NEW GENERALIZED RESULTS ON GRUSS-TYPE FRACTIONAL INTEGRAL INEQUALITIES INVOLVING THE CAPUTO–FABRIZIO INTEGRAL OPERATOR
By
Bhawesh Khatri1 and Meena Kumari Gurjar2,*
1,2Department of Mathematics and Statistics, J.N.V. University, Jodhpur, Rajasthan, India–342001
Email: bhawesh13197@gmail.com, meenanetj@gmail.com *Corresponding author
(Received: February 01, 2026, In format: February 14, 2026; Revised: April 05, 2026; Accepted June 20, 2026)
DOI: https://doi.org/10.58250/jnanabha.2026.56114
Abstract
The Gr¨uss inequality is a well-known result in mathematical analysis that gives an upper estimate for the difference between the integral of the product of two functions and the product of their individual integrals. Its generalizations in various mathematical contexts, including fractional calculus. In this paper, we present a comprehensive generalization of certain Gr¨uss-type integral inequalities and other related integral inequalities using the Caputo-Fabrizio fractional integral operator. Specifically, we extend and generalize classical integral inequalities to obtain new results within the framework of fractional calculus. Unlike traditional fractional operators, the CaputoFabrizio operator is characterized by a non-singular exponential kernel that provides a more regular and physically meaningful approach to fractional integration. By utilizing this operator into the analysis, we derive new generalized versions of the classical Gr¨uss inequality.
2020 Mathematical Sciences Classification: 26A33, 26D15, 26D10, 47G20, 45P05.
Keywords and Phrases: Young’s inequality, Gr¨uss inequality, weighted Gr¨uss inequality, preGr¨uss inequality, Caputo–Fabrizio fractional integral operator.