On Chebyshev and Riemann-Liouville Fractional Inequalities in q-Calculus

Main Article Content

Stephen. N. Ajega-Akem
Mohammed M. Iddrisu
Kwara Nantomah

Abstract

This paper presents some new inequalities on Fractional calculus in the context of q-calculus. Fractional calculus generalizes the integer order differentiation and integration to derivatives and integrals of arbitrary order. In other words, Fractional calculus explores integrals and derivatives of functions that involve non-integer order(s). Quantum calculus (q-Calculus) on the other hand focuses on investigations related to calculus without limits and in recent times, it has attracted the interest of many researchers due to its high demand of mathematics to model complex systems in nature with anomalous dynamics. This paper thus establishes some new extensions of Chebyshev and Riemann-Liouville fractional integral inequalities for positive and increasing functions via q-Calculus.

Keywords:
Chebyshev inequality, riemann-liouville, fractional calculus, q-Calculus

Article Details

How to Cite
Ajega-Akem, S. N., Iddrisu, M. M., & Nantomah, K. (2019). On Chebyshev and Riemann-Liouville Fractional Inequalities in q-Calculus. Asian Research Journal of Mathematics, 15(2), 1-10. https://doi.org/10.9734/arjom/2019/v15i230144
Section
Original Research Article

References

Chebyshev PL. Sur les expressions approximatives des integrales denies par les autres prises entre les mmes limites. Proc. Math. Soc. Charkov. 1882;93-98.

Dahmani Z. Some results associate with fractional integrals involving the extended chebyshev functional. Acta Univ. Apulens. 2011;27:217-224.

Lakoud AG, Aissaouinew F. Chebyshev type inequalities for double integrals. Journal Math. Inequali. 2011;5(4):453-462.

Set E, Dahmani Z, Mumcu I. New extensions of chebyshev type inequalities using generalized katugampola integrals via polya-szeg inequality,. IJOCTA. 2018;8(2):137-144.

Usta F, Sarikaya MZ, Budak H. Some new chebyshev type integral inequalities via fractional integral operator with exponential kerne. Research Gate. 2017;3-5.

Jackson F. On q-denite integrals. Quart. J. Pure and Appl. Math. 1910;41:1193-203.

Nantomah K. Generalized holders and minkowskis inequalities for jacksons q-integral and some applications to the incomplete q-gamma function. Hindawi Abstract and Applied Analysis. 2017;6. (ID 9796873)

Iddrisu MM. q-steensens inequality for convex functions. International Journal of Mathematics and its Applications. 2018;6(2-A):157-162.

Nantomah K, Iddrisu MM, Okpoti CA. On a q-analogue of the nielsens function. 2018, 163171 ISSN:234. Int. J. Math. and Appl. 2018;6(2a):165.

Freihet A, Hasan S, Al-Smadi M, Gaith M, Momani S. Construction of fractional power series solutions to fractional stiff system using residual functions algorithm. Advances in Difference Equations. 2019;(1):95.

Momani S, Arqub OA, Freihat A, Al-Smadi M. Analytical approximations for fokker-planck equations of fractional order in multistep schemes. Applied and Computational Mathematics.;15(3):319-330.

Oney S. The jackson integral. Int. J. Math. And Appl; 2007.

Sofonea DF. Some new properties in q-calculus. General Mathematics. 2008;16(1):47-54.

Kilbas AA. Hadamard-type fractional calculus. J. Korean Math. Soc. 2001;38(6):1191-1204.

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional dierential equations. North-Holland Mathematical Studies, Elsevier, Amsterdam. 2006;204-245.

Belarbi S, Dahmani Z. On some new fractional integral inequalities. J.Inequal. Pure Appl. Math. 2009;10(3):86.

Annaby M, Mansour S. q-fractional calculus and equations. Lecture Notes; 2012.
Available:http://www.springer.com/978-3-642-30897-0

Hasan S, Al-Smadi M, Freihet A, Momani S. Two computational approaches for solving a fractional obstacle system in hilbert space. Advances in Dierence Equations. 2019(1):55.

Tariboon J, Ntouyas SK, Sudsutad W. Some new riemann-liouville fractional integral inequalities. International Journal of Mathematics and Mathematical Sciences; 2014.