CERTAIN NEW HERMITE-HADAMARD TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA FRACTIONAL INTAGRALS

Size: px
Start display at page:

Download "CERTAIN NEW HERMITE-HADAMARD TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA FRACTIONAL INTAGRALS"

Transcription

1 Aville online: Ferury 4, 8 Commun. Fc. Sci. Univ. Ank. Ser. A Mth. Stt. Volume 68, Numer, Pge DOI:.5/Commu_89 ISSN 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

2 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:

3 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

4 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

5 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 +..

6 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.

7 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

8 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 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, ] İşcn, İ., Hermite-Hdmrd type inequlitie or hrmoniclly convex unction, Hcet. J. Mth. Stt. Doi:.567/HJMS

9 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, ] 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, ] Srıky, M.Z., Set, E., Yldız, H. nd Bşk, N., Hermite-Hdmrd inequlitie or rctionl integrl nd relted rctionl inequlitie, Mth. Comput. Model. 579-, ] 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/ ] Set, E. nd Gözpınr, A., Hermite-Hdmrd Type Inequlitie or convex unction vi generlized rctionl integrl opertor, Topol. Alger Appl, ] Set, E., Akdemir, A.O. nd Çelik, B., On Generliztion o Fejér Type Inequlitie vi rctionl integrl opertor, ReerchGte, ] Set, E. nd Çelik, B., On generliztion relted to the let ide o Fejér inequlite vi rctionl integrl opertor, ReerchGte, 3] Set, E., Choi, J. nd Çelik, B., Certin Hermite-Hdmrd type inequlity involving generlized rctionl integrl opertor, RACSAM, Doi:.7/ ] 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, Current ddre: Erhn Set: Deprtment o Mthemtic, Fculty o Science nd Art, Ordu Univerity, Ordu, Turkey E-mil ddre: erhnet@yhoo.com ORCID Addre: 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: Current ddre: Necl Korkut: Deprtment o Mthemtic, Fculty o Science nd Art, Ordu Univerity, Ordu, Turkey E-mil ddre: neclkrkt63@gmil.com ORCID Addre:

On New Inequalities of Hermite-Hadamard-Fejer Type for Harmonically Quasi-Convex Functions Via Fractional Integrals

On New Inequalities of Hermite-Hadamard-Fejer Type for Harmonically Quasi-Convex Functions Via Fractional Integrals X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey On New Ineulities of Hermite-Hdmrd-Fejer Type for Hrmoniclly Qusi-Convex Functions Vi Frctionl Integrls Mehmet Kunt * nd İmdt İşcn Deprtment

More information

Hermite-Hadamard type inequalities for harmonically convex functions

Hermite-Hadamard type inequalities for harmonically convex functions Hcettepe Journl o Mthemtics nd Sttistics Volume 43 6 4 935 94 Hermite-Hdmrd type ineulities or hrmoniclly convex unctions İmdt İşcn Abstrct The uthor introduces the concept o hrmoniclly convex unctions

More information

On new Hermite-Hadamard-Fejer type inequalities for p-convex functions via fractional integrals

On new Hermite-Hadamard-Fejer type inequalities for p-convex functions via fractional integrals CMMA, No., -5 7 Communiction in Mthemticl Modeling nd Applictions http://ntmsci.com/cmm On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions vi rctionl integrls Mehmet Kunt nd Imdt Iscn Deprtment

More information

The Hadamard s inequality for quasi-convex functions via fractional integrals

The Hadamard s inequality for quasi-convex functions via fractional integrals Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz

More information

Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via fractional integrals

Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via fractional integrals NTMSCI 4, No. 3, 39-53 6 39 New Trends in Mthemticl Sciences http://d.doi.or/.5/ntmsci.6337 Hermite-Hdmrd-Fejér type ineulities or hrmoniclly conve unctions vi rctionl interls Imdt Iscn, Mehmet Kunt nd

More information

Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral

Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral DOI 763/s4956-6-4- Moroccn J Pure nd Appl AnlMJPAA) Volume ), 6, Pges 34 46 ISSN: 35-87 RESEARCH ARTICLE Generlized Hermite-Hdmrd-Fejer type inequlities for GA-conve functions vi Frctionl integrl I mdt

More information

Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a; b]; (1 6 a < b)

Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a; b]; (1 6 a < b) Lypunov-type inequlity for the Hdmrd frctionl boundry vlue problem on generl intervl [; b]; ( 6 < b) Zid Ldjl Deprtement of Mthemtic nd Computer Science, ICOSI Lbortory, Univerity of Khenchel, 40000, Algeri.

More information

ON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR

ON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR Krgujevc ournl of Mthemtics Volume 44(3) (), Pges 369 37. ON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR H. YALDIZ AND M. Z. SARIKAYA Abstrct. In this er, using generl clss

More information

Research Article Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions

Research Article Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions Hindwi Pulishing Corportion Journl of Applied Mthemtics Volume 4, Article ID 38686, 6 pges http://dx.doi.org/.55/4/38686 Reserch Article Fejér nd Hermite-Hdmrd Type Inequlities for Hrmoniclly Convex Functions

More information

Generalized Hermite-Hadamard Type Inequalities for p -Quasi- Convex Functions

Generalized Hermite-Hadamard Type Inequalities for p -Quasi- Convex Functions Ordu Üniv. Bil. Tek. Derg. Cilt:6 Syı: 683-93/Ordu Univ. J. Sci. Tech. Vol:6 No:683-93 -QUASİ-KONVEKS FONKSİYONLAR İÇİN GENELLEŞTİRİLMİŞ HERMİTE-HADAMARD TİPLİ EŞİTSİZLİKLER Özet İm İŞCAN* Giresun Üniversitesi

More information

Some estimates on the Hermite-Hadamard inequality through quasi-convex functions

Some estimates on the Hermite-Hadamard inequality through quasi-convex functions Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper

More information

arxiv: v1 [math.ca] 28 Jan 2013

arxiv: v1 [math.ca] 28 Jan 2013 ON NEW APPROACH HADAMARD-TYPE INEQUALITIES FOR s-geometrically CONVEX FUNCTIONS rxiv:3.9v [mth.ca 8 Jn 3 MEVLÜT TUNÇ AND İBRAHİM KARABAYIR Astrct. In this pper we chieve some new Hdmrd type ineulities

More information

New Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity

New Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity Punjb University Journl of Mthemtics (ISSN 116-56) Vol. 45 (13) pp. 33-38 New Integrl Inequlities of the Type of Hermite-Hdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science,

More information

New general integral inequalities for quasiconvex functions

New general integral inequalities for quasiconvex functions NTMSCI 6, No 1, 1-7 18 1 New Trends in Mthemticl Sciences http://dxdoiorg/185/ntmsci1739 New generl integrl ineulities for usiconvex functions Cetin Yildiz Atturk University, K K Eduction Fculty, Deprtment

More information

GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED s-convex IN THE SECOND SENSE

GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED s-convex IN THE SECOND SENSE Journl of Alied Mthemtics nd Comuttionl Mechnics 6, 5(4), - wwwmcmczl -ISSN 99-9965 DOI: 75/jmcm64 e-issn 353-588 GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES

More information

Generalized Hermite-Hadamard type inequalities involving fractional integral operators

Generalized Hermite-Hadamard type inequalities involving fractional integral operators Setetl.Journl of Ineulities nd Alictions 7 7:69 DOI.86/s366-7-444-6 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,

More information

Bulletin of the. Iranian Mathematical Society

Bulletin of the. Iranian Mathematical Society ISSN: 07-060X Print ISSN: 735-855 Online Bulletin of the Irnin Mthemticl Society Vol 3 07, No, pp 09 5 Title: Some extended Simpson-type ineulities nd pplictions Authors: K-C Hsu, S-R Hwng nd K-L Tseng

More information

f (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)

f (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1) TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS

More information

ON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS. 1. Introduction. f(a) + f(b) f(x)dx b a. 2 a

ON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS. 1. Introduction. f(a) + f(b) f(x)dx b a. 2 a Act Mth. Univ. Comenine Vol. LXXIX, (00, pp. 65 7 65 ON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR Abstrct. In this pper, we estblish some

More information

Integral inequalities for n times differentiable mappings

Integral inequalities for n times differentiable mappings JACM 3, No, 36-45 8 36 Journl of Abstrct nd Computtionl Mthemtics http://wwwntmscicom/jcm Integrl ineulities for n times differentible mppings Cetin Yildiz, Sever S Drgomir Attur University, K K Eduction

More information

Hermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave

Hermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave Applied Mthemticl Sciences Vol. 9 05 no. 5-36 HIKARI Ltd www.m-hikri.com http://d.doi.org/0.988/ms.05.9 Hermite-Hdmrd Type Ineulities for the Functions whose Second Derivtives in Absolute Vlue re Conve

More information

NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX

NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX Journl of Mthemticl Ineulities Volume 1, Number 3 18, 655 664 doi:1.7153/jmi-18-1-5 NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX SHAHID

More information

Hermite-Hadamard and Simpson-like Type Inequalities for Differentiable p-quasi-convex Functions

Hermite-Hadamard and Simpson-like Type Inequalities for Differentiable p-quasi-convex Functions Filomt 3:9 7 5945 5953 htts://doi.org/.98/fil79945i Pulished y Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: htt://www.mf.ni.c.rs/filomt Hermite-Hdmrd nd Simson-like Tye Ineulities for

More information

Some Hermite-Hadamard type inequalities for functions whose exponentials are convex

Some Hermite-Hadamard type inequalities for functions whose exponentials are convex Stud. Univ. Beş-Bolyi Mth. 6005, No. 4, 57 534 Some Hermite-Hdmrd type inequlities for functions whose exponentils re convex Silvestru Sever Drgomir nd In Gomm Astrct. Some inequlities of Hermite-Hdmrd

More information

Hadamard-Type Inequalities for s-convex Functions

Hadamard-Type Inequalities for s-convex Functions Interntionl Mthemtil Forum, 3, 008, no. 40, 965-975 Hdmrd-Type Inequlitie or -Convex Funtion Mohmmd Alomri nd Mlin Dru Shool o Mthemtil Siene Fulty o Siene nd Tehnology Univeriti Kebngn Mlyi Bngi 43600

More information

Some new integral inequalities for n-times differentiable convex and concave functions

Some new integral inequalities for n-times differentiable convex and concave functions Avilble online t wwwisr-ublictionscom/jns J Nonliner Sci Al, 10 017, 6141 6148 Reserch Article Journl Homege: wwwtjnscom - wwwisr-ublictionscom/jns Some new integrl ineulities for n-times differentible

More information

A Companion of Ostrowski Type Integral Inequality Using a 5-Step Kernel with Some Applications

A Companion of Ostrowski Type Integral Inequality Using a 5-Step Kernel with Some Applications 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

More information

ON COMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE CONVEX WITH APPLICATIONS

ON COMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE CONVEX WITH APPLICATIONS Miskolc Mthemticl Notes HU ISSN 787-5 Vol. 3 (), No., pp. 33 8 ON OMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE ONVEX WITH APPLIATIONS MOHAMMAD W. ALOMARI, M.

More information

A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL. 1. Introduction. f(x)dx a

A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL. 1. Introduction. f(x)dx a Journl of Frctionl Clculus nd Applictions, Vol. 4( Jn. 203, pp. 25-29. ISSN: 2090-5858. http://www.fcj.webs.com/ A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL VAIJANATH L. CHINCHANE

More information

Ostrowski Grüss Čebyšev type inequalities for functions whose modulus of second derivatives are convex 1

Ostrowski Grüss Čebyšev type inequalities for functions whose modulus of second derivatives are convex 1 Generl Mthemtics Vol. 6, No. (28), 7 97 Ostrowski Grüss Čebyšev type inequlities for functions whose modulus of second derivtives re convex Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Abstrct In this pper,

More information

EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Sayı: 3-1 Yıl:

EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Sayı: 3-1 Yıl: EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Syı: 3- Yıl: 9-9 NEW INEQUALITIES FOR CONVEX FUNCTIONS KONVEKS FONKSİYONLAR İÇİN YENİ EŞİTSİZLİKLER Mevlüt TUNÇ * ve S. Uğur KIRMACI Kilis 7 Arlık Üniversitesi,

More information

RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES THROUGH GENERALIZED (m, h 1, h 2 )-PREINVEXITY

RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES THROUGH GENERALIZED (m, h 1, h 2 )-PREINVEXITY ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 7 345 37 345 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES THROUGH GENERALIZED m h h -PREINVEXITY Cheng Peng Chng Zhou Tingsong Du Deprtment

More information

AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir

AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An

More information

Some New Inequalities of Simpson s Type for s-convex Functions via Fractional Integrals

Some New Inequalities of Simpson s Type for s-convex Functions via Fractional Integrals Filomt 3:5 (7), 4989 4997 htts://doi.org/.98/fil75989c Published by Fculty o Sciences nd Mthemtics, University o Niš, Serbi Avilble t: htt://www.m.ni.c.rs/ilomt Some New Ineulities o Simson s Tye or s-convex

More information

Calculus of variations with fractional derivatives and fractional integrals

Calculus of variations with fractional derivatives and fractional integrals Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl

More information

On some refinements of companions of Fejér s inequality via superquadratic functions

On some refinements of companions of Fejér s inequality via superquadratic functions Proyecciones Journl o Mthemtics Vol. 3, N o, pp. 39-33, December. Universidd Ctólic del Norte Antogst - Chile On some reinements o compnions o Fejér s inequlity vi superqudrtic unctions Muhmmd Amer Lti

More information

Hermite-Hadamard Type Fuzzy Inequalities based on s-convex Function in the Second Sense

Hermite-Hadamard Type Fuzzy Inequalities based on s-convex Function in the Second Sense Mthemtic Letter 7; 3(6): 77-8 http://wwwciencepulihinggroupcom/j/ml doi: 648/jml7364 ISSN: 575-53X (Print); ISSN: 575-556 (Online) Hermite-Hdmrd Type Fuzzy Inequlitie ed on -Convex Function in the Second

More information

INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei

INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV

More information

Revista Colombiana de Matemáticas Volumen 41 (2007), páginas 1 13

Revista Colombiana de Matemáticas Volumen 41 (2007), páginas 1 13 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

More information

A short introduction to local fractional complex analysis

A short introduction to local fractional complex analysis A short introduction to locl rctionl complex nlysis Yng Xio-Jun Deprtment o Mthemtics Mechnics, hin University o Mining Technology, Xuhou mpus, Xuhou, Jingsu, 228, P R dyngxiojun@63com This pper presents

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.

More information

Improvement of Grüss and Ostrowski Type Inequalities

Improvement of Grüss and Ostrowski Type Inequalities Filomt 9:9 (05), 07 035 DOI 098/FIL50907A Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://wwwpmfnicrs/filomt Improvement of Grüss nd Ostrowski Type Inequlities An Mri

More information

Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals

Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, 355 366 Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for

More information

WENJUN LIU AND QUÔ C ANH NGÔ

WENJUN LIU AND QUÔ C ANH NGÔ AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous

More information

ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES

ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES Volume 1 29, Issue 3, Article 86, 5 pp. ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES SOUMIA BELARBI AND ZOUBIR DAHMANI DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MOSTAGANEM soumi-mth@hotmil.fr zzdhmni@yhoo.fr

More information

A Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications

A Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications Applied Mthemticl Sciences, Vol. 8, 04, no. 38, 889-90 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.988/ms.04.4 A Generlized Inequlity of Ostrowski Type for Twice Differentile Bounded Mppings nd Applictions

More information

Keywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.

Keywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula. Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using

More information

NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a

NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS MOHAMMAD ALOMARI A MASLINA DARUS A AND SEVER S DRAGOMIR B Abstrct In terms of the first derivtive some ineulities of Simpson

More information

AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION

AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION

More information

Some Improvements of Hölder s Inequality on Time Scales

Some Improvements of Hölder s Inequality on Time Scales DOI: 0.55/uom-207-0037 An. Şt. Univ. Ovidius Constnţ Vol. 253,207, 83 96 Some Improvements of Hölder s Inequlity on Time Scles Cristin Dinu, Mihi Stncu nd Dniel Dănciulescu Astrct The theory nd pplictions

More information

A basic logarithmic inequality, and the logarithmic mean

A basic logarithmic inequality, and the logarithmic mean Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu

More information

INEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX

INEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX INEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI A, M. DARUS A, AND S.S. DRAGOMIR B Astrct. In this er, some ineulities of Hermite-Hdmrd

More information

M. A. Pathan, O. A. Daman LAPLACE TRANSFORMS OF THE LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS

M. A. Pathan, O. A. Daman LAPLACE TRANSFORMS OF THE LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS DEMONSTRATIO MATHEMATICA Vol. XLVI No 3 3 M. A. Pthn, O. A. Dmn LAPLACE TRANSFORMS OF THE LOGARITHMIC FUNCTIONS AND THEIR APPLICATIONS Abtrct. Thi pper del with theorem nd formul uing the technique of

More information

On some inequalities for s-convex functions and applications

On some inequalities for s-convex functions and applications Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 htt://wwwjournlofineulitiesndlictionscom/content/3//333 R E S E A R C H Oen Access On some ineulities for s-convex functions nd lictions Muhmet Emin

More information

Dynamics and stability of Hilfer-Hadamard type fractional differential equations with boundary conditions

Dynamics and stability of Hilfer-Hadamard type fractional differential equations with boundary conditions Journl Nonliner Anlyi nd Appliction 208 No. 208 4-26 Avilble online t www.ipc.com/jn Volume 208, Iue, Yer 208 Article ID jn-00386, 3 Pge doi:0.5899/208/jn-00386 Reerch Article Dynmic nd tbility of Hilfer-Hdmrd

More information

Bounds for the Riemann Stieltjes integral via s-convex integrand or integrator

Bounds for the Riemann Stieltjes integral via s-convex integrand or integrator ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 6, Number, 0 Avilble online t www.mth.ut.ee/ct/ Bounds for the Riemnn Stieltjes integrl vi s-convex integrnd or integrtor Mohmmd Wjeeh

More information

CHEBYSHEV TYPE INEQUALITY ON NABLA DISCRETE FRACTIONAL CALCULUS. 1. Introduction

CHEBYSHEV TYPE INEQUALITY ON NABLA DISCRETE FRACTIONAL CALCULUS. 1. Introduction Frctionl Differentil Clculus Volume 6, Number 2 (216), 275 28 doi:1.7153/fdc-6-18 CHEBYSHEV TYPE INEQUALITY ON NABLA DISCRETE FRACTIONAL CALCULUS SERKAN ASLIYÜCE AND AYŞE FEZA GÜVENILIR (Communicted by

More information

On the Co-Ordinated Convex Functions

On the Co-Ordinated Convex Functions Appl. Mth. In. Si. 8, No. 3, 085-0 0 085 Applied Mthemtis & Inormtion Sienes An Interntionl Journl http://.doi.org/0.785/mis/08038 On the Co-Ordinted Convex Funtions M. Emin Özdemir, Çetin Yıldız, nd Ahmet

More information

The Hadamard s Inequality for s-convex Function

The Hadamard s Inequality for s-convex Function Int. Journl o Mth. Anlysis, Vol., 008, no. 3, 639-646 The Hdmrd s Inequlity or s-conve Function M. Alomri nd M. Drus School o Mthemticl Sciences Fculty o Science nd Technology Universiti Kebngsn Mlysi

More information

Research Article On The Hadamard s Inequality for Log-Convex Functions on the Coordinates

Research Article On The Hadamard s Inequality for Log-Convex Functions on the Coordinates Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 29, Article ID 28347, 3 pges doi:.55/29/28347 Reserch Article On The Hdmrd s Inequlity for Log-Convex Functions on the Coordintes

More information

Some inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions

Some inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions Avilble online t www.tjns.com J. Nonliner Sci. Appl. 8 5, 7 Reserch Article Some ineulities of Hermite-Hdmrd type for n times differentible ρ, m geometriclly convex functions Fiz Zfr,, Humir Klsoom, Nwb

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics MOMENTS INEQUALITIES OF A RANDOM VARIABLE DEFINED OVER A FINITE INTERVAL PRANESH KUMAR Deprtment of Mthemtics & Computer Science University of Northern

More information

A General Dynamic Inequality of Opial Type

A General Dynamic Inequality of Opial Type Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn

More information

ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), Kristína Rostás

ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), Kristína Rostás ARCHIVUM MAHEMAICUM (BRNO) omu 47 (20), 23 33 MINIMAL AND MAXIMAL SOLUIONS OF FOURH ORDER IERAED DIFFERENIAL EQUAIONS WIH SINGULAR NONLINEARIY Kritín Rotá Abtrct. In thi pper we re concerned with ufficient

More information

Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics ICTAMI 2003, Alba Iulia

Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics ICTAMI 2003, Alba Iulia Proceedings o the Interntionl Conerence on Theor nd Applictions o Mthemtics nd Inormtics ICTAMI 2003, Al Iuli CARACTERIZATIONS OF TE FUNCTIONS WIT BOUNDED VARIATION Dniel Lesnic Astrct. The present stud

More information

Parametrized inequality of Hermite Hadamard type for functions whose third derivative absolute values are quasi convex

Parametrized inequality of Hermite Hadamard type for functions whose third derivative absolute values are quasi convex Wu et l. SpringerPlus (5) 4:83 DOI.8/s44-5-33-z RESEARCH Prmetrized inequlity of Hermite Hdmrd type for functions whose third derivtive bsolute vlues re qusi convex Shn He Wu, Bnyt Sroysng, Jin Shn Xie

More information

An inequality related to η-convex functions (II)

An inequality related to η-convex functions (II) Int. J. Nonliner Anl. Appl. 6 (15) No., 7-33 ISSN: 8-68 (electronic) http://d.doi.org/1.75/ijn.15.51 An inequlity relted to η-conve functions (II) M. Eshghi Gordji, S. S. Drgomir b, M. Rostmin Delvr, Deprtment

More information

0 N. S. BARNETT AND S. S. DRAGOMIR Using Gruss' integrl inequlity, the following pertured trpezoid inequlity in terms of the upper nd lower ounds of t

0 N. S. BARNETT AND S. S. DRAGOMIR Using Gruss' integrl inequlity, the following pertured trpezoid inequlity in terms of the upper nd lower ounds of t TAMKANG JOURNAL OF MATHEMATICS Volume 33, Numer, Summer 00 ON THE PERTURBED TRAPEZOID FORMULA N. S. BARNETT AND S. S. DRAGOMIR Astrct. Some inequlities relted to the pertured trpezoid formul re given.

More information

INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION

INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION BAI-NI GUO AND FENG QI Abstrct. In the rticle, using the Tchebycheff s integrl inequlity, the suitble properties of double integrl nd

More information

On Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex

On Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex Mly J Mt 34 93 3 On Hermite-Hdmrd tye integrl ineulities for functions whose second derivtive re nonconvex Mehmet Zeki SARIKAYA, Hkn Bozkurt nd Mehmet Eyü KİRİŞ b Dertment of Mthemtics, Fculty of Science

More information

Research Article On New Inequalities via Riemann-Liouville Fractional Integration

Research Article On New Inequalities via Riemann-Liouville Fractional Integration Abstrct nd Applied Anlysis Volume 202, Article ID 428983, 0 pges doi:0.55/202/428983 Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment

More information

LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER

LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER S. S. DRAGOMIR ;2 Astrct. In this pper we otin severl new logrithmic inequlities for two numers ; minly

More information

RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE

RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17 (2009, 3 12 RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE GEORGE A. ANASTASSIOU Abstrct. Here re estblished vrious tight probbilistic

More information

GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)

GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b) GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions

More information

ON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality

ON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality Krgujevc Journl of Mthemtics Volume 40( (016, Pges 166 171. ON A CONVEXITY PROPERTY SLAVKO SIMIĆ Abstrct. In this rticle we proved n interesting property of the clss of continuous convex functions. This

More information

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 + Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

More information

The Modified Heinz s Inequality

The Modified Heinz s Inequality Journl of Applied Mthemtics nd Physics, 03,, 65-70 Pulished Online Novemer 03 (http://wwwscirporg/journl/jmp) http://dxdoiorg/0436/jmp03500 The Modified Heinz s Inequlity Tkshi Yoshino Mthemticl Institute,

More information

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.

Review on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones. Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5. - 5.3) Remrks on the course. Slide Review: Sec. 5.-5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description

More information

Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that

Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under

More information

Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions

Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions Applied Mthetic 0 9-96 doi:0.436/.0.079 Pulihed Online Octoer 0 (http://www.scirp.org/journl/) Eitence nd Uniquene of Solution for Frctionl Order Integro-Differentil Eqution with Non-Locl nd Glol Boundry

More information

HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α, m)-convex

HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α, m)-convex HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α -CONVEX İMDAT İŞCAN Dertent of Mthetics Fculty of Science nd Arts Giresun University 8 Giresun Turkey idtiscn@giresunedutr Abstrct:

More information

Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

More information

n-points Inequalities of Hermite-Hadamard Type for h-convex Functions on Linear Spaces

n-points Inequalities of Hermite-Hadamard Type for h-convex Functions on Linear Spaces Armenin Journl o Mthemtics Volume 8, Number, 6, 38 57 n-points Inequlities o Hermite-Hdmrd Tpe or h-convex Functions on Liner Spces S. S. Drgomir Victori Universit, Universit o the Witwtersrnd Abstrct.

More information

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

More information

Chapter 8.2: The Integral

Chapter 8.2: The Integral Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

More information

LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS

LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS Electronic Journl of Differentil Equtions, Vol. 2017 (2017), No. 139, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR

More information

Euler-Maclaurin Summation Formula 1

Euler-Maclaurin Summation Formula 1 Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,

More information

HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS

HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS FLAVIA CORINA MITROI nd CĂTĂLIN IRINEL SPIRIDON In this pper we estblish some integrl inequlities

More information

Improvements of some Integral Inequalities of H. Gauchman involving Taylor s Remainder

Improvements of some Integral Inequalities of H. Gauchman involving Taylor s Remainder Divulgciones Mtemátics Vol. 11 No. 2(2003), pp. 115 120 Improvements of some Integrl Inequlities of H. Guchmn involving Tylor s Reminder Mejor de lguns Desigulddes Integrles de H. Guchmn que involucrn

More information

Geometrically Convex Function and Estimation of Remainder Terms in Taylor Series Expansion of some Functions

Geometrically Convex Function and Estimation of Remainder Terms in Taylor Series Expansion of some Functions Geometriclly Convex Function nd Estimtion of Reminder Terms in Tylor Series Expnsion of some Functions Xioming Zhng Ningguo Zheng December 21 25 Abstrct In this pper two integrl inequlities of geometriclly

More information

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :

x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x  0 : Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given

More information

Integral inequalities

Integral inequalities Integrl inequlities Constntin P. Niculescu Bsic remrk: If f : [; ]! R is (Riemnn) integrle nd nonnegtive, then f(t)dt : Equlity occurs if nd only if f = lmost everywhere (.e.) When f is continuous, f =.e.

More information

RGMIA Research Report Collection, Vol. 1, No. 1, SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIA

RGMIA Research Report Collection, Vol. 1, No. 1, SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIA ttp//sci.vut.edu.u/rgmi/reports.tml SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIABLE MAPPINGS AND APPLICATIONS P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS Astrct. Some generliztions of te Ostrowski

More information

Lecture 14 Numerical integration: advanced topics

Lecture 14 Numerical integration: advanced topics Lecture 14 Numericl integrtion: dvnced topics Weinn E 1,2 nd Tiejun Li 2 1 Deprtment of Mthemtics, Princeton University, weinn@princeton.edu 2 School of Mthemticl Sciences, Peking University, tieli@pku.edu.cn

More information

CLASSROOM NOTE Some new mean value theorems of Flett type

CLASSROOM NOTE Some new mean value theorems of Flett type Interntionl Journl of Mthemticl Eduction in Science nd Technology 014 http://dxdoiorg/101080/000739x01490457 CLASSROOM NOTE Some new men vlue theorems of Flett type Chenggun Tn nd Songxio Li Deprtment

More information

EXISTENCE OF SOLUTIONS TO INFINITE ELASTIC BEAM EQUATIONS WITH UNBOUNDED NONLINEARITIES

EXISTENCE OF SOLUTIONS TO INFINITE ELASTIC BEAM EQUATIONS WITH UNBOUNDED NONLINEARITIES Electronic Journl of Differentil Eqution, Vol. 17 (17), No. 19, pp. 1 11. ISSN: 17-6691. URL: http://ejde.mth.txtte.edu or http://ejde.mth.unt.edu EXISTENCE OF SOLUTIONS TO INFINITE ELASTIC BEAM EQUATIONS

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics ON LANDAU TYPE INEQUALITIES FOR FUNCTIONS WIT ÖLDER CONTINUOUS DERIVATIVES LJ. MARANGUNIĆ AND J. PEČARIĆ Deprtment of Applied Mthemtics Fculty of Electricl

More information

WHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) =

WHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) = WHEN IS A FUNCTION NOT FLAT? YIFEI PAN AND MEI WANG Abstrct. In this pper we prove unique continution property for vector vlued functions of one vrible stisfying certin differentil inequlity. Key words:

More information

MonotonicBehaviourofRelativeIncrementsofPearsonDistributions

MonotonicBehaviourofRelativeIncrementsofPearsonDistributions Globl Journl o Science Frontier Reserch: F Mthemtics nd Decision Sciences Volume 8 Issue 5 Version.0 Yer 208 Type : Double lind Peer Reviewed Interntionl Reserch Journl Publisher: Globl Journls Online

More information