On Multiplicative Fractional Operators of Hadamard and Katugampola Types in G-Calculus and Related Hermite–Hadamard Inequalities

Abstract

This paper explores the extension of classical fractional operators to the framework of G-calculus, a non-Newtonian calculus in which differentiation and integration are defined via multiplicative analogs of their classical counterparts. We begin by recalling key concepts from both fractional calculus and G-calculus. Next, we revisit the recently introduced multiplicative Riemann–Liouville fractional operators and extend the multiplicative Riemann–Liouville fractional derivative to arbitrary order α>0. Building on this foundation, we introduce multiplicative versions of the Hadamard and Katugampola fractional integrals and derivatives. Finally, we establish Hermite–Hadamard inequalities for both newly defined integrals.