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

Necmettin Alp; Mehmet Zeki Sarıkaya

Research Article

Year 2017, Volume: 5 Issue: 2, 146 - 159, 15.10.2017

Necmettin Alp Mehmet Zeki Sarıkaya

Abstract

References

[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] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, JIPAM. J. Inequal. Pure Appl. Math. 2009., 10: Article ID 86.
[3] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 2010, 9: 493497.
[4] T. Ernst, A method for q-calculus. J. Nonlinear Math. Phys. 10 (4), 487525 (2003).
[5] H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl. 2004, 47: 281300. 10.1016/S0898-1221(04)90025-9.
[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) 171215.
[7] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.
[8] M. A. Noor, K. I. Noor and M. U. Awan, Some Quantum estimates for HermiteHadamard inequalities, Appl. Math. Comput. 251, 675679 (2015).
[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] H. Ogunmez and U.M. Ozkan, Fractional quantum integral inequalities, J. Inequal. Appl. 2011., 2011: Article ID 787939
[11] W. Sudsutad and S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, Jour. Math Ineq. Volume 9, Number 3 (2015), 781793.
[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] J. Tariboon and S. K. Ntouyas, Quantum integral inequalities on nite intervals, J. Inequal. Appl. 2014, 2014:121.

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

Year 2017, Volume: 5 Issue: 2, 146 - 159, 15.10.2017

Necmettin Alp Mehmet Zeki Sarıkaya

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] 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] S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, JIPAM. J. Inequal. Pure Appl. Math. 2009., 10: Article ID 86.
[3] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 2010, 9: 493497.
[4] T. Ernst, A method for q-calculus. J. Nonlinear Math. Phys. 10 (4), 487525 (2003).
[5] H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl. 2004, 47: 281300. 10.1016/S0898-1221(04)90025-9.
[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) 171215.
[7] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.
[8] M. A. Noor, K. I. Noor and M. U. Awan, Some Quantum estimates for HermiteHadamard inequalities, Appl. Math. Comput. 251, 675679 (2015).
[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] H. Ogunmez and U.M. Ozkan, Fractional quantum integral inequalities, J. Inequal. Appl. 2011., 2011: Article ID 787939
[11] W. Sudsutad and S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, Jour. Math Ineq. Volume 9, Number 3 (2015), 781793.
[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] J. Tariboon and S. K. Ntouyas, Quantum integral inequalities on nite intervals, J. Inequal. Appl. 2014, 2014:121.

There are 13 citations in total.

Details

Subjects	Engineering
Journal Section	Articles
Authors	Necmettin Alp Mehmet Zeki Sarıkaya
Publication Date	October 15, 2017
Submission Date	October 6, 2017
Acceptance Date	October 12, 2017
Published in Issue	Year 2017 Volume: 5 Issue: 2

Cite

APA	Alp, N., & Sarıkaya, M. Z. (2017). A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL. Konuralp Journal of Mathematics, 5(2), 146-159.
AMA	Alp N, Sarıkaya MZ. A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL. Konuralp J. Math. October 2017;5(2):146-159.
Chicago	Alp, Necmettin, and Mehmet Zeki Sarıkaya. “A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL”. Konuralp Journal of Mathematics 5, no. 2 (October 2017): 146-59.
EndNote	Alp N, Sarıkaya MZ (October 1, 2017) A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL. Konuralp Journal of Mathematics 5 2 146–159.
IEEE	N. Alp and M. Z. Sarıkaya, “A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL”, Konuralp J. Math., vol. 5, no. 2, pp. 146–159, 2017.
ISNAD	Alp, Necmettin - Sarıkaya, Mehmet Zeki. “A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL”. Konuralp Journal of Mathematics 5/2 (October 2017), 146-159.
JAMA	Alp N, Sarıkaya MZ. A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL. Konuralp J. Math. 2017;5:146–159.
MLA	Alp, Necmettin and Mehmet Zeki Sarıkaya. “A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL”. Konuralp Journal of Mathematics, vol. 5, no. 2, 2017, pp. 146-59.
Vancouver	Alp N, Sarıkaya MZ. A NEW DEFINITION AND PROPERTIES OF QUANTUM INTEGRAL WHICH CALLS $\overline{q}$-INTEGRAL. Konuralp J. Math. 2017;5(2):146-59.

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