A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL

Abstract

In this paper, we present a new definition of $q$-integral by using trapezoid pieces and we name second sense $q$-integral which is showed $ \overline{q}$-integral and we give some results and properties of $\overline{ q}$-integral. Finaly, we establish some new $\overline{q}$-Hermite-Hadamard type inequalities for convex functions.

Keywords

• Hermite-Hadamard's inequalities,convex functions,q-integrals

References

1. [1] N. Alp, M. Z. Sarikaya, M. Kunt and ·I. ·I¸scan, q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, Journal of King Saud University - Science, 2016, dx.doi.org/10.1016/j.jksus.2016.09.007.
2. [2] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, JIPAM. J. Inequal. Pure Appl. Math. 2009., 10: Article ID 86.
3. [3] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 2010, 9: 493–497.
4. [4] T. Ernst, A method for q-calculus. J. Nonlinear Math. Phys. 10 (4), 487–525 (2003).
5. [5] H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl. 2004, 47: 281–300. 10.1016/S0898-1221(04)90025-9.
6. [6] J. Hadamard, Etude sur les propri´ et´ es des fonctions enti´ eres et en particulier dune fonc- tion consider´ ee par Riemann, J. Math. Pures Appl. 58 (1893) 171–215.
7. [7] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.
8. [8] M. A. Noor, K. I. Noor and M. U. Awan, Some Quantum estimates for Hermite–Hadamard inequalities, Appl. Math. Comput. 251, 675–679 (2015).

9. [9] M. A. Noor, K. I. Noor and M. U. Awan, Quantum Ostrowski inequalities for q-di¤ erentiable convex functions, J. Math. Inequlities, (2016).
10. [10] H. Ogunmez and U.M. Ozkan, Fractional quantum integral inequalities, J. Inequal. Appl. 2011., 2011: Article ID 787939
11. [11] W. Sudsutad and S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, Jour. Math Ineq. Volume 9, Number 3 (2015), 781–793.
12. [12] J. Tariboon, S. K. Ntouyas, Quantum calculus on …nite intervals and applications to impul- sive di¤ erence equations, Adv. Di¤er. Equ. 2013, 2013:282.
13. [13] J. Tariboon and S. K. Ntouyas, Quantum integral inequalities on …nite intervals, J. Inequal. Appl. 2014, 2014:121.