Some New Boole-Type Inequalities via Modified Convex Functions with Their Applications and Computational Analysis

Talha Anwar, Abdul Mateen, Hela Elmannai, Muhammad Aamir Ali and Loredana Ciurdariu ()
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Talha Anwar: School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand
Abdul Mateen: School of Mathematical Sciences, Ministry of Education, Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China
Hela Elmannai: Department of Information Technology, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Muhammad Aamir Ali: School of Mathematics, Hohai University, Nanjing 210098, China
Loredana Ciurdariu: Department of Mathematics, Politehnica University of Timișoara, 300006 Timișoara, Romania

Mathematics, 2025, vol. 13, issue 21, 1-21

Abstract: In numerical analysis, the Boole’s formula serves as a pivotal tool for approximating definite integrals. The approximation of the definite integrals has a big role in numerical methods for differential equations; in particular, in the finite volume method, we need to use the best approximation of the integrals to obtain better results. This paper presents a rigorous proof of integral inequalities for first-time differentiable s -convex functions in the second sense. This paper has two main goals. The first is that the use of s -convex function extends the results for convex functions which cover a large class of functions and the second is the best approximation. To prove the main inequalities, we drive integral identity for differentiable functions. Then, with the help of this identity, we prove the error bounds of Boole’s formula for differentiable s -convex functions in the second sense. Some new midpoint-type inequalities for generalized convex functions are also given which can help us in finding better error bounds for midpoint integration formulas compared to the existing ones. Moreover, we provide some applications to quadrature formulas and special means for the real numbers of these newly established inequalities. Furthermore, we present numerical examples and computational analysis that show that these newly established inequalities are numerically valid.

Keywords: Boole’s formula-type inequality; modified convex function; quadrature formulae; midpoint formula; error bounds (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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