Some integral inequalities involving \(q\)-\(h\) fractional integral operator
Volume 38, Issue 4, pp 535--545
https://dx.doi.org/10.22436/jmcs.038.04.08
Publication Date: February 14, 2025
Submission Date: October 07, 2024
Revision Date: November 25, 2024
Accteptance Date: January 16, 2025
Authors
A. Abbas
- Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan.
F. Fiyaz
- Department of Mathematics, University of Jhang, Jhang 35200, Pakistan.
S. Mubeen
- Department of Mathematics, Baba Guru Nanak University, Nankana Sahib 39100, Pakistan.
A. Khan
- Department of Mathematics and Science, Prince Sultan University, P.O. Box 66833, 11586 Riyadh.
T. Abdeljawad
- Department of Mathematics and Science, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia.
- Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, 602105, Tamilnadu, India.
- Center for Applied Mathematics and Bioinformatics (CAMB), Gulf University for Science and Technology, Hawally, 32093, Kuwait.
- Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa.
Abstract
In this work, we extend several well-known classical inequalities by applying a newly defined \(q\)-\(h\) operator over finite intervals. Specifically, we generalize the Cauchy-Schwarz integral inequality for double integrals, Grüss integral inequality, Korkine identity, and Grüss-Čebyšev integral inequality. These generalizations provide tighter bounds and enhanced applicability in the framework of quantum calculus. The \(q\)-\(h\)-integral, which combines features of the \({q}\)-integral and \({h}\)-integral, serves as a unifying tool to connect and extend existing results. Furthermore, we examine special cases to demonstrate the broader scope of these inequalities. Our findings highlight the versatility of the \(q\)-\(h\)-operator in refining and expanding the mathematical framework of integral inequalities in quantum calculus.
Share and Cite
ISRP Style
A. Abbas, F. Fiyaz, S. Mubeen, A. Khan, T. Abdeljawad, Some integral inequalities involving \(q\)-\(h\) fractional integral operator, Journal of Mathematics and Computer Science, 38 (2025), no. 4, 535--545
AMA Style
Abbas A., Fiyaz F., Mubeen S., Khan A., Abdeljawad T., Some integral inequalities involving \(q\)-\(h\) fractional integral operator. J Math Comput SCI-JM. (2025); 38(4):535--545
Chicago/Turabian Style
Abbas, A., Fiyaz, F., Mubeen, S., Khan, A., Abdeljawad, T.. "Some integral inequalities involving \(q\)-\(h\) fractional integral operator." Journal of Mathematics and Computer Science, 38, no. 4 (2025): 535--545
Keywords
- \(q\)-\(h\)-Integral
- \(q\)-\(h\)-integral inequalities
- Cauchy-Schwarz integral inequality
- Grüss-Čebyšev integral inequality
- Grüss integral inequality
MSC
References
-
[1]
T. A. Aljaaidi, D. B. Pachpatte, Some Grüss-type inequalities using generalized Katugampola fractional integral, AIMS Math., 5 (2019), 1011–1024
-
[2]
A. Alkhazzan, J. Wang, Y. Nie, H. Khan, J. Alzabut, A stochastic SIRS modeling of transport-related infection with three types of noises, Alex. Eng. J., 76 (2023), 557–572
-
[3]
G. A. Anastassiou, Intelligent Mathematics: Computational Analysis, Springer-Verlag, Berlin (2011)
-
[4]
S. S. Dragomir, Some integral inequalities of Gr¨ss type, RGMIA Res. Rep. Collect., 1 (1998), 95–111
-
[5]
S. S. Dragomir, N. T. Diamond, Integral inequalities of Grüss type via Pólya-Szegö and Shisha-Mond results, RGMIA Res. Rep. Collect., 5 (2002), 1–9
-
[6]
H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl., 47 (2004), 281–300
-
[7]
K. Jangid, S. D. Purohit, R. Agarwal, On Gr¨ss type inequality involving a fractional integral operator with a multi-index Mittag-Leffler function as a kernel, Appl. Math. Inf. Sci., 16 (2022), 269–276
-
[8]
M. B. Jeelani, A. M. Saeed, M. S. Abdo, K. Shah, Positive solutions for fractional boundary value problems under a generalized fractional operator, Math. Methods Appl. Sci., 44 (2021), 9524–9540
-
[9]
M. Jleli, B. Samet, On some integral inequalities in Quantum calculus, J. Funct. Spaces, 2020 (2020), 10 pages
-
[10]
V. G. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York (2002)
-
[11]
H. Khan, J. Alzabut, H. Gulzar, O. Tunç, S. Pinelas, On system of variable order nonlinear p-Laplacian fractional differential equations with biological application, Mathematics, 11 (2023), 17 pages
-
[12]
H. Khan, J. Alzabut, A. Shah, Z.-Y. He, S. Etemad, S. Rezapour, A. Zada, On fractal-fractional waterborne disease model: A study on theoretical and numerical aspects of solutions via simulations, Fractals, 31 (2023), 16 pages
-
[13]
A. McD. Mercer, An improvement of the Grüss inequality, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), 4 pages
-
[14]
N. Minculete, L. Ciurdariu, A generalized form of Grüss type inequality and other integral inequalities, J. Inequal. Appl., 2014 (2014), 18 pages
-
[15]
D. S. Mitrinovi´c, J. E. Peˇcari´c, A. M. Fink, Classical and new inequalities in analysis, Kluwer Academic Publishers Group, Dordrecht (1993)
-
[16]
G.-H. Peng, Y. Miao, A note on Grüss type inequality, Appl. Math. Sci. (Ruse), 3 (2009), 399–402
-
[17]
D. Shi, G. Farid, B. A. I. Younis, H. Abu-Zinadah, M. Anwar, A unified representation of q-and h-integrals and consequences in inequalities, Axioms, 13 (2024), 12 pages
-
[18]
W. Shatanawi, A. Boutiara, M. S. Abdo, M. B. Jeelani, K. Abodayeh, Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative, Adv. Differ. Equ., 2021 (2021), 19 pages
-
[19]
J. Sousa, D. S. Oliveira, E. C. de Oliveira, Gr¨ss-type inequality by means of a fractional integral, arXiv preprint arXiv:1705.00965, (2017), 16 pages
-
[20]
J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013 (2013), 19 pages
-
[21]
J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 1–13
