Some New Inequalities Involving Generalized Erdélyi-Kober Fractional q-integral Operator

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1 Applied Mathematical Sciences, Vol. 9, 5, no. 7, HIKARI Ltd, Some New Inequalities Involving Generalized Erdélyi-Kober Fractional q-integral Operator Junesang Choi Department of Mathematics, Dongguk University Gyeongju 78-74, Republic of Korea Daniele Ritelli School of Economics, Management and Statistics Department of Statistics, University of Bologna via Belle Arti 4, 46 Bologna, Italy Praveen Agarwal Department of Mathematics Anand International College of Engineering Jaipur-33, India Copyright c 5 Junesang Choi, Daniele Ritelli and Praveen Agarwal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract During the past four decades and longer, the subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has provided several potentially useful tools for solving differential, integral and integro-differential equations, and various other problems involving special functions of mathematical physics as well as their extensions (q-extensions) and generalizations in one and more variables. Here, in this paper, we aim to establish some new and potentially useful inequalities involving generalized Erdélyi-Kober fractional q-integral operator of the two parameters of deformation and

2 3578 Junesang Choi, Daniele Ritelli and Praveen Agarwal due to Gaulué [], by following the similar process used by Gaulué [3] and Dumitru and Agarwal [5]. Relevant connections of the results presented here with those earlier ones are also pointed out. Primary 6D, 6D5; Sec- Mathematics Subject Classification: ondary 6A33, 5A3 Keywords: Gamma function; q-gamma function; Integral inequalities; Generalized q-erdélyi-kober fractional integral operator; q-erdélyi-kober fractional integral operator Introduction and Preliminaries Throughout this paper, N, Z, R, and C denote the sets of positive integers, integers, real numbers, and complex numbers, respectively, N := N } and Z := Z \ N. The enormous success of the theory of integral inequalities involving various fractional integral operators has stimulated the development of a corresponding theory in q-fractional integral inequalities (see, e.g., [3, 4, 6, 7, 8, 9,, 6, 8]). In this paper, we are aiming at presenting some new and potentially useful inequalities involving generalized Erdélyi-Kober fractional q- integral operator of the two parameters of deformation and due to Gaulué [], by following the same lines used by Gaulué [3] and Baleanu and Agarwal [5]. We also point out relevant connections of the results presented here with those earlier ones. For our purpose, we recall the following definitions (see, e.g., [7, Section 6]) and some earlier works. The q-shifted factorial (a; q) n is defined by (n = ) (a; q) n := n k= ( a q k ) (n N), where a, q C and it is assumed that a q m (m N ). The q-shifted factorial for negative subscript is defined by We also write (a; q) n := ( a q ) ( a q ) ( a q n ) (a; q) := () (n N ). () ( ) a q k (a, q C; q < ). (3) k=

3 Inequalities involving fractional q-integral operator 3579 It follows from (), () and (3) that (a; q) n = which can be extended to n = α C as follows: (a; q) α = where the principal value of q α is taken. (a; q) (a q n ; q) (n Z), (4) (a; q) (a q α ; q) (α C; q < ), (5) It is noted that Jackson [4] was the first to develop q-calculus in a systematic way. The q-calculus was also developed in the recent monograph [5]. The q-derivative of a function f(t) is defined by D q f(t)} := d q f(qt) f(t) f(t)} =. (6) d q t (q )t For q, the q-derivative (6) is easily seen to yield the usual derivative, that is, lim D qf(t)} = d q dt f(t)}, if we assume that f(t) is a differentiable function. The function F (t) is a q-antiderivative of f(t) if D q F (t)} = f(t). It is denoted by f(t) d q t. (7) The Jackson integral of f(t) is defined, formally, by f(t) d q t := ( q)t q j f ( q j t ), (8) which can be easily generalized in the Stieltjes sense as follows: f(t) d q g(t) = j= f ( q j t ) ( g ( q j t ) g ( q j+ t )). (9) j= Suppose that < a < b. The definite q-integral is defined as follows: b f(t) d q t := ( q)b q j f ( q j b ) () j=

4 358 Junesang Choi, Daniele Ritelli and Praveen Agarwal and b a f(t) d q t = b f(t) d q t a A more general version of () in the Stieltjes sense is given by b f(t) d q g(t) = f(t) d q t. () f ( q j b ) ( g ( q j b ) g ( q j+ b )). () j= The notation [z] q is defined by [z] q := qz q = qz q (z C; q C \ }; q z ). (3) A special case of (3) when z N is [n] q = qn q = + q + + qn (n N), (4) which is called the q-analogue (or q-extension) of n N, since lim [n] q q = lim ( ) + q + + q n = n. q The classical Gamma function Γ(z) (see, e.g., [7, Section.]) was introduced by Leonhard Euler in 79 while he was trying to extend the factorial n! = Γ(n + ) (n N ) to real numbers. The q-analogue of n! is then defined by if n =, [n] q! := (5) [n] q [n ] q [] q [] q if n N, ( from which the q-binomial coefficient or the Gaussian polynomial analogous to n ) k is defined by [ ] n := k q [n] q! [n k] q! [k] q! (n, k N ; k n). (6) The [n] q! can be rewritten as follows: ( ) q k+ ( q) n ( q k++n ) = (q; q) ( q) n := Γ (q n+ q (n + ) ( < q < ). ; q) k= (7) Replacing n by a in (7), Jackson [4] defined the q-gamma function Γ q (a) by Γ q (a) := (q; q) (q a ; q) ( q) a ( < q < ). (8)

5 Inequalities involving fractional q-integral operator 358 The q-analogue of (t a) n is defined by the polynomial (n = ), (t a) n q := (t a) (t q a) (t q n a) (n N). ( a ) =t n t ; q (n N ). n We recall more definitions which will be needed in the sequel. (9) Definition.. Let f and g be two real-valued functions defined on an interval I R which are integrable on I. Then we say that f and g are synchronous on I if, for each x, y I, the following inequality holds: (f(x) f(y)) (g(x) g(y)). () Similarly f and g are asynchronous on I if, for any x, y I, the inequality in () is reversed, that is, (f(x) f(y)) (g(x) g(y)). () Definition.. A real-valued function f(t) (t > ) is said to be in the space C λ (λ R) if there exists a real number p > λ such that f(t) = t p φ(t), where φ(t) C(, ). A function f(t) (t > ) is said to be in the space C n λ (n R) if f (n) C λ. Definition.3. Let < q <, R(β), R(µ) >, and η C. Then a q-analogue of generalized Erdélyi-Kober fractional integral Iq α,β,η is defined by (see []): where I η,µ,β q f(t)} = β t β(η+µ) Γ q (µ) t = β( /β )( q) µ (t β τ β q) µ τ β(η+) f(τ) d q τ, k= (q µ ; q) k (q; q) k q k(η+) f(tq k/β ), () [ ( y ) ] (x y q) ν := x ν x q n ( ) y. (3) x q n+ν n= For f(t) = t λ in (), we get the known result [3, p. 8, Eq. (3)] I } [ ] q η,µ,β t λ Γ q (η + + λ/β) = β β Γ q (µ + η + + λ/β) tλ, (4) where R(β) >, R(µ) >, R(η + + λ/β) >, η C, and < q <. q

6 358 Junesang Choi, Daniele Ritelli and Praveen Agarwal Definition.4. Let < q <, R(µ) > and η C. Then the q-analogue of the Kober fractional integral operator is given by (see []) Iq η,µ f(t)} = t η µ Γ q (µ) t = ( q) µ (t τ q) µ τ η f(τ) d q τ k= (q µ ; q) k (q; q) k q k(η+) f(tq k ). (5) Remark.5. It is easy to see that Γ q (µ) > and (q µ ; q) k > (6) for all µ > and k N. If f : [, ) [, ) is a continuous function. Then it is seen that, in the second equality in (), each term being nonnegative, for all β, µ > and η R. I η,µ,β q f(t)} (7) Likewise the Kober q-integral operator (5) is also nonnegative, that is, for all µ > and η R. I η,µ q f(t)} (8) Generalized Erdélyi-Kober q-integral Inequalities Here we present five q-integral inequalities involving the generalized Erdélyi- Kober q-integral () stated in Theorems. to.8 below. Theorem.. Let f, g be two synchronous functions on [, ) and h, u : [, ) (, ) be continuous. Then the following inequality holds true: I ζ,ν,δ + I ζ,ν,δ I ζ,ν,δ + I ζ,ν,δ u(t)} I η,µ,β u(t) h(t)} I η,µ,β u(t) g(t)} I η,µ,β u(t) f(t) g(t) h(t)} + I ζ,ν,δ u(t) f(t) h(t)} I η,µ,β u(t) f(t) g(t) } + I ζ,ν,δ u(t) f(t) h(t)} + I ζ,ν,δ u(t) g(t)} + I ζ,ν,δ u(t) f(t) g(t)} I η,µ,β u(t) h(t)} u(t) f(t) g(t) h(t)} I η,µ,β u(t) f(t)} I η,µ,β u(t)} u(t) g(t) h(t)} u(t) g(t) h(t)} I η,µ,β u(t) f(t)}, (9) for all t >, < <, < <, µ >, ν >, β >, δ >, η >, and ζ >.

7 Inequalities involving fractional q-integral operator 3583 Proof. By using Definition. with h >, we get (f(τ) f(ρ)) (g(τ) g(ρ)) (h(τ) + h(ρ)) for all τ, ρ [, ), (3) that is, f(τ)g(τ)h(τ) + f(ρ)g(ρ)h(τ) + f(τ)g(τ)h(ρ) + f(ρ)g(ρ)h(ρ) f(τ)g(ρ)h(τ) + f(ρ)g(τ)h(τ) + f(ρ)g(τ)h(ρ) + f(τ)g(ρ)h(ρ). It is easy to see from (3) that ( t β τ β )µ = tβ(µ ) k= (3) [ ] (τ/t) β q k >, (3) (τ/t) β q k+µ for all β >, < <, µ >, and τ, t R with < τ < t. Let F (η, µ, β; t, τ; ; u) := β t β(η+µ) (t β τ β ) µ τ β(η+) u(τ). (33) Γ q (µ) Then we find from (3) that F (η, µ, β; t, τ; ; u) > under the conditions in (3) and the function u : [, ) (, ). Now, multiplying both sides of (3) by F (η, µ, β; t, τ; ; u) and taking the integration of the resulting inequality with respect to τ from to t, and using (), we get I η,µ,β u(t) f(t) g(t) h(t)} + f(ρ) g(ρ)i η,µ,β + f(ρ)g(ρ)h(ρ)i η,µ,β + f(ρ)h(ρ)i η,µ,β u(t)} g(ρ)i η,µ,β u(t) h(t)} + h(ρ)i η,µ,β u(t) f(t) h(t)} + f(ρ)i η,µ,β u(t) g(t)} + g(ρ)h(ρ)i η,µ,β u(t)f(t)}. u(t) f(t) g(t)} u(t) g(t) h(t)} (34) Next, multiply both sides of (34) by F (ζ, ν, δ; t, τ; ; u) which is positive under the conditions in (9), and integrating the resulting inequality with respect to ρ from to t, and applying Definition.3, we are led to the desired result (9). This completes the proof of Theorem.. Theorem.. Let f, g be two synchronous functions on [, ) and h, u, l : [, ) (, ) be continuous. Then the following inequality holds true: I ζ,ν,δ + I ζ,ν,δ I ζ,ν,δ + I ζ,ν,δ l(t)} I η,µ,β l(t) h(t)} I η,µ,β l(t) g(t)} I η,µ,β u(t) f(t) g(t) h(t)} + I ζ,ν,δ l(t) f(t) h(t)} I η,µ,β u(t) f(t) g(t) } + I ζ,ν,δ u(t) f(t) h(t)} + I ζ,ν,δ u(t) g(t)} + I ζ,ν,δ l(t) f(t) g(t)} I η,µ,β u(t) h(t)} l(t) f(t) g(t) h(t)} I η,µ,β l(t) f(t)} I η,µ,β u(t)} u(t) g(t) h(t)} l(t) g(t) h(t)} I η,µ,β u(t) f(t)}, (35) for all t >, < <, < <, µ >, ν >, β >, δ >, η >, and ζ >.

8 3584 Junesang Choi, Daniele Ritelli and Praveen Agarwal Proof. To prove the above result, multiplying both sides of (34) by F (ζ, ν, δ; t, τ; ; l), where F (ζ, ν, δ; t, τ; ; l) > under the conditions in (3) and the function l : [, ) (, ). Then integrating the resulting inequality with respect to ρ from to t, and using (), we are led to the desired result (35). This completes the proof of Theorem. Remark.3. It may be noted that the inequalities in (9) and (35) are reversed if the involved functions are asynchronous. The special case of (35) when u = l is easily seen to reduce to the result in Theorem.. Theorem.4. Let f, g, h be three continuous functions on [, ) and u : [, ) (, ) be a continuous function satisfying the following inequality: ψ f(x) Ψ, φ g(x) Φ and ω h(x) Ω (36) for some φ, ψ, ω, Φ, Ψ, Ω R, and for all x [, ). Then, for all t >, < <, < <, µ >, ν >, β >, δ >, η >, and ζ >, the following inequality holds true: I η,µ,β u(t) f(t) g(t) h(t)} Iq ζ,ν,δ u(t)} + Iq η,µ,β u(t) h(t)} Iq ζ,ν,δ u(t) f(t) g(t)} +I η,µ,β I η,µ,β I η,µ,β I η,µ,β u(t) g(t)} I ζ,ν,δ u(t) g(t) h(t)} I ζ,ν,δ u(t) f(t) g(t)} I ζ,ν,δ u(t)} I ζ,ν,δ u(t) f(t) h(t)} + I η,µ,β u(t) f(t)} I η,µ,β u(t) h(t)} Iq η,µ,β u(t)} (Ψ ψ)(φ φ)(ω ω). Proof. We find from (36) that, for all τ, ρ, u(t) f(t) } I ζ,ν,δ u(t) f(t) h(t)} I ζ,ν,δ u(t)} I ζ,ν,δ g(t) h(t) u(t)} u(t) g(t)} u(t) f(t) g(t) h(t)} f(τ) f(ρ) Ψ ψ, g(τ) g(ρ) Φ φ, h(τ) h(ρ) Ω ω. (37) This implies where A(τ, ρ) (Ψ ψ) (Φ φ) (Ω ω), (38) A(τ, ρ) := (f(τ) f(ρ)) (g(τ) g(ρ)) (h(τ) h(ρ)) = f(τ)g(τ)h(τ) + f(ρ)g(ρ)h(τ) + f(τ)g(ρ)h(ρ) + f(ρ)g(τ)h(ρ) f(τ)g(ρ)h(τ) f(ρ)g(ρ)h(ρ) f(τ)g(τ)h(ρ) f(ρ)g(τ)h(τ), (39) for all τ, ρ [, ). Multiplying both sides of (39) by F (η, µ, β; t, τ; ; u), where F (η, µ, β; t, τ; ; u) > under the conditions in (3) and the function

9 Inequalities involving fractional q-integral operator 3585 u : [, ) (, ), and taking -integration of the resulting identity with respect to τ from to t and using (), we get t = I η,µ,β F (η, µ, β; t, τ; ; u) A(τ, ρ) d q τ + f(ρ)h(ρ)i η,µ,β u(t) f(t) g(t) h(t)} + f(ρ)g(ρ)i η,µ,β f(ρ)i η,µ,β u(t) g(t) } h(ρ)i η,µ,β u(t) h(t)} + g(ρ)h(ρ)i η,µ,β u(t) f(t) g(t)} g(ρ)i η,µ,β u(t) g(t) h(t)} f(ρ)g(ρ)h(ρ)i η,µ,β u(t)}. u(t) f(t)} u(t) f(t) h(t)} (4) Next, multiply both sides of (4) by F (ζ, ν, δ; t, τ; ; u), where F (ζ, ν, δ; t, τ; ; u) > under the conditions in (3) and the function u : [, ) (, ), then integrating the resulting equality with respect to ρ from to t, and using (), we have t t = I η,µ,β + I η,µ,β I η,µ,β I η,µ,β F (η, µ, β; t, τ; ; u) F (ζ, ν, δ; t, τ; ; u) A(τ, ρ) d q τ d q ρ u(t) f(t) g(t) h(t)} I ζ,ν,δ u(t) g(t)} I ζ,ν,δ u(t) g(t) h(t)} I ζ,ν,δ u(t) f(t) g(t)} I ζ,ν,δ u(t)} + I η,µ,β u(t) f(t) h(t)} + I η,µ,β u(t) f(t)} I η,µ,β u(t) h(t)} I η,µ,β u(t) h(t)} I ζ,ν,δ u(t) f(t) } I ζ,ν,δ u(t) f(t) h(t)} I ζ,ν,δ u(t) f(t) g(t)} g(t) h(t) u(t)} u(t) g(t)} u(t)} I ζ,ν,δ u(t) f(t) g(t) h(t)}. (4) Now it is easy to see from the inequality (38) that the quantity in (4) is bounded by the resulting one obtained by applying the two integrations to the right-hand side of (38), that is, the last expression of (37). The proof is complete. Theorem.5. Let f, g, and h be three continuous functions on [, ), and u, l : [, ) (, ) be two continuous functions satisfying the following inequality: ψ f(x) Ψ, φ g(x) Φ and ω h(x) Ω (4) for some φ, ψ, ω, Φ, Ψ, Ω R and for all x [, ). Then, for all t >, < <, < <, µ >, ν >, β >, δ >, η >, and ζ >, the following inequality holds true: η,µ,β u(t) f(t) g(t) h(t)} Iq ζ,ν,δ l(t)} + Iq η,µ,β u(t) h(t)} Iq ζ,ν,δ l(t) f(t) g(t)} +I η,µ,β I η,µ,β I η,µ,β I η,µ,β u(t) g(t)} I ζ,ν,δ u(t) g(t) h(t)} I ζ,ν,δ u(t) f(t) g(t)} I ζ,ν,δ u(t)} I ζ,ν,δ l(t) f(t) h(t)} + I η,µ,β l(t) f(t)} I η,µ,β l(t) h(t)} Iq η,µ,β l(t)} (Ψ ψ)(φ φ)(ω ω). u(t) f(t) } I ζ,ν,δ u(t) f(t) h(t)} I ζ,ν,δ u(t)} I ζ,ν,δ g(t) h(t) l(t)} l(t) g(t)} l(t) f(t) g(t) h(t)} (43)

10 3586 Junesang Choi, Daniele Ritelli and Praveen Agarwal Proof. Multiply both sides of (4) by F (ζ, ν, δ; t, τ; ; l), where F (ζ, ν, δ; t, τ; ; l) > under the conditions in (3) and the function l : [, ) (, ) and then integrate the resulting inequality with respect to ρ from to t, and use (). Then a similar argument as in the proof of the inequality (37) is easily seen to yield the desired result (43). Remark.6. The special case of (43) when u = l is easily seen to reduce to the result in Theorem.4. We recall the following definition to give more inequalities involving the q-integral operator. Definition.7. If a real-valued function f has domain D(f) contained in R, we say that f satisfies a Lipschitz condition if there exists a constant L > such that f(x) f(y) L x y (44) for all points x, y in D(f). It is noted that the Lipschitz condition is an important tool in obtaining various famous inequalities in the literature and involves many other inequalities (for a very recent work, see [] and the references therein). Theorem.8. Let f, g : [, ) R satisfy Lipschitz conditions with constants L and L, respectively, and u : [, ) (, ) be a continuous function. Then the following inequality holds true: Iq η,µ,β f(t) g(t) u(t)} Iq ζ,ν,δ u(t)} + Iq η,µ,β u(t)} Iq ζ,ν,δ f(t) g(t) u(t)} I η,µ,β L L [ I η,µ,β I η,µ,β f(t) u(t)} I ζ,ν,δ t u(t) } I ζ,ν,δ t u(t)} I ζ,ν,δ g(t) u(t)} I η,µ,β u(t)} t u(t)} + I η,µ,β g(t) u(t)} I ζ,ν,δ f(t) u(t)} u(t)} Iq ζ,ν,δ t u(t) } ] (45) for all t >, < <, < <, µ >, ν >, β >, δ >, η >, and ζ >. Proof. We find from the assumption that, for all τ, ρ [, ), L L (τ ρ) B(τ, ρ) L L (τ ρ), (46) where B(τ, ρ) : = (f(τ) f(ρ))(g(τ) g(ρ)) = f(τ) g(τ) f(τ) g(ρ) f(ρ) g(τ) + f(ρ) g(ρ)

11 Inequalities involving fractional q-integral operator 3587 and (τ ρ) = τ τ ρ + ρ. Now, multiplying both sides of (46) by F (η, µ, β; t, τ; ; u) (which is positive under the given conditions) and applying the -integration to the resulting double inequality with respect to τ from to t. Then multiplying both sides of the last resulting double inequality by F (ζ, ν, δ; t, ρ; ; u) (which is positive under the given conditions) and applying the -integration to the resulting double inequality with respect to ρ from to t, after a little simplification, we are led to the desired result. 3 Special Cases and Concluding Remarks We conclude our present investigation by remarking further that we can present a large number of special cases of our main inequalities in Theorems -5. We first illustrate the special cases of Theorems, 3, and 5 when u(t) = t λ as in Corollaries,, and 3. Corollary 3.. Let f, g be two synchronous functions on [, ), h : [, ) (, ). Then the following inequality holds true: [ ] Γ q(ζ + + λ/δ) δ δ Γ q (ν + ζ + + λ/δ) tλ Iq η,µ,β t λ f(t) g(t) h(t) } + Iq ζ,ν,δ t λ f(t) g(t) } Iq η,µ,β t λ h(t) } + Iq ζ,ν,δ t λ h(t) } Iq η,µ,β t λ f(t) g(t) } [ ] Γ q(η + + λ/β) + β β Γ q (µ + η + + λ/β) tλ Iq ζ,ν,δ t λ f(t) g(t) h(t) } t λ g(t) } I η,µ,β t λ f(t) h(t) } + I ζ,ν,δ t λ f(t) } I η,µ,β I ζ,ν,δ + I ζ,ν,δ t λ f(t) h(t) } I η,µ,β t λ g(t) } + I ζ,ν,δ t λ g(t) h(t) } t λ g(t) h(t) } Iq η,µ,β t λ f(t) }, (47) for all t >, < <, < <, µ >, ν >, β >, δ >, η >, ζ >, η + + λ/β > and ζ + + λ/δ >. Corollary 3.. Let f, g, h be three continuous functions on [, ) and satisfying the following inequality: ψ f(x) Ψ, φ g(x) Φ and ω h(x) Ω (48) for some φ, ψ, ω, Φ, Ψ, Ω R, and for all x [, ). Then, for all t >, < <, < <, µ >, ν >, β >, δ >, η >, ζ >, η + + λ/β >, and ζ + + λ/δ >, the following inequality

12 3588 Junesang Choi, Daniele Ritelli and Praveen Agarwal holds true: [ ] Γ q(ζ + + λ/δ) δ δ Γ q (ν + ζ + + λ/δ) tλ Iq η,µ,β t λ f(t) g(t) h(t) } + Iq η,µ,β t λ h(t) } Iq ζ,ν,δ t λ f(t) g(t) } + Iq η,µ,β t λ g(t) } Iq ζ,ν,δ t λ f(t) h(t) } + Iq η,µ,β t λ f(t) } Iq ζ,ν,δ t λ g(t) h(t) } Iq η,µ,β t λ g(t) h(t) } Iq ζ,ν,δ t λ f(t) } Iq η,µ,β t λ f(t) h(t) } Iq ζ,ν,δ t λ g(t) } Iq η,µ,β t λ f(t) g(t) } Iq ζ,ν,δ t λ h(t) } [ ] Γ q(η + + λ/β) β β Γ q (µ + η + + λ/β) tλ Iq ζ,ν,δ t λ f(t) g(t) h(t) } [ ] [ ] Γ q(η + + λ/β) β δ β δ Γ q (µ + η + + λ/β) Γ (ζ + + λ/δ) Γ q (ν + ζ + + λ/δ) tλ (Ψ ψ)(φ φ)(ω ω). (49) Corollary 3.3. Let f, g : [, ) R satisfy Lipschitz conditions with constants L and L, respectively. Then the following inequality holds true: δ [/δ] q Γ q (ζ + + λ/δ) Γ q (ν + ζ + + λ/δ) tλ I η,µ,β t λ f(t) g(t) } Γ q (η + + λ/β) + β [/β] q Γ q (µ + η + + λ/β) tλ Iq ζ,ν,δ t λ f(t) g(t) } Iq η,µ,β t λ f(t) } Iq ζ,ν,δ t λ f(t) } t λ g(t) } I η,µ,β t λ g(t) } I ζ,ν,δ L L β δ [/β] q [/δ] q t λ+ [ Γ q (η + + (λ + )/β) Γ q (ζ + + λ/δ) Γ q (µ + η + + (λ + )/β) Γ q (ν + ζ + + λ/δ) Γ q (η + + (λ + )/β) Γ q (ζ + + (λ + )/δ) Γ q (µ + η + + (λ + )/β) Γ q (ν + ζ + + (λ + )/δ) + Γ (η + + λ/β) Γ q (ζ + + (λ + )/δ) ] Γ q (µ + η + + λ/β) Γ q (ν + ζ + + (λ + )/δ) (5) for all t >, < <, < <, µ >, ν >, β >, δ >, η >, ζ >, η + + λ/β >, and ζ + + λ/δ >. The formula (4) is further specialized as follows (see [3, p. 83, Eq. (3)]): [ ] Iq η,µ,β Γ q (η + ) K} = β K, (5) β Γ q (µ + η + ) q

13 Inequalities involving fractional q-integral operator 3589 where K is a constant, R(β) >, R(µ) >, R(η) >, and < q <. Setting u(t) = and = in those results of Theorem and Corollary and using (5) is seen to yield the known results [3, p. 84, Eq. (37)] and [3, p. 86, Eq. (4)], respectively. Also we briefly consider some other consequences of the results derived in the previous sections. Following Gaulué [3], the operator () would reduce immediately to the extensively investigated Riemann-Liouville and Kober type fractional integral operators, respectively, given by the following relationships (see also [] and []): I η,µ q f(t)} = Iq η,µ, f(t)} = t (η+µ) Γ q (µ) (η, µ C; R(µ) > ) t (t τ q) µ τ η f(τ) d q τ (5) and I µ q f(t)} = t µ I,µ, q f(t)} = Γ q (µ) (µ C; R(µ) > ). t (t τ q) µ f(τ) d q τ (53) Setting u(t) =, β = δ =, = = q in the result of Theorem and using (5) is seen to give the known fractional q-integral inequalities [3, p. 86, Eq. (43)]. The special cases of the results of Theorems -5 when β = δ = and the limit q is taken are seen to provide, respectively, the known inequalities due to Baleanu and Agarwal [5]. Setting β = δ = λ = η = ζ = and = = q in the result of Corollary and using (53) is seen to yield the known fractional q-integral inequalities due to Gaulué [3, p. 87, Eq. (44)] and Öǧümez and Özkam [6, p. 5, Eq. (3.)]. Further taking the limit q in those results just obtained gives the known result Belarbi and Dahmani[6, p. 88, Eq. 6]. The special case of Theorem when u(t) =, β = δ =, ζ = η = and = = q and formulas (5) and (53) are used yields the known result Gaulué [3, p. 87, Eq. (45)] and Sulaiman [8, p. 456, Eq. (3.)]. As we have seen in this section, the results presented here are of general character and potentially useful in deriving various q-inequalities in the theory of fractional q-integral operators and inequalities in the theory of fractional integral operators.

14 359 Junesang Choi, Daniele Ritelli and Praveen Agarwal Acknowledgements. This research was, in part, supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology of the Republic of Korea (Grant No. -5). This work was supported by Dongguk University Research Fund. References [] R. P. Agrawal, Certain fractional q-integrals and q- derivatives, Proc. Camb. Philos. Soc. 66 (969), [] W. A. Al-Salam, Some fractional q-integrals and q- derivatives, Proc. Edin. Math. Soc. 5 (966), [3] G. A. Anastassiou, Advances on Fractional Inequalities, Springer Briefs in Mathematics, Springer, New York, [4] G. A. Anastassiou, q-fractional Inequalities, CUBO A Mathematical Journal 3() (), [5] D. Baleanu and P. Agarwal, Certain inequalities involving the fractional q-integral Operators, Abstr. Appl. Anal. 4 (4), Article ID 3774, pages. [6] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, Int. Journal of Math. Analysis 4(4) (), [7] J. Choi and P. Agarwal, Some new Saigo type fractional integral inequalities and their q-analogues, Abstr. Appl. Anal. 4 (4), Article ID 5796, pages. [8] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 9 (), [9] Z. Dahmani, O. Mechouar and S. Brahami, Certain inequalities related to the Chebyshev s functional involving a type Riemann-Liouville operator, Bull. Math. Anal. Appl. 3(4) (),

15 Inequalities involving fractional q-integral operator 359 [] S. S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pure Appl. Math. 3(4) (), [] S. S. Dragomir, Some Lipschitz type inequalities for complex functions, Appl. Math. Comput. 3 (4), [] L. Gaulué, Generalized Erdélyi-Kober fractional q-integral operator, Kuwait J. Sci. Eng. 36(A) (9), 34. [3] L. Gaulué, Some results involving generalized Eedélyi-Kober fractional q-integral operators, Revista Tecno-Cientfica URU 6(4), [4] F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 4 (9), [5] V. Kac and P. Cheung, Quantum Calculus, Springer, New York,. [6] H. Öǧünmez and U. M. Özkan, Fractional quantum integral inequalities, J. Inequal. Appl., Article ID , 7 pp. [7] H. M. Srivastava and J. Choi, Zeta and q-zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York,. [8] W. T. Sulaiman, Some new fractional integral inequalities, J. Math. Anal. () (), 3 8. Received: March 7, 5; Published: April 3, 5