Fractional Boole's inequalities for twice differentiable functions for Riemann-Liouville fractional integrals

Shehzadi A., Budak H., Haider W., Mateen A., Chen H.

JOURNAL OF APPLIED MATHEMATICS AND COMPUTING, vol.71, no.4, pp.5781-5800, 2025 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 71 Issue: 4
  • Publication Date: 2025
  • Doi Number: 10.1007/s12190-025-02465-5
  • Journal Name: JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, ABI/INFORM, Aerospace Database, Communication Abstracts, Compendex, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.5781-5800
  • Keywords: Boole’s formula type inequality, Convex function, Error bounds, Twice differentiable
  • Kocaeli University Affiliated: Yes

Abstract

Boole's type inequalities for first-order differentiable convex functions have been extensively studied, particularly in the context of numerical integration and fractional calculus. However, such inequalities for twice-differentiable functions remain relatively unexplored. This paper addresses this gap by establishing a novel Boole's identity via the Riemann-Liouville fractional integral for twice-differentiable functions. Twice-differentiable functions allow for the use of second-order derivatives, which provide more precise information about the curvature and behavior of functions compared to first-order derivatives. This leads to sharper inequalities and better numerical approximations. Based on this identity, several fractional Boole's type inequalities are derived, where the absolute value of the second derivatives is convex. The results include detailed error bounds, highlighting the effectiveness of the proposed approach in practical computations. Illustrative examples are provided to validate the theoretical findings and demonstrate their computational applicability.