Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis

Mateen, Abdul; Bin-Mohsin, Bandar; Tipu, Ghulam Hussain; Shehzadi, Asia

doi:10.3390/math13172874

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Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis

¹

School of Mathematical Sciences, Ministry of Education Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China

²

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

³

Department of Mathematics, Shanghai University and Newtouch Center for Mathematics of Shanghai University, Shanghai 200444, China

⁴

Newtouch Center for Mathematics, Shanghai University, Shanghai 200444, China

⁵

School of Mathematics and Statistics, Central South University, Changsha 410083, China

^*

Authors to whom correspondence should be addressed.

Mathematics 2025, 13(17), 2874; https://doi.org/10.3390/math13172874

Submission received: 19 July 2025 / Revised: 27 August 2025 / Accepted: 4 September 2025 / Published: 5 September 2025

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Abstract

This paper presents new integral inequalities for differentiable generalized convex functions in the second sense, with a focus on improving the accuracy of Weddle’s formula for numerical integration. The study is motivated by the following three key factors: the generalization of convexity through s-convex functions, the enhancement of the approximation quality, particularly as

s \to 0^{+}

, and the effectiveness of Weddle’s formula in cases where Simpson’s

1 / 3

rule fails. An integral identity is derived for differentiable functions, which is then used to establish sharp error bounds for Weddle’s formula under s-convexity. Numerical examples and comparative tables demonstrate that the proposed inequalities yield significantly tighter bounds than those based on classical convexity. Applications to numerical quadrature highlight the practical utility of the results in computational mathematics.

Keywords: Weddle formula-type inequality; quadrature formulas; error bounds; convex functions; s-convex function

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MDPI and ACS Style

Mateen, A.; Bin-Mohsin, B.; Tipu, G.H.; Shehzadi, A. Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis. Mathematics 2025, 13, 2874. https://doi.org/10.3390/math13172874

AMA Style

Mateen A, Bin-Mohsin B, Tipu GH, Shehzadi A. Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis. Mathematics. 2025; 13(17):2874. https://doi.org/10.3390/math13172874

Chicago/Turabian Style

Mateen, Abdul, Bandar Bin-Mohsin, Ghulam Hussain Tipu, and Asia Shehzadi. 2025. "Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis" Mathematics 13, no. 17: 2874. https://doi.org/10.3390/math13172874

APA Style

Mateen, A., Bin-Mohsin, B., Tipu, G. H., & Shehzadi, A. (2025). Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis. Mathematics, 13(17), 2874. https://doi.org/10.3390/math13172874

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis

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