Generalized Hermite-Hadamard type inequalities involving fractional integral operators

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1 Setetl.Journl of Ineulities nd Alictions 7 7:69 DOI.86/s R E S E A R C H Oen Access Generlized Hermite-Hdmrd tye ineulities involving frctionl integrl oertors Erhn Set, Muhmmed Aslm Noor,3, Muhmmed Uzir Awn 4* nd Abdurrhmn Gözinr * Corresondence: wn.uzir@gmil.com 4 Dertment of Mthemtics, Government College University, Fislbd, Pkistn Full list of uthor informtion is vilble t the end of the rticle Abstrct In this rticle, new generl integrl identity involving generlized frctionl integrl oertors is estblished. With the hel of this identity new Hermite-Hdmrd tye ineulities re obtined for functions whose bsolute vlues of derivtives re convex. As conseuence, the min results of this er generlize the existing Hermite-Hdmrd tye ineulities involving the Riemnn-Liouville frctionl integrl. MSC: 6A33; 6D; 6D5; 33B Keywords: Hermite-Hdmrd ineulity; convex function; Hölder ineulity; frctionl integrl oertor Introduction nd reliminries During the lst century the theory of convexity hs emerged s n interesting nd fscinting field of mthemtics. It lys ivotl role in otimiztion theory, functionl nlysis, control theory nd economics etc. Afunctionf : I R R is sid to be convex if the ineulity f tx ty tf x tf y holds for ll x, y I nd t, ]. The following ineulity is so-clled clssicl Hermite-Hdmrd tye ineulity for convex functions. Let f : I =, b] R R be convex function nd, b I with < b, then b f b b f x dx f fb.. This ineulity is one of the most useful ineulities in mthemticl nlysis. For new roofs, noteworthy extension, generliztions nd numerous lictions on this ineulity, see, e.g., 3] where further references re given. The reltionshi between theory of convexity nd theory of ineulities hs motivted mnyresercherstostudythesetheoriesindeth.asconseuenceofthisfctseverl ineulities hve been obtined vi convex functions; see ]. The Authors 7. This rticle is distributed under the terms of the Cretive Commons Attribution 4. Interntionl License htt://cretivecommons.org/licenses/by/4./, which ermits unrestricted use, distribution, nd reroduction in ny medium, rovided you give rorite credit to the originl uthors nd the source, rovide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge of The history of frctionl clculus cn be trced bck to the letter of L Hositl to Leibniz in which he inuired him bout the nottion he ws using for the nth derivtive of the liner function f x =x, Dn x Dx n. L Hositl sked the uestion: wht would the result be if n =. Leibniz relied: An rent rdox, from which one dy useful conseuences will be drwn. Nowdys frctionl clculus hs become owerful tool in mny brnches of mthemtics. Sriky et l. 4] used the definitions of Riemnnn-Liouville integrls nd develoed new generliztion of Hermite-Hdmrd ineulity. This result insired mny reserchers to study this re. For more detils, nd for recent results nd recently found roerties concerning this oertor one cn consult 4 ]. We need some definition nd mthemticl reliminries of frctionl clculus theory for using in this study s follows. Definition. Let f L, b]. The Riemnn-Liouville integrls J α f nd Jα b f of order α > with redefinedby nd J α f x= Ɣα J α b f x= Ɣα x b x x t α f t, x >, t x α f t, x < b, resectively. Here Ɣt is the Gmm function nd its definition is Ɣt = e x x t dx. It is to be noted tht J f x =J b f x =f x;inthecseofα =, the frctionl integrl reduces to the clssicl integrl. In 3], Zhu et l. estblished new identity for differentible convex mings vi the Riemnn-Liouville frctionl integrl. Lemm. 3] Let f :, b] R be differentible ming on, b with < b. If f L, b], then the following eulity for frctionl integrls hold: where Ɣα J b α α f b Jα b f ] b f = b ktf t tb t α t α] f t tb ],., < t kt=,, < t. Using the bove identity, they gve the following result for the Riemnn-Liouville frctionl integrl.

3 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge 3 of Theorem. 3] Let f :, b] R be differentible ming on, b with < b. If f is convex on, b], then the following frctionl ineulity for frctionl integrls holds: Ɣα J b α α f b Jα b f ] b f b α 3 f f b ]..3 4α α In 4], Rin introduced clss of functions defined formlly by Fρ,λ,,... x=fρ,λ x= k= k Ɣρk λ xk ρ, λ >; x < R,.4 where the coefficients kk N = N {} re bounded seuence of ositive rel numbers nd R is the set of rel numbers. With the hel of.4, Rin 4] nd Agrwl et l. 5] defined the following left-sided nd right-sided frctionl integrl oertors, resectively: J ρ,λ,;w ϕ x x= x t λ Fρ,λ wx t ρ ] ϕt x > >,.5 J ρ,λ,b ;w ϕ b x= t x λ Fρ,λ wt x ρ ] ϕt < x < b,.6 x where λ, ρ >,w R nd ϕt is such tht the integrl on the right side exists. Recently some new integrl ineulities involving this oertor hve ered in the literture see, e.g.,5 ]. It is esy to verify tht Jρ,λ,;w ϕxndj ρ,λ,b ;w ϕx re bounded integrl oertors on L, b, if M := F ρ,λ wb ρ ] <..7 In fct, for ϕ L, b, we hve J ρ,λ,;w ϕx Mb λ ϕ.8 nd J ρ,λ,b ;w ϕx Mb λ ϕ,.9 where b ϕ := ϕt. Here, mny useful frctionl integrl oertors cn be obtined by secilizing the coefficient k. For instnce the clssicl Riemnn-Liouville frctionl integrls J α nd Jα b of order α follow esily by setting λ = α, = nd w =in.5nd.6.

4 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge 4 of Motivted by the work in 3 5], firstly, we will rove generliztion of the identity given by Zhu et l. using generlized frctionl integrl oertors. Then we will give some new Hermite-Hdmrd tye ineulities, which re generliztions of the results in 3] to the cse λ = α, = nd w =. Our results cn be viewed s significnt extension nd generliztion of the reviously known results. Results nd discussions In this section, we derive our min results. For the ske of simlicity, we denote L f, b; w; J:= b λ J ρ,λ,b ;w f J ρ,λ, ;w f b ] F ρ,λ wb ρ ] b f Lemm. Let f :, b] R be differentible ming on, b with < b. If f L, b], then the following eulity for generlized frctionl integrl oertors holds: where { b L f, b; w; J= ktf t tb Fρ,λ kt= wb ρ ], < t, Fρ,λ wb ρ ], < t, ρ, λ >,w R. Proof It suffices to note tht t λ F ρ,λ wb ρ t ρ] f t tb t λ Fρ,λ wb ρ t ρ] f t tb },. I = F ρ,λ wb ρ ] f t tb F ρ,λ wb ρ ] f t tb t λ F ρ,λ wb ρ t ρ] f t tb t λ F ρ,λ wb ρ t ρ] f t tb := I I I 3 I 4.. Chnging vribles with x = t tb,weget I = Fρ,λ wb ρ ] f t tb b = F ρ,λ wb ρ ] b f b f ],

5 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge 5 of I = F ρ,λ wb ρ ] f t tb = F ρ,λ wb ρ ] b Integrting by rts, we hve I 3 = Anlogously f f b ]. t λ F ρ,λ wb ρ t ρ] f t tb = b tλ Fρ,λ wb ρ t ρ] f t tb t λ F b ρ,λ wb ρ t ρ] f t tb = b F ρ,λ wb ρ ] f b b x λ x ρ ] f x Fρ,λ wb ρ b b b b dx I 4 = = b F ρ,λ wb ρ ] f b t λ F ρ,λ wb ρ t ρ] f t tb b λ J ρ,λ,b ;w f.. = b tλ Fρ,λ wb ρ t ρ] f t tb t λ F b ρ,λ wb ρ t ρ] f t tb = b F ρ,λ wb ρ ] f b b x λ b x ρ ] f x Fρ,λ wb ρ b b b b dx = b F ρ,λ wb ρ ] f b λ J ρ,λ, ;w f b..3 Substituting the resulting eulities into eulity., we hve I = b F ρ,λ wb ρ ] b f b λ J ρ,λ,b ;w f J ρ,λ, ;w f b ]..4 Thus, multilying both sides by b,theresultisobtined. Remrk. Choosing λ = α, = nd w = in Lemm., eulity. reducesto eulity..

6 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge 6 of Theorem. Let f :, b] R be differentible function on, b] with < b. If f is convex on, b, then the following ineulity for generlized frctionl integrl oertors holds: L f, b; w; J b 4 F ρ,λ w b ρ ] f f b ],.5 where k=k λ ρk 3, λρk ρ, λ >,w R, s, ]. Proof Using Lemm. nd convexity of f,wehve Lf, b; w; J b { F ρ,λ wb ρ ] f t tb Fρ,λ wb ρ ] f t tb b t λ Fρ,λ wb ρ t ρ] f t tb t λ Fρ,λ wb ρ t ρ] f t tb } { Fρ,λ wb ρ ] t f t f b k= b k w k b ρk Ɣρk λ t λρk t λρk t f t f b t λρk t λρk t f t f b ] } f f b f f b = b k= k= k w k b ρk Ɣρk λ t t λρk t λρk { f f b t λρk t λρk t t λρk t t λρk tt λρk t λρk } k w k b ρk Ɣρk λ ] f f b ] λ ρk λρk

7 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge 7 of = b 4 k= k w k b ρk λ ρk 3 f f b ] Ɣρk λ λρk = b 4 F ρ,λ w b ρ ] f f b ],.6 using the fcts tht t t λρk t λρk = tt λρk t λρk = λρk λ ρk λ ρk λ ρk λρk nd t λρk t λρk t = t λρk t t λρk = λ ρk λ ρk λρk λ ρk 3 λ ρk λ ρk λρk. Thus the roof is comleted. Remrk. Choosing λ = α, = nd w =intheorem.,ineulity.5reduces to ineulity.3. Theorem. Let f :, b] R be differentible function on, b with < b. If f is convex nd >with =,then the following ineulity for generlized frctionl integrl oertors holds: Lf, b; w; J b F ρ,λ w b ρ ] f f F ρ,λ w b ρ ] f 3 f b ] 3 f f b ] ] F 3 ρ,λ w b ρ ] f f b ],.7 where k=k λ ρk 3 k=k ρ, λ >nd w R. ], λρk 4 λ ρk ], λρk

8 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge 8 of Proof By using Lemm.,wehve L f, b; w; J b { F ρ,λ wb ρ ] f t tb Fρ,λ wb ρ ] f t tb t λ Fρ,λ wb ρ t ρ] f t tb t λ Fρ,λ wb ρ t ρ] f t tb } { b k w k b ρk f t tb Ɣρk λ k= k= k w k b ρk Ɣρk λ t λρk t λρk f t tb t λρk t λρk f t tb ] }..8 Using the well-known Hölder ineulity nd convexity of f we get f t tb f f..9 Thus t λρk t λρk f t tb = t λρk t λρk ] f t tb t λρk t λρk ] t f t f ] λ ρk λρk ] f 3 f b ]. 8 8 ] nd t λρk t λρk f t tb t λρk t λρk ] f t tb t λρk t λρk ] t f t f ] = λ ρk λρk ] 3 f f b ],. 8 8 where we used tht A B A B for ny A B nd in.nd.. ]

9 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge 9 of Let =3 f, b = f b, = f, b =3 f b.here< <for >.We use the fct tht n k b k s k= n n s k b s k. k= k= For s <,,, 3,..., n, b, b, b 3,...,b n. Combining the ineulities. with.weobtin nd t λρk t λρk f t tb t λρk t λρk f t tb λ ρk λρk ] 8 3 f f b ] f 3 f b ] λ ρk λ ρk = 4 λ ρk f f λρk λρk λρk ] 3 f f b ] 8 ] 4 f f b ] 8 ] f f b ]. f f b ]..3 Thus utting the ineulities.9,.nd.3 in.8, the roof is comleted. Corollry. Choosing λ = α, = nd w =in Theorem., ineulity.7 becomes the following ineulity: Ɣα J b α α f b Jα b f ] b f b { f f ] ] f 3 f b ] 3 f f b ] } α α b 4 α α ] f f b ]..4 3 Conclusion In this er, we hve obtined new frctionl integrl identity. Utilizing this new identity s n uxiliry result, we hve obtined some new vrints of Hermite-Hdmrd tye

10 Setetl.Journl of Ineulities nd Alictions 7 7:69 Pge of ineulities. The results derived in this er become nturl generliztions of clssicl results. It is exected tht the interested reder my find useful lictions of these results nd conseuently this er my stimulte further reserch in this re. Acknowledgements The uthors re thnkful to the nonymous referee for his/her vluble comments nd suggestions. The uthors re lesed to cknowledge the suort of Distinguished Scientist Fellowshi Progrm DSFP, King Sud University, Riydh, Sudi Arbi. Third uthor is thnkful to HEC, Pkistn for SRGP roject -985/SRGP/R&D/HEC/6. Cometing interests The uthors declre tht they hve no cometing interests. Authors contributions ES, MAN, MUA nd AG worked jointly. All the uthors red nd roved the finl mnuscrit. Author detils Dertment of Mthemtics, Fculty of Science nd Arts, Ordu University, Ordu, Turkey. Dertment of Mthemtics, King Sud University, Riydh, Sudi Arbi. 3 Dertment of Mthemtics, COMSATS Institute of Informtion Technology, Prk Rod, Islmbd, Pkistn. 4 Dertment of Mthemtics, Government College University, Fislbd, Pkistn. Publisher s Note Sringer Nture remins neutrl with regrd to jurisdictionl clims in ublished ms nd institutionl ffilitions. Received: 4 Februry 7 Acceted: 3 June 7 References. Drgomir, SS, Perce, CEM: Selected Toics on Hermite-Hdmrd Ineulities nd Alictions. RGMIA Monogrhs. Victori University. Mitrinović, DS, Lcković,IB:Hermitendconvexity.Aeu.Mth.8, Set, E, Özdemir, ME, Srıky, MZ: Ineulities of Hermite-Hdmrd s tye for functions whose derivtives bsolute vlues re m-convex. AIP Conf. Proc. 39, Srıky, MZ, Set, E, Yldız, H, Bşk, N: Hermite-Hdmrd s ineulities for frctionl integrls nd relted frctionl ineulities. Mth. Comut. Model. 57, Dhmni, Z: New ineulities in frctionl integrls. Int. J. Nonliner Sci. 94, Dhmni, Z, Tbhrit, L, Tf, S: New generliztions of Grüss ineulity using Riemnn-Liouville frctionl integrls. Bull. Mth. Anl. Al. 3, Gorenflo, R, Minrdi, F: Frctionl clculus: integrl nd differentil eutions of frctionl order. In: Frctls nd Frctionl Clculus in Continuum Mechnics, Sringer, Wien Noor, MA, Cristescu, G, Awn, MU: Generlized frctionl Hermite-Hdmrd ineulities for twice differentible s-convex functions. Filomt 94, Srıky, MZ, Yıldı rım, H: On Hermite-Hdmrd tye ineulities for Riemnnn-Liouville frctionl integrls. Miskolc Mth. Notes 7, Set, E, Srıky, MZ, Özdemir, ME, Yıldı rım, H: The Hermite-Hdmrd s ineulity for some convex functions vi frctionl integrls nd relted results. J. Al. Mth. Stt. Inform., Set, E: New ineulities of Ostrowski tye for ming whose derivtives re s-convex in the second sense vi frctionl integrls. Comut. Mth. Al. 63, Set, E, İşcn, İ, Srıky, MZ, Özdemir, ME: On new ineulities of Hermite-Hdmrd-Fejer tye for convex functions vi frctionl integrls. Al. Mth. Comut. 59, Zhu, C, Feckn, M, Wng, J: Frctionl integrl ineulities for differentible convex mings nd lictions to secil mens nd midoint formul. J. Al. Mth. Stt. Inform. 8, Rin, RK: On generlized Wright s hyergeometric functions nd frctionl clculus oertors. Est Asin Mth. J., Agrwl, RP, Luo, M-J, Rin, RK: On Ostrowski tye ineulities. Fsc. Mth. 4, Set, E, Gözınr, A: Some new ineulities involving generlized frctionl integrl oertors for severl clss of functions. AIP Conf. Proc. 833, doi:.63/ Set, E, Gözınr, A: Hermite-Hdmrd tye ineulities for convex functions vi generlized frctionl integrl oertors. ReserchGte. htts:// 8. Set, E, Akdemir, AO, Çelik, B: On generliztion of Fejér tye ineulities vi frctionl integrl oertor. ReserchGte. htts:// 9. Set, E, Çelik, B: On generliztion relted to the left side of Fejér s ineulites vi frctionl integrl oertor. ReserchGte. htts:// Set, E, Choi, J, Çelik, B: A new roch to generlized of Hermite-Hdmrd ineulity using frctionl integrl oertor. ReserchGte. htts:// Ust, F, Budk, H, Srıky, MZ, Set, E: On generliztion of trezoid tye ineulities for s-convex functions with generlized frctionl integrl oertors. Filomt cceted. Yldız, H, Srıky, MZ: On the Hermite-Hdmrd tye ineulities for frctionl integrl oertor. ReserchGte. htts://