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Keywords = Boole’s quadrature rule

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25 pages, 518 KB  
Article
Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications
by Saad Ihsan Butt, Mohammed Alammar and Youngsoo Seol
Fractal Fract. 2026, 10(1), 49; https://doi.org/10.3390/fractalfract10010049 - 12 Jan 2026
Abstract
The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex [...] Read more.
The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. Full article
(This article belongs to the Section General Mathematics, Analysis)
20 pages, 1314 KB  
Article
Upper Bounds for the Remainder Term in Boole’s Quadrature Rule and Applications to Numerical Analysis
by Muhammad Zakria Javed, Muhammad Uzair Awan, Bandar Bin-Mohsin and Savin Treanţă
Mathematics 2024, 12(18), 2920; https://doi.org/10.3390/math12182920 - 20 Sep 2024
Cited by 4 | Viewed by 1450
Abstract
In the current study, we compute some upper bounds for the remainder term of Boole’s quadrature rule involving convex mappings. First, we build a new identity for first-order differentiable mapping, an auxiliary result to establish our required estimates. We provide several upper bounds [...] Read more.
In the current study, we compute some upper bounds for the remainder term of Boole’s quadrature rule involving convex mappings. First, we build a new identity for first-order differentiable mapping, an auxiliary result to establish our required estimates. We provide several upper bounds by utilizing the identity, convexity property, and bounded property of mappings and some well-known inequalities. Moreover, based on our primary findings, we deliver applications to the means, quadrature rule, special mappings, and non-linear analysis by developing a novel iterative scheme with cubic order of convergence. To the best of our knowledge, the current study is the first attempt to derive upper bounds for Boole’s scheme involving convex mappings. Full article
(This article belongs to the Special Issue Mathematical Inequalities and Fractional Calculus)
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