A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL. 1. Introduction. f(x)dx a
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1 Journl of Frctionl Clculus nd Applictions, Vol. 4( Jn. 203, pp ISSN: A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL VAIJANATH L. CHINCHANE AND DEEPAK B. PACHPATTE Abstrct. In this pper, we estblish certin integrl inequlities for the Chebyshev functionl in cse of synchronous function, using the Hdmrd frctionl integrl. Consider the functionl T (f, g := b f(xg(xdx. Introduction ( ( b f(xdx b g(xdx, (. where f nd g re two integrble functions which re synchronous on [, b], (i.e(f(x f(y(g(x g(y 0 for ny x, y [, b], given in [7]. Mny reserchers hve studied (. nd number of inequlities ppered in literture see [,3,9-2]. The min objective of this pper is to estblish some frctionl inequlities for (., using Hdmrd frctionl integrls. Recently mny uthors hve studied integrl inequlities on frctionl clculus using Riemnn-Liouville, Cputo derivtive, see [2,5,6,8]. The necessry bckground detils re given in the book by A.A.Kilbs [2,p.0-8], nd S.G.Smko et l. [8,p ]. 2. Preliminries In this section we give some preliminries nd bsic proposition used in our subsequent discussion. Here we give some definitions of Hdmrd derivtive nd integrl s in [4, p.59-7]. Definition 2.. The Hdmrd frctionl integrl of order α R + of function f(x, for ll x > is defined s where = 0 e u u α du.,x f(x = x ln( x t α f(t dt t, ( Mthemtics Subject Clssifiction. 26D0, 26D33. Key words nd phrses. Hdmrd frctionl integrl, Hdmrd frctionl derivtive nd frctionl integrl inequlity. Submitted Aug., 202. Published Jn.,
2 26 VAIJANATH L. CHINCHANE AND DEEPAK B. PACHPATTE JFCA-203/4 Definition 2.2. The Hdmrd frctionl derivtive of order α [n, n, n Z +, of function f(x is given s follows. HD α,xf(x = Γ(n α (x d dx n x ln( x t n α f(t dt t. (2.2 From bove definitions, we see the difference between Hdmrd nd Riemnn- Liouville frctionl derivtive nd integrls s kernel in the Hdmrd integrl hs the form of ln( x t insted of the form of (x t, which is involves both in the Riemnn-Liouville nd Cputo integrl. The Hdmrd derivtive hs the opertor(x d dx n, whose construction is well suited to the cse of the hlf-xis nd is invrint reltion to diltion [8, p.330], while the Riemnn-Liouville derivtive hs the opertor ( d dx n. We give some imge formuls under the opertor (2. nd (2.2, which would be used in the derivtion of our min result. Proposition 2. (4. If 0 < α <, the following reltion hold: respectively.,x (ln xβ = Γ(β + α (ln xβ+α, (2.3 HD α,x(ln x β = Γ(β α (ln xβ α, (2.4 Next, we will introduce the weighted spce C γ,ln [, b], Cδ,γ m [, b] of function f on the finite intervl [, b], if γ (0 Re(γ <, n < α n, then C γ,ln [, b] := {f(x : ln( x γ f(x C[, b], f cr = ln( x f(x c}, C 0,ln [, b]=c[, b] nd Cδ,ln m [, b] := {g(x : (δn g(x C γ,ln [, b], g n cr,ln= k=0 (δk g c + (δ n g cr,ln }, δ = x d dx. For the convenience of estblishing the result, we give the semigroup property ( H D α,x ( HD β,x f(x = H D (α+β,x f(x. ( Min Result Theorem 3.. Let f nd g be two synchronous function on [0, [. Then for ll t > 0, α > 0, we hve,t Γ(α + (fg(t (ln t α ( HD,t α f(t( HD,t α g(t. (3. Proof: Since f nd g re synchronous on [0, [ for ll τ 0, ρ 0, we hve From (3.2, (f(τ f(ρ(g(τ g(ρ 0, (3.2 f(τg(τ + f(ρg(ρ f(τg(ρ + f(ρg(τ, (3.3 Multiplying both side of (3.3 by (ln( t τ α τ, which is positive becuse τ (0, t, t > 0. Then integrting the resulting identity with respect to τ over (,t, we obtin ln( t τ α f(τg(τ dτ τ + ln( t τ α f(τg(ρ dτ τ + ln( t τ α f(ρg(ρ dτ τ ln( t τ α f(ρg(τ dτ τ, (3.4
3 JFCA-203/4 ON SOME FRACTIONAL INTEGRAL INEQUALITIES 27 consequently, we get HD,t α (fg(t + f(ρg(ρ g(ρ ln( t τ α f(τ dτ τ + f(ρ ln( t τ α dτ τ ln( t τ α g(τ dτ τ, (3.5 HD,t α (fg(t + f(ρg(ρ HD,t α ( g(ρ HD,t α f(t + f(ρ HD,t α g(t. (3.6 multiplying both side of (3.6 by (ln( t ρ α ρ, which is positive becuse ρ (0, t, t > 0. Then integrting the resulting identity with respect to ρ over (,t, we obtin,t (fg(t hence H D α,t f(t ln( t ρ α dρ,t ( ln( t ρ α g(ρ dρ,t g(t ln( t ρ α f(ρg(ρ dρ ρ ln( t ρ α f(ρ dρ ρ, (3.7 HD,t α (fg(t HD,t α ( + H D,t α ( HD,t α (fg(t H D,t α f(t HD,t α g(t + H D,t α g(t HD,t α f(t, (3.8 we get HD,t α ( [ 2 H D,t α (fg(t] 2 H D,t α f(t g(t, (3.9 This ends the proof of Theorem 3.. Theorem 3.2. Let f nd g be two synchronous function on [0, [, then for ll t > 0, α > 0, β > 0 we hve (ln t β D α (ln tα,t (fg(t + D β,t Γ(β + H Γ(α + (fg(t H D,t α f(t HD β,t g(t H (3.0 + H D,t α g(t HD β,t f(t. Proof: To prove bove Theorem multiplying eqution (3.6 by (ln( t ρ β ρ, which is positive becuse ρ (0, t, t > 0. Then integrting resulting identity with respective ρ over to t, we obtin,t (fg(t H D α,t f(t (ln( t ρ nd this ends the proof of Theorem 3.2. β dρ,t ( (ln( t ρ β g(ρ dρ,t g(t,t (ln( t ρ β f(ρg(ρ dρ ρ (ln( t ρ β f(ρ dρ ρ, Remrk 3.. Applying Theorem 3.2 for α = β, we obtin Theorem 3.. (3.
4 28 VAIJANATH L. CHINCHANE AND DEEPAK B. PACHPATTE JFCA-203/4 Theorem 3.3. Let (f i i=,2,...n be positive incresing function on [0, [, then for ll t > 0, α > 0, we hve ( n HD,t α f i (t [ HD,t α (] n n. HD,t α f i(t. (3.2 i= Proof: We prove this Theorem by induction. Clerly, for n =, HD,t α f (t H D,t α f (t, for ll t > 0, α > 0. for n = 2, pplying eqution (3., we obtin,t (f f 2 [,t (],t f (t H D α,t f 2(t. (3.3 Suppose tht by induction hypothesis n HD,t α ( i= f i (t [ n HD,t α (] 2 n i= i=,t f i(t for ll t > 0, α > 0. (3.4 Now, since (f i i=,2,...n re positive incresing function, then ( n i= f i(t is n incresing function, therefore we cn pply Theorem 3. to the function n i= f i = g, f n = f, we obtin,t ( n (f i (t i=,t [ [ n i=,t (] H f i f n n,t (] H D α,t ( i= (t H D α,t (g.f(t D α,t g(t,t f(t [,t (] [ [,t (] n This completes the proof of Theorem 3.3. f i (t H D α,t f n n,t (] 2 n ( n HD,t α f i(t. i= References i=,t f i(t H D α,t f n (3.5 [] G.A.Anstssiou, Frctionl Differentition Inequlities, Springer, New York, [2] A.A.Kilbs, H.M.Srivstv nd J.J.Trujillo, Theory nd Appliction of Frctionl Differentil Equtions, Elsevier, Amersterdm, [3] B.G.Pchptte, A note on Chebyshev-Gruss type inequlities for diffrerentil function, Tmsui Oxf. J. Mth. Sci., 22(0(2006, [4] D.Blenu, J.A.T.Mchdo nd C.J.Luo, Frctionl Dynmic nd Control, Springer, 202, pp [5] I.Podlubny, Frctionl Differentil Equtions, Acdemic Press, New York, 999. [6] K.S.Miller nd B.Ross, An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions, Wiley, New York, 993. [7] P.L.Chebyshev, Sur les expressions pproximtives des integrles definies pr les utres prise entre les memes limites, Proc. Sco. Chrkov, 2(882, [8] S.G.Somko, A.A.Kilbs nd O.I.Mrichev, Frctionl Integrl nd Derivtive Theory nd Appliction, Gordon nd Brech, Switzerlnd, 993. [9] S.M.Mlmud, Some complements to the Jensen nd Chebyshev inequlities nd problem of W.Wlter, Proc. Amer. Mth. Soc., 29, 9 (200,
5 JFCA-203/4 ON SOME FRACTIONAL INTEGRAL INEQUALITIES 29 [0] S.Mrinkoric, P.Rjkovic nd M.Stnkovic, The inequlities for some q-integrls, Comput. Mth. Appl., 56(2008, [] S. Belrbi nd Z. Dhmni, On some new frctionl integrl inequlity, J. Inequl. Pure nd Appl. Mth., 0(3(2009, Art.86, 5 pp. [2] S.S.Drgomir nd C.E.Perce, Selected topics in Hermit-Hdmrd inequlity, Monogrphs: Victori University, Vijnth L. Chinchne Deprtment of Mthemtics, Deogiri Institute of Engineering nd Mngement Studies Aurngbd-43005, INDIA. E-mil ddress: chinchne85@gmil.com Deepk B. Pchptte Deprtment of Mthemtics, Dr. Bbsheb Ambedkr Mrthwd University, Aurngbd , INDIA. E-mil ddress: pchptte@gmil.com
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