Fractional Quantum Integral Inequalities

The aim of the present paper is to establish some fractional -integral inequalities on the specific time scale, , a nonnegative integer, where , and .

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.

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].

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.

The authors thank referees for suggestions which have improved the final version of this paper.

Affiliations

1. Department of Mathematics, Faculty of Science and Arts, Kocatepe University, 03200, Afyonkarahisar, Turkey

   • Hasan Öğünmez
   •  & UmutMutlu Özkan

Corresponding author

Correspondence to UmutMutlu Özkan.

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