New Inequalities in Fractional Integrals
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1 ISSN (prin), (online) Inernionl Journl of Nonliner Science Vol.9(21) No.4,pp New Inequliies in Frcionl Inegrls Zoubir Dhmni Zoubir DAHMANI Lborory of Pure nd Applied Mhemics, Deprmen of Mhemics,Fculy of SESNV,Universiy of Mosgnem, Algeri (Received 9 June 29, cceped 18 Ocober 29) Absrc: In his pper, we use he Riemnn-Liouville frcionl inegrl o presen recen resuls on frcionl inegrl inequliies. By considering he exended Chebyshev funcionl in he cse of synchronous funcions, we esblish wo min resuls. The firs one dels wih some inequliies using one frcionl prmeer. The second resul concerns ohers inequliies using wo frcionl prmeers. Keywords: Chebyshev inequliy; Frcionl inegrl inequliies; Riemnn-Liouville frcionl inegrl 1 Inroducion Le us consider he funcionl T (f, g, p, q) := q(x)dx p(x)f (x) g (x) dx + p(x)dx q(x)f (x) g (x) dx ( ) ( b ) ( q(x)f (x) dx b ) ( p(x)g (x) dx b ) p(x)f (x) dx b q(x)g (x) dx, (1) where f nd g re wo inegrble funcions on, b nd p, q re posiive inegrble funcions on, b. If f nd g re synchronous on, b ( i.e. (f(x) f(y))(g(x) g(y)), for ny x, y, b ), hen T (f, g, p, q) ( see 6, 8 ). The sign of his inequliy is reversed if f nd g re synchronous on, b ( i.e. (f(x) f(y))(g(x) g(y)), for ny x, y, b ). For p(x) = q(x), x, b, we ge he Chebyshev inequliy 3. In 9, Osrowski esblished he following generlizion of he Chebyshev inequliy: If f nd g re wo differenible funcions, synchronous on, b, p is posiive inegrble funcion on, b nd f (x) m, g (x) r, for x, b, hen If f nd g re synchronous on, b, hen T (f, g, p) := T (f, g, p, p) mrt (x, x ; p). (2) T (f, g, p) mrt (x, b x; p). If f nd g re wo differenible funcions on, b, p is posiive inegrble funcion on, b nd f (x) M, g (x) R, for x, b, hen T (f, g, p) MRT (x, x ; p). (3) Mny reserchers hve given considerble enion o he funcionl T (f, g, p) nd number of exensions, generlizions nd vrins hve ppered in he lierure, see 1, 2, 4, 7 nd he references given herein. The min purpose of his pper is o use he Riemnn-Liouville frcionl inegrl o esblish some new frcionl inegrl inequliies using he exended Chebyshev funcionl (1). E-mil ddress: zzdhmni@yhoo.fr Copyrigh c World Acdemic Press, World Acdemic Union IJNS /378
2 494 Inernionl Journl of NonlinerScience,Vol.9(21),No.4,pp Descripion of he Frcionl Clculus In he following, we will give he necessry noion nd bsic definiions. More deils, one cn consul 5,1. Definiion 1 A rel vlued funcion f(), > is sid o be in he spce C μ, μ R if here exiss rel number p > μ such h f() = p f 1 (), where f 1 () C(, )). Definiion 2 A funcion f(), > is sid o be in he spce C n μ, n R, if f (n) C μ. Definiion 3 The Riemnn-Liouville frcionl inegrl operor of order α, for funcion f C μ, (μ 1) is defined s J α f() = 1 ( τ)α 1 f(τ)dτ; α >, >, J f() = f(), (4) where := e u u α 1 du. For he convenience of esblishing he resuls, we give he semigroup propery: which implies he commuive propery J α J β f() = J α+β f(); α, β, (5) J α J β f() = J β J α f(). (6) For he expression (4), when f() = μ we ge noher expression h will be used ler: J α μ = Γ(μ + 1) Γ(α + μ + 1) α+μ ; α >, μ > 1, >. (7) Remrk 1 In wh follows we shll consider he rel vlued funcions defined on he spce C μ, (μ 1). 3 Min Resuls Theorem 2 Le f nd g be wo synchronous funcions on, ) nd le r, p, q :, ), ). Then for ll >, α >, we hve: 2J α r() J α p()j α (qfg)() + J α q()j α (pfg)() + 2J α p()j α (q)())j α (rfg)() J α r() J α (pf)()j α (qg)() + J α (qf)()j α (pg)() + J α p() J α (rf)()j α (qg)()+ J α (qf)()j α (rg)() + J α q() J α (rf)()j α (pg)() + J α (pf)()j α (rg)(). Lemm 3 Le f nd g be wo synchronous funcions on, ) nd le v, w :, ), ). Then for ll >, α >, we hve: J α v()j α (wfg)() + J α w()j α (vfg)() J α (vf)()j α (wg)() + J α (wf)()j α (vg)(). (9) Proof. Since he funcions f nd g re synchronous on, ), hen for ll τ, ρ, we hve ( )( ) f(τ) f(ρ) g(τ) g(ρ). (1) Therefore f(τ)g(τ) + f(ρ)g(ρ) f(τ)g(ρ) + f(ρ)g(τ). (11) Muliplying boh sides of (11) by ( τ)α 1 v(τ), τ (, ), we ge (8) IJNS emil for conribuion: edior@nonlinerscience.org.uk
3 Z. Dhmni: New Inequliies in Frcionl Inegrls 495 Inegring (12) over (, ), we obin: ( τ) α 1 ( τ)α 1 v(τ)f(τ)g(τ) + v(τ)f(ρ)g(ρ) ( τ) α 1 v(τ)f(τ)g(ρ) + ( τ)α 1 v(τ)f(ρ)g(τ). (12) Consequenly 1 1 ( τ) α 1 v(τ)f(τ)g(τ)dτ + 1 ( τ) α 1 v(τ)f(ρ)g(ρ)dτ ( τ)α 1 v(τ)f(τ)g(ρ)dτ + 1 ( τ)α 1 v(τ)f(ρ)g(τ)dτ. (13) So we hve g(ρ) J α 1 (vfg)() + f (ρ) g (ρ) ( τ) α 1 v(τ)dτ Γ (α) ( τ)α 1 v(τ)f (τ) dτ + f(ρ) ( τ)α 1 v(τ)g (τ) dτ. J α (vfg)() + f (ρ) g (ρ) J α (v)() g (ρ) J α (vf)() + f (ρ) J α (vg)(). (15) (14) Now muliplying boh sides of (15) by ( ρ)α 1 w(ρ), ρ (, ), we obin: ( ρ) α 1 w(ρ)j α ( ρ)α 1 (vfg)() + w(ρ)f (ρ) g (ρ) J α (v)() Γ (α) Γ (α) ( ρ) α 1 w(ρ)g (ρ) J α (vf)() + ( ρ)α 1 w(ρ)f (ρ) J α (vg)(). (16) Inegring (16) over (, ), we ge: J α (vfg)() J α (vf)() ( ρ) α 1 w(ρ)dρ + J α (v)() w(ρ)f(ρ)g(ρ)( ρ) α 1 dρ ( ρ)α 1 w(ρ)g(ρ)dρ + J α (vg)() ( ρ)α 1 w(ρ)f(ρ)dρ. (17) Therefore J α (w)()j α (vfg)() + J α (v)()j α (wfg)() J α (vf)()j α (wg)() + J α (wf)()j α (vg)(), (18) nd his ends he proof of Lemm 3. Proof of Theorem 2: Proof. Puing v = p, w = q nd using Lemm 3, we cn wrie: J α (p)()j α (qfg)() + J α (q)()j α (pfg)() J α (pf)()j α (qg)() + J α (qf)()j α (pg)(). (19) Muliplying boh sides of (19) by J α (r)(), we obin: J α (r)() J α (p)()j α (qfg)() + J α (q)()j α (pfg)() J α (r)() J α (pf)()j α (qg)() + J α (qf)()j α (pg)(). (2) Puing v = r, w = q nd using gin Lemm 3, we ge: J α (r)()j α (qfg)() + J α (q)()j α (rfg)() J α (rf)()j α (qg)() + J α (qf)()j α (rg)(). (21) Muliplying boh sides of (21) by J α (p)(), we ge: J α (p)() J α (r)()j α (qfg)() + J α (q)()j α (rfg)() J α (p)() J α (rf)()j α (qg)() + J α (qf)()j α (rg)(). (22) IJNS homepge: hp://
4 496 Inernionl Journl of NonlinerScience,Vol.9(21),No.4,pp Wih he sme rgumens s before, we cn obin: J α (q)() J α (r)()j α (pfg)() + J α (p)()j α (rfg)() J α (q)() J α (rf)()j α (pg)() + J α (pf)()j α (rg)(). (23) The required inequliy (8) follows on dding he inequliies (2,22,23). Our second resul is: Theorem 4 Le f nd g be wo synchronous funcions on, ) nd le r, p, q :, ), ). Then for ll >, α >, β >, we hve: J α r() J α q()j β (pfg)() + 2J α p()j β (qfg)() + J β q()j α (pfg)() + J α p()j β (q)()) + J β p()j α (q)() J α (rfg)() J α r() J α (pf)()j β (qg)() + J β (qf)()j α (pg)() + (24) J α p() J α (rf)()j β (qg)() + J β (qf)()j α (rg)() + J α q() J α (rf)()j β (pg)() + J β (pf)()j α (rg)(). Remrk 5 Applying Lemm 3 for α = β, we obin Theorem 2 nd for α = β = 1, p(x) = q(x) = r(x) = 1, for ny x,, we obin he Chebyshev inequliy on,, (see 3). To prove Theorem 4, we need he following lemm: Lemm 6 Le f nd g be wo synchronous funcions on, nd le v, w :,,. Then for ll >, α >, we hve: J α v()j β (wfg)() + J β w()j α (vfg)() J α (vf)()j β (wg)() + J β (wf)()j α (vg)() (25) Proof of Lemm 6: Proof. Muliplying boh sides of (15) by ( ρ)β 1 w(ρ), ρ (, ), we obin: ( ρ) β 1 w(ρ)j α (vfg) () + J α (v) () ( ρ) β 1 Inegring (26) over (, ), we obin J α (vfg)() J α (vf)() ( ρ)β 1 w(ρ)f (ρ) g (ρ) w(ρ)g (ρ) J α (vf) () + ( ρ)β 1 w(ρ)f (ρ) J α (vg) (). ( ρ) β 1 w(ρ)dρ + J α v() ( ρ)β 1 w(ρ)g (ρ) dρ + J α (vg)() Lemm 6 is hus proved. Proof of Theorem 4: Proof. Using Lemm 6 wih v = p, w = q, we cn wrie: w(ρ)f (ρ) g (ρ) ( ρ) β 1 dρ ( ρ)β 1 w(ρ)f (ρ) dρ. (26) (27) J α (p)()j β (qfg)() + J β (q)()j α (pfg)() J α (pf)()j β (qg)() + J β (qf)()j α (pg)(). (28) Muliplying boh sides of (28) by J α (r)(), we obin: J α (r)() J α (p)()j β (qfg)() + J β (q)()j α (pfg)() J α (r)() J α (pf)()j β (qg)() + J β (qf)()j α (pg)(). Using Lemm 6 wih v = r, w = q nd hen muliplying boh sides of (29)by J α (p)(), we obin: J α (p)() J α (r)()j β (qfg)() + J β (q)()j α (rfg)() J α (p)() J α (rf)()j β (qg)() + J β (qf)()j α (rg)(). (29) (3) IJNS emil for conribuion: edior@nonlinerscience.org.uk
5 Z. Dhmni: New Inequliies in Frcionl Inegrls 497 Wih he sme rgumens, we cn ge: J α (q)() J α (r)()j β (pfg)() + J β (p)()j α (rfg)() J α (q)() J α (rf)()j β (pg)() + J β (pf)()j α (rg)(). (31) The inequliy (24) follows on dding he inequliies (29,3,31). Remrk 7 The inequliies (8) nd (24) re reversed in he following cses:. The funcions f nd g synchronous on, ). b. The funcions r, p, q re negive on, ). c. Two of he funcions r, p, q re posiive nd he hird one is negive on, ). References 1 M Bierncki: Sur une inglie enre les inegrles due Tchebysheff, Ann. Univ. Mrie Curie-Sklodowsk, 1(1951)(5): H Burkill, L Mirsky: Commens on Chebysheff s inequliy, Period. Mh. Hungr. 6( 1975): P L Chebyshev: Sur les expressions pproximives des inegrles dfinies pr les ures prises enre les memes limie. Proc. Mh. Soc. Chrkov. 2(1882): I Gvre: On Chebyshev ype inequliies involving funcions whose derivives belog o L p spces vi isonic funcionl. J. Inequl. Pure nd Appl. Mh. 7(26)(4): R Gorenflo, F Minrdi: Frcionl clculus: inegrl nd differenil equions of frcionl order. Springer Verlg, Wien J C Kung: Applied inequliies. Shndong Sciences nd T echnologie Press. (Chinese)24. 7 S Mrinkovic, P Rjkovic, M Snkovic: The inequliies for some ypes q-inegrls. Compu. Mh. Appl. 56(28): D S Mirinovic: Anlyic inequliies. Springer Verlg. Berlin A M Osrowski: On n inegrl inequliy. Aequions Mh. 4(197): I Podlubni: Frcionl Differenil Equions. Acdemic Press, Sn Diego IJNS homepge: hp://
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