MULTIPLICATIVE FRACTIONAL BULLEN-TYPE INEQUALITIES IN THE FRAMEWORK OF $ G $-CALCULUS

Abdelghani Lakhdari; Yazeed Alkhrijah; Borhen Louhichi; Aicha Ben Makhlouf; Hüseyin Budak

doi:10.11948/20250333

2026 Volume 16 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(5): 2458-2473

Next Article Previous Article

Abdelghani Lakhdari, Yazeed Alkhrijah, Borhen Louhichi, Aicha Ben Makhlouf, Hüseyin Budak. MULTIPLICATIVE FRACTIONAL BULLEN-TYPE INEQUALITIES IN THE FRAMEWORK OF $ G $-CALCULUS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2458-2473. doi: 10.11948/20250333

Citation:

Abdelghani Lakhdari, Yazeed Alkhrijah, Borhen Louhichi, Aicha Ben Makhlouf, Hüseyin Budak. MULTIPLICATIVE FRACTIONAL BULLEN-TYPE INEQUALITIES IN THE FRAMEWORK OF $ G $-CALCULUS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2458-2473. doi: 10.11948/20250333

MULTIPLICATIVE FRACTIONAL BULLEN-TYPE INEQUALITIES IN THE FRAMEWORK OF $ G $-CALCULUS

1.
Department CPST, National Higher School of Technology and Engineering, Annaba 23005, Algeria
2.
Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Umuttepe Campus, Kocaeli 41001, Türkiye
3.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
4.
Department of Electrical Engineering, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
5.
Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
6.
Department of Computer Science, College of Computer, Qassim University, Buraydah, 52571, Qassim, Saudi Arabia

Author Bio: Email: a.lakhdari@ensti-annaba.dz(A. Lakhdari); Email: ymalkhrijah@imamu.edu.sa(Y. Alkhrijah); Email: blouhichi@imamu.edu.sa(B. Louhichi); Email: a.benmakhlouf@qu.edu.sa(A. Ben Makhlouf)
Corresponding author: Email: hsyn.budak@gmail.com(H. Budak)
Fund Project: Y. Alkhrijah was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2603)

Abstract

Abstract

This paper introduces a multiplicative analogue of the Bullen quadrature rule and develops a suitable notion of convexity tailored to the $ G $-calculus framework. Building on these foundations, we derive a new fractional identity in the multiplicative setting, which serves as a key enabler for establishing Bullen-type inequalities via multiplicative Riemann-Liouville fractional integrals. This work integrate fractional calculus with multiplicative analysis for the study of integral inequalities, thereby proposing a novel pathway within non-Newtonian mathematical systems. Our results advance the theory of generalized calculus and open promising directions for future investigations into multiplicative fractional inequalities.
- $G$-calculus /
- multiplicative fractional integrals /
- multiplicative Bullen rule /
- $GG$-convex functions /
- $GA$-convex functions
MSC: 26D10, 26A51, 26D15, 11U10

FullText(HTML)

References

[1]	G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., 2007,335, 1294–1308. doi: 10.1016/j.jmaa.2007.02.016 CrossRef Google Scholar
[2]	U. Baş, A. Akkurt, A. Has and H. Yıldırım, Multiplicative Riemann-Liouville fractional integrals and derivatives, Chaos, Solitons and Fractals, 2025,196, 116310. doi: 10.1016/j.chaos.2025.116310 CrossRef Google Scholar
[3]	U. Baş, A. Akkurt and H. Yıldırım, Multiplicative fractional integrals and derivatives for two-variable functions, J. Comput. Appl. Math., 2026, 476, Paper No. 117144, 16 pp. Google Scholar
[4]	A. E. Bashirov, E. M. Kurpınar and A. Özyapıcı, Multiplicative calculus and its applications, J. Math. Anal. Appl., 2008,337(1), 36–48. doi: 10.1016/j.jmaa.2007.03.081 CrossRef Google Scholar
[5]	K. Boruah and B. Hazarika, Application of geometric calculus in numerical analysis and difference sequence spaces, J. Math. Anal. Appl., 2017,449(2), 1265–1285. doi: 10.1016/j.jmaa.2016.12.066 CrossRef Google Scholar
[6]	K. Boruah and B. Hazarika, $G$-calculus, TWMS J. Appl. Eng. Math., 2018, 8(1), 94–105. Google Scholar
[7]	H. Budak, F. Ertuǧral, M. A. Ali, C. C. Bilişik, M. Z. Sarikaya and K. Nonlaopon, On generalizations of trapezoid and Bullen type inequalities based on generalized fractional integrals, AIMS Math., 2023, 8(1), 1833–1847. doi: 10.3934/math.2023094 CrossRef Google Scholar
[8]	T. Du, C. Luo and Z. Cao, On the Bullen-type inequalities via generalized fractional integrals and their applications, Fractals, 2021, 29(7), 2150188. doi: 10.1142/S0218348X21501887 CrossRef Google Scholar
[9]	A. Fahad, S. I. Butt, B. Bayraktar, M. Anwar and Y. Wang, Some new Bullen-type inequalities obtained via fractional integral operators, Axioms, 2023, 12(7), 691. doi: 10.3390/axioms12070691 CrossRef Google Scholar
[10]	S. G. Georgiev and K. Zennir, Multiplicative Differential Calculus., Vol. I, Chapman & Hall/CRC, Boca Raton, FL, 2022. Google Scholar
[11]	M. Grossman, Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foundation, Massachusetts, 1983. Google Scholar
[12]	M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972. Google Scholar
[13]	H. R. Hwang, K. L. Tseng and K. C. Hsu, New inequalities for fractional integrals and their applications, Turkish J. Math., 2016, 40(3), 471–486. Google Scholar
[14]	A. Lakhdari, M. B. Almatrafi, W. Saleh, B. Meftah and H. Budak, On multiplicative beta-convexity and associated Katugampola fractional inequalities: Hermite–Hadamard and parametrized Newton–Cotes results, Modern Physics Letters B, 2026, 40(10), 2650053. doi: 10.1142/S0217984926500533 CrossRef Google Scholar
[15]	A. Lakhdari, M. A. Alqudah, F. Jarad, H. Budak and T. Abdeljawad, On multiplicative fractional operators of Hadamard and Katugampola types in $G$-Calculus and related Hermite–Hadamard inequalities, Fractals, 2026. https://doi.org/10.1142/S0218348X26500726. Google Scholar
[16]	A. Lakhdari, M. U. Awan, H. Budak, B. Meftah and S. S. Dragomir, An expanded analysis of multiplicative integral inequalities in $G$-calculus, Math. Slovaca, 2026. https://doi.org/10.1515/ms-2026-0008. Google Scholar
[17]	A. Lakhdari and W. Saleh, Multiplicative fractional Hermite–Hadamard-type inequalities in $G$-calculus, Mathematics, 2025, 13(21), 3426. doi: 10.3390/math13213426 CrossRef Google Scholar
[18]	A. Lakhdari, W. Saleh, H. Budak, B. Meftah and F. Jarad, Multiplicative tempered fractional integrals in $G$-calculus and associated Hermite–Hadamard-type inequalities, Fractals, 2026. https://doi.org/10.1142/S0218348X26500246. Google Scholar
[19]	B. Meftah and S. Samoudi, Some Bullen-Simpson type inequalities for differentiable $s$-convex functions, Math. Morav., 2024, 28(1), 63–85. doi: 10.5937/MatMor2401063M CrossRef Google Scholar
[20]	C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl., 2000, 3(2), 155–167. Google Scholar
[21]	M. Z. Sarikaya, On the some generalization of inequalities associated with Bullen, Simpson, midpoint and trapezoid type, Acta Univ. Apulensis Math. Inform., 2023, 73, 33–52. Google Scholar
[22]	R. Sassane, A. Lakhdari and B. Meftah, On Bullen-type inequalities for fractional integrals with exponential kernels, Sahand. Commun. Math. Anal., 2025, 22(2), 31–46. Google Scholar
[23]	D. Stanley, A multiplicative calculus, Primus Ⅸ, 1999, 4,310–326. Google Scholar
[24]	D. Zhao, M. A. Ali, H. Budak and Z. Y. He, Some Bullen-type inequalities for generalized fractional integrals, Fractals, 2023, 31(4), 2340060. doi: 10.1142/S0218348X23400601 CrossRef Google Scholar