Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures
Abstract
1. Introduction
2. Preliminaries
2.1. Convexity and Related Integral Inequalities
2.2. Fractional Integral Operators
2.3. Superquadraticity and Related Integral Inequalities
- 1.
- .
- 2.
- If is differentiable at as well as and then .
- 3.
- is convex and as well as if .
2.4. Fundamental Properties of Multiplicative Calculus
- 1.
- 2.
- 3.
- 4.
- 5.
- 1.
- 2.
- 3.
- 4.
- 5.
3. Fractional Inequalities for Multiplicative Superquadratic Function via Multiplicatively A-B Fractional Operator
4. Applications
4.1. Special Means
4.2. Modified Bessel Functions
4.3. Moment of Random Variables
4.4. Implications
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Jallani, G.; Butt, S.I.; Khan, D.; Seol, Y. Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures. Fractal Fract. 2025, 9, 617. https://doi.org/10.3390/fractalfract9090617
Jallani G, Butt SI, Khan D, Seol Y. Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures. Fractal and Fractional. 2025; 9(9):617. https://doi.org/10.3390/fractalfract9090617
Chicago/Turabian StyleJallani, Ghulam, Saad Ihsan Butt, Dawood Khan, and Youngsoo Seol. 2025. "Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures" Fractal and Fractional 9, no. 9: 617. https://doi.org/10.3390/fractalfract9090617
APA StyleJallani, G., Butt, S. I., Khan, D., & Seol, Y. (2025). Refined Hermite–Hadamard Type Inequalities via Multiplicative Non-Singular Fractional Integral Operators and Applications in Superquadratic Structures. Fractal and Fractional, 9(9), 617. https://doi.org/10.3390/fractalfract9090617