- Research Article
- Open Access
Fractional Quantum Integral Inequalities
© H. Öğünmez and U. M. Özkan. 2011
- Received: 10 November 2010
- Accepted: 16 February 2011
- Published: 10 March 2011
The aim of the present paper is to establish some fractional -integral inequalities on the specific time scale, , a nonnegative integer , where , and .
- Positive Integer
- Real Number
- Exponential Function
- Specific Time
- Similar Calculation
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 .
for all , .
for all , , and .
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.
In this section, we will state our main results and give their proofs.
and the proof is complete.
The following result may be seen as a generalization of Theorem 3.1.
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 ).
We prove this theorem by induction.
for all , .
for all , .
and this ends the proof.
The authors thank referees for suggestions which have improved the final version of this paper.
- Atıcı FM, Eloe PW: Fractional -calculus on a time scale. Journal of Nonlinear Mathematical Physics 2007,14(3):341–352.View ArticleMathSciNetMATHGoogle Scholar
- Belarbi S, Dahmani Z: On some new fractional integral inequalities. Journal of Inequalities in Pure and Applied Mathematics 2009,10(3, article 86):-5.Google Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar
- Denton Z, Vatsala AS: Fractional integral inequalities and applications. Computers & Mathematics with Applications 2010,59(3):1087–1094.MathSciNetView ArticleMATHGoogle Scholar
- Anastassiou GA: Multivariate fractional Ostrowski type inequalities. Computers & Mathematics with Applications 2007,54(3):434–447. 10.1016/j.camwa.2007.01.024MathSciNetView ArticleMATHGoogle Scholar
- Anastassiou GA: Opial type inequalities involving fractional derivatives of two functions and applications. Computers & Mathematics with Applications 2004,48(10–11):1701–1731. 10.1016/j.camwa.2003.08.013MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP: Certain fractional -integrals and -derivatives. Proceedings of the Cambridge Philosophical Society 1969, 66: 365–370. 10.1017/S0305004100045060View ArticleMathSciNetMATHGoogle Scholar
- Al-Salam WA: Some fractional -integrals and -derivatives. Proceedings of the Edinburgh Mathematical Society. Series II 1966, 15: 135–140. 10.1017/S0013091500011469MathSciNetView ArticleMATHGoogle Scholar
- Rajković PM, Marinković SD, Stanković MS: A generalization of the concept of -fractional integrals. Acta Mathematica Sinica (English Series) 2009,25(10):1635–1646. 10.1007/s10114-009-8253-xMathSciNetView ArticleMATHGoogle Scholar
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