Certain Inequalities Involving Generalized Erdélyi-Kober Fractional q-Integral Operators

In recent years, a remarkably large number of inequalities involving the fractional q-integral operators have been investigated in the literature by many authors. Here, we aim to present some new fractional integral inequalities involving generalized Erdélyi-Kober fractional q-integral operator due to Gaulué, whose special cases are shown to yield corresponding inequalities associated with Kober type fractional q-integral operators. The cases of synchronous functions as well as of functions bounded by integrable functions are considered.

The inequality in (2) is reversed if and are asynchronous on [ , ]; that is, for any , ∈ [ , ]. If ( ) = ( ) for any , ∈ [ , ], we get the Chebyshev inequality (see [1]). Ostrowski [4] established the following generalization of the Chebyshev inequality. If and are two differentiable and synchronous functions on [ , ] and is a positive integrable function on [ , ] with | ( )| ≥ and | ( )| ≥ for ∈ [ , ], then we have Here, it is worth mentioning that the functional (1) has attracted many researchers' attention mainly due to diverse applications in numerical quadrature, transform theory, probability, and statistical problems. Among those applications, the functional (1) has also been employed to yield a number of integral inequalities (see, e.g., [5][6][7][8][9][10][11]). The study of the fractional integral and fractionalintegral inequalities has been of great importance due to the fundamental role in the theory of differential equations. In recent years, a number of researchers have done deep study, that is, the properties, applications, and different extensions of various fractional -integral operators (see, e.g., [12][13][14][15][16]).
The purpose of this paper is to find -calculus analogs of some classical integral inequalities. In particular, we will find -generalizations of the Chebyshev integral inequalities by using the generalized Erdélyi-Kober fractional -integral operator introduced by Galué [17]. The main objective of this paper is to present some new fractional -integral inequalities involving the generalized Erdélyi-Kober fractional -integral operator. We consider the case of synchronous functions as well as the case of functions bounded by integrable functions. Some of the known and new results are as follows, as special cases of our main findings. We emphasize that the results derived in this paper are more generalized results rather than similar published results because we established all results by using the generalized Erdélyi-Kober fractionalintegral operator. Our results are general in character and give some contributions to the theory -integral inequalities and fractional calculus.

Preliminaries
In the sequel, we required the following well-known results to establish our main results in the present paper. The -shifted factorial ( ; ) is defined by where , ∈ C and it is assumed that ̸ = − ( ∈ N 0 ). The -shifted factorial for negative subscript is defined by We also write It follows from (8), (9), and (10) that which can be extended to = ∈ C as follows: where the principal value of is taken. We begin by noting that F. J. Jackson was the first to develop -calculus in a systematic way. For 0 < < 1, the -derivative of a continuous function on [0, ] is defined by and if The Jackson integral of ( ) is thus defined, formally, by which can be easily generalized as follows: Suppose that 0 < < . The definite -integral is defined as follows: A more general version of (18) is given by The classical Gamma function Γ( ) (see, e.g., [18, Section 1.1]) was found by Euler while he was trying to extend the factorial ! = Γ( + 1)( ∈ N 0 ) to real numbers. The - The Scientific World Journal 3 can be rewritten as follows: Replacing by − 1 in (22), Jackson [19] defined the -Gamma function Γ ( ) by The -analogue of ( − ) is defined by the polynomial More generally, if ∈ R, then Then a generalized Erdélyi-Kober fractional integral , , for a real-valued continuous function ( ) is defined by (see, [17]) Definition 2. A -analogue of the Kober fractional integral operator is given by (see, [20]) ( > 0; ∈ C; 0 < < 1) .

(27)
Remark 3. It is easy to see that for all > 0 and ∈ N 0 . If : [0,∞) → [0,∞) is a continuous function, then we conclude that, under the given conditions in (26), each term in the series of generalized Erdélyi-Kober -integral operator is nonnegative and thus for all , > 0 and ∈ C.
On the same way each term in the series of Koberintegral operator (27) is also nonnegative and thus for all > 0 and ∈ C.

Inequalities Involving a Generalized Erdélyi-Kober Fractional -Integral Operator for Synchronous Functions
This section begins by presenting two inequalities involving generalized Erdélyi-Kober -integral operator (26) stated in Lemmas 4 and 5 below.
for all , > 0 and ∈ C.
Proof. Let and be two continuous and synchronous functions on [0, ∞). Then, for all , ∈ (0, ) with > 0, we have or, equivalently, Now, multiplying both sides of (33) by integrating the resulting inequality with respect to from 0 to , and using (26), we get  Next, multiplying both sides of (34) by integrating the resulting inequality with respect to from 0 to , and using (26), we are led to the desired result (31). Proof. Multiplying both sides of (34) by which remains nonnegative under the conditions in (35), integrating the resulting inequality with respect to from 0 to , and using (26), we get the desired result (35).
for all , > 0 and ∈ C.
Proof. By setting = and V = in Lemma 4, we get Finally, we find that the inequality (42) follows by adding the inequalities (44) and (45), side by side.
for all > 0 and ∈ C.  The Scientific World Journal Remark 12. If we take = 0 and = 1 in Theorem 6 and = = 0 and = = 1 in Theorem 7, then we obtain the known results due to Dahmani [21].
As special cases of Theorems 13, we obtain the following results.
To prove ( )-( V), we use the following inequalities: