- TAMKANG JOURNAL OF MATHEMATICS Volume 5, Number, 7-5, June doi:5556/jkjm555 Avilble online hp://journlsmhkueduw/ - - - GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS MARCELA V MIHAI AND DANIEL ALEXANDRU ION Absrc Some Hermie-Hdmrd ype ineuliies re provided We del wih funcions whose derivives in bsolue vlue re convex or concve By defining wo cumulive gps which enble us o generlize known resuls in he frmework of Riemnn- Liouville frcionl clculus, we open new perspecive on he clssic semen of he ineuliy Inroducion The Hermie-Hdmrd ineuliy ses h if funcion f : [,b] R is convex hen b f f xdx Boh ineuliies hold in he reversed direcion if f is concve f f b, H H This ineuliy received gre del of enion in he ls decde insnce, mny generlizions pplicions being obined See [], [], [], [6], [8], [] he references herein Of specil ineres o us is he following improvemens of he Hermie-Hdmrd ineuliy, h cn be found in he monogrph [7], p 5: b f [ f 3 b f 3b ] f xdx, LH H f xdx [ ] f f b b f f f b, R H H The purpose of he presen pper is o esblish new Hermie-Hdmrd ype ineuliies wihin Riemnn-Liouville frcionl clculus Unlike he clssicl cse, he funcions under enion re no ssumed convex or concve, bu his fc is sked for he bsolue vlue Received November 8, 3, cceped December 3, 3 Mhemics Subjec Clssificion 6A5 Key words phrses Convex funcion, Hermie-Hdmrd ineuliy, Riemnn-Liouville frcionl inegrls 7
8 MARCELA V MIHAI AND DANIEL ALEXANDRU ION of heir derivives Under hese circumsnces we will prove he exisence of wo srings of ineuliies refining he ineuliies LH H R H H In wh follows we will consider only rel-vlued funcions defined on inervls [, b] wih < b, n is n odd number Le f L [,b] be n inegrble funcion J α f, of orderα>, ched o f re defined respecively by b The Riemnn-Liouville inegrls J α f J α f x= Γα J α b f x= Γα x x x α f, for x>, x α f, for x < b Here,Γα= e α is he Gmm funcion We mke he convenion J f x= J b f x= f x The heory of Riemnn-Liouville frcionl inegrls cn be found in he book [3] Min resuls As bove, we ssume h [,b] is compc subinervl of [, f : [,b] R is n inegrble funcion We define he cumulive o he lef α,n gp of f by he formul n / L α,n,b= f k= Γα n kbk J α n k bk n f α n / k= [ J α n kbk n f n kbk In he priculr cse whereα= n= 3 we hve L,3,b = [ f 3 b f 3b n k b k ] ] f, so he cumulive o he lef gp L,3,b esimes he precision of he righ h side ineuliy in LH H f xdx [ f 3 b f 3b ]
GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 9 The cumulive o he lef gps L α,n,b hve he sme mening, relive o higher order refinemens of LH H The following echnicl lemm provides suible formul for esiming L α,n,b in bsolue vlue: Lemm We hve L α,n,b= n / [ α f n kbk n k b k k= α f n k bk n kbk Proof Pu I k = I k = α f n kbk n k b k, ] α f n k bk n kbk By using he inegrion by prs he subsiuions n kbk n k b k u =, n k bk n kbk v =, we infer h n kbk I k I k = f Γα [ n k b k J α f n kbk n ] n kbk J α f n k bk n The proof is compleed α We re now in posiion o se prove he following resul: Theorem Assume h f : [,b] R is differenible funcion such h f is convex on [,b] Then Lα,n,b n / k= [ α 5α n kbk αα f
MARCELA V MIHAI AND DANIEL ALEXANDRU ION n k b k αα f ] α n k bk α f Proof Using Lemm he convexiy of f we obin Lα,n,b n /[ n kbk f α k= n k b k f α n k bk f α n kbk ] f α The proof ends fer srighforwrd compuion in he righ h side erm The Be funcion is defined by he formul Bx, y= x y for x, y > Our nex resul is s follows: Theorem Assume h f : [,b] R is differenible funcion such h f is convex on [,b] for some exponen > Then Lα,n,b where p = n / { p k= αp n kbk [ f n k b k f ] ] p [ α B p, α [ n k bk f n kbk f Proof According o Lemm Hölder s ineuliy, we hve Lα,n,b n / k= [ α p p ] }
GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS f n kbk n k b k α p p f n k bk n kbk ] Since f is convex on [,b], we hve: f n kbk n k b k [ n kbk f n k b k f ] f n k bk n kbk [ n k bk f n kbk f ] A simple compuion shows h he proof is compleed αp = αp, α p = α B p, α Theorem 3 Assume h f : [,b] R is differenible funcion such h f is convex on [, b] for some exponen Then he following ineuliy holds: Lα,n,b α p α { n / [ n kbk f n k b k k= α f α [ n k bk f α3 n kbk α f where p = Proof Using Lemm he power men ineuliy, we hve Lα,n,b α f n / k= [ α p n kbk n k b k ] ] },
MARCELA V MIHAI AND DANIEL ALEXANDRU ION α p α f n k bk n kbk ] Since f is convex on [,b], we hve: α f n kbk n k b k n kbk α f n k b k αα f α f n k bk n kbk α n k bk α f α 3α n kbk αα f This complees he proof of he heorem Theorem Assume h f : [,b] R is differenible funcion such h f is concve on [,b] for some exponen > Then where p = Lα,n,b α B n / k= [ p, α /p n k bk f αp /p ] n k bk 3 f, Proof From Lemm Hölder s inegrl ineuliy for > p =, we hve Lα,n,b n / k= [ α p p f n kbk n k b k α p p f n k bk n kbk ]
GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 3 Since f is concve on [,b], we infer from Jensen s ineuliy for concve funcions h f n kbk n k b k n kbk n k b k f = n k bk f In he sme mnner, f n k bk n kbk n k bk 3 f Using we complee he proof αp = Our nex resul is s follows: αp, α p = α B p,, α Theorem 5 Assume h f : [,b] R is differenible funcion such h f is concve on [,b] for some exponen > Then Lα,n,b α n /[ α f k= α α α f α Proof From Lemm we hve Lα,n,b n / k= [ n kbk n k b k α n k bk α3 n kbk α α f n kbk n k b k α f n k bk n kbk Since f is concve, by Jensen s ineuliy we obin Lα,n,b n / [ α I k= α ] I, ] ]
MARCELA V MIHAI AND DANIEL ALEXANDRU ION where n kbk n k b k I = f α α = α f n kbk n k b k α α I = f n k bk n kbk α α = α f n k bk α3 n kbk α α he proof is compleed Remrk Forα= in he Theorems,, 3, respecively 5, we recover he resuls sed in [9, Theorems 6-] Also, forα= in Lemm, we ge [9, Lemm ] Remrk For n = 3 in he Theorems,, 3, respecively, we recover he resuls sed in [5, Theorems -] Also forα= in Lemm, we ge [5, Lemm ] We end our pper by considering he cumulive o he righ α,n gp defined by he formul n /[ ] n k b k n k bk R α,n,b= f f k= α n / [ n kbk Γα J α n kb k f k= n ] n k bk J α f n kbk, n In he priculr cse whereα= n= 3 we hve R,3,b = [ ] f f b b f f, so he cumulive o he righ gp R,3,b esimes he precision of he lef h side ineuliy in R H H, f xdx [ ] f f b b f Using he bove echniues, one cn prove compnions of ll he resuls we proved for he cumulive o he lef α,n gp The sring poin is he following formul for compuing R α,n,b
GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 5 Lemm We hve R α,n,b= n / [ k= α f α f n k b k n kbk n kbk n k bk Using Lemm, one cn prove vrious esimes of R α,n,b such s Rα,n,b n /[ n k b k k= α f α α n kbk αα f α 3α n k bk αα f ] ] References [] M Bessenyei, The Hermie Hdmrd Ineuliy on Simplices, Americn Mhemicl Monhly, 5 8, 339 35 [] S S Drgomir C E M Perce, Seleced Topic on Hermie-Hdmrd Ineuliies Applicions, Melbourne Adelide, December, [3] R Gorenflo F Minrdi, Frcionl Clculus: Inegrl Differenil Euions of Frcionl Order, Springer Verlg, Wien, 997 [] H Kvurmci M Avci, M E Özdemir, New ineuliies of Hermie-Hdmrd ype for convex funcions wih pplicions, rxiv: 6593v [5] M Mihi, Some Hermie-Hdmrd ype ineuliies obined vi Riemnn-Liouville frcionl clculus submied [6] F-C Miroi C I Spiridon, Hermie-Hdmrd ype ineuliies of convex funcions wih respec o pir of usi-rihmeic mens, Mh Rep, [7] C P Niculescu L-E Persson, Convex Funcions heir Applicions A Conemporry Approch, CMS Books in Mhemics vol 3, Springer-Verlg, New York, 6 [8] C P Niculescu, The Hermie-Hdmrd ineuliy for log-convex funcions, Nonliner Anlysis 75, 66 669 [9] M Emin Özdemir, A Ekinci A Akdemir, Some new inegrl ineuliies for funcions whose derivives of bsolue vlues re convex concve, RGMIA Reserch Repor Collecion, 5, Aricle 8, pp [] S W sowicz A Wikowski, On some ineuliy of Hermie-Hdmrd ype, Opuscul Mh, 3, 59 6 Deprmen of Mhemics, Universiy of Criov, Sree A I Cuz 3, Criov, RO-585, Romni E-mil: mrcelmihi58@yhoocom Deprmen of Mhemics, Universiy of Criov, Sree A I Cuz 3, Criov, RO-585, Romni E-mil: dn_lexion@yhoocom