Aville online: Ferury 4, 8 Commun. Fc. Sci. Univ. Ank. Ser. A Mth. Stt. Volume 68, Numer, Pge 6 69 9 DOI:.5/Commu_89 ISSN 33 599 http://communiction.cience.nkr.edu.tr/index.php?eriea CERTAIN NEW HERMITE-HADAMARD TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA FRACTIONAL INTAGRALS ERHAN SET, M. EMIN ÖZDEMIR, AND NECLA KORKUT A. The oject o thi pper i to otin certin Hermite-Hdmrd type integrl inequlitie involving generl cl o rctionl integrl opertor nd the rctionl integrl opertor ith exponentil kernel y uing hrmoniclly convex unction.. I Let rel unction e deined on ome nonempty intervl I o rel line R. The unction i id to e convex on I i inequlity tx + ty tx + ty hold or ll x, y I nd t, ]. We y tht i concve i i convex. Convexity i n importnt concept in mny rnche o mthemtic. In prticulr, mny importnt integrl inequlitie re ed on convexity umption o certin unction. For exmple, the olloing mou inequlity i one o them. Let : I R R e convex unction deined on the intervl I o rel numer nd, I ith <. The olloing inequlity hold: + xdx +.. It irtly dicovered y Ch. Hermite ] in 88 in the journl Mthei. But thi inequlity nohere mentioned in the mthemticl literture nd not idely knon Hermite reult. E.F. Beckench rote tht thi reult proven y J. Hdmrd in 893. In 974, D.S. Mitrinovic ound Hermite note in Mthei. Thi inequlity knon Hdmrd inequlity i no commonly reerred the Hermite-Hdmrd inequlity. Hermite-Hdmrd inequlity i Received y the editor: My 3, 7,Accepted: Septemer 5, 7. Mthemtic Suject Cliiction. 6A33, 6D, 6D5, 33B. Key ord nd phre. Hrmoniclly convex unction, Hermite-Hdmrd inequlity, rctionl integrl opertor. c 8 A nkr U niverity. Communiction Fculty o Science Univerity o Ankr-Serie A M themtic nd Sttitic. Communiction de l Fculté de Science de l Univerité d Ankr-Série A M themtic nd Sttitic. 6
6 ERHAN SET, M. EMIN ÖZDEMIR, AND NECLA KORKUT plying very importnt role in ll the ield o mthemtic. Thu uch inequlitie ere tudied extenively y mny reercher nd numer o the pper hve een ritten on thi inequlity providing ne proo, noteorthy extenion, generliztion nd numerou ppliction. In recent yer, one more dimenion h een dded to thi tudie, y introducing vriou integrl inequlitie involving rctionl integrl opertor like Riemnn-Liouville, Hdmrd, Erdelyi-Koer, Ktugmpol rctionl opertor nd rctionl opertor ith exponentil kernel. A dierent cl o the convexity i introduced y İşcn the olloing: Deinition. 3] Let I R/{} e rel intervl. A unction : I R i id to e hrmoniclly convex, i xy tx + ty. tx + ty or ll x, y I nd t, ]. I the inequlity in. i reerved, then i id to e hrmoniclly concve. In 3], İşcn etlihed the olloing inequlitie hich i dierent verion o Hermite-Hdmrd inequlity. Theorem. Let : I R/{} R e hrmoniclly convex unction nd, I ith <. I L, ] then the olloing inequlitie hold: x + + x..3 We need to recll ome deinition nd knon reult. Deinition. Let L, ]. The Riemnnn-Liouville rctionl integrl J+ nd J o order > ith re deined y nd J +x Γ J x Γ x x x t tdt, x > t x tdt, x < repectively. Here Γt i the Gmm unction nd it deinition i Γt e x x t dx. It i to e noted tht J+x J x x. In the ce o, the rctionl integrl reduce to the clicl integrl. For more detil nd propertie concerning the rctionl integrl opertor, e reer, or exmple, to the ork 6, 8]. İşcn nd Wu 4], recently, uing Riemnn-Liouville rctionl integrl, preented Hermite-Hdmrd integrl inequlitie or hrmoniclly convex unction ollo:
CERTAIN NEW FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 63 Theorem. Let : I, R e unction uch tht L, ] here, I ith <. I i hrmoniclly convex unction on, ], then the olloing inequlitie or rctionl integrl hold: + Γ + ith > nd gx /x. + { J og + J og } +.4 Ltely, Kirne nd Toreek 5], hve introduced ne cl o rctionl integrl hich re ummrized ollo: Deinition 3. Let L, ]. The rctionl integrl I nd I o order, re deined y I x x { exp } x d, x > nd repectively. I, then I x lim I x x { exp } x d, x d, lim I x x x < d. Thereore the opertor I nd I re clled rctionl integrl o order. Moreover, ecue lim exp x δx, then lim I x x, lim I x x. In 7], Rin introduced cl o unction deined ormlly y Fρ,λx σ F σ,σ,... σk ρ,λ x Γρk + λ xk ρ, λ > ; x < R,.5 k here the coei cient σk k N N {} i ounded equence o poitive rel numer nd R i the et o rel numer. With the help o.5, Rin 7] nd Agrl et l. ] deined the olloing let-ided nd right-ided rctionl integrl opertor repectively, ollo: J σ ρ,λ,+; ϕ x x x t λ Fρ,λx σ ϕtdt x > >,.6 J σ ρ,λ, ; ϕ x t x λ Fρ,λt σ x ρ ]ϕtdt < x <,.7 x
64 ERHAN SET, M. EMIN ÖZDEMIR, AND NECLA KORKUT here λ, ρ >, R nd ϕt i uch tht the integrl on the right ide exit. In recently ome ne integrl inequlitie involving thi opertor hve ppered in the literture ee, e.g., ],9]-5]. It i ey to veriy tht Jρ,λ,+; σ ϕx nd J ρ,λ, ; σ ϕx re ounded integrl opertor on L,, i In ct, or ϕ L,, e hve nd here M : F σ ρ,λ+ ρ ] <..8 J σ ρ,λ,+;ϕx M λ ϕ.9 J σ ρ,λ, ;ϕx M λ ϕ,. ϕ p : ϕt p dt Here, mny ueul rctionl integrl opertor cn e otined y pecilizing the coei cient σk. For intnce the clicl Riemnn-Liouville rctionl integrl J+ nd J o order ollo eily y etting λ, σ nd in.6 nd.7. Here, motivted y the ork in 4],5],7], e im t etlihing certin ne Hermite-Hdmrd type inequlitie ocited ith the rctionl integrl opertor ith exponentil kernel nd generl cl o rctionl integrl opertor y uing hrmoniclly convex unction. Relevnt connection o the reult preented here re lo pointed out.. M R Firtly, e ill preent Hermite-Hdmrd inequlitie or hrmoniclly convex unction vi rctionl integrl opertor ith exponentil kernel. We henceorth in denote, A or,. Theorem 3. Let :, ] R e unction ith < nd L, ]. I i hrmoniclly convex unction on, ], then the olloing inequlitie or rctionl integrl opertor ith exponentil kernel hold: + exp A] I og p. + I og ] +. Proo. Since i hrmoniclly convex unction on, ], e hve or ll x, y, ] xy x + y. x + y For x t+ t, y t+ t, e otin
CERTAIN NEW FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 65 + t+ t + t+ t.. Multiplying oth ide o. y exp At, then integrting the reulting inequlity ith repect to t over, ], e get exp Atdt + { } exp At dt + exp At dt. t + t t + t Hence, e otin exp A A + { exp { + exp } { exp } d + { exp } ] d ] I og + I og } ] d d here gx x nd the irt inequlity i proved. For the proo o the econd inequlity in., e irt note tht i i hrmoniclly convex unction then, or t, ], it yield t + t t + t nd t + t. t + t By dding thee inequlitie e hve t + t + t + t +..
66 ERHAN SET, M. EMIN ÖZDEMIR, AND NECLA KORKUT Then multiplying oth ide o. y exp At, nd integrting the reulting inequlity ith repect to t over, ], e otin exp At dt + exp At dt t + t t + t + ] exp Atdt. Uing the imilr rgument ove e cn ho tht ] I og + I og + ]. exp A So, the proo i completed. No, uing generl rctionl integrl opertor introduced y Rin 7] nd Agrl et l. ], e ill prove Hermite-Hdmrd inequlitie or hrmoniclly convex unction. Theorem 4. Let :, ] R e unction uch tht L, ], here, I ith <. I i hrmoniclly convex unction on, ], then the olloing inequlitie or rctionl integrl opertor hold: + λ Fρ,λ+ σ + here λ >, gx x. Proo. For t, ], let x o yield + ] ρ] J ρ,λ, ;og + J ρ,λ, +;og t+ t, y t+ t.3 t+ t. The hrmoniclly convexity + t+ t Multiplying oth ide o.4 y t λ Fρ,λ σ reulting inequlity ith repect to t over, ], e otin t λ Fρ,λ σ + t λ Fρ,λ σ + t λ Fρ,λ σ ρ ρ..4 ρ, then integrting the ρ ] t ρ dt t + t t + t dt dt.
CERTAIN NEW FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 67 nd The olloing integrl clculted y uing.5, e hve t λ F σ ρ,λ t λ F σ ρ,λ+ k t λ F σ ρ,λ+ k σk k Γρk + λ λ ρ dt ρ ] t ρ ρk λ λ σk k Γρk + λ k k F σ ρ,λ+ t λ k σk k Γρk + λ σk k Γρk + λ + σk k ρk ] t ρk dt Γρk + λ ρk ρk ρ ], t λ+ρk dt dt λ t + t ρk ] ρk σk k ] ρk Γρk + λ ρ ] t ρ ρk A conequence, e otin k λ F σ ρ,λ t + t ρk λ σk k Γρk + λ k λ λ F σ ρ,λ dt ] ρk λ ρk ] ρ ] d d d d λ d ρ ] d. λ λ
68 ERHAN SET, M. EMIN ÖZDEMIR, AND NECLA KORKUT F σ ρ,λ+ ρ ] + λ J + ρ,λ, ;og + J ρ,λ, +;og here gx x nd the irt inequlity i proved. For the proo o the econd inequlity in.3, e irt note tht i i hrmoniclly convex unction, then or t, ], e hve + +..5 t + t t + t Then multiplying oth ide o.5 y t λ Fρ,λ σ reulting inequlity ith repect to t over, ], e otin + t λ F σ ρ,λ t λ F σ ρ,λ + ] ρ ] t ρ ρ t + t t + t t λ Fρ,λ σ Uing the imilr rgument ove e cn ho tht λ ] J + ρ,λ, ;og + J ρ,λ, +;og F σ ρ,λ So, the proo i completed. ρ + ]. ] ρ nd integrting the dt dt ρ dt. Remrk. I in Theorem 4, e get λ, σ,, then the inequlitie.3 ecome the inequlitie.4. Remrk. I in Theorem 4, e get λ, σ,, then the inequlitie.3 ecome the inequlitie.3. R ] Agrl, R.P., Luo M.J. nd Rin, R.K., On Otroki type inequlitie, Fciculi Mthemtici, 4 6, 5-7. ] Hermite, C., Sur deux limite d une integrle deinie, Mthei, 3, 8 883. 3] İşcn, İ., Hermite-Hdmrd type inequlitie or hrmoniclly convex unction, Hcet. J. Mth. Stt. Doi:.567/HJMS.443759.
CERTAIN NEW FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITIES 69 4] İşcn, İ. nd Wu, S., Hermite-Hdmrd type inequlitie or hrmoniclly convex unction vi rctionl integrl, Applied Mthemtic nd Computtion 38, 4 37-44. 5] Kirne, M. nd Toreek B.T., Hermite-Hdmrd, Hermite-Hdmrd-Fejer, Drgomir- Agrl nd Pchptte type inequlitie or convex unction vi rctionl integrl, rxiv:7.9v mth.fa] 6. 6] Kil, A.A., Srivtv, H.M. nd Trujillo, J.J., Theory nd Appliction o Frctionl Di erentil Eqution, Elevier, Amterdm 6. 7] Rin, R.K., On generlized Wright hypergeometric unction nd rctionl clculu opertor, Et Ain Mth. J., 5, 9-3. 8] Srıky, M.Z., Set, E., Yldız, H. nd Bşk, N., Hermite-Hdmrd inequlitie or rctionl integrl nd relted rctionl inequlitie, Mth. Comput. Model. 579-, 3 43-47. 9] Set, E. nd Gözpınr, A., Some ne inequlitie involving generlized rctionl integrl opertor or everl cl o unction, AIP Conerence Proceeding, 833, 38 7; doi:.63/.498686. ] Set, E. nd Gözpınr, A., Hermite-Hdmrd Type Inequlitie or convex unction vi generlized rctionl integrl opertor, Topol. Alger Appl, 5 7 55 6. ] Set, E., Akdemir, A.O. nd Çelik, B., On Generliztion o Fejér Type Inequlitie vi rctionl integrl opertor, ReerchGte, http://.reerchgte.net/puliction/345467. ] Set, E. nd Çelik, B., On generliztion relted to the let ide o Fejér inequlite vi rctionl integrl opertor, ReerchGte, http://.reerchgte.net/puliction/36586. 3] Set, E., Choi, J. nd Çelik, B., Certin Hermite-Hdmrd type inequlity involving generlized rctionl integrl opertor, RACSAM, Doi:.7/3398-7-444-. 4] Ut, F., Budk, H., Srıky M.Z. nd Set, E., On generliztion o trpezoid type inequlitie or -convex unction ith generlized rctionl integrl opertor, Filomt, ccepted. 5] Yldız H. nd Srıky, M.Z., On the Hermite-Hdmrd type inequlitie or rctionl integrl opertor, ReerchGte, http://.reerchgte.net/puliction/398475. Current ddre: Erhn Set: Deprtment o Mthemtic, Fculty o Science nd Art, Ordu Univerity, Ordu, Turkey E-mil ddre: erhnet@yhoo.com ORCID Addre: http://orcid.org/-3-364-5396 Current ddre: M. Emin Özdemir: Deprtment o Elementry Eduction, Fculty o Eduction, Uludğ Univerity, Bur, Turkey E-mil ddre: eminozdemir@uludg.edu.tr ORCID Addre: http://orcid.org/--599-94x Current ddre: Necl Korkut: Deprtment o Mthemtic, Fculty o Science nd Art, Ordu Univerity, Ordu, Turkey E-mil ddre: neclkrkt63@gmil.com ORCID Addre: http://orcid.org/-3-94-567x