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Improved error limits for numerical integration via some perturbed trapezoidal inequalities.

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Title: Improved error limits for numerical integration via some perturbed trapezoidal inequalities.
Authors: DÖNMEZ DEMİR, Duygu1 duygu.donmez@cbu.edu.tr, ŞANAL, Gülsüm2
Source: Sigma: Journal of Engineering & Natural Sciences / Mühendislik ve Fen Bilimleri Dergisi. Feb2026, Vol. 44 Issue 1, p309-317. 9p.
Subjects: Numerical integration, Integral inequalities, Approximation error, Mathematical optimization, Convex functions
Abstract: Integral inequalities prove extremely effective in obtaining error bounds for numerical integration formulas, which are particularly useful in optimization problems. This study presents reduced results of perturbed trapezoidal inequalities derived using the triangle inequality, Hölder’s inequality, and power mean inequalities for previously constructed nth-order differentiable s-convex and tgs-convex functions. These results help determine error limits for the trapezoidal rule and the remaining term of the midpoint formula in numerical integration. The study shows that these reduced inequalities yield better error limits. Additionally, an example involving a tgs-convex function defined under certain conditions demonstrates the practical application of these theoretical results. [ABSTRACT FROM AUTHOR]
Copyright of Sigma: Journal of Engineering & Natural Sciences / Mühendislik ve Fen Bilimleri Dergisi is the property of Sigma: Journal of Engineering & Natural Sciences / Mühendislik ve Fen Bilimleri Dergisi and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Integral inequalities prove extremely effective in obtaining error bounds for numerical integration formulas, which are particularly useful in optimization problems. This study presents reduced results of perturbed trapezoidal inequalities derived using the triangle inequality, Hölder’s inequality, and power mean inequalities for previously constructed nth-order differentiable s-convex and tgs-convex functions. These results help determine error limits for the trapezoidal rule and the remaining term of the midpoint formula in numerical integration. The study shows that these reduced inequalities yield better error limits. Additionally, an example involving a tgs-convex function defined under certain conditions demonstrates the practical application of these theoretical results. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Sigma: Journal of Engineering & Natural Sciences / Mühendislik ve Fen Bilimleri Dergisi is the property of Sigma: Journal of Engineering & Natural Sciences / Mühendislik ve Fen Bilimleri Dergisi and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.14744/sigma.2026.1983
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        Text: English
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        PageCount: 9
        StartPage: 309
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      – SubjectFull: Numerical integration
        Type: general
      – SubjectFull: Integral inequalities
        Type: general
      – SubjectFull: Approximation error
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      – SubjectFull: Mathematical optimization
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      – SubjectFull: Convex functions
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              Text: Feb2026
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              Y: 2026
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