Research Article Generalized Fractional Integral Inequalities for Continuous Random Variables

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1 Journl of Proiliy nd Sisics Volume 2015, Aricle ID , 7 pges hp://dx.doi.org/ /2015/ Reserch Aricle Generlized Frcionl Inegrl Inequliies for Coninuous Rndom Vriles Adullh Akkur, Zeynep Kçr, nd Hüseyin Yildirim Deprmen of Mhemics, Fculy of Science nd Ars, Universiy of Khrmnmrş Süçü İmm, Khrmnmrş, Turkey Correspondence should e ddressed o Adullh Akkur; dullhm@gmil.com Received 5 Ocoer 2014; Acceped 9 Decemer 2014 AcdemicEdior:Z.D.Bi Copyrigh 2015 Adullh Akkur e l. This is n open ccess ricle disriued under he Creive Commons Ariuion License, which permis unresriced use, disriuion, nd reproducion in ny medium, provided he originl work is properly cied. Some generlized inegrl inequliies re eslished for he frcionl expecion nd he frcionl vrince for coninuous rndom vriles. Specil cses of inegrl inequliies in his pper re sudied y Brne e l. nd Dhmni. 1. Inroducion Inegrl inequliies ply fundmenl role in he heory of differenil equions, funcionl nlysis, nd pplied sciences. Imporn developmen in his heory hs een chieved in he ls wo decdes. For hese, see [1 8 nd he references herein. Moreover, he sudy of frcionl ype inequliies is lso of vil impornce. Also see [9 13 for furher informion nd pplicions. The firs one is given in [14; in heir pper, using Korkine ideniy nd Holder inequliy for doule inegrls, Brne e l. eslished severl inegrlinequliiesfor he expecione(x) nd he vrince σ 2 (X) of rndom vrile X hving proiliy densiy funcion (p.d.f.) f : [, R.In[15 17 he uhors presened new inequliies for he momens nd for he higher order cenrl momens of coninuous rndom vrile. In [17, 18 Dhmni nd Mio nd Yng gve new upper ounds for he sndrd deviion σ(x), for he quniy σ 2 (X)( E(X)) 2, [,,ndforhel p solue deviion of rndom vrile X. Recenly, Ansssiou e l. [9 proposed generlizion of he weighed Mongomery ideniy for frcionl inegrls wih weighed frcionl Peno kernel. More recenly, Dhmni nd Niezgod [17, 19 gve inequliies involving momens of coninuous rndom vrile defined over finie inervl. Oher ppers deling wih hese proiliy inequliies cn e found in [ In his pper, we inroduce new conceps on generlized frcionl rndom vriles. We oin new generlized inegrl inequliies for he generlized frcionl dispersion nd he generlized frcionl vrince funcions of coninuous rndom vrile X hving he proiliy densiy funcion (p.d.f.) f:[, R y using hese conceps. Our resuls re exension of [12, 14, Preliminries Definiion 1 (see [23). Le f L 1 [,. The Riemnn- Liouville frcionl inegrls J α f(x) nd Jα f(x) of order α 0redefined y J α x 1 [f (x) = J α 1 [f (x) = x (x ) α 1 f () d x>, (1) ( x) α 1 f () d x<, (2) respecively, where Γ(α) = e u u α 1 du is Gmm funcion 0 nd J 0 f(x) = J0 f(x) = f(x). We give he following properies for he J α : J α Jβ [f () =J αβ [f (), α 0, β 0, (3) J α Jβ [f () =J β J α [f (), α 0, β 0.

2 2 Journl of Proiliy nd Sisics Definiion 2 (see [24). Consider he spce L p,k (, ) (k 0, 1 p < ) of hose rel-vlued Leesgue mesurle funcions f on [, for which f L p,k(,) 1/p =( f (x) p x k dx) <, 1 p<, k 0. Definiion 3 (see [24). Consider he spce X p c (, ) (c R, 1 p < ) ofhoserel-vluedleesguemesurle funcions f on [, for which f X p c nd for he cse p= f X c =( xc f (x) p dx 1/p x ) <, (1 p<, c R) (4) (5) = ess sup [x c f (x), c R. (6) x In priculr, when c = (k 1)/p (1 p<,k 0)he spce X p c (, ) coincides wih he L p,k(, )-spce nd lso if we ke c = (1/p) (1 p< ) he spce X p c (, ) coincides wih he clssicl L p (, )-spce. Definiion 4 (see [24). Le f L 1,k [,. The generlized Riemnn-Liouville frcionl inegrls f(x) nd f(x) of orders α 0nd k 0redefined y (k1)1 α f (x) = (k1)1 α f (x) = x (x k1 k1 ) α 1 k f () d ( k1 x k1 ) α 1 k f () d x x>, (7) > x. (8) Here Γ(α) is Gmm funcion nd J 0,k f(x) = J 0,k f(x) = f(x).inegrlformuls(7) nd (8) reclledrighgenerlized Riemnn-Liouville inegrl nd lef generlized Riemnn- Liouville frcionl inegrl, respecively. Definiion 5. The frcionl expecion funcion of orders α 0nd k 0, for rndom vrile X wih posiive p.d.f. f defined on [,, is defined s E X,α () := [f () = (k1)1 α ( k1 τ k1 ) α 1 τ k1 f (τ) dτ, <. Inhesmewy,wedefinehefrcionlexpecionfuncion of X E(X)y wh follows. (9) Definiion 6. The frcionl expecion funcion of orders α 0, k 0,nd<, for rndom vrile X E(X), is defined s E X E(X),α () := (k1)1 α ( k1 τ k1 ) α 1 (τ E(X)) τ k f (τ) dτ, where f:[, R is he p.d.f. of X. For =, we inroduce he following concep. (10) Definiion 7. The frcionl expecion of orders α 0, <,ndk 0, for rndom vrile X wih posiive p.d.f. f defined on [,, is defined s E X,α = (k1)1 α ( k1 τ k1 ) α 1 τ k1 f (τ) dτ. (11) For he frcionl vrince of X,we inroducehe following wo definiions. Definiion 8. The frcionl vrince funcion of orders α 0, <,ndk 0, for rndom vrile X hving p.d.f. f:[, R, is defined s σ 2 X,α := Jα,k [( E(X))2 f () = (k1)1 α ( k1 τ k1 ) α 1 (τ E(X)) 2 τ k f (τ) dτ, (12) where E(X) := τf(τ)dτ is he clssicl expecion of X. Definiion 9. The frcionl vrince of order α 0,for rndom vrile X wih p.d.f. f:[, R, is defined s σ 2 X,α = (k1)1 α ( k1 τ k1 ) α 1 (τ E(X)) 2 τ k f (τ) dτ. We give he following imporn properies. (13) (1) If we ke α=1nd k=0in Definiion 5,weoin he clssicl expecion E X,1 = E(X). (2) If we ke α = 1 nd k = 0 in Definiion 7, we oin he clssicl vrince σ 2 X,1 =σ2 (X) = (τ E(X)) 2 f(τ)dτ. (3) If we ke k = 0 in Definiions 5 9, weoin Definiions in [17. (4) For α > 0,hep.d.f.f sisfies J α [f() = ( ) α 1 /Γ(α). (5) For α = 1, we hve he well known propery J α [f() = 1.

3 Journl of Proiliy nd Sisics 3 3. Min Resuls Theorem 10. Le X e coninuous rndom vrile hving p.d.f. f:[, R.Then () for ll <, α 0,ndk 0, [f ()σ2 X,α () (E X E(X),α()) 2 f 2 [ [ (k1) 1 α ( k1 k1 ) α Γ (α1) [2k2 (14) If, in (18),wekep() = f() nd g() = h() = k1 E(X), (, ),henwehve (k1) 2 2α =2 ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 f(τ) f(ρ)(τ k1 ρ k1 ) 2 τ k ρ k f (τ) dτ dρ [f ()Jα,k [f () (k1 E(X)) 2 ( [)2, 2[ f() (k1 E(X)) 2. (19) provided h f L [, ; () he inequliy [f ()σ2 X,α () (E X E(X),α ()) (k1 k1 ) 2 ( [)2 (15) is lso vlid for ll <, α 0,ndk 0. Proof. Le us define he quniy for p.d.f. g nd h: H (τ, ρ) := (g (τ) g (ρ)) (h (τ) h (ρ)) ; τ, ρ (, ), <, α 0. (16) Tking funcion p:[, R, muliplying (16) y (( k1 τ k1 ) α 1 /Γ(α))p(τ)τ k,τ (,), nd hen inegring he resuling ideniy wih respec o τ from o,wehve (k1) 1 α ( k1 τ k1 ) α 1 p (τ) H (τ, ρ) τ k f (τ) dτ = [pgh () h(ρ)jα,k [pg () g(ρ) [ph () g (ρ) h (ρ) Jα,k [p (). (17) Similrly, muliplying (17) y (( k1 ρ k1 ) α 1 /Γ(α))p(ρ)ρ k, ρ (, ), nd inegring he resuling ideniy wih respec o ρ over (, ),wecnwrie On he oher hnd, we hve (k1) 2 2α ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 f(τ) f(ρ)(τ k1 ρ k1 ) 2 τ k ρ k f (τ) dτ dρ f 2 [ 2 (k1)1 α ( k1 k1 ) α Γ (α1) [2k2 [ 2( [)2. Thnks o (19) nd (20), we oin pr () of Theorem 10. For pr (), we hve (k1) 2 2α ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 (20) f(τ) f(ρ)(τ k1 ρ k1 ) 2 τ k ρ k f (τ) dτ dρ sup (τk1 ρ k1 ) τ,ρ [, 2 [J α,k f ()2 (k1) 2 2α ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 =( k1 k1 ) 2 [ f ()2. (21) =2 2 p(τ) p (ρ) H (τ, ρ) τ k ρ k f (τ) dτ dρ [p ()Jα,k [pgh () [pg ()Jα,k [ph (). (18) Then, y (19) nd (21), we ge he desired inequliy (14). We give lso he following corollry. Corollry 11. Le X e coninuous rndom vrile wih p.d.f. f defined on [,.Then

4 4 Journl of Proiliy nd Sisics (i) if f L [,,henfornyα 0nd k 0,onehs ( k1 k1 ) (α 1) σ 2 X,α E2 X,α f 2 ( k1 k1 ) 2α2 k1 [ Γ (α1) Γ (α3) (( k1 ) α1 ) ; Γ (α1) [ (22) (ii) he inequliy ( k1 k1 ) (α 1) σ 2 X,α E2 X,α 1 2 [ ( k1 k1 ) 2α [ (23) is lso vlid for ny α 0nd k 0. Remrk 12. (r1) Tking α=1nd k=0in (i) of Corollry 11, we oin he firs pr of Theorem 1 in [14. (r2) Tking α=1nd k=0in (ii) of Corollry 11, we oin he ls pr of Theorem 1 in [14. We will furher generlize Theorem 10 y considering wo frcionl posiive prmeers. Theorem 13. Le X e coninuous rndom vrile hving p.d.f. f:[, R. Then one hs he following. () For ll <, α 0, β 0,ndk 0, [f ()σ2 X,β () Jβ,k [f ()σ 2 X,α () 2(E X E(X),α ())(E X E(X),β ()) f 2 [ [ f 2 [ [ (k1) 1 α ( k1 k1 ) α Γ (α1) (k1) 1 β ( k1 k1 ) β 2( where f L [,. () The inequliy [f ()σ2 X,β Γ(β1) [)(Jβ,k J β,k [ 2k2 [), () Jβ,k [f ()σ 2 X,α () [2k2 2 (24) Proof. Using (15),we cn wrie (k1) 2 α β Γ(β) = ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 [p ()Jβ,k p(τ) p (ρ) H (τ, ρ) τ k ρ k f (τ) dτ dρ [pgh ()J β,k [ph ()Jβ,k [pg () [p () [pgh () J β,k [ph () [pg (). (26) Tking p() = f() nd g() = h() = k1 E(X), (, ), in he ove ideniy, yields (k1) 2 α β Γ (β) = We hve lso (k1) 2 α β Γ(β) ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 p(τ) p(ρ)(τ k1 ρ k1 ) 2 τ k ρ k f (τ) dτ dρ [f ()Jβ,k [f () ( k1 E(X)) 2 J β,k [f () [f () (k1 E(X)) 2 2 [f () (k1 E(X)) J β,k [f() ( k1 E(X)). f 2 [ [ ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 p(τ) p(ρ)(τ k1 ρ k1 ) 2 τ k ρ k f (τ) dτ dρ (k1) 1 α ( k1 k1 ) α Γ (α1) J β,k [ 2k2 (k1)1 β ( k1 k1 ) β Γ(β1) [2k2 (27) 2(E X E(X),α ())(E X E(X),β ()) ( k1 k1 ) 2 ( [)(Jβ,k [) (25) 2( [)(Jβ,k [). (28) is lso vlid for ny <, α 0, β 0,ndk 0. Thnks o (27) nd (28), we oin ().

5 Journl of Proiliy nd Sisics 5 Toprove(),weusehefchsup τ,ρ [, (τ k1 ρ k1 ) 2 =( k1 k1 ) 2.Weoin Corollry 16. Le f e he p.d.f. of X on [,. Thenforny α 0nd k 0,onehs (k1) 2 α β Γ(β) ( k1 τ k1 ) α 1 ( k1 ρ k1 ) α 1 ( k1 k1 ) (α 1) σ 2 X,α (E X E(X),α ()) (k1 k1 ) 2α. (34) f(τ) f(ρ)(τ k1 ρ k1 ) 2 τ k ρ k f (τ) dτ dρ ( k1 k1 ) 2 ( [)(Jβ,k [). And, y (27) nd (29),wege(25). (29) Remrk 14. (r1) Applying Theorem 13 for α=β,weoin Theorem 10. We give lso he following frcionl inegrl resul. Remrk 17. Tking α=1in Corollry 16,weoinTheorem 2of[14. We lso presen he following resul for he frcionl vrince funcion wih wo prmeers. Theorem 18. Le f e he p.d.f. of he rndom vrile X on [,.Thenforll<, α 0, β 0,ndk 0,onehs [f ()σ2 X,β () Jβ,k [f ()σ 2 X,α () Theorem 15. Le f e he p.d.f. of X on [,.Thenforll<, α 0,ndk 0,onehs [f ()σ2 X,α () (E X E(X),α ()) (k1 k1 ) 2 ( [)2. Proof. Using Theorem 1 of [25, we cn wrie Jα,k [p ()Jα,k [pg2 () ( [pg ())2 1 4 (Jα,k [p())2 (M m) 2. (30) (31) Tking p() = f() nd g() = k1 E(X), (, ), hen M= k1 E(X) nd m= k1 E(X).Hence,(30) llows us o oin 0 [f ()Jα,k [f () (k1 E(X)) 2 ( [f() (k1 E(X)) 2 ) (Jα,k [f())2 ( k1 k1 ) 2. (32) 2( k1 E(X))( k1 E(X)) [f ()Jβ,k [f () ( k1 k1 2E(X)) ( [f ()(E X E(X),β ()) J β,k [f ()(E X E(X),α ())). Proof. Thnks o Theorem 4 of [25, we cn se h [ [p ()Jβ,k 2 [pg 2 ()J β,k [pg() Jβ,k [pg() 2 [(M (J β,k (J β,k [p () Jα,k [pg ()) [p () [pg2 () [pg () mj β,k [p ()) [pg () mj β,k [p ()) (MJ β,k [p() J β,k [pg()) 2. (35) (36) This implies h [f ()σ2 X,α () (E X E(X),α ()) 2 Theorem 15 is hus proved. 1 4 (k1 k1 ) 2 ( [)2. (33) In (35),wekep() = f() nd g() = k1 E(X), (, ). We oin [ [f ()Jβ,k [f () ( k1 E(X)) 2 J β,k [f () [f () (k1 E(X)) 2 For =, we propose he following ineresing inequliy. 2 [f() (k1 E(X)) J β,k [f() ( k1 E(X)) 2

6 6 Journl of Proiliy nd Sisics [(M (J β,k ( [f () Jα,k [f () (k1 E(X))) [f () ( k1 E(X)) mj β,k [f ()) [f () (k1 E(X)) m [f ()) (MJ β,k [f() J β,k [f() ( k1 E(X))) 2. (37) Comining (27) nd (37) ndkinginoccounhefch he lef-hnd side of(27) is posiive, we ge [f ()Jβ,k [f () ( k1 E(X)) 2 J β,k [f () [f () (k1 E(X)) 2 2 (M (J β,k ( Therefore, [f () (k1 E(X)) J β,k [f () ( k1 E(X)) [f () Jα,k [f () (k1 E(X))) [f () ( k1 E(X)) mj β,k [f ()) [f () (k1 E(X)) m [f ()) (MJ β,k [f () J β,k [f () ( k1 E(X))). [f ()Jβ,k [f () ( k1 E(X)) 2 J β,k [f () [f () (k1 E(X)) 2 M( [f ()(E X E(X),β ()) J β,k [f () (E X E(X),α ())) m( [f ()(E X E(X),β ()) J β,k [f () (E X E(X),α ())). (38) (39) Susiuing he vlues of m nd M in (33), hensimple clculion llows us o oin (35). Theorem 18 is hus proved. To finish, we presen o he reder he following corollry. Corollry 19. Le f e he p.d.f. of X on [,. Thenforll <, α 0,ndk 0,heinequliy σ 2 X,α () (k1 E(X))( k1 E(X)) [f () is vlid. ( k1 k1 2E(X))E X E(X),α () (40) Conflic of Ineress The uhors declre h here is no conflic of ineress regrding he pulicion of his pper. References [1 N. S. Brne, P. Cerone, S. S. Drgomir, nd J. Roumeliois, Some inequliies for he expecion nd vrince of rndom vrile whose PDF is n-ime differenile, Journl of Inequliies in Pure nd Applied Mhemics, vol.1,no.2,ricle21, [2 P. L. Cheyshev, Sur les expressions pproximives des inegrles definies pr les ures prises enreles mêmes limies, Proceedings of he Mhemicl Sociey of Khrkov, vol.2,pp.93 98, [3 S. S. Drgomir, A generlizion of Grussís inequliy in inner produc spces nd pplicions, Journl of Mhemicl Anlysis nd Applicions,vol.237,no.1,pp.74 82,1999. [4D.S.Mirinović, J. E. Pečrić, nd A. M. Fink, Clssicl nd New Inequliies in Anlysis, vol.61ofmhemics nd is Applicions, Kluwer Acdemic Pulishers Group, Dordrech, Dordrech, The Neherlnds, [5 B. G. Pchpe, On mulidimensionl Gruss ype inegrl inequliies, Journl of Inequliies in Pure nd Applied Mhemics,vol.32,pp.1 15,2002. [6 F.Qi,A.J.Li,W.Z.Zho,D.W.Niu,ndJ.Co, Exensionsof severl inegrl inequliies, Journl of Inequliies in Pure nd Applied Mhemics,vol.7,no.3,pp.1 6,2006. [7 F. Qi, Severl inegrl inequliies, Journl of Inequliies in Pure nd Applied Mhemics,vol.1,no.2,pp.1 9,2000. [8 M. Z. Sriky, N. Akn, nd H. Yildirim, On weighed Čeyšev-Grüss ype inequliies on ime scles, Journl of Mhemicl Inequliies,vol.2,no.2,pp ,2008. [9 G. A. Ansssiou, M. R. Hooshmndsl, A. Ghsemi, nd F. Mofkhrzdeh, Mongomery ideniies for frcionl inegrls nd reled frcionl inequliies, Journl of Inequliies in Pure nd Applied Mhemics,vol.10,no.4,pp.1 6,2009. [10 G. A. Ansssiou, Frcionl Differeniion Inequliies, Springer Science, [11 S. Belri nd Z. Dhmni, On some new frcionl inegrl inequliies, Journl of Inequliies in Pure nd Applied Mhemics,vol.10,no.3,pp.1 12,2009. [12 Z. Dhmni, New inequliies in frcionl inegrls, InernionlJournlofNonlinerScience,vol.9,no.4,pp , [13 Z. Dhmni, On Minkowski nd Hermie-Hdmd inegrl inequliies vi frcionl inegrion, Annls of Funcionl Anlysis,vol.1,no.1,pp.51 58,2010. [14 N. S. Brne, P. Cerone, S. S. Drgomir, nd J. Roumeliois, Some inequliies for he dispersion of rndom vrile whose PDF is defined on finie inervl, Journl of Inequliies in Pure nd Applied Mhemics,vol.2,no.1,pp.1 18,2001. [15 P. Kumr, Momen inequliies of rndom vrile defined over finie inervl, Journl of Inequliies in Pure nd Applied Mhemics,vol.3,no.3,pp.1 24,2002. [16 P. Kumr, Inequliies involving momens of coninuous rndom vrile defined over finie inervl, Compuers nd Mhemics wih Applicions, vol.48,no.1-2,pp , 2004.

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