Integration
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1 Abstrct nd Applied Anlysis Volume 202, Article ID , 0 pges doi:0.55/202/ Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment of Mthemtics, Fculty of Science nd Arts, Düzce University, Düzce, Turkey 2 Deprtment of Mthemtics, Fculty of Science nd Arts, Afyon Koctepe University, Afyon, Turkey Correspondence should be ddressed to Mehmet Zeki Sriky, srikymz@gmil.com Received 9 August 202; Accepted 6 October 202 Acdemic Editor: Ciprin A. Tudor Copyright q 202 M. Z. Sriky nd H. Ogunmez. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. We etend the Montgomery identities for the Riemnn-Liouville frctionl integrls. We lso use these Montgomery identities to estblish some new integrl inequlities. Finlly, we develop some integrl inequlities for the frctionl integrl using differentible conve functions.. Introduction The inequlity of Ostrowski gives us n estimte for the devition of the vlues of smooth function from its men vlue. More precisely, if f :, b R is differentible function with bounded derivtive, then f ftdt [ b/22 f 4 2,. for every, b. Moreover, the constnt /4 is the best possible. For some generliztions of this clssic fct see 2, pges by Mitrinović et l. A simple proof of this fct cn be done by using the following identity 2. If f :, b R is differentible on, b with the first derivtive f integrble on, b, then Montgomery identity holds f ftdt P, tf tdt,.2
2 2 Abstrct nd Applied Anlysis where P, t is the Peno kernel defined by t, t<, P, t : t b, t b..3 Recently, severl generliztions of the Ostrowski integrl inequlity re considered by mny uthors; for instnce, covering the following concepts: functions of bounded vrition, Lipschitzin, monotonic, bsolutely continuous, nd n-times differentible mppings with error estimtes with some specil mens together with some numericl qudrture rules. For recent results nd generliztions concerning Ostrowski s inequlity, we refer the reder to the recent ppers 3 0. In this pper, we etend the Montgomery identities for the Riemnn-Liouville frctionl integrls. We lso use these Montgomery identities to estblish some new integrl inequlities of Ostrowski s type. Finlly, we develop some integrl inequlities for the frctionl integrl using differentible conve functions. Lter, we develop some integrl inequlities for the frctionl integrl using differentible conve functions. From our results, the weighted nd the clssicl Ostrowski s inequlities cn be deduced s some specil cses. 2. Frctionl Clculus Firstly, we give some necessry definitions nd mthemticl preliminries of frctionl clculus theory which re used further in this pper. For more detils, one cn consult, 2. Definition 2.. The Riemnn-Liouville frctionl integrl opertor of order α 0with 0is defined s J α f Γα J 0 f f. t α ftdt, 2. Recently, mny uthors hve studied number of inequlities by using the Riemnn- Liouville frctionl integrls, see 3 6 nd the references cited therein. 3. Min Results In order to prove some of our results, by using different method of proof, we give the following identities, which re proved in 3. Lter, we will generlize the Montgomery identities in the net theorem. Lemm 3.. Let f : I R R be differentible function on I with, b I < b nd f L, b,then f Γα b α J α fb J α ( P2, bfb ) J α ( P2, bf b ), α, 3.
3 Abstrct nd Applied Anlysis 3 where P 2, t is the frctionl Peno kernel defined by t b α Γα, t<, P 2, t : t b b α Γα, t b. 3.2 Proof. By definition of P, t, we hve ΓαJ α ( P, bf b ) b t α P, tf tdt ( ) t b t α f tdt [ b t α t f tdt ( ) t b b t α f tdt b t α f tdt 3.3 I I 2. Integrting by prts, we cn stte I b t α t ft [ α b t α 2 t b t α ftdt b α f α b t α 2 t ftdt b t α ftdt, 3.4 nd similrly, I 2 b t α ft b α b t α ftdt b α f α b t α ftdt. 3.5 Adding 3.4 nd 3.5, weget ΓαJ α ( P, bf b ) { b α f α b t α 2 t ftdt } α b t α ftdt b t α ftdt. 3.6
4 4 Abstrct nd Applied Anlysis If we dd nd subtrct the integrl α b tα 2 t bftdt to the right-hnd side of the eqution bove, then we hve ΓαJ α ( P, bf b ) { b α f α b t α ftdt b α f α b t α 2 P, tftdt } b t α 2 P, tftdt 3.7 b t α ftdt b α f Γα Jα fb ΓαJ α ( P, bfb ). Multiplying both sides by b α,weobtin J α ( P2, bf b ) f Γα b α J α fb J α ( P2, bfb ), 3.8 nd so f Γα b α J α fb J α ( P2, bfb ) J α ( P2, bf b ). 3.9 This completes the proof. Now, we etend Lemm 3. s follows. Theorem 3.2. Let f : I R R be differentible function on I with f L, b, then the following identity holds: 2λf Γα ( ) α b α J α fb λ f b J α ( P3, bfb ) J α ( P3, bf b ), α, 3.0
5 Abstrct nd Applied Anlysis 5 where P 3, t is the frctionl Peno kernel defined by t λ λb b α Γα, t<, P 3, t : t λb λ b α Γα, t b, 3. for 0 λ. Proof. By similr wy in proof of Lemm 3., we hve ΓαJ α ( P3, bf b ) b t α P 3, tf tdt [ Γαb α b t α t λ λbf tdt b t α t λb λf tdt Γαb α J J Integrting by prts, we cn stte J b α λ λbf α f α b t α 2 t λ λbftdt b t α ftdt, 3.3 nd similrly, J 2 b α λb λf α b t α 2 t λ λbftdt b t α ftdt. 3.4 Thus, by using J nd J 2 in 3.2,weget3.0 which completes the proof. Remrk 3.3. We note tht in the specil cses, if we tke λ 0inTheorem 3.2, then we get 3. with the kernel P 2, t.
6 6 Abstrct nd Applied Anlysis Theorem 3.4. Let f :, b R be differentible on, b such tht f L, b, where<b. If f M for every, b nd α, then the following inequlity holds: ( ) Γα α 2λf b α J α fb λ f J α ( P3, bfb ) b { M α b α[ 2λ α 2 λ α λ αα [ b 2α b } α. 3.5 Proof. From Theorem 3.2, weget ( ) Γα α 2λf b α J α fb λ f J α ( P3, bfb ) b b t α P 3, t f t dt Γα [ b α b t α t λ λb f t dt b t α t λb λ f t dt { Mb α b t α t λ λb dt } b t α t λb λ dt Mb α {J 3 J 4 }. 3.6 By simple computtion, we obtin J 3 b t α t λ λb dt λb λ b t α λb λ tdt b t α t λb λdt λb λ b [ α 2 λ α b α λ αb λα, αα αα 3.7
7 Abstrct nd Applied Anlysis 7 nd similrly J 4 b t α t λb λ dt λ λb b t α λ λb tdt b t α t λ λbdt λ λb 3.8 2λα α αα b α αb λα. αα By using J 3 nd J 4 in 3.6,weobtin3.5. Remrk 3.5. If we tke λ 0inTheorem 3.4, then it reduces Theorem 4. proved by Anstssiou et l. 3. So, our results re generliztions of the corresponding results of Anstssiou et l. 3. Theorem 3.6. Let f :, b R be differentible conve function on, b nd f L, b. Then for ny, b, the following inequlity holds: [ ( ) b 2 α αα f α b α b 2 α α b f Γα b α J α fb J α ( P2, bfb ) f, α. Proof. Similrly to the proof of Lemm 3., we hve 3.9 f Γα b α J α fb J α ( P2, bfb ) [ b α b t α t f tdt b t α f tdt Since f is conve, then for ny, b we hve the following inequlities: f t f for.e. t,, 3.2 f t f for.e. t, b If we multiply 3.2 by b t α t 0, t,, α nd integrte on,,weget b t α t f tdt b t α t f dt [ α b α αb α f αα, 3.23
8 8 Abstrct nd Applied Anlysis nd if we multiply 3.22 by b t α 0, t, b, α nd integrte on, b, welsoget b t α f tdt b t α f dt b α f α Finlly, if we subtrct 3.24 from 3.23 nd use the representtion 3.20 we deduce the desired inequlity 3.9. Corollry 3.7. Under the ssumptions Theorem 3.6 with α, one hs [ b 2 f 2 2 f ftdt f The proof of Corollry 3.7 is proved by Drgomir in 6. Hence, our results in Theorem 3.6 re generliztions of the corresponding results of Drgomir 6. Remrk 3.8. If we tke b/2incorollry 3.7, weget 0 [ ( ) ( ) b b f f ( ) b ftdt f Theorem 3.9. Let f :, b R be differentible conve function on, b nd f L, b. Then for ny, b, the following inequlity holds: Γα b α J α fb J α ( P2, bfb ) f [ αα α b 2 f b ( α b α α b 2 α b ), α Proof. Assume tht f nd f b re finite. Since f is conve on, b, then we hve the following inequlities: f t f for.e. t,, 3.28 f t f b for.e. t, b. 3.29
9 Abstrct nd Applied Anlysis 9 If we multiply 3.28 by b t α t 0, t,, α nd integrte on,, we hve b t α t f tdt b t α t f dt [ α b α αb α f αα, 3.30 nd if we multiply 3.29 by b t α 0, t, b, α nd integrte on, b, we lso hve b t α f tdt b t α f bdt b α f α b. 3.3 Finlly, if we subtrct 3.30 from 3.3 nd use the representttion 3.20 we deduce the desired inequlity Corollry 3.0. Under the ssumptions Theorem 3.9 with α, one ftdt f 2 [ b 2 f b 2 f The proof of Corollry 3.0 is proved by Drgomir in 6. So, our results in Theorem 3.9 re generliztions of the corresponding results of Drgomir 6. Remrk 3.. If we tke b/2incorollry 3.0, weget 0 ( ) b ftdt f 2 [ f 8 b f References A. M. Ostrowski, Über die bsolutbweichung einer differentiebren funktion von ihrem integrlmitelwert, Commentrii Mthemtici Helvetici, vol. 0, pp , D. S. Mitrinović, J. E. Pečrić, nd A. M. Fink, Inequlities Involving Functions nd Their Integrls nd Derivtives, vol. 53, Kluwer Acdemic Publishers, Dordrecht, The Netherlnds, P. Cerone nd S. S. Drgomir, Trpezoidl-type rules from n inequlities point of view, in Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, pp , Chpmn & Hll/CRC, Boc Rton, Fl, USA, J. Duondikoete, A unified pproch to severl inequlities involving functions nd derivtives, Czechoslovk Mthemticl Journl, vol. 5, no. 26, pp , S. S. Drgomir nd N. S. Brnett, An Ostrowski type inequlity for mppings whose second derivtives re bounded nd pplictions, RGMIA Reserch Report Collection, vol., pp , S. S. Drgomir, An Ostrowski type inequlity for conve functions, Univerzitet u Beogrdu. Publikcije Elektrotehničkog Fkultet. Serij Mtemtik, vol. 6, pp. 2 25, Z. Liu, Some compnions of n Ostrowski type inequlity nd pplictions, Journl of Inequlities in Pure nd Applied Mthemtics, vol. 0, no. 2, rticle 52, 2 pges, M. Z. Sriky, On the Ostrowski type integrl inequlity, Act Mthemtic Universittis Comenine, vol. 79, no., pp , 200.
10 0 Abstrct nd Applied Anlysis 9 M. Z. Sriky, On the Ostrowski type integrl inequlity for double integrls, Demonstrtio Mthemtic, vol. 45, no. 3, pp , M. Z. Sriky nd H. Ogunmez, On the weighted Ostrowski-type integrl inequlity for double integrls, Arbin Journl for Science nd Engineering, vol. 36, no. 6, pp , 20. R. Gorenflo nd F. Minrdi, Frctionlclculus: Integrl nd Differentible Equtions of Frctionl Order, Springer, Wien, Austri, S. G. Smko, A. A. Kilbs, nd O. I. Mrichev, Frctionl Integrls nd Derivtives Theory nd Appliction, Gordn nd Brech Science, New York, NY, USA, G. Anstssiou, M. R. Hooshmndsl, A. Ghsemi, nd F. Moftkhrzdeh, Montgomery identities for frctionl integrls nd relted frctionl inequlities, Journl of Inequlities in Pure nd Applied Mthemtics, vol. 0, no. 4, rticle 97, 6 pges, S. Belrbi nd Z. Dhmni, On some new frctionl integrl inequlities, Journl of Inequlities in Pure nd Applied Mthemtics, vol. 0, no. 3, rticle 86, 5 pges, Z. Dhmni, L. Tbhrit, nd S. Tf, Some frctionl integrl inequlities, Nonliner Science Letters, vol. 2, no., pp , Z. Dhmni, L. Tbhrit, nd S. Tf, New inequlities vi Riemnn-Liouville frctionl integrtion, Journl of Advnced Reserch in Scientific Computing, vol. 2, no., pp , 200.
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