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Fractal Fract
Volume 10
Issue 1
10.3390/fractalfract10010049
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Article

Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications

by
Saad Ihsan Butt
1,
Mohammed Alammar
2 and
Youngsoo Seol
3,*
1
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
2
Applied College, Shaqra University, Shaqra 11961, Saudi Arabia
3
Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea
*
Author to whom correspondence should be addressed.
Fractal Fract. 2026, 10(1), 49; https://doi.org/10.3390/fractalfract10010049
Submission received: 3 December 2025 / Revised: 31 December 2025 / Accepted: 8 January 2026 / Published: 12 January 2026
(This article belongs to the Section General Mathematics, Analysis)
Versions Notes

Abstract

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature.
Keywords: fractional calculus; convex functions; Boole’s inequality; Riemann–Liouville fractional integrals; majorization theory; Boole’s quadrature rule; q-diagamma function fractional calculus; convex functions; Boole’s inequality; Riemann–Liouville fractional integrals; majorization theory; Boole’s quadrature rule; q-diagamma function

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MDPI and ACS Style

Butt, S.I.; Alammar, M.; Seol, Y. Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications. Fractal Fract. 2026, 10, 49. https://doi.org/10.3390/fractalfract10010049

AMA Style

Butt SI, Alammar M, Seol Y. Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications. Fractal and Fractional. 2026; 10(1):49. https://doi.org/10.3390/fractalfract10010049

Chicago/Turabian Style

Butt, Saad Ihsan, Mohammed Alammar, and Youngsoo Seol. 2026. "Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications" Fractal and Fractional 10, no. 1: 49. https://doi.org/10.3390/fractalfract10010049

APA Style

Butt, S. I., Alammar, M., & Seol, Y. (2026). Fractional Integral Estimates of Boole Type: Majorization and Convex Function Approach with Applications. Fractal and Fractional, 10(1), 49. https://doi.org/10.3390/fractalfract10010049

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