Research Article | Open | Published:
Fractional Quantum Integral Inequalities
Journal of Inequalities and Applicationsvolume 2011, Article number: 787939 (2011)
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  serves as a bridge between the fractional -calculus in the literature and the fractional -calculus on a time scale , where , and .
Belarbi and Dahmani  gave the following integral inequality, using the Riemann-Liouville fractional integral: if and are two synchronous functions on , then
for all , .
Moreover, the authors  proved a generalized form of (1.1), namely that if and are two synchronous functions on , then
for all , , and .
Furthermore, the authors  pointed out that if are positive increasing functions on , then
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 .
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
If there is no confusion concerning , we will denote by . For a function , the nabla -derivative of is
for all . The -integral of is
The fundamental theorem of calculus applies to the -derivative and -integral; in particular,
and if is continuous at 0, then
Let , denote two time scales. Let be continuous let be -differentiable, strictly increasing, and . Then for ,
The -factorial function is defined in the following way: if is a positive integer, then
If is not a positive integer, then
The -derivative of the -factorial function with respect to is
and the -derivative of the -factorial function with respect to is
The -exponential function is defined as
Define the -Gamma function by
The fractional -integral is defined as
3. Main Results
In this section, we will state our main results and give their proofs.
Let and be two synchronous functions on . Then for all , , we have
Since and are synchronous functions on , we get
for all , . By (3.2), we write
Multiplying both side of (3.3) by , we have
Integrating both sides of (3.4) with respect to on , we obtain
Hence, we have
Multiplying both side of (3.7) by , we obtain
Integrating both side of (3.8) with respect to on , we get
and the proof is complete.
The following result may be seen as a generalization of Theorem 3.1.
Let and be as in Theorem 3.1. Then for all , , we have
By making similar calculations as in Theorem 3.1 we have
Integrating both side of (3.12) with respect to on , we obtain
Thus, (3.11) holds for all , , , so the proof is complete.
The inequalities (3.1) and (3.11) are reversed if the functions are asynchronous on (i.e., , for any ).
Let be positive increasing functions on . Then for any , we have
We prove this theorem by induction.
Clearly, for , we have
for all , .
For , applying (3.1), we obtain
for all , .
Since are positive increasing functions, then is an increasing function. Hence, we can apply Theorem 3.1 to the functions , . We obtain
Taking into account the hypothesis (3.17), we obtain
and this ends the proof.
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The authors thank referees for suggestions which have improved the final version of this paper.