Some trapezoid and midpoint type inequalities for newly defined quantum integrals

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-2021-01-0013

Keywords:

Hermite-Hadamard inequality, q-integral, Trapezoid, Midpoint, Quantum calculus, Convex function

Abstract

In this paper, we first obtain prove two new identities for the quantum integrals. Then we establish Trapezoid and Midpoint type inequalities for quantum integrals defined by Bermudo et al. in [3]. The inequalities in this study generalize some results obtained in earlier works.

Author Biography

Hüseyin Budak, Düzce University.

Dept. of Mathematics, Faculty of Science and Arts.

References

N. Alp, M. Z. Sarikaya, M. Kunt and I. Işcan, “q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions”, Journal of King Saud University Science, vol. 30, no. 2, pp. 193-203, Apr. 2018, doi: 10.1016/j.jksus.2016.09.007

N. Alp and M. Z. Sarikaya, “Hermite Hadamard’s type inequalities for co-ordinated convex functions on quantum integral”, Preprint, Dec. 2018. [On line]. Available: https://bit.ly/2XzUush

S. Bermudo, P. Kórus, and J. Nápoles. Valdés, “On q–Hermite-Hadamard inequalities for general convex functions”, Acta mathematica hungarica, vol. 162, no. 1, pp. 364-375, Oct. 2020, doi: 10.1007/s10474-020-01025-6

S. S. Dragomir and C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications. Melbourne: RGMIA Monographs, Victoria University, 2000. [On line]. Available: https://bit.ly/3nLdEpA

S. S. Dragomir and R. P. Agarwal, “Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula”, Applied mathematics letters, vol. 11, no. 5, pp. 91-95, Sep. 1998, doi: 10.1016/S0893-9659(98)00086-X

T. Ernst, The history of q-calculus and new method. Uppsala: Uppsala University, 2000. [On line]. Available: https://bit.ly/2LQkdK5

T. Ernst, A comprehensive treatment of q-calculus. Basel: Birkhäuser, 2012, doi: 10.1007/978-3-0348-0431-8

F. H. Jackson, “On a q-definite integrals”, The quarterly journal pure applications mathematics, vol. 41, pp. 193-203, 1910. [On line]. Available: https://bit.ly/35DhdYy

V. Kac and P. Cheung, Quantum calculus. New York, NY: Springer, 2002, doi: 10.1007/978-1-4613-0071-7

U. S. Kirmaci, “Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula”, Applied mathematics and computation, vol. 147, no. 1, pp. 137-146, Jan. 2004, doi: 10.1016/S0096-3003(02)00657-4

M. A. Noor, K. I. Noor, and M. U. Awan, “Some quantum estimates for Hermite-Hadamard inequalities”, Applications mathematics computation, vol. 251, pp. 675-679, Jan. 2015, doi: 10.1016/j.amc.2014.11.090

M. A. Noor, K. I. Noor, and M. U. Awan, “Some quantum integral inequalities via preinvex functions”, Applications mathematics computation, vol. 269, pp. 242-251, Oct. 2015, doi: 10.1016/j.amc.2015.07.078

M. Noor, K. Noor, and M. Awan, “Quantum Ostrowski inequalities for q-differentiable convex functions”, Journal of mathematical inequalities, vol. 10, no. 4, pp. 1013-1018, 2016, doi: 10.7153/jmi-10-81

J. E. Pĕcarić, F. Proschan, and Y. L. Tong, Convex functions, partial orderings and statistical applications. Boston, MA: Academic Press, 1992.

W. Sudsutad, S. K. Ntouyas, and J. Tariboon, “Quantum integral inequalities for convex functions”, Journal mathematics inequalities, vol. 9, no. 3, pp. 781-793, 2015, doi: 10.7153/jmi-09-64

J. Tariboon and S. K. Ntouyas, “Quantum calculus on finite intervals and applications to impulsive difference equations”, Advances difference equations, vol. 282, Art ID. 282, Nov. 2013, doi: 10.1186/1687-1847-2013-282

H. Zhuang, W. Liu, J. Park, “Some quantum estimates of Hermite-Hadmard inequalities for quasi-convex functions”, Mathematics, vol. 7, no.2, Art ID. 152, Feb. 2019, doi: 10.3390/math7020152

Published

2021-01-16

How to Cite

[1]
H. Budak, “Some trapezoid and midpoint type inequalities for newly defined quantum integrals”, Proyecciones (Antofagasta, On line), vol. 40, no. 1, pp. 199-215, Jan. 2021.

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Section

Artículos