Local Fractional Perspective on Weddle’s Inequality in Fractal Space
Abstract
1. Introduction
Yang Local Fractional Calculus
- ,
- ,
- ,
- .
- How can the error terms of Weddle’s rule be established for local fractional differentiable mappings? Mathematically,
- How different generalized classes of mappings can be used to evaluate the bounds of the above inequality?
- What are the applications of the proposed bounds?
- How can the bounds of several local fractional integrals be found incorporated with the proposed inequalities?
2. Fractal Estimates of Weddle’s Inequality
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- ,
- .
3. Applications
3.1. Applications to Means
- 1.
- The generalized arithmetic mean:
- 2.
- The generalized weighted arithmetic mean:
- 3.
- The generalized log--mean:
3.2. Error Bounds
3.3. Applications to Probability
- It is a known fact that various definite integrals cannot be evaluated through analytical techniques neither in classical nor in Yang local fractional calculus. They are approximated through Newton–Cotes schemes. In the aforementioned sections, we have provided the error bounds of definite integrals approximated through generalized Weddle’s incorporated with various classes of mappings. Our results provide the upper bounds of error terms for first-order local differentiable mapping. Additionally, from these bounds one can compute the upper bounds of definite integrals. For this, we apply Theorem 7 for over with . Then we get . Note that this bound can be refined by increasing the order of differentiability. Through the application of our proposed inequalities, one can establish the bounds and relation between special functions like q-digamma function, modified Bessel functions and Beta functions. For further detail, consult [29,30,31].
- Adopting the technique of [32], several new iterative methods to solve non-linear equations can be obtained. To discuss the convergence and dynamic analysis of proposed methods will be a new challenge for researchers.
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Wang, Y.; Asif, U.; Awan, M.U.; Javed, M.Z.; Khan, A.G.; Bin-Asfour, M.; Albalawi, K.S. Local Fractional Perspective on Weddle’s Inequality in Fractal Space. Fractal Fract. 2025, 9, 662. https://doi.org/10.3390/fractalfract9100662
Wang Y, Asif U, Awan MU, Javed MZ, Khan AG, Bin-Asfour M, Albalawi KS. Local Fractional Perspective on Weddle’s Inequality in Fractal Space. Fractal and Fractional. 2025; 9(10):662. https://doi.org/10.3390/fractalfract9100662
Chicago/Turabian StyleWang, Yuanheng, Usama Asif, Muhammad Uzair Awan, Muhammad Zakria Javed, Awais Gul Khan, Mona Bin-Asfour, and Kholoud Saad Albalawi. 2025. "Local Fractional Perspective on Weddle’s Inequality in Fractal Space" Fractal and Fractional 9, no. 10: 662. https://doi.org/10.3390/fractalfract9100662
APA StyleWang, Y., Asif, U., Awan, M. U., Javed, M. Z., Khan, A. G., Bin-Asfour, M., & Albalawi, K. S. (2025). Local Fractional Perspective on Weddle’s Inequality in Fractal Space. Fractal and Fractional, 9(10), 662. https://doi.org/10.3390/fractalfract9100662