On some new inequalities involving generalized Erd\'elyi-Kober fractional $q$-integral operator

In the present investigation, we aim to establish some inequalities involving generalized Erd\'elyi-Kober fractional $q$-integral operator of the two parameters of deformation $q_{1}$ and $q_{2}$ due to Gaulu\'e, by following the same lines used by Baleanu and Agarwal in their recent paper. Relevant connections of the results presented here with those earlier ones are also pointed out.


Introduction and Preliminaries
Throughout this paper, N, R, C, and Z − 0 denote the sets of positive integers, real numbers, complex numbers, and nonpositive integers, respectively, and N 0 := N ∪ {0}. The enormous success of the theory of integral inequalities involving various fractional integral operators has stimulated the development of a corresponding theory in qfractional integral inequalities (see, e.g., [2,3,5,6,7,8,9,13,15]). In this paper we aim to present some inequalities involving generalized Erdélyi-Kober fractional q-integral operator of the two parameters of deformation q 1 and q 2 due to Gaulué [10], by following the same lines used by Baleanu and Agarwal [4] in their recent paper. Relevant connections of the results presented here with those earlier ones are also pointed out.
The q-shifted factorial (a; q) n is defined by where a, q ∈ C and it is assumed that a = q −m (m ∈ N 0 ).
We begin by noting that Jackson [11] was the first to develop q-calculus in a systematic way, this topic is also developed in the recent monograph [12].
The q-derivative of a function f (t) is defined by For q → 1 we obtain the definition of differential function, being: It is denoted by The Jackson integral of f (t) is thus defined, formally, by which can be easily generalized in the Stieltjes sense as follows: (1.11) A more general version of (1.10) is given by The classical Gamma function Γ(z) (see, e.g., [14,Section 1.1]) was introduced by Leonhard Euler in 1729 while he was trying to extend the factorial n! = Γ(n+ 1) (n ∈ N 0 ) to real numbers. The q-factorial function (n) q ! (n ∈ N 0 ) of n! defined by (n) q ! := 1 (n = 0), can be rewritten as follows: (1 − q) −n := Γ q (n + 1) (0 < q < 1). (1.14) Replacing n by a − 1 in (1.14), Jackson [11] defined the q-Gamma function Γ q (a) by The q-analogue of (t − a) n is defined by the polynomial (1.16) Definition 3. Let 0 < q < 1, f ∈ C λ . Then for ℜ(β), ℜ(µ) > 0, and η ∈ C we define a generalized Erdélyi-Kober fractional integral I α,β,η q as follows (see, [10]) : (1.17) Definition 4. Let 0 < q < 1, f ∈ C λ . Then for ℜ(µ) > 0 and η ∈ C a q-analogue of the Kober fractional integral operator is given by (see [1]) is a continuous function. Then we conclude that, under the given conditions in (1.17), each term in the series of generalized Erdélyi-Kober q-integral operator is nonnegative defined by for all µ > 0 and η ∈ C.
On the same way each term in the series of Kober q-integral operator (1.18) is also nonnegative defined by for all µ > 0 and η ∈ C.

Generalized Erdélyi-Kober q-integral Inequalities
In this section, we present six q-integral inequalities, which are the core of our research, involving the generalized Erdélyi-Kober q-integral (1.17) stated in Theorem 1 to 6 below. Before we recall, as stated in [6,p. 1,Eq. 2] therein, that if f, g are two real function defined and integrable on a real interval (I ≡ I ∈ [a, b]) , we say that f and g are synchronous on I if for each x, y ∈ I the following inequality holds: Similarly f and g are asynchronous on I if for any x, y ∈ I the inequality is reversed, that is: is a continuous function, then the following inequality holds true: for all t > 0, µ, ν, β, δ > 0 and η, ζ ∈ C.
Proof: Let f, g and h are three continuous and synchronous functions on [0, ∞) and (2.1) is satisfied. Then for all τ, ρ ≥ 0, we have Now, multiplying both sides of (2.5) by and integrating the resulting inequality with respect to τ from 0 to t, and using (1.17), we get Next, multiply both sides of (2.6) by which remains nonnegative under the conditions in (1.20) and integrating the resulting inequality with respect to ρ from 0 to t, and using (1.17), we are led to the desired result (2.3). This complete the proof of Theorem 1.
Proof: To prove the above result, multiplying both sides of (2.6) by which remains nonnegative under the conditions in (1.20) and integrating the resulting inequality with respect to ρ from 0 to t, and using (1.17), we are led to the desired result (2.7). This complete the proof of Theorem 2.

Remark 1. It may be noted that the inequalities in (2.3) and (2.7)
are reversed if functions f, g and h are asynchronous. It is also easily seen that the special case u = v of (2.7) in Theorem 2 reduces to that in Theorem 1.
Proof: Since f, g and h are three continuous and synchronous functions on [0, ∞), for all τ, ρ ≥ 0, the inequality (2.1) is satisfied. we have from (2.8): (2.10) Let us define the function

11)
Multiplying both sides of (2.11) by taking q-integration of the resulting inequality with respect to τ from 0 to t and using (1.17), we get: (2.12) Next, multiply both sides of (2.12) by then q-integrate of the resulting inequality with respect to ρ from 0 to t, and use (1.17), so that (2.13) Finally, by using (2.10) on to (2.13), we arrive at the desired result (2.9), involve in Theorem 3, after a little simplification.
Proof: Multiplying both sides of (2.12) by and taking the q-integration of the resulting inequality with respect to ρ from 0 to t with the aid of Definition 1 and then applying (2.10) on the resulting inequality, we get the desired result (2.15).

Remark 2.
It is easily seen that the special case u = v of (2.15) in Theorem 4 reduces to that in Theorem 3.
Proof: Let us define the following relations for all τ, ρ ∈ [0, ∞) : where, A(τ, ρ) is given by (2.11). Then, by setting: First, we multiplying both sides of (2.19) by respectively, and taking the q-integration of the resulting inequality with respect to τ and ρ from 0 to t with the aid of Definition 1 and then applying (2.10) and (2.18) on the resulting inequality, we get the desired result (2.16). This complete the proof of Theorem 5.
Remark 3. It is easily seen that the special case u = v of (2.20) in Theorem 6 reduces to that in Theorem 5.

Concluding Remarks
We can present a large number of special cases of our main inequalities in Theorems 1 to 6. Here we give only one example: Setting β = 1 in Theorem 1-6, we get the known results given by Baleanu and Agarwal [4]. Therefore, we arrived at the conclusion that our present investigation are general in character and useful in deriving various q-inequalities in the theory of fractional q-integral operators.