- TAMKANG JOURNAL OF MATHEMATICS Volume 46, Number, 67-73, March 25 doi:.5556/j.tkjm.46.25.586 Available online at http://journals.math.tku.edu.tw/ - - - + + ON CERTAIN NEW CAUCHY-TYPE FRACTIONAL INTEGRAL INEQUALITIES AND OPIAL-TYPE FRACTIONAL DERIVATIVE INEQUALITIES ARIF M. KHAN, AMIT CHOUHAN AND SATISH SARASWAT Abstract. The aim of this paper is to establish several new fractional integral and derivative inequalities for non-negative and integrable functions. These inequalities related to the etension of general Cauchy type inequalities and involving Saigo, Riemann-Louville type fractional integral operators together with multiple Erdelyi-Kober operator. Furthermore the Opial-type fractional derivative inequality involving H-function is also established. The generosity of H-function could leads to several new inequalities that are of great interest of future research.. Introduction In last few years the fractional integral and derivative inequalities and their applications have been addressed etensively by several authors like Anastassiou [, 2], Beesack [3], Handley et al. [8], Opial [] etc., by using the Riemann Liouville fractional integrals and Opial type fractional Derivatives. Researchers have great interest in this field due to vast applications of these inequalities in fractional differential equation in establishing the uniqueness of the solution of initial value problems, giving upper bounds to their solutions. In the presented paper authors established certain theorems based on general Cauchy type inequality, involving Riemann-Liouville fractional integral operators, Saigo operator and multiple Erdely-Kober operator, then finally in last section Opial type fractional derivative inequality involving H-function is established, which is capable of yielding various results in the theory of Opial type integral inequalities. 2. Preliminaries and definitions results. In this section, we will present some definitions that will be used in the proof of our main Received July 2, 23, accepted April 5, 24. 2 Mathematics Subject Classification. 26D, 26A33. Key words and phrases. Fractional inequalities, Riemann-Louville fractional operators, Erdely-Kober operators, opial fractional inequalities. Corresponding author: Amit Chouhan. 67
68 ARIF M. KHAN, AMIT CHOUHAN AND SATISH SARASWAT Definition 2.. If a and b are positive real numbers, satisfying a + b and f and g are monotonic functions defined on, ), then Cauchy s general inequality [9] is defined as a f ) + bg y ) [ f ) ] a [ g y )] b 2.) Definition 2.2. The standard Riemann-Liouville fractional integral [2] of order α C for function f :, ) R is given as I α f t) I α f t) Γα) provided that integral on the right-hand side converges. t ) α f )d, R α)> 2.2) Definition 2.3. Saigo [] defined the fractional integration operator as follows: I α,β,η f ) α β Γα) t) α 2F α + β, η;α; t )f t)dt 2.3) where α C,Re α) >,βand ηare real numbers and f ) is a real valued and continuous function defined on the interval, ). For β α, in 2.3), we get Riemann - Liouville operator 2.2). Definition 2.4. Erdelyi [6] defined the space of functions Cα for arbitrary real number α with set of real valued function C [, ), as follows: where α min k m βτ k) with m z +,β >, k,...,m;τ,τ 2, τ 3,...,τ m be arbitrary real numbers. } Cα {f ) q g ); q < α, g ) C [, ) 2.4) Definition 2.5. Erdelyi [6] also defined the linear space of function C α for arbitrary real numbers α, with set of real valued functions C [, ), as follows: where α numbers. { } C α f ) p f ) : p > α, f ) C [, ) 2.5) ma k m [ β γ k + ) ], m z +,β >,k,...,m;γ,γ 2,γ 3,...,γ m be arbitrary real Definition 2.6. A multiple Erdelyi-Kober operator of Riemann - Liouville type is defined in the form, [6] I γ k),δ k ) β k),λ k ),m f ) Hm,m m, t γ k + δ k + β k, β k ) m γ k + λ k, λ k ) m f t)dt 2.6)
ON CERTAIN NEW CAUCHY-TYPE FRACTIONAL 69 where m z + ;β k,λ k,δ k,γ k >,k,...,m;γ,γ 2,γ 3,...,γ m be arbitrary real numbers. Furthermore, m m k λ k k β k and f ) C α, α ma { k m λk γk + )}. Here H m,n p, q[] is H-function due to Fo [7], introduced and defined via a Mellin-Barnes type integral for integers m,n, p, q such that m q, n p, for a i,b j C with C, the set of comple numbers and for α i,β j R +, ), i,2,..., p; j,2..., q ), as [ Hp,q m,n z) Hp,q m.n z a i,α i ),p b j,β j ),q ] Hp,q m,n s) z s ds 2.7) 2πi L with H m.n p,q s) H m.n p,q [ ai,α i ) ] m,p s j Γb j + β j s) n i Γ a i α i s) b j,β j ) p,q in+ Γa i + α i s) q j m+ Γ b j β j s) 2.8) Asymptotic epansions and analytic continuations of the H-function have been discussed by Braaksma [4]. Definition 2.7. Canavati [5] defined the generalized v-fractional Riemann-Liouville integral for, [a,b] such that, is fied, v, for the function f C [a,b]), as follows I v f ) ) t) v f t)dt, b. 2.9) Γv) Further the generalized v-fractional derivative [5] of f over [,b] is given as D v f : DI α f n) f n) : D n f ). 2.) where n [v] the integral part, v > and α v n < α < ). Furthermore, Anastassiou [] defined subspace C v [a,b])of C n [a,b]) for f C v [a,b]); as follows: C v [a,b]) { f C n [a,b]) : I α Dn f C [,b]) }. 2.) 3. Cauchy type fractional integral inequalities Theorem. If a and b are positive real numbers satisfying a +b and f and g are monotonic functions defined on [, ), then for all α,β C, Re α) >,Re β ) >, t >, the following inequality holds at β Γ β + ) I α f t)+ bt α Γα + ) I β g t) I β g b t) I α f a t) 3.)
7 ARIF M. KHAN, AMIT CHOUHAN AND SATISH SARASWAT Proof. Multiplying both side of Cauchy general inequality 2.) by t )α Γα) with respect to, between the limits to t, we get and integrating a Γα) t ) α f )d + b Γα) g y ) [ g y )] b Γα) By using 2.2), we obtained t ) α [ f ) ] a d t ) α d ai α f t) + bg y ) I α ) [g y ) ] b I α [f t)] a 3.2) multiplying both sides of equation 3.2) by t y)β Γβ) and integrating w.r.t.to y, between the limits to t and finally by virtue of 2.2), we obtained inequality 3.). Remark. For α β in equation 3.), we obtained ai α f t) + bi α g t) Γ + α) t α I α f a t) I α g b t). 3.3) Theorem 2. If a and b are positive real numbers satisfying a +b and f and g are monotonic functions defined over [, ), then for all α,β,η,γ C,Re α) >,Re γ ) >, t > the following inequality holds Γ δ + σ) a Γ δ)γ + γ + σ ) t δ I α,β,η Γ β + η ) f t) + b Γ β ) Γ + α + η ) t β I γ,δ,σ g t) I α,β,η f a t) I γ,δ,σ g b t) 3.4) Proof. Multiplying both sides of equation 2.) by t α β Γα) t ) )α 2F α + β, η;α; t then integrating w.r.t. from to t and by virtue of 2.3), we have ai α,β,η f t) + bg y ) I α,β,η ) g b y ) I α,β,η f a t) now multiplying both sides of above equation by t γ δ t y) γ Γγ) 2F γ + δ, σ,γ; y t ) then integrating w.r.t. y from to t and by virtue of 2.3), we obtained inequality 3.4). Remark 2. If α γ,β δ and η σ, inequality 3.4) becomes ai α,β,η f t) + bi α,β,η g t) Γ ) ) β Γ + α + η t +β Γ β + η ) I α,β,η f a t) I α,β,η g b t). 3.5) Remark 3. Putting β α, in inequality 3.4) we obtained inequality 3.3).
ON CERTAIN NEW CAUCHY-TYPE FRACTIONAL 7 Theorem 3. If a and b are positive real numbers, satisfying a+b and f ), g ) C α,m,n z + ;λ k,β k,δ k,γ k > where, β k,λ k,δ k,γ k be arbitrary real numbers with m m λ k k then the following inequality holds k ai γ k),δ k ) [ ] γ β k)λ k ),m f t) I k)δ k ) I γ k)δ k ) β k)λ k ),m,α ma β k k m [ λk γk + )] β k)λ k ),n ) + bi γ k)δ k ) β k)λ k ),m ) I γ k)δ k ) [ ] β k)λ k ),n g t) [ ] ai γ f t) k)δ k ) [ ] b β k)λ k ),n g t) 3.6) where k,...,m for operator I γ k),δ k ) β k)λ k ),m [.] and k,...,n for operator I γ k),δ k ) β k)λ k ),n [.]. Proof. Multiply both side of general ) Cauchy inequality 2.) by m t H m,m m, γ k + δ k + β k, β k ) t m γ k + λ k, then integrating with the limits to t and using equation λ k 2.6), we get ai γ k)δ k ) β k)λ k ),m f t) + bg y ) I γ k)δ k ) β k)λ k ),m ) [ g y) ] b γ I k)δ k ) [ ] a β k)λ k ),m f t) now multiply both side by t H n, to t, we obtained 3.6). y n,n t γ k + δ k + β k, β k ) n γ k + λ k, λ k ) n and integrating with the limits 4. Opial type fractional derivative inequalities Theorem 4. Let f C v [a, b]), v and f i) ),i,,...,n, n [v];, [a,b] :. Let p, q > such that p + q, then the following inequality holds f w) D v f ) w) 2 /q ) pv p+2)/p dw Γv) pv p + ) pv p + 2 ) ) /p [ t w+α H t M,N+ σ ω,σ), ) )) q a j,α j ),P dt] 2 q P+,Q+ b j,β j, ω α,σ) ) where, f ) w H M,N σ a j,α j),p. P,Q b j,β j),q,q 4.) Proof. Let f ) w H M,N P,Q ) σ a j,α j),p t) v D b j,β j) v,q Γv) f t) dt
72 ARIF M. KHAN, AMIT CHOUHAN AND SATISH SARASWAT on setting [ t) v ] ) /p p [ dt D v Γv) f ) ] /q qdt t) ) pv p+)/p [ D v Γv) pv p + ) /p f ) ] /q qdt t) 4.2) ) z ) D v f t)) q dt, z ) ) 4.3) then d d z ) z ) D v f ) q, ) i.e., D v f ) ) z )) /q ω+α H M,N+ σ ω,σ),a j,α j),p P+,Q+ b j,β j),q ω α,σ) now by virtue of 4.2) and 4.4), we have f w) D v f w) Γv) ω ) pv p+)/p [{ ω pv p + ) /p ω, further integrating over [, ], we get D v f } ) /q qdt t) z ω)], f w) D v f w)dw Γv) pv p + ) /p w ) pv p+)/p z w).z w) ) /q dw ) /p Γv) pv p + ) /p w ) pv p+) dw z w) z w)dw ) pv p+2)/p Γv) [ pv p + ) pv p + 2) ] /p now using equation 4.3), we obtained f w) D v f w) 2/q )].[z 2 /q 2 /q ) pv p+2)/p Γv) [ pv p + ) pv p + 2) ] /p finally using equation 4.4), we arrived at 4.). [ D v f t) ) qdt ) /q ] 2/q 4.4) References [] G. A. Anastassiou, Opial type inequalities involving Riemann - Liouville fractional derivatives of two functions with applications, Mathematical and Computer Modelling, 483) 28), 344-374. [2] G. A. Anastassiou,Multivariate Landau fractional inequalities, Journal of Fractional Calculus and Applications, 8) 2), -7. [3] P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc., 43) 962), 47-475. [4] B. L. J. Braaksma, Asymptotic epansions and analytical continuations for a class of Barnes - integrals, Compositio Math. 5 964), 239-34.
ON CERTAIN NEW CAUCHY-TYPE FRACTIONAL 73 [5] J. A. Canavati, The Riemann- Liouville integral, Nieuw Archief Voor Wiskunde, 5) 987), 53-75. [6] A. Erdely,On fractional integration and its application to the theory of Hankel transform, Q. J. Math. Oford Series, 442) 94), 293-33. [7] C. Fo, Integral transforms based upon fractional integration, In Proc. Cambridge Philos. Soc, 59, 963), 63-7. [8] G. D. Handley, J.J.Koliha and J. Pecaric, Hilbert Pachpatte type integral inequalities, J. Math. Anal. Appl., 257 2) 238-25. [9] Q. A. Ngo, D. D. Thang, T. T.Dat and D. A. Tuan, Notes on integral inequality, J. Inequal. Pure and Appl. Math, 74) 26). [] Z. Opial, Surune inégalité, Ann. Polon. Math., 8 96), 29-32. [] M. Saigo, A remark on integral operators involving the Guass hypergeometric function, Rep. College General Ed., Kyushu Univ., 978), 35-43. [2] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon et al. 993). Department of Mathematics, JIET, JIET Group of Institutions, Jodhpur 3422, India. Department of Mathematics, JIET-SETG, JIET Group of Institutions, Jodhpur 3422, India. E-mail: amit.maths@gmail.com Department of Mathematics, Govt. College Kota, Kota 324, India.