Jourl o Pure d Alied Mthetics: Advces d Alictios Volue 7 Nuer 7 Pges -7 Aville t htt://scietiicdvces.co.i DOI: htt://dx.doi.org/.864/j_7745 HERMITE-HADAMARD TYPE FRACTIONAL INTEGRAL INEQUALITIES FOR GENERALIZED ( -PREINVEX FUNCTIONS ARTION KASHURI d ROZANA LIKO Dertet o Mthetics Fculty o Techicl Sciece Uiversity Isil Qeli Vlor Ali e-il: rtiokshuri@gil.co rozliko86@gil.co Astrct I the reset er y usig ew idetity or rctiol itegrls soe ew estites o geerliztios o Herite-Hdrd tye ieulities or the clss o geerlized ( -reivex uctios vi Rie-Liouville rctiol itegrl re estlished. At the ed soe lictios to secil es re give.. Itroductio The ollowig ottio is used throughout this er. We use I to deote itervl o the rel lie R ( d I to deote the Mthetics Suject Clssiictio: Priry 6A5; Secodry 6A33 6D7 6D 6D5. Keywords d hrses: Herite-Hdrd ieulity Holder s ieulity ower e ieulity Rie-Liouville rctiol itegrl s-covex i the secod sese -ivex P-uctio. Received Noveer 7 6 6 Scietiic Advces Pulishers
ARTION KASHURI d ROZANA LIKO iterior o I. For y suset K R K is used to deote the iterior o K R is used to deote geeric -diesiol vector sce. The oegtive rel uers re deoted y R [. The set o itegrle uctios o the itervl [ ] is deoted y L [ ]. The ollowig ieulity ed Herite-Hdrd ieulity is oe o the ost ous ieulities i the literture or covex uctios. Theore.. Let I o rel uers d holds: : I R R e covex uctio o itervl I with <. The the ollowig ieulity ( ( ( x dx. (. The ollowig deiitio will e used i the seuel. Deiitio.. The hyergeoetric uctio F ( ; c; z is deied y F ( ; c; z c t ( t ( zt dt β( c or c > > d z < where β ( x y is the Euler et uctio or ll x y >. I recet yers vrious geerliztios extesios d vrits o such ieulities hve ee otied (see [] [] [3]. For other recet results cocerig Herite-Hdrd tye ieulities through vrious clsses o covex uctios (see [9] d the reereces cited therei lso (see [8] d the reereces cited therei. Frctiol clculus (see [9] d the reereces cited therei ws itroduced t the ed o the ieteeth cetury y Liouville d Rie the suject o which hs ecoe ridly growig re d hs oud lictios i diverse ields rgig ro hysicl scieces d egieerig to iologicl scieces d ecooics.
HERMITE-HADAMARD TYPE FRACTIONAL 3 Deiitio.3. Let L [ ]. The Rie-Liouville itegrls J d J o order > with re deied y x > J ( x Γ( ( x t ( t dt x d J ( x Γ( > x e u ( t x ( t dt x where Γ ( u du. Here J ( x J ( x ( x. I the cse o the rctiol itegrl reduces to the clssicl itegrl. Due to the wide lictio o rctiol itegrls soe uthors exteded to study rctiol Herite-Hdrd tye ieulities or uctios o dieret clsses (see [9]-[6] d the reereces cited therei. Now let us recll soe deiitios o vrious covex uctios. Deiitio.4 (see [3]. A oegtive uctio sid to e P-uctio or P-covex i : I R R is ( tx ( t y ( x ( y x y I t [ ]. Deiitio.5 (see [6]. A uctio i the secod sese i : R R is sid to e s-covex or ll y R λ [ ] s ( λ x ( λ y λ ( x ( λ ( y (. x d s ( ]. It is cler tht -covex uctio ust e covex o R s usul. The s-covex uctios i the secod sese hve ee ivestigted i (see [6]. s
4 ARTION KASHURI d ROZANA LIKO R Deiitio.6 (see [4]. A set K is sid to e ivex with resect to the ig η : K K R i x tη( y x K or every x y K d t [ ]. Notice tht every covex set is ivex with resect to the ig η ( y x y x ut the coverse is ot ecessrily true. For ore detils lese see ([4] [5] d the reereces therei. Deiitio.7 (see [7]. The uctio deied o the ivex set K R is sid to e reivex with resect η i or every x y K d t [ ] we hve tht ( x tη( y x ( t ( x t ( y. The cocet o reivexity is ore geerl th covexity sice every covex uctio is reivex with resect to the ig η ( y x y x ut the coverse is ot true. Motivted y these results the i o this er is to estlish soe geerliztios o Herite-Hdrd tye ieulities usig ew idetity give i Sectio or geerlized ( -reivex uctios vi Rie-Liouville rctiol itegrl. I Sectio 3 soe lictios to secil es re give. These geerl ieulities give us soe ew estites or Herite-Hdrd tye rctiol itegrl ieulities.. Mi Results Deiitio. (see []. A set K is sid to e -ivex with resect to the ig η : K K ( ] R or soe ixed ( ] i x tη( y x K holds or ech x y K d y t [ ]. Rerk.. I Deiitio. uder certi coditios the ig η ( y x could reduce to η ( y x. R
HERMITE-HADAMARD TYPE FRACTIONAL 5 Deiitio.3 (see []. Let K e oe -ivex set with resect to η : K K ( ] R d : I R cotiuous icresig uctio. For R : K R d y ixed s ( ] i s ( ( x tη( ( y ( x ( t ( ( x t ( ( y (. is vlid or ll x y K t [ ] the we sy tht ( x is geerlized ( s -reivex uctio with resect to η. Rerk.4. For s i Deiitio.3 we sy tht ( x is geerlized ( -reivex uctio with resect to η. Rerk.5. I Deiitio.3 it is worthwhile to ote tht the clss o geerlized ( s -reivex uctio is geerliztio o the clss o s-covex i the secod sese uctio give i Deiitio.5. Throughout this er we deote A ( x; η λ µ s η( ( ( x η( ( ( { λ ( ( x η( ( ( x ( λ ( ( x } η( ( ( x η( ( ( Γ( η( ( ( { µ ( ( x ( µ ( ( x η( ( ( x } [ J ( ( ( ( ( ( ( x J ( ( ( ( ( ( ( x ] x η x x η x where x r ( r or r [ ]. I this sectio i order to rove our i results regrdig soe Herite-Hdrd tye ieulities or geerlized ( -reivex uctio vi rctiol itegrls we eed the ollowig les:
6 ARTION KASHURI d ROZANA LIKO Le.6. Let : I R e cotiuous icresig uctio. Suose K R e oe -ivex suset with resect to η : K K ( ] R or y ixed ( ] d let ( ( K < with η( ( (. Assue tht : K R is dieretile uctio o [ ]. The or y λ µ [ ] d > the K d L ( ( ollowig idetity holds: A ( x; η λ µ η( ( ( x η( ( ( ( t µ ( ( x tη( ( ( x dt η( ( ( x η( ( ( ( λ t ( ( x tη( ( ( x dt (. where x r ( r or r [ ]. Proo. A sile roo o the eulity (. c e doe y erorig itegrtio y rts i the itegrls ro the right side d chgig the vrile. The detils re let to the iterested reder. For the silicities o ottios let ( t dt ( t t dt ( t dt. (.3
HERMITE-HADAMARD TYPE FRACTIONAL 7 Le.7. For we hve ( ( < <. ; ; ( ( ( < <. ; ; (c ( ( ( β < < β. ; ; ; ; F Proo. These eulities ollows ro strightowrd couttio o deiite itegrls. Now we tur our ttetio to estlish ew itegrl ieulities o Herite-Hdrd tye or geerlized ( -reivex uctios vi rctiol itegrls. Usig Les.6 d.7 the ollowig results c e otied or the corresodig versio or ower o the solute vlue o the irst derivtive:
8 ARTION KASHURI d ROZANA LIKO Theore.8. Let : I R e cotiuous icresig uctio. Suose A R e oe -ivex suset with resect to η : A A ( ] R or y ixed ( ] d let ( ( A < with η( ( (. Assue tht : A R is dieretile uctio o A. I is geerlized ( -reivex uctio o [ ( ( ] > the or y λ µ [ ] d > the ollowig ieulity or rctiol itegrls holds: A ( x; η λ µ η( ( ( x η( ( ( ( λ ( ( ( ( x η( ( ( x ( ( ( ( x (. ( ( ( µ η (.4 Proo. Suose tht >. Usig Les.6 d.7 the ct tht is geerlized ( -reivex uctio roerty o the odulus d Hölder s ieulity we hve A ( x; η λ µ η( ( ( x λ t η( ( ( ( ( x tη( ( ( x dt η( ( ( x t η( ( ( µ ( ( x tη( ( ( x dt
HERMITE-HADAMARD TYPE FRACTIONAL 9 η( ( ( x η( ( ( λ t dt ( ( x tη( ( ( x dt η( ( ( x η( ( ( t µ dt ( ( x tη( ( ( x dt η( ( ( x η( ( ( ( λ ( ( t ( ( x t ( ( dt η( ( ( x η( ( ( ( µ ( ( t ( ( x t ( ( dt η( ( ( x η( ( ( ( λ ( ( ( ( x η( ( ( x ( ( ( ( x (. ( ( ( µ η The roo o Theore.8 is coleted.
ARTION KASHURI d ROZANA LIKO Corollry.9. Uder the coditios o Theore.8 i we choose µ λ x d ( ( ( ( ( x y x y η we get the ollowig geerlized Herite-Hdrd tye ieulity or rctiol itegrls: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Γ J J ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. (.5 Theore.. Let R I : e cotiuous icresig uctio. Suose R A e oe -ivex suset with resect to η A A : ( ] R or y ixed ( ] d let ( ( A < with ( ( (. η Assue tht R A : is dieretile uctio o. A I is geerlized ( -reivex uctio o ( ( [ ] the or y [ ] µ λ d > the ollowig ieulity or rctiol itegrls holds:
HERMITE-HADAMARD TYPE FRACTIONAL A ( x; η λ µ η( ( ( x η( ( ( ( λ [ ( λ ( ( ( ( λ ( λ ( ( x ] η( ( ( x η( ( ( ( µ [ ( µ ( ( ( ( µ ( µ ( ( x ]. (.6 Proo. Suose tht. Usig Les.6 d.7 the ct tht is geerlized ( -reivex uctio roerty o the odulus d the well-kow ower e ieulity we hve A ( x; η λ µ η( ( ( x λ t η( ( ( ( ( x tη( ( ( x dt η( ( ( x t η( ( ( µ ( ( x tη( ( ( x dt η( ( ( x η( ( ( λ t dt λ t ( ( x tη( ( ( x dt η( ( ( x η( ( ( t µ dt t µ ( ( x tη( ( ( x dt
ARTION KASHURI d ROZANA LIKO ( ( ( ( ( ( ( λ η η x ( ( ( ( ( ( ( ( [ ] x λ λ λ ( ( ( ( ( ( ( µ η η x ( ( ( ( ( ( ( ( [ ]. x µ µ µ The roo o Theore. is coleted. Corollry.. Uder the coditios o Theore. i we choose µ λ x d ( ( ( ( ( x y x y η we get the ollowig geerlized Herite-Hdrd tye ieulity or rctiol itegrls: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Γ J J ( ( ( ( ( ( ( ( ( ( ( ( ( (. (.7
HERMITE-HADAMARD TYPE FRACTIONAL 3 3. Alictios to Secil Mes I the ollowig we give certi geerliztios o soe otios or ositive vlued uctio o ositive vrile. Deiitio 3. (see [7]. A uctio M : R R is clled e uctio i it hs the ollowig roerties: ( Hoogeeity: M ( x y M ( x y or ll >. ( Syetry: M ( x y M ( y x. (3 Relexivity: M ( x x x. (4 Mootoicity: I x x d y y the M ( x y M ( x y. (5 Iterlity: i{ x y} M ( x y x{ x y}. We cosider soe es or ritrry ositive rel uers β ( β. ( The rithetic e: A β : A( β. ( The geoetric e: G : G( β β. (3 The hroic e: H : H( β. β (4 The ower e: Pr : ( β r Pr β. r r r
4 ARTION KASHURI d ROZANA LIKO (5 The idetric e: I β : I( β e β β; β. (6 The logrithic e: β L : L( β ; β β. l β l (7 The geerlized log-e: L : L ( β β ( ( β ; R \ { } β. (8 The weighted -ower e: M iu i u i u u where > ( i with. i u i i It is well-kow tht L is ootoic odecresig over R with L : L d L : I. I rticulr we hve the ollowig ieulity H G L I A. Now let d e ositive rel uers such tht <. Cosider the uctio M : M ( ( ( : [ ( ( η( ( ( ] [ ( ( η( ( ( ] R which is oe o the ove etioed es d : I R e cotiuous icresig uctio thereore oe c oti vrious ieulities usig the results o Sectio or these es s ollows: Relce η ( ( y ( x with η ( ( y ( x d settig η ( ( ( M ( ( ( η( ( ( x M ( ( ( x η( ( ( x M ( ( ( x x A or vlue i (.4 d (.6 oe c oti the ollowig iterestig ieulities ivolvig es: i
HERMITE-HADAMARD TYPE FRACTIONAL 5 M ( ( ( x { λ ( ( x M ( ( ( x ( λ ( ( x } M ( ( ( M ( ( ( x M ( ( ( Γ( M ( ( ( { µ ( ( x ( µ ( ( x M ( ( ( x } [ ] J ( ( ( ( ( ( ( x J ( ( ( ( x M x x M x ( ( ( x M ( ( ( x M ( ( ( ( λ ( ( ( ( x M ( ( ( x M ( ( ( ( ( ( ( x ( µ (3. M ( ( ( x { λ ( ( x M ( ( ( x ( λ ( ( x } M ( ( ( M ( ( ( x M ( ( ( Γ( M ( ( ( { µ ( ( x ( µ ( ( x M ( ( ( x } [ ] J ( ( ( ( ( ( ( x J ( ( ( ( x M x x M x ( ( ( x M ( ( ( x M ( ( ( ( λ [ ( λ ( ( ( ( λ ( λ ( ( x ] M ( ( ( x M ( ( ( ( µ [ ( µ ( ( ( ( µ ( µ ( ( x ]. (3.
6 ARTION KASHURI d ROZANA LIKO Lettig M ( ( x ( y A G H P I L L M x y A i (3. d (3. we get the ieulities ivolvig es or rticulr choices o dieretile geerlized ( -reivex uctio. The detils re let to the iterested reder. Reereces r [] A. Kshuri d R. Liko Ostrowski tye rctiol itegrl ieulities or geerlized ( s -reivex uctios Aust. J. Mth. Al. Al. 3( (6 - Article 6. [] T. S. Du J. G. Lio d Y. J. Li Proerties d itegrl ieulities o Hdrd- Siso tye or the geerlized (s -reivex uctios J. Nolier Sci. Al. (6 3-36. [3] S. S. Drgoir J. Pečrić d L. E. Persso Soe ieulities o Hdrd tye Soochow J. Mth. (995 335-34. [4] T. Atczk Me vlue i ivexity lysis Nolier Al. 6 (5 473-484. [5] X. M. Yg X. Q. Yg d K. L. Teo Geerlized ivexity d geerlized ivrit ootoicity J. Oti. Theory Al. 7 (3 67-65. [6] H. Hudzik d L. Mligrd Soe rerks o s-covex uctios Aeutioes Mth. 48 (994 -. [7] R. Pii Ivexity d geerlized covexity Otiiztio (99 53-55. [8] H. Kvurci M. Avci d M. E. Özdeir New ieulities o Herite-Hdrd tye or covex uctios with lictios rxiv: 6.593v [th. CA] ( -. [9] W. Liu W. We d J. Prk Herite-Hdrd tye ieulities or MT-covex uctios vi clssicl itegrls d rctiol itegrls J. Nolier Sci. Al. 9 (6 766-777. [] W. Liu W. We d J. Prk Ostrowski tye rctiol itegrl ieulities or MT-covex uctios Miskolc Mtheticl Notes 6( (5 49-56. [] Y. M. Chu G. D. Wg d X. H. Zhg Schur covexity d Hdrd s ieulity Mth. Ieul. Al. 3(4 ( 75-73. [] X. M. Zhg Y. M. Chu d X. H. Zhg The Herite-Hdrd tye ieulity o GA-covex uctios d its lictios J. Ieul. Al. ( Article ID 5756 ges. [3] Y. M. Chu M. A. Kh T. U. Kh d T. Ali Geerliztios o Herite- Hdrd tye ieulities or MT-covex uctios J. Nolier Sci. Al. 9(5 (6 435-436.
HERMITE-HADAMARD TYPE FRACTIONAL 7 [4] M. Tuç Ostrowski tye ieulities or uctios whose derivtives re MT-covex J. Cout. Al. Al. 7(4 (4 69-696. [5] W. J. Liu Soe Siso tye ieulities or h-covex d ( -covex uctios J. Cout. Al. Al. 6(5 (4 5-. [6] F. Qi d B. Y. Xi Soe itegrl ieulities o Siso tye or GA -covex uctios Georgi Mth. J. (5 (3 775-788. [7] P. S. Bulle Hdook o Mes d their Ieulities Kluwer Acdeic Pulishers Dordrecht 3. g