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 Received 23 My, 29; ccepted 24 June, 29 Communicted by G. Anstssiou ABSTRACT. In this pper, using the Riemnn-Liouville frctionl integrl, we estblish some new integrl inequlities for the Chebyshev functionl in the cse of two synchronous functions. Key words nd phrses: Frctionl integrl inequlities, Riemnn-Liouville frctionl integrl. 2 Mthemtics Subject Clssifiction. 26D1, 26A33. Let us consider the functionl [1]: 1.1 T f, g := 1 b b 1. INTRODUCTION f x g x dx 1 b b 1 f x dx b b g x dx, i.e. fx where f nd g re two integrble functions which re synchronous on [, b] fygx gy, for ny x, y [, b]. Mny reserchers hve given considerble ttention to 1.1 nd number of inequlities hve ppered in the literture, see [3, 4, 5]. The min purpose of this pper is to estblish some inequlities for the functionl 1.1 using frctionl integrls. 2. DESCRIPTION OF FRACTIONAL CALCULUS We will give the necessry nottion nd bsic definitions below. For more detils, one cn consult [2, 6]. Definition 2.1. A rel vlued function ft, t is sid to be in the spce C µ, µ R if there exists rel number p > µ such tht ft = t p f 1 t, where f 1 t C[, [. Definition 2.2. A function ft, t is sid to be in the spce C n µ, n R, if f n C µ. The uthors would like to thnk professor A. El Frissi for his helpful. 139-9

2 SOUMIA BELARBI AND ZOUBIR DAHMANI Definition 2.3. The Riemnn-Liouville frctionl integrl opertor of order α, for function f C µ, µ 1 is defined s 2.1 J α ft = 1 J ft = ft, t τ α 1 fτdτ; α >, t >, where := e u u α 1 du. For the convenience of estblishing the results, we give the semigroup property: 2.2 J α J β ft = J α+β ft, α, β, which implies the commuttive property: 2.3 J α J β ft = J β J α ft. From 2.1, when ft = t µ we get nother expression tht will be used lter: 2.4 J α t µ = Γµ + 1 Γα + µ + 1 tα+µ, α > ; µ > 1, t >. 3. MAIN RESULTS Theorem 3.1. Let f nd g be two synchronous functions on [, [. Then for ll t >, α >, we hve: 3.1 J α Γα + 1 fgt J α ftj α gt. t α Proof. Since the functions f nd g re synchronous on [, [, then for ll τ, ρ, we hve 3.2 fτ fρ gτ gρ. Therefore 3.3 fτgτ + fρgρ fτgρ + fρgτ. Now, multiplying both sides of 3.3 by t τα 1, τ, t, we get 3.4 t τ α 1 fτgτ + t τα 1 fρgρ Then integrting 3.4 over, t, we obtin: 3.5 1 Consequently, t τ α 1 fτgτdτ + 1 1 3.6 J α fgt + f ρ g ρ 1 g ρ t τα 1 fτgρ + t τ α 1 fρgρdτ t τ α 1 fτgρdτ + 1 t τ α 1 dτ t τ α 1 f τ dτ + f ρ t τα 1 fρgτ. t τ α 1 fρgτdτ. t τ α 1 g τ dτ. J. Inequl. Pure nd Appl. Mth., 13 29, Art. 86, 5 pp. http://jipm.vu.edu.u/

FRACTIONAL INTEGRAL INEQUALITIES 3 So we hve 3.7 J α fgt + f ρ g ρ J α 1 g ρ J α ft + f ρ J α gt. Multiplying both sides of 3.7 by t ρα 1, ρ, t, we obtin: 3.8 t ρ α 1 J α fgt + t ρα 1 f ρ g ρ J α 1 Now integrting 3.8 over, t, we get: t ρα 1 g ρ J α ft + t ρα 1 f ρ J α gt. 3.9 J α fgt Hence t ρ α 1 dρ + J α 1 J α ft fρgρt ρ α 1 dρ t ρ α 1 gρdρ + J α gt 3.1 J α fgt 1 J α 1 J α ftj α gt, nd this ends the proof. The second result is: t ρ α 1 fρdρ. Theorem 3.2. Let f nd g be two synchronous functions on [, [. Then for ll t >, α >, β >, we hve: 3.11 t α Γ α + 1 J β t β fgt + Γ β + 1 J α fgt J α ftj β gt + J β ftj α gt. Proof. Using similr rguments s in the proof of Theorem 3.1, we cn write 3.12 t ρ β 1 J α fg t + J α 1 t ρβ 1 f ρ g ρ By integrting 3.12 over, t, we obtin 3.13 J α fgt nd this ends the proof. t ρ β 1 dρ + J α 1 J α f t t ρβ 1 g ρ J α f t + f ρ g ρ t ρ β 1 dρ t ρ β 1 g ρ dρ + J α g t t ρβ 1 f ρ J α g t. t ρ β 1 f ρ dρ, Remrk 1. The inequlities 3.1 nd 3.11 re reversed if the functions re synchronous on [, [ i.e. fx fygx gy, for ny x, y [, [. Remrk 2. Applying Theorem 3.2 for α = β, we obtin Theorem 3.1. The third result is: J. Inequl. Pure nd Appl. Mth., 13 29, Art. 86, 5 pp. http://jipm.vu.edu.u/

4 SOUMIA BELARBI AND ZOUBIR DAHMANI Theorem 3.3. Let f i,...,n be n positive incresing functions on [, [. Then for ny t >, α >, we hve n n 3.14 J α f i t J α 1 1 n J α f i t. Proof. We prove this theorem by induction. Clerly, for n = 1, we hve J α f 1 t J α f 1 t, for ll t >, α >. For n = 2, pplying 3.1, we obtin: J α f 1 f 2 t J α 1 1 J α f 1 t J α f 2 t, for ll t >, α >. Now, suppose tht induction hypothesis 3.15 J α f i t J α 1 2 n J α f i t, t >, α >. Since f i,...,n re positive incresing functions, then f i t is n incresing function. Hence we cn pply Theorem 3.1 to the functions f i = g, f n = f. We obtin: n 3.16 J α f i t = J α fg t J α 1 1 J α f i t J α f n t. Tking into ccount the hypothesis 3.15, we obtin: n 3.17 J α f i t J α 1 1 J α 1 2 n J α f i tj α f n t, nd this ends the proof. We further hve: Theorem 3.4. Let f nd g be two functions defined on [, + [, such tht f is incresing, g is differentible nd there exists rel number m := inf t g t. Then the inequlity 3.18 J α fgt J α 1 1 J α ftj α gt mt α + 1 J α ft + mj α tft is vlid for ll t >, α >. Proof. We consider the function h t := g t mt. It is cler tht h is differentible nd it is incresing on [, + [. Then using Theorem 3.1, we cn write: 3.19 J α g mt f t J α 1 1 J α ft J α gt mj α t Hence J α 1 1 J α ftj α gt m J α 1 1 t α+1 J α ft Γ α + 2 J α 1 1 J α ftj α mγ α + 1 t gt J α ft Γ α + 2 J α 1 1 J α ftj α gt mt α + 1 J α ft. 3.2 J α fgt J α 1 1 J α ftj α gt mt α + 1 J α ft+mj α tft, t >, α >. Theorem 3.4 is thus proved. J. Inequl. Pure nd Appl. Mth., 13 29, Art. 86, 5 pp. http://jipm.vu.edu.u/

FRACTIONAL INTEGRAL INEQUALITIES 5 Corollry 3.5. Let f nd g be two functions defined on [, + [. A Suppose tht f is decresing, g is differentible nd there exists rel number M := sup t g t. Then for ll t >, α >, we hve: 3.21 J α fgt J α 1 1 J α ftj α gt Mt α + 1 J α ft + MJ α tft. B Suppose tht f nd g re differentible nd there exist m 1 := inf t f x, m 2 := inf t g t. Then we hve 3.22 J α fgt m 1 J α tgt m 2 J α tft + m 1 m 2 J α t 2 J α 1 1 J α ftj α gt m 1 J α tj α gt m 2 J α tj α ft + m 1 m 2 J α t 2. C Suppose tht f nd g re differentible nd there exist M 1 := sup t f t, M 2 := sup t g t. Then the inequlity 3.23 J α fgt M 1 J α tgt M 2 J α tft + M 1 M 2 J α t 2 J α 1 1 J α ftj α gt M 1 J α tj α gt M 2 J α tj α ft + M 1 M 2 J α t 2. is vlid. Proof. A: Apply Theorem 3.1 to the functions f nd Gt := gt m 2 t. B: Apply Theorem 3.1 to the functions F nd G, where: F t := ft m 1 t, Gt := gt m 2 t. To prove C, we pply Theorem 3.1 to the functions F t := ft M 1 t, Gt := gt M 2 t. REFERENCES [1] P.L. CHEBYSHEV, Sur les expressions pproximtives des integrles definies pr les utres prises entre les mêmes limites, Proc. Mth. Soc. Chrkov, 2 1882, 93 98. [2] R. GORENFLO AND F. MAINARDI, Frctionl Clculus: Integrl nd Differentil Equtions of Frctionl Order, Springer Verlg, Wien 1997, 223 276. [3] S.M. MALAMUD, Some complements to the Jenson nd Chebyshev inequlities nd problem of W. Wlter, Proc. Amer. Mth. Soc., 1299 21, 2671 2678. [4] S. MARINKOVIC, P. RAJKOVIC AND M. STANKOVIC, The inequlities for some types q- integrls, Comput. Mth. Appl., 56 28, 249 2498. [5] B.G. PACHPATTE, A note on Chebyshev-Grüss type inequlities for differentil functions, Tmsui Oxford Journl of Mthemticl Sciences, 221 26, 29 36. [6] I. PODLUBNI, Frctionl Differentil Equtions, Acdemic Press, Sn Diego, 1999. J. Inequl. Pure nd Appl. Mth., 13 29, Art. 86, 5 pp. http://jipm.vu.edu.u/