Error Bounds of Boole-Type Inequalities for Caputo Fractional Operator with Their Computational Analysis and Applications

Abstract

This paper establishes Boole-type inequalities for n-times differentiable convex functions within the framework of fractional calculus, utilizing the Caputo fractional operator to generalize classical results. To achieve this, a novel integral identity is first established using the Caputo fractional integral, which serves as a foundational tool for deriving several new Boole-type inequalities. The study extends these inequalities to encompass broader classes of functions, including bounded and Lipschitzian functions, employing fractional integrals to derive refined results. Key contributions include the establishment of generalized error bounds for higher-order fractional Boole’s formula and their explicit dependence on the fractional differentiation of order

\alpha

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}

. The principal advancement of this work lies in deriving distinct inequality classes corresponding to function differentiability for different values of n through the parametric selection of the positive integer n. The present research provides sharp error estimates for Boole’s rule by employing Hölder’s inequality and the power mean inequality. Numerical examples and graphical analysis are provided to illustrate the practical significance of the results. Furthermore, applications to special functions such as the Mittag-Leffler function and the

\mathbf{q}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textbf{q}}$$\end{document}

-polygamma function, revealing their effectiveness in handling fractional-order models involving monotonic or convex behavior are presented. These approaches can advance the understanding of Boole’s formula and enhance its error bounds within fractional frameworks.

Keywords

Boole-type inequalities / Caputo fractional operator / Error bounds / Convex function / Lipschitzian functions / Bounded functions / 26D10 / 26D15 / 26A51

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