Multiplicative view point analysis of Hahn calculus and their applications to inequality theory

Muhammad Nasim Aftab; Saad Ihsan Butt; Mohammed Alammar; Youngsoo Seol; Muhammad Nasim Aftab; Saad Ihsan Butt; Mohammed Alammar; Youngsoo Seol

doi:10.3934/math.20251337

AIMS Mathematics

2025, Volume 10, Issue 12: 30478-30506. doi: 10.3934/math.20251337

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Research article

Multiplicative view point analysis of Hahn calculus and their applications to inequality theory

1.
Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
3.
Applied College, Shaqra University, Shaqra, Saudi Arabia
4.
Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea

Received: 09 October 2025 Revised: 09 December 2025 Accepted: 15 December 2025 Published: 25 December 2025
MSC : 05A30, 26D10, 11D57, 26D15, 39A70, 33E20

Hahn multiplicative calculus is the generalization of quantum multiplicative ($ \mathit{q} $-multiplicative) calculus. In this manuscript, we defined novel definitions for derivative and definite integral called left Hahn multiplicative derivative and definite integral in the Hahn multiplicative calculus. In addition, we derived fundamental results for this newly defined integral. Furthermore, we constructed left Hahn multiplicative Hermite-Hadamard inequalities. Additionally, we defined new definitions for the derivative and definite integral in the Hahn calculus, which enabled us to define further definitions in Hahn multiplicative calculus called the right Hahn multiplicative derivative and definite integral. Moreover, we construct the power rule of the newly defined definite integral in the Hahn calculus, which assisted us in deriving the right Hahn multiplicative Hermite-Hadamard inequalities. Finally, we gave an application of the newly established Hermite-Hadamard inequalities through an example wherein it could be seen that these inequalities were crucial for finding the lower and upper bounds of the range of those functions whose Hahn multiplicative definite integrals were very difficult to find.
- Hahn multiplicative calculus,
- Hahn calculus,
- multiplicative calculus,
- Hermite-Hadamard inequality
Citation: Muhammad Nasim Aftab, Saad Ihsan Butt, Mohammed Alammar, Youngsoo Seol. Multiplicative view point analysis of Hahn calculus and their applications to inequality theory[J]. AIMS Mathematics, 2025, 10(12): 30478-30506. doi: 10.3934/math.20251337

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Abstract

Hahn multiplicative calculus is the generalization of quantum multiplicative ($ \mathit{q} $-multiplicative) calculus. In this manuscript, we defined novel definitions for derivative and definite integral called left Hahn multiplicative derivative and definite integral in the Hahn multiplicative calculus. In addition, we derived fundamental results for this newly defined integral. Furthermore, we constructed left Hahn multiplicative Hermite-Hadamard inequalities. Additionally, we defined new definitions for the derivative and definite integral in the Hahn calculus, which enabled us to define further definitions in Hahn multiplicative calculus called the right Hahn multiplicative derivative and definite integral. Moreover, we construct the power rule of the newly defined definite integral in the Hahn calculus, which assisted us in deriving the right Hahn multiplicative Hermite-Hadamard inequalities. Finally, we gave an application of the newly established Hermite-Hadamard inequalities through an example wherein it could be seen that these inequalities were crucial for finding the lower and upper bounds of the range of those functions whose Hahn multiplicative definite integrals were very difficult to find.

References

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