1. Introduction

The study of fractional -calculus in [1] serves as a bridge between the fractional -calculus in the literature and the fractional -calculus on a time scale , where , and .

Belarbi and Dahmani [2] gave the following integral inequality, using the Riemann-Liouville fractional integral: if and are two synchronous functions on , then

(1.1)

for all , .

Moreover, the authors [2] proved a generalized form of (1.1), namely that if and are two synchronous functions on , then

(1.2)

for all , , and .

Furthermore, the authors [2] pointed out that if are positive increasing functions on , then

(1.3)

for any , .

In this paper, we have obtained fractional -integral inequalities, which are quantum versions of inequalities (1.1), (1.2), and (1.3), on the specific time scale , where , and . In general, a time scale is an arbitrary nonempty closed subset of the real numbers [3].

Many authors have studied the fractional integral inequalities and applications. For example, we refer the reader to [4–6].

To the best of our knowledge, this paper is the first one that focuses on fractional -integral inequalities.

2. Description of Fractional -Calculus

Let and define

(2.1)

If there is no confusion concerning , we will denote by . For a function , the nabla -derivative of is

(2.2)

for all . The -integral of is

(2.3)

The fundamental theorem of calculus applies to the -derivative and -integral; in particular,

(2.4)

and if is continuous at 0, then

(2.5)

Let , denote two time scales. Let be continuous let be -differentiable, strictly increasing, and . Then for ,

(2.6)

The -factorial function is defined in the following way: if is a positive integer, then

(2.7)

If is not a positive integer, then

(2.8)

The -derivative of the -factorial function with respect to is

(2.9)

and the -derivative of the -factorial function with respect to is

(2.10)

The -exponential function is defined as

(2.11)

Define the -Gamma function by

(2.12)

Note that

(2.13)

The fractional -integral is defined as

(2.14)

Note that

(2.15)

More results concerning fractional -calculus can be found in [1, 7–9].

3. Main Results

In this section, we will state our main results and give their proofs.

Theorem 3.1.

Let and be two synchronous functions on . Then for all , , we have

(3.1)

Proof.

Since and are synchronous functions on , we get

(3.2)

for all , . By (3.2), we write

(3.3)

Multiplying both side of (3.3) by , we have

(3.4)

Integrating both sides of (3.4) with respect to on , we obtain

(3.5)

So,

(3.6)

Hence, we have

(3.7)

Multiplying both side of (3.7) by , we obtain

(3.8)

Integrating both side of (3.8) with respect to on , we get

(3.9)

Obviously,

(3.10)

and the proof is complete.

The following result may be seen as a generalization of Theorem 3.1.

Theorem 3.2.

Let and be as in Theorem 3.1. Then for all , , we have

(3.11)

Proof.

By making similar calculations as in Theorem 3.1 we have

(3.12)

Integrating both side of (3.12) with respect to on , we obtain

(3.13)

Thus, (3.11) holds for all , , , so the proof is complete.

Remark 3.3.

The inequalities (3.1) and (3.11) are reversed if the functions are asynchronous on (i.e., , for any ).

Theorem 3.4.

Let be positive increasing functions on . Then for any , we have

(3.14)

Proof.

We prove this theorem by induction.

Clearly, for , we have

(3.15)

for all , .

For , applying (3.1), we obtain

(3.16)

for all , .

Suppose that

(3.17)

Since are positive increasing functions, then is an increasing function. Hence, we can apply Theorem 3.1 to the functions , . We obtain

(3.18)

Taking into account the hypothesis (3.17), we obtain

(3.19)

and this ends the proof.