TAMKANG JOURNAL OF MATHEMATICS Volume 46, Number 1, 67-73, March 2015 doi:10.5556/j.tkjm.46.2015.1586 -
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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 extension 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.
1. Introduction In last few years the fractional integral and derivative inequalities and their applications have been addressed extensively by several authors like Anastassiou [1, 2], Beesack [3], Handley et al. [8], Opial [10] 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 In this section, we will present some definitions that will be used in the proof of our main results. Received July 20, 2013, accepted April 15, 2014. 2010 Mathematics Subject Classification. 26D10, 26A33. Key words and phrases. Fractional inequalities, Riemann-Louville fractional operators, Erdely-Kober operators, opial fractional inequalities. Corresponding author: Amit Chouhan.
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ARIF M. KHAN, AMIT CHOUHAN AND SATISH SARASWAT
Definition 2.1. If a and b are positive real numbers, satisfying a + b = 1 and f and g are monotonic functions defined on (0, ∞), then Cauchy’s general inequality [9] is defined as ( ) [ ]a [ ( )]b a f (x) + bg y = f (x) g y
(2.1)
Definition 2.2. The standard Riemann-Liouville fractional integral [12] of order α ∈ C for function f : (0, ∞) → R is given as 1 Γ (α)
α I α f (t ) = I 0+ f (t ) =
∫ 0
t
(t − x)α−1 f (x) d x, R (α) >0
(2.2)
provided that integral on the right-hand side converges. Definition 2.3. Saigo [11] defined the fractional integration operator as follows: α,β,η I 0+ f
x −α−β (x) = Γ(α)
∫
x 0
t (x − t )α−1 2 F 1 (α + β, −η; α; 1 − ) f (t )d t x
(2.3)
where α ∈ C, Re (α) > 0, βand ηare real numbers and f (x) is a real valued and continuous function defined on the interval (0, ∞). For β = −α, in (2.3), we get Riemann - Liouville operator (2.2). Definition 2.4. Erdelyi [6] defined the space of functions C α∗∗ for arbitrary real number α∗ ′
with set of real valued function C [0, ∞), as follows: { } ′ C α∗∗ = f (x) = x q g (x) ; q < α∗ , g (x) ∈ C [0, ∞) where α∗ =
mi n 1≤k≤m (βτk ) with m
(2.4)
∈ z + , β > 0, k = 1, . . . , m; τ1 , τ2, τ3 , . . . , τm be arbitrary real num-
bers. Definition 2.5. Erdelyi [6] also defined the linear space of function C α for arbitrary real num′
bers α, with set of real valued functions C [0, ∞), as follows: { } ′ C α = f (x) = x p f˜ (x) : p > α , f˜(x) ∈ C [0, ∞) where α =
( max 1≤k≤m [−β γk
(2.5)
) + 1 ], m ∈ z + , β > 0, k = 1, . . . , m; γ1 , γ2 , γ3 , . . . , γm be arbitrary real
numbers. Definition 2.6. A multiple Erdelyi-Kober operator of Riemann - Liouville type is defined in the form, [6]
I
1 (γk ),(δk ) f (x) = (βk ),(λk ),m x
)m ¯( ¯ 1 1 γ + δ + 1 − , ¯ k k βk βk 1 m,0 t ¯ ( )m f (t ) d t Hm,m 1 1 x ¯¯ γk + 1 − λk , λk
∫
x 0
1
(2.6)
ON CERTAIN NEW CAUCHY-TYPE FRACTIONAL
69
where m ∈ z + ; βk , λk , δk , γk > 0, k = 1, . . . , m; γ1 , γ2 , γ3 , . . . , γm be arbitrary real numbers. Furthermore,
m 1 m 1 ∑ ∑ = k=1 λk k=1 βk { ( )} max and f (x) ∈ C α , α = 1≤k≤m −λk γk + 1 . Here H m,n p, q [x] is H-function due to Fox [7], in-
troduced and defined via a Mellin-Barnes type integral for integers m, n, p, q such that 0 ≤ m ≤ q, 0 ≤ n ≤ p, for a i , b j ∈ C with C, the set of complex numbers and for αi , β j ∈ R + = ( ) (0, ∞) , i = 1, 2, . . . ., p; j = 1, 2 . . . , q , as ] [ ¯ ∫ 1 ¯ ( a i , αi )1,p m,n m.n H m,n (s) z −s d s H p,q = (2.7) ≡ H z (z) ¯ p,q 2πi L p,q ( b j , β j )1,q with [ m.n m.n H p,q (s) ≡ H p,q
∏m ∏n ] (a i , αi )1,p ¯¯ j =1 Γ(b j + β j s) i =1 Γ(1 − a i − αi s) ¯s = ∏p ∏q ( b j , β j )1,q Γ(a i + αi s) Γ(1 − b j − β j s) i =n+1
(2.8)
j =m+1
Asymptotic expansions 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 x, x 0 ∈ [a, b] such that x = x 0 , x 0 is fixed, v = 1, for the function f ∈ C ([a, b]) , as follows ∫ x ( x0 ) 1 I v f (x) = (x − t )v−1 f (t ) d t , x 0 = x = b. (2.9) Γ (v) x0 Further the generalized v-fractional derivative [5] of f over [x 0 , b] is given as ( ) x0 D xv0 f := D I 1−α f (n) f (n) := D n f .
(2.10)
where n= [v] the integral part, v > 0 and α = v − n (0 < α < 1). Furthermore, Anastassiou [1] defined subspace C xv0 ([a, b]) of C n ([a, b]) for f ∈ C xv0 ([a, b]); as follows:
{ } x0 C xv0 ([a, b]) = f ∈ C n ([a, b]) : I 1−α D n f ∈ C 1 ([x 0 , b]) .
(2.11)
3. Cauchy type fractional integral inequalities Theorem 1. If a and b are positive real numbers satisfying a + b = 1 and f and g are monotonic ( ) functions defined on [0, ∞) , then for all α, β ∈ C , Re (α) > 0, Re β > 0, t > 0, the following inequality holds
at β bt α ( ) I α f (t )+ I β g (t ) ≥ I β g b (t) I α f a (t ) Γ (α + 1) Γ β+1
(3.1)
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ARIF M. KHAN, AMIT CHOUHAN AND SATISH SARASWAT
Proof. Multiplying both side of Cauchy general inequality (2.1) by
(t −x)α−1 Γ(α)
and integrating
with respect to x, between the limits 0 to t, we get ∫ ( ) t b f (x) d x + g y (t − x)α−1 d x (t − x) Γ(α) 0 0 ∫ t [ ( )]b 1 [ ]a ≥ g y (t − x)α−1 f (x) d x Γ(α) 0
a Γ(α)
∫
t
α−1
By using (2.2), we obtained ( ) ( ) aI α f (t ) + bg y I α (1) ≥ [g y ]b I α [ f (t )]a multiplying both sides of equation (3.2) by
(t −y)β−1 Γ(β)
(3.2)
and integrating w.r.t.to y, between the lim-
its 0 to t and finally by virtue of (2.2), we obtained inequality (3.1). Remark 1. For α = β in equation (3.1), we obtained aI α f (t ) + bI α g (t ) = Γ (1 + 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 = 1 and f and g are monotonic ( ) functions defined over [0, ∞), then for all α, β, η, γ ∈ C , Re (α) > 0, Re γ > 0, t > 0 the following inequality holds ( ) Γ 1−β+η Γ (1 − δ + σ) γ,δ,σ −δ α,β,η ( ) t I 0+ f (t ) + b ( ) ( ) t −β I 0+ g (t ) a Γ (1 − δ) Γ 1 + γ + σ Γ 1−β Γ 1+α+η α,β,η a
≥ I 0+
γ,δ,σ b
f (t ) I 0+
(3.4)
g (t )
Proof. Multiplying both sides of equation (2.1) by
t −α−β Γ(α) (t
) ( − x)α−1 2 F 1 α + β, −η; α; 1 − xt then
integrating w.r.t. x from 0 to t and by virtue of (2.3), we have α,β,η
aI 0+
( ) α,β,η ( ) α,β,η f (t ) + bg y I 0+ (1) ≥ g b y I 0+ f a (t ) γ−1
now multiplying both sides of above equation by
t −γ−δ (t −y) Γ(γ)
y 2 F 1 (γ + δ, −σ, γ; 1 − t )
then inte-
grating w.r.t. y from 0 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 α,β,η aI 0+ f
α,β,η (t ) + bI 0+ g
( ) ( ) Γ 1 − β Γ 1 + α + η t +β α,β,η a α,β,η ( ) I 0+ f (t ) I 0+ g b (t ) . (t ) ≥ Γ 1−β+η
Remark 3. Putting β = −α, in inequality (3.4) we obtained inequality (3.3).
(3.5)
ON CERTAIN NEW CAUCHY-TYPE FRACTIONAL
71
Theorem 3. If a and b are positive real numbers, satisfying a+b = 1 and f (x) , g (x) ∈ C α , m, n ∈ z + ; λk , βk , δk , γk > 0 where, βk , λk , δk , γk be arbitrary real numbers with m 1 m 1 ∑ ∑ [ ( )] = , α = max −λk γk + 1 1≤k≤m k=1 λk k=1 βk
then the following inequality holds aI
] (γ )(δ ) ] (γk ),(δk ) [ (γ )(δ ) (γ )(δ ) [ f (t ) I βk (λk ),n (1) + bI βk (λk ),m (1) I βk (λk ),n g (t ) β (λ ),m ( k) k ( k) k ( k) k ( k) k ]a (γk )(δk ) [ ]b (γk )(δk ) [ ≥ I β (λ ),m f (t ) I β (λ ),n g (t ) ( k) k ( k) k
where k = 1, . . . , m for operator I
(3.6)
(γk ),(δk ) (γ ),(δ ) [.] and k = 1, . . . , n for operator I βk (λ k),n [.]. (βk )(λk ),m ( k) k
Proof. Multiply both side of general )m ¯Cauchy inequality (2.1) by ¯( ¯ ¯ 1 1 ¯ γ + δk + 1 − βk , βk ¯ 1 m,0 x ¯ k( 1 )m ¯ then integrating with the limits 0 to t and using equation ¯ t Hm,m t ¯ 1 1 ¯ ¯ γk + 1 − λ , λ k
k
1
(2.6), we get
( ) (γ )(δ ) [ ]b (γ )(δ ) [ ]a (γk )(δk ) f (t ) + bg y I βk (λk ),m (1) ≥ g (y) I βk (λk ),m f (t ) (βk )(λk ),m ( k) k ( k) k )n ¯ ¯( ¯ ¯ 1 1 γ + δ + 1 − , ¯ k k βk βk 1 ¯¯ 1 n,0 y ¯ )n now multiply both side by t Hn,n t ¯ ( ¯ and integrating with the limits ¯ ¯ γk + 1 − λ1k , λ1k aI
1
0 to t , we obtained (3.6). 4. Opial type fractional derivative inequalities
Theorem 4. Let f ∈ C xv0 ([a, b]) , v = 1 and f (i ) (x 0 ) = 0, i = 0, 1, . . . , n − 1, n = [v] ; x, x 0 ∈ [a, b] : x = x 0 . Let p, q > 1 such that ∫
x¯ x0
1 p
+ q1 = 1, then the following inequality holds
2−1/q (x − x 0 )(pv−p+2)/p ( )( ) 1/p 0 Γ (v) ( pv − p + 1 pv − p + 2 ) ( ¯ ))q ] q2 [∫ ( ) ( x ¯ w+α M ,N +1 σ ¯ (−ω, σ) , a j , α j 1,P ) dt t HP +1,Q+1 t ¯ ( × b j , β j 1,Q , (−ω − α, σ) x0
¯ ¯( ¯ ) ¯ f (w)¯ ¯ D v f (w)¯ d w ≤
¯ ( ¯ )¯ ¯ M ,N σ ¯ (a j ,α j )1,P ¯ ¯ ¯ ¯. where, f (x) = x ¯ HP,Q x ¯ b ,β ( j j )1,Q ¯ w
Proof. Let ¯ ( ¯ )¯ ∫ x ¯ ¯ ¯ ¯ ¯ ¯ f (x)¯ = ¯x w H M ,N x σ ¯ (a j ,α j )1,P ¯ ≤ 1 (x − t )v−1 | D xv0 f (t ) |d t ¯ ¯ (b j ,β j )1,Q ¯ Γ (v) P,Q x0
(4.1)
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ARIF M. KHAN, AMIT CHOUHAN AND SATISH SARASWAT
≤
1 Γ (v)
(∫
x[ x0
(x − t )v−1
]p
)1/p (∫ dt
1 (x − x 0 )(pv−p+1)/p = Γ (v) (pv − p + 1)1/p on setting
(∫ z (x) =
then
d ′ d x z (x) = z (x) =
x x0
(∫
¯ ] ¯D v f ¯ (t ) q d t x0
x [¯ x0
¯ ] ¯D v f ¯ (t ) q d t x0
x [¯ x0
)1/q
)1/q (4.2)
) (|D xv0 f |(t ))q d t , (z (x 0 ) = 0)
(4.3)
(¯ v ¯ )q ¯D f ¯ (x) , i.e., x0
( ¯ ) ¯ v ¯ ¯ ¯D f ¯ (x) = (z ′ (x))1/q = x ω+α H M ,N +1 x σ ¯ (−ω,σ),(a j ,α j )1,P x0 ¯ (b j ,β j )1,Q (−ω−α,σ) P +1,Q+1
(4.4)
now by virtue of (4.2) and (4.4), we have ¯ ¯¯ ¯ ¯ f (w)¯ ¯D v f ¯ (w) = x0
1 (ω − x 0 )(pv−p+1)/p Γ (v) (pv − p + 1)1/p
[{∫
} ]1/q ¯ ) ¯D v f ¯ (t ) q d t × z ′ (ω) , x0
ω (¯ x0
x 0 = ω = x, further integrating over [x 0 , x], we get ∫ x ¯ ¯¯ ¯ ¯ f (w)¯ ¯D v f ¯ (w) d w x0 x0 ∫ x ( )1/q 1 (pv−p+1)/p ′ = (w − x ) z .z (w) dw (w) 0 Γ(v) (pv − p + 1)1/p x0 (∫ x )1/p (∫ x )1/q 1 ′ (pv−p+1) d w − x z z (w)d w ≤ (w ) (w) 0 Γ (v) (pv − p + 1)1/p x0 x0 =
Γ(v)
[(
(x − x 0 )(pv−p+2)/p [z (x)]2/q ) ]1/p . 21/q pv − p + 1 (pv − p + 2)
now using equation (4.3), we obtained ¯ ¯¯ ¯ ¯ f (w)¯ ¯D v f ¯ (w) = x0
(pv−p+2)/p
2−1/q (x − x 0 ) [( ) ]1/p Γ (v) pv − p + 1 (pv − p + 2)
[∫
¯ ) ¯D v f ¯ (t ) q d t x0
x (¯ x0
]2/q
finally using equation (4.4), we arrived at (4.1). References [1] G. A. Anastassiou, Opial type inequalities involving Riemann - Liouville fractional derivatives of two functions with applications, Mathematical and Computer Modelling, 48(3) (2008), 344-374. [2] G. A. Anastassiou,Multivariate Landau fractional inequalities, Journal of Fractional Calculus and Applications, 1(8) (2011), 1-7. [3] P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc., 104(3) (1962), 470-475. [4] B. L. J. Braaksma, Asymptotic expansions and analytical continuations for a class of Barnes - integrals, Compositio Math. 15 (1964), 239-341.
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[5] J. A. Canavati, The Riemann- Liouville integral , Nieuw Archief Voor Wiskunde, 5(1) (1987), 53-75. [6] A. Erdely,On fractional integration and its application to the theory of Hankel transform, Q. J. Math. Oxford Series, 44(2) (1940), 293-303. [7] C. Fox, Integral transforms based upon fractional integration, In Proc. Cambridge Philos. Soc, 59, (1963), 63-71. [8] G. D. Handley, J.J.Koliha and J. Pecaric, Hilbert – Pachpatte type integral inequalities, J. Math. Anal. Appl., 257 (2001) 238-250. [9] Q. A. Ngo, D. D. Thang, T. T.Dat and D. A. Tuan, Notes on integral inequality, J. Inequal. Pure and Appl. Math, 7(4) (2006). [10] Z. Opial, Surune inégalité, Ann. Polon. Math., 8 (1960), 29-32. [11] M. Saigo, A remark on integral operators involving the Guass hypergeometric function, Rep. College General Ed., Kyushu Univ., 11 (1978), 135-143. [12] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon et al. (1993).
Department of Mathematics, JIET, JIET Group of Institutions, Jodhpur 342002, India. Department of Mathematics, JIET-SETG, JIET Group of Institutions, Jodhpur 342002, India. E-mail:
[email protected] Department of Mathematics, Govt. College Kota, Kota 324001, India.
