Fractional Quantum Hermite-Hadamard Type Inequalities

Abstract

In this paper, Riemann-Liouville fractional quantum Hermite-Hadamard type inequalities are proved. Also, two identities for continuous functions in the form of Riemann-Liouville fractional quantum integral type are obtained. By using these identities, some Riemann-Liouville fractional quantum trapezoid and midpoint type inequalities for convex functions are given. The results of this paper generalize the results given in earlier works.

Keywords

• Quantum derivative,Quantum integral,Riemann-Liouville fractional integrals,Fractional quantum integral,Hermite-Hadamard type inequalities,Convex functions

References

1. [1] Annaby M.H., Mansour Z.S., q- Fractional Calculus and Equations, Springer, Heidelberg, (2012).
2. [2] Alp N., Sarıkaya M.Z., Kunt M., ˙Is¸can ˙I., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, Journal of King Saud University –Science, 30(2) (2018) 193-203.
3. [3] Dragomir S.S., Agarwal R. P., Two Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Trapezoidal Formula, Appl. Math. Lett., 11(5) (1998) 91-95.
4. [4] Kırmacı, U. S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004) 137-146.
5. [5] Kac V., Cheung P., Quantum Calculus, Springer, New York, (2002).
6. [6] Kunt M., İşcan, İ, Erratum: Quantum integral inequalities for convex functions, Researchgate, DOI: 10.13140/RG.2.1.3509.1441, (2016).
7. [7] Kunt M., İşcan, İ., Erratum: Some quantum estimates for Hermite-Hadamard inequalities, Researchgate, DOI: 10.13140/RG.2.1.4076.4402, (2016).
8. [8] Kunt M., Karapınar D., Turhan S.,˙Is¸can ˙I., The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions, Mathematica Slovaca, 69 (4), 773-784, 2019

9. [9] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations, Elsevier, Amsterdam (2006).
10. [10] Noor M. A., Noor K. I., Awan M. U., Some quantum estimates for Hermite–Hadamard inequalities, App. Math. Comput., 251 (2015) 675–679.
11. [11] Pearce C. E. M., Pecaric J., Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13 (2000) 51-55.
12. [12] A. W. Roberts, D. E. Varberg, Convex functions, Academic Press, New York (1973.)
13. [13] Sarıkaya M. Z., Set E., Yaldız H., Bas¸ak N., Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Computer Mod., 57(2013) 2403-2407.
14. [14] Sudsutad W., Ntouyas S. K., Tariboon J., Quantum integral inequalities for convex functions, J. Math. Inequal., 9(3) (2015) 781-793.
15. [15] Sudsutad W., Ntouyas S. K., Tariboon J., Integral inequalities via fractional quantum calculus, J. Inequal. Appl., 81(2016) 1-15.
16. [16] Tariboon J, Ntouyas S. K., Quantum integral inequalities on finite intervals, J. Inequal. Appl. 121 (2014) 1-13.
17. [17] Tariboon J, Ntouyas S. K., Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Difference Equ. 282 (2013) 1-19.
18. [18] Tariboon J.,Ntouyas S. K., Agarwal P., New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations, Adv. Diff. Equ. 18(2015) 1-19.
19. [19] Zhang, Y. Z., Du, T-S., Wang, H., Shen, Y-J., Different types of quantum integral inequalities via (a;m)-convexity, J. Inequal. Appl., 264(2018) 1-24.