Some ($p$,$q$)-integral Inequalities

Authors

  • İlker Genç Türk Kırıkkale University Department of Mathematics 71450, Yahsihan, Kırıkkale, Turkey

DOI:

https://doi.org/10.5644/SJM.20.02.06

Keywords:

inequalities, (p,q)-derivative, (p,q)-integral

Abstract

In this paper, we obtain a (p,q)-analogue of an open problem represented by Q. A. Ngo et al. in the paper, Notes on an integral inequality, J. Inequal. Pure and Appl. Math., 7(4) (2006), Art. 120, by using analytic and elementary methods in (p,q)-calculus.

 

Statistics

Abstract: 271  /   PDF: 75

 

References

[1] A. Aral and V. Gupta, Applications of (p,q)-gamma function to Szasz Durrmeyer operators, Publ. Inst. Math. (Beograd) (N.S.), 102 (2017), 211–220.

[2] N. Arunrat, K. M. Nakprasit, K. Nonlaopon, J., Tariboon and S. K. Ntouyas, On Fejer Type Inequalities via (p,q)-calculus, Symmetry, 13(6)(2021), 953.

[3] L. Bougoffa, Note on an open problem, JIPAM. J. Inequal. Pure Appl. Math., 8 (2)(2007), Art. 58.

[4] K. Boukerrioua and A. Guezane-Lakoud, On an open question regarding an integral inequality, JIPAM. J. Inequal. Pure Appl. Math., 8 (3) (2007), Art. 77.

[5] K. Brahim, On some q-integral inequalities, JIPAM. J. Inequal. Pure Appl. Math., 9 (4)(2008), Art. 106.

[6] I. M. Burban, Two-parameter deformation of the oscillator algebra and (p,q)-analog of twodimensional conformal field theory, J. Nonlinear Math. Phys., 2 (1995), 384–391.

[7] I. M. Burban and A. U. Klimyk, (p,q)-differentiation, (p,q)-integration and (p,q)-hypergeometric functions related to quantum groups, Integral Transform. Spec. Funct., 2 (1994), 15–36.

[8] R. Chakrabarti and R. Jagannathan, A (p,q)-oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen., 24 (1991), 711–718.

[9] U. Duran, M. Acikgoz, A. Esi, and S. Araci, A note on the (p,q)-Hermite polynomials, Appl. Math. Inf. Sci., 12 (2018), 227–231.

[10] ˙I. Genc¸turk, ¨ Some Feng Qi type (p,q)-integral inequalities, Journal of Balikesir University Institute of Science and Technology, 23(1)(2021), 366–376. (Turkish).

[11] ˙I. Genc¸turk, ¨ Boundary value problems for a second-order (p,q)-difference equation with integral conditions, Turkish J. Math., 46(2)(2022), 499–515.

[12] M. N. Hounkonnou, J. Desir ´ e and B. Kyemba, ´ R(p,q)-calculus: Differentiation and integration, SUT J. Math., 49 (2013), 145–167. 248 ˙ILKER GENC¸ TURK ¨

[13] W. J. Liu, C. C. Li and J. Dong, On an open problem concerning an integral inequality, JIPAM. J. Inequal. Pure Appl. Math., 8 (3) (2007), Art. 74.

[14] W. J. Liu, G.S. Cheng and C. C. Li, Further development of an open problem concerning an integral inequality, JIPAM. J. Inequal. Pure Appl. Math., 9(1) (2008), Art.14.

[15] W. Luangboon, K. Nonlaopon, J. Tariboon and S. K. Ntouyas, On Simpson type inequalities for generalized strongly preinvex functions via (p,q)-calculus and applications, AIMS Mathematics, 6(9)(2021), 9236–9261.

[16] W. Luangboon, K. Nonlaopon, J. Tariboon and S. K. Ntouyas, Simpson-and Newton-Type Inequalities for Convex Functions via (p,q)-calculus, Mathematics, 9(12)(2021), 1338.

[17] T. F. Mori, ´ A general inequality of Ngo-Thang-Dat-Tuan type ˆ , JIPAM. J. Inequal. Pure Appl. Math., 10(1) (2009), Art. 10.

[18] M. D. Nasiruzzaman, A. Mukheimer and M. Mursaleen, Some Opial-type integral inequalities via (p,q)-calculus, J. Inequal. Appl., 2019 (2019), 1–11.

[19] Q. A. Ngo, D. Thang, T. Dat and D. Tuan, Notes on an integral inequality, JIPAM. J. Inequal. Pure Appl. Math., 7(4)(2006), Art. 120.

[20] J. Prabseang, K. Nonlaopon, J. Tariboon and S. K. Ntouyas, (p,q)-Hermite–Hadamard Inequalities for Double Integral and (p,q)-Differentiable Convex Functions, Axioms, 8(2)(2019), 68.

[21] J. Prabseang, K. Nonlaopon, J. Tariboon and S. K. Ntouyas, Refinements of Hermite-Hadamard Inequalities for Continuous Convex Functions via (p,q)-calculus, Mathematics, 9(4)(2021), 446.

[22] P. N. Sadjang, On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas, Results Math., 73(1) (2018), 1–21.

[23] S. Thongjob, K. Nonlaopon, J. Tariboon and S. K. Ntouyas, Generalizations of some integral inequalities related to Hardy type integral inequalities via (p,q)-calculus, J. Inequal. Appl. , 2021(1) (2021), 1–17.

[24] M. Tunc¸ and E. Gov, ¨ Some integral inequalities via (p,q)-calculus on finite intervals, Filomat, 35(5) (2021), 1421–1430.

[25] M. Tunc¸ and E. Gov, ¨ (p,q)-integral inequalities, RGMIA Res. Rep. Coll., 19 (2016), Art. 97, 1–13.

[26] F. Wannalookkhee, K. Nonlaopon, J. Tariboon and S. K. Ntouyas, On Hermite-Hadamard type inequalities for coordinated convex functions via (p,q)-calculus, Mathematics, 9(7)(2021), 698.

Downloads

Published

14.04.2025

How to Cite

Genç Türk, İlker. (2025). Some ($p$,$q$)-integral Inequalities. Sarajevo Journal of Mathematics, 20(2), 241–248. https://doi.org/10.5644/SJM.20.02.06

Issue

Section

Articles